Queueing Systems: Theory

QUEUEING SYSTEMS VOLUME J: THEORY Leonard Kleinrock Prof essor Computer Science Dep artment S chool of Engineering and ...

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QUEUEING SYSTEMS VOLUME J: THEORY Leonard Kleinrock Prof essor Computer Science Dep artment S chool of Engineering and Applied Sci ence University of California, Los Angeles

A Wiley-Interscience Publication

John Wiley & Sons New Yo rk • Chiches te r > Brisbane • Toronto

.I

J

" Ah, ' All thina s come to th ose who wait.' They come, but cjten come too late." From Lady Mary M. Curr ie: Tout Vient

a Qui Sait

Attendre (1890)

27947

Co pyr ight © 197 5 , by J oh n Wiley & So ns , Inc. All rights reserved . Publ ished simulta neo usly in Canada. Reproduction or translation of any part of th is work beyond that

perm itted by Sections 107or 108of 'he 1976 United States Cop yright Ac t wi thout the permission of the copyrig ht owne r is unlawful. Requests fo r perm ission or furthe r in fo rmatio n should be addressed to the Permissio ns Depart ment , Jo hn wi ley & Son s. Inc .

Library of Congress Cataloging in Publication Data: K leinrock , Leon ard . Queuein g systems.

"A Wiley-Iru erscien ce pu blicat io n." CONTE NTS : v . I. Th eor y. I. Que ueing theory. I. T itle. T57.9 . K 6

5 19.8'2

IS BN 0-47 1-49110- 1

13 1415

74-98 46

Preface

How much time did yo u waste waiting in line this week ? It seems we cann ot escape frequent delays, and they are getting progressively worse ! In this text we study the phen omena of sta nding , waiting, and serving, and we call this study queueing theory . An y system in which arrivals place demands upon a finite-cap acity resource may be termed a queueing system. In particular, if the arri val times of these demand s are unpredictable , or if the size of these demands is unpredictable , then conflicts for the use of the resource will ari se and queu es of waiting customers will form . The lengths of these queue s depend upon two aspects of the flow pattern : first, they depend upon the average rate at which demands are placed upon the resource; an d second, they depend upon the statistical flu ctuations of this rate. Certainly, when the average rate exceeds the cap acity, then the system breaks down and unbounded queue s will begin to form ; it is the effect of this average overload which then dominates the growth of queue s. Ho wever, even if the average rate is less than the system capacity, then here, too , we have the forma tion of queues due to the sta tistical fluctu ati on s and spurts of arrivals that may occur; the effect of these var iatio ns is greatly magnified when the average load approaches (but does not necessarily exceed) that of the system cap acity. The simplicity of these queueing structure s is decepti ve, a nd in our studies we will often find ourselves in deep ana lytic waters. Fortunately, a familiar and fund amental law of science perme ate s our queueing investigations . This law is the conservation of flow, which states that the rate at which flow increases within a system is equal to the difference between the flow rat e int o and the flow rate out of tha t system. Thi s observation permits us to write down the basic system equa tions for rath er co mplex structures in a relativel y easy fashion . The pu rpose of this book , then , is to present the theory of queue s at the first-year gradua te level. It is assumed that the student has been expose d to a first co urse in probabi lity theory ; however, in Appendi x II of this text we give a pr obability theory refresher and state the basic pr inciples that we shall need. It is also helpful (but not necessary) if the student has had some exposure to tran sform s, alth ough in this case we presen t a rat her com plete vii

viii

PREFACE

transform theory refresher in Appendix I. The student is advised to read both appendices before proceeding with the text itself. Whereas our material is presented in the language of mathematics, we do take great pains to give as informal a presentation as possible in order to strike a balance between the abstractions usually encountered in such a study and the basic need for understanding and applying these tools to practical systems. We feel that a satisfactory middle ground has been established that will neither offend the mathematician nor confound the practitioner. At times we have relaxed the rigor in proofs of uniqueness , existence, and convergence in order not to cloud the main thrust of a presentation. At such times the reader is referred to some of the other books on the subject. We have refrained from using the dull "theorem-proof" approach; rather, we lead the reader through a natural sequence of steps and together we "discover" the result. One finds that previous presentations of this material are usually either too elementary and limited or far too elegant and precise, and almost all of them badly neglect the applications; we feel that the need for a book such as this, which treads the boundary inbetween, is necessary and useful. This book was written over a period of fiveyears while being used as course notes for a one-year (and later a two-quarter) sequence in queueing systems at the University of California, Los Angeles. The material was developed in the Computer Science Department within the School of Engineering and Applied Science and has been tested successfully in the most critical and unforgiving of all environments, namely, that of the graduate student. This text is appropriate not only for computer science departments , but also for departments of engineering, operations research, mathematics, and many others within science, business , management and planning schools . In order to describe the contents of this text, we must first describe the very conven ient shorthand notation that has been developed for the specification of queueing systems. It basically involves the three-part descriptor A/B/m that denotes an m-server queueing system, where A and B describe the interarrival time distribution and the service time distribution , respectively. A and B take on values from the following set of symbols whose interpretation is given in terms of distributions within parentheses: M (exponential) ; E (r-stage Eriangian); HR (R-stage hyperexponential); D (deterministic) ; G (general). Occasionally, some other specially defined symbols are used . We sometimes need to specify the system's storage capacity (which we denote by K) or perhaps the size of the customer population (which we denote by M) , and in these cases we adopt the five-part descriptor A/B/m/K /M; if either of these last two descriptors is absent, then we assume it takes on the value of infinity. Thus, for example, the system D/M /2/20 is a two-server system with constant (deterministic) interarrival times, with exponentially distributed service times, and with a system storage capacity of size 20. T

PREFA CE

ix

Th is is Volume I (Theory) of a two-volume series, the second of which is devoted to computer applications of this theory. The text of Volume I (which consists of four parts ) begins in Chapter I with an intr oducti on to queuein g systems, how they fit into the general scheme of systems of flow, and a discussion of how one specifies and evaluates the performance of a queueing system. Assuming a knowledge of (or after reviewing) the mater ial in Appendices I and II, the reader may then proceed to Chapter 2, where he is warned to take care! Section 2.1 is essential and simple. However, Sections 2.2, 2.3, and 2.4 are a bit "heavy" for a first reading in queueing systems, and it would be quite reasonable if the reader were to skip these sections at this point, proceeding directly to Section 2.5, in which the fundamental birthdeath process is introduced and where we first encounter the use of a-transform s and Laplace tran sforms . Once these preliminaries in Part I are estab lished one may proceed with the elementary queueing theory presented in Par t II. We begin in Ch apter 3 with the general equilibrium solut ion to birthdeath processes and devote most of the chapter to providin g simple yet importa nt examples. Chapter 4 genera lizes this treatment , and it is here where we discuss the method of stages and prov ide an intr oduction to networks of Mar kovian queue s. Whereas Part II is devoted to algebraic and tr ansform oriented calculations , Part III returns us once again to probabilistic (as well as tran sform) agruments. This discussion of intermediate queueing theory begins with the important M/G /I queue (Chapter 5) and then proceeds to the dual G/M /I queue a nd its natural generalization to the system G/M /m (Chapter 6). Th e material on collective mark s in Ch apter 7 develops the probabilistic interpretation of tran sforms . Finally, the advanced mat erial in Part IV leads us to the queue G/G /I in Chapter 8; th is difficult system (whose mean wait cann ot even be expressed simply in term s of the system parameters) is studied thro ugh the use of the spectral solution to Lindley's integral equation. An approximati on to the precedence structure among chapters in these two volumes is given below. In this diagram we have represented chapters in Volume I as numb ers enclosed in circles and have used small squares for Volum e II. The shading for the Volume I nodes indicates an appropri ate amount of mater ial for a relatively leisurely first cour se in queu eing systems that can easily be accompl ished in one semester or can be comfortably handled in a one-qua rter co urse. The shading of Chapter 2 is meant to indicate that Sections 2.2- 2.4 may be omitted on a first reading, and the same applies to Sections 8.3 and 8.4. A more rapid one-semester pace and a highly accelerated one-quarter pace would include all of Volume I in a single cour se. We close Volume I with a summary of important equations, developed thr oughout the book, which are grouped together according to the class of queu eing system involved ; th is list of results then serves as a "handbook " for later use by the reader in co ncisely summarizing the principal results of this text. The results

X

PREFACE

o

o

Volume I

Vo lume II

2

5

are keyed to the page where they appear in order to simplify the task o f locating the explanatory material associated with each result. Each chapter contains its own list of references keyed alphabetically to the author and year; for example, [KLEI 74] would reference this book . All equations of importance have been marked with the symbol - , and it is these which are included in the summary of important equations. Each chapter includes a set of exercises which , in some cases, extend the material in that chapter ; the reader is urged to work them out.

XII

PREFACE

the face of the real world's complicated models, the mathematicians proceeded to ad vance the field of queueing theory rapidly and elegantl y. The frontiers of this research proceeded into the far reache s of deep a nd complex mathematics. It was soo n found that the really intere sting model s did not yield to solution and the field quieted down considerably. It was mainly with the advent of digital computers that once again the tools of queueing theory were brought to bear on a class of practical problems, but thi s time with great success. The fact is that at present, one of the few tools we have for an alyzing the performance of computer systems is that of queuein g the ory , and this explains its popularity am ong engineers and scientists today. A wealth of new problems are being formulated in terms of this theory and new tools and meth ods are being developed to meet the challenge of these problems. Moreover, the application of digital computers in solving the equations of queuein g theory has spawned new interest in the field. It is hoped that thi s two-volume series will provide the reader with an appreciation for and competence in the methods of analysis and application as we now see them . I take great pleasure in closing th is Preface by acknowledgin g those indi vidual s and institutions that made it possible for me to brin g this book int o being . First , I would like to thank all tho se who participated in creatin g the stimulating environment of the Computer Science Department at UCLA, which encouraged and fostered my effort in this directi on. Acknowledgment is due the Advanced Research Projects Agency of the Department of Defense , which enabled me to participate in so me of the most exciting and ad vanced computer systems and networks ever developed . Furthermore , the John Simon Guggenheim Foundation provided me with a Fellowship for the academic year 1971 -1 972, during which time I was able to further pursue my investigati ons. Hundreds of students who have passed through my queueingsystems courses have in major and minor ways contributed to the creation of this book , and I am happy to ackn owledge the specia l help offered by Arne Nilsson, Johnny Wong, Simon Lam, Fouad Tobagi, Farouk Kam oun, Robert Rice, and Th omas Sikes. My academic and profes sional collea gues have all been very suppo rtive of th is endeavour. To the typi sts l owe all. By far the lar gest port ion of this book was typed by Cha rlo tte La Roche , and I will be fore ver in her debt. To Diana Skoc ypec and Cynthia Ellm an I give my deepest thanks for carrying out the enormous task of. proofreading and correction-making in a rapid , enthusiastic, and suppo rt ive fash ion. Others who contributed in major ways are Barbara Warren , Jean Dubinsky, Jean D'Fucci , and Gloria Roy. l owe a great debt of thanks to my fam ily (and especially to my wife, Stella) who have stood by me and supported me well beyond the call of duty or marriage contract. Lastl y, I would certainly be remiss in omitting an ackn owledgement to my ever-faithful dictatin g machine, which was constantly talking back to me. LEONARD KLEI NROCK

March, 1974

Contents VOLUME I PART

I: PR ELIMINARIES

Chapter 1 Queueing Sys tems

1.1. Systems of Flow . 1.2. The Specification and Measure of Queueing Systems Chapter 2 2. 1. 2.2 . 2.3 . 2.4 . 2.5 .

PART

So me Imp or tant Rand om Processes Notatio n a nd Structu re fo r Basic Que ueing Systems Definition and Classification of Stochastic Processes Discrete-Time Markov C hains Co nti nuo us-Time Mar kov Ch ain s . Birth-Death Processes.

3 3 8 10 10 19 26 44 53

II: ELEMENTARY Q UE UEING THEORY

Chapter 3

Birth-Death Queueing Sys tems in Equilibrium

3.1. Gener al Eq uilibrium So lution 3.2. M/M/I: The Classical Q ueueing System . 3.3 . Discouraged Arrivals 3.4. M / M/ro: Resp on sive Se rvers (Infinite N umber of Server s) 3.5. M/M /m: The m-Server Case. 3.6. M /M /I /K : Finite Storage 3.7. M/ M/m/m : m-Server Loss Syste ms . 3.8. M/M /I IIM : Finite Custome r Population-Single Server 3.9. M/M / roIIM : Finite Cu sto mer Po pulation- " Infinite" Numbe r of Servers 3.10. M /M /m/K /M : Fi nite Population, m-Serve r Case , Finite Storage

89 90 94 99 101 102 103 105 106 107 108 X III

xiv

CONTEN TS

Chapter 4 Markovian Queues in Equilibrium 4.1. 4.2. 4.3. 4.4. 4.5. 4.6. 4.7. 4.8. PART

The Equilibrium Equ at ions . The Method of Stages- Erlangian Distribution E . The Queue M/Er/1 The Queue ErlM /I Bulk Arri val Systems Bulk Service Systems Series-Parallel Stages : Generalization s Networks of Markovian Queues

I 15 11 5 119 126 130 134 137 139 147

III: INTERMEDIATE QUEUEING THEORY

Chapter 5 The Queue M /G/I

167

5. 1. The M/G/I System 168 5.2. The Paradox of Residu al Life: A Bit of Renewal Theory . 169 5.3. The Imbedded Markov Chain 174 177 5.4. The Transition Probabilities . 5.5. The Mean Queue Length . 180 191 5.6. Distribution of Number in System . 196 5.7. Distribution of Waiti ng Time 206 5.8. The Busy Peri od and Its Durat ion . 5.9. The Numbe r Served in a Busy Period 216 5.10. From Busy Periods to Waitin g Times 219 5. 11. Combin at orial Method s 223 226 5.12. T he Tables Inte grodifferential Equation . Chapter 6 The Queue G/M /m

241

6. 1. Transition Prob abilit ies for the Imbedded Markov Chain (G/M /m ) 241 6.2. Conditi onal Distributi on of Queue Size . 246 6.3. Cond itional Distribution of Waiting Time 250 251 6.4. The Queue G/M /I 6.5. T he Queue G/M /m 253 6.6. Th e Queue G /M /2 256 Chapter 7 The Method of Collective Ma rks

261

7. 1. The Mar king of Customers 7.2. The Catastrophe Proce ss

26 I 267

CONTDITS

PART

XV

IV: ADVANCED MATERIAL

Chapter 8 8. 1. 8.2. 8.3. 8.4 .

The Queue GIGII

275

Lin dley's I ntegra l Equa tio n Spect ra l Sol ution to Lindley' s In tegra! Eq uatio n Ki ngman 's Algebra for Queues T he Idle Tim e a nd D uali ty

275 283 299 304 319

Epilogue Appendix I : Transform Theory Refresher: z-Transforrn and Laplace Transform

..

1.1. Why Transforms ? 1.2. The z-T ra nsform . 1.3. Th e La place T ran sfo rm 1.4. Use of Tran sforms in the Solution of D ifference a nd Differen tia l Equa tions

321 327 338 355

Appendix II: Probability Theory Refresher

II. I. R ules of th e G ame 11.2. Rand om Va riables I1.3. Expectatio n 11.4. Transfo rms, Generating Funct ion s, a nd Ch aracteristic F un ctio ns . 11 .5. In equal ities a nd Limit Theorems 11.6. St oc hastic Processes

363 368 377 38 1 388 393

Glossary of Notation

396

Summary of Important Results

400

Index

4 11

-. xvi

CONTENTS

VOLUME 1/ Chapter I

A Queuein g The ory Primer

I. Notation

2. 3. 4. 5. 6. 7. 8. 9. 10.

Gene ra l Results Markov, Birth-Death, and Poisson Processes The M /M / l Que ue The MI M l m Q ueueing System Markovian Que ueing Networks The M / G/l Queue The GIMII Queue T he GI Mlm Queue The G/G /l Que ue

Chapter 2

Bound s, Inequalitie s and Approximations

I. The Heavy-Traffic Approximation

2. 3. 4. 5. 6. 7. 8. 9. 10.

Chapter 3

An Upper Bound for the Average Wait Lower Bounds for the Average Wait Bounds on the Tail of the Waiting Time Distribution Some Remarks for GIGlm A D iscrete Approximation The Fluid Approximation for Queues Diffusion Processes Diffusion Approximation for MIG II The R ush-H our Approximation

Priority Queueing

I . The Model 2. An Approach for Calculating Average Waiting Times 3. The Delay Cycle, Generalized Busy Periods, and Waiting T ime Distributions 4. Conservation Laws 5. The Last-Come- First-Serve Queueing Discip line

CONTENTS

6. 7. 8. 9.

Head- of-the-Lin e Priorities Ti me-Dependent Prior ities Opt imal Bribin g for Qu eue Position Service-Tim e-Dep end ent Disciplines

Chapter 4

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.

Definitions and Models Distribu tion of Att ained Service Th e Batch Pro cessing Algorithm The Round-Robin Scheduling Algorithm The Last-Com e-First-Serve Schedul ing Algorithm Th e FB Schedul ing Algorithm Th e Mul tilevel Processor Sharing Schedulin g Algor ithm Selfish Schedu ling Algo rithms A Co nservation Law for T ime-Shared Systems Ti ght Bou nds on the Me an Response T ime F inite Popul ation Mod els Mult iple-R esour ce Mod els Mod els for Multiprogramming Remote Terminal Acce ss to Computers

Chapter 5

I. 2. 3. 4. 5. 6. 7. 8. 9. 10. II . 12.

Computer Time-Sharing and Multiaccess Systems

Computer-Communication Networks

Resource Sharin g Some Contrasts and Trade-Off's Networ k Structures and Packet Switchin g The ARPANET-An Operational Descripti on of an Existing Network Definitions, the Model, and the Problem Statement s Delay An alysis Th e Capacity Assignment Problem Th e Traffic Flow Assignm ent Problem The Capac ity and Flow Assignment Problem Some Topological Co nside rations-Applicatio ns to the ARPAN ET Satellite Packet Switchin g Grou nd Radio Packet Switching

xvii

xvi ii

CONTENTS

Chapter 6

1. 2. 3. 4. 5. 6. 7. 8. 9.

Computer-Communicatio n N etworks Measurement, Flow Control and ARP ANET Traps

Simulatio n and Routing Ea rly AR PANET Measur ements Flow Co ntro l Lockups, Degradations a nd Traps Network T hro ughput One Week of ARPA NET Data Line Overhead in the ARPANET Recent Changes to the Flow Co ntrol Procedure T he Cha llenge of the Futu re

Glossary Summary of R esults In dex

QUEUEING SYSTEMS VOLUME I: THEORY

PART

I

PRELIMINARIES

It is difficult to see the forest for the trees (especially if one is in a mob rather than in a well-ordered queue). Likewise, it is often difficult to see the impact of a collection of mathematical results as you try to master them; it is only after one gains the understanding and appreciation for their application to real-world problems that one can say with confidence that he understands the use of a set of tools . The two chapters contained in this preliminary part are each extreme in opposite directions. The first chapter gives a global picture of where queueing systems arise and why they are important. Entertaining examples are provided as we lure the reader on. In the second chapter, on random processes, we plunge deeply into mathematical definitions and techniques (quickly losing sight of our long-range goals); the reader is urged not to falter under this siege since it is perhaps the worst he will meet in passing through the text. Specifically, Chapter 2 begins with some very useful graphical means for displaying the dynamics of customer behavior in a queueing system. We then introduce stochastic processes through the study of customer arrival, behavior, and backlog in a very general queueing system and carefully lead the reader to one of the most significant results in queueing theory, namely, Little's result, using very simple arguments. Having thus introduced the concept of a stochastic process we then offer a rather compact treatment which compares many well-known (but not well-distinguished) processes and casts them in a common terminology and notation, leading finally to Figure 2.4 in which we see the basic relationships among these processes; the reader is quickly brought to realize the central role played by the Poisson process because of its position as the common intersection of all the stochastic processes considered in this chapter. We then give a treatment of Markov chains in discrete and continuous time; these sections are perhaps the toughest sledding for the novice, and it is perfectly acceptable ifhe passes over some of this material on a first reading. At the conclusion of Section 2.4 we find ourselves face to face with the important birth-death processes and it is here

2

PRELIMINARIES

where things begin to take on a relationship to physical systems once again . In fact , it is not unreasonable for the reader to begin with Section 2.5 of thi s chapter since the treatment following is (almost) self-contained from there throughout the rest of the text. Only occasionally do we find a need for the more deta iled material in Sections 2.3 and 2.4. If th e reader perseveres through Chapter 2 he will have set the stage for the balance of the textb ook.

I

Que ue ing Systems

One of life' s more disagreea ble act ivities, namel y , waiting in line, is the delightful subject of thi s book. One might reasonably ask, " Wha t does it profit a man to study such unpleasant phenomena 1" The an swer , of course, is that through understanding we gain compassion, and it is exactl y this which we need since people will be wa iting in lon ger and lon ger queues as civilizat ion progresses, an d we must find ways to toler ate the se unpleasant situa tio ns. Think for a moment how much time is spent in one's daily acti vities waiting in some form of a queue: waiting for breakfast ; sto pped at a traffic light ; slowed down on the highways and freewa ys ; delayed at th e en tran ce to o ne's parking facility; queued for access to an elevat or ; sta nding in line for the morn ing coffee; holding the telephone as it rin gs, and so o n. The list is endless, and too often also are the queues. The orderliness of queues varies from place to place ar ound the world. Fo r example, the English are terribly susceptible to formation of o rderly queues, whereas so me of the Mediterranean peopl es con sider th e idea ludicrous (have yo u ever tried clearing the embarkation pr ocedure at the Port of Brindisi 1). A common sloga n in the U.S. Army is, "Hurry up and wait." Such is the nature of the phenomena we wish to study. 1.1.

SYSTEMS O F FLOW

Queueing system s represent an example of a much bro ad er class of interest ing dynamic systems, which, for con venience, we refer to as " systems of flow." A flow system is one in which so me commodity flows, moves, o r is tran sferred through one or more finite-capacity channels in orde r to go from o ne point to another. For example , con sider the flow of a uto mobi le tra ffic t hr ough a road network, or the transfer of good s in a railway system, o r the st reami ng of water th rough a dam , or the tr ansmission of telephone or telegraph messages, or the passage of customers through a superma rket checko ut co unter, or t he flow of computer pr ogram s t hrou gh a time-shar ing computer system. In these examples the commodities are the a uto mobiles, the goo ds, the water, the telephone o r telegraph messages, th e customers, and the programs, respecti vely ; the channel or channels a re th e road network, 3

4

QUEUE ING SYSTEMS

the railway net wor k, the dam , the teleph one or telegraph networ k, the supermarke t checkout counter, and the computer processing system, respectively. The " finite capacity" refers to the fact th at the channel can satisfy the demands (placed upon it by the commodity) at a finite rate only. It is clear that the ana lyses of man y of these systems requ ire analytic tools drawn from a variety of discipline s and , as we shall see, queueing the ory is ju st one such disciplin e. When one an alyzes systems of flow, they naturally break int o two classes : steady and unsteady flow. The first class con sists of those systems in which the flow proceeds in a predictable fashion . Th at is, the qu antity of flow is exactly known and is const ant over the int erval of interest; the time when tha t flow appears at the channel, and how much of a demand that flow places upon the channel is known and consta nt. These systems are trivial to an alyze in the case of a single channel. For example , consider a pineapple fact ory in which empty tin cans are being transported along a conveyor belt to a point at which they must be filled with pineapple slices and must then proceed further down the conveyo r belt for addi tional operatio ns. In this case, assume that the cans arrive at a constant rate of one can per second and that the pineap ple-filling operation takes nine-tenths of one second per can . The se numbers are constant for all cans and all filling operations. Clearly thi s system will funct ion in a reliable and smooth fashion as long as the assumptions stated above continue to exist. We may say th at the arrival rate R is one can per second and the maximum service rate (or capacity) Cis 1/0.9 = 1.11111 . .. filling operations per second . The example above is for the case R < c. However , if we have the condition R > C, we all know wha t happens : cans and /or pineapple slices begin to inundate and overflow in the factory! Thus we see that the mean capacity of the sys tem must exceed the average flow requirements if chaotic congestion is to be avoided ; this is true for all systems of flow. Th is simple observation tells most of the sto ry. Such systems a re of little interest theoretically. T he more interesting case of stea dy flow is that of a net work of cha nnels. For stable flow, we obviously require that R < C on each cha nnel in the networ k. However we now run int o some serio us combinat orial problem s. For example , let us consider a rail way networ k in the fictitious lan d of Hatafla. See Fig ure 1.1. The scena rio here is that figs grown in the city of Abra must be transported to the destination city of Cadabra , makin g use of the railway netwo rk shown. Th e numbe rs on each chann el (sectio n of railway) in Figure 1.1 refer to the maximum numb er of bushels of figs which that cha nnel can handle per day. We are now co nfro nted with th e following fig flow problem: How man y bushels of figs per day can be sent from Ab ra to Cadabra and in wha t fashion sha ll this flow of figs take place ? The answer to such questions of maximal " traffic" flow in a variety of networ ks is nicely

1.1. Zeus

8

SYST EMS OF FLOW

5

Nonabel

Cadabra

Abra

Sucsamad

6

Oriac

Figure 1.1 Maximal flow prob lem. sett led by a well-kno wn result in net work flow theory referred to as the max-flow-min-cut theorem. To state this theo rem, we first define a cut as a set of channel s which, once removed from the network , will separate all possible flow from the origin (Abra) to the destination (Cadabra). We define the capacity of such a cut to be the total fig flow that can travel acro ss that cut in th e direction from origin to destination . For exa mple, one cut con sists of the bran ches from Ab ra to Zeus, Sucsam ad to Zeus , and Sucsamad to Oriac ; the cap acit y of this cut is clearly 23 bushels of figs per day. The max-flowmin-cut the orem states th at the maximum flow that can pass bet ween an origin and a destin ation is the minimum capacity of all cuts. In our example it can be seen th at the maximum flow is therefore 21 bu shels of figs per day (work it out). In general, one must consider all cut s that sepa rate a given origin and destination. This computation can be enormously time consuming. Fortunately, there exists an extremely powerful method for finding not only what is the maximum flow, but also which flow pattern ach ieves th is maximum flow. This procedure is known as the labeling algorithm (d ue to Ford and F ulkerson [FORD 62]) a nd is efficient in tha t th e computational requ irement grows as a small power of the number of nodes ; we present the algor ithm in Volume II , Ch apt er 5. In additio n to maximal flow problems, one can pose nume rou s other interesting and worthwhile questions regarding flow in such networks. For example , one might inq uire int o the minimal cost network which will suppo rt a given flow if we assign costs to each of the channels. Also , one might as k the sa me questions in network s when more than one origin and dest inati on exist. Co mplicating ma tters further, we might insist that a given netwo rk suppo rt flow of various kind s. for example, bushels of figs, carton s of cartrid ges and barrel s of oil. This multic omm od ity flow problem is an extremely difficult one, and its solution typically requires consi de rable computati onal effort. The se and numerous other significant problem s in networ k flow theory are addressed in the comprehensive text by Frank and Frisch [FRAN 71] and we shall see them aga in in Volume II , Chapter 5. Network flow theory itself requires met hod s from gra ph the ory, combinator ial

6

QUEUEING SYSTEMS

mathematics, optimization theory, mathematical programming, and heuristic programming. The second class into which systems of flow may be divided is the class of random or stochastic flow problems. By this we mean that the times at which demands for service (use of the channel) arrive are uncertain or unpredictable, and also that the size of the demands themselves that are placed upon the channel are unpredictable. The randomness, unpredictability, or unsteady nature of this flow lends considerable complexity to the solution and understanding of such problems. Furthermore, it is clear that most real-world systems fall into this category. Again, the simplest case is that of random flow through a single channel; whereas in the case of deterministic or steady flow discussed earlier in which the single-channel problems were trivial, we have now a case where these single-ehannel problems are extremely challenging and, in fact , techniques for solution to the single-channel or single-server problem comprise much of modern queueing theory . For example, consider the case of a computer center in which computation requests are served making use of a batch service system. In such a system, requests for computation arrive at unpredictable times, and when they do arrive, they may well find the computer busy servicing other demands. If, in fact, the computer is idle, then typically a new demand will begin service and will be ru~ until it is completed. On the other hand, if the system is busy, then this job will wait on a queue until it is selected for service from among those that are waiting. Until that job is carried to completion, it is usually the case that neither the computation center nor the individual who has submitted the program knows the extent of the demand in terms of computational effort that this program will place upon the system ; in this sense the service requirement is indeed unpredictable. A va riety of natural questions present themselves to which we would like intelligent and complete answers . How long, for example , maya job expect to wait on queue before entering service? How many jobs will be serviced before the one just submitted? For what fraction of the day will the computation center be busy? How long will the intervals of continual busy work extend? Such questions require answers regarding the probability of certain periods and numbers or perhaps merely the average values for these quantities. Additional considerations, such as machine breakdown (a not uncommon condition), complicate the issue further; in this case it is clear that some preemptive event prevents the completion of the job currently in service. Other interesting effects can take place where jobs are not serviced according to their order of arrival. Time-shared computer systems, for example, employ rather complex scheduling and servicing algorithms, which , in fact , we explore in Volume II, Chapter 4. The tools necessary for solving single-channel random-flow problems are

1.1.

SYSTEMS O F FLOW

7

conta ined and described withi n q ueue ing th eory, to which much o f th is text devote s itself. Th is requires a back ground in pr obability th eory as well as a n understanding o f complex variables and so me of the usual tr an sformcalculus meth ods ; th is material is re viewed in Appendices I and II. As in the case of determin istic flow , we may enla rge our sco pe of p robl ems to that of networks of channels in which random flow is encountered. An exa mple of such a system would be that of a computer network. Such a system con sists of computers connected together by a set of communica tion lines where the capacity o f the se lines for carryin g information is finite. Let us return to the fictiti ous land of Hatafla and assume that the railway net work considered earli er is now in fact a computer net work. Assume th at user s located a t Abra requ ire computational effort on the facility at Cad abra. The particular times a t which these requests are made are themselves unpredictable , and th e commands or instructio ns that describe these requests are also of unpredictable len gth . It is the se commands which mu st be transmitted to Cadabra over our communication net as message s. When a message is inserted int o the netw ork a t Abra , and after an appropriate deci sion rule (referred to as a routing procedure) is accessed, then th e message proceeds through the netw ork a long so me path. If a port ion of this pa th is busy, a nd it may well be, then the message must queue up in front of the bu sy channel and wait for it to bec ome free. Const ant decisions must be made regarding the flow of messages "and routing procedures. Hopefully, the message will eventually emerge at Cadabra, the computation will be performed , and the results will then be inserted into the network for delivery back at Abra. It is clear th at the problems exemplified by our computer net wor k involve a variety of extremely complex qu eueing problems, as well as networ k flow and deci sion problems. In a n earlier work [KLEI 64] the auth or addressed him self to certain as pects of the se questions. We de velop the an alysis of the se systems lat er in Volume II , Chapter 5. Having thus classified * systems of flow , we hope th at the reader understands where in the genera l scheme of things the field of queueing the ory ma y be placed . The method s from thi s the ory a re central to a nalyzing most stochas tic flow problems, an d it is clear from a n examina tion of the current litera ture that the field and in particular its applications a re growing in a viable a nd purposeful fashion. • The classifica tion described a bove places qu eueing systems within the class of systems of flow. This approach identifies and emph asizes the fields o f applicatio n for queu eing theory. A n a lterna tive a pproa ch wo uld ha ve been to place queueing theory as belongi ng to the field of app lied stochas tic processes ; this classifica tion would have emphasized the mat hema tical structure of queueing theory ra ther than its a pplica tions. Th e poin t of view taken in th is two-volume book is the form er one, namely. with a pplica tion of the theory as its major goal rat her than extension of the math emat ical for mal ism a nd results.

8

QUEU EIN G SYSTEMS

1.2. THE SPECIFICATION AND MEASURE OF QUEUEING SYSTEMS In o rder to completely specify a queuein g system, o ne mu st iden tify the stochas tic processes that describe the arriving stream as well as th estructure and di sciplin e of the service facility. Generally, the arri val pr ocess is described in term s o f the p robability distribution o f the interarrical times of custo mers a nd is den oted A (t) , where* A (t )

= P[time between arrivals ~

t]

(I.I )

The as sumption in most of queueing theory is that th ese interarrival time s are independent, identically distributed random variables (a nd, therefore, the strea m of arrivals forms a stationary renewal process ; see Chapter 2). Thus, onl y the distribution A (t) , which de scribes the time between a rrivals, is usually of significa nce. The second sta tistica l quantity th at mu st be described is the am ount of demand the se arrivals place upon the channel; thi s is usuall y referred to as the service tim e whose probability distribution is den oted by B(x) , that is, B(x) = P[service time ~ x ] ( 1.2) Here service time refers to the length of time th at a customer spends in the ser vice facility . N ow regarding the st ructure and discipline of the service facility , one must spec ify a variety of additio na l qu antities. One o f the se is the extent of storage capacity available to hold waiting customers a nd typically thi s quantit y is described in term s of the variable K ; often K is taken to be infinite. An additional specification involves the number of service stations ava ilable, and if more th an one is available, then perhaps the distribution o f service time will differ for each , in which case the distribution B(x) will include a su bscript to indicate that fact. On the other hand, it is so metimes the ca se that the arriving strea m con sists of more th an one identifiable class of customers ; in such a case the interarrival distributi on A (t ) as well as the service distr ibut ion B(x) may each be characteristic of each cla ss and will be identified aga in by use of a subscrip t o n these d istr ibution s. An other importa nt structura l descripti on o f a queueing system is th at of the queueing discipline; thi s describes the order in which customers a re taken from the queue a nd a llo wed int o service . For example, so me sta nda rd queueing disciplines are first-co mefirst-serve (F C FS), last-come-first-serve (LCFS), a nd random o rder of service. When the arriving customers are distin guishable according to gro ups, then we encou nter the case of priority queueing disciplines in which priority • Th e notat ion P[A] denotes, as usua l. the " pro bability of the event A,"

1.2.

THE SPECIFICATION AND ~I EASUR E OF QUEUEING SYSTntS

9

among gro ups may be esta blished. A further sta tement regarding the availability of the service facility is also necessary in case t he service faci lity is occasio na lly requ ired to pay attention to other ta sks (as, fo r example, its own breakdown). Beyo nd th is, q ueue ing systems may enjoy custo mer behavio r in the fo rm of defections from th e qu eue, j ock ey ing a mo ng th e man y qu eues, balking before ent ering a queue, bribing fo r queue positi on , cheating for q ueue po sition, a nd a variety of o the r interesting a nd not-unexpected humanlike cha rac terist ics. We will encounter these as we move th rough t he text in an o rderly fashion (first-come-fi rst-serve ac co rding to page nu mber). No w tha t we ha ve indi ca ted how one must specify a queueing system, it is appropriate t hat we ide nti fy the meas ures of performance a nd effectiveness th a t we sha ll obtai n by ana lysis. Basicall y, we a re int erested in the waiting time for a custo mer, the number of customers in the system, th e length of a busy period (the contin uo us interva l d uring which th e serve r is busy), the length of an idle period , a nd th e cu rrent 1I'0rk backlog expressed in un its of tim e. All t hese quant ities a re ra ndom varia bles a nd thus we seek th eir complete p rob a bilistic desc rip tion (i.e., their proba bility dist ribu tion fu nction ). Us ually , ho wever , to give th e distribution functio n is to give more th an one can easi ly make use of. Consequ en tly, we often settle fo r the first few mo ments (mean , var iance, etc.). Ha ppily, we shall begin with simp le co nside rations a nd de velop the tools in a st raigh tforwa rd fashio n , paying a tte ntio n to th e essential details of a na lysis. In t he followi ng pages we will enco unter a va riety of simple qu eueing problems, simple at least in the sense of description and usually rather so phistica ted in term s of so lution. However , in o rde r to do t his pr op erly, we first devote our efforts in the following chapter to desc ribing some of t he imp orta nt ra ndo m processes that ma ke up the a rriva l a nd service processes in o ur q ueueing systems. REFERENCES FORD 62 Ford, L. R. and D. R. Fu lkerson, Flows in Networks, Princeton University Press (Princeton, N.J.), 1962. FRAN 71 Frank . H. and I. T. Frisch, Communication. Transmission , and Transportation Ne twork s, Addison-Wesley (Reading, Mass.), 1971. KL EI 64 Kleinrock, L. . Communication Nets ; Stochastic Message Flow and Delay . McGraw-Hili (New York), 1964 , out of print. Reprinted by Dover (New York ), 1972.

2

Some Important Random Processes*

We assume that the read er is familiar with the basic elementary notions, terminology, and concepts of probability theo ry. Th e particular aspects of that theory which we requ ire are presented in summary fashion in Appendix II to serve as a review for th ose readers desi ring a quick refresher and remin de r; it is recommended that the material therein be reviewed, especially Sectio n 11.4 on transform s, generating functions , and characteristic function s. Included in Appendix " are the fo llowing important definitions, concep ts, and results :

•

• • • • •

2.1.

Sample space, events , and probability. Conditional probability, statistical inde pendence, the law of total probability, and Bayes' theorem. A real random va riable, its pro babili ty dist ribution function (PD F), its probability density func tion (pdf), and their simple properties. Events related to random variables and their p robabilities. Joint dist ribu tion functions. Functions of a random variable and t heir de nsity functions. Expectation . Laplace transforms , generating functions, and characteristic functi on s and their relationships and p ropertics.t Inequalities and limit theorems . Definition of a stochastic process.

NOTATI ON AND ST RUCTU RE FOR BASIC QUEUEING SYSTEMS

Before we plun ge headlong into a step-by-step development of queueing theory from its elementary not ions to its inte rmediate and then finally to some ad vanced material , it is impo rtant first that we understan d the basic • Section s 2.2, 2.3, a nd 2.4 may be skipped on a first read ing. [ is a transform theor y refresher. This materia l is a lso essential to the proper under standing of this text.

t Appendix 10

2.1.

NOTAn ON AND STRUC TU RE FO R BASIC QUEUEING SYSTEMS

II

·stru cture o f queues. Also, we wish to provide the read er a glimpse as to where we a re head ing in th is journ ey. It is o ur purpose in thi s sectio n to define so me notation , both sy mbo lic and gra p hic, and then to introduce one o f the ba sic sto chas t ic pr oce sses that we find in queueing systems. F urth er , we will deri ve a simple but significa nt result, which relates so me first moments of impo rta nce in these systems. In so doin g, we will be in a positi on to define the quantities a nd processes th at we will spend man y pag es study ing later in th e text. The syste m we co nsider is the very general queueing syste m G/G/m ; recall (fro m the Preface) th a t thi s is a system whose interarrival time di stribution A (I) is completel y a rbitra ry a nd who se service time di stribution B (x) is a lso completely arbitrary (a ll interar rival tim es and serv ice time s are assum ed to be inde pe ndent o f each o ther). The syste m ha s m servers and order of service is also quite a rbitra ry (in particular , it need not be first-come-first-serve). We focu s a ttentio n o n th e flow o f customers as the y arri ve , pass throu gh , a nd eventuall y lea ve thi s syste m: as such, we choose to number the cu stomers with the subsc rip t n a nd define C ,. a s foll ows : C n denotes the nth custom er to enter the system

(2. 1)

Thus, we may portray o ur system as in Figure 2.1 in which the box represents t he qu eu eing syste m a nd the flow of cust omers both in a nd o ut of the system is shown . One can immediately define so me rand om processes of int ere st. For example , we a re int erested in N( I) where * N(I) ~ number of cust om ers in the system at time

I

(2.2)

An oth er stochastic process o f interest is th e un finished wo rk V( I) th at exists in the system a t tim e I , th at is, V( I) ~ the unfinished work in th e system a t time I ~ th e rem a inin g t ime required to empty the system of a ll cu stomers pre sent a t time I

(2.3)

Whenever V( I ) > 0 , then the system is sa id to be bu sy , a nd o nly when V( I) = 0 is th e syste m sai d to be idle . The durati on and loca tio n of the se busy an d idl e peri ods a rc al so qu antiti es of int ere st.

. ,J; ~

Oueoei nq system

;I -

-

-

-

-

-

-

----'

Figure 2. 1 A general queueing system. • The notation

~

is to be read as "equals by defi nition."

12

SOME IMPOR TANT RANDOM P ROC ESSES

The det ail s of the se sto chastic processes may be ob served first by defining the foll owing va riables and then by displayin g the se va riab les on a n appropriate time diagram to be d iscussed belo w. We begin with the definition s. Recalling that the nth cu stomer is den oted by Cn. we define his a rriva l time to the queuein g system as Tn

~ a rriva l time for C n

(2.4)

(2. 5)

interarrival time s a re drawn from the dis-

P[t" ~ t]

=

A(t)

(2.6)

which is independent of n. Similarly , we define the service time for COl as X n ,;;

serv ice time for C n

(2.7)

and from our assumptions we have P[Xn ~ x ] = B(x)

'(2.8;

The sequences {t n } a nd {x n } may be th ought of as input va ria bles for OUI queueing system ; the way in which the system handles the se cust omers give: rise to queues and waiting times th at we mu st now defin e. Thus, we de fine t h. waiting time (time spent in the queue) * as . II' n

~ wa itin g ti me (in qu eue) fo r C;

(2.9

The total time spent in the system by COl is the sum of his waiti ng time a n, service time, which we denote by

s; ~ system time (q ueue plus service) for COl

= II' n + x"

(2. 1(

Thus we have defined for the nth cu stomer his a rriva l time, " his" intera rri v. time , his service time, his waiting time , a nd his system t ime. We find * T he terms " waiting ti me" and " queueing time" have conflicting defini tions within tl body of queueing-theory literatu re. T he fo rmer so metimes refers to the tota l time spent system. an d the latter then refers to the total time spent on queue ; however . these tv definitions are occasio na lly reversed . We a ttem pt to remove tha t confusion by defini waiting a nd queu eing time to be the sa me quant ity. name ly. the time spent wa iting ' queu e (but not being served ); a more appropriate term perhaps would be " wasted tim' Th e tota l time spent in the system will be referred to as "sys tem time" (occasiona lly kn o a s " flow time" ).

2.1.

NOTATION AND STR UCT UR E FOR BASIC QUE UEIN G SYSTEMS

13

expedien t at th is point to elaborate somewhat further on notation. Let us con sider the interarrival time In once again . We will have occasion to refer to the limiting random varia ble i defined by I- =~

I'im n-e cc

1n

(2.11)

which we den ote by I n -+ i. (We ha ve already requ ired that the interarrival times In have a distribution independent of n, but this will not necessaril y be the case with many other random variables of interest.) The typical notation for the probab ility distribut ion function (PDF) will be (2.12) and for the limiting PDF P[i ~ I] = A (I)

r

s

e

This we denote by A n(l) -+ A(t ) ; of course, for the interarrival time we have as sumed that A n(l ) = A (I) , which gives rise to Eq. (2.6). Similarly, the probability den sity function (pdf) for t n a nd i will be an(l) and aCt), respectively, and will be den oted as an(t) -+ aCt). Finally, the Laplace transform (see Appendix II) o f the se pdf's will be denoted by An *(s) and A *(s), respecti vely, with the obvio us notation A n*(s ) -+ A *(s) . The use of the letter A (a nd a) is meant as a cue to remind the reader that they refer to the interarrival time . Of .course, the moments of the interarrival time are of interest a nd the y will be denoted as follows * : E[l n]';;'i.

I) d

))

al it he in vo

ng on

-vn

(2.13)

Acc ording to our usual notation , the mean interarrival time for random variable will be given] by i in the sense that i. -+ i. As i, which is the average interarrival time between customers, frequently in o ur equ ati on s that a special notation ha s been follows : _ - S(y ),

time for C,

sn(Y) -+ s(y),

S n"(s)

-+

S *(s) (2.21)

All th is notation is self-eviden t except perh ap s for th e occas ional special symbo ls used for the first moment and occasionally the higher mom ent s of th e random va ria bles invol ved (tha t is, th e use of the sym bo ls )" a, Il, b, IV, a nd T ). The reader is, at thi s poin t , directed to the Gl ossary for a complete set of not ati on used in thi s bo ok . With the above not ation we now suggest a time-diagram notation for queues, which pe rmits a gra p hica l view of the dynamics o f ou r queueing system and a lso provides the det a ils of the underlying stochastic p roce sses. This diagram is shown in Figure 2.2. Thi s particu lar figure is show n for a

2.1.

NOTATION A:--ID STR UCTURE FOR BASIC Q UEUEI N G SYST EMS

s. C ll _ 1

'I C.

cr. ,!'.

15

C n +2

Servicer

C~ T.

Queue

Cn +1

C:"'+2

Tlme v-c-e--

T/u 2

'n +1

(11 +2

Cn

en +1

Cn -t2

Figure 2.2 Time-diagram nota tion for queues. first-co me-first-serve order of service, but it is e~sy to see ho w the figure may also be made to represent any order of service. In this time diagram the lower horizontal time line rep resents the queue and the upper hori zontal time line represents the service facility; moreove r, the diagram shown is fo r the case of a single server, although this too is easily generalized. An arrow approaching the q ueue (or service) line from below indicates that an arrival has occurred to the queue (or service facility) . Arrows emanating from the line indicate the departure of a customer from the queue (or service facility). In this figu re we see th at customer C n+1 arrives before customer Cn enters service; o nly when C ; departs from service may Cn+l ente r service and , of course, the se two events occur simultaneously. Notice that when Cn + 2 enters the system he finds it empty and so immediately proceeds through an empty queue directly int o the service facility . In th is diagram we have also sho wn th e waiting time and the system time for C n (note that 1\'n+2 = 0). Thus, as time proceed s we can identify the number of cu stomers in the syste m N(t), the unfini shed work Vet) , and also the idle a nd busy period s. We will find much use for thi s time -diagram notation in what follows. In a genera l que ueing system one expects that when the number of customers is lar ge then so is the waiti ng time. One manifestation of thi s is a very simple relati on ship between the mean number in the queueing system, the mean a rriva l rate of customers to that system, and the mean system time for customers. It is our purpose next to deri ve th at relati on ship a nd thereby familiarize ourselves a bit further with th e underlying behavior of the se system s. Referring back to Figure 2.1 , let us position ourselves at the input of the queueing system and count ho w man y customers ent er as a function of time . We denote this by ot(/) where

ot(t) ~ number of arrivals in (0, t )

(2.22)

16

SOM E IMPORTA NT RA NDOM PROCESSES

12, -- - - - - - - - - - -- - - - - - - - -- ----, 11 10 ~

E

B

'5 c

'0

" ~ ~

9 8 7 6 5

4 3 2 1

o L----"'="'" Time

l

Figure 2.3 Arrivals and departures. Alternat ively, we may position ourselves at the outp ut of the queuei ng system and count the number of departures that leave; thi s we den ot e by bet) ~ number of dep ar ture s in (0, r)

(2.23)

Sample functions for these two stochastic processes are sho wn in Figur e 2.3. Clearly N( t), the number in the system at time t, must be given by N( t )

= «(r) -

bet)

On the other hand, the tot al are a bet ween these two curves up to some point , say t , repr esents t he tota l time all customers have spe nt in the system (measur ed in unit s of customer-seconds) during the int erval (0, t) ; let us denote this cumulative area by yet ). M oreover, let At be defined as the average arrival rate (custo mers per second) during the interval (0, t); th at is, (2.24) We may define T, as the system time per customer averaged over all custome rs in the interval (0, t); since yet ) repre sents the accumulated customer- seconds up to time t , we may divide by the number of arriva ls up to that poin t to ob tain yet) Tt = (X( t) Lastly, let us define NI as the average nu mber of custome rs in the qu eueing system during the interval (0, r): this may be obtained by dividin g the accumulated number of customer-seconds by the total interval length t

2.1.

NOTA TION AND STRUCTURE FOR BASIC QUE UEING SYSTEMS

17

thusly

_

y(t)

N,= -

t

From the se last three equ ations we see

N, = A,T, Let us no w assume that our queueing system is such that the following limits exist as t -.. CtJ: A = lim J. ,

,-'"

T=lim T,

,- '"

Note that we are using our former definitions for A and T representing the average customer arrival rate and the average system time , respectively. If these last two limits exist, then so will the limit for N" which we denote by N now representing the average number of customers in the system ; that is,

N= AT

- (2.25)

Thi s last is the result we were seeking and is known as Little's result . It states that the average number of customers in a queueing system is equal to the average arrical rate of customers to that system, times the average time spent in that sys tem. * The above proof does not depend upon any specific assumption s regarding the arrival distribution A (r) or the service time distribution B(x) ; nor does it depend upon the number of servers in the system or upon the particular queueing discipline within the system. Th is result existed as a " folk the orem" for man y years ; the first to establish its validity in a formal way was J. D. C. Little [LITT 61] with some later simpl ifications by W. 'S . Jewell [JEWE 67) . S. Eilon [EILO 69 ) and S. Stidh am [STID 74). It is important to note that we have not precisely defined the boundary around o ur queueing system. For exampl e, the box in Figure 2.1 co uld apply to the entire system com pose d of qu eue an d serv er , in whic h case Na nd T as defined refer to qu ant ities for the entire syste m; on the othe r hand , we co uld have co nsidered the bound ar y of the queueing sys tem to co ntai n o nly the q ueue itsel f, in which case the relationsh ip wou ld have been No = AW - (2.26) where No represents the average number of customers in the queue and , as defined ea rlier, W refers to the avera ge time spent waitin g in the queue. As a third possible alterna tive the queueing system defined could have surrounded • An intuitive proof of Little' s result depends on the observa tio n that an arriving customer sho uld find the sa me average number, N, in the system as he leaves behind upon his departure. Thi s latter quantity is simp ly the arrival ra te A times his a verage time in system , T.

18

SOME I11PORTANT RANDOM PROCESSES

only the server (or servers) itself; in th is case our equation would have reduced to R. = AX (2.27) where R. refers to the average number of customers in the service facility (or facilities) and x, of course, refers to the average time spent in the service box. Note that it is always true that T = x+ W _ (2.28) The queuein g system could refer to a specific class of customers, per haps based on priority or some other attribute of this class, in which case the same relationship would apply. In other words, the average arri val rate of custo mers to a "queueing system" times the average time spent by customers in that "system" is equal to the average number of customers in the " system ," regardless of how we define that " system." We now discuss a basic parameter p, which is comm only referred to as the utilization fa ctor. The utilization factor is in a fundamental sense really the ratio R /C, which we introduced in Chapter I. It is the rat io of the rate at which "work" enter s the system to the maximum rat e (capacity) at which the system can perform this work; the work an arri ving customer brings into the system equals the number of seconds of service he requires. So, in the case of a single-server system, the definition for p becomes p ,;; (average arrival rate of customers) X (average service time) = AX _ (2.29) Thi s last is true since a single-server system has a maximum capacity for doin g work , which equals I sec/sec and each ar riving customer brings an am ount of work equal to x sec ; since, on the average, ..1. customers ar rive per second , then }.x sec of work are brought in by customers each second that passes, on th e average. In the case of multiple servers (say, III servers) the definition remains the same when one considers the ratio R/C , where now th e work capacity of the system is III sec/sec; expressed in terms of system parameters we then have a -AX p = _ (2.30) m Equat ions (2.29) and (2.30) apply in the case when the maximum service rat e is independent of the system sta te; if th is is not the case, then a more careful definition must be pro vided. The rate at which work enters the system is sometimes referred to as the traffic intensity of the system and is usually expressed in Erlangs ; in single-server systems, the utilizat ion facto r is equal to the traffic inten sity whereas for (m) multiple servers, the tr affic intensity equal s mp . So long as 0 ~ p < I, then p may be interpreted as p

=

E[fraction of busy servers1

(2.3I)

2.2.

DEFINITIO N AND CLASSIFICATION OF STOCHASTIC PROCESSES

19

[In the case of a n infinite number of servers, the ut ilizati on fact or p plays no impor ta nt part, and instead we are interested in the number of busy servers (and its expectati on).] Indeed , for the system GIGII to be stable , it must be th at R < C, that is, o ~ p < I. Occasionally, we permit the case p = 1 with in the ran ge of sta bility (in particul ar for the system 0 /0 /1). Stability here once again refer s to the fact that limiting distributions for all random vari ables of interest exist , and that all customers are eventually served. In such a case we may carry out the following simple calcul ation . We let 7 be an arbitrarily long t ime interval ; during this interval we expect (by the law of large numbers) with probability 1 that the number of arrivals will be very nearly equal to .AT. M ore over , let us define Po as the probability that the server is idle at some randomly selected time . We may, therefore, say that during the interval 7, the server is busy for 7 - TPO sec, and so with pr obability I , the number of customers served during the interval 7 is very nearly (7 - 7po)fx . We may now equate the number of arri vals to the number served during thi s int erval, which gives, for lar ge 7,

Thus, as 7 ->- 00 we ha ve Ax = I - Po; using Defin iti on (2.29) we finall y ha ve the important conclusion for GfG /l p =

1 - P»

(2.32)

The interpretati on here is that p is merely the fracti on of time the server is bu sy ; thi s suppo rts the conclusion in Eq. (2.27) in which Ax = p was shown equal to the averag e number of customers in the service facilit y. Thi s, then , is a rapid look at a n overall queueing system in which we ha ve exp osed so me of the ba sic sto chast ic processes, as well as some of the impo rta nt de finition s a nd notation we will enc ounter. More over , we have establi shed Little's result , which permits us to calcul ate the average number in the system once we have calculated the average time in the system (or vice versa). N ow let us move on to a more careful study of the imp ortant stochas tic processes in our queueing systems. 2.2 *.

DEFINITION AND CLASSIFICATION OF STOCHASTIC PROCESSES

At the end of Appendix II a definiti on is given for a stochas tic process, which in essence states that it is a famil y of random vari able s X (t) wher e the • The reader may choos e to skip Sections 2.2, 2.3, and 2.4 a t this point a nd move directly to Section 2.5. He may then refer to this materi al only as he feels he needs to in the balan ce of the text.

20

SOME IMPORTANT RAN DOM PRO CESSES

random variables are "indexed" by the time parameter I. F or example, the number of pe ople .sitting in a movie theater as a funct ion o f time is a stochastic process, as is also the atmospheric pressure in that movie the ater as a functi on of time (at least those functi on s may be m odeled as stoc has tic processes). Often we refer to a stochastic process as a random process. A rand om process may be thought of as describing the moti on of a particle in some space. The classification o f a random process depends up on three quantities: the slate space; the index (lim e) parameter; and the statistical dependencies among the random va ria bles X (I ) for different va lues o f the index parameter t. Let us discuss each o f these in order to provide the general framework for random processes. Fir st we con sider the state space. The set of possible va lues (o r states) th at X (I) may take on is called its state space. Referring to our analogy with regard to the motion o f a particle , if the positions th at particle may occupy a re finite or countable, then we say we have a discrete-state process, often referred to as a chain. The state space for a cha in is usually the set o f inte gers {O, 1,2, .. .}. On the other hand, if the permitted positions o f the particle are over a finite or infinite continuous interval (or set of such intervals), then we say that we ha ve a cont inuous-state process. Now for the index (lime) parameler . If the permitted times at which ch an ges in positi on may take place are finite o r countable, then we say we hav e a discrele-(tim e) param eter process; if these changes in positi on may occur an ywhere within (a set of) finite or infinite intervals on the time axis, then we say we hav e a continuous-parame ter process. In the former case we o fte n write X n rather than X(I) . X n is often referred to as a random or stochas tic sequellce whereas X (I) is often referred to as a random or stochas tic process . The truly distinguishing feature of a stochas tic p roce ss is th e relati on ship of the random va ria bles X (I) or X n to other members of the sa me famil y. As d efined in Appendi x II , one mu st specify the co mplete j oint dis trib ution function among the random variables (which we ma y th ink of as vecto rs den oted by the use of boldface) X = [X(t l ), X( I.) , . . .J, namel y, Fx(x; t) ~ P[X(t I) ~ Xl ' · .. , X( l n) ~ xn l

(2.33)

for a ll x = (Xl' X., . . . , X n ) , t = (II> I. , ... , In), and 11. As menti oned there, thi s is a formidable ta sk ; fortunately , many interesting sto chastic processes perm it a simpler description. In any ca se, it is the funct ion Fx(x ; t) th at really de scribes the dependencies a mo ng the random va ria bles of th e stoc has tic process. Below we de scribe some of the usual type s o f sto chas tic pr ocesses th at a re ch aracterized by different kinds of dependency relati on s am on g their rand om va riables. We provide thi s cla ssificati on in order to give t he read er a global view of this field so that he may better understand in which particular

2.2.

DEFI~ 1TI0N AND C LASSIFICAT ION OF STOCH ASTI C PROC ESSES

21

region s he is o pera ting a s we proceed with our st udy of queueing theory and it s related sto chas tic pr ocesses. (a) Stationary Processes. As we discuss at the ver y end of Appendix II, a sto chas tic process X (I ) is sa id to be sta tiona ry if Fx(x ; t) is inv ari ant to shifts in time for a ll va lues o f its arguments; th at is , given an y con stant T the following must hold : FX(x ; t

+ T) = Fx (x ; t)

(2.34)

where the notati on t + T is defined as the vector ( 11 + T , 12 + T, . •. , I n + T) . An associated noti on , that o f wide-sense stationarity, is identified with the random process X (I) if merely both the first and second moments are independent of the location o n the time a xis, th at is , if E[X(I )] is independent of I and if E[X(I)X(I + T)] depends only upon T and not upon I. Observe that all st ati onary processes are wide-sen se stati onary, but not conversely. The theory o f sta tio na ry rand om pr oce sses is, a s o ne might expect, simp ler th an that for nonstationary processes. (b) Independent Processes. The simplest a nd most tr ivial sto chas tic process to con sider is the random seq uence in which {X n } forms a set of independent random variables, that is , the j oint pdf defined for o ur sto chastic proce ss in Appendix .II mu st fact or in to the product, thusly

(2.35) In th is case we are stretching th ings somewhat by calling such a sequence a rand om proce ss since there is no stru cture or dependence among the random variables. I n the case of a continuous random process, such an independent pr oce ss may be defined, and it is commonl y referred to as " white noise" (an example is the time derivative of Brownian motion). (c) Markov Processes. In 1907 A. A. Mark ov published a paper [MARK 07] in which he defined and investigated the properties of what are now kn own as Mark ov processes . In fact, what he created was a simp le and highly useful form o f dependency amon g the random vari ables forming a stochastic process, which we now describe. A Mark ov proces s with a di screte state space is referred to as a Markov chain. The d iscrete-time Markov chain is the easiest to conceptualize and understand. A set of random variables {X n } forms a Markov chain if the pr obability that the next va lue (sta te) is X n +1 depends onl y up on the current value (state) X n and not upon any previous va lues. Thus we have a random sequence in which the dependency extends backwards one unit in time. That

22

SOME IMPOR TANT RANDO~I PROCESSES

is, the way in which the entire past histo ry affects the future of the process is completely summarized in the current value of the process. In t he case of a discrete-time Markov chain the instants when state changes may occur are preord ained to be at the integers 0, 1,2, . . . , n, . . . . In the case of the continuous-time Markov chain, however, the transition s between states may take place at any instant in time. Thu s we are led to consider the rand om variable that describe s how long the proce ss remain s in its curr ent (discrete) state before making a tr ansition to some ot her state . Because the Markov pr operty insists that the past history be compl etely summarized in the specification of the current state, then we are not free to requ ire th at a specification also be given as to how long the proce ss has been in its current sta te ! Th is imposes a heavy con straint on the distribution of time that the process may remain in a given state. In fact , as we shall see in Eq. (2.85), "this state time must be exponent ially distributed. In a real sense, then , the exponential distribution is a continuous distribution which is "rn emoryless" (we will discuss this not ion a t considerable length later in this chapter). Similarl y, in the discrete-time Markov chain , the process may remain in the given state for a time that must be geome trically distributed ; this is the only discrete pr obab ility mass funct ion that is memoryless. This memoryless property is requi red of all Markov cha ins and restri cts the generality of the processes one would like to cons ider . Expressed analytically the Marko v property may be written as P[X(tn+l) = x n+1 X (t n) = Xn, X( t n_1) = Xn_l>' . . ,X(t l) = = P[X(t n+l) = x n+1 I X (t n) = xnl 1

xtl (2.36)

where t 1 < t 2 < . .. < I n < t n + 1 and X i is included in some discrete sta te space. The consideration of Markov processes is central to the study of queueing theory and much of this text is devoted to th at study. Therefore , a good porti on of thi s chapter deals with discrete-and continuous-time Mar kov chain s. (d) Birth-death Processes. A very important special class of Mar kov chains has come to be known as the birth-death process. The se may be either discrete-or continuou s-time processes in which the defining condit ion is that sta te transition s take place between neighboring sta tes only. That is, one may ch oose the set of integer s as the discrete state space (with no loss of generality) and then the birth-death process require s that if X n = i, then Xn+l = i - I, i, or i + I and no other. As we shall see, birth -death processes have played a significant role in the development of queueing the ory. Fo r the moment, however , let us proceed with our general view of stoc hastic processes to see how each fits int o the gener al scheme of thin gs.

2.2.

DEF INITlO N AN D CLASSIFIC ATI ON OF STOCHASTI C PR OCESSES

23

(e) Semi-Markov Processes. We begin by discussing discrete-time semi-Ma rkov proce sses. The discrete-time Mark ov chain had the propert y that at every unit inter val on the time axis the process was required to make a transition from the current state to some other state (possibly back to the same state). The transition probabilities were completely arbitrary; however , the requirement that a transition be made at every unit time (which really came ab out because of the Markov property) leads to the fact that the time spent in a sta te is geo metrically distributed [as we shall see in Eq. (2.66)]. As mentioned earlier, this impo ses a strong restriction on the kind s of processes we may consider. If we wish to relax that restriction , namel y, to permi t an arbitra ry distribution of time the proce ss may remain in a sta te, then we are led directly into the notion of a discrete-time semi-Markov process; specifically, we now perm it the times between state transitions to obey an arbitrary probability distribution. Note , however, that at the instants of state tran sition s, the process behaves just like an ordinary Markov chain and, in fact , at th ose instants we say we have an imbedded Markov chain. Now the definition of a continu ous-time semi-Markov pr ocess follows directly. Here we permit state transitions at any instant in time . However, as opposed to the Mar kov process which required an exponentially distributed time in state, we now permit an arbitrary distribution . Thi s then affords us much greater generality, which we are happy to employ in our study of queueing systems. Here , again , the imbedded Markov process is defined at those instants of state transition. Certainly, the class of Markov processes is contained within the class of semi-Markov processes. (f) Random Walks. In the stud y of random processes one often encounters a process referred to as a random walk . A random walk may be th ought of as a particle moving am ong sta tes in some (say, discrete) sta te space. What is of interest is to identify the location of the particle in that state spac e. Th e salient feature c f a rand om walk is th at the next positio n th e pr ocess occupies is equal to the previ ous position plu s a random variable whose value is drawn independently from an arbitrary distribution ; thi s distribution , however, does not change with the sta te of the pro cess. * Th at is, a sequence of random variables {S n} is referred to as a random walk (sta rting at the origin) if

S n = X,

+ X z + ... + X n

n

=

I, 2, . . .

(2.37)

where So = 0 and X" X z , .. . is a seq uence of independent rand om varia bles with a comm on distribution. The inde x n merely counts the nu mber of sta te transitions the process goes through ; of course , if the instants of the se tran sition s are taken from a discrete set, then we ha ve a discrete-time random

* Except perhap s at some bound ary states.

24

SOME IMPOR TANT RAND OM PR OCESSES

walk, whereas if they a re taken from a continuum , th en we have a con tinu oustime random walk. In any case , we assume th at the interval between these tr an sition s is distributed in an a rbitra ry way a nd so a random walk is a special case of a semi-Ma rkov process. * In the case when the co mmon distribution for X n is a discrete distribution , th en we ha ve a discrete- stat e random wal k ; in thi s case the transiti on probability Pi; of goi ng from sta te i to stat e j will depend only up on th e differenc e in indices j - i (which we den ote by q;_;). An exa mple of a continuous-t ime rand om walk is tha t of Brownian mot ion ; in the disc rete-time case a n exa mple is th e total number of head s observed in a seq uence of indepe ndent coin tosses. A random walk is occasionally referred to as a process with " independent increments." (g) Renewal Processes. A renewal proce ss is rela ted] to a random walk. However, the interest is not in followin g a pa rticle am ong man y sta tes but rather in counting transitions th at take place as a functi on of time . T ha t is, we co nsider the real time axi s on which is laid ou t a sequence of points ; the distribution of time between adj acent point s is an a rbitrary common distribution and each point corresponds to an instant of a state tra nsition. We ass ume tha t the process begins in sta te 0 [i.e., X(O) = 0] a nd increases by unity at each transiti on ep och ; th at is, X (t) eq uals the number of sta te tran siti on s that ha ve taken place by t. In thi s sense it is a special case of a rand om walk in which q, = I and q; = 0 for i ~ I. We may think of Eq. (2.37) as describin g a rene wal pr ocess in which S ; is the random va riable den ot ing the time a t which the nt h tr an siti on tak es place. As earl ier , the seq uence {Xn } is a set of inde pende nt identically distributed random variab les where X n now represent s th e time bet ween the (n - I)th a nd nth tr an sition. One sho uld be . careful to distinguish the interpretati on of Eq. (2.37) when it ap plies to renewal pr ocesses as here and when it a pplies to a random walk as ea rlier. The d ifference is that here in the renewal process th e equ at ion describes the time of the nth renewal or tran sition , whereas in the rand om walk it describes the state of the pr ocess and the tim e between sta te tr a nsitions is some ot he r rand om va ria ble. An impo rta nt example of a renewal process is th e set of a rrival insta nts to th e G /G/m queue. In th is case, X n is identified with the interarrivaI time . • Usually, the distribution of time between intervals is of lillieconcern in a random walk; emphasis is placed on the value (position) S n after n transitions. Often, it is assumed that this distribution of interval time is memoryless, thereby making the randomwalk a special case of Markov processes; we are more generous in our definition here and permit an arbitrary distribution. t It may be considered to be a special case of the random walk as defined in (f) above. A renewal process is occasionally referred to as a recurrent process.

2.2.

DEFINITION AND CLASSIfi CATIO N OF STOCHASTIC PRO C ESSES

S Prj

IT MP Pi j arbitrary

~l

25

P

arbitrary arbitrary

RW

n.: IT

qj - i

arbi trary

RP q, =

1

ITarbitrary

Figure 2.4 Relationships among the interesting random processes. SMP : SemiMarkov process; MP: Markov process; RW: Random walk; RP: Renewal process; BD: Birth-Death Process. So there we have it-a self-con sistent classification of some interesting stoc hastic processes. In order to aid the reader in understanding the relationship amo ng Markov .pr ocesses, semi-Markov processes, a nd the ir special cases, we have prepared the diagram of Figure 2.4, which sho ws thi s relationship for discrete-state systems. The figure is in the form of a Venn diagram. Moreover , the symbol Pii denotes the probability of making a tran sition next to sta te j given that the process is currently in state i. Also , fr den ote s the distribution of time between transitions; to say th at "fr is mernoryless" implies that if it is a discrete-time process, thenfr is a geometric distribution , whereas if it is a continuous-time process, then fr is an exponential distributi on . Furthermore, it is implied that fr may be a functi on both of the current a nd the next state for the pr oce ss. The figure shows that birth-death processes form a subset of Markov pr ocesses, which them selves form a sub set of the class of semi-Markov processes. Similarl y , renewal processes form a subset of random walk pr ocesses which also are a subset of semi-Ma rko v processes. Moreover , there are some renewal processes that may also be classified as birth-death

26

SOME IMPORTANT RANDOM PROCESSES

processes. Similarl y, those Markov processes for wh ich PH = q j - i (tha t is, where th e tran sit ion probabilities depend only up on the di fference of the indices) ove rla p those random walks whercj', is mem or yless. A rand om wa lk for which [, is memo ryless and for which q j- i = 0 when Ij - i/ > I ove rlaps the class of birth-death processes. If in addition to thi s last requirement our rando m walk has q, = I , then we have a process th a t lies a t the intersectio n of all five of the processes show n in th e figure. This is referred to as a "pure birth" pr ocess ; alt ho ugh /, must be. memoryless, it may be a distribution which depend s up on the sta te itself, ii ]; is independent of the sta te (th us giving a con stant " birth rate" ) then we have a process th at is figura tively and literally at the "center" of the study of stochastic pr ocesses a nd enjoys the nice properties o f each ! This very special case is referred to as th e Poisson p rocess and plays a major role in queueing the ory. We shall de velop its properties later in thi s chapter. So much for the classificati on of stochas tic processes at this poin t. Let us now elab orate up on the defin ition and properties of discrete-state Markov processes. Thi s will lead us naturally int o some of th e elementa ry qu euein g systems. Some of the required th eory beh ind the mo re sop histicated continuou s-stat e Markov processes will be developed later in th is work as the need a rises. We begin with the simp ler discrete -state , discrete-time Markov chains in the next section and foll ow th at with a section on discrete-state, continu ou s-t ime Mark ov chains.

2.3. DISCRETE-TL\tIE MARKOV CHAINS' As we ha ve said , Markov processes may be used to describe the motion of a particle in so me space. We no w con sider discrete-t ime Mar kov chai ns, which pe rm it the particle to occu py discrete positions and permi t transiti on s between these position s to tak e place only a t discre te time s. We present the elements of t he th eor y by carrying along th e following contemp orary exa mple. Con sider the hipp ie who hitchhikes from city to city acr oss the country. Let X n den ote the city in which we find our hippie at noon on da y n. When he is in so me particular city i, he will accept the first ride leavi ng in the evening from th a t city. We assume that the tr avel tim e betwee n any two cit ies is negligible . Of co urse, it is possible th at no ride comes alo ng, in which case he will remain in city i un til the next evening. Since vehicles head ing for va rio us neighboring cities come alo ng in some un predictable fash ion , the hippie's posi tio n at so me time in th e fut ure is clearly a rand om variable. It turns out t hat th is random variable may properly be described throu gh the use of a Markov cha in. • See footnote on p. 19.

DISCR ETE- Tl ~IE MARK OV CHA IN S

2.3.

27

We hav e the foll owing definition D EFINITIO N : The seq uence of random variables Xl' X 2 , •• • forms a d iscrete-time Markov chain if for all n (n = I, 2, . . .) and a ll possible values of the random variables we have (for i 1 < i 2 < . .. < in) that

P[X.

= j I Xl =

iI' X 2 = i 2, .. . , X . _1 = in_I]

= P[X. = j

IX . _

1

= in_I]

- (2.38)

In term s of our example, this defin ition merely states that the city next to be visited by the hippie depends only upon the city in which he is currently located and not up on all the pre vious cities he has visited . In th is sense the memory of the random p rocess, or Markov chain , goes baek only to the mo st recent position of the particle (hippie). When X . = j (the hipp ie is in cit y j on day 11), then the system is said to be in state E ; at time n (or at the nth step) . T o get our hippie started on day 0 we begin with some in itial prob ab ilit y distribution P [X o = j] . The expression on the right side of Eq. (2.38) is referred to as the (o ne-step) transition probability and gives the conditional p robability of making a transition from state E i . _ 1 at step n - I to sta te E; at the nth step in the proces s. It is clear that if we are given the initial state probability distributi on and the transition probabilities, then we can uniquely find the probabi lity of bei ng in various states at time n [see Eqs. (2.55) and (2.56) below]. If it turns out that' the transition probabilities are independent of n, then we have what is referred to as a hom ogeneous Markov chain and in th at case we make the further definition Pi; ,;; P[X.

= j IX n_ 1 =

i]

(2.39)

which gives the probability of going to sta te E; on the next step, given that we a re currently at sta te i. What fo llows refers to homogene ous Mark ov ch ain s only. These chain s are such tha t thei r transiti on probabilities are statio na ry with time ": therefore, given the current city o r state (pun) the p robability of various states III steps into the future depends only upon m and not up on the current time; it is expedient to define the m-step tr ansition probabilities as p:i ) £ P[X . +m = j I X n = i] - (2.40 ) From the Markov property given in Eq. (2.38) it is easy to establ ish the following recursive formula for calculating plj ): ( m)

Po

=

,, £.,

<m- i )

Pik

Pki

III

= 2,3 , . ..

(2.41)

k

This equation me rely says that if we a re to travel from E, to E; in m steps, • No te that although this is a Marko v process with sta tionary transitions, it need 1101 be a stationary random process.

28

SOME IMPORTA NT RANDOM PROCESSES

then we must do so by first traveling from E i to some state E k in m - I steps and then from Ek to E, in one more step ; the probability of the se last two independent events (remember this is a Markov chain) is the pr oduct of the probability of each and if we sum this product over all pos sible intermediate states Ek , we arrive at p:i ). We say that a Markov chain is irreducible* if every state can be reached from every other state; that is, for each pair of states (E , and Ej ) there exists an integer mo (which may depend upon i and j) such that (mol >

PH

0

Further, let A be the set of all states in a Markov chain. Then a subset of states Al is said to be closed if no one-step transition is possible from any state in Al to any state in Ale (the complement of the set AI). If Al consi sts of a single state, say E i , then it is called an absorbing state; a necessary and sufficient condition for E, to be an absorbing state is P« = I. If A is closed and does not contain any proper subset which is closed , then we have an irreducible Markov chain as defined above. On the other hand , if A contains proper subsets that are closed, then the chain is said to be reducible. If a closed subset of a reducible Markov chain contains no closed subsets of itself, then it is referred to as an irreducible sub-M arkov chain; the se subchains may be studied independently of the other sta tes. It may be that our hippie prefers not to return to a previously visited city. However, due to his mode of travel thi s may well happen, a nd it is imp ortant for us to define th is quantity. Accordingly , let

f i n) ~ P [first return to E , occurs n steps after leaving EjJ It is then clear that the probability of our hippie ever returning to city j is given by co

fj

=

2: f~n ) =

P[ever returning to E;]

n_ 1

It is now possible to classify sta tes of a Markov cha in according to the value obtained for /;. In particular, if!j = I then sta te E, is said to be recurrent ; if on the other hand ,/; < I, then sta te E , is said to be transient. Furthermore, if the o nly possible steps at which our hippie can return to sta te E , a re y , 2y , 3y , . . . (where y > 1 and is the largest such integer), th en sta te E, is said to be periodic with peri od y; if y = 1, then E, is aperiodic. Con sidering sta tes for which/; = 1, we may then define th e mean recurrence tim e of E, as 1'V[ j

='" 2:a:> nf \n)

(2.42)

n =l

• Man y of the intcresti ng Markov chains which one encounters in queueing theory are irreducible.

2.3.

DISCRETE -TIM E MARK OV CH AINS

29

This is me rely the average time to return to E;. With thi s we may then classify sta tes even further. In particular, if M ; = 00 , then E; is said to be recurrent null , whereas if M ; < 00, then E; is said to be recurrent nonnull. Let us define 71"Jnl to be the pr ob ability of finding the system in state E; at the nth step, that is, 71"jnl ;; P[X n = j] _ (2.43) We may now state (without proof) two important the orems. The first comments o n the set of sta tes for an irreducible Markov chain.

Theorem 1 The states of an irreducible Mark ov chain are either all transient or all recurrent nonnull or all recurrent null. If p eriodic, then all states have the same period y. Assum ing th at our hippie wanders fore ver , he will pass through the various cities o f the nation many times , and we inquire as to whether o r not there exists a stationary probability distribution {71";} describing his probability of being in cit y j a t some time arbitrarily far into the future . [A pr ob ability distribu tion P; is said to be a stationary distribution if when we choose it for our initial state distribution (that is, 71"JOI = Pi) then for all n we will ha ve 71"Jnl = P;.] Solvin g for {71";} is a mo st important part of the an alysis of Markov chains. Our second theorem addresses itself to thi s question .

Theorem 2 In an irreducible and aperiodic homogeneous Mark ov chain the lim iting probabilities TTj

= lim 1T ~ n )

(2.44)

n -",

alway s exist and are independent of the initial state probability distr ibution. .M oreover, either (a) all states are transient or all states are recurrent null in which cases 71"; = 0 f or all j and there ex ists no sta tio na ry distribution, or (b) all states are recurrent nonnull and then 71"; > 0 f or all j , in which case the set {1T;} is a stationary probability distribution and Tr j

1

=-

(2.45)

AI ;

In this case the quantities 7T j are uniquely determined through the fo llowing equations

1 1T j

=I =

71",

(2.46)

L 1Ti P U

(2.47)

30

SOME IMPORTANT RA NDOM PROC ESSES

We now introduce the notion of ergodicity. A state E; is said to be ergod ic if it is aperiodic, recurrent, and nonnull; that is, if;; = I, M ; < co, and y = I. If all states of a Markov chain are ergodic, then the Mark ov chain itself is said to be ergodic. Moreover, a Markov chain is said to be ergod ic if the probability distribution {r.)"J} as a function of n always converge s to a limitin g stationary distribution {7T;} , which is independent of the initial state distribution. It is easy to show that all states of ajinite* aperiodic irreducible Markov chain are ergodic. Moreover, among Foster's criteria [FELL 66] it can be shown that an irreducible and aperi odic Markov chain is ergodic if the set of linear equations given in Eq. (2.47) has a nonnull solution for which L;17T;1 < co. The limiting probabilities {7T;}, of an ergodic Markov chain are often referred to as the equilibrium probabilities in the sense that the effect of the initial state distribution 7T)0 1 has disappeared. By way of example, let's place the hippie in our fictitious land of Hatafla , and let us consider the network given in Figure 1.1 of Chapter I. In order to simplify thi s example we will assume that the cities of Nonabel , Cadabra, and Oriac have been bombed out and that the resultant road network is as given in Figure 2.5. In this figure the ordered links represent permi ssible directions of road travel ; the numbers on these links repre sent the probability (Pi;) that the hippie will be picked up by a car travelin g over that road, given that he is hitchhiking from the cit y where the arrow eman ate s. Note that from the city of Sucsamad our hippie has probability 1/2 of remaining in that city until the next day. Such a diagram is referred to as a state-transition diagram. The parenthetical numbers following the cities will henceforth be used instead of the city names. Zeus

(1)

Abra (0 )

Figure 2.5 A Markov chain. • A finite Mar kov chain is one with a finite number of states. If an irre ducible Mark ov cha in is of type (a) in Theorem 2 (i.e., recur rent null or transient ) then it ca nno t be finite .

2.3.

DISCRETE-TIME MARKO V CHAINS

31

In order to continue our example we now define, in genera l, the transition probability matrix P as consisting of elements Pu , that is, - (2.48) If we further define the probability vector

1t

as (2.49)

then we may rewrite the set of relati ons in Eq. (2.47) as re

=

1tP

- (2.50)

F or our exa mple shown in Figure 2.5 we ha ve

3

0

P=

4

1 4

0

1 4

1 4

~l

~J

a nd so we may so lve Eq. (2.50) by conside rin g the three equation s deri vable from it, th at is,

3

7T1

= -

4

7TO

+

0 1T1

+ -1 4

7T2

(2.51)

131 = 1To + - 1T 1 + - 7T., - 4 4 2 -

7T.

N ote from Eq . (2.51) that the first of the se three equ ati on s equ als the negat ive sum of the seco nd a nd third , indicating th at there is a linear dependence am ong them. It alway s will be the case th at o ne of the eq ua tions will be linea rly de penden t on the others, and it is therefore necessary to int roduce the addition al con servati on relat ionship as given in Eq. (2.46) in order to solve the system. In ou r example we then requi re (2.52) Thus the sol utio n is obtai ned by simultane ously so lving any two of the

32

SOME IMPORTANT RAN DO~l PROCESSES

equations given by Eq. (2.51) along with Eq. (2.52). Solving we obtain 170

= -1 =

17,

= -7 =

5

0.20 0.28

25

172

(2.53)

= -13 = 0.52 25

Thi s gives us the equilibrium (stationary) state probabilities. It is clear that this is an ergodic Markov ch ain (it is finite and irreducible). Often we are interested in the transient behavior of the system. The transient beha vior involves solving for 17) n ), the probability of finding our hippie in city j at time II. We also define the probability vector at time II as 1t

1n ) ~ [17 1n ) 171n) 17(n) o , 1 , 2 , • • •]

- (2.54)

Now using the definition of tran sition pr obabil ity and makin g use of Definition (2.48) we have a method for calculatin g 1t1l) expressible in term s of P and the initial state distribution 1t101• That is, 1tlll

= 1t101P

Similarly , we may calculate the state probabilitie s at the second step by n ( 2)

= 1tlllp

From this last we can then generalize to the result II

=

1,2, ...

_ (2.55)

=

I , 2, . ..

- (2.56)

which may be solved recurs ively to obt ain II

Equation (2.55) gives the general method for calculatin g the state probabilities steps int o a process, given a tran sition pr obability mat rix P and an initial state vector 1t1O ). From our earlier definitions , we have the stationary probability vector

II

1t

= lim 1t( n )

. assumin g the limit exists. (From Theorem 2, we know that this will be th e case if we have an irreducible aperiodic homogeneous Markov chain.)

2.3.

DISCRETE-TIME MARKOV CHAINS

33

Then , from Eq . (2.55) we find

and so 7t

=

7tP

which is Eq. (2.50) again. Note that the solution for 7t is independent of the initial state vecto r. Applying this to our example , let us assume that our hippie begins in the city of Ab ra at time 0 with probability I, th at is 7t(0 )

=

[1 ,0,0]

(2.57)

From thi s we may calculate the sequence of values 7t( n ) and the se are given in the chart below. The limitin g value 7t as given in Eq . (2.53) is also entered in this chart. n

0

7T~n )

I 0 0

(n )

7T1 (n )

7T2

0 0.75 0.25

2

3

4

co

0.250 0.062 0.688

0.187 0.359 0.454

0.203 0.254 0.543

0.20 0.28 0.52

We may alternati vely have chosen to assume th at the hippie begins in the city of Zeu s with pr obability I , which would give rise to the init ial sta te vecto r 7t (O) = [0, I, 0] (2.58) and which result s in the following table: n ( n)

7T0

o o

(n )

I

( n)

o

7T,

7T2

0.25

o

0.75

2

3

4

0.187 0.375 0.438

0.203 0.250 0.547

0.199 0.289 0.512

0.20 0.28 0.52

Similarly , beginning in the city of Sucsamad we find 7t(0)

n

0

7T~n)

0 0

(n)

7T1 ( n)

7T2

I

0.25 0.25 0.50

= [0, 0, I]

(2.59)

2

3

4

0.187 0.313 0.500

0.203 0.266 0.531

0.199 0.285 0.516

0.20 0.28 0.52

From these calculations we may make a number of observati on s. First, we

34

SOME IMPORTANT RA NDOM PR OCESSES

see th at after only four steps the quantities 11";"1 for a given value of i are a lmost identic al regardless of the city in which we began . The rapid ity with which these quantities converge, as we shall soo n see, depends up on the eigenvalue s of P. In all cases, however , we o bserve th at the limiting values at infinity are rapidly approached and, as stated earlier, are independent of the init ial positi on of the particle. In order to get a bett er ph ysical feel for what is occurri ng, it is instructive to follow the probabilities fo r the vari ous states of the Mark ov chain as time evo lves. T o this end we introduce the noti on of baricentric coordinates, which are extremely useful in portraying probabil ity vecto rs. Consider a pr obabil ity vecto r with N components (i.e., a Markov process with N sta tes in o ur case) and a tetrahedron in N - I dimensions. In our example N = 3 and so o ur tetrahedron becomes an equil ateral triangle in two dimen sions. In genera l, we let the height of thi s tetrah edr on be unity. Any pr obability vecto r 1t 1n l may be repre sented as a point in this N - I space by identifying eac h component of tha t pr obability vecto r wit h a distance from one face of the tetrahedron . Th at is, we mea sure from face j a distance equal to the pr oba bility assoc iated with th at component 11"~"); if we do this for each face and th erefore for each compon ent , we will specify o ne point within th e tetr ahedr on and that point co rrectly identifies our prob ab ility vecto r. Eac h unique prob ability vecto r will map into a un ique point in th is spa ce, and it is easy to determine .the pr obability measure from its locati on in th at space. In our exa mple we may plot the three initial sta te vecto rs as given in Eqs . (2.57)(2.59) as show n in Figure 2.6. The numbers in parentheses represen t which pr oba bility compon ents a re to be measu red from the face associa ted with th ose nu mbers. Th e initial state vecto r corresponding to Eq. (2.59), for

[a, 0 , I J

T

Height = 1

[0. 1, OJ

(2)

1

[ 1. 0 , OJ

Figure 2.6 Representation of the convergence of a Markov chain.

2.3.

DISCR ETE-Tl~[E MARKOV C HAINS

35

example, will appea r at the apex of the triangle a nd is indicated as such . In our ea rlier calc ulation s we followed the progress of o ur pr ob ab ility vecto rs beginning with three initial state pr obability vectors . Let us no w follow th ese path s simultan eo usly and obse rve, for example, that the 'vector [0, 0, I] , following Eq . (2.59) , moves to the po sition [0.25,0.25,0.5) ; the vecto r [0, 1, 0] moves to the position [0.25 ,0,0.75], and the vecto r [1, 0 ,0] moves to th e position [0,0.75,0.25]. Now it is clear th at had we sta rted with an initial sta te vector anywhere within the ori ginal equil ateral tri angle, that point would have been mapped into t he interior of the sma ller trian gle, which now joins the three point s ju st referred to and which represent possible positions of the original state vectors . We note fr om t he figure th at thi s new tri angle is a shru nken version of the ori ginal triangle. If we now continue to map these three points into the seco nd step of the pr ocess as given by th e three cha rts above, we find an even sma ller trian gle inter ior to both the first and the second tri angles, and thi s region represents the possible locati ons of any origina l stat e vecto r after tll'O steps int o th e pr ocess. Clearl y , this shrinking will co ntinue unti l we reach a convergent point. Thi s con vergent point will in the limit be exactl y that given by Eq. (2.53)! Thu s we can see the way in which the possible position s of our pr o bability vecto rs move around in o u r space. The calculation of the transient respon se 1t l n ) from Eqs . (2.55) or (2.56) is extremely ted ious if we desire more than just the first few term s. In o rde r to o btai n the general solution, we often resort to tran sform meth od s. Below we demonstrate this meth od in general and th en a pply it to o ur hippie hitchhikin g example. Thi s will give us an opportunity to apply the z-transform calcul ation s tha t we have introduced in Append ix I. * Our point of departure is Eq . (2.55) . That equation is a difference equ ati on among vecto rs. T he fact th at it is a difference equ ati on suggests the use of c-tra nsfor ms as in App end ix I, and so we naturally define th e following vector tran sform (t he vecto rs in no way interfere with ou r tran sform approach except th at we must be car eful when taki ng inverses) : co

II(z) 4,

L: 1t l n lZ n

(2.60)

n= O

Th is tran sform will certa inly exist in the unit disk , th at is, Izi ..,:; I. We no w a pply the transfor m method to Eq . (2.55) over its ran ge of app licatio n (11 = 1,2 , . . . ,); thi s we do by first multiplying that equ ati on by c :' a nd th en sum ming from I to infinity, thu s

L:"" 1t( n )z " = L:oo 1t( n-llp;; " n= l

* The

n= l

ste ps involved in ap plying this meth od are summa rized on pp . 74-5 of th is chap ter.

36


We have now reduced our infinite set of difference equations to a single algebraic equation. FolIowing through with our meth od we mu st now try to identify our vecto r transform D (z). Our left-h and side contains all but the initial term o f this transform and so we ha ve

The parenthetical term o n the right-ha nd side of thi s last equation is rec0llt nized as D (z) simply by cha nging the index of summati on. Thus we find D (=:) -

= zD(z)P

7t(O)

z is merel y a scalar in th is vector equation and may be moved freely a cross vectors and matrices. Solving this matrix equation we immed ia tely come up with a general solution for our vector transform : (2.6 1) where 1 is the identity matrix and the (-I) notation implies the matrix in verse . If we can invert this equation , we will ha ve, by the uniqueness of transforms, the transient solution; that is, using the double-headed, d oublebarred arrow notation as in Appendix I to denote tr an sform pairs, we have D (z)

~ 7t (n )

=

7t(O)p n

(2.62)

In thi s last we have taken advantage of Eq . (2.56). Comparing Eqs . (2.6 1) and (2.62) we have th e obvious transform pair

[I - zP j- l ~ P "

I

l'

- (2.63)

Of course P " is precisely what we are looking for in order to obta in our transient solutio n since thi s will directly give us 7t( n l from Eq . (2.56). All that is required , therefore, is that we form the matrix inverse indi cated in Eq. (2.63). In gene ral this bec omes a rather complex ta sk when the number of sta tes in our Markov chain is at all lar ge. Nevertheless, th is is one formal procedure for ca rrying ou t the transient a na lysis. Let us apply these techn iques to our hipp ie hit chhiking example. Recall that the transiti on probability matrix P was given by

0

P= I

4 I

4

3

I

4

4

0 I 4

3 4 I

2

2.3.

37

DISCRET E-TIME MARKOV CHAINS

First we must form

1

I

3

-- z

- z

4

4

3

- - z 4

1 - zP = 1 -- z 4

1- 1. z 2

Next, in order to find the inverse of this matrix we must form its determinant thus : det (I - zP) = 1 -

1. z _ :L Z2 _ l.. Z3 2

16

16

which factors nicely into

+~ zr

det(I- ZP)=(I- Z)(1

It is ea sy to show that z = I is always a root of the determinant for an irreducible Markov chain (and, as we shall see, gives rise to our equilibrium solution). We now proceed with the calculation of the matrix inverse using the usual methods to arrive at

[I _ ?P j- t -

=

1 (1 - z) [I

+ (1/4)zf

1-

1. Z _1- z 2 2

x

1 4-

16

I . 16 -

- "'+- .,.I - z

4

+ -1

16

Z2

3 5 . - z - - z-

1. z _l.. Z2

-1 z 4 3 - z

2

4

4

1-

16

1z

-t

4

16

3 z2 +16

1

+ -9

Z2

16

+ -1

. z-

16

_1- z 2 16

Having found the matrix inverse, we are now faced with finding the inverse transform of thi s matrix which will yield P ", This we do as usu al by carrying out a partial fraction expansion (see Appendix I) . The fact that we have a matrix presents no problem ; we merely note that each element in the matrix is itself a rati onal function of z which must be expanded in partial fraction s term by term . (This task is simplified if the matrix is written as the sum of three matrices: a constant matrix ; a constant matrix times z ; and a constant matrix times Z2.) Since we have three roots in the denom inator of o ur rational functions we expect th ree terms in our partial fraction expansion. Carrying

38

SOME IMPORTA NT RANDOM PROCESSES

out this expansion and separating the three terms we find

[I - =P ]-

I

1/25 [5

7

= -- 5 1- z

7

5

7

-~ -~]

13] 1/5 [0 13 + (1 + =/4)2 0

13

2

0

+

1/25 [20

-5 1 + =/4 -5

-2

33 8 -17

-53]

-3 22

(2.64)

We observe immediately from this expansion that the matrix associated with the root (l - e) gives precisely the equilibrium solution we found by direct methods [see Eq . (2.53)]; the fact that each row of this matrix is identical reflects the fact that the equilibrium solution is independent of the initial state. The other matrices associated with roots greater than unity in absolute value will always be what are known as differential matrices (each of whose row s must sum to zero). Inverting on z we finally obtain (by our tables in Appendix I)

P"

~

7

-8 13] 1 1 n[O 13 +:5 (n + 1)(- 4) 0 2

7

13

7

;,[:

0

1( 1)"[ -5 338 --53] 3

+--4. 25

20

-5

-17

2

-~]

-2

n = 0, 1, 2, . .. (2.65)

22

This is then the complete solution since application o f Eq. (2.56) directly gives 7t ( n ) , which is the transient solution we were seeking. N ote th at for II = 0 we obtain the identity matrix whereas for II = I we mu st, of course, obtain the transition probability matrix P. Furthermore, we see that in thi s case we have two transient matrices, which deca y in the limit leaving only the con stant matrix representing our equilibrium solution. When we think ab out the decay of the transient, we are reminded of the shrinking triangles in Figure 2.6. Since the transients decay at a rate related to the characteristic values (one over the zeros of the determinant) we therefore expect the permitted positions in Figure 2.6 to decay with II in a similar fashi on . In fact, it can be sh own that these triangles shrink by a con stant factor each time II increases by 1. This shrinkage factor for any Markov process can be shown to be equ al to the absolute value of the product of the characteristic values of its tr ansition probability matrix; in our example we have characteristic value s equal to 1, 1/4, 1/4 . Their product is 1/16 and thi s indeed is the fact or by which the area of our triangles decreases each time II is increased.

2.3.

39

DIS CRET E-TIM E MARK OV CHA INS

This method of tran sform a nalysis is extended in two excellent vo lumes by Howard [HO WA 71] in wh ich he treat s such problems and disc usses additional a pp roaches such as the flow- graph method of ana lysis. Throughout thi s discussion of di screte-time M arkov chains we have not explicitly addressed ourselves to the memoryless property* o f the time that the syst em spe nds in a given state. Let us now prove that the nu mber of time units th at the system spends in the sa me sta te is geome trically d istributed ; the geometric distribution is the unique discrete memoryless di stribution . Let us assume the system ha s just entered state E; It will rem a in in this sta te at the next step with probability Pii; similarly, it will leave th is sta te a t the next step with probability I - Pu- If indeed it d oes remain in thi s sta te at the next step, then the probability of its remaining fo r an additi onal step is again Pu»and sim ila rly the conditional probability of its lea ving at thi s seco nd step is given by I - Pu - And so it goe s. Furthermore , due to the Marko v property the fact that it has remained in a given state for a kn own number of ste ps in n o way affects the probability that it leaves at the next step. Since the se probabilities a re independent , we may then write P[s y stem rem ains in E i for exactly m addition al steps given that it has

just en tered E i] = (I - Pii)Pii

ffl

(2.66)

This, of course, is the geometric distribution as we cla ime d . A simila r argument will be given later for the continuous-time Markov chain. So far we have concerned ourselves principally with homogeneou s Markov processes. Recall that a homogeneous M arkov ch ain is one for which the tr ansit ion prob abilities a re ind ependent of time. Amon g the qu antities we were able to calculate wa s the m-step transiti on pr ob ab ility p\il, which gave the probability of passing from state E, to state E, in m steps ; the recursive formula for thi s calculation wa s given in Eq . (2.4 1). We now wish to take a mo re gene ra l point of view and permit the transit ion prob ab ilities to depend u pon ti me. We intend to deri ve a relation ship not unlike Eq . (2.4 1), whic h will form our point of depa rture for many further developments in the a pp lica tion o f Markov pr oce sses to queueing p roblems. F or th e time bei ng we con tinue to res trict ourselves to d iscrete-t ime , discre te- stat e Ma rkov cha ins . Generalizin g the homogene ou s definition for the mu ltistep tr an sition prob ab ilities given in Eq. (2.40) we now define _ (2.67) which gives the probability that the system will be in state E, at step n, given • The memoryless prope rty is discussed in some detail la ler.

40

SOME IMPORTANT RANDOM PR OCESSES

n

q

In

Time step

Figure 2.7 Sample paths of a stochastic process. th at it was in state E; at step m, where 11 ~ m. As discussed in the hom ogeneou s case, it certainly must be true that if our proces s goes from state E; at time m to state E; at time 11, then a t some intermediate time q it must have passed through some state Ek • This is depicted in Figure 2.7. In this figure we have shown four sample paths of a stochastic process as it moves from state E ; at time m to state E; at time 11. We have plotted the state of the process vertically and the discrete time steps horizontally. (We take the liberty of dra wing continuous curves rather than a sequence of points for convenience.) Note that sa mple paths a and b both pass through state Ek at timeq, whereas sample paths c and d pass through other intermediate sta tes at time q. We are certa in of one thing only , namely , that we must pass through some intermediate state at time q. We may then express poem , 11) as the sum of probabiliti es for all of these (mutually exclusive) intermediat e states; that is,

I

(2.68)

p;lm, n) = L P[X n = j , X. = k X m = i] k

I

,

for m ~ q ~ 11. Thi s last equation must hold for any stochastic process (not necessarily Markovian) since we are considering all mutually exclusive and exha ustive possibilities. From the definition of conditional probability we may rewrite this last equation as

.1

Pilm , n) = LP[X. = k

I x; =

i]P[X n =j

Ix ; =

i, X . = k ]

(2.69)

k

Now we invoke the Markov property and observe that P[ X n

= i, X. = k] = P[ X n = j

I x, =

k]

Appl ying this to Eq . (2.69) and makin g use of our definition in Eq. (2.67) we finally arrive at poem, 11)

= L p;k(m , q)Pk,(q , 11 ) k

- (2.70)

2.3.

DIS CR ETE-TIM E MARK OV CHA INS

41

for m ::; q ::; n. Equation (2.70) is known as the Chapman-Kolmogorov equat ion for discrete-time Mark ov proce sses. Were this a homogeneous Mark ov chain then from the definition in Eq. (2.40) we would have the relat ionship p ;;(m,n) = p\;- ml and in the casewhenn = q + I our ChapmanKolmogoro v equat ion would reduce to our earlier Eq . (2.41). The ChapmanKolmogoro v equat ion states that we can partition any n - m step tran sition pr obab ility into the sum of products of a q - m and an n - q step tran sition probability to and from the inte rmediate states that might have been occupied at some time q within the interval. Indeed we are permitted to choose any part itioning we wish, and we will take adv ant age of th is shortly. It is conven ient at thi s point to write the Chapman-Kolmogorov equation in matrix form . We have in the past defined P as the matrix containing the elements PH in the case of a homogeneous Mark ov chain. Since these quant ities may now depend upon time, we define p en) to be the one-step tran sition probability matrix at time n , that is, p en) ~ [p;; (n, n

+ I)]

- (2.71)

Of course, p en) = P if the chain is homogeneous. Also, for the homogeneous case we found that the n-step transition pr obab ility matrix was equal to P ". In the nonhomogeneous case we must make a new definiti on and for this purpose we use the symbol H (m, n) to denote the following multistep transition pr obabil itymatrix: H (m, n) ~ [Pi j(m, n)]

- (2.72)

Note that H (n, n + I) = p en) and th at in the homogeneous case H (m, m + n) = P ". With the se definitions we may then rewrit e the Chapman Kolmogorov equation in matri x form as H (m, n) = H(m , q)H(q, n)

_ (2.73)

for m ::; q ::; n. To complete the definiti on we require th at H (II, II) = I , where I is the identity matri x. All of the matrice s we are considering are square matrices with dimensionality equal to the number of states of the Mark ov chain . A solution to Eq. (2.73) will co nsist of expressing H (m, n) in term s of the given matrices p en). As mentioned abov e, we a re free to choose q to lie anywhere in the interval between m and II. Let us begin by choosing q = n - I. In th is case Eq . (2.70) becomes (2.74) p;;(m, II) = 2 Piim, n - I)Pk;(n - I , n) k

which in matri x form may be written as H (m, n)

=

H (m, n -

I)P (n - I)

- (2.75)

42

soxn;

IMPORTA NT RANDOM PROCESSES

Equ ati ons (2.74) and (2.75) are known as the fo rward Chaprnan-Kolrnogorov equations for discrete-time Markov chains since they a re writt en at the for ward (most recent time) end of the interv al. On the other hand, we could ha ve chosen q = m + I, in which case we obtain pilm, n)

= 2: Pik(m , m + I)Pk,(m + 1, II)

(2.76)

k

whose matri x form is

H (m, II)

=

P (m)H (m

+ 1, n)

- (2.77)

The se last two are referred to as the backward Chapman-Kolmogorov equations since they occur at the backward (oldest time) end of the interval. Since the forward and backward equations both describe the same discretetime Markov chain, we would expect their solution s to be the same, and indeed this is the case. The gener al form of the solution is H (m , n) = P (m)P (m

+ I) . .. P(I! -

I)

m::;, I! -

I

- (2.78)

That this solves Eqs. (2.75) and (2.77) may be establi shed by direct substitution . We observe in the homogeneou s case that this yields H (m , n) = p n- m as we have seen earlier. By similar arguments we find that the time -dependent probab ilities {1rl n l } defined earlier may now be obtained through the following equation :

-

whose solution is 7t(n+!1 = 7t(oIP (O)P (I ) .. . P (II)

_ (2.79)

The se last two equations corre sp ond to Eqs. (2.55) and (2.56), respectively, for the hom ogeneous case. The Chapman-Kolmogorov equations give us a mean s for describing the time-dependent probabilities of man y interesting queu eing systems that we develop in later chapters. * Before leaving discrete -time Markov chains, we wish to introduce the special case of discrete time birth-death processes. A birth-death process is an example of a Mark ov proces s that may be thought of as modelin g chan ges in the size of a popul ation. In what follows we say that the system is in state Ek when the popul ation consists of k members. We further assume th at chan ges in popul ati on size occur by at most one; th at is, a " birt h" will chan ge the popul ati on' s size to one greater, whereas a "death" will lower the popul at ion size to one less. In consider ing birth -death processes we do not perm it multiple birth s or bulk disasters; such possibilities will be con sidered

* It is clear fro m this develop ment that a ll Mar kov processes must sati sfy the Chapma nKolmogorov equatio ns. Let us note, however , that a ll proc esses that sa tisfy the Cha pmanKolmogorov equation are not necessarily Mark ov processes; see . for exam ple. p. 203 of [PAR Z 62].

,

2.3.

DISCR ETE-TIM E MARKOV CHAINS

43

later in the text and correspond to rand om walks . We will con sider the Mar kov chain to be hom ogene ous in that the transition probabilities P i; do not change with time; howe ver , cert ainly the y will be a functi on of the state of the system. Thus we"have that for our discrete-time birth-death process j= i - I

PH =

{ I-h. d, - d,

b,

j=i j

(2.80)

=i+I ot herwise

0

Here d, is the pr obability that at the next time step a single death will occur, driving the population size down to i - I , given that the population size now is i. Similarly, b, is the probability that a single birth will occur, given th at the current size is i, thereby dri ving the populati on size to i + I at the next time step. I - b, - d, is the probability that neither of these event s will occur and that at the next time step the population size will not change. Onl y these three possibilities are permitted. Clearly do = 0, since we can have no deaths when there is no one in the populati on to die. However, contrary to intuition we do permit b o > 0; this correspond s to a birth when there are no members in the population. Whereas this may seem to be spo ntaneo us generation, or perhaps divine creation , it does provide a meaningful model in term s of queueing the ory. The model is as follows : The population corresponds to the custo mers in th e queueing system ; a death corresponds to a customer departure from that system; and a birth corresponds to a customer arrival to th at system. Thus we see it is perfectly feasible to ha ve a n arrival (a birth) to an empty system ! The sta tiona ry pr obab ility tran sition matrix for the general birth-death pr ocess t hen appears as follows : I- bo d,

bo I - b, - d ,

0

d'!,

0

0

0

0

0

0

b,

0

0

0

0

0

I - bz-dz b,

0

0

0

0

di

I- bi - d,

b,

0

p =

0

...

If we are dealing with a finite cha in, then the last row of thi s matri x would be [00 . . . 0 ds I - dsL which illustrates the fact th at no births are permitted when the populati on has reached its maximum size N . We see th at th e P

f

44

SOME IMPORTA NT RANDOM PRO CESSES

matrix has nonzero terms only along the main diagonal and along the diagonals directly above and below it. This is a highly specialized form for the transition probability matrix, and as such we might expect that it can be solved . To solve the birth-death process means to find the solution for the state probabilities 1t(n l . As we have seen, the general form of solution for these probabilities is given in Eqs. (2.55) and (2.56) and the equation that describes the limiting solution (as n -- 00) is given in Eq , (2.50). We also demonstrated earlier the z-transform method for finding the solution. Of course, due to this special structure of the birth-death transition matrix, we might expect a more explicit solution. We defer discussion of the solution to the material on continuous-time Markov chains , which we now investigate.

2.4.

CONTINUOUS-TIME MARK OV CHAINS'

Ifwe allow our particle in motion to occupy positions (take-on values) from a discrete set, but permit it to change positions or states at any point in time, then we say we have a continuous-time Markov chain. We may continue to use our example of the hippie hitchhiking from city to city, where now his transitions between cities may occur at any time of day or night. We let X(f) denote the city in which we find our hippie at time f. X(f) will take on values from a discrete set, which we will choose to be the ordered integers and which will be in one-to-one correspondence with the cities which our hippie may visit. In the case of a continuous-time Markov chain, we have the following definition : DEFINITION: The random process X(f) forms a continuous-time Markov chain if for all integers n and for any sequence f" f 2 , •• • , f n+l such that f 1 < f 2 < ... < f n +1 we have

P[X(tn+l)

= j I X(tl) =

ii' X(t2)

=

;2' .. . , X(t n)

= P[X(tn+l)

=

= j

in]

I X(t n) =

in]

(2.81)

This definitiont is the continuous-time version of that given in Eq . (2.38). The interpretation here is also the same, namely, that the future of our hippie's travels depends upon the past only through the current city in which we firid him. The development of the theory for continuous time parallels that for discrete time quite directly as one might expect and, therefore, our explanations will be a bit more concise . Moreover, we will not overly concern • See footnote on p. 19.

t An alternate definition for a discrete-state continuous-time Markov process is that the following relation must hold : P[X(t)

= j I X(T)

for

TI

S

T

S

T2

< I] = P[X(I ) = j I X h

)]

,

2.4.

45

CONTINUOUS-TIME MARKOV CHA INS

ourselves wit h some of the deeper que stions of con vergence of limits in passing fro m discrete to continuous time ; for a car eful treatm ent the reader is referred to [PARZ 62, FELL 66]. Earl ier we stated for any Markov process that the time which the process spends in any state must be " memoryless" ; th is implies th at the discrete-time Mark ov chain s mu st have geo metrically distributed state time s [which we have already pr oved in Eq. (2.66)] and that continuous-t ime Ma rkov chai ns must ha ve exponentially distributed sta te time s. Let us now prove thi s last sta tement. F or thi s purpose let T i be a random variable that repre sent s the time which the process spends in state Ei . Recall the Markov pr operty which sta tes that the way in which the past trajectory of the process influence s the future de velopment is completely specified by giving the cur rent sta te of the process. In particular, we need not specify how long the pr ocess ha s been in its curren t state. This mean s th at the remaining time in E, mu st have a distribution that depends only upon i and not up on how lon g the pr ocess has been in E i . We may write th is in the followin g fo rm :

+ 1 It , > s] =

P['Ti > s

h(l)

where h(l ) is a function only of the additional time 1 (and not of the expended time s) *. We may rewrite thi s conditional probability as follows :

P['Ti

> 5 + 1 I'Ti > 5] =

_ P~ ['T.!..i..:..>_ 5 --:+----'I'_'T!...i :::.. >_5-,]

P['Ti > 5]

> 5 + I] P[Ti > 5] > s + 1 implie s the

P[Ti

Thi s last step follows since the event 'T i event Rewritin g this last equati on and introducing h(l ) o nce again we find

Setting s

P['T i > s

Ti

> s.

(2.82) > s]h( l) = and observing that P['T i > 0] = I we have immed iately th at

°

+ I] =

I

P[Ti > I]

P[Ti

=

h(l )

Using thi s last equ ati on in Eq. (2.82) we then obtain

P['Ti

> 5 + I] =

°

P['Ti > 5 ]P['T,

> I]

(2.83)

for s, 1 ~ 0. (Setti ng s = 1 = we aga in requ ire P[Ti > 0] = I.) We now show that the only continuous distribution satisfying Eq. (2.83) is the • T he symbo l s is used as a time variab le in this section on ly an d should not be confuse d with its use as a transform varia ble elsewhere.

46

SOME IMPOR TANT RANDOM PROC ESSES

exponential distributi on. First we have, by definition , the following general relati onsh ip:

d

-

dt

(P[T i

> tJ) = -d

dt

=

( I - Ph

~

(2.84)

- JT,(t )

where we use the notation!T.(t) to den ote the pdf for tiate Eq . (2.83) with respect to s, yieldin g dPh

t])

> s + t]

----"-'------" = - JT (s) P [T i

ds

•

Ti.

Now let us

diff~~en

> t]

where we have taken advantage ofEq . (2.84). Dividing both sides by P[T i and setting s = 0 we have dP[T, P[T i

> t] = > t]

If we integrate this last from 0 to

I

> I]

- /,(0) ds T,

.

we obtain

or P[Ti

> t] = e- f T, (O) '

Now we use Eq. (2.84) again to obtain the pdf for

JT,( t)

Ti

as

= JT,(O)e- Ir ,lOlt

(2.85)

which hold s for I ~ O. There we have it: the pdf for the time the process spends in state E, is exponentially distributed with the parameter ;;,(0), which may depend upon the state E,. We will have much more to say abo ut this exponential distribution and its imp ortance in Mark ov processes sho rtly. In the case of a discrete-time hom ogeneous Mar kov chai n we defined the transition probabilities as Pis = P[Xn = j I X n _ 1 = i] and also the m-step transiti on probabilities as p~j ) = P[X n+ m = j I X n = i] ; th ese quant ities were independent of n due to the homogeneity of the Markov chain. In the case of the nonhomogeneous Markov chain we found it necessary to identify points along the time axis in an absolute fashion and were led to th e import ant tr ansition probability definition Pii(m , n) = P[Xn = j I X m = i]. In a completel y analogous way we must no w define for our continuous-time Markov chain s the following time-d ependent transition probability: p,;(s, t) ~ P[X(t)

=j

I X es) = i]

- (2.86)

where XCI) is the position of the particle at time I ~ s. Ju st as we considered three successive time instant s m ~ q ~ n for the discrete case, we may

,

2.4.

CONTINUOUS-TIME ~IARKOV C HAINS

47

con sider the following three successive time instants for our continuous time chain s ~ 1I ~ I . We may then refer back to Figure 2.7 and iden tify so me sample paths for what we willnow consider to be a continuous-time Mark ov chain; the critic al observa tion once again is that in passing from sta te E, at time s to state E , at time t, the process must pass through some intermed iate stat e E. at the intermediate time 1I. We then proce ed exactly as we did in derivin g Eq. (2.70) and arrive at the followi ng Chapman-Kolmogoro v equa tio n for continuous-time Markov chains: (2.87) where i.] = 0, I, 2, .. . . We may pu t thi s eq uation into matri x form if we first define the matrix con sisting of elements Pii(S, t) as

H (s , t) ~ [Pii(S, t»)

- (2.88)

Then the Chapman-Kolmogorov equation becomes .

H (s, t) = H (s, lI)H (u, t)

- (2.89)

[We define H (/, t) = I, the identity matrix.] In the case of a homogeneous discrete-time Markov chain we found th at the mat rix equ ation 1t = 1tP ha d to be investigated in ord er to determ ine if the chain was ergodic, and so on ; also, the tran sient solution in the nonhomogeneous case could be determi ned from 1t1n+1l = 1tIOIP (O)P (l ) . .. p en), which was given in terms of the time-dependent transition probabilities Pii(m, n) . For the continuous-time Markov chain the one-step tran sition probab ilities are replaced by the infinit esimal rates to be defined below; as we sha ll see they are given in te ~m s of the time derivat ive of P'i(S, t) as t -->- s. What we wish now to do is to form the continuous-time ana log of the for ward and backward equations. So far we have reached Eq. (2.89), which is ana logo us to Eq. (2.73) in the discrete-time case. We wish to extract the a nalog for Eqs. (2.74)-(2.77), which sho w both the term-by-term and matrix form of the for ward and backward equ ati ons , respecti vely. We choose to do th is in the case of the forwa rd equation, for example, by sta rting with Eq. (2.75), namely, H (m, n) = H (m , n - I )P (n - I), and allowing the unit time interval to shrink toward zero . T o this end we use thi s last equation an d form the following difference :

H (m, n) - H(m, n - I)

= H (m, n

-

= H (m, n -

I )P (n -

I) - H (m, n -

I)[P(n - I) - I]

I)

(2.90)

We must now con sider some limits. Ju st as in the discrete case we defined p en) = H(n, n + I), we find it con venient in this continuous-time case to

,

48

SOME I:'IPORTANT RANDOM PROCESSES

define th e following matrix: P(t ) ~ [p ;;(t , t

+ 6.t»)

- (2.9 1)

Furthermore we identify the matrix H (s , t) as the limit of H (m , n) as our time interval shrinks; similarly we see that the limit of p en) will be p et ). Retu rn ing to Eq. (2.90) we now divide both sides by the time step, which we denote by !'It, an d take the limit as 6.t ->- O. Clea rly then the left-hand side limits to the derivative, resulting in

a H(s , t)

at

=

H(s, t)Q(t)

~

s

t

- (2.92)

where we have defined the matri x Q (t ) as the following limit: Q(t) = lim p et) - I - (2.93) "'-0 6.t T his matrix Q (t ) is kn own as the infin itesimal generator of the transition matrix fu nction H (s , t). Ano the r more descriptive name for Q (t ) is t he transit ion rat e matrix ; we will use both names interchangeably. The elements of Q (t ), which we denote by q;;(t ), a re the rates that we referred to earlier.

Th ey are defined as follows : .

q;;(t)

.

p;;(t, t

= 11m

+ 6.t)

- 1

(2.94)

D.t

41-0

. P"J( t, t + 6. t) q;;(t) = 11m 41 -0 6.t

i

,e j

(2.9 5)

T hese limits have the following inte rpretation. If the system at time t is in state E ; then the probability that a transition occurs (to any state other than E i) during the inte rval (t, t + D.t) is given by -qii(t) 6.t + o(6.t). * Thus we may say th at - q;;(t ) is the rat e at which the process de par ts from sta te E ; when it is in that sta te. Similarly, given that th e system is in sta te E, at time t, the co nditional probabil ity tha t it will make a transition from this state to state E; in the time interval (t, 1 + 6.t) is given by q;;(I) 6.1 + o(6.t) . Thus • As usual, the notation o(~t) denotes allY functio n that goes to zero with !!'t faster than !!'t itself, that is , lim oeM) ~,- o !it

=

0

More generally, one states that the function g et) is o(y (t )) as t - t l if j"

t~~,

I I get) ye t)

=

See also Chapter 8, P: 284 for a definition of 0(').

0

,

2.4.

CON TINUOU S- TIME MARK OV CH AINS

49

q;i(t) is the rate at which the pr ocess moves from E ; to E i , given th at th e system is currently in the sta te E;. Since it is a lways true that I i p;;(s, I) = I then we see th at Eq s. (2.94) and (2.95) imply that for all i

(2.96)

T hus we have interp reted the terms in Eq. (2.92); th is is nothing more than the for ward Chapman -Kolmo gorov equ ation for the continuou s-time Ma rkov ch a in. In a sim ilar fas hion, beginning with Eq . (2.77) we may deri ve the back ward Ch apman -Kolm ogor ov equ ati on

aH~;, t)

= - Q(s)H (s, t)

s

~

t

- (2.97)

The for ward and backward matrix equations j ust deri ved may be expressed through their indi vidu al terms as follows. The forward equation gives us [with t he addi tio na l condition that the pa ssage to the limit in Eq. (2.95) is uniform in i for fixed j] (2.98) Th e initial sta te E, a t t he initia l time s affect s the solution of thi s set of differential equ ati on s only through the initial condition s

pil s, s)

= {~

if j = i if j ~ i

From the bac kwar d matrix equation we obtain (2.99) T he "i nitial" co ndi tio ns for th is equation are

pilt , t) =

{~

if i = j if i ~ j

These equ ati on s [(2.98) and (2.99)] uniquely determine the tr an siti on p rob abilities p,ieS, t) and mu st, of course , a lso satisfy Eq. (2.87) as well as the ini tial condition s. In matrix not ati on we may exhibit the solution to the forw ard a nd backward Eqs. (2.92) and (2.97), respectively, in a stra ightfo rwa rd manner ; the

,

50

SO~[E I ~I PORTANT RAN DOM PROCESSES

result is' H(s, I)

=

exp

[fQ(U) duJ

- (2.10 0)

We observe that thi s so lutio n also sa tisfies Eq. (2.89) and is a continuou s-time an alog to the discrete-time so lut ion given in Eq. (2.78) . N ow fo r the sta te p rob ab ilit ies the mselves : In a na logy with 7Tl n ) we now define 7T;(t) ~ P[X(I) = jj - (2. 101) as well as the vecto r of the se probabilitie s

n et ) ~ [1TO(t ), 7T I (t),

7T2

(1), . .. j

(2.102)

If we are given the initial state d istr ibution n CO) then we can so lve for the t ime-dependen t sta te probabilitie s from

n et)

=

n(O)H(O, t)

(2.103)

where a general solutio n may be seen from Eq . (2. 100) to be

n (l)

=

n CO) exp

[1'

Q(u) duJ

- (2. 104)

This corresponds to the discrete-time solution given in Eq , (2.79). The mat rix differential equ ation corresp onding to Eq. (2.103) is easi ly seen to be dn( l ) -

dl

,

= n (I)Q (t)

This last is simila r in form to Eq . (2.92) a nd ma y be expr essed in terms of its elemen ts as (2. 105) The sim ilarity between Eq s. (2. 105) an d (2.98) is not accidental. The latt er de scr ibes th e pr obab ility th at t he process is in sta te E; at time t given that it was in state E; at time s. The fo rmer merel y gives the probability that the system is in state E; a t time t ; information as to whe re the proce ss began is given in the initial state probability vecto r n CO). If indeed 7T k (O) = I for k = i a nd 7T k(O) = 0 for k ,t= i, then we are sta ting for sure th at the system was in state E ; at ti me O. In th is case 1T;(I) will be identically eq ua l to Pu(O, I). Both form s for thi s probability are often used ; th e form Pu(s, I) is used whe n • Th e expression e P ' where P is a squa re mat r ix is defined as the following matrix po wer series: e Pl

=I+

PI

+

(2

p 2_ 2!

+

(3

p 3_ 3!

+ .. .

i

.I

CONTINUOUS-TIME ~IAR KOV C HAINS

2.4 .

51

we wa nt to specifica lly sho w the initial state ; th e form 7T;(I) is used wh en we ch oose to neglect o r imply the initial sta te. We now con sider the case where o ur continuous-time Marko v chain is homogeneous. In this ca se we drop .the dependence upon time and ad opt the foll owing notation : (2. 106) Pi' (I) ~ po(s , S + I)

qij ~ qij(l ) H (I)

=.l .l =

i,j = 1, 2, . . .

+ I) =

H (s, s

[pi,(I»)

(2 .107)

... (2.108)

Q Q(I) = [qij) (2.109) In this ca se we may list in rapid o rder the corresponding results. Fir st , the Chapman-Kolmogorov equations become

Pij(S + I)

= L Pik(S)Pk,(l ) k

and in matrix form*

H (s

+ I) =

H (s)H(I)

The forward and backward equation s be come , respectively ,

dpi,(l ) - d- = q jiPi](l ) I

-

-

,

+ ,L, qk;Pik(l)

(2.1 10)

.~ ~j

and (2 . 111)

,

and in matrix form th ese bec ome , respect ively ,

dH (I)

-

dl

=

H (I)Q

- (2 . 112)

=

QH(I)

- (2. 113)

and

dH(I)

-

dl

with the comm on init ial cond it ion H (O) given by

H (I)

=

=

I. The so lutio n for thi s ma trix is

eO t

-

N ow for the sta te probabilit ies them sel ves we have th e d ifferenti al eq ua tio n

d7T,(t) - d - = q ji 7T j ( l) t

+ L qkj7Tk(l)

(2.114)

k:;:.j

which in matrix form is

d7t(I) -

dl

= 7t(I)Q

• The corresponding discrete-time result is simply p m+n = p mpn.

-

52

SO~IE 1 ~IPO RTA l" T RA NDO~l PROC ESSES

Fo r an irreduci ble hom ogeneou s Mark ov chain it can be shown that the follow ing limits a lways exist and a re independent of the initi al sta te o f th e ch a in , name ly, lim Pil(t)

=

TT j

1-",

This set { TTj } will fo rm the lim iting sta te p robab ility di stribut ion . For a n e rgodi c M arkov ch a in we will ha ve the furth er limit , whic h will be ind ependen t of th e in itia l d istr ibu tion, nam el y, lim TT /t )

=

TT j

I- x

This limit ing di stribution is given uniquely as the so lutio n of th e follo win g system o f linear equati on s :

«.» , + 2: q kj TTk

= 0

(2 . 115)

k* i

In matrix fo rm th is la st equati on may be expressed as 1tQ

= 0

- (2. 116)

wher e we ha ve used t he ob viou s notati on 1t = [TTO' TTl ' 71'2, • •• ] . This la st equati on coupled with th e probabi lity con ser vati on relat ion , namely , (2 . 117)

uniquely gives us o ur limit ing sta te pr ob abilities. We compar e th e Eq . (2: 1 16) wit h ou r ea rlier eq ua tio n for d iscre te-time Mar kov chai ns, nam ely , 1t = 1tP : here P wa s th e matrix of tr an siti on probabilities. wh erea s th e infinitesima l genera tor Q is a matri x of tr ansition rates. T his comp letes our d iscu ssion o f di scr ete-state Markov cha ins. In th e table o n pp . 402-403 . we su m ma rize the maj or result s for the four eases conside red he re. For..a furt her di scu ssion , t he reader is referred to [BHA R 60] . . ~~u s sed di screte-state Mark ov ch ain s (bo th in di scret e a nd continu ou s time) it would see m natural th at we next co nsider co ntinuoussta te Mar kov pr oce sses. This we will not d o , but rat her we pos tpo ne co nside ra tio n o f such mat er ial un til we require it [viz.• in C ha pter 5 we co nsid er Takacs' in teg ro d ifferentia l equ ati on for M/G/I . a nd in C ha p ter 2 (Vo lume II) we devel op the Fokk er-Planck eq ua t io n for use in th e diffu sion a pp rox ima tio n fo r qu eues]. On e wo uld furt her expect th at foll owing the stud y o f Ma r kov proce sses. we wou ld then investi gate ren ewal processes , random wa lks, a nd fina lly, semi- Ma r kov p rocesses. Here too, we choose to postpone such di scu ssion s until the y are need ed later in the te xt (e.g ., th e d iscu ssi on in C ha pter 5 o f M ar kov ch ain s imbedded in semi- Ma rkov processes).

r-

2.5.

BIRTH -DEATH PROCESSES

53

Inde ed it is fair to say that much of the balance of this textb ook depend s upon addi tional mate ria l from the theory of stochas tic pr ocesses and will be developed as needed. Fo r the -time being we choose to specialize the results we have obtaine d from t he co ntinuous-time Mar kov chai ns to the class of birth-death pr ocesses, which , as we have fore warned , playa majo r role in queu eing systems anal ysis. Th is will lead us directl y to the imp ortant Poisson p rocess.

2.5.

BIRTH-DEATH PROCESSES

Earli er in this chapter we said that a birth-death proce ss is the special case of a Markov process in which transitions from state E k are permitted only to neigh borin g states Ek+l ' Ek , and E k _ l • Thi s restr iction permits us to carry the solution much further in many cases. The se processes turn out to be excellent mod els for all of the material we will study under elementary queueing th eory in Chapt er 3, and as such form s our point of departure for th e st udy of queuein g systems. The discrete-time birth-death process is of less inte rest to us than the conti nuous-time case, and , therefore , discrete-time birth-death processes a re not considered explicitly in the following development ; needless to say , an almost parallel treatment exists for that case. More over, tran sitions of the form from state E ; back to E ; are of direct intere st only in th e discrete-t ime Markov chai ns ; in the continuou s-time Mar kov chai ns, t he rate at which the process returns to the state th at it currentl y occupies is infinite, and the as tute reader sho uld have observed that we very-carefully subtracted this term out of our definition for q u (t ) in Eq. (2.94). Th erefore , our main interest will focu s on continuous-time birth-death processes with discrete state space in which transition s only to neighboring states Ek-'- I or Ek _ 1 from sta te E, a re per mitted. * Ea rlier we described a birth-death process as one that is appro priate for modelin g changes in the size of a population. Indeed, when the pr ocess is said to be in sta te E k we will let th is denote the fact that the popul at ion at that time is of size k. Moreover , a transition from E k to Ek+l will signify a " birth " with in the popul ation, whereas a tran sition from Ek to Ek _ 1 will denote a "deat h" in the popul ati on . Thu s we consider chan ges in size of a populati on where tr ansitions from sta te E k take place to nearest neighbors only. Regarding the nature of birt hs and deaths, we int rod uce the not ion of a birth rate i.k , which describes the

* Thi s is true in the one-dimensio nal case. Later, in Chap ter 4, we consider multidimensiona l systems for which the sta tes are described by discrete vectors, a nd then each state has two neighbors in each dimension . For example , in the two-dim ensional case, the sta te descriptor is a cou plet (k t , k,) denoted by £ k, .k, whose four neighbo rs are £k, - t .k" £k"k, - I ' £k,+1.k,- and £k , . k, ~I'

,

54

SOM E IMP ORTANT RA NDOM PR OCESSES

ra te at which births occur when the population is of size k. Similarly , we define a death rate fl k>. which is the rate at which deaths occur when the population is of size k. Note that these birth and death rate s are independent oft ime and depend only on Ek ; thus we ha ve a continuous-time hom ogeneous Markov chain of the birth-death type. We ad opt this special notat ion since it lead s us directly into the queueing system notation; note that, in term s of our earlier definit ions, we have a nd fl k = Qk ,k-l

Th e nearest-nei ghbor condition requires that qkj = 0 for Ik - jl > I. Moreover, since we have pre viously shown in Eq. (2.96) that L.;qkj = 0, then we require

q kk

= _ (flk + Ak)

(2.118)

Thus our infinitesimal generator for the general hom ogeneou s birth -death process takes the form -,10

)'0

0

0

0

fll

-()., + fll)

Ai

0

0

0

fl .

i'2

0

0

0

Q=

-(A.

+ P.)

fl 3

- (;'3 +

fl 3)

,13

,

L

Note that except for the main , upper, and lower diagonals, all term s are zero. T o be more explicit, the assumptio ns we need for the birth-death process are th at it is a hom ogeneous Markov chain X (t ) on the sta tes 0 , I , 2, . . . , that births and death s are independent (this follows directl y from the Markov pr operty), and B, :

P [exactly I birth in (r, t

+ nt ) I k

in populat ion]

= ;'k n t + o( nl) D1 :

P[exactly I death in (t , t

+ nt ) I k

in population]

= Pk nt + o(nt) B.:

P[exactly 0 birth s in (r, t

+ nt) Ik

in population] = I - ;'k n l

D. :

P[e xactly 0 deaths in (I, t

+ o(nt)

+ nt ) I k in population] = I - Pk nl

+ o(n t)

2.5.

BIRTH-DEATH PROC ESSES

55

Fr om these assumptions we see that multiple births, multiple deaths, or in fact, both a birth and a death in a small time interval are prohibited in the sense that each such multipleevent is of order o (~t). Wh at we wish to solve for is the probabil ity that the population size is k: at some time t ; th is we denote by' Pk(t ) ~ P[X(t )

=

k]

(2.119)

Thi s calculation could be carried out directly by using our result in Eq. (2.114) for 7T J(t) and our specific values for q i j ' However , since the deriva tion of these equation s for the bir th-death process is so straightforward and follows from first principles, we choose not to use the heavy machine ry we developed in the previou s section , which tend s to cam ouflage the simplicity of the basic approach, but rather to rederive them below. The reader is encouraged to identify the parallel steps in this development and compare them to the more general steps taken earlier. Note in term s of our previous definiti on th at Pk(t) = 7T k(t). Moreover , we are " suppressing" the initial condition s temporarily, and will introduce them only when required . We begin by expre ssing the Chapman-Kolmogoro v dynam ics, which are quite trivial in this case. In particular, we focus on the possible motions of our particle (that is, the number of members in our population) during an interval (t , t + ~t) . We will find ourselves in state E k at time t + ~t if one of the three follo wing (mutually exclusive and exhau stive) eventualities occurred: f

1.

2. 3.

that we had k in the population at time t and no state chan ges occur red; that we had k - 1 in the population at time t and we had a birth during the interval (t , t + ~t); that we had k + 1 members in th e populati on at time t and we had one death during the interval (t, t + ~t).

Th ese three cases ar e portrayed in Figure (2.8). T he p robability for the first of these possibilities is merely the probability Pk(t) that we were in st ate E k at time f time s the probability hk(~f) that we moved from state Ek to state E, (i.e., had neither a birth nor a death) durin g the next ~f seconds ; thi s is rep resented by the first term on the right-hand side ofE q . (2.120) below. T he second and th ird terms on the right-hand side of th at equ ati on correspond , respectivel y, to the second and third cases listed ab ove. We need no t concern ourselves specifically with transition s fr om states other than neare st neighb or s to state E k since we have assumed that such transitions in a n interval of • We use X (r) here to denote the num ber in system at time I to be consistent with the use of for ou r genera l stochastic process. Cer ta inly we cou ld have used N(t) as defined earlier; we use N (t) outside of this chapter. X (l )

"/

56

SOME IMPORTANT RA NOmr PR OCESSES

Time

Figure 2.8 Possible transitions into Ek •

duration !:;.t are of order o(!:;. t). Thus we may write Pk(t

+ Ar) =

Pk(t)Pk.k(!:;.t)

+ Pk_1(t)Pk_l.k(!:;.t ) + Pk+ (t)Pk+l.k(!:;.t) + oeM) k~l 1

(2.120)

We may add the three probabilitie s ab ove since these events are dearly mutually exclusive. Of course, Eq. (2.120) only make s sense in the case for k ~ I , since clearly we could no t have had - I members in the population . For the case k = 0 we need the special boundary equati on given by Po(t

+ !:;.t) = Po(t)Poo(!:;.t ) + P, (t )PlO (!:;.t ) + o(!:;.t) k = 0

(2.12 1)

Furthermore, it is also clear for all values of t th at we must conserve our probability, and this is exp ressed in th e following equati on : (2.122) T o solve the system represented by Eqs. (2.120)-(2. 122) we must make use of our assumptions B" D B2 , and D 2 , in order to evaluate the coefficients "

2.5.

57

BIRTH- DEATH PROCESSES

in these equ ati on s. Carrying out thi s opera t ion our eq uati on s convert to Pk(t

+ llt ) = Pk(t)[l .; i'k llt + o(ll t)][ l - flk llt + o (llt) ] + Pk_1(t)[i'k_l llt + o (llt )] + Pk+l(t )[Pk+lllt + o(llt) ] + o( llt ) k ~ I Po(t + llt) = Po(t)[l - i,o llt + o (llt )] + P1(t)[fll llt + o (ll t )] + o( ll t) k = 0

(2.123)

(2. 124)

In Eq. (2.124) we ha ve used the assumption that it is imposs ible to ha ve a death when the population is of size 0 (i.e., flo = 0) and the assumption that o ne indeed can have a birt h when the populat ion size is 0 (i,o ~ 0). Expanding the right-hand side of Eq s. (2.123) and (2.124) we ha ve Pk(t Po(t

+ llt ) =

+ ,Uk) lltP.(t) + Ak-i ll tPk_l( t) + flk+l ll tPk+l(t) + o(llt ) + llt ) = Po(t ) - i.o lltPo(t ) + fllllIP l(t) + o(ll t) Pk(l ) - (I'k

k ~ 1

k=O

If we now subtract Pk(t) from both sides of each equation and divide by llt, we have the following : .

Po(t

+ Ill ) III

Po(l )

=

- i.oPo(t )

+ fllP l(t ) + o( ll t)

k

=0

(2.126)

III

T ak ing th e limit as llt a pp roac hes 0 we see th at the left-hand sides of Eq s. (2.125) a nd (2.126) represent the formal derivati ve of Pk(t ) with respect to t and also that the ter m o (ll l) jll l goes to O. Con sequ ently, we have the result ing equation s : k ~ l

- (2.127)

k =O The set of equations given by (2.127) is clearly a set of different ial-difference equati on s and represent s the dynamics of our probabil ity system ; we

58


recognize them as Eq. (2.114) and their solution will give the behavio r of Pk(t ). It remains for. us to solve them. (No te that t his set was obtai ned by

essentially using the Chapman- Ko lrnogorov equations.) In order to solve Eqs. (2.127) for the time-dependent behavior Pk(t) we now require our initial cond ition s: that is, we must specify Pk(O) for k = 0, I, 2, . . . . In addi tio n, we further require that Eq . (2.122) be satisfied. Let us pa use temp orarily to describe a simple inspect ion technique for finding the differenti al-difference equa tions given ab ove. We begin by observing that an alternate way for disp laying the information contained in the Q matri x is by means of the state-transition-rate diagram . In such a diagram the sta te Ek is represented by an ova l surro unding the number k. Each nonzero infinitesimal rate q j j (the elements of the Q matrix) is represented in the sta te-transition-ra te diagram by a directed branch point ing from E, to E , and label ed with the value q j j ' Fur thermo re, since it is clear that the terms a long the main diagonal of Q cont ain no new informa tion [see Eqs. (2.96) and (2.118)] we do not include the "self"-loop from E, back to E j • Thus the sta te-transition-rate diagram for the genera l birt h-death pro cess is as shown in Figure 2.9. In viewing this figure we may tru ly think of a pa rticle in motion moving among the se states; the branches identify the per mitted transitions and th e bra nch labels give the infinitesimal rates at which th ese transitions take place. We emph asize that the labels on the ordered link s refer to birth and dea th rates a nd not to probabilities. If one wishes to con vert these labels to proba bilities, one must multiply eac h by the quant ity dt to obtain the probabili ty of such a transition occurring in the next interval of time whose duration is dt , In t hat case it is also necessary to put self-loops on each -state indicating the prob ab ility that in the next interval of time dt the system remains in the given state . No te that t he sta te-transition-rate diagra m contains exactly the sa me informati on as does the tr ansition-rate matrix Q . Co ncentra ting on state E k we observe that one may en ter it only from state E k_1 or from sta te Ek+l an d similarly one leaves state E k only by entering sta te Ek - 1 or sta te Ek + 1 • From this picture we see why such processes are referr ed to as "nearest-neighbo r" birt h-deat h processes . Since we a re considering a dynamic situatio n it is clea r that the difference between the rate a t which the system ent ers Ek and the ra te at which the system leaves E k must be equal to the rate of change of "flow" into that state . This

Figure 2.9 State-transition-rate diagram for the birth-d eath process.

2.5.

DlRTH-DEAT H PR OCESSES

59

notion is crucial an d prov ides for us a simple intu itive mea ns fo r writing d own the equ ati on s of motion for the probabil ities Pk(t) . Specifically, if we focus up on sta te E k we observe th at the rat e at which probability " flows" into this state at time t is given by Fl ow rate into Ek

=

Ak_1Pk_1(t ) + flk+IPk+l (t)

whereas th e flow rate out of that state at time t is given by Fl ow rate out of Ek

=

(I'k + flk)Pk(t )

Clearly the difference between th ese two is the effective probability flow rate into this sta te, that is,

dPk(t )

~ =

_ Ak_1 Pk_1( t)

+ flk+l Pk+l(t) -

U k + fJk)Pk(t )

- (2.128)

But thi s is exactly Eq . (2. l27) ! Of course, we ha ve not a tte nded to th e details for the bound ar y state Eo but it is easy to see that the rate argument ju st given lead s to the correct equa tio n fo r k = O. Ob ser ve that each ter m in Eq . (2.128) is of the form : pr obability of bein g in a particular state at tim e t multiplied by the infinitesimal rate of leaving that state. It is clear that wha t we have done is to draw an imaginary boundary surrou nd ing sta te Ek and hav e calcul ated th e pr obability flow rates cr ossing th at boundary , where we place opposite signs . on flows entering as oppos ed to leaving ; thi s tot al computatio n is th en set equa l to the time derivati ve of the prob ability flow rate into that sta te. Actu ally there is no reason for selecting a single sta te as the "system" for wh ich the flow equ ati on s mu st hold . In fact one may encl ose an y number of sta tes wit hin a contour a nd th en write a flow equ ati on for all flow crossing ' th at boundary. Th e only d an ger in de aling with such a con glomerate set is th a t one may write down a dependent set of equ ati ons rather than an independent set; on the other hand , if one systema tically encloses each sta te sing ly a nd writes d own a con servation law for each, then one is guaran teed to have a n independent set o f equ a tions for the syste m with the qu ali fication th at the co nservatio n of prob ability given by Eq. (2. 122) mu st also be a pplied. * T hus we have a simple inspection techn iqu e for a rriving at the equa tions of moti on for the birth-death proce ss. As we sha ll see lat er th is ap proa ch is perfectly suita ble for other M ar kov pr ocesses (includi ng sem i- . M arkov p rocesses) a nd will be used extensively ; the se observa tio ns also lead us to the no tion of globa l and local balan ce equ ati on s (see C ha pter 4). At thi s point it is imp ortant for the reader to recognize and accept the fact that the birth-death pr ocess descr ibed abov e is capa ble of pr ovidin g the • When the number of states is finite (say. K states) then any set of K - I single-node sta te equations will be indepe ndent. T he addi tio nal equatio n needed is Eq . (2.122).

:

60

SOME IMPORTANT RANDOM PRO CESSES

framework for di scussing a large number of imp ortant an d interesting problems in queueing th eory. The direct solu tion for a ppropriate specia l case s of Eq. (2. 127) provides for us the tr an sient behavior of the se queueing systems and is of less int erest to this book than th e equilibrium or stea dystate beh avior of qu eues. * However, for purposes of illustrati on a nd to elaborate fur ther up on th ese equation s, we now con sider so me imp ortant examples. The simplest system to con sider is a pure birth system in which we assume fl k = 0 for all k (note th at we ha ve now entered the next- to- innermost circle in Figure 2.4!) . Moreover , to simplify the problem we will assume th at Ak = A for all k = 0, 1,2, . ... (N o w we have ent ered the innermost circle! We therefore expect some marvelou s properties to emerge.) Substituting thi s into our Eq s. (2.127) we have d Pk(l) - - = - APk(l) dl

+ APk_,(I)

k?:

1

(2.129) dPo(t) = - APo(l ) dl

k=O

For simplicity we assume that the system beg ins at time 0 with 0 members, that is,

k=O k,.,O

(2.130)

Sol ving for Po(l) we ha ve immed iately p oet) = e- At

Inserting thi s last int o Eq. (2.129) for k = I result s in dP,(t) = - AP,( t) dt

+ u:"

The so lution to th is d ifferenti al equati on is clearly P,(t)

=

J,te- AI

Continuing by induction, then , we finally have as a solution to Eq. (2.129) (}.t)k P (I ) = - - e k

k!

AI

k ?: 0, I ?: 0

_ (2.131)

This is the celebrated Poisson di stribution. It is a pure birth pr ocess with constant birth rat e A and gives rise to a sequence of birth ep och s which a re • Transien t behavior is discussed elsewhere in this text, nota bly in Chapte r 2 (Vol. II). For a n excellent trea tment the reader is referred to [COH E 69].

2.5.


61

said to constitute a Poisson process. Let us study the Poisson process more carefully and show its relat ionship to the exponential distribution. The Poisson process is central to much of elementary and intermediate queue ing theory and is widely used in their development. T he special position of this process comes about for two reasons. First , as we have seen, it is the "innermost circle" in Figure 2.4 and, therefore, enjoys a number of mar velous and simplifying anal ytical a nd probabilistic properties ; this will become und eniably apparent in our subsequent development. The second reason for its great import ance is that , in fact, nume rous natu ral physical and organic processes exhibit behavior that is probably meanin gfully modeled by Poisson pr ocesses. For example , as Fry [FRY 28] so graphically point s out, one of the first observations of the Poisson process was that it properly represented the number of army soldiers killed due to being kicked (in the head ?) by their horses. Other examples include the sequence of gamma rays emitting from a rad ioact ive part icle, and the sequence of times at which telephone calls a re originated in the teleph one network . In fact , it was shown by Palm [PALM 43] and Khinchin [KHIN 60] that in many cases the sum ofa large number of independent stationary renewal processes (each with an arbitrary distribution of renewal time) will tend to a Poisson process. Thi s is an imp ortant limit the orem and explain s why Poisson pr ocesses appear so often in nature where the aggregate effect of a large number of individual s or particles is under observa tion. " Since this development is intended for our use in the study of queueing systems, let us immediately adopt queueing notation and also conditi on ourselves to d iscussing a Poisson process as the arrival of customers to some queueing facility rather than as the birth of new members in a population. Thus ,l is the average rate at which the se customer s arrive . With the"initial condition in Eq, (2.130), PkV) gives the pr obability th at k arrivals occur during the time interva l (0, I). It is intuitively clear , since the average arrival rate is ,l per second , that the average number of a rrivals in an inte rval of length I must be AI. Let us carry out the calculation of this last intuitive statement. Defining K as the number of arr ivals in this interval of length I [previously we used a(I)] we have co

E[K] =

L kPk(t)

k ~O

-ll ~

= e

(i.t) k

L. k - -

k!

k _O

= e- 1 1 1

· ( ,lt)k

k _l

(k - I )!

-. t ] is j ust the probability that po et) . Therefore, we ha ve A(t)

=

110

> t] arrivals occur in (0, r), that is,

I - p oet)

and so from Eq. (2.131), we obtain th e PD F (in the Po isson case) A (t )

=

I - e:"

t~O

(2.138)

Di fferen tiating, we obta in the pdf aC t) = i.e- 1 t

t~O

(2. 139)

This is th e well-known exponential di stribution ; its pdf a nd PD F a re sho wn in Figure 2.12. W hat we ha ve show n by Eq s. (2. 138) a nd (2.139) is tha t for a Poisson arrival pr ocess, the time bet ween arri vals is exponentially d istributed ; thus we say that the Po isson arrival process ha s exp onential interarrival times.

I

I

l

66

SO~(E 1~IPORTANT RANDOM PROCESSES

1--------- _

(b ) PDF

Figure 2.12 Exponential distribution. The most amazing characteristic of the exponential distribution is that it has the remarkable memoryless property, which we introduced in our discussion of Markov processes. As the name indica tes, the past history of a random variable that is distributed exponentially plays no role in predicting its future; precis ely, we mean the following. Consider that an arrival has ju st occurred at time O. If we inquire as to what our feeling is regarding the distribution of time until the next arrival , we clearly respond with the pdf given in Eq. (2. 139). Now let some time pa ss, say, to sec, during which no arrival occurs. We may at this point in time again ask , " Wha t is the probability that the next arrival occurs t sec from now?" This que stion is the sa me question we asked at time 0 except we now know th at the time between arrivals is a t least to sec. To an swer the second que stion , we carry out the following calcul ati on s :

P [i ~ t

P[t o < i < t + 10J + to Ii> toj = --'-"--'--_-='----'---""

P[t

=

P[i

> toJ

< t + toJ - P[i < toJ P[i > toJ

Due to Eq . (2.138) we then ha ve

and so

P[i ~ t

+ to I i > toJ =

1 - e- "

(2.140)

This result sho ws that the distribution of rem aining time until the next a rr ival , given that to sec has elap sed since the last a rrival, is iden tically equal to the uncondition al distribution of intera rrival time . The imp act of this stat ement is that our probabilistic feeling regard ing the time unt il a future a rrival occurs is inde pendent of how lon g it has been since th e last arr ival occurred. Th at is, the future of an exponentially distributed rand om vari able

... I I

2. 5.


67

Figure 2. I3 The memoryless propert y of the exponential distribution. is independent of the pa st history of that variable and th is distribution remains constant in time . The exponential distribution is the Dilly continuou s distribution with thi s pr operty. (In the case of a discrete random vari able we have seen th at the geo metric distribution is the only discrete distr ibution with th at same property.) We may further a pprecia te the nature of this memo ryless property by co nsidering Figu re 2. I3. In this figure we show the exp onential den sity x«:". Now given that to sec has elap sed , in o rder to calculate the den sity funct ion for the time until the next arrival, what one must do is to take that portion of the density function lying to the right of the point to (show n shad ed) an d recognize that thi s region represent s our pr obabili stic feeling regardi ng the future ; the portion of the den sity functi on in the to to is pa st history and involves no more uncert ainty. In interval from o rder to make the shaded regio n into a bona fide den sity function, we must magnify it in o rder to increase its total area to unity ; the appropriate magnification tak es place by dividing the functi on rep resenting the tail of thi s distribution by the are a of the shaded region (which must , of course, be P[ J > tD. T his opera tion is identical to the opera tion of creati ng a cond itional distributi on by dividing a joint distributi on by the probability of the condition . Thus the sha ded region magnifies into the seco nd ind icated curve in Figure 2.13. This new functi on is an exact replica of the origina l density funct ion as shown from time 0, except that it is shifted to sec to the right. No other den sity functi on has the pr operty th at its tail everywhere possesses the exact same sha pe as the entire density functio n. We now use the memoryless pr operty of the exponential distribution in order to close th e cir cle regarding the relati ons hip between th e Poisson and exp onential distrib uti ons. Eq uatio n (2.140) gives an expres sion for the PDF of the inte rarrival time cond itioned on the fact th at it is at least as lar ge as to. Let us positio n ou rselves at time to and as k for the probability that the next

°

68

SOME I~I PORTANT RA NDOM PRO C ESSES

arrival occurs within the next !::>.f sec. From Eq . (2.140) we have P[i ~ 1 +!::>.I Ii> 1 ] = I - e- a t 0

0

[ 1 - I. !::>.I

= 1-

=

(A !: >. t)"

+ 2"!" - ..-]

A!::>.I + O(!::>. I)

(2. 14 1)

Equation (2.14 1) tells us, given that a n arrival has not yet occurred, th at the prob ability of it occurring in the next interval of length !::>.I sec is A!::>.t + O(!::>.I). But thi s is exactly assumption B[ from the opening paragraphs of thi s section. Furthermore, the probability of no a rrival in the interval (to, to + !::>.t) is calculated as

P[i

> 10 + !::>.t Ii> 10 ] =

1 - P[i ~ 10 + !::>.I = 1 - (1 - e- w ) = e- a t

=

• 1 - I. !::>. I

= 1 - I.!::>.I

I i > 10 ]

(A !::>.I)" +- - .. .

2!

+ O(!::>.I)

This corroborates assumpti on B2 • Furthermore, P[2 or more arrival s in (10' 10 = 1-

+ !::>.I)]

P[none in (to, 10

= 1 - [1 = 0(!::>.1)

1.!::>.1

+ !::>.I)] -

+ O(!::>.I)] -

Pron e in (to, 10

[I.!::>.1

+ !::>.I)]

+ O(!::>.I)]

This corroborates the "multiple-birth" assumption . Our conclusion , then , is tha t the ass umption of exp onentially distributed interarri val time s (which are independent one from the other) implies that we ha ve a Poi sson proce ss, which implies we have a con stant birth rate. The con verse implication s a re also true. This relationship is shown gra phica lly in Figur e 2.14 in which the symb ol t is given by J. 6>t [+ o(6)t)] and thus i. itse lf must be the average rate of arrivals ; thus the average time between arrivals must be l{i.. In orde r to evaluate the variance, we first calculate the second moment for the interarrival time as follows:

2

70

SOME I~IPORTA NT RANDOM PROCESSES

Thus the variance is given by

a/ =

E[(i)"] -

vr

=:2- (1Y and so 1

9

(2.143)

a( = ::; ) .-

As usual, these two moments could more easily have been calculated by first considering the Laplace transform of the probability density functi on for this random variable. The notati on for the Laplace transform of the interarrival pdfis A *(s) . In thi s special case of the exponential distribution we then have the followin g : A*(s)

~ r)~-'la(t) dt =

1""e- "i.e-J.' dt

and so A *(s)

= -)'-

(2.144)

s +}.

Equation (2.144) thus gives the Laplace transform for the exponential density functi on . Fr om Appendix II we recognize that the mean of thi s density function is given by f = _ dA*(s) ds

=

(s

I ,~ O

I

i.

+ if ,_0

I

i. The second moment is also calculated in a similar fashion: 2

E[(i)2] =

d A *(s) ds

-2-

2i.

I ,~ O

I

= (s + }.)3 ,~O 2

2.5.

BIRTH- DEATH PR OCESSES

71

an d so

Thus we see the ease with which moments can be calculated by making use of tr an sforms. No te also, th at the coefficien t of variation [see Eq . (II .23)] fo r the exp onential is (2.145) It will be of further interest to us later in the text to be able to calculate the pdf for the time interval X required in o rder to collect k arrivals from a Poisson pro cess. Let us define th is random variable in terms of the random va riables In where In = time between nth and (n - I)th arrival (where the " zeroth" arrival is assumed to occur a t time 0). Thus

We define f x(x) to be the pdf for this random vari able . From Appendix II we sho uld immedia tely recognize that the density of X is given by the con volu tion of the den sities on each of th e I,:S , since they are indep endently distri buted. Of course, thi s con voluti on operatio n is a bit lengthy t o carry out, so let us use our further result in Appendix II, which tells us that th e Lapl ace tr an sform of th e pdf for the sum of independent random va ria bles is equ al to the product of the Laplace transforms of the den sity for each . In our case each I n ha s a comm on exponential distribution and therefor e the Laplace transform for th e pdf of X will merel y be the kth po wer of A *(s) where A *(s) is given by Eq. (2.144); that is, defining

X*( s)

=

f'

e-SXfx(x) d x

for the Lapl ace tr an sform of the pdf of o ur sum, we have

X *(s) = [A *(S)]k thus

X *(s)

J

= ( -?-)k s +?

(2.146)

72

SOME IM PORT A;-,/T RA;-,/DOM PRO CESSES

We must now invert thi s tr ansform. Fortunately, we identify the needed transform pair as entry. 10 in Table 1.4 of Appendix I. Thus the density funct ion we are lo oking for , which describes the time required to o bserve k arrivals, is given by

fx(x)

=

A(Ax )k-I .x (k _ I)! e-'

x ~ O

(2.147)

This family o f density functi ons (one for each va lue of k) is referred to as the family of Erlang distributions. We will have con siderable use for thi s famil y later when we di scu ss the method of stages, in Chapter 4. So much for the Poisson arrival process and its relati on to the exponential di stribution. Let us now return to the birth-death equations a nd consider a m ore genera l pure birth process in which we perm it state-dependent birth rates Ak (for the Poisson process , we had Ak = A). We once again insist that th e de ath rates fl.k = O. From Eq . (2.127) thi s yield s the set of equations

dPk(t)

-- = dt

dPo(t)

- - = dt

- AkPk(t)

_

+ Ak_IPk_l(t )

k~1

(2 .148)

k=O

-)'oPo(t)

Again, let us a ssume the initial di stribution as given in Eq. (2. 130), which states that (with probability one) the population begin s with 0 members a t time O. Solving for poet) we have

p oet)

=

e- Ao'

The general solution * for Pk(t) is given bel ow with a n explicit expression for the first two va lues of k:

Pk(t )

= e-Ak' [i'k_Ii'Pk_I(X)eAkX d x + Pk(O)]

k

= 0, 1,2, . ..

(2 . 149)

i.o(e-Ao' _ e- AI') P I( t) = -=-----'

i' l

i·o A" A,' P (t ) = . i,o)'I. [e- - :- A" _ e- 2 Al "'0 A2 - 1.1 i' 2 - Ao -

e-Ao'J

As a th ird exam pie of the time-dependent solution to the birth-death equations let us consider a pure death process in which a populat ion is ini tia ted with , say, N members and all that can happen to thi s population is that members die; none a re born. Thus Ak = 0 for all k , and fl.k = fl. ~ 0 • The validity of this solution is easily verified by substituting Eq. (2.149) into Eq. (2.148).

2.5.

BIRTH -D EATH PROCESSES

73

for k = I, 2, . . . , N. For this constant de ath rate process we have d Pk(t) = dl

-flPk(t )

+ flPk +l(t )

O

P(z, t) = 2,Pk(t)Zk k=O

(c)

and find the partial differential equation that P(z, t ) must sat isfy. Show that the solution to this equ ation is

P(z, t) = exp (;

(1- e-")(z - 1»)

with the initial condition PoCO) = I. Comparing the solution in part (c) with Eq . (2.134), give the . expression for Pk(t) by inspection . (e) F ind the limiting values for the se probabilities as t ---+ 00 .

(d)

2.13.

Consider a system in which the birth rate decre ases and the death rate increases as the number in the system k increases, th at is,

(K - k )A

Ak

={

k~K

o

k > K

Write down the differential-difference equation s for

Pk(t) = P[k in system a t time t]. 2.14.

Consider the case of a linear birth-death process in which Ak = k A a nd P.k = kp:

Find the partial-differential equation th at must be satisfied by P(z, t ) as defined in Eq . (2. 153). (b) Assuming that the population size is one a t time zero, show that the funct ion that satisfies the equation in part (a) is

(a)

EXERCIS ES

(c)

Expanding P(z, t) in a power series show that

Pk(t) poet) (d) (e)

2.15.

83

= =

[l ....:. ex:(t)][1 - ,8(t)][,8(t)]k-1 ex:(t)

k

=

1, 2, . . .

and find «(r) and ,8(t). Find the mean and variance for the number in system at time t . Find the limiting probability that the population dies out by time t for t ->- 00 .

Consider a linear birth-death process for which Ak = kA + ex: and flk = ku, (a) Find the differential-difference equations that must be satisfied by Pk(t). (b) From (a) find the partial-differential equation that must be satisfied by the time-dependent transform defined as eo

P(z, t) = ~ Pk(t)Zk k -O

(c)

What is the value of P(I, t)? Give a verbal interpretation for the expression

-

a

N(t) = lim - P(z, t)

(d)

(e)

2.16.

.-1az

Assuming that the population size begins with i members at time 0, find an ordinary differential equation for N(t) and then solve for N(t) . Consider the case A = fl as well as A ,,-6 fl. Find the limiting value for N(t) in the case A < fl (as t --+ (0).

Consider the equations of motion in Eq, (2.148) and define the Laplace transform

P/(s)

=

i'"

Pk(t)e- $/ dt

For our initial condition we will assume poet) = I for t = 0. Transform Eq. (2.148) to obtain a set of linear difference equations in {Pk *(s)}. (a) Show that the solution to the set of equations is k- l

IT )' i P/( s) =

-k-,-, i -,-" O_-

IT (s + Ai) i= O

(b)

From (a) find p.(t) for the case Ai =

).

(i

=

0, I, 2, ...).

84


2.17.

Consider a time interval (0, t) during which a Poisson process generates arrivals at an average rate A. Deri ve Eq . (2.147) by con siderin g the two events: exactly k - I arrivals occur in the interval (0, t - tl.t) and the event that exactly one arrival occur s in the interval (t - tl.t, r). Considering the limit as tl.t ->- 0 we immedi ately arrive at our desired result.

2.18.

A barber opens up for business at t = O. Customers arrive at random in a Poisson fashion ; that is, the pdf of interarrival time is aCt) = Ae- lt • Each haircut takes X sec (where X is some random variable). Find the probability P that the second arriving customer will not have to wait and also find W, the average value of his waiting time for the two following cases : i. X = c = constant. ii. X is exponentially distributed with pdf: b( x)

2.19.

= pe-~'

At t = 0 customer A places a request for service and finds all m servers busy and n other customers waiting for service in an M/M /m queueing system . All customers wait as long as necessary for service, waiting customers are served in order of arri val, and no new requests for service are permitted after t = O. Service times are assumed to be mutually independent, identical , exponentially distributed rand om variables, each with mean duration I Ip. (a) Find the expected length of time custome r A spends waiting for service in the queue. (b) Find the expected length of time from the arrival of customer A at t = 0 until the system become s completely empty (all cust omer s complete service). (c) Let X be the order of completion of service of customer A ; that is, X = k if A is the kth customer to complete service afte r t = O. Find P[X = k] (k = 1,2 , . .. , m + n + I). (d) Find the probability that customer A completes service before the custo mer immediately ahead of him in the queue. (e) Let 1V be the amount of time custo mer A waits for service. Find P[lV > x ] .

2.20. In thi s problem we wish to proceed from Eq. (2. 162) to the transient solution in Eq. (2.163). Since P* (z , s) must converge in the region [z] ~ 1 for Re( s) > 0 , then, in this region, the zero s of the denom inator in Eq. (2.162) mu st also be zeros of the numerat or. (a) Find those two values of z th at give the denominator zeros, and denote them by ct1 (s) , ~ (s) where 1ct2 (s) 1 < Ict 1 (s)l.

I

i

I ]

EXERCISES

85

Using Rouche's theo rem (see Appendix I) show that the denominator of P* (z, s ) has a single zero within the unit disk Izl ~ 1. (c) Req uiring that the -numerator of P *(z , s) vanish at z = CX2(S) from our earlier considerations, find an explicit expression for Po* (s). (d) Write P* (z, s) in terms of cx,(s) = cx, and cx 2 (s) = cx 2 • Then show that this equati on may be reduced to (b)

P*(z s)

= (z' + ~zH + .. . + CX2') + cx;+l/(I

,

- CX2)

Acx,(1 - Z/CX1)

(e)

Usin g the fact that Icx 21 < I and that CX 1CX2 = P./A show th at the inversion on z yields the followi ng expression for Pk *( s), which is the Laplace transform for our tra nsient probabilities Pk(t):

(f)

In what follows we take advantage of property 4 in Table 1.3 and also we make use of the following transform pair :

.

k pkl _t-lf k(al)-=-

[s+

";S2 -

2.A.

4A.u]-k

where p and a are as defined in Eqs. (2. 164), (2.165) and where f k(x) is the modified Bessel function of the first kind of order k as defined in Eq . (2.166). Using these facts and the simple relations among Bessel function s, namel y,

show th at Eq. (2.163) is the inverse transform for the expression show n in part (e). 2.21.

The rand om variables Xl' X 2 , • •• , Xi' . . . are independent. identically distributed random variables each with density f x (x) and characteristic funct ion x (u) = E[~ Uxl . Consider a Poisson process N (t) with parameter A which is independent of the rand om variables Xi' Consider now a second random process of the form ..\" (t)

X(t)

=

LXi i =l

86


This second random process is clearly a family of staircase functions where the jumps occur at the discontinuities of the random process N(I); the magnitudes of such jumps are given by the random variables Xi. Show that the characteristic function of this second random process is given by ePXW(II) = 2.22.

e .lt[4>.rlul -ll

Passengers and taxis arrive at a service point from independent Poisson processes at rates )., p. , respectively. Let the queue size at time I be q" a negative value denoting a line of taxis, a positive value denoting a queue of passengers. Show that, starting with qo = 0, the distribution of q, is given by the difference between independent Poisson variables of means At, ut. Show by using the normal approximation that if), = p., the probability that -k ~ q, ~ k is, for large 1)(41TAt)-1/2. t, (2k

+

I I

I

,I I

PART

II

ELEMENTARY QUEUEING THEORY

Elementary here means that all the systems we consider are pure Markovian and, therefore, our state description is convenient and manageable. In Part I we developed the time-dependent equations for the behavior of birth-death processes ; here in Chapter 3 we address the eq uilibrium solut ion for these systems. The key eq uation in th is chapte r is Eq. (3.11), and the balance of the material is the simple application of that formula. It , in fact , is no more than the solution to the equation 1t = 1tP deri ved in Chapter 2. The key tool used here is again that which we find throughout the text, namely, the calculation of flow rates across the bou ndaries of a closed system. In the case of equilibrium we merely ask that the rate of flow into be equal to the rate of flow out of a system. The application of these basic results is more than just an exercise for it is here that we first obtain some equations of use in engineering and designing queueing systems. The classical M IM II queue is studied and some of its important performance measures are evaluated. More comple x models involving finite storage, multiple servers, finite customer population, and the like, are developed in the balance of this chapter. In Chapter 4 we leave the birth-death systems and allow more general Markovian queue s, once again to be studied in equilibrium. We find that the technique s here are similar to our earlier ones, but find that no general solution such as Eq. (3.11) is available ; each system is a case unt o itself and so we are rapidly led into the solutions of difference equations, which force us to look carefully at the method of z-transforms for these solutions. The ingenious method of stages introduced by Erlang is considered here and its generality discussed. At the end of the chapter we introduce (for later use in Volume II) networks of Markovian queues in which we take exquisite ad vantage of the memoryle ss propertie s that Mark ovian queues provide even in a network environment. At this point , however, we have essent ially exhausted the use of the memoryless distribution and we must depa rt from that crutch in the following parts.

87

3 Birth-Death Queueing Systems in Equilibrium

In the pre vious chapter we studied a variety of stochastic pr ocesses. We indicated that Markov processes play a fundamenta l role in the study of queueing systems, and after presenting the main results from that theory, we then considered a special form of Markov pr ocess known as the birthdeath process. We also showed that birth-death processes enjoy a most convenient property, namely, that the time between births and the time between deaths (when the system is nonempty) are each exponentially distributed. * We then developed Eq. (2.127), which gives the basic equa tions of moti on for the general birth-death process with stationary birth and death rates.] The solution of this set of equations gives the transient beha vior of the queueing process and some importa nt special cases were discussed earlier. In th is chapter we stud y the limiting form of these equations to obtain the equilibrium behavi or of birth-death queueing systems. The importance of elementary queueing theory comes from its histori cal influence as well as its ability to describe behavior that is to be found in more complex queueing systems. The methods of analysis to be used in this chapter in large part do not carryover to the more involved queueing situations; nevertheless, the obta ined results do provide insight into the basic behavior of many of these other queueing syste ms. It is necessary to keep in mind how the birth-death process describes queueing systems. As an example , consider a doctor's office made up of a waiting room (in which a queue is allowed to form, unfortu nately) and a service facility consisting of the doctor's examination room. Each time a patient ente rs the waiting room from outside the office we conside r this to be an arrival to the queueing system; on the other hand, this arrival may well be considered to be a birth of a new member of a population, where the population consists of all patients present. In a similar fashion, when a patient leaves • Th is comes directly fro m the fact that they are Markov processes.

t In addit ion to these equations, one requires the con servat ion relat ion given in Eq. (2.122) and a set of initia l conditions {Pk(OJ}.

89

90

BIRTH-DEATH QUEUEING SYSTEMS IN EQUILIBRIUM

the office after being treated, he is considered to be a departure from the queueing system; in terms of a birth-death process this is considered to be a death of a member of the population . We have considerable freedom in constructing a large number of queueing systems through the choice of the birth coefficients Ak and death coefficients flk, as we shall see shortly. First, let us establish the general solution for the equilibrium behavior.

3.1.

GENERAL EQUILIBRIUM SOLUTION

As we saw in Chapter 2 the time-dependent solution of the birth-death system quickly becomes unmanageable when we consider any sophisticated set of birth-death coefficients. Furthermore, were we always capable of solving for Pk(t) it is not clear how useful that set of functions would be in aiding our understanding of the behavior of these queueing systems (too much information is sometimes a curse!). Consequently, it is natural for us to ask whether the probabilities Pk(t) eventually settle down as t gets large and display no more "transient" behavior. This inquiry on our pa rt is analogous to the questions we asked regarding the existence of 1Tk in the limit of 1Tk (t ) as t ->- CfJ. For our queueing studies here we choose to denote the limiting probability as Pk rather than 1Tk> purely for convenience. Accordingly, let Pk ~ lim Pk(t)

(3.1)

,-'"

where Pk is interpreted as the limiting probability that the system contains k members (or equivalently is in state Ek ) at some arbitrary time in the distant futu re. The question regarding the existence of these limiting probabilities is of concern to us, but will be deferred at this point until we obtain the general steady-state or limiting solution. It is important to understand that whereas Pk (assuming it exists) is no longer a function of t, we are not claiming that the process does not move from state to state in this limiting case; cert ainly, the number of members in the population will change with time, but the long-run probability of finding the system with k members will be properly described by P» Accepting the existence of the limit in Eq. (3. I), we may then set lim dPk(t)! dt as t ->- CfJ equal to zero in the Kolmogorov forward equations (of motion) for the birth-death system [given in Eqs. (2.127)] and immediately obtain the result

o= o=

+ flk)Pk + )'k-IPk-l + flHIPHl -AoPo + fllPl

-(At

k ~ 1

=0

(3.2)

(3.3) The annoying task of providing a separate equation for k = 0 may be overcome by agreeing once and for all that the following birth and death k

3.1.

91

GENERAL EQUILIBRI UM SOLUTION

coefficients are identically equal to 0 :

A_1 flo

== A_ 2 = A_3 = = fL-1 = fL-2 =

=0 =0

Furthermore , since it is perfectly clear that we cannot have a negative number of members in our population, we will , in most cases , adopt the co nvention that P- 1 = P-2 = P- 3 = .. . = 0 Thus, for all value s of k , we may reformulate Eqs. (3.2) and (3.3) into the following set of difference equations for k = .. . , - 2, -I, 0, 1, 2, . ..

o=

-

(Ak

+ fLk)Pk + Ak - 1Pk- 1 + f.lk+1Pk+1

(3.4)

We also require the conservation relation co

(3.5)

2.Pk = 1 k= O

Recall from the previ ous chapter that the limit given in the Eq. (3. 1) is independent of the initial conditions. Ju st as we used the state-transition-rate diagram as an inspection technique for writing down the equations of motion in Chapter 2, so may we use the sa me concept in writing down the equilibrium equations [Eq s. (3.2) and (3.3)] directly from that d iagram. In thi s equilibrium case it is clear that flow mu st be conserved in the sense that the input flow must equal the output flow from a given state. For example, if we look at Figure 2.9 once again and concentrate on sta te E k in equilibrium, we observe that and

In equilibrium the se two mu st be the same and so we ha ve immediatel y (3.6)

But this last is ju st Eq . (3.4) again! By inspection we have established the equilibrium difference equations f or our system . The sa me comments apply here as applied earlier regard ing the conser vat ion of flow across any closed boundary ; for example, rather than surrounding each sta te a nd writing d own its equation we could choose a sequence of boundaries the first of wh ich surrounds Eo , the second of which surrounds Eo and E and so on , each time add ing the next higher-numbered state to get a new "boundary . In such a n exa mple the kth boundary (which surrounds sta tes Eo, E . .. , Ek _ 1) would

"

92

BIRTH-D EATH QUEUEI NG SYSTEMS IN EQUILIBRIUM

lead to the following simple conservation of flow relationship: (3.7)

Ak-IP k-I = flkPk

This last set of equations is equivalent to drawing a vertical line separating adjacent states and equating flows across this boundary ; this set of difference equations is equivalent to our earlier set. The solution for P» in Eq. (3.4) may be obtained by at least two methods. One way is first to solve for P, in terms of Po by considering the case k = O. that is. PI

A

= - O Po

(3.8)

fll

We may then consider Eq. (3.4) for the case k

o= o= o= and so

=

I and using Eq. (3.8) obtain

+ fl,)P, + AoPo + fl2P2 o -(AI + flI) A Po + AoPo + fl2P2

-(J' I

flI

-

AIAo

-

flI

+ AoPo + fl2P2

Po - J..,po

P2

AOA I

=-

fllfl2

(3.9)

Po

If we examine Eqs. (3.8) and (3.9) we may justifiably guess that the general solution to Eq. (3.4) must be

(3.10)

fllfl 2 . . . flk

To validate this assertion we need merely use the inductive argument and apply Eq. (3.10) to Eq, (3.4) solving for PHI' Carrying out this operation we do. in fact . find that (3.10) is the solution to the general birth-death process in this steady-state or limiting case. We have thus expressed all equilibrium probabilities P» in terms of a single unknown constant Po: k-I }... Pk = Po I1' i=O

k=0.1,2• .. .

- (3.II )

I' i+l

(Recall the usual convention that an empty product is unity by definition.) Equation (3.5) provides the additional condition that allows us to determine Po; thus, summing over all k , we obtain I <X>

I +~

k -I

A

I1-'

k= l i = O

Pi+l

- (3.12)

3.1.

GENERAL EQUILIBRI UM SOL UTION

93

Th is "product" solution for P» (k = 0 , I , 2, . . .) simply obt ained , is a principal equati on in elementary queueing theory and, in fact, is the point of dep arture for all of our further solutions in this chapter. A second easy way to obtain the solution to Eq. (3.4) is to rewrite that equ at ion as follows : (3.13) Defining (3.14) we have from Eq. (3. 13) that (3.15) Clearl y Eq. (3.15) implies that

gk = constant with respect to k However, since A._I

= Po =

(3.16)

0, Eq. (3.14) gives

g_1

=

0

and so the constant in Eq. (3.16) must be O. Setting gk equal to 0, we immediately obtain from Eq , (3.14) P HI

A.k

= - - Pk

(3.17)

P HI

Solving Eq. (3.17) successively beginning with k = 0 we obtain the earlier solution , namely, Eqs. (3.11) and (3.12). We now address ourselve s to the ex istence of the steady-state probabilities P» given by Eqs. (3.11) and (3. 12). Simply stated, in order for those expression s to repre sent a probability distribution, we usually require that P» > O. Thi s clearly places a condition upon the birth and death coefficients in those equations. Essentially, what we are requiring is that the system occasionally empties ; that this is a condition for stability seems quite reasonable when one interprets it in terms of real life situations. * More precisely , we may classify the possibilities by first defining the two sums (3.18)

(3.19) • It is easy to construct counterexa mples to th is case. and so we requ ire the precise argument s which follow.

j

94

BIRTH-DEATII QUEUE ING SYSTEMS IN EQU ILIBRIUM

All states E k of our birth-death process will be ergodic if and only if Ergodic :

5,

< IX)

52 =

IX)

On the other hand , all states will be recurrent null if and only if Recurrent null:

5, 52

= =

IX) IX)

Also, all states will be transi ent if and only if Transient :

5,= 52

IX)

< IX)

It is the ergodic case that gives rise to the equilibrium probabilities {fk} and that is of most interest to our studies. We note that the condition for ergodicity is met whenever tbesequence {).,JPk} remains below unit y from so me k onwards, that is, if there exists some k o such th at for all k ~ k o we have

Ak < I (3.20) Pk We will find this to be true in most of the queueing systems we study. We are now ready to apply our general solution as given in Eqs. (3. 11) and (3. 12) to some very important special cases. Before we launch headlong into that discussion. let us put at ease those readers who feel that the birthdeath constraints of permitting on ly nearest-ne ighb or tran sition s are too confining. It is true that the solution given in Eqs. (3. 1I) and (3. 12) applies only to neare st-ne ighb or birth-death processes. H owever . rest assured that t he equilibrium meth ods we have described can be extended to more general than neare st-neighbor system s ; these generalizat ions a re co nside red in Chapter 4. 3.2.

M/M/1: THE CLASSICAL QUEUEING SYSTEM

As mention ed in Chapter 2, the celeb rated MIM II queue is the simplest nontrivial interestin g system and may be described by selecti ng the birth death coefficient s as follows :

k = 0, 1,2, .. . k=I ,2,3 • . . .

That is, we set all birth * coefficient s equal to a constant A and all death * • In this case. the average intera rrival time is f = 1/). a nd the average service time is l /p; this follows since t and i are both exponen tially distributed .

i =

---3.2. M /M /I: A

THE CLASSIC AL Q UEU EING SYST EM

95

A

~ ... Figure 3.1 State-transition-rate diagram for M/M /I.

coefficients equ al to a constant /-l . We further assume that infinite queueing space is provided and that customers a re served in a first-come-first-served fashion (although this last is not necessary fo r man y of our results). For thi s important example the sta te-transition-rate diagram is as given in Figure 3.1. Applying these coefficients to Eq . (3. 11) we have k- l }.

Pk = Po II -

'-0 /-l

or (3.21) The result is immediate. The condi tion s for our system to be ergodic (and , therefore, to ha ve an equilibrium solution P» > 0) are that S, < CXJ and So = CXJ; in this case the first condition becomes

The series On the left-hand side of the inequality will converge if and only if Af/-l < I. The second conditi on for ergodicit y becomes

T his last condition will be satisfied if Af/-l ::;; I ; thus the necessary and sufficient condition for ergodicity in the M IMII queue is simply}. < /-l. In order to solve for Po we use Eq. (3.12) [or Eq . (3.5) as suits the reader] and obtai n

96

BIRTH-DEATH QUEUEI NG SYSTEMS IN EQUILIBRI UM

The sum conver ges since).

< fl and so 1 1

+

Alp1 - ;./fl

Thus

A

Po = 1 - (3.22) PFrom Eq. (2.29) we have p = Alfl' From our stability conditi ons, we therefore require that 0 ~ p < 1; note that this insures that Po > O. From Eq. (3.21) we have, finally,

Pk = (I - p)pk

k

= 0, 1,2, .. .

- (3.23)

Equation (3.23) is indeed the solution for the steady-state probability of finding k customers in the system. * We make the important observation that P» depends upon). and fl only through their ratio p. The solution given by Eq. (3.23) for this fundamental system is graphed in Figure 3.2 for the case of p = 1/2. Clearly, thi s is the geometric distribution (which shares the fundamental memoryless property with the exponential distribution). As we develop the behavior of the MIMII queue, we shall continue to see that almost all of its important pr obabil ity distributions are of the memoryless type. An important measure of a queuein g system is the average number of customers in the system N. This is clearly given by

= (I

co

- p),I kp k k= O

Using the trick similar to the one used in deriving Eq. (2.142) we have

a

<Xl

N= (I-p ) p - I l ap k- O

= (1 -

a

1

p) p - - -

N =P1- p

ap 1 - p - (3.24)

• If we inspect the transient solution for M/M /l given in Eq. (2.163), we see the term p)pk ; the reader may verify that , for p < 1, the limit of the transient solution agrees with our solution here.

(I -

3.2. MIM/l :

TIlE CLASSICAL QUEUEING SYSTEM

97

l- p

~

(l -p)p

(1_p)p2

o

Figure 3.2 The solution for P« in the system MIMI!. The behavior of the expected number in the system is plotted in Figure 3.3. By similar methods we find that the va riance of the number in the system is given by 00

G,v" = Z(k -

R)"Pk

k- O

- (3.25) We may now appl y Little's result directly from Eq . (2.25) in order to obtain

o Figure 3.3 The average number in the system MIMI!.

98


o p ~

Figure 3.4

Average delay as a funcIion of p for M IMI!.

T , the average tim e spent in the sy stem a s follows :

N

T=-

;.

T=

C~)G) IIp.

T=-l-p

- (3.26)

Th is d ependence o f a verage t ime on the utilizati on factor p is shown in Figure 3.4. The va lue obtained by T when p = 0 is exactly the av e rage servi ce time expected by a cu st omer; th at is, he spe nds n o time in queue a nd IIp. sec in se rvi ce o n the av erage. The beha vior given by Eqs . (3.24) and (3.26) is rather dramatic. As p a p p roac hes unity , both the a ve ra ge number in the sys tem a nd the average tim e in the sys te m gro w in an unbounded fa shi on. * Bot h th ese q uantities have. a

• We observe at p = I that the system behavior is unstable ; this is not surprising if one recalls that p < I was our condition for ergodicity. What is perhaps surprising is that the behavior of the average number iii and of the average system time T deteriorates so badly as p - I from below; we had seen for steady flow systems in Chapter I that so long as R < C (which corresponds to the case p < I) no queue formed and smooth, rapid flow proceeded through the system. Here in the M IMI! queue we find this is no longer true and that we pay an extreme penalty when we attempt to run the system near (but below) its capacity. The

3.3.

DISCOURAGED ARRI VALS

99

simple pole at p = 1. This type oj behavior with respect to p as p approaches 1 is characteristic of almost every queueing system one. CUll encounter. We will see it aga in in M IGII in Chap ter 5 as well as in the heavy traffic beh avior of G/Gjl (and also in the tight bounds on G/Gfl behavior) in Volume II, Chapter 2. Another interesting quantity to calculate is the pr obability of finding at least k customers in the system : P[~k in system]

= I'" Pi i =k

= I'"( l

- p)pi

i- k

P[~k in system] = pk

- (3.27)

Thus we see that the probability of exceeding some limit on the number of customers in the system is a geometrically decrea sing function of that number and decays very rapidly. With the tools at hand we are now in a position to develop the probability density function for the time spent in the system. However, we defer that development until we treat the more general case of M/Gfl in Chapter 5 [see Eq. (5.118)]. Meanwhile, we proceed to discuss numerous other birthdeath queues in equil ibrium . 3.3.

DISCOURAGED ARRIVALS

This next example considers a case where arrivals tend to get discouraged when more and more people are present in the system. One possible way to model this effect is to choose the birth and death coefficients as follows : ex

)'k

= - -

k = 0, 1,2, . . .

ft k

= ft

k = 1, 2, 3, . . .

k+1

We are here assumin g an harmonic discouragement of arriv als with respect to the number present in the system. The state-transition-rate diagram in thi s intuitive explanation here is that with ra ndo m flow (e.g., M/M/! ) we get occasional bursts of traffic which temporarily overwhelm the server ; while it is still true tha t the server will be idle on the average I - p = Po of the time this average idle time will not be distri buted uniform ly within sma ll time interva ls but will onl y be true in the long run . On the ot her hand , in the steady flow case (which cor respond s to our system D/D /! ) the system idle time will be distributed quite uniforml y in the sense that a fter every service time (of exactly I II' sees) there will be an idle time of exactly (J f).) - ( l ip.) see. Thus it is the cariability in bot h the interarrival time and in the service time which gives rise to the disastro us beha vior near p = 1; any reduction in the var iati on of either rand om varia ble will lead to a reduct ion in the average wait ing time , as we shall see again and aga in.

100


ex

ex!2

cxlk

cxI(k + l )

~ ... Figure 3.5 State-transition-rate diagram for discouraged arrivals. ca se is as shown in Figure 3.5. We apply Eq . (3.11) immediately to obtain

iox p; oc/(i + 1) II :..:!...OC-'-2

k -1

Pk = Po

Pk

=

po(~r

(3.28)

fl

;- 0

:!

(3.29)

Solving for Po from Eq, (3.12) we have

1/[1 + I (~)k l.-J

Po =

fl

k-1

Po

=

k!

e~!p

From Eq. (2.32) we have therefore , p

=

1-

e -a /

p

(3.30)

N ote that the ergodic condition here is merely ocl fl Eq. (3.29) we ha ve the final solution Pk

= (OC~)k e- ' !p

k

=

< 00 .

Going back to

0, 1, 2, . . .

(3.31)

We thus ha ve a Poisson distribution for the number of customers in the system of discouraged arrivals! From Eqs. (2.131) and (2.132) we have that the expected number in the system is oc N = -

fl In o rder to calculate T , the average time spent in the system , we may use Little's result agai n. For this we require A, which is dir ectly calculated from p = AX = Al fl ; thus from Eq. (3.30)

). =

flP

= fl(l

-

e -o / P )

Using this* and Little's result we the n obta in

T

=

(3.32)

oc

fl"( I - e- ' /P)

• No te that this result could have been obtained from ;' verify this last calculat ion.

= L kJ.kPk- The

reader should

3.4. M/M / oo:

3.4.

RESPONSIVE SERVERS (I NFINITE N UMBER OF SERVERS)

101

M/M/ oo: RESPONSIVE SERVERS (INFINITE NUMBER OF SERVERS)

Here we con sider the case that may be interpreted either as th at of a responsive server who accelera tes her service rate linearly when more custom ers a re wait ing or may be interpreted as the case where there is always a new clerk or server ava ilable for each arriving custo mer. In particular , we set k

=

0, 1,2

k = 1,2.3,

. .

Here the state-t ransition-ra te dia gram is th at shown in Figure 3.6. G oing dire ctly to Eq. (3.11) for the solution we obtain

i,

k -1

Pk = Po .~ ITo (.I + I )f-l

(3.33)

Need we go a ny further? The reader should compare Eq . (3.33) with Eq. (3.28). These two are in fact equi valent for ex = i., and so we immedi atel y have the solutions for P« and N, P,

=

(}'If-l)k

- Al p

----z! e

k = 0. 1. 2•. ..

(3.34)

N=! f-l

Her e. too . the ergodic condition is simply i.l f-l < 00. It a ppea rs then that a system of d iscouraged arriva ls beh aves exactly the same as a system th at includ es a resp onsive server. H owever, Little 's result provid es a different (and simp ler) form for T here th an th at given in Eq . (3.32) ; thus I

T=-

f-l

This answer is, of co urse. obvio us sinc e if we use t he interpreta tion where each a rriving customer is gra nted his own server. then his tim e in system will be merely his service time which clearl y equ als l il t o n the ave rage.

x

x

11

211

~ ... Figure 3.6 State-transition-rate diagram for the infinite-server case M/M/oo.

r:

102

BIRTH-D EATH QUEUEING SYSTEMS IN EQU ILIBRIUM

3.5. M/M/m: THE m-SERVER CASE Here again we consider a system with an unlimited waiting room and with a constant arrival rate A. The system provides for a maximum of m servers. This is within the reach of our birth-death formulation and leads to

I' k

=

I.

=

k

0, 1,2, . . .

flk = min [kfl , mfl]

= {kfl

0 ~

ks

m

m ~ k

mft

From Eq. (3.20) it is easily seen that the condition for ergodicity is A/mft < I. The state-transition-rate diagram is shown in Figure 3.7. When we go to solve for P» from Eq. (3.11) we find that we must separate the solution into two parts, since the dependence of ftk upon k is also in two parts. Accordingly, for k ~ m , k-l A Pk = Po II (. 1) '- 0

~

fl

po(;r~!

=

Similarly, for k

+

I

(3.35)

m,

A

m-l

k- l

A

Pk = Po II . II '-0 (I + l)ft ;-m mft

Ak

I

= Po ( ft) m! mk- m

(3.36)

Collecting together the results from Eqs. (3.35) and (3.36) we have (mp)k

POI:! Pk = {

- (3.37)

(p) kmm

k~

Po--m!

m

where

A

p = mfl

This expression for A

p

0, m

(a)

= 1,2, . ..

S( m)) < ~ (A/p,)k «": < c(m)) p, k! p, k -m

(b)

3. 10.

c(m,;)

Here we consider an M/M/l queue in di screte time where time is segmented into intervals of length q sec each. We assume th at event s can only occur at the ends of the se discrete time intervals. In particul ar the probability of a single arrival at the end of such an interval is given by Aq and the pr obability of no arrival at that point is I - i.q (thus at most o ne arrival may occur). Similarly the dep arture pr ocess is such th at if a customer is in service during an interval he will co mplete service at the end of that interval with pr obability I - (J or will require at least one more interval with pr obability (J. (a) Derive the for m for a(l ) and b(x) , the intera rrival time and service time pdf's, respecti vely. (b) Assuming FCFS , write down the equilibrium equa tions th at govern the behavior of Pk = P[k customers in system at the end of a discrete time interval] where k includ es an y arrivals who

EXER CISES

(c) 3.11.

have occur red at the end of this interval as well as any customers who are about to leave at this point. Solve for the expected value of the number of customers at these points.

Consider an M/M/I system with "feedback"; by this we mean that when a customer departs from service he has probability a of rejoining the tail of the queue after a random feedback time, which is exponentially distributed (with mean 1/1' sec) ; on the other hand , with probability I - a he will depart forever after completing service. It is clear that a customer may return many times to the tail of the queue before making his eventual final departure. Let hi be the equilibrium probability that the re are k customers in the "system" (that is, in the queue and the service facility) and that there are j customers in the process of returning to the system . (a) Write down the set of difference equations for the equilibrium probabilities hi' (b)

Defining the double z-transform 00

=~

PCZ1, 2:2)

C

" Z2 -

- ) apCZh Z2) + {oCI A

""' I

00

~ Pk;ZtkZ/

1.-= 0 j= O

show that aZ2

+ ,u[1 -

-

Zl

)

a ~J}PCZ1, Z2)

I - a ZI

-1

=,u (c)

3.12.

113

[

a I -I - Zl

a ZOJ -: P(O, Z2) "" I

By taking advantage of the moment-generating properties of our z-tra nsforms, show th at the mean number in the " system" (queue plus server) is given by p!(l - p) and that the mean number returning to the tail of the queue is given by ,uap!y , where p = A/ Cl - o),u.

Consider a " cyclic queue" in which 1\ I. We a re still faced with the numerator summation th at contains the un kn own probabilities Pi ; we mu st now appeal to the second footnote in step 5 o f our ztransform procedure (see Chapter 2, pp. 74-75), which takes ad vantage of the observa t ion th at the a-tra nsform of a prob ability distributio n mu st be analytic in the range [z] < I in the following way. Since P(z) mu st be b ounded in the range Izi < I [see Eq. (II .28») and since the denominator has r - I zeroes in thi s ran ge , then certainly the numerator must also ha ve ~e roes a t the same r - I points. The numerator consists of tw o factors ; the first of the form (I - zr) all o f whose zeroes have a bso lute value equal to unity; and the seco nd in the form of a su mmation . Consequently , the "compensating" zero es in the numerator mu st come from the summa tion itself (the summa tio n is a pol ynomial of de gree r - I and therefore has exactly r - I zero es). These observa tio ns, th erefore, perm it us to equate the numerator sum to the denominator (after its two roots a t z = I and z = Zo a re factored o ut) as follows: r pzr+l _ ( I + r p)zr + 1 r- I ; ...!----'---'----'-----'-- = K L P;z ( I - z)( 1 - z/zo) ;- 0 wh ere K is a con stant to be evalu at ed bel ow. This computation permits us to re write Eq . (4.35) as

P(z) =

__

('-I _ -_z--'r)'-----_

K (I - z)( 1 - * 0)

But sin ce P( I ) = I we find th at

K = r/(I - Ilzo) a nd so we ha ve ( I - z')( 1 - I/zo) P( z) = - ' - - ---'---'--"'r (1 - z)( 1 - z/ zo)

(4.36)

We now kno w a ll there is to kn ow about the pole s a nd zeroes of P(z) ; we a re, therefore, in a position to make a partial fracti on expansion so th at we may invert o n z. U nfo rtunately P(z) as expressed in Eq . (4.36) is not in th e pr oper form for the partial fraction exp an sion , since the numerat or degre e is not less th an the den om inat or de gree. However , we will take adva ntage of property 8 in Table I.l of Append ix I , wh ich sta tes that if F(z) -ce- i nthe n

!

4.4.

THE QUEUE

E,/ MfJ

133

z' F(z) / "_,, where we recall that the notation indica tes a transform pair. With thi s observation then , we carry out the following pa rti al fraction expan sion P(z)

= (I

-

ZJ [ _I_/r- + _-_ I-'/_ ..rz-'--0J 1 - z/zo

I - z

If we den ote the inverse transform of the quantity in sq ua re br ackets by /; then it is clear that the inverse transform for P(z) must be (4.37)

P; = /; - /;_,

By inspection we see th at

,

(1

j; =

(I -

ZO;-I)

j ~ O

r

o

jO

- (4.42)

We note that thi s distribution for number of customers is geo me tric with a slightly mod ified first term. We could a t this point calcul ate the waiting time dis tr ibutio n , but we will postpone th at unt il we study th e system G/M /l in Cha pter 6.

134

MARKOVIAN QU EUES IN EQUILIBRIUM

4.5. BULK ARRIVAL SYSTEMS In Section 4.3 we studied the system M/ET{I in which each customer had to pass through r stages of service to complete his total service. The key to the solution of that system was to count the number of service stages remaining in the system, each customer contributing r stages to that number upon his arrival into the system. We may look at the system from another point of view in which we consider each "customer" arrival to be in reality the arrival of r customers. Each of these r customers will require only a single stage of service (that is, the service time distribution is an exponential *). Clearly, these two points of view define identical systems : The former is the system M{ET{1 and the latter is an M /M/I system with "bulk" arrivals of size r. In fact, if we were to draw the state-transition-rate diagram for the number of customers in the system, then the bulk arrival system would lead to the diagram given in Figure 4.6; of course, that diagram was for the number of stages in the system M/ET{I. As a consequence, we see that the generating function for the number of customers in the bulk arrival system must be given by Eq . (4.26) and that the distribution of number of customers in the system is given by Eq. (4.29) since we are equating stages in the original system to customers in the current system. Since we are .considering bulk arrival systems, we may as well be more generous and permit other than a fixed-size bulk to arrive at each (Poisson) arrival instant. What we have in mind is to permit a bulk (or group) at each arrival instant to be of random size where

gi ~ P[bulk size is i]

(4.43)

(As an example, one may think of random-size families arriving at the doctor's office for individual vaccinations.) As usual, we will assume that the arrival rate (of bulks) is i.. Taking the number of customers in the system as our state variable, we have the state-transition-rate diagram of Figure 4.8. In this figure we have shown details only for state E k for clarity. Thus we find that we can enter Ek from any state below it (since we permit bulks of any size to arrive); similarly, we can move from state E k to any state above it, the net rate at which we leave Ek being i.g, + i.g. + ... = AL;';., gi = A. If, as usual we define Pk to be the equilibrium probability fer the number of customers in the system, then we may write down the following equilibrium • To make the correspondence complete. the parameter for this exponential distribution should indeed be ru, However, in the following development, we will choose the parameter merely to be Il and recall this fact whenever we compare the bulk arrival system to the system M/ETII.

4.5.

BULK ARRIVAL SYSTEMS

135

Figure 4.8 The bulk arrival state-transition-rate diagram. equations using our inspection method:

(A + fl)Pk = flPk+l

k-l

+ iL PiAgk- i =O

k

~

1

(4.44)

Apo = flPl (4.45) Equation (4.44) has equated the rate out of state Ek (the left-hand side) to the rate into that state, where the first term refers to a service completion and the second term (the sum) refers to all possible ways that arrivals may occur and drive us into state Ek from below. Equation (4.45) is the single boundary equation for the state Eo. As usual, we shall solve these equations using the method of z-tra nsforms ; thus we have 00

(A + fl) L PkZk = k =l

'

Jl

GO

Q)

-Z kL"",lPk+l i

+

1

k- l

+ L L PiAgk_iZk

(4.46)

k =l i =O

We may interchange the order of summation for the double sum such that GO

k- l

0, which would then have allowed zero-size bulks to arrive, and this would have put self-loops in our state-transition diagram corresponding to null arrivals. Had we done so, then the definition for G(z) would have ranged from zero to infinity, and everything we say belowapplies for this case as well.

'I

136

MAR KOVIAN QUEUES IN EQU ILI BRIUM

Extra cting these transform s from Eq. (4.46) we have

(), + fl )[P(z) -

Po] =

~ [P(z)

- Po - P1z]

+ AP(Z)G(z)

N ote that the product P(z)G(z) is a manifestat ion of prop erty II in Ta ble I. I of Appendi x 1, since we have in effect formed the tran sform of the convoluti on of the sequence {Pk} with th at of {gk} in Eq. (4.44). Appl ying the bound ary equation (4.45) and simplifying, Wi: have P(z) =

fl Po(1 - z) fl(1 - z) - ).z(l - G(z)]

To eliminate Po we use P(I) = I; direct application yields the indeterminate form % and so we must use L'Hospital' s rule , which gives po = I - p. We obtain .. _,-,fl(,---I_----'p--'-)(OI_ - -_z.. :. .)_ P(z) = - (4.49) fl (1 - z) - AZ(l - G(z)]

Th is is the final solution for th e transform of number of customers in the bulk arrival M/M /l system. Once the sequence {gk} is given, we may then face the pr oblem of inverting this transform . One may calculate the mean and variance of the numb er of customers in the system in term s of the system parameters directl y from P(z) (see Exercise 4.8). Let us note that the appropriate definition for the utilization fact or p must be carefully defined here. Recall that p is the average arrival rate of customers times the average service time. In our case, the average arri val rate of customers is the product of the average a rrival rate of bulks and the average bulk size. From Eq. (1I.29) we have immediately that the average bulk size must be G'(I). Thus we naturally conclude that the appropriate definition for p in this system is ).G'( l) p= -

(4.50)

fl It is instructive to consider the special case where all bulk sizes are the same, namely, k

=

k

~ r

r

Clearly, this is the simplified bul k system discussed in the beginning of thi s section ; it correspond s exactly to the system M/Er/l (where we must make the minor modification as indicated in our earlier footn ote that fl must now be replaced by rfl) . We find immediately that G(z) = ZT and after substituting this into our solution Eq . (4.49) we find that it correspo nds exactly to our earlier solution Eq. (4.26) as, of course, it must.

4.6.

BU LK SERVICE SYSTEMS

137

4.6. BULK SERVICE SYSTEMS In Section 4.4 we studied the system ErIM/I in which arrivals were considered to have passed through r stages of "arrival." We fou nd it expedient in th at case to take as our state variable the number of " arri val stages" that were in the system (where each fully arrived customer still in the system contributed r stages to that count) . As we found an analogy between bulk arrival systems and the Erlan gian service systems of Section 4.3, here also we find an anal ogy between bulk service systems and the Erlangian arri val systems stud ied in Section 4.4. Thu s let us consider an M/M /I system which provides service to groups of size r . Th at is, when the server become s free he will accept a "bulk" of exactly r custo mers from the queue and administer service to them collectively; the service time for thi s group is drawn from an exponential distribution with parameter fl.. If , up on becoming free, the server finds less than r customers in the queue , he then wait s until a total of r accumulate and then accepts them for bulk service, and so on. * Customers a rrive from a simple Poisson proce ss, at a rate A, one at a time. It should be clear to the reader th at this bulk service system and the ErlM /1 are identical. Were we to draw the state-transition-rate diagram for the number of customers in the bulk service system, then we would find exactly the diagram of Figure 4.7 (with the parameter rA replaced by A; we must account for this parameter chan ge, however, whenever we compare our bulk service system with the system Er/ M/I). Since the two systems are equivalent , then the solution for the distribution of number of customers in the bulk service system must be given by Eq. (4.4 1) (since stages in the original system correspond to customers in the current system). It certainly seems a waste for our server to remain idle when less th an r customers are available for bulk service. Therefore let us now consider a system in which the server will, upon becomin g free, accept r custo mers for bulk service if they are available, or if not will accept less th an r if any are available. We take the number of customers in the system as our state variable and find Figure 4.9 to be the state-transition-rate diagram. In this figure we see that all stat es (except for sta te Eo) behave in the same way in that they are entered from their left-hand neighbor by an arrival, and from their neighb or r units to the right by a gr oup departure, and they are exited by eith er a n arrival or a group departure ; on the other hand, state Eo can be entered fro m anyone of the r states immedi ately to its right and can be exited only by an a rrival. These considerati ons lead directly to the following set of equat ions for the equ ilibrium pr obability P» of finding k customers in • For exam ple. the shared taxis in Israel do not (usually) depart unt il they have collected a full load of custo mers, all of whom receive service simultaneously.

138

MARKOVIAN QUEUES IN EQUILIBRIUM

I'

I'

Figure 4.9 The bulk servicestate-transition-rate diagram. the system: (l

+ P.)h = P.Pk+r + lh-I k ~ I lpo = fl(PI + P2 + ... + Pr)

(4.51)

Let us now apply our z-transform method; as usual we define a>

P(z) = ~p,tZk k _O

We then multiply by z\ sum, and then identify P(z) to obtain in the usual way (l

+ p.)[P(z) -

Po]

= ;[P(Z) -

ktp~] + lzP(z)

Solving for P(z) we have r

P(z)

=

+ fl)Pozr (l + fl)zr + p.

fl ~ p,tZ' - (l

~kc= -,,O -----}.Zr+1 _

From our boundary Eq. (4.51) we see that the negative term in the numerator of this last equation may be written as - zr(}.po

+ p.Po) =

r

-flZr~Pk k= O

and so we have r -I

~

P(z) =

Pk(Zk - zr)

k_O

rpzr+l - (1

+ rp) zr +

(4.52)

where we have defined p = Afflr since, for this system, up to r customers may be served simultaneously in an interval whose average length is l /fl sec. We

1

4.7.

139

SERIES-PARALLEL STAGES: GENERALIZATIONS

immediately observe that the denominator of this last equation is precisely the same as in Eq. (4.35) from our study of the system ET/M /I. Thus we may give the same arguments regarding the location of the denominator roots; in particular, of the r + I denominator .zeroes, exactly one will occur at the point z = I, exactly r - I will be such that [z] < I, and only one will be found, which we will denote by zo, such that IZol > I. Now let us study the numerator of Eq. (4.52). We note that this is a polynomial in z of degree r . Clearly one root occurs at z = I. By arguments now familiar to us, P(z) must remain bounded in the region Izi < I, and so the r - I remaining zeroes of the numerator must exactly match the r - 1 zeroes of the denominator for which Izi < I; as a consequence of this the two polynomials of degree r - I must be proportional, that is, T-1

K

I

PtCzk - ZT)

k= O

1- z

rp zT+l - (1

+ rp)zT + 1

(1 - z)(1 - Z/Zo)

Taking advantage of this last equation we may then cancel common factors in the numerator and denominator of Eq. (4.52) to obtain 1

P(z) = - - " - - - K(I - z/zo) The constant K may be evaluated in the usual way by requiring that P(I) which provides the following simple form for our generating function:

P(z) = 1 - I/zo 1 - z/ zo

=

I,

(4.53)

This last we may invert by inspection to obtain finally the distribution for the number of customers in our bulk service system

k = 0, 1,2, . . .

-(4.54)

Once again we see the familiar geometric distribution appear in the solution of our Markovian queueing systems!

4.7. SERIES-PARALLEL STAGES : GENERALIZATIONS How general is the method of stages studied in Section 4.3 for the system M/Er/l and studied in Section 4.4 for the system ErfM /I? The Erlangian distribution is shown in Figure 4.5 ; recall that we may select its mean by appropriate choice of 11 and may select a range of standard deviations by adjusting r. Note, however, that we are restricted to accept a coefficient of

140

MAR KOVI AN QUEUES IN EQU ILI BRIU M

varia tio n that is less than th a! of the exponenti al distributi on [from Eq. (4.14) we see that Co = IIJ r wherea s for r = I the exponenti al gives C b = I] and so in some sense Erlang ian rand om variables are " mo re regular" than exponent ial variables. Thi s situation is cert ainly less than completely general. One dire ction for generalizatio n would be to remove the restriction that one of our two basic queueing distributi on s must be exponential ; tha t is, we certa inly could consider the system ErJErJ I in which we have an ra-stage Erlangian distributi on for the interarr ival time s and an rb-stage Erlan gian distribution for the service times . * On the other hand , we could atte mpt to generali ze by broadening the class of distributions we consider beyond that of the Erlangian. Thi s we do next. We wish to find a stage-type arran gement that gives larger coefficient s of va riation than the exponential. One might consider a generalizatio n of the r-stage Erlangi an in which we permit each stage to have a differ ent service rate (say, the ith stage has rate fl ,). Perhaps this will extend the ran ge of C, ab ove unit y. In this case we will ha ve instead of Eq. (4. 15) a Lapl ace tran sform for the service-time pdf given by B*(s) -

(~)(~) s + fll S + fl2

...

(~) S + fl r

(4.55)

The service time density hex) will merely be the con volution of r exponen tial den sities each with its own parameter fl i. The squa red coefficient of variati on in this case is easily shown [see Eq. (11.26), Appendix II] to be

But for real a, ~ 0, it is always tru e th at I i a/ ~ (I i a;)2 since the right hand side contains the left-hand side plus the sum of all the nonn ega tive cros s term s. Ch oosing a, = I lfl;, we find that C b2 ~ I. Thu s, unfortuna tely, no gener alization to larger coefficients of variation is obtained this way. We previou sly found that sending a customer th rough a n increas ing sequen ce of faster exponential stages in series tended to reduce the vari abil ity of the service time , and so o ne might expect that sending him through a parallel arra ngement would increase the variability. Thi s in fact is tru e. Let us therefore con sider the two-sta ge parallel service system show n in Figure 4.10. The situation may be contrasted to th e service st ructure shown in Figure 4.3. In Figure 4.10 an entering customer approaches the lar ge oval (which represents the service facility) from the left. Upo n entry into the • We co nsider this short ly.

4.7.

SERI ES-PA RALL EL STAGES: GENE RALIZATIONS

141

Service facility

Figure 4.10 A two-stage parallel server H 2 • facility he will procee d to serv ice stage I with probabil ity 0( , or will proceed to service stag e 2 with pr ob ab ility 0(2' where 0( , + 0(2 = 1. He will then spend an exponentially distributed interval of time in the ith such stage who se mean is I{fl i sec. After th at interval the customer departs and o nly then is a new cu stomer allowed int o the serv ice fac ility. It is clear fro m th is des cription tha t the service time pdf will be given by x ~ O

a nd also we ha ve

B*(5) = O(, ......f:!..!5

+ fl,

+

0(2

~ 5

+ fl 2

Of cou rse th e more genera l case with R parallel stages is sh own in F igure 4.11. (Co nt ras t th is with F igure 4.4.) In thi s case, as always , at most one cu stomer at any one time is permitted within the large oval representing the service facility. Here we ass ume that

(4.56) Clearly, R

b(x) =

!

O(iflie-#' z

x ~ O

- (4.57)

i= l

a nd

B*(5) =

!R lX i - /l-'-i-' 5 + /l i

The pdf given in Eq. (4.57) is referred to as the hyperexp onent ial distribution a nd is denoted by H R. Hopefully, the coefficient of var iati on (C b ,;; a Ji) is now grea ter th an unity and the refore repre sent s a wider va ria tio n than

142

~1ARKOVIA N QUEU ES IN EQUILIBRIUM

R

Service facility

Figure 4.11 The R-stage parallel server HR. that of the exponential. Let us prove thi s. From Eq. (II.26) we find immediatel y that _

x =

R C1.

1: i = l }li i

Forming the square of the coefficient of variation we then have

(4.58)

Now, Eq . (II. 35), the Cauchy-Schwarz ineq uality , may also be expre ssed as follo ws (fo r ai' b, real): (4.59)

4.7.

SERIES-PARALLEL STAGES: GENERALIZ ATIONS

143

Figure 4.12 State-transition-rate diagram for M/H 2 / I.

(T his is often referred to as the Cauchy inequality.) lfwe mak e th e asso ciatio n ai = J CJ. i , hi = J "- J,u,, then Eq. (4.59) shows

(I CJ.i)2~ (ICJ.') (I CJ.~) I

Jli

I

t

Pi

But from Eq, (4.56) the first factor on the right-hand side of thi s inequ alit y is ju st unity; thi s result along wit h Eq. (4.58) permits us to write

- (4.60) which pr oves the desired result. One might expect t hat an a nalysis by the method of stages exists for the systems M/H rt/I , H rt/M fI, H R a / H rtb fI , and thi s is indeed true. The rea son th at the ana lysis can proceed is that we may take account of the nonexponential character of the service (or arrival) facilit y merely by specifying which stage within the service (or arri val) facility the customer currentl y occupies. Thi s inform at ion along with a sta tement regarding the number of customers in the system creates a Mark ov chain , which may then be studied much as was done earlier in this chapt er. For exa mple, the system M/H 2 /l would have the sta te-tra nsitio n-rate diag ram show n in Figure 4.12. In this figure the designati on k, implies th at the system contains k customers and that the customer in service is locat ed in stage i (i = I , 2). T he transitions for higher numbered sta tes are ide ntica l to the transitions between states I, and 2,. We are now led directly int o the foll owing genera lization of series stages and parallel stages ; specifica lly we are free to combine series and par allel

• 144


r,

2

Service facility

Figure 4.1 3 Series-parallel server. stages into arbitrarily complex structures such as shown in Figure 4.13. This diagram shows R parallel "stages," the ith "stage" consisting of an restage series system (i = 1,2, .. . , R); each stage in the ith series branch is an exponential service facility with parameter rill- i. It is clear that great generality can be built into such series-parallel systems. Within the service facility one and only one of the multitude of stages may be occupied by a customer and no new customer may enter the large oval (repre senting the service facility) until the previous customer departs. In all cases, however, we note that the state of the service facility is completely contained in the specification of the particular single stage of service in which the customer may currently be found . Clearly the pdf for the service time is calculable directly as above to give b() x

=

~

£- rt.i ,~l

ri/1. i(r ,ll-ix )' ,- l - r . e Cr, - 1)! I

·x I

x~O

(4.61)

and has a tran sform given by (4.62) One further way in which we may generalize our series-parallel server is to remove the restriction that each stage within the same series branch has the same service rate (rill- i); if indeed we permit the jth series stage in the ith

4.7.

SERIES-PARALLEL STAGES : GENER ALIZA TION S

145

",

",

Service facilit y

Figure 4.1 4 Another stage-type server. parallel branc h to have a service rate given by {-I;;, then we find tha t the Laplace tr ansform of the service time density will be generalized to 8 *(5)

r, ( = LR «, IT .uzu.: II

;- 1

;- 1

5

+ {-IH

)

(4.63)

These genera lities lead to rather comple x system equations. Another way to create the series-parallel effect is as follows. Consider the service facility shown in Figure 4.14. In thi s system there are r service stages only one of which may be occupied at a given time. Cust omers enter from the left and depart to the right. Before entering the ith stage an independent choice is made such that with probability f3; the customer will proceed into the ith exponential service stage and with probability cx; he will depart from the system immediately ; clearly we require f3; + cx ; = I for i = I , 2, ... , r, After compl eting the rth stage he will depart from the system with pr obability 1. One may immediately write down the Laplace transform of the pdf for the system as follows: (4.64)

where (l(r +l = I. One is tempted to consider more general transiti ons among stages th an that shown in thi s last figure ; for example, rather th an choosing only between immed iate departure and entry int o the next stage one might co nsider feedb ack or feedforward to ot her stages. Cox [COX 55] has shown that no furth er generality is introduced with this feedback and feedforwa rd concept over that of the system shown in Figure 4.14. It is clear that each of these last three expressions for B *(s) may be rewritten as a rati onal funct ion of s, that is, as a rati o of polynomials in s. The position s of the poles (zeroes of the denominator polynomial) for B *(s) will of necessity be located on the negative real axis of the complex s-plane. This is not quite as general as we would like, since an arb itrary pdf for

146

MARKOVIAN QU EU ES IN EQUILIBRIUM

service time may have poles located anywhere in the negati ve half s-plane [that is, for Re(s) < 0]. Cox [COX 55] has studied this pr oblem and suggests that complex values for the exponential parameters rill . be permitted ; the ar gument is that whereas this correspond s to no physically realizable exponential stage, so long as we provide poles in complex conju gate pai rs then the entire service facility will have a real pdf, which corresponds to the feasible cases. If we permi t complex-conjugate pair s of poles th en we have complete generality in synthesizing any rational functi on of s for our service-time tran sform B *(s). In addition, we have in effect outlined a meth od of solving these systems by keeping track of the state of the service facility. Moreover , we can similarly construct an interarrival time distri buti on from seriesparallel stages, and thereby we are capable of con siderin g any G/G/ I system where the distributions have transform s that are rational function s of s. It is further true that any nonrati onal functi on of s may be approx imated arbitrarily closely with rational functi ons. * Thus in pr inciple we have solved a very general problem. Let us discuss this meth od of solution. Th e sta te descript ion clearly will be the number of customers in the system, the stage in which the arriving cust ome r finds himself within the (stage-type) arriving box and the stage in which the cust omer finds himself in service. Fr om thi s we may draw a (horribly complicated) state-transition dia gram . Once we have this diagram we may (by inspect ion) write down the equilibrium equations in a rather straightfo rward manner ; th is large set of equ ati on s will typ ically have many bound ary conditions. H owever, these equ ati on s will all be linear in the unknown s and so the solution meth od is straightforward (albeit extremely tedi ou s). What more natural setup for a computer solutio n could one ask for ? Ind eed , a digital co mputer is extremely adept at solving large sets of linear equ ati ons (such a task is much eas ier for a digital computer to handle than is a sma ll set of nonlinear equ ations). In carrying out the digital solution of this (typically infinite) set of linear equa tions, we must redu ce it to a finite set; thi s can only be done in an ap pro ximate way by first deciding at what point we ar e satisfied in truncatin g the seq uence Po ,PI> p", .. . . Then we may solve the finite set and perh ap s extrap olate the • In a rea l sense, then, we are faced with an approximation pro blem ; how may we "best" app roximate a given dist ribution by one tha t has a rat iona l tra nsform. If we a re given a pdf in numerical form then Prony' s method IWHI T 44] is one acceptable procedure. On the other hand, if the pdf is given analytica lly it is difficult to describe a genera l proced ure for suita ble approxi mation. Of course one wou ld like to make these approximati ons with the fewest number of stages possib le. We comment that if one wishes to fit the first an d second moment s of a given distributi on by the method of stages then the number of stage s canno t be significantly less than I / Cb" ; unfortun ate ly, this implies that when the distribut ion tends to concentrate ar ound a fixed value, then the num ber of stage s required grows ra ther quickly.

4.8.

NET WOR KS O F MARKOVIAN QUEUES

147

solution to the infinite set; all this is in way of ap proximation and hopefull y we are able to carry out the .computation far enough so that the neglected terms a re indeed negligible. One must not overemphas ize the usefulne ss of this pr ocedure ; this solutio n meth od is not as yet a utomated but does at least in principl e provide a meth od of approach. Other anal ytic meth od s for handling the more comple x qu eueing situatio ns are discussed in the balance o f this book.

4.8. NETIVORKS OF MA RKOVIAN QUEUES We ha ve so far considered Markovian systems in which each customer was demanding a single service operation from the system. We may refer to this as a "single-node" system. In this section we are concerned with multiple-node systems in which a customer req uires service at more than one sta tion (node). Thus we may think of a network of nodes, each of which is a service center (perha ps wit h multiple servers at some of the nodes) and each wit h storage room for queue s to form. Customers enter the system at va rious points, queue for service, and up on dep arture from a given node then pr oceed to some other node, there to receive additional service. We are now describ ing the last category of flow system discussed in Cha pter I , namely , stochastic flow in a network . A number of new considerat ion s emerge when one considers networ ks. For example, the to pological structure of the network is important since it describes the perm issible transition s between nodes. Also the path s taken by individua l customers must someho w be described. Of great significance is the nature of the stochastic flow in term s of the basic stochastic pr ocesses describ ing tha t flow ; for example , in the case of a tandem queue where custo mers departing fro m node i immediate ly enter node i + I , we see that the interdeparture times from the fo rmer generate the interarrival time s to the latter. Let us for the moment con sider the simple two-node tandem networ k shown in Figu re 4.15. Each ova l in th at figure describes a qu eueing system consisting of a queue and server(s) ; within each oval is given t he node number. (It is import ant not to confuse the se physical net work diagram s with the abstract state-transition-rate diagram s we ha ve seen earli er.) For the moment let us assume tha t a Poisson process generates the arrivals to the system at a rate i., all of which enter node one ; further assume th at node one consists of a single expon en tial server at rate p . Thus node one is exactly an M/M /l queu eing system. Also we will assume that node two has a single

·

8f---t--~Of--··--

Figure 4.15 A two-node tandem network.

148

MARKOVIAN QUEUES IN EQUI LIBRIUM

exponential server also of rate p,. The basic que stion is to solve for the interarrival time distribut ion feeding node two ; th is certainly will be equivalent to the interdeparture time distribution from node one . Let d (t ) be the pdf describing the interdeparture process from node one and as usual let its Laplace transform be denoted by D*(s). Let us now calculate D*(s). When a customer departs from node one either a second customer is ava ilable in the queue and ready to be taken into service immed iately or the queue is empt y. In the first case, the time until this next customer departs from node one will be distributed exactly as a service time and in that case we will have D* (s ) l node one nouempty =

B*(s)

On the other hand , if the node is empty upon th is first customer's departure then we must wait for the sum of two intervals, the first being the time until the second customer arrives and the next being his service time ; since these two intervals are independently distributed then the pdf of the sum must be the convoluti on of the pdf's for each. Certainly then the tran sform of the sum pdf will be the pr oduct of the transforms of the individual pdfs and so we have

A

=- -

D*(S)l nod e o ne empty

s +A

B*(s)

where we have given the explicit expression for the tran sform of the interarrival time densit y. Since we ha ve an expo nential server we may also write B*(s) = p,/ (s + p, ); furthermore , as we shall discuss in Ch apter 5 the probability of a departure leaving behind an empty system is the same as the probability of a n a rrival finding an empty system, namely, I - p. T his permits us to write down the unc onditi onal transform for the inte rdepa rture time density as D*(s)

=

p) D*(S)lnode one

(I -

+ pD*(S)lnode one none m ptv

empty

Using our abo ve calculati ons we then have D*(s)

=

(I _

p)(~)(_P ) + p(---.f!- ) S +A

s+ p

s+ p

A little algebra gives D*(s)

=~

(4.65)

S +A

and so the interdeparture time distributi on is given by D (t ) = I -

e-).'

t~ O

-

4.8.

NETWO R KS OF MARK OVIAN QUEUES

149

T hus we find the remar kable conclu sion that the interdeparture times are expo nentia lly distribut ed with t he same parameter as the interarrival times! In other words (in the case of a stable sta tionary queueing system), a Poisson pr ocess driving an exponential server generate s a Poisson process for departures. This startling result is usually referred to as Burk e's theorem [BURK 56]; a number of others also studied the pr oblem (see, for example, the discussion in [SAAT 65]). In fact , Burke' s theorem says more, namely, that the steady-sta te output of a stable M/M /m queu e with input parameter Aand service-time parameter flo for each of the m cha nnels is in fact a Poisson process at the same rate A. Burke also established that the output process was independent of the other processes in the system. It has also been sho wn tha t the M/M /m system is the only such FCFS system with this pro perty. Returning no w to Figure 4.15 we see therefore that node two is dri ven by an independent Poisson arrival process and therefore it too beha ves like an M/M fJ system and so may be analyzed independently of node one. In fact Burke's the orem tells us that we may connect many multiple- server nodes (each server with exponential pdf) together in a feedfor ward * network fashion and still preserve th is node-by-node decomp osition . Jack son [JACK 57) addressed himself to this question by considering an arbitrar y net work of queue s. The system he studied consists of N nodes where the it h node consists of m , exponential servers each with par ameter fIo i; fur ther the ith node receives arrivals from outside the system in the form of a Poisson process at rate Y i' Th us if N = I then we have an M/M /m system. Upon leaving the ith node a customer then proceeds to the jth node with probability r ii ; this formul ati on permits the case where r« ~ O. On the other ha nd, aft er completing service in the ith node the proba bility that the customer departs from the netwo rk (never to return again) is given by I - Li'.:,l r ii . We must calculate the total ave rage arriva l rate of customers to a given node. T o do so, we must sum the (Poisson) ar rivals from out side the system plu s arrivals (no t necessarily Poisson) from all intern al nodes; that is, den oting th e total average a rrival rate to node i by j' i we easily find that this set of par ameters must sa tisfy the following equ at ions : S

Ai =

r, + L

}1i i

i=I , 2, .. . , N

- (4.66)

j= l

I n order for all nod es in this syste m to represent ergodic Ma rkov cha ins we require that i'i < m ill i for all i; aga in we cau tio n the read er not to confuse t he nodes in this discussion with the system states of each node from our • Specifically we do not permit feedba ck pat hs since this may dest roy the Poisso n nature of the feedback depart ure stream. In sp ite of this, the following discussion of Ja ckson's work points ou t that even networks with feedback are such that the individua l node s behave as if they were fed totall y by Poisson arrivals, when in fact they are not.

l ISO

MA RKO VIAN QU EUES I N EQU ILIBRIUM

previous discussion s. What is amazing is th at Jackson was a ble to show that each node (say the it h) in the netw ork beh aves as if it were an independent M/M /m system with a Poisson input rate A,. In general, the total input will not be a Poisson process. The state variable for th is N-n ode system consists of the vecto r (k ,. k 2 • • • • , k s) . where k ; is the number of cu stomers in the ith nod e [including the customer (s) in service]. Let the equili brium pr o ba bility asso ciated with this sta te be den oted by ptk.; k, • . . . , k s ). Similarl y we den ot e the marginal d istribution of findi ng k , customer s in the ith node by p .(k,). Jackson was abl e to show th at the joint distri bu tion for all nodes factored into the pr oduct of each of the mar ginal distribution s. th at is, - (4.67)

I,

and ' pi (k ,) is given as the solutio n to the classical M/M / m system [see. for example, Eqs . (3.37)-(3.39) with the obvious chan ge in not ation ]! This last result is commonly referred to as Jack son's theorem . On ce agai n we see the "product" form of solution for Mark o vian queues in equ ilibriu m. A mod ificat ion of Jack son 's network of queues was con sidered by G ordon and Ne well [GORD 67]. The modification th ey investiga ted was th at of a closed Mark ovian netw ork in the sense that a fixed and finite number of cust omers, say K , are con side red to be in the system and a re trapped in that system in the sense th at no o thers may enter and none of the se may leave : this cor responds to Jack son's case in which ~;:. \ r ij = I and Yi = 0 for all i. (A n interestin g example of thi s class of systems know n as cyclic queues had been con sidered earli er by K oenigsberg [KO EN 58]; a cyclic queue is a tandem q ueue in which the last stage is conn ected bac k to the first.) In the general case co nsidered by G ord on and Ne well we do not quite expect a pr oduct soluti on since there is a dep end ency a mo ng the element s of the sta te vecto r (k\ . k, • . . . • k s ) as foll ows : S

I ki =

K

(4.68)

i= l

As is the case for Jackson 's model we ass ume that this discre te-state Ma rkov pr ocess is irred ucible and therefor e a unique equ ilibrium pr o bability distribution exists for p(k\ . k" . . . , k s ). In thi s mo del, however , th ere is a finite num ber of sta tes; in particular it is easy to see that the num ber of dist ingui shable states of th e system is eq ual to the nu mber of ways in which o ne can place K custom ers a mo ng th e N nodes. and is eq ua l to the binomial coefficient (

N

+K N -

I)

-

I

4.8.

NETWORKS OF MARKOVIAN QUEUES

151

The following equations desc ribe the behavior of the equilibrium distribution of custo mers in this closed syste m and may be written by inspection as

.v

P(/(I, /(2' . . . , /(x)

2 0k.- ICf..( /(;)fl i

i= 1

s s

=

I

IOkj_ICf.,(k i

+ l)!t i ' ij p(k

k2 ,

l,

• • • ,

k, -

1, . . . , k ,

i = l r "", l

+ I , ... , ks ) (4.69)

where the discrete unit step-funct ion defined in Appendix I ta kes the for m k = 0, 1, 2, . . .

'" {I0

(4.70) k< O and is included in' the eq uilibri um equations to indicate the fact that the service rate must be zero when a given node is empty ; furthe rmore we define Ok =

Cf.i(k i) =

k. '

{11l

k 1·< -

nI ,,

i

which merely gives th e number of cust omers in service in th e ith node when there a re k, custo mers a t th at nod e. As usual the left-h a nd side of Eq . (4.69) des cribes the flow of 'pro bability out of sta te (k l , k 2 , • • • , k".) whereas the right-hand side acco unts for the flow of probability into that state from neighboring states. Let us proceed to write down the solution to these equation s. We define the function (li(k i) as follows :

k< , - m ,· Consider a set of numbers {Xi}' which are solutio ns to the foliowing set of linea r equations : N

# iXi

= L p j x jr ji

i = 1,2, . . . , lV

(4.71)

;=1

Note that thi s set of equations is in t he sa me form as 1t = 1tP whe re now th e vector 1t may be co nsidered to be (fl,x" . . . , flsxs) and the elemen ts of the matrix P a re conside red to be the elements rij. * Since we assume th at the • Again the reader is caut ioned that, on the one hand, we have been con sidering Markov cha ins in which the quantities Pij refer to the transition probabilities among the possible slates that the system may take on, wherea s, on the other hand, we have in this section in additi on been considering a network of queuein g systems in which the prob ab ilities r ij refer to tran sition s that customers make between nodes in tha t network .

•

152

MARKOVIA N Q UEUES IN EQUILIBRIUM

matrix of transition probabilities (whose elements are " i) is irreducible, then by our previous studies we know that there must be a solution to Eqs. (4.71), all of whose components are positive; of course, they will only be determined to within a multiplicati ve constant since there are only N - I independent equ ati ons there . With these definitions the solution to Eq. (4.69) can be shown to equ al - (4.72) where the norm alization constant is given by N

G(K)

x .k j

= L II - 'k e .,! ' _ 1

fliCk ,)

(4.73)

Here we imply that the summation is taken over all state vectors k ~ (k" . . . , k N ) that lie in the set A, and this is the set of all state vectors for which Eq. (4.68) holds. Thi s then is the solution to the closed finite queueing network pr oblem, and we observe once aga in that it has the product form. We may expose the pr oduct formulati on somewhat further by co nsidering the case where K ~ 00 . As it turns out, the quantities x;/m, are critical in this calculation ; we will assume that there exists a unique such rati o that is largest and we will ren umber the nodes such that x,!m, > x;/m, (i,e I). It can then be shown that pik, k 2 , . . • , k N ) ~ 0 for any state in which k, < 00 . Thi s implies that an infinite number of customers will form in node one , and th is node is often referred to as the " bottleneck" for the given network . On the other hand , however, the marginal distribution p (k 2 , • • • , k,v) is well-defined in the limit and takes the form (4.74) Thus we see the pr oduct solution directly for this marginal distribution and , of cour se, it is similar to Jackson's theorem in Eq. (4.67); note that in one case we have an open system (one that permit s external a rrivals) and in the other case we have a closed system. As we shall see in Chapter 4, Volume II , th is model has significant applications in time-shared and multi-access computer systems. Jack son (JACK 63] earlier con sidered an even more genera l open queue ing system, which includes the closed system just considered as a special case. The new wrinkles introduced by Jackson a re, first , that the customer arrival proce ss is permitted to depend up on the total number of customers in the system (using this, he easily creates closed network s) and, second, that the service rate at any node may be a function of the number of cust omers in that node. Thus defining S(k) ~ k , + k, + . .. + k»

4.8.

NETWO RKS OF MARK OVIAN QUE UES

153

we the n permit the tota l a rrival rate to be a function of S(k) when the system sta te is given by the vecto r k. Similarl y we define the exp onential service rat e a t node i to be Ilk, when there are k , cu stome rs at that nod e (includ ing th ose in ser vice). As earlier, we ha ve the node transiti on probabilities ' ij (i , j = 1,2 , . . . , N) wit h the following additional definitions : '0, is the probability th at the next externally generated arrival wiII enter the network at node i ; ' i .N +l is the probability that a cu stomer leaving node i departs from the system ; and 'O, N +l is the probability th at the next arrival will require no service from the system and leave immediately upon arrival. Thus we see that in this case y, = 'Oiy(S(k», where y(S(k» is the total external arrival rate to the system [conditioned on the number of customers S (k) at the moment] from our external Poisson process. It can be seen that the prob ability o f a customer arriving at node i l and then passing through the node sequence i 2 , i 3 , . • . , in and then departing is given by ' oil' I,,',,i,' " " . _ l i . 'i • •V+l ' Rather than seek the solution of Eq . (4.66) for the traffic rates, since the y are funct ion s of the total number of cu stomer s in the system we rather seek the solution for the following equivalent set : N

e,

= '0' + 1_21 ej'ji

(4.75)

[In the case where the arrival rates are independent of the number in the system then Eqs. (4.66) and (4.75) differ by a multiplicative factor eq ual to the total arrival rate of customers to the system.] We assume th at the solution to Eq. (4.75) exists, is unique , and is such that e, ~ 0 for all i; th is is equ ivalent to assuming that with prob ability I a cu stomer' s j ourney throu gh the netwo rk is of finite length . e, is, in fact , the expected number of times a customer will visit nod e i in passing through the netw ork. Let us define the time-dependent state probabilities as Pk(t ) = P[system (vecto r) state a t time t is k]

(4.76)

By our usual methods we may write down the differential-difference equations go vern ing these probabilities as follows:

..v

J.V

+ 2 Ilk.+l' "N+IPk (i+)(t) + 2 i= l

I

1.V

2 1lk+l r jiP kl,.il (t)

i = l j= l

(4.77)

J

i /:- j

where terms a re omitted when any component of the vector a rgument goes negative ; k (i-) = k except for its ith component, which takes o n the value

a:

154

MARKOV IAN QUEUES IN EQU ILI BRIUM

k, - 1; k (i+) = k except for its ith comp onent , which takes on the value k , + I; and k (i,j) = k except that its ith comp onent is k , - I and its jth component is k , + I where i ~ j . Complex as this notati on appears its interpretat ion sho uld be rather straightforward for the reader. Jackson shows that the equilibrium distribution is unique (if it exists) an d de fines it in our earl ier notati on to be lim Pk(t ) g Pk g pt k, k 2 , •• • , k N) as t ->- 00. In order to give the equilibrium solution for Pk we must unfortunately define the following furt her notation :

gII

K- l

F(K )

y(S(k»

K = 0, 1, 2, . ..

(4.78)

S lk ) ~ O

N

ki

II II .5-

f( k) ';'

(4.79)

1"", 1 ij = l f-l; i

H(K )

g

I f(k )

(4.80)

k e..l

G

g{K~/(K)H(K)

if the sum con verges

(4.81)

otherwise where the set A shown in Eq . (4.80) is the same as that defined for Eq. (4.73). In ter ms of these definiti ons then Jackson's more general theorem states that if G < 00 then a unique equilibrium-state prob ability distribution exists for the general state-dependent networks and is given by Pk =

1

-

G

f( k) F(S( k»

(4.82)

Again we detect the product form of solutio n. It is also possible to show that in the case when arrivals are independent of the total number in the system [that is, y g y( S(k» ) then even in the case of state-dependent service rates Jack son's first the orem applies, namely, that the jo int pdf fact ors into the produc t of the individual pd f' s given in Eq. (4.67). In fact PiCk;) tu rns out to be the same as the probabi lity distribut ion for the nu mber of customers in a single-node system where arriv a ls come from a Poisson pr ocess at rate y e; and with the sta te-dependent service rates fl., such as we ha ve derived for our general birth-death process in Chapter 3. Thu s one impact of Jackson's second theorem is that for the constant-arrival-rate case, the equilibrium prob abili ty distributions of number of customer s in the system at individ ual

4.8.

NETW O RKS OF MARKOVIAN Q UEUES

155

centers are independent of other centers; in addition, each of these distri but ions is identical to the weil-known single-node service center with the sa me pa ra meters. * A remar kable result! This last theo rem is perhap s as far as one can got with simple Markovian networks, since it seems to extend Burke' s theo rem in its most genera l sense. When one relaxes the Mar kovian assumpti on on arrivals and/o r service times, then extreme complexity in the inter depar ture process arises not only from its marginal distri butio n, but also from its lack of independence on othe r state variables. These Markovian queuein g network s lead to rath er depr essing sets of (linear) system equ ations ; this is due to the enormous (yet finite) sta te descripti on. It is indeed remar kable that such systems do possess reasonably straightforward solutions. The key to solution lies in the observation that these systems may be repr esented as Mark ovian population processes, as neatly described by Kingman [KI NG 69) and as recently pursued by Chandy [CHAN 72). In particular , a Mar kov popu lation process is a continuous-time Markov cha in over the set of finite-dimen sional sta te vectors k = (k 1 , k 2 , • • • , k s ) for which transitions are permitted only between sta tesf : k a nd k (i+) (an external ar rival at node i) ; k and k (i- ) (an external departure from node i) ; and k and k(i ,j ) (an internal tra nsfer from node ito nodej). Kingman gives an elegant discussion of the interesting classes and properties of these processes (using the notion and properties of reversible Markov chai ns). Chandy discusses so me of these issues by observing that the equilibrium pr obabi lities for the system sta tes obey not only the global-balance equati ons that we have so far seen (and typica lly which lead to product-form solutions) bu t also that this system of equati ons may be decomposed into many sets of smaller systems of equations, each of which is simpler to solve. Th is tran sfor med set is referred to as the set of " local" -balance equa tions , which we now proceed to discuss. The concep t of local balance is most valuab le when one deals with a network of queu es. H owever, the concept does apply to single-node Mar kovian queues, and in fact we have already seen an example of loca l balan ce at pla y. • Thi s model also permit s one to handle the closed queueing systems studied by Gordon a nd Newell. In order to crea te the constant tot al number of customers one need merely set y (k ) = 0 for k ~ K an d y( K - I) = co, where K is the fixed number one wishes to conta in within the system. In order to keep the node tran sition probabilities iden tica l in the open and closed systems, let us denote the former as earlier by r;; and the latter now by rii' : to mak e th e limit of Jackson' s genera l system equivalent to the closed system of Gordon an d Newell we then require r;;' = ri; + (r i .N+l)(rU;)' t In Chapter 4, Volume II , we describe some recent result s that do in fact exte nd the model to han dle different customer classes and different service disciplines at each node (permitting. in some ca ses, more genera l serv ice-time distributions). t Sec the definitions following Eq. (4.77).

....

156


Node l

Node 2

Nod e 3

Figure 4.16 A simple cyclic network example: N = 3, K = 2. Let us recall the global -balance equations (the flow-conservati on equations) for the general birth-death process as exemplified in Eq. (3.6). Thi s equation was obtained by balancing flow into and out of state Ek in Figure 2.9. We also commented at that time that a different boundary could be considered across which flow must be conserved , and this led to the set of equations (3.7). These latter equations are in fact local-balance equations and have the extremely interesting property that they match terms from the left-hand side of Eq. (3.6) with corre spondin g terms on the right-hand side ; for example , th e term Ak- 1Pk-l on the left-hand side of Eq. (3.6) is seen to be equal to fl kPk on the right-h and side of that equation directl y from Eq. (3.7), and by a second application of Eq. (3.7) we see that the two remainin g term s in Eq. (3.6) must be equal. Thi s is precisely the way in which local balance operates, namely, to observe that certain sets of term s in the global-balance equation must balanc e by themselves giving rise to a number of "Iocal " -balance equations. The significant observation is that, if we are dealing with an ergodic Markov process, then we know for sure that ther e is a uniqu e solution for the equilibrium probabilities as defined by the generic equati on 7t = 7tP. Second , if we decomp ose the global-balance equations for such a process by mat ching terms of the large global-balance equations into sets of smaller local-balance equations (and of cour se account for all the terms in the global balanc e), then any solutio n satisfied by this large set of local-b alance equations must also satisfy the global-balance equations; the converse is not generally true. Th us any solution for the local-balance equations will yield the unique solution for our Mark ov process. In the interesting case of a network of queues we define a local-balance equation (with respect to a given network state a nd a network node i) as one that equates the rate of flow out of that network stat e due to the depar ture of a customer from node i to the rate of flow into that network state due to the ar rival of a customer to node i, * Thi s notion in the case of networks is best illustra ted by the simple example shown in Figure 4.16. Here we show the case of a three-node network where the service rate in the ith node is given as • When service is nonexponential but rather given in term s of a stage-type service distr ibut ion , then one equates ar rivals to and departures from a given stage of service (ra ther than to and from the node itself).

:r

[

4.8.

NETWORKS OF MARKOVIAN QUEUES

157

Figure 4.17 State-transition-rate diagram for example in Figure 4.16. fl i a nd is independent of the number of customers at th at node ; we assume there are exactly K = 2 customers circulating in this closed cyclic netw ork .

Clearly we have ' 13 = '32 = ' 21 = I and ' if = 0 otherwise. Our state description is mer ely the triplet (k l , k 2 , k 3 ) , where as usual k , gives the number of custome rs in node i and where we require, of course, that k 1 + k 2 + k 3 = 2. For thi s net work we will therefore have exactly N (

+K N -

I

I) 6 =

states wit h sta te-tra nsitio n rat es as show n in Figure 4.17. Fo r this system we have six glo bal-ba lan ce equations (o ne of which will be redu ndant as usual; the extra cond ition come s from the con servation of probability); th ese a re fllp(2 , 0, 0)

= p2p( l,

1,0)

(4.83)

fl2P(0, 2, 0)

= P3P(0, I , I)

(4.84)

fl3P(0, 0,2) = PIP (I , 0, I)

(4.85)

+ fl 2P(l , 1,0) = P2P(0 , 2, 0) + fl3P(I, 0, I) fl2P(0, I, I) + fl3P(0, I , I) = P3P(0 , 0, 2) + fl IP (I , 1, 0) fl IP(! ,O, I) + p 3p(l, 0 , I) = P2P(0 , I , I) + PIP(2 , 0, 0)

(4.86)

fllp( l, 1,0)

(4.87) (4.88)

158

MARKOVIAN QUEU ES IN EQUILIBRI UM

Each of these glob al-balance equ ati ons is of the form whereby the left-hand side repre sents the flow out of a state and the right-hand side represents the flow int o that sta te. Equations (4.83)-(4.85) are already local-balance equations as we shall see; Eqs. (4.86)-(4.88) have been written so th at th e first term on the left-hand side of each equation balances the first term on the right-hand side of the equ ation, and likewise for the seco nd term s. Thus Eq . (4.86) gives rise to the following local-balance equations: PIP(1, 1,0)

= p ,p(O, 2, 0)

(4.89)

p,p(l , 1,0)

= P3P(I, 0,

(4.90)

I)

Note, for example, th at Eq . (4.89) takes the rate out of sta te (I , 1,0) due to a departure from node I and equates it to the rate into that state due to arrivals at node I; similarly , Eq . (4.90) doe s likewise for departures and arrivals at node 2. This is the principle of local balance and we see therefor e that Eqs . (4.83)-(4.85) are already of this form . Thus we genera te nine local -balance equ ations* (four of which mu st therefore be redundan t when we con sider the con servation of probability), each of which is extremely simple and therefore permits a stra ightfo rward solution to be found. If thi s set of equations do es indeed have a solution, then they certainly guarantee that the global equations are satisfied and therefore that the solution we have found is the unique solution to the original global equ ati ons. The read er may easily verify the following solution : \,

,

p(l , 0, I) = fil - P( 2,0,0) fi 3

pel, 1,0)

= fil p(2, 0, 0) fi'

_ (/l l) 2

p(O, 1, 1) -

?

p(_ , 0, 0)

fl 2fl3 p(O , 0,

2)= (~rp(2, 0, 0)

fl l)2p( 2, 0, 0) p(O, 2, 0) = ( ;:. p(2 , 0, 0) = [1

+ PI + fl3

!!:l

u«

+ (fll )2 + (fl l)2+ fl 2P3

fl 3

1

(fll)1 fl 2

(4.91)

Had we allowed all possible transitions among nodes (rather th an the cyclic behavior in this example) then the state-transition-rate dia gram would have • The reade r should write them o ut directly from Figure 4.17.

4.8.

159

NETWORKS OF MARKOV IAN QUEUES

Figure 4.1 8 State-transition-rate diagram showing local balance (N

=

3, K

=

4).

perm itted transitions in both directions where now only unidire ction al transition s are perm itt ed ; however, it will always be true that only tr ansitions t o neare st-nei ghb or states (in thi s two-d imensional dia gram ) are permitted so that such a diagram can always be drawn in a planar fashion . For example, had we allowed four customer s in a n arbitra rily conn ected three-node network , then the state-transition-rate di agram would have been as shown in Figure 4.18. In t his diagram we repr esen t possible tran siti ons between nodes by an undirected branch (representing two one-way branches in opposi te directions). Also , we have collected together sets of branches by joinin g the m with a heavy line, and these are mean t to repr esent branches whose cont ributi ons appear in the same local-balance equ ati on . Th ese diagrams can be extended to higher dimensions when the re a re more than three nodes in the system. In particular , with four nodes we get a tetrahedron (that is, a threedimensional simplex). In general, with N nodes we will get an (N - 1)dimensional simplex with K + 1 nodes along each edge (where K = number of customers in the closed system). We note in these diagram s that all node s lying in a given straight line (pa ral!el to any base of the simplex) maintai n one comp onent of the sta te vector at a constant value and that this value increases or decreases by un ity as one moves to a parallel set of nodes. The localbalan ce equ ati ons are identi fied as balancing flow in th at set of bran ches that conn ects a given node on one of these constant lines to all other nodes on that constant line adjacent and parallel to this node , and th at decreases by unity that component that had been held con stant. In summa ry, then , the

160

MA RK O VIAN QUEUES IN EQU ILI BR IU M

local- bal ance equ ati on s a re tr ivial to write down, a nd if one can succeed in findin g a solution that satisfies them , then one has found the solut ion to the globa l-bala nce equati on s as well! As we see, most of the se Markovian ne tworks lead to rather complex systems of linear equations. Wall ace and Rosenberg [WALL 66] propose a numerical so lutio n metho d for a large class of the se equation s which is computati on ally effi cient. They di scuss a computer program, which is designed to evaluate the equilibrium probability distribution s of state variables in very large finite Mark ovian queueing net works. Specifically, it is designed to so lve the equilibrium equ ati on s of the form given in Eqs. (2.50) a nd (2. 116), namely , 7t = 7tP and 7tQ = O. The procedure is of the "power-iteration type" such th at if7t (i) is the ith iterate then 7t(i + I) = 7t(i)R is the (i + I)th iterate ; the matrix R is either equal to the matri x GtP + (I - Gt) I (where a: is a scalar) or equal to the matrix ~ Q + I (where ~ is a scalar and I is the identity matrix), depending up on wh ich of the two above equation s is to be solved . The sca la rs a: and ~ a re ch osen carefully so as to give a n efficient con vergence to the solution of the se equations. The speed of solution is quite remarkable and the reader is referred to [WALL 66] and its references for further det ails. Thus ends our study of purely Markovian systems in equilibrium. The unify ing feature throughout Chapters 3 and 4 has been that these systems give rise to product-type so lutions; one is therefore urged to look for solution s of thi s for m wheneve r Mark ovian queueing system s are enc ountered. In the next chapter we permit either A (t) or B (x) (but not both) to be of arbitrary form , requiring the other to rem ain in exponential form .

REFERENCES BURK 56

Burke, P. J., " The Output of a Queueing System," Operations Research, 4, 699-704 (1966).

CHAN 72 Chandy, K. M., " T he Analysis and Solutions for General Queueing Networks," Proc. Sixth Annual Princeton Conference on Information Sciences and Systems , Princeton University, March 1972. Cox, D. R., " A Use of Complex Probabilit ies in the Theory of StoCOX 55 chastic Processes," Proceeding s Cambridge Philosophical Socie ty, 51,313-31 9 (1955).

GORD 67 Gordon , W. J. and G. F. Newell, " Closed Queueing Systems with Exponential Servers," Operations Research, 15, 254-265 (1967). JACK 57 Jackson, J . R., "Networks of Waiting Lines," Operations Research,S, 518-521 (1957).

JACK 63 KING 69

Jack son , J. R., "Jobshop-Like Queueing Systems," Manag ement S cience , 10, 131 -142 (1963). Kingman, J. F. C., "Markov Population Processes," Journal of Applied Probability, 6, 1-18 (1969).

EXER CISES

161

KOEN 58 Koenigsberg, E., " Cyclic Queues," Operations Research Quarterly, 9, 22-35 (1958).

SAAT 65

Saaty, T. L., "Stochastic Network Flows: Advances in Networks of Queues," Proc. Symp. Congestion Theory, Univ, of North Carolina Press, (Chapel Hill), 86-107, (1965). WALL 66 Wallace, V. L. and R. S. Rosenberg, "Markovian Models and Numerical Analysis of Computer System Behavior," AFIPS Spring Joint Computer Confe rence Proc., 141-148, (1966).

WHIT 44

Whittaker, E. and G. Robinson, The Calculus ofObservations, 4th ed., Blackie (London), (1944).

EXERCISES 4.1.

Consider the Markovian queueing system shown below. Branch labels are birth and death rates. Node labels give the number of customers in the system.

Solve for Pk: Find the average number in the system. For). = fl , what values do we get for parts (a) and (b)? Try to interpret these results. (d) Write down the transition rate matrix Q for this pr oble m and give the matri x equation relating Q to the probabilitie s found in part (a) .

(a) (b) (c)

4.2.

Consider an Ek/En/1 queueing system where no queue is permitted to form. A customer who arrives to find the service facility busy is " lost" (he departs with no service). Let Ei j be the system state in which the "arriving" customer is in the ith arrival stage and the cust omer in service is in the jth service stage (no te that there is always so me customer in the arrival mechanism and that if there is no customer in the service facility, then we let j = 0). Let I lk), be the average time spent in any arrival stage and I lnfl be the average time spent in any service stage. (a) Draw the state tr ansition diagram showing all the transition rat es.

(b) Write down the equilibrium equation for E;j where I

o f( x) dy dx

Jz=y

m1

This immediately gives the final result:

! (y)

=

I - F(y)

- (5.10)

nit

This is our second result. It gives the density of residu al life in terms of the common distribution of interval length and its mean. * Let us express thi s last result in terms of transforms. Using our usual transform notation we have the following correspondences:

,, .

f (x)-=- F*(s) !(x)-=- I *(s) Clearl y, all the random va riables we ha ve been d iscu ssing in th is section are nonnegati ve, and so the relationsh ip in Eq . (5. 10) may be tr ansformed directly by use of entry 5 in Table 1.4 and entry 13 in Table 1.3 to give

r es) = 1 - F*(s)

- (5.11 )

Sni t

It is now a tri vial ma tte r to find the moments of residual life in terms of th e moments of the lifetimes themselves. We denote the nth moment of the lifetime by m; a nd th e Ilth mom ent of the residual life by r n ' that is, nJ " ,:;,

E[(Tk - Tk_t )" ]

r, ,:;, E[Y"]

(5.12) (5.13)

U sing our momen t formula Eq. (1I.26), we may di ffer ent iate Eq . (5.11) to obta in the moments of residu al life. As s ->- 0 we obtai n indeterminate for ms • It may a lso be show n that th e limiting pdf for age (%0) is the sa me as for residual life ( Y) given in Eq. (5.10).

5.2.

TH E PARADOX OF RESIDUAL LI FE: A BIT OF RENEWAL THEORY

173

which may be evaluated by means of L 'Hospital's rule ; this computation gives the moments of residual life as ln n+ 1 = ---'''-'-''--

r n

(n

+

I)m l

-(5 .14)

This important formula is most often used to evaluate 'I ' the me an residual life, which is found equal to

- (5.15) and ma y also be expressed (J 2 ~ m z - m 12) to give

In

terms of the lifetime variance (denoted by

ml

(J2

'I = -2 +-2m

(5.16)

l

This last form shows that the correct answer to the hippie paradox is m, /2, half the mean interarrival time , only if the variance is zero (regula rly spaced arrivals); however, for the Poisson arrivals, m l = IIA a nd (J2 = IfJ.2, giving '1 = IIA = mt> which confirms our earlier solution to the hippie paradox of residual life. Note that mtl2 ~ 'I and 'I will gr ow without bound as (J2 ->- 00. The result for the mean residual life (' I) is a rather counterintuitive result; we will see it appear again and again. Before lea vin g renewal theory we take this opportunity to qu ote so me other useful results. In the lan guage of renewal theory the age-d ependent fa ilure rate rex) is defined as the instantaneous rate at which a component will fail given th a t it has already attained a n age of x ; th at is, , (x) dx ~ P[x < lifetime o f component ~ x + dx I lifetime > z ], From firs t principles, we see that this conditional density is

f(x) rex ) = 1 _ F(x )

- (5.1 7)

where once again f (x) and F(x) refer to the common di stribution o f component lifetime. The renewal fun ction H (x) is defined to be

H (x) ~ E[number of renewals in an interval o f len gth xl

(5.18)

and the renewal density h ex) is merely the renewal rate at time x defined by

hex) ~ dH( x ) dx

(5.19)

174

TIl E QUEUE

M/G/!

Renewal theory seems to be obsessed with limit theorems, and one of the important results is the renewal theorem, which states that lim hex) z ..... ex)

=....!.

(5.20)

nIl

Thi s merely says that in the limit one cannot identify when the rene wal process began, and so the rate at which components are renewed is equal to the inverse of the average time between renewal s (m.). We note that hex) is not a pdf; in fact, its integral diverges in the typical case. Ne vertheless, it does possess a Laplace transform which we denote by H *(s). It is easy to show that the following relationship exists between this transform and the transform of the underlying pdf for renewals, namely :

H*(s)

=

F*(s) 1 - F*(s)

(5.21)

Thi s last is merely the transform expression of the integral equation ofrenewal theory, which may be written as

hex)

= f(x) + fh(X

- t)f(t) dt

(5.22)

More will not be said about renewal theory at this point. Again the reader is urged to consult the references mentioned above.

5.3. THE IMBE DDED MARKOV CHAIN We now consider the method of the imbedded Markov chain and apply it to the M/G /I queue . The fundamental idea behind this method is that we wish to simplify the description of state from the two-dimensional description [N(t), Xo(t)] into a one-dimensional description N(t) . If indeed we are to be successful in calculating future values for our state variable we must also impl icitly give, along with this one-dimensional description of the number in system, the time expended on service for the customer in service. Furthermore (and here is the crucial point), we agree that we may gain this simplification by looking not at all points in time but rather at a select set of points in time . Clearly, these special epochs must have the property that, if we specify the number in the system at one such point and also provide future inputs to the system , then at the next suitable point in time we can again calculate the number in system ; thus somehow we must implicitly be specifying the expended service for the man in service. How are we to identify a set of points with this property? There are many such sets. An extremely convenient set of points with this property is the set of departure instants from service . It is

5.3.

TH E IMBEDDED MARKOV CHAI N

175

clear if we specify the number of customers left behind by a departing customer that we can calculate this same quantity at some point in the future given only the additional inputs to the system. Certainly, we have specified the expended service time at these instants: it is in fact zero for the customer (if any) currently in service since he has just at that instant entered service !* (There are other sets of point s with this property, for example, the set of points that occur exactly I sec after customers enter service; if we specify the number in the system at these instants, then we are capable of solving for the number of customers in the system at such future instants of time. Such a set as j ust described is not as useful as the departure instants since we must worry about the case where a customer in service does not remain for a duration exceeding I sec.) The reader sho uld recognize that what we are describing is, in fact , a semiMarkov process in which the state transitions occur at customer departure instants. At these instants we define the imbedded Markov chain to be the number of customers present in the system immediately following the departure. The transition s take place only at the imbedded points and form a discrete-state space . The distribution of time between state transitions is equal to the service time distribution R(x) whenever a departure leaves behind at least one cust omer, whereas it equals the convolution of the interarrivaltime distribution (expo nentially distributed) with b(x) in the case that the departure leaves behind .an empty system. In any case, the behavi or of the chain at these imbedded points is completely describable as a Markov process, and the results we have discus sed in Chapter 2 are appl icable. Our approach then is to focus attention upon departure instants from service and to specify as our state variable the numb er of customers lef t behind by such a departing customer. We will proceed to solve for the system behavior at these instants in time. F ortunately, the solution at these imbedded Markov points happens also to provide the solution for all points in time. t In Exercise 5.7 the reader is asked to rederive some MIG/! results using the method of supplementary variables; this method is good at all points in time and (as it must) turns out to be identical to the results we get here by using the imbedded Mark ov chain approach. This proves once again that our solution • Mo reover ~ we assume that no service has been expended o n any other custome r in the queue. t This happ y circu mstance is due to the fact that we have a Poisson input and therefore (as shown in Section 4.1) an ar riving custome r ta kes wha t am ou nts to a " random" look at the system. Furthermore, in Exercise 5.6 we ass ist the reader in proving that the limiting distribution for the number of customers left behind by a depart ure is the same as the limiting distrib ution of custome rs found by a new arrival for a ny system that change s state by unit step values (positive or negati ve); th is result is true for arb itrary arriva l- and arbitrary service-time distributions ' Thu s. for MJG/I. a rrivals. depa rtu res, and random observers all see the same distr ibution of number in the system.

176

THE QUEUE

M IG II

is good for all time. In the following pages we establi sh results for the queuelength distribution, the. waiting-time distribution, and the busy-peri od distribution (all in terms of transforms); the waiting-time and busy-peri od durati on results are in no way restricted by the imbedd ing we have described . So even if the other methods were not available, these results would still hold and would be unconstrained due to the imbedding pr ocess. As a final reassurance to the reader we now offer an intuitive ju stificati on for the equivalence between the limiting distributions seen by departures and arrivals. Taking the state of the system as"the number of customers therein, we may observe the changes in system sta te as time evolves ; if we follow the system state in continuous time, then we observe that these chan ges are of the nearest-neighb or type. In particular, if we let Ek be the system state when k cust omers are in the system , then we see that the only tran sition s from this state a re Ek --+ E k+l and E k --+ E k _ 1 (where this last can only occur if k > 0). Thi s is den oted in Figure 5.2. We now make the observati on that the number of transitions of the type E k --+ E k+l can differ by at most one from the number of transitions of the type E k+l --+ E k . The form er corre spond to customer arri vals and occur at the arriv al instants ; the latter refer to customer dep artures and occur a t the dep arture instants. After the system has been in opera tion for an arbitrarily long time, the number of such transitions upward must essentially equal the number of transition s down ward. Since th is upand-down motion with respect to E k occurs with essenti ally the same frequ ency, we may therefore conclude that the system states found by arrivals must have the sa me limitin g distribution (rk ) as the system sta tes left behind by departures (which we denote by dk ) . Thu s, if we let N(I) be the numb er in the system at time I, we may summarize our two conclu sions as follows: 1.

F or Poisson arrivals, it is alway s true that [see Eq. (4.6)] P[N(t) = k] = P[arrival at time t finds k in system] th at is, (5.23)

2.

If in any (perhaps non-Markovian) system N( I) makes only discontinuous chan ges of size (plus or minus) one , then if either one of the following limiting distributions exists, so does the other and they are equal (see Exercise 5.6) : . I"k ,;;

lim P[arrival at

t

finds k custom ers in system]

t - ",

dk

,;;

lim P[departure a t

1 leaves

k custom ers behind]

t - ",

- (5.24) Thus, for M/G /l,

-

5.4.

THE TRANS IT ION PRO BABILITIES

177

Figure 5.2 State transitions for unit step-change systems. Our a pproac h for the balance o f thi s chapter is first to find the mean number in system, a result referred to as the P ollaczek-Khinch in mean-value formula . * F ollowin g that we obta in the genera ti ng functi on for the distribution of number of custo mers in the system and then the tran sform for both the waiting-time and total system-time distributions. These last transform results we sha ll refe r to as Pollaczek-Khinchin tr an sform equ ations. * Furthermore, we so lve for the transform of the bu sy-pe riod durati on a nd for the number served in the busy pe riod; we then show how to derive waitin g-t ime results from the bu sy-period a na lysis. Lastly, we deri ve the Takacs integrodifferentia l equ ation for the unfinished work in the system. We begin by defining so me notation and identifying the transiti on probabilities associa ted with ou r im bedded Markov chain.

5.4. THE TRAl'1SmON PROBABILITIES We have already discussed the use of customer departure instants as a set of imb edded points in the time axis; at these instants we define the imbedded Markov cha in as the number of customers left behind by the se departures (th is forms our imbedded Markov chain). It should be clear to the reader th at th is is a co mpl ete sta te description since we kn ow for sure that zero service ha s so far been expended on the customer in service and th at the time since the last arrival is irr elevant to the future devel opment of the process, since th e interarriva l-time distribu ti on is mem oryless. Ea rly in Ch a pter 2 we introduced some symbo lical and gra p hical not at ion ; we as k th at the reader refresh his understand ing of Figure 2.2 and th at he recall the following de finition s : C n represents th e nth customer to enter the system r ; = arrival time of C; t ; = T n - T n _ 1 = intera rrival time between C n _ 1 and C; X n = service time for C n In addition, we int roduce two new random va ria bles of consider ab le interes t :

qn = number of cu stomers left behind by departure of Cn from service V n = nu mber of customers a rriving during the service of C n • There is considerable disagreement within the queueing theory literature regarding the names for the mean-value and transform equations. Some authors refer to the mean-value expression as the Pollaczek-Kh inchin formula, whereas others reserve that term for the transform equations. We attempt to relieve that confusion by adding the appropriate adjectives to these names.

r-

r i

178

TH E QUE UE

MIG/!

We are interested in solving for the distributi on of q", namely , Piq; = kj , which is, in fact , a t ime-dependent probability ; its limiting distribution (as II ->- co) corresponds to elk' which we know is equ al to Pk> the basic d istribution discussed in Chapters 3 and 4 previously. In carrying out that so lution we will find that th e n umber of a rriving cu stomers V n plays a crucial ro le. As in Chapter 2, we find that the tr an sition probabilities descr ibe our Markov chain ; thu s we define the one-step transiti on pr ob abilities

Pi; ~ P[qn+!

= j Iq. = i]

(5.25)

Since the se tr an srnons are observed only at departures, It IS clear th at qn+J < qn - I is an impossible situa tio n ; on the other hand , q,,+! ~ q. - I is po ssible for all values due to the arrivals V n +!. It is easy to see that the matrix of transiti on probabilities P = [Pi;] (i,j = 0 , 1,2 , . ..) take s the following form :

,

P=

I

,j

eL.

eLl

eL 2

eLa

eL.

eL J

eL 2

eL a

0

eL.

eLl

(X2

0

0

eL.

eLl

0

0

0

eL.

I; where eLk

~ P[v. +!

=

k]

(5.26)

For example, the jth component of the first row o f thi s matri x gives the prob ability th at the previou s customer left behind a n emp ty system and that during th e service of C n + l exactly j customers a rriv ed (a ll of who m were left behind by the dep arture of C n+\); similarly, for other than the first row, the entry Pi; for j ~ i - I gives the probability that exac tly j - i + I customers a rr ived during the service peri od for C,,+I> give n tha t C" left behind exactly i customers ; of these i customers one was ind eed C "+ 1 and thi s acc ounts for the + I term in th is last co mp uta tio n. The sta te-tra nsitionprobability dia gram for th is Markov ch ain is show n in F igure 5.3, in which we show only trans iti on s o u t of E i . Let us now calc ulat e eLk' We ob serve first o f all th at the a rriva l pr ocess (a Poisson process at a rate of A customers per seco nd) is ind ependen t of the sta te of the queueing system . Similarl y, x"' the service time for C", is independent

5.4. TH E TRA NSITIO N PROBABILITI ES

179

ao

Figure 5.3 State-transition-probabilit y diagram for the M/G/I imbedded Mar kov Chain.

of 11 and is distributed according to B( x). Therefore, Vn, the number of arrivals during the service time X n depends on ly upon the durati on of X n and not upon 11 at all. We may therefore dispense with the subscripts on V n and x n , repl acin g them with the random variables u a nd x so that we ma y write P[x n ::;; x] = P[x ::;; x] = B(x) and P[vn = k ] = P[ u = k] = (f.k . We may now proceed with the calcu lati on of (f.k . We have by the law of tot al prob ability (f.k

= P[u = k] =

f'

P[u = k, x

< x ::;; x + dx] dx

By condition al probabilities we furthe r have

(f.k

=

f'

P[u = k

Ix =

x ]b(x) dx

(5.27)

where again b(x) = dB (x)/dx is the pdf fo r service time. Since we have a Poisson arrival process, we may replace the pr ob abil ity bene ath th e int egral by the expre ssion given in Eq . (2.131), t ha t is, (f.k

=

i'" o

(}.X)k - e- l' b( x ) d x

k!

(5.28)

Thi s the n completely specifies the transition pr obability matrix P . We note that since (f.k > 0 for all k ~ D it is possible to reach all o ther sta tes from a ny given state ; thu s o ur Markov cha in is irreducible (a nd a period ic). More over , let us make ou r usual definition : p

=

AX

a nd point out th at thi s Markov chain is ergodi c if p < 1 (unless specified otherwise, we sha ll assume p < I below) . T he stationary pro ba bilities may be obtained from the vector equ ati on p = pP where p = [Po, p" P2' . . .] whose kth component Pk ( = .-ik ) is

180

TH E QUEUE

M/G/l

merely the limiting probability that a departing customer will leave behind k customers, namely, Pk

=

P[q

=

k]

(5.29)

In the following section we find the mean value E[q] and in the section following that we find the z-transform for h .

5.5. THE MEAN QUEUE LENGTH In this section we derive the Pollaczek-Khinchin formula for the mean value of the limiting queue length. In particular, we define

q = lim qn

(5.30)

which certainly will exist in the case where our imbedded chain is ergodic. Our first step is to find an equation relat ing the random variable qn+l to the random variable qn by considering two cases. The first is shown in Figure 5.4 (using our time-diagram notation) and corre spond s to the case where C; leaves behind a nonempty system (i.e., qn > 0). Note that we are assuming a first-come-first-served queueing discipline, alth ough this assumption only a ffects waiting times a nd not queue lengths or busy periods. We see from Figure 5.4 that qn is clearly greater than zero since C n+l is already in the system when C n departs. We purposely do not show when customer Cn +2 arr ives since th at is unimportant to our developing argument. We wish now to find an expression for q n +l ' the number of customers left behind when C n+l dep arts. Th is is clearly given as equ al to qn the numb er of customers present when C; departed less I (since customer C n+l departs himself) plus the number of customers that arri ve during the service interval Xn +l ' Thi s last term is clearly equal to Dn+l by definition and is shown as a "s et" of arri vals

Q' H-l left

q. lef t

behind

behind Serv er--------.----~.-------¥---

T ime~

Queue - - r - - - - - - ' - - - - - - - . - . L - - - - - - - : - - -

'---v---J

Cn "

~ v n. l arrive

F igure 5.4

Case whe re q« > O.

5.5.

r--

Server

c.

lSI

THE MEAN QUE UE LENG TH

x

, . , --","

~ q." left behind Ti m e ~

C,, +I

Qu eue ---,r-----'----~c_-----+__--

t

C.

t

C,,.1

~ V IJ+ l

~

arri ve

Figure 5.5 Case where qn = O.

in the diagram. Thus we have

qn

>0

(5.31)

Now consider the secon d case where qn = 0, that is, our departing customer leaves behind an empty system; this is illustrated in Figure 5.5. In this case we see that qn is clea rly zero since e n+! has not yet arrived by the time C n departs. T hus qn+!, the number of customers left behi nd by the depar ture of C n +1 , is merely equal to the number of arrivals d urin g his service time. Thus (5.32) qn = 0 Collectin g together Eq. (5.31) and Eq. (5.32) we have qn > 0

qn = 0

(5.33)

It is convenient at thi s point to introduce D. k , the shifted discrete step function

k = 1,2, . . . k~O

(5.34)

which is related to the discrete step functi on Ok [defined in Eq . (4.70)] through D. k = 15k _ I , Applying thi s definition to Eq, (5.33) we may now write the single definin g equation for qn+l as - (5.35)

182

TH E QUEUE

M/G/I

Equation (5.35) is the key equation for the st udy of M/GfI systems. It remain s for us to extract from Eq . (5.35) the mean value * for qn' As usual, we concern ourselves not with the time-dependent behavior (which is inferred by the subscript II) but rather with the limiting distribution for the rand om variable qn, which we den ote by g. Accordingly we assume that the jth moment of qn exists in the limit as II goes to infinity independent of II , namely , (5.36) lim E[q /] = EW] n -e cc

(We are in fact requiring ergodicity here.) As a first attempt let us hope that forming the expectation of both sides of Eq. (5.35) and then takin g the limit as II - . cc will yield the average value we are seekin g. Proceeding as described we have

Using Eq . (5.36) we have, in the limit as II ---+ co, E[g] = E[g] - E[Ll q ] + E[ v] Alas, the expectation we were seeking drops out of this equation, which yield s instead . (5.37) E[6 .] = E[ v] What insight does this last equ at ion provide us ? (No te that since v is the number of arrivals during a customer's service time , which is independent of II , the ind ex on u; could have been dropped even before we went to the limit.) We have by definiti on that E[ v]

=

average number of arri vals in a service time

Let us now interpret the left-hand side of Eq . (5.37). By definiti on we may calcul ate this directly as 00

E[6;;]

= .26kP[g =

k]

k= O

= 6 oP[ g = 0]

+ 6,P[g =

1]

+ ...

• We could a t this point pr oceed to the next section to obtain the (z-tra nsform of the) limit ing distribution for numbe r in system and from that expression evaluate the avera ge number in system. Instead , let us calculate the average number in system directly from Eq . (5.35) following the method of Kendall [KENO 51] ; we choose to car ry out this extra work to dem onstrate to the student the simplicity of the a rgument.

5.5.

T H E MEAN QUEUE LENGTH

183

But , from the definition in Eq . (5.34) we may rewr ite this as

E [D.. ] = O{P[q

= OJ) +

I{P [q > OJ}

or

E [D.. ] = P[ q > 0]

(5.38)

Since we a re dealin g with a single-server system , Eq. (5.38) may also be writte n as (5.39) E [D..] = P[busy system] An d from o ur defin ition of the ut ilizat ion factor we furt her ha ve (5.40)

P [busy system] = p

as we had o bserved* in Eq . (2.32). Thus from Eq s. (5.37), (5.39), and (5.40) we con clude tha t - (5.4 1) E [v] = p We thus have the perfe ctly reason able conclusion that the expected number of arrivals pe r service inte rval is eq ual to p (= ix). For stability we of co urse require p < I , a nd so Eq . (5.4 1) ind ica tes that customers must arrive more slowly th an the y can be served (on the average). We now return to the ta sk of solving for the expected va lue of q. Forming the first mo me nt of Eq . (5.35) yielded interesti ng resul ts but fai led to give the des ired expectati on. Let us now a ttem pt to find th is average value by first squaring Eq . (5.35) and then ta king expectati on s as follows : (5.42) From o ur de finition in Eq. (5.34) we ha ve (D. o )" = D.o" an d also Applyi ng this to Eq. (5.42) a nd taking expecta tio ns ,we have

qn

D. o"

= q n'

In this eq ua tion, we hav e t he expec ta tion of the product of two random variab les in the last two terms . Howeve r, we o bserv e that L'n+l [the nu mber of a rriva ls du ring the (11+ I)th service int er val] is inde penden t of q" (th e number of customers left behind by e n)' Conseq uent ly, the last two expec ta tions may each be written as a prod uct of the expectations. Taking the limit as n goes to infinity, an d using our limit ass umptions in Eq . (5.36), we have

o=

E[D..]

+ E[v']

- 2E[q]

+ 2E[q]E[v] -

2E[D. q]E[v]

* Fo r any M/G fl systcm , we see tha t P [g = 0] = I - P [q > 0] = 1 - p and so P[ ncw customer need 1101 queue] = I - p. Th is agrees with our ear lier observation for G IG I I.

• 184

THE QUEUE

M/G /l

We now make use of Eqs. (5.37) and (5.4 1) to obtain, as an intermedi at e result for the expectation of g, E - _ [q] - P

+

E[i?] - E[ o] 2(1 - p)

(5.43 )

The only unknown here is E[v2 ] . Let us solve not only for the second moment of 0 but, in fact , let us describe a meth od for obta ining all the moments, Equati on (5.28) gives an expression for (Xk = P[ o = k]. From this exp ression we should be able to calculate the moments. However, we find it expedient first to define the z-tra nsform for the random variable 0 as -

.:l

.6.

V(z) = E[z"] =

00

I

P[o =

k] Zk

(5.44)

k= O

Forming V(z) from Eqs. (5.28) and (5.44) we have

'" r

I

V(z) =

k~ O

. -(h)k e-AXb(x) d x Zk

0

k!

Our summation and integral are well behaved , and we may interchange the order of these two operations to obtain V(z) =

l

ro

e- AX

I (Axzt) - - b(x ) d x

• ( co

o

k -O

k!

= L X> e- AXe AXZb(x ) dx =

r

e-IA-A=lxb(x) dx

(5.45)

At thi s point we define (as usual) the Laplace transform B*(s) for the service time pdf as B*(s)

~ LX> e- SXb(x) d x

We note that Eq. (5.45) is of this form , with the complex variable s replaced by i. - }.z, and so we recognize the impo rtan t result th at V(z)

=

B*(Je - h )

- (5.46)

Thi s last equation is extremely useful and rep resents a relati onship between the z-transform of the probability distribution of the random variable 0 and the Laplace transform of the pdf of the ra ndom variable x when the Laplace transform is evaluated at the critical point Je - h. The se two rand om variables are such that ii rep resents the number of arrivals occurring du ring the

5.5.

THE MEAN QUEUE LENGTH

185

inte rval i where the arrival pr ocess is Poi sson at an average rate of Aarrivals per seco nd. We will sho rtly have occa sion to incorp orate thi s interpretati on of Eq. (5.46) in our further results. F rom Appendix II we note th at vari ou s derivati ves of z-tra nsforms evaluated for z = I give the various moments of the rand om varia ble under considerati on. Similarl y, the appropriate deriv ati ve of the Laplace transform evaluated at its ar gument s = 0 also gives rise to moments. In particular, from th at appe ndix we recall that B*(k\ O)

~ dkB*(s) I = (-I )kE[Xk] k

(5.47)

~ d V(z) I

(5.48)

ds

V(ll(1)

,_0

dz

V(2)(1)

= E[ ii]

:- 1

~ d'V~z) I = dz"

E[ii 2 ]

-

E[ii]

(5.49)

:~l

In order to simplify the nota tion for the se limitin g derivat ive opera tions, we have used the more usual superscript notation with the argument replaced by its limit. Furthermore, we now resort to the overb ar notat ion to denote expected value of the random variable below that bar. t Thus Eqs. (5.47)(5.49) become B*Ckl(O)

=

( - I )kx"

V(ll(1) = iJ

V(2l( l)

(5.50) (5.51)

= v' -

iJ

(5.52)

Of course, we must also have the con servati on of probability given by B*(O)

=

V(1)

=

I

(5.53)

We now wish to exploit the relationship given in Eq . (5.46) so as to be able to obtai n th e moment s of the random variable ii from the expre ssion s given in Eqs. (5.50)-(5.53). Thus from Eq . (5.46) we have d V(z)

dB *(}. - AZ)

dz

dz

-- -

(5.54)

t Recall from Eq. (2.19)tha t E [x nk ] _ x k = bk (ra ther tha n the more cumbersome nota tion (ilk which one might expect). We ta ke the sa me liberties with vand ij, namely, (if = ;;; and (fj)k = qk.

186

TH E QUEUE

M/G /l

Thi s last may be calculated as dB*(A ~ k ) dz

=

(dB *,(i. - , AZ)) (d (i. - i.Z)) d( /. - I.Z) dz , dB *( y)

= - A-

(5.55)

-

dy

where y =

A- ;.z

(5.56)

Setting Z = 1 in Eq. (5.54) we have

=

V(ll(1)

But from Eq . (5.56) the case Z

_ A dB *(y ) dy

I

:~1

= 1 is the case y = 0, a nd so we have

VOI(I) = - AB*(l)(O)

(5.57)

From Eqs. (5.50), (5.51), and (5.57), we fina lly have ij

=

i3:

(5.58)

But Ax is ju st p and we have once again established that which we knew from Eq . (5.41), namely, ij = p . (This certainl y is encouraging.) We may continue to pick up higher moments by differentiating Eq. (5.54) once again to obtain d 2 V(z) d 2 B*(A - k) -(5.59) 2 2 dz

dz

U sing the first derivati ve of B *(y ) we now for m its second der ivative as follows : d

2B* (). - i.z) dz 2

=

.!!-[_;. dz

=

dB *(y)] dy .

_A(d2B*~!J))(dY) dy-

dz

or d 2B*(}. - i.z) d z2

, 2

=

d 2B*( y )

I.

d y'

Setting z equal to 1 in Eq. (5.59) and using Eq . (5.60) we have

(5.60)

5.5.

T HE MEAN QU EU E LENGTH

187

T hus, from ea rlier results in Eqs. (5.50) and (5.52), we obtain - (5.61) We have thus fina lly solved for v'. Thi s clearly is the quantity requ ired in order to evaluate Eq. (5.43). If we so desired (and with suita ble ener gy) we could continue this differentiati on game a nd extract additional moment s of iJ in term s of the moments of i; we prefer not to yield to that temptati on here. Returning to Eq . (5.43) we apply Eq . (5.61) to obtain ij

=

P+

j. 2 2 X

2-(1.:.:... · _"'-p)

(5.62)

T his is the result we were after ! It expresses the average queue size at customer departure instants in terms of known quantities, namel y, the utilizati on factor (p = AX), }., and x' (the second moment of the service-time distr ibuti on). Let us rewr ite thi s result in terms of C; = Gb'/{x)', the squared coefficient of variat ion for service time :

__ + ' (1 + Cb' ) q -

p

P 2(1 - p)

- (5.63)

Thi s last is the extremely well-known formula for the average number of custome rs in an M/G lI system and is comm only* referred to as the PollaczekKhinchin (P- K ) mean-value f ormula. Note with emphasis th at thi s average dep end s only up on the fi rst ruo moments (x and x' ) of the service-time dis tribution. Moreover , observe that ij gro ws linearly with the variance of the service-time distribution (or, if you will, linearly with its squ ared coefficient of variation). T he P-K mean -value formula provides a n expre ssion for ij that represent s the average number of customers in the system at departure instants ; however, we alr eady know that this also repre sents the average number at the arriva l instan ts and, in fact , at all point s in time. We already have a not ati on for the average number of customers in the system, namely iii, which we introduced in Chapter 2 and have used in pre viou s chapters; we will continue to use the iii notat ion outside of this chapt er. Furthermore, we have defined iii. to be th e average nu mber of custo mers in the queue (no t coun ting the customer in service). Let us take a moment to develop a relati onship between these two quan tities. By definiti on we have

-0- '" N = "2: kP[ ij k= O

• See footnote on p. t 77.

=

k]

(5.64)

188

TH E QUEU E

MIGI I

Similarly we may calculate the ave rage queue size by subtracting unity from this pre viou s calculation so long as there is at least o ne customer in the system, that is (no te the iowe r lim it) , " ",

Nq

= I(k

- I )P[q

=

k)

k= l

This easily gives us

I'" P[q = k)

'"

Nq = I kP [q = k) -

k= l

k= O

But the second sum is merely p and so we have the result

Nq = N -

- (5.65)

p

This simple formula gives the general relationship we were seeking. As an example of the P- K mean-val ue for mula , in the case of an MIMfI system, we have that the coefficient of va riati on for the exponential distributi on is uni ty [see Eq . (2. 145»). Thus for this system we have

__ + q -

p

2 (2) P 2(1 - p)

or q= -pI - P

MIMII

(5.66)

Equati on (5.66) gives the expected number of cust omers left behind by a departi ng custome r. Compare thi s to t he expression for the average number of customers in a n MIMfI system a s give n in Eq . (3.24). They a re identical and lend va lidit y to our ea rlier statemen ts that th e meth od of the imbedded Markov cha in in the MIGfI case gives rise to a so lution that is good a t all points in time. As a second example , let us con sider the service-time distributi on in which service time is a con stant a nd equ al to x. Such systems are de scribed by the notation MIDII , as we ment ioned earlier. In th is case clea rly C b 2 = 0 a nd so we have

__ +

q-

P

2

1

P 2(1 -

p)

ij = - p- - --,P_1- P 2( 1 - p)

- (5.67)

MIDII

Thus the MIDfI system has p 2 /2(1 - p) fewer customers o n the a verage than the MIMI I system, demonstrating the earlier sta tement th at ij increases with the vari ance of the service-time distribution .

5.5.

T HE MEAN QUEUE LENGTH

I S9

Service faci Iity

Figure 5.6 The M/H 2 /1 example. F or a th ird example, we consider an M /H 2/l system in wh ich x ~ O

(5.6S)

That is, the service facility consists of two parallel service stages, as shown in Fi gure 5.6. N ot e that A is also the arrival rate, as usual. We may immediately ca lculate x = 5/(S).) a nd (Jb 2 = 31 /(64.12) , which yield s C/ = 31/25. Thus

--

q -

p

P"( 2.24) +..:........:_-

2(1 - p)

p

O.12p 2

I- p

I -p

= --+-Thus we see t he (small) increase in ij for the (sma ll) increase in C;2 over th e va lue of un ity for M/M / 1. We note in this example th at p is fixed a t p = i.x = 5/S; th erefore, ij = 1.79, whereas for M /M/l a t thi s va lue of p we get ij = 1.66. We have introduced thi s M /H 2/l example here since we intend to carry it (a nd the M/M/ I exa mple) thr ou gh our MIG/l discussion. The main result o f th is sect ion is th e Pollaczek -Khinchin fo rm ula fo r the mean number in system, as given in Eq . (5.63). This result bec omes a special case of ou r results in the next sect io n , but we feel th at its development has been useful as a pedagogical device. Moreover , in ob tai ning th is res u lt we established the ba sic equation for MIG/I given in Eq . (5.35) . We a lso obtai ned the ge nera l relati on ship between V( z) a nd B*(5) , as given in Eq. (5.46); from t his we a re a ble to obtai n the moments for the number o f a rr ivals during a service interval.

190

TH E QU EUE

M IGII

We have not as yet derived an y results regarding time spent in the system ; we are now in a positi on to do so . We recall Little's result:

This result relates the expected number of customers iii in a system to 1 , the arrival rate of customers and to T, their average time in the system. For MIGII we have deri ved Eq . (5.63), which is the expected number in the system at customer departure instants. We may therefore appl y Little's result to this expected number in order to obtain the average time spent in the system (queue + service) . We know that ij als o represents the average number of customers found at random , and so we may equate ij = iii. Thus we have _

+ C.2 )

• (1

N=p+p·

2(1 - p)

=1T

Solving for T we have

T =

px(1

+ C; )

x + -'--'--'----"-'2(1 - p)

(5.69)

This last is easily interpreted. The average total time spent in system is clearly the average time spent in service plus the average time spent in the queue. The first term above is merely the average service time and thu s the seco nd term mu st represent the average queueing time (which we den ote by W). Thus we have th at the average queueing time is px(l

+ C;)

W = '---''-----''---'2(1 - p )

or Wo W=-I-p

- (5.70)

where W o ~ i0/2; W o is the average remaining service time for th e cust omer (if an y) found in service by a new arrival (work it out using the mean residu al life formula). A particularly nice normalization fact or is now apparent. Consider T, the average time spent in system. It is natural to comp are this time to x, the average service time required of the system by a cust omer. Thus the ratio Tlx expre sses the ratio of time spent in system to time required of the system and repre sents the factor by which the system inconvenie nces

5.6.

DISTRIB UTI O N OF NU MBER IN SYSTEM

191

customers due to the fact that they are sharing the system with other customers. If we use this normalization in Eqs. (5.69) and (5.70), we arrive at the following, where now time is expre ssed in units of average service intervals: T

-

x ·

W

-

x

+ p (1 + C b ) 2

= 1

=

p

2(1 - p)

(l

+C

2 b )

2(1 - p)

_ (5.71)

_ (5.72)

Each of these last two equations is also referred to as the P-K mean-value formula [along with Eq . (5.63)]. Here we see the linear fashi on in which the statistical fluctuati ons of the input processes create delay s (i.e., I + C b 2 is the su m of the squared interarrivai-time and service-time coeffici ents of variation). Further, we see the highly nonlinear dependence of delays upon the average load p . Let us now comp are the mean normalized queueing time for the systems" M /M /l and M /D fl ; these have a squared coefficient of variation Cb 2 equal to I and 0, respectively. Applying this to Eq. (5.72) we ha ve W

x W

x

-

P

(I - p) P

2(1 - p)

MIM II

_ (5.73)

M IDII

_ (5.74)

Note that the system with constant service time (M /D/l) has half the average waitin g time of the system with exponentially distributed service time (M / M {l) . Thus, as we commented earlier, the time in the system and the number in the system both grow in proportion to the vari an ce of the service-time distribution . Let us now proceed to find the distribution of the number in the system. 5.6.

DISTRIBUTION OF NUMBER IN SYSTEM

In the previ ous sections we characterized the M IGII queueing system as a n imbedded Markov chain and then established the fundamental equation (5.35) repeated here : (5.75) By forming the average of this last equation we obtained a result regarding the utilizati on factor p [see Eq . (5.41)]. By first squaring Eq. (5.75) and then • Of less interest is our highly specialized MjH zll example for which we obtain 1.12pj(1 - pl.

W j;;; =

192

TH E QUEUE

M fG fl

takin g expectati on s we were able to obtain P-K formulas that gave the expected number in the system [Eq. (5.63)] and the norm alized expected time in the system [Eq. (5.71)]. If we were now to seek the second moment of the number in the system we could obtain this quantity by first cubing Eq. (5.75) and then taking expectations. In thi s operation it is clear that the expectation E[f] would cancel on both sides of the equation once the limit on n was taken ; thi s would then leave an expression for the second moment of g. Similarly, all higher moments- can be obtained by raisin g Eq. (5.75) to successively higher powers and then forming expectations. * In this section, however, we choose to go after the distribution for qn itself (actually we consider the limiting random variable g). As it turns out, we will obtain a result which gives the z-transforrn for this distribution rather than the distributi on itself. In principle, these last two are completely equivalent; in practice, we sometimes face great difficulty in inverting from the z-tra nsform back to the distribution . Nevertheless, we can pick off the moments of the distributi on of g from the z-transforrn in extremely simple fashion by making use of the usual properties of transforms and the ir deri vatives. Let us now proceed to calculate the a-transform for the probability of finding k customers in the system immediately following the departure of a customer. We begin by defining the z-transform for the random va riable qn as (5.76) From Appendix II (and from the definition of expected value ) we have that thi s z-transform (or probability generating functi on) is also given by Qn(z) ~ E[z·n]

(5.77)

Of interest is the z-transform for our limiting random variable Q(z)

'" = lim Qn(z) = 2: P[g = n -e cc

k ]Zk

=

-

E[z"]

g: (5.78)

""= 0

As is usual in these definit ions for tr an sform s, the sum on the right-hand side of Eq. (5.76) converges to Eq . (5.77) only within some circle of co nvergence in the z-plane which defines a ma ximum value for [z] (certai nly [a] ~ I is allowed). The system M fG fl is characterized by Eq. (5.75). We therefore use both sides of thi s equ at ion as an exponent for z as follows :

• Specifically, th e k th power leads to an expression for Erqk- 'j that involves the first k momen ts of service time.

5.6:

DISTRIB UTIO N OF NU MBER IN SYSTEM

193

Let us now take expectations: E[z·" ,] = E[z· .--

l. From the P-K tra nsfo rm

( 1 - p)(1 - z)[8 + 7(1 - z)] + 7(1 - z) - 4z(2 - z)(3 - z)

F actoring the den ominator a nd canceling the commo n term (I - z) we ha ve

( 1 - p)(1 - (7{15)z] Q(z)

=

[1 _ (2/5 )z][1 - (2/3)z]

We now exp and Q(z) in partial fraction s, which gives

1{4 Q(z) = (I - p) ( I _ (2/5)z

+I

3{ 4 ) _ (2/3)z

This la st may be inverted by inspection (by now the reader sho uld rec ogni ze the sixth entry in Table 1.2) to give P.

=

P[ij

=

k]

=

(1 -

p>[~(~r+ ~(~n

(5.9 1)

Lastl y , we note th at the value for p ha s a lready been calculated a t 5/8 , and so for a final soluti on we have k

= 0, 1,2, . . .

(5.92)

It sho u ld not surprise us to find thi s su m of geo metric terms for our so lutio n. Further examples will be found in the exerci ses. F or now we terminate th e d iscussion of how many cu st omers are in the system a nd proceed with the calculati on of how long a cu st omer spends in the system .

5.7.

DISTRIBUTION OF WAITING TIME

Let us n ow set out to find the distribution of time sp ent in th e system a nd in the queu e. These particul ar qu ant ities are rather easy to obta in fr om o ur earl ier principal result, nam ely, the P-K tr an sform eq ua tion (a nd as we ha ve sa id , lead to expression s which sha re th at nam e). Note th at the order in which cu st omers receive serv ice has so far not affected our results. No w, however, we mu st use our ass u mptio n th at the order of service is first-co mefirst- ser ved . In o rder to pr oceed in the sim plest possibl e fashi on , let us re-examine the deri va tion of th e foll owing equat ion :

V(z) = B* (i. - k)

(5.93)

5.7.

DISTR IBUTION OF WA ITING TI ME

197

Time------;;.... Ououo -

,

-;.., _ _

\.~----,v~---'}

~.

"n

arrive

Figure 5.7 Derivation of V(z)

=

B* (i. - i.z).

In Figure 5.7, the reader is reminded of the structure from which we obtained this equation. Recall that V (z) is the z-transform of the number of customer arrivals in a particular inter val, where the arrival proce ss is Poisson at a rate A cust omers per second. The particular time interval involved happens to be the service interval for C n; this interval has distribution B(x) with Laplace t ransform B *(s). Th e deri ved relati on between V(z) and B* (s) is given in Eq. (5.93). The imp ortant observation to make now is that a relationship of this form must exist between any two random variables where the one identifies the number of customer arrivals from a Poisson process and the other describes the time interval over which we are co unting these customer arri vals. It clearly makes no difference what the interpretation of this time interval is, only that we give the distribution of its length ; in Eq. (5.93) it ju st so happens that the interval involved is a service interv al. Let us now d irect our attention to Figure 5.8, which concentrates on the tim e spent in the sys tem for C n' In th is figure we have traced the history of C n' The interval labeled lI" n ident ifies the time from when C; enters the queue until that customer leaves the queue and enters service; it is clearly the waiting time in queue for C n' We have also identified the service time X n for C n' We may thu s

Tj me~

\'-----~ ~-----'

~

q"

arrive

Figure 5.8 Derivation of Q (z ) = S * (i. - i.:;).

198

TH E QUEUE

M/G/I

identify the total time spen t

i ll

sy stem

Sn

for CO' (5.94)

We have earlier defined gn as the number of customers left beh ind upon the departure of Cn' In considering a first-come-first-served system it is clear th at all those customers present upon the arri val of C n must depart before he d oes; consequently, those customer s that C; leaves behind him (a total of gn) must be precisely th ose who arri ve durin g his stay in th e system. Th us, referring to Figure 5.8, we may identify those customers who arrive du ring the time interval s; as bein g our previously defined rand om variab le gn' Th e reader is now asked to comp are Figures 5.7 and 5.8. In bot h cases we have a Poisson arrival process at rate I customers per second. In Figure 5.7 we inqu ire into the number of arrivals (un) during the interval whose durat ion is given by X n ; in Figure 5.8 we inquire int o the number of arrivals (gn) during an interval whose durati on is given by S n' We now define the distribut ion for the total time spent in system for C; as Sn(Y) ~ P[sn ~ y]

(5.95)

Since we are assuming ergodicity, we recognize immediat ely that the limit of this distribution (as n goes to infinity) must be independent of II . We deno te this limit by S(y) and the limiting rand om varia ble by s [i.e., Sn(Y) ->- S(y ) and s; ->- s]. Thus S (y ) ;; P[s ~ y] (5.96) Finally, we define the Lap lace transform of the pdf for total time in system as • S *(s) ;;

f'

e- ' · dS( y)

= E[e~S]

(5.97)

With these definitions we go back to the analogy between Figures 5.7 and 5.8. Clearly, since Un is an alogous to gO' then V( z) must be analogous to Q(z), since each describes the generating functi on for the respective nu mber distribution. Similarly, since X n is analogous to S n , then B *(s) must be anal ogous to S *(s). We ha ve therefore by dir ect analogy from Eq. (5.93) t hat t Q(z) = S* (i. - }.z) (5.98) Since we already have an explicit expression for Q(:) as given in the P-K transform equat ion , we may therefore use that with Eq . (5.98) to give an explicit expression for S * (s) as S *(,1. _ ,1.z) = B*(}. _ ,1.:) (l - p)(l - z) B*(,1. - }.z) - z

(5.99)

t Thi s can be der ived directly by the unco nvinced reader in a fashion similar to tha t which led to Eqs. (5.28) and (5.46).

5.7.

199

DI ST RI BUTI O N O F W AITI NG TIME

Thi s last equat ion is just crying for the o bvio us change of va ria ble which gives

= = I -~ A Making thi s chan ge of variable in Eq. (5.99) we then have

5 *(s) = B*(s)

s( 1 - p) s - A + AB*(s)

- (5.1 00)

Equat ion (5.100) is the desired exp licit expression for the Lapl ace transfor m of the distribution of total time spent in the M IGII system. It is given in terms of known quantities derivable from the initial statement of the pr oblem [namely, the specificati on of the servi ce-time distribution B( x ) and the par ameters A a nd x ). This is the second of the three equ at ion s th at we refer . to as the P-K tra nsform equ ati on. Fr om Eq. (5. 100) it is tr ivial to deri ve the Laplace tr an sform of the distr ibution of wai ting time , which we sha ll den ote by W*(s). We define th e PDF for e n's waiting time (in queue) to be W n(y), th at is, W n(y) ~ P[w n ~ y )

Furthermore , we define the limit ing quantities (as n ->- co) , Wn(y) and W n ->- Iii, so th at W(y ) ~ P[I~' :-:; y )

(5. 101) ->-

W(y)

(5.102)

The corresponding Laplace transform is

JV *(s)

~ L "e- s• dW(y)

= E[e- ';;;)

( 5. 103)

F ro m Eq . (5.94) we may de rive the dist ributio n of l~' from the d istribut ion of s and x (we drop subscri pt notation now since we a re con sidering equ ilibrium behavior). Since a customer' s service time is independent of his qu eueing tim e, we hav e th at s, the time spent in system for some customer, is the sum of two independent random vari abl es: l~' (his queueing time) and x (his service time). T hat is, Eq. (5.94) has the limiting for m (5.104) As derived in Appendix II the Laplace transform of the pdf of a random vari able that is itself the sum of two independent rand om vari able s is equal to the prod uct of the Lapl ace transforms for th e pdf of ea ch. Con sequently, we have 5 *(s) = W*(s) B*(s)

200

TH E QUEUE

M/G/I

Thus fr om Eq. (5.100) we obtain immed iat ely that

W *(s) =

s( 1 - p) s - A + AB*(s)

- (5.105)

Thi s is the desired expre ssion for the Laplace tran sform of the queu eing (waiting)-time distribution. Here we have the third equ ati on that will be referred to as the P-K transform equation . Let us rewrite the P-K transform equation for waitin g time as follows:

*

1- p

W (s)

=

.

1- p

[I - B*(S)]

(5.106)

_

sx

We reco gnize the bracketed term in the denominator of thi s equation to be exactly the Laplace transform associated with the density of residual service time from Eq. (5.1I). Using our special notation for residual den sities and . the ir tr ansform s, we define

B*(s) ;; 1 - B*(s) SX and are therefore permitted to write

(5.107)

* I - p W (s) - ------,'-----

(5.108)

- I - pB*(s)

Thi s observa tion is trul y amazi ng since we recognized at the outset that the problem with the M/Gfl analysis was to take account of the expended service time for the man in service. Fr om that investigat ion we found that the residual service time remain ing for the customer in service had a pdf given by b(x) , whose Laplace transform is given in Eq. (5.107). In a sense ther e is a poetic ju stice in its appearance a t thi s point in the final solution. Let us follow Benes [BENE 56] in inverting this transform in term s of these residu al service time den sities. Equation (5.108) may be expanded as the following power series : co (5.109) W*(s) = ( I - p)2: l[B*(s)]k P O

From Appendix I we know that the kth power of a Lapl ace tran sfor m corresponds to the k-fold con volution of the inverse tran sform with itself. As in Appendix I the symbol 0 is used to denote the conv oluti on opera to r, and we no w choose to den ote the k-fold convoluti on of a funct ion f (x ) with itself by the use of a parenthetical subscript as follows : d

f (k)(X) = ,f (x) 0 f ( x )0 .. ·0 f( x)

~

k-fold convo lut ion

( 5.110)

5.7.

DISTRIBUTION OF WAITING T IME

20 1

Us ing this notation we may by inspection invert Eq. (5.109) to obtai n the waiting-time pdf, which we de note by w(y) ~ dW(y)/dy; it is given by w(y)

'" (I =L

- p)pk bCkl(y)

(5.111)

k=O

Thi s is a most intriguing result! It state s th at the waiting time pdf is given by a weigh ted sum of conv olved residual service time pdf' s. The interesting observatio n is that the weightin g factor is simply (I - p)pk, which we now recognize to be th e pro bab ility distribution for the number of custo mers in an M/M /l system . Tempting as it is to try to give a physical explanation for th e simp licity of this result and its relation to M/M /I , no satisfactory, int uitive explan at ion has been found to explain th is dramatic form. We note that the contributio n to the waitin g-time den sity decreases geometrically with p in thi s series. Thu s, for p not especially close to unit y, we expect the high-o rde r terms to be of less an d less significance, and one pract ical application of this equ ati on is to provide a rapidly converging approximatio n to the density of waiting time. So far in th is section we have esta blished two principle results, namely, the P-K transfor m equatio ns for time in system and time in queue given in Eq s. (5.100) and (5. 105), respectively. In the previous section we have already given the first moment of these two rand om variable s [see Eqs. (5.69) and (5.70)]. We wish now to give a recurrence formula for the moments of t he waiting time. We denote the kth moment of the waitin g time E [wk ], as usual , by Irk. Takacs [TAKA 62b] has show n that if X i+! is finite, then so also are Iii, \\,2, . . . , Wi; we now adopt our slightly simp lified notati on for the ith moment of service time as follows : hi ~ x'. Th e Tak acs recurr ence for mula is •

k

T"" wk = - I.' I - P i='

(k)

b

~.'

i+ 1 ---IV

i (i

+

(5.112)

I)

where \\ ,0 ~ 1. Fr om this formula we may write down the first couple of moments for waiting time (and note that the first moment of waiting time agrees with the P-K formula): sb ; lii (= IV) = (5.113) 2(1 - p) -; • 3 (5.114) IV- = 2( + - }.b ---"-3(1 - p)

l,'r

In orde r to obtain similar moments for the total time III system, that is, s", we need merely take ad vant age of Eq. (5. 104) ; from this equ ation we find

E[5 k ] , which we denote by

(5.115)

202

THE QUEUE M /GfI

Using the bin omi al expansion and the ind ependence bet ween wai ting time and service time for a given customer, we find

? = i (~)Wk-ibi i=O

(5.11 6)

I

Thus calculating the moments of the wa iting time from Eq . (5.112) a lso permits us to calcul ate the moments of time in system from this las t equation. In Exercise 5.25 , we drive a relati on ship bet ween Sk and the mom ent s o f the number in system; the simplest of these is Little's result, a nd the others are useful genera liza tio ns. At the end of Section 3.2, we promised the reader th at we wo uld de velop the pd f for the time spent in the system for an M IM II queueing system. We are now in a position to fulfill th at promise. Let us in fact find both the distribution of waiting time and distribution of system time for cu stomers in M /M /I . Usi ng Eq. (5.87) for the system M /M fI we may calculate S*(s) from Eq. (5. 100) as follows : S*(s)

=

1-

p

(s

+ p ) L; -

s( l - p)

A + Ap/(s

S*(s) = p (1 - p) . s + p( l - p)

+ p)

p ) e- P(l - p) u

MIM II

(5.117)

y ~O

M IM I 1

- (5.118)

~

M IMII

- (5.1l 9)

y

e-p(l- p) u

0

Simil arly, from Eq, (5.105) we may obtain W* (s) as

W*(s) =

s( 1 - p)

s - A + i.p/(s (s s

+ It)

+ p )(1 - p) + (p - ;.)

= (I

_ p)

+

T hi (

to '

reF Re (5. frc d i: tir th (5 m SI

(5.120)

Before we ca n invert thi s we mu st place the right-ha nd side in proper form , namely, where the numerat or polyn omi al is of lower degree th an the denomi nator. We d o this by d ividin g out the constant term a nd ob ta in

W*(s)

Imp F ro

The cor responding PDF is given by S(y) = I -

ApF

]

Th is equat ion gives the Laplace tr an sform of th e pdf for time in the system which we den ote , as usu al , by s(y) ~ dS(y) ldy. Fortunately (as is usual with the case M /M /I) , we recogni ze the inver se of thi s tr an sform by inspection. Thus we have immediat ely that s(y) = p( 1 -

Th is we I

H

• p

;.(1 - p) s + p(l - p)

(5.12 1)

a t'

5.7.

DISTRIBUTION OF WAITING TIME

203

I" (y )

o

y

Figure 5.9 Waiting-time distribution for MIMI !. This exp ression gives the Lap lace transform for the pdf of waiting time which we denote, as usual , by w(y) ~ dW(y)/dy. From ent ry 2 in Table 1.4 of Appendix I, we recogn ize that the inverse transform of ( I - p) mu st be a n impulse at the origin ; thus by inspection we have w(y) = (1 - p)uo(y)

+ A(I

- p)e-· 11- p l •

y ~0

M/M /I

- (5.122)

From this we find the PDF of waiting time simply as W(y) = 1 - pe-·(l-pl .

y ~ 0

M/M/I

- (5.123)

This distribution is sh own in Fi gure 5.9. Ob serve that the probability of not queueing is merely I - p ; compare tbi s to Eq . (5.89) fo r the probabil ity that if = O. Clearly, they are the same; both represent the probability of not queueing. This also was found in Eq. (5.40). Recall further th at the mean no rmalized queueing time was given in Eq. (5.73); we obtain the same answer, of course, if we calcu late thi s mea n value fr om (5.123). It is interesting to note for M/MjI that a ll of tbe interestin g distribution s a re mem oryless: this applies not only to the given interarrival time and service time processes, but also to th e distribution of the number in the system given by Eq . (5.89), the pdf of time in the system given by Eq . (5.119), and the pdf of waiting time* given by Eq . (5.122). It turns out th at it is possible to find the density given in Eq . (5.118) by a more direct calculation , and we display this method here to indicate it s simplicity. O ur point of departure is our early result given in Eq . (3.23) for the p rob ability o f finding k cu stomers in system up on arrival , namely ,

h

=

(I - p)pk

(5.124)

• A simple exponential form for the tail of the waiting-t ime distribution (that is, the probabilities associated with long waits) can bederived for thesystem M/G /1. We postpone a discussion of this asymptotic result until Chapter 2, Volume II, in which we establish this result for the more general system GIG/I.

204

THE QUEUE

M/G/I

We repeat agai n that thi s is the same expression we foun d in Eq. (5.89) and we know by now that this result app lies for all po ints in time. We wish to form t he Lapl ace transform of the pdf of total time in the system by considering thi s Lapl ace transform conditioned on the number of customer s found in th e system upon arrival of a new customer. We begin as generally as possible a nd first consider the system M IGII. In particular , we define the condit iona l d istribution

I = P[customer's total

S (y k )

I

time in system j; y he finds k in system upon his arrival]

We now define the Lapl ace transform of this conditional density

I

Jo e- sv d5(y I k )

t. ( ""

5 *(s k ) =

(5.125)

Now it is clear that if a customer finds no one in system upon his a rrival, then he must spend an amount of time in the system exactly equal to his own service time , and so we have S *( s I 0)

=

B *(s )

On the other hand , if our arriving customer finds exactly one customer ahead of him , then he remains in the system for a time equal to the time to finish the man in service, plu s his own service time; since these two int ervals are independent, then the Laplace transform of the density of this sum must be the product of the Lapl ace tr ansform of each density, giving S *(s I I)

=

8 *(s)B *(s )

where B *(s) is, again, the tran sform for the pdf for residual service time. Similarly, if our arriving customer finds k in front of him , then his total system time is the sum of the k service times associated with each of t hese customer s plus his own service time. Th ese k + I rand om variable s are all independent, and k of them are dra wn from the same distributio n S ex). Thus we have the k-fold product of B *(s) with B*(s) giving

I =

5 *(s k )

[B*(s)jk8*(s)

(5. 126)

Equ ati on (5.126) hold s for M IG II. Now for our M/M /I problem , we have that B* (s) = I'-/ (s + 1'-) and, similarly, for B*(s) (memoryless); thus we have

I = ( -I'--)k+'

5*(s k )

s + 1'-

(5.127)

5.7.

DISTRIB UTION OF WAITI NG TI ME

205

I

In order to obtain S *(s) we need merel y weight the transform S *(s k ) with the pr obability P» of our customer finding k in the system upon his arrival, namely , cc

S*(s)

= L 5*(s I k)Pk k=O

Substituting Eqs. (5.127) and (5.124) into this last we have

S *(s)

co ( =L -P-)k+l(I

k~O S + P

s

!l(I -

p)

+ p(I

- p)

- p)p'

(5.128)

We recogni ze that Eq . (5.128) is identical to Eq . (5.117) and so the remaining steps leading to Eq . (5.118) follow immediately. This demonstration of a simpler method for calcul ating the distribution of system time in the MIMII queu e demon strates the followin g import ant fact: In the development of Eq. (5. 128) we were required to consider a sum of random variables, each distributed by the same exponential distributi on ; the number of terms in that sum was itself a rand om "variab le distributed geometrically. What we fou nd was t hat this geomet rical weighting on a sum of identically dis tributed exponential random vari ables was itself expo nential [see Eq . (5.118)]. This result is true in general, namely , that a geometric sum of exponential random variables is itself exponentially distributed. Let us now carry out the calculations for our M/H./I example. Using the expr ession for B*(s) given in Eq. (5.90), and applying this to the P-K transform equation for waiting-time den sity, we have

*

4_s-,-(I_----'-p - -'--)("-s--:+'----'l)-'-(s_+ ..:...-2_l )' --_ - 4(s - l )(s + l )(s + 2).) + 8).3 + 7l 2 s

W (s) -

Thi s simplifies up on fact oring the den ominator , to give

*

I_-----'-p.:..o )(s--,+_).-,-,)(:.. . s.-: +_ 2_),.:. .) [s + (3/2)l ][s + (112»).]

.0....(

W (s) = -

Once again, we must divide numerator by den ominator to reduce the degree of the numerator by one, giving

*

W (s)

= (I

- p)

] + _ ).-:,(_1 _--,-p-,--)['--s.-:+_ (,---51,---4,---»).-,--[s + (3/2»)'][s + ( 1/2»).]

We may now carry out our partial-fr acti on expansion:

W*(s)

= (1 -

1 p>[ + s

}./4

3}./4

]

+ (3/2»). + s + ( 1/2) )'

== 206

MIG/!

THE QUEUE

T his we may now invert by inspection to o bta in the pd f for waiting time (a nd recalling that p = 5/8) : . 3 () wy () = - u y

8

3). -(3! 21". +e + -9), e- (I ! 2)".

0

32

32

y;:::O

(5. 129)

This complete s o ur d iscussion o f the waiting-time an d system-time d istr ibution s fo r M/G/1. We now introduce the bu sy peri od , an imp ortant stochas tic process in queueing systems.

5.8. THE BUSY PERIOD AN D ITS DURATION We now ch oo se to study queueing systems from a different po int of view. We make the observation tha t the system passes through a lternating cycles o f busy peri od , idle peri od , busy pe riod, idle period, and so on . Our purpose in this section is to deri ve the distribution for the length of th e idle peri od a nd the length of the busy peri od for the M/G/) queue. As we a lrea dy understand , the pertinent sequences of rand om va ria bles that drive a queueing system a re the instants of arri val a nd the seq uence of service times. As usual let

C; 7"n

In Xn

=

the nth customer

= arrival time o f C;

=

7"n -

7"n _I

=

interarrival time betwee n C n _ I and C;

= serv ice time for C;

We now recall the imp o rtant sto chastic process V( I) as de fined in Eq . (2.3) : V(t) ~ the unfini shed work in the system at time I ~ the rem aining time req uired to empty the system of all

customers present a t time

I

This functi on V (I ) is appropriately refe rred to as the unfinished work at time I since it represents the interval of time th a t is required to empty the system completely if no new customers are ail owed to enter a fter the insta nt I . Th is funct ion is sometimes referred to as the "vi rtua l" waiting time a t time I since, for a first-c ome-first-served system it repre sents how lon g a (virtual) cu stome r would wait in queue if he entered at time I ; however , thi s waitin gtim e inte rpretation is goo d only for first-c ome-first-served disciplines, whereas the un finished work interpretation applies for all discipline s. Beh avior of this functi on is extremely important in understand ing qu euein g systems when one stud ies them from the point of view of the bu sy peri od . Let us refer to Figure 5.1Oa , which shows the fashi on in which bu sy pe riods alternate with id le pe riods. The busy-pe riod duration s a re denoted by Y" Y 2 , Y3 , • •• and the idle period du rations by I" 12 , • • •• Cu st omer C,

5.8.

207

TH E BUSY PERIOD AN D ITS DURATION

U(I)

(a>

~ I~

I

r-----;y,~ 11-+ yd 1

21 6

TH E QU EUE

M IGII

The busy peri od pdf given in Eq . (5.145) is much more complex than we would have wished for this simplest of interesting queuein g systems ! It is ind icati ve of the fact that Eq. (5. I37) is usually unin vertible for more general service-time distributions. As a seco nd exampl e, let' s see how well we can do with our M/H 2 /1 example. Using the expression for B* (s) in our funct ional equat ion for the busy period we get G*(s) = 8). 2 + 7).[s + }. - )'G*(s)] 4[s + A - }.G*(s) + A][S + A - }.G*(s) + 2A] which lead s dire ctly to the cubic equation 4[G * (S)]3 - 4(2s

+ 5)[G* (s)J2 + (4s + 20s + 31 )G* (s) 2

(15

+ 7s) = 0

Th is last is not easily solved and so we stall at this po int in our attempt to invert G* (s). We will return to the functional equati on for the busy period when we discuss pri orit y queueing in Chapt er 3, Volume II. Th is will lead us to the concept of a delay cycle, which is a slight generalization of the busy-period analysis we have j ust carried out and greatly simplifies priority queueing calculations. 5.9. THE NUMBER SERVED IN A BUSY PERIOD In th is section we discuss the distribution of the number of customers served in a busy period. Th e development parallels that of the previou s section very closely, both in the spirit of the der ivation and in the nature of the result we will obtain. Let N b p be the number of customers served in a busy period . We are interested in its probab ility d istribu tion Indefined as

In =

P[ N b p

=

II]

(5.148)

The best we can do is to obt ain a functi onal equati on for its z-transform defined as (5.149) The term for II = 0 is omitted from this definitio n since at least one customer must be served in a busy peri od. We recall that the random var iable ii repre sent s the number of arrivals during a service peri od and its z-transform V(z) obeys the equation deri ved earlier, namely , V( z)

=

B*(A - Az)

(5.150)

Proceedin g as we did for the durati on of the busy period , we condition our argument on the fact that ii = k , that is, we assume that k customers arrive

5.9.

THE NUMBER SERVED IN A BUSY PERIOD

217

during the service of C1 • Moreover, we recognize immediately that each of these arrivals will generate a. sub-busy period and the number of customers served in each of these sub-busy periods will have a distribution given by fn. Let the random variable M, denote the number of customers served in the ith sub-busy period. We may then write down immediately £[zSbP

I iJ =

k] = £[z1+J1I+.1[,+ · · .+.11,]

and since the M, are independent and identically distributed we have

I iJ = k]

£[ZSbP

k

= z II £[z ·lI,] i= 1

But each of the M i is dist ributed exactly the same as N b p and, therefore, E[ZSb P

I iJ = k] = z[F(z)]k

Removing the condition on the number of arrivals we have 00

F(z)

= L E[z.Y bP I iJ =

k]P[iJ

=

k]

k= O 00

= z

LP[iJ =

k][F(zW

k=O

From Eq, (5.44) we recognize this last summation as V(z) (the z-transform associated with iJ) with transform variable F(z); thus we have (5.151)

F(z) = zV [F(z)]

But from Eq. (5.150) we may finally write F(z)

=

Z8*[A - ).F(z)]

- (5.152)

This functional equation for the z-transform of the number served in a busy period is not unlike the equation given earlier in Eq . (5.137). From this fundamental equation we may easily pick off the moments for the number served in a busy period. We define the kth moment of the number served in a busy period as 11k • We recognize then h1

=

Flll(l)

= 8*(1)(0)[- AF(1)(I)]

+ 8*(0)

Thus which immediately gives us 1 h1 = - -

1- p

- (5.153)

218

TH E QUEUE

M/G/I

We further recogni ze . F(2)(l ) = h2 - hI Carrying o ut thi s computation in the usual way , we obtain the second moment and va ria nce of the number ser ved in the busy period: J1. =

2p(1 -

-

Uk

2

=

p)

(I -

+ A x +-1 p)3 1- p 2

2

p(l - p) + A2 ? (1 _ p)3

~--'-':"""';'---

- (5.154) - (5.155)

As an example we again use the simple case of the M/M jl system to solve for F(z) from Eq. (5.152). Carrying thi s out we find

F(z) = z

/l

+ A- AF(z) + A)F(z) + /lz =

/l AF2( z) - (/l Solving,

F(Z)=!..±1'[I-(I2p

0

1 J

4pz )1/ ( 1 + p)"

(5.156)

Fortunately, it turns o ut that the equatio n (5.156) can be inverted to obtain prob ab ility of having n served in the busy peri od:

In' the

- (5.157) As a seco nd example we con sider the system M/D/1. For thi s system we have hex ) = uo(x - x) and from entry three in Table 1.4 we ha ve immediately that B*(s) = e- ' z U sing thi s in our functi onal equ ati on we obta in

F(z)

= z e- Pe pF ( z )

where as usual p = AX. It is convenient to make the substitution u a nd H (u) = pF(z), which th en permits us to rewrite Eq. (5. 158) as

(5.158)

=

z pe: "

u = H(u)e-ll(u) The solutio n to th is equ ation may be obta ined [RIOR 62) a nd then our original fun ction may be evaluated to give

F(z) = i (np )n-I n= l 11!

e-n pz n

5.10.

FROM BUSY PERIODS TO WAITING TIMES

219

From this power series we recognize immediately that the distribution for the number served in the MIDII busy period is given explicitly by n- l

In

=

) ( .!!.f!....-e- np Il

!

- (5.159)

Fo r the case of a constan t service time we know tha t if the busy period ser ves II customers then it must be of durat ion nii , and therefore we may immediately write down the solution for the MIDfI busy-period dist ribution as [V/il( n p) n-l

G(y)

=L- n= l

It

!

e- np

- (5.160)

where [ylx] is the largest integer not exceedi ng ylx. 5.10.

FRO M BUS Y PERIODS TO WAITING TIMES

We had mentioned in the ope ning paragraphs of this chapter that waiting times could be ob tai ned from the busy-period analysis. We are now in a position to fulfill tha t claim. As the reader may be aware (and as we shall show in Chapter 3, Volume II), whereas the distribution of the busy-period duration is independent of the queueing discipline, the distribution of waiting time is strongly de pendent upon order of service. Therefore, in this section we consider on ly first-come-first-served MIG/! systems. Since we restrict ourse lves to thi s discipline , the reordering of customers used in Section 5.8 is no longer permitted. Instead, we must now decompose the busy period into a sequence of interva ls whose length s are dep endent random variables as follows. Co nsider Figure 5.12 in which we show a single busy period for the first-come-first-served system [in terms of the unfinished work Vet)]. Here we see that customer C, initiates the busy peri od upon his arrival at time T , • The first interva l we cons ider is his service time Xl> which we denote by X o ; during this interval mo re custome rs arrive (in this case C2 and C 3 ) . All those customers who arrive during X o are served during the next interva l, whose duration is X, and which equals the sum of the service times of all a rrivals du ring Xi> (in this case C2 and C 3 ) . At the expiration of X" we then create a new inte rval of duration X 2 in which all customers arriving during X , are served , and so on. Thus Xi is the length of time required to service all those customers who arrive during the previous interval whose duration is Xi_l' If we let n i denote the nu mber of customer arriva ls du ring the interval Xi' the n n i customers arc served during the interval Xi+l' We let no equal the number of custome rs who arrive du ring X o (the first customer's service time).

220

T HE Q UEUE

MiGfl

U(t )

c, OL....-+_ _--->:,---_

_

---'':-_~

~

_

_.._:I~._

'~X" ' +-x.--+~+-x,-+x.~ Figure 5.1 2 The busy period: first-come-first-served. Thus we see that Y, the duration of the total busy period , is given by

""

Y = LXi ;= 0

where we permit the possibility of an infinite sequence of such inter vals. Clearly, we define Xi = 0 for those intervals that fall beyond the termination of this busy period ; for p < I we know that with pr obability I there will be a finite i o for which Xio (and all its successors) will be O. Furthermore, we know that Xi+! will be the sum of », service interv als [each of which is distributed as B(x)].

We now define Xi(y) to be the PDF for Xi' that is,

"

X i(y) = P[X i

:s;;

y]

and the correspondi ng Lapl ace transform of the assoc iated pdf to be

X,*(s)

~ 1"" e-

SV

dXi(y)

= E[e- ' X'] We wish to derive a recurrence relati on am ong the X,*( s). Th is derivat ion is muc h like that in Section 5.8, which led up to Eq. (5. 137). Th at is, we first condition our transform sufficiently so that we may write it down by inspection; the cond itions are on the interval length X i _ l and on the number of

I

5. 10.

221

FROM BUSY PERIODS TO WAITI NG T IMES

a rri vals n i - 1 during that interval, that is, we may write

I

E [e- 'X ; X i - -I -- Y, " i - l -- n ] -- [B*(s)]" Thi s last follows from our con volution property leading to the multiplicati on of tr an sforms in t he case when the va ria bles are independent ; here we have n independent service times, all with identical distributions. We may uncondition first on n : 00 (A )" E[e- ' x ; I X i_1 = Y] = I ..JL e-J.V[B*(s)]" n=O n ! and next on Y:

Clearly , the left-hand side is X;*(s); evaluating the sum on the right-hand side lead s us to

X i*(s)

=

f.-0 OO

•

e- [J.-J.B (, )]. dXi_1(y )

Thi s integra l is recogni zed as the tran sform of the pd f for Xi-I> na mely,

X i*(s)

=

Xi~l [A

- AB*(s)]

(5.161)

Thi s is the first step. We now condition our calculations on the event that a new (" tagged") arriva l occurs during the busy period and , in particular, while the busy peri od is in its ith interval (of duration X ;). From our ob servations in Secti on 4.1 , we kn ow th at Poisson arrivals find the syste m in a given sta te with a pro bab ility equ al to the equilibrium probability of th e system bein g in th at state. N ow we kn ow that if the system is in a busy period, then the fracti on of time it spends in th e interval of du rati on Xi is given by E [Xi]1E[ Y] (t his can be made rigorous by renewal the ory a rguments). Con sider a custom er who arrives during an interval of duration X i. Let his waiting time in system be de not ed by IV; it is clear that th is wait ing time will equal the sum of the remaining time (residua l life) of the ith interva l plus the sum of the service times of all j obs who arr ived before he did during the ith interval. We wish to calculat e E [e- ';;' i], which is the tr an sform of the waiti ng time pdf fo r a n a rrival during the ith interval; again , we perform thi s calculation by cond ition ing on the three variables Xi' Y i (defined to be the residu al life of th is ith interval) a nd on N , (defined to be t he number of a rrivals during the it h interval but pri or to our customer's arrival-that is, in the interval Xi - Yi). Thus, using our co nvo lutio n property as before , we may write

I

E[e- 'WI i , X i

=

y, Y,

=

v' , s,

=

/I]

=

e- '" [B*(s)r

222

THE QUEUE

M /G/l

N ow sinc e we ass ume th at n cu st omers have arrived during an interval of duration y - y' we uncondition on N , as follows: E[e: .;;;

I I.,

X i = y , Yt

-_

y' ' ] -_

e-'"

~

.L. n :o O

[A(Y - y' )] n e-,t(' - " ' [B*(s) ]n n!

= e- s J/' - l (V- lI' )+ A( lI- Y' ) B - ( s )

(5. 162)

We ha ve a lready observed that Y i is the residual life of the lifetime X i' Equation (5.9) gives the joint density for the residual life Yand lifetime X; in that equation Yand X play the roles of Y i and X; in our problem. Therefore, replacing/ ex) dx in Eq. (5.9) by dXi(y) a nd noting that y and y' ha ve replaced x and y in that development , we see that the j oint density for X i and Y i is given by dXJy) dy'/E[Xi] for 0 ::s; y' ::s; y ::s; 00 . By means of this joint density we may remove the condition on Xi and Y i in Eq . (5.162) to ob ta in

E[e- ' ;;; I i]

= =

r'" r'

e-['-HW·(, »).~-P-AB·( ')l> dX / y) dy' /E[ X ;]

Jy=o J JI'= O

'" 1._0

[e- " - e-[ ,t-,tB'(,»).]

[- s

+ A-

AB* (5)]E [X ;]

dX(y) •

These la st integrals we recognize a s tr an sforms a nd so

I

E[e- ' ;;; i] = X/(5) - X/(J. - ;.8*(5» [- 5 + I. - 1.8*(5) ]E[X ;] But now Eq . (5.161) permits us to rew rite the seco nd o f th ese tr an sforms to ob ta in .

- ,W I I].

E [e

X7+1(5) - X ;*(5) [5 - I. + A8*(5)] £[ X i ]

= ----"-'-'--'-----'---'-'---

Now we may rem o ve the cond ition o n our arriva l entering during the ith interval by weighting th is la st ex pression by the probability th at we have formerly expressed for the occurre nce of th is event (still condition ed on o ur ar riva l en tering during a bu sy per iod) , a nd so we have

E[e- ' WI enter in b usy period ] =

I E[e- ";;; I i] E[X E [ Y]

I

i]

i- O

[5 _"I.

1

:L '" [v* ,i \ · ... ( S )

+ I."8*( 5)]E[ Y] .'_- 0

, ,1

-

X .*(s)] ,

5.11.

CO MBINATOR IAL METHO DS

223

Th is last sum nicely collap ses to yield 1 - X o*(s) since Xi*(s) = I for those inte rvals beyond the busy period (recall X i = 0 for i ~ i o) ; also , since X o = x" a service time, then X o*(s) = B *(s ) , and so we arrive at

I

E[e-S;;; enter

.

In

b usy peno . d] =

1 - B*(s) + }.B*(s)]E[Y ]

[s - }.

Fr om pre viou s con sider ation s we know that the probability of an a rrival ente ring during a busy per iod is merely p = Ax (and for sure he mu st wait for service in such a case); further, we may evaluate the average length of the busy peri od E[ Y] either from our pre vious calcul ati on in Eq . (5. 141) o r from elementary considerations ' to give E [Y] = 'i/ (l - p). Thus, unc onditioning on an arrival finding th e system bu sy, we finally have E[e- SW]

= (I -

-

p)E[e- SWI en ter in idle period]

+ pE[e- s"- Ienter in busy period]

[1 - B*(s)](1 - p)

= ( 1 - p)

+ p [s _ A + AB*(s)]'i

= ----'s'-'-(I=--------'---p)~

(5.163) s - A + AB*(s) Voila! T his is exactl y the P-K tran sform equation for waiting time , namel y, W *(s) ~ E[e- siD ] given in Eq. (5. 105). Thus we have shown how to go from a busy-period analysis to the calcul ation of waiting time in the system. Thi s meth od is rep orted up on in [CO NW 67] and we will have occasio n to return to it in Chapter 3, Volu me 11 .

S.U.

COMBINATORIAL METH ODS

We had menti oned in the opening remarks of this chapter th at consideration of rand om walks and combinat ori al meth ods was applica ble to the study of th e M/G!I qu eue. We take thi s oppo rtunity to ind icate so me asp ects of th ose methods. In Figure 5.13 we have reproduced Vet) from Figur e 5.1Oa. In additio n, we have indic at ed th e " ra ndom walk" R (t) , which is the same as Vet) excep t th at it does not satura te at zero but rat her co ntinues to decline at a rat e of I sec/sec below the hori zontal axis ; of course, it too tak es vertica l j umps at the custo mer-arriva l insta nts. We intro d uce th is diagram in orde r to define wha t are known as ladder indices. The kth (descending) ladder index • The following simple argument ena bles us to ealculate E[ Y]. In a long interval (say, I) the server is busy a fraction p of the time. Each idle per iod in M /G /l is of average length I f}. sec and therefore we expect to have ( I - p)I /(l I).) idle periods. This will also be the number of busy periods, approxi mately; therefore, since the time spent in busy perio ds is pI , the average durat ion of each must be pl l ).I( 1 - p) = :el(l - p) . As I ~ 00 , this ar gument becomes exact.

224

TH E Q UEUE

M /G !I

Figure 5.13 The descending ladder indices. is defined as the instant when the random walk R (t) rises from its kth new minimum (and the value of this minimum is referred to as the ladder height). In Figure 5.13 the first three ladder indices are indicated by heavy dots. Fluctuation theory concerns itself with the distribution of such ladder indices and is amply discussed both in Feller [F ELL 66] and in Prabhu [PRAB 65] in which they consider the applications of that theory to queueing proce sses. Here we merely make the obse rvation that each ladder index identifies the arrival instants for those customers who begin new busy p eriods a nd it is th is observation that makes them interesting for queuein g theory. More over, whenever R (t ) drops below its previous ladder height then a busy peri od terminates as shown in Figu re 5.I3. Thu s, between the occurrence of a ladder index and the first time R (t) drops below the corresponding ladder height, a busy period ensues and both R (t) and U(t ) have exactly the same shape, where the former is shifted down from the latter by an am ount exactly equal to the accumulated idle time since the end of the first busy peri od . One sees that we a re quickly led into meth ods from combinatorial theory when we deal with such indices. In a similar vein, Tak acs has successfully applied combinatorial theory to the study of th e busy period. He consider s this subject in depth in his book [TAKA 67] on combinatorial methods as applied to queuein g theory and develops, as his cornerstone , a generali zati on of the classical ballot theorem. The classical ballot theorem concerns itself with the counting of votes in a. two-way conte st involving candidate A and candidate B. Ifwe assume th at A scores a votes and B scores b votes and that a ;:::: mb , where m is a nonnegati ve integer and if we let P be the probability that through ou t the

/

5.11.

COMBINATORIAL METHODS

225

counting of votes A continually leads B by a factor greater than m and further , if all possible sequences of voting records are equally likely, then the classical ballot theorem states that a - mb P = =-----:..:.:::: (5.164)

a+b

This theorem originated in 1887 (see [TAKA 67] for its history). Takacs generalized thi s theorem and phrased it in terms of cards drawn from an urn in the following way. Consider an urn with Il cards, where the cards are marked with the nonnegative integers k I , k 2 , • • • , k ; and where n

L k, =

k ~

Il

i= l

(that is, the ith card in the set is marked with the integer k ;). Assume that all cards are drawn without replacement from the urn. Let o; (r = I, . . . , Il) be the number on the card drawn at the rth drawing. Let

Il

Nr =

VI

+ V2 + .. . + V

T

r = I, 2, . .. ,

11

NT is thus the sum of the numbers on all cards drawn up through the rth draw. Takacs' generalization of the classical ballot theorem states that -

P[N T

< r for

all r

=

1,2, .. . , 11]

11 - k =-

(5.165)

11

The proof of this theorem is not especially difficult but will not be reproduced here . Note the simplicity of the theorem and , in particular, that the probability expressed is independent of the particular set of integers k; and depends only upon their sum k . We may identify o; as the number of customer arrivals during the service of the rth customer in a busy period of an fovl /G/l queueing system. Thus FlT + I is the cumulative number of arrivals up to the conclusion of the rth customer's service during a busy period . We are thus involved in a race between FlT + I and r : As soon as r equals FlT + I then the busy period must terminate since, at this point, we have served exactly as many as have arrived (including the customer who initiated the busy period) and so the system empties. If we now let N b P be the number of customers served in a busy period it is possible to apply Eq . (5.165) and obtain the following result [TAKA 67]:

P[N b p

= III = -1 P[Nn =

11 -

I]

(5.166)

Il

It is easy to calculate the probability on the right-hand side of this equation

since we have Po isson arrivals: All we need do is condition this number of

226

TH E QUEUE

M/G/I

arriva ls on th e durati on of the busy period , multiply by the p robabi lity that 11 service interva ls will, .in fact , su m to thi s length and then integrate ove r all p ossible lengths. Thus

P[N n

= II -

I]

=

( '" (AV)' - l -.l . e "bl.l(y) dy . 0 (II - I)!

(5 . 167)

where bl. l(y) is the n-fold convolution of bey) with it self [see Eq . (5. 110)] a nd repre sen ts th e pd f for the sum of n independent random varia bles , where each is drawn from th e co mmon den sity bey). Thus we a rr ive at a n expl icit expression for the pr obability d istr ibution for the number served in a bu sy period:

P[N b p = II] =

'" I O

(;ly)n-l - .l" - - e bl. l(y) dy

- (5.168)

il!

We may go further and ca lcula te G(y) , the distributi on of the bu sy period, by integrating in Eq . (5.168) o nly up to so me point y (ra ther than 00) and then summing over a ll p ossible numbers served in the bu sy per iod , th at is,

and so , G(y )

=

"I I

co

O .~ l

e"

.lz(i·X)· - l

- - b1nl(X) d x II !

- (5.169)

Thus Eq . (5.169) is a n exp licit expression in terms of known quantities for the distribution of the bu sy period a nd in fact may be used in place of the expression given in Eq . (5. 137), the Lapl ace tr ansform of dG(y)/dy. Th is is the expre ssion we had p ro mis ed earlier, a ltho ug h we ha ve expressed it as a n infinite summati on ; nevertheles s, it does pr ovide the ability to a pproxi ma te the busy-peri od distribution numericall y in a ny given situa tion. Similarl y, Eq . (5.168) gives an explicit expression for the number served in th e bu sy period . The reader may have o bserved th at ou r study of the busy per iod has reall y been th e study of a transient phenomenon a nd thi s is one of the reasons th at t he de velopment bogged d own . In the next sectio n we con sider certain aspects of the transient solution for M/G/I a bit fur th er.

5.12.

THE TAKACS INT EGRO DIF F ERENTIAL EQUATION

In th is section we ta ke a cl oser look at the un finished wo rk and de rive the forward Kolm ogoro v equation for its time-dependent beh a vior. A mom en t' s re flection will reveal the fact th at th e unfini shed wo rk U(t) is a co nti nuoustim e continuou s-state Mark ov pr ocess that is subject to di scont inu ou s

!

5.12.

227

THE TAKACS INT EGRODlffERE NTIAL EQU ATION

chan ges. It is a Markov process since the entire past history of its motion is summarized in its current value as far as its future behavior is concerned. That is, its ver tical discont inuities occur at instants of customer arrivals and for M/G/l these a rrivals form a Poisson pr ocess (therefore, we need not know how lon g it ha s been since the last arrival), and the current value for Vet) tells us exactly how much work remains in the system at each instant. We wish to deri ve the probability distribution funct ion for Vet), given its initial value at time t = O. Accordingly we define F(w, t ; wo) ';; P[U(t ) ::;;

wi U(O) =

wo]

(5.170)

This notation is a bit cumbersome and so we choose to suppress the initial value of the unfinished work a nd use the shorthand notation F(w, t) ~ F(w, I ; 1"0) with the understand ing that the init ial value is 11'0' We wish to relate the probability F(w, t + D.I) to its possible values at time I. We observe that we can reach th is sta te from I if, on the one hand, there had been no arri vals during this increment in time [which occurs with probab ility I - AD.t + o (D.t)] and the unfinished work was no larger than II' + D.I a t time t : or if, on the other hand , there had been an arrival in this int erval [with probabil ity AD.t + o( D.t) ] such th at the unfinished work at time I, plus the new increment of work brought in by this customer, together do not exceed I". These ob servati on s lead us to the followin g equation : F(w, 1+ D.I) = ( 1 - A D. I)F(w

+ D.I , I) + AD.t

aF( x , I) B(w - x ) -d x + O(D. I) (5.171 ) x~o ax w

J

Clearly, (a F(x , t) jax) dx ~ dFt», t) is the pr obability that at time I we have x < Vet ) ::;; x + dx. Expanding our distribution functi on on its first vari able we have aF(w, t) D.I + O(D.I ) F(w + D.I, I) == F(w, t) + aw Using thi s expan sion for the first term on the right-hand side of Eq . (5.171) we obtain F(w, t

+ D. t) =

F(w, t)

+ aF(w, I) D.I _

AD.t[ F(W, t)

aw

+ aF(w, t) D.tJ aw

+ i. D.tL : oB(W -

x) dxF(x , I)

+ O(D. I)

Subtracting F(w, r), dividing by D.t , and passing to the limit as D.t finally ob tain the Taka cs integrodifferential equation for V( t) : . aF(w, t)

aF(w, t)

at

ow

--'----'- =

.

- i.F(w, t)

I

+A

W

B(w - x) dxF(x, t)

x-o

--->

0 we

- (5.172)

I

228

M/G/I

TIl E QUEUE

T ak ac s [TAKA 55] deri ved thi s equation for the more genera l case of a nonhom ogene ou s Poisson process, namely , where th e a rriva l rat e .1.(1)depends up on I. He sho wed t ha t this equ ation is good for almost all W ~ 0 an d 1 ~ 0 ; it d oes 1/01 hold a t th ose w a nd 1 for which of(lV, 1)/OlV has an accumulati on of probability (na mely, an impulse) . This occurs , in particular, a t 1\' = 0 a nd would give rise to the term F(O , I)uo(w) in of(lV, I)/OW, whereas no other term in the equation contains such an impulse. We may gai n more information from the Takac s integr odifferential equation if we transform it on the variable W (a nd not on t) ; thus using the tr an sform variable I' we define

W *'(r, I)

~fo~ e-

TW

dF w(w, I)

(5.173)

We use t he notation (*.) to denote transformation on the first , but not the second a rgument. The symbo l Wis ch osen since, as we shall see, lim W*'(r, I) = W *(r) as 1 ->- 00 , which is our former tr ansform for the waitin g-time ' pdf [see, for example, Eq . (5. 103)]. Let us examine the tran sform of each term in Eq, (5. 172) sepa ra tely. First we note th at since F(w, I) = S~ '" d Fi», I), then from entry 13 in Table 1.3 o f Appendix I (a nd its footnote) we mu st ha ve

.

'" F(w, I)e

- TW

J.o

dw =

W*'(r, I)

+ F(O- , I)

--'---'-----'-~ I'

and , sim ilarl y, we ha ve

J.

'" B(w)e

- TW

dw =

B*---,,(--,_ r)---,+,-B(,O~-) I'

0-

H owever , since the unfini shed work and the ser vice time are both nonnegat ive random varia bles , it mu st be that F(O-, I) = B (O-) = 0 a lways . We rec ogni ze th at th e last term in the T ak acs inte grodifferential equa tion is a con volution between B(w) an d of(W,I)/O W, a nd therefore th e tr an sform o f th is co nvolution (includi ng the con stant multiplier A) mu st be (by properties 10 a nd 13 in that sa me tabl e) }.W* ·(r, I)[B *(r) - B (O- )]Ir = }.lV*·(r, I)B*(r)/r. N ow it is clear that the tr an sform for the term of(w, I)/OW will be W* '( r, I) ; but thi s tra nsfo rm includes F«(j+- , I), the tr ansform of the impulse locat ed a t the o rigin for thi s partial deri vative, and since we kn ow th at the T ak acs int egr od ifferential equati on does not contain that impulse it mu st be subtracted out. Thus, we ha ve from Eq . (5.172), I )ow*'(r , I)

(r

01

= IV

*, ( I',

+

I) - F(O , r) -

i.W *'(r, I) r

W *'(r, I)B*(r) + A-~~--'--'r

';

I I

(5.I 74)

I

J

5.12.

229

THE TAKACS INTEG RODl FFERENTIAL EQUATION

which may be rewritten as

oW *"(r, t) " * *" + --o-'-t-'---'- = [r - A + .1.B (r) ]W (r, t) - rF(O , I)

(5.175)

Takacs gives the solution to thi s equ ati on {p, 51, Eq . (8) in [TAKA 62b]}. We may now transfor m on o ur seco nd vari able 1 by first defining the double transform

I"

(5.176)

1

(5.177)

F**(r, s) =t. J o e-' tW *"(r, t) dt We also need the definiti on

r,*(s) ~

00

e- stF(O+, t) dl

We may now transform Eq . (5.175) usin g the tran sform pr operty given as entry II in Table I.3 (and its foot note) to obtain

sF**(r, s) - W*"(r ,O-)

+ .1.B*(r)]F**(r , s) -

=

[r - ;.

=

W*"(r, 0-) - rFo*(s) s - r + ). - .1.B*(r)

rF o*(s)

From thi s we obta in

F**(r, s)

(5.178)

The unknown funct ion Fo*(s) may be determined by insisting th at the transform F**(r, s) be an alytic in the region Re (s) > 0, Re (r) > O. Thi s implies th at the zeroes of the numerator and denominator must coinc ide in th is region ; Benes [BEN E 56] has shown th at in th is region 1] = 1](s) is the un ique root of the denominator in Eq. (5.178). Thus W *'(7J , o--) = 1]Fo*(s) and so (writing 0-- as 0), we have

F**(r s) = W*"(r, O) - (r!7J)W*"(1] . 0) , s - r + A- i.B*(r)

(5.179)

Now we recall that V (O) = IVO with probability one, and so from Eq . (5. 173) we have W *' (r ,O) = e-r u: o. Thus F**(r , s) takes the final form

F**(r, s)

=

(rl1])e-~ Wo

;.B*(r) - i.

- e- rwo

+r-

s

- (5.180)

We will return to this equati on later in Ch apter 2, Volume II , when we d iscuss the diffusion ap proxi matio n. For now it beh ooves us to investigate the steady-sta te value of these functions ; in particular, it can be shown that F(w, t) has a limit as t ->- 00 so long as p < I , and thi s limit will be independent of the initi al co ndition

I

230

TH E QUEUE

M/G/ l

F (O, w) : we d en ot e this .lirnit by F (lI') = lim F ( lI', r) as t ---+ CIJ, a nd from Eq . (5. 172) we find th at it mu st sa tisfy the following equ at ion:

d F(w)

-- = dw

l WB(w -

U(w) - A

x) d F( x )

(5. 181)

=0

Furtherm ore , for p < 1 then W *(r ) ~ lim W *' (r , t) as t ---+ CIJ will exist and be independent of-the init ial distribution . Taki ng the tr an sform o f Eq. (5.181) we find as we did in deri ving Eq . (5. 174) + W*( r) - F (O )

i. W *(r)

).B*(r )W* (r )

r

r

= -- _ _

....:....:...---='-'

where F (O+) = lim F (O+, t) as t ---+ CIJ and equals the p robability that the unfini shed wo rk is zero. Th is last may be re written to give rF(O+)

W*(r ) = - - ---'--'-r - ). + ).B*(r)

H owe ver , we require W* (O) = 1, which requ ires th a t the unkn own consta nt p. Finally we ha ve

F (O+) ha ve a va lue F (O+) = I -

W* (r)

= r -

r(1 - p) i. AB*(r)

+

(5 . 182)

which is exactly the Pollaczek-Khinchin transform equation for wa iting tim e as we pr omi sed! This completes our discu ssion of the system M/G/l (fo r the time bein g). Next we con sider the "companion " system, G /M /m. REFERENCES BENE 56 cONW67 COX 55

COX 62 FELL 66

GAVE 59

Benes, V. E., " On Que ues with Poisson Arrivals," Annals of M athematical Statistics, 28, 670-6 77 (1956). Co nway, R. W., W. L. Maxwell, and L. W. Miller, Theory ofScheduling , Addison-Wesley (Reading , Mass.) 1967. Cox, D. R., " The Analysis of No n-Markovian Stochastic Processes by the Inclusion of Supplementary Variables," Proc. Camb. Phil. Soc . (M ath. and Phy s. S ci.), 51,433-441 (1955). Cox, D. R., Renewal Theory , Methuen (London) 1962. Feller, W., Probability Theory and its Applications Vol. II , Wiley (New York), 1966. Gaver, D. P., Jr ., "Imbedded Mar kov Cha in Analysis of a WaitingLine Process in Continu ous Time," Annals of Mathematical S tatistics 30, 698-720 (1959).

I

EXERCISES

231

HEND 72

Henderson, W., " Alterna tive Approaches to the An alysis of the M/G /I and G/M /I Queues," Operations Research, 15,92-101 (1972). KEIL 65 Keilson , J ., " The Role of Green's Fun ction s in Conge stion The ory ," Proc. Symp osium 0 11 Conge stion Theory , U niv. of No rth Carolina Press, 43- 71 (1965). KEND 51 Kend all, D. G ., "Some Probl ems in the The ory of Que ues," Journal of the Royal Statistical Society , Ser. B, 13, 151-1 85 (1951). KEND 53 Kendall, D. G ., "Stochastic Processes Occurring in the Theory of Queues and the ir Analysis by the Method of the Imbedded Markov Chain," Annals of Math ematical St atistics, 24, 338-354 (1953). KHIN 32 Khinchin , A. Y. , " Ma thema tical The ory of Stati onary Queues," Mat . Sbornik, 39, 73-84 (1932). Lindle y, D. Y., "The Theory of Queues with a Single Server ," Proc. LIND 52 Cambridge Philosophical Society, 48, 277-289 (1952). PALM 43 Palm, C.; "Intensitatschwankungen im Fernsprechverkehr," Ericsson Technics, 6,1 -189 (1943). Pollaczek, F., "Uber eine Aufgab e dev Wahrscheinlichkeitstheori e," POLL 30 I-II Mat h. Ze itschrift., 32, 64--100, 729- 750 (1930). PRAB 65 Prabhu, N. U., Queues and Inventories, Wiley (New York) 1965. RIOR 62 Riordan , J. , Stochastic Service Sy stems, Wiley (New York) 1962. Smith , W. L., " Renewal Theory and its Ramifications ," Journal of the SMIT 58 Royal Statistical Society, Ser . B, 20, 243-302 (1958). TAKA 55 Takacs, L. , "Investigation of Wait ing Time Problems by Redu ction to Markov Processes," Acta Math Acad. Sci. Hung ., 6,101 -129 (1955). TAKA 62a Tak acs, L., Introduction to the Theory of Queues, Oxford University Press (New Yor k) 1962. TAKA 62b Takacs, L., " A Single-Server Queue with Poisson Input ," Operations Research, 10, 388-397 (1962). TAKA 67 Takacs, L. , Combinatorial M ethods in the Theory of Stoch astic Processes, Wiley (New York) 1967.

EXERCISES 5.1.

Prove Eq . (5. 14) from Eq. (5.11) .

5.2.

Here we derive t he residual lifetime density j(x) di scu ssed in Section 5.2 . We u se th e notation o f Fi gure 5.1. (a) O bservin g that the event { Y ::S; y } can o ccur if a nd only if t < T k ::s; t y < T k+l for so me k , show th at

+

t ,(y) ~ pry <XI

= k~l

:s;

y It]

(1+.

J,

[L - F(t

+y

-

x) ] dP[Tk

:s;

x]

I

232

TH E QUEUE

(b)

M jGjl

Observing that Tk :s; x if and only if oc(x) , the number of " arrivals" in (0, x): is at least k , that is, P[Tk :s; x ] = P[oc(x) ;::: k] , show that <Xl

L Ph :s;

'"

x ] = L kP [oc(x)

k= l

(c)

=

k]

k= l

For large x , the mean-value expression in (b) is x /m I. Let F(y) = lim Ft(y) as t ->- 00 with corresponding pdf f ey). Show that we now have

• fey)

=

_I_-_F--.:-(y::..:.)

mi

5.3.

Let us rederive the P- K mean-val ue formula (5.72). (a) Recognizing that a new arri val is delayed by one service time for each queued customer plus the residual service time of the customer in service, write an expression for W in terms of R., p, X, (lb' and P[w > 0). (b) Use Little 's result in (a) to obtain Eq. (5.72).

5.4.

Replace I Q(I) = V(I) constant.

p

in Eq. (5.85) by an unknown constant and show that p for this

= I easily gives us the correct value of I -

5.5.

From Eq. (5.86) form Q(l)(I) and show that it gives the expression for q in Eq. (5.63). Note that L'Hospital's rule will be required twice to remove the indeterminacies in the expression for Ql1l(I). (b) From Eq, (5.105), find the first two moments of the waiting time and compare with Eqs. (5.113) and (5.114).

5.6.

We wish to prove that the limiting probability rk for the number of customers found by an arrival is equal to the limiting probability d k for the number of customers left behind by a departure, in any queueing system in which the state changes by unit step values only (positi ve or negative). Beginning at t = 0, let X n be those instants when N( t) (the number in system) increases by one and Yn be those instants when N (t) decrease s by unity, n = I , 2, .. . . Let N (x n- ) be denoted by OC n and N (Yn+) by f3 n. Let N(O) = i. (a) Sho w that if f3n H :s; k , then OC n+ k+1 :s; k . (b) Show that if OC n+ k+l k, then f3n+i :s; k . (c) Show that (a) and (b) must therefore give, for any k,

(a)

s

lim P [f3 n :s; k] = lim P[oc n which estab lishes that rk = dk •

s

k]

1/

233

EXERCISES

5.7.

In this 'exercise, we explore the method of supplementary variables as applied to the M/G/I . queue . As usual , let Pk(t) = P[N (t ) = k]. Moreover, let Pk(t , x o) dx o = P[N (t ) = k, X o < Xo(t ) ~ X o + dx o] where Xo(t) is the service already received by the customer in service at time t. (a) Show th at oPo(t -) = ot

- APo(t )

+ I'" Pl(t , xo)r(x o) dx o 0

where

h(x o)

rex) -

_....:......:e.....-

1 - B(xo)

o -

(b)

Let h = lim Pk(t) as t -.. ~ and h (x o) = lim Pk(t , x o) as From (a) we have the equilibrium result Apo

=

t -..

co,

l'"

Pl(XO)r(x o) dx o

Show the following equilibrium results [where Po(x o) ~ 0]: °Pk(XO) (i) - oXo

= - [}. + r(xO)]pk(xO) + APk_l(XO)

(ii)

piO)

=

(iii)

Pl(O) =1"'P2(x o)r(xo) d x o + Apo

1'"

Pk+l(XO)r(x o) dx o

k

k

~

1

>1

(e) The four equatio ns in (b) determine the equilibrium probabilities when comb ined with an appropriate norm alizat ion equation. In term s of po and hex,,) (k = 1, 2, . ..) give this norm alizati on equation. (d) Let R (z, x o) = 2::1 h (XO)Zk. Show that oR(z, x o)

---''-..:--=

and

ox o

= [}.z -

zR(z, O) =

(e)

1'"

• A -

r(xo)]R(z, x o)

r(xo)R(z , x o) dx o + ;.z(z - l) po

Show that t he solution for R (z, x o) from (d) mu st be R( z, x o) = R( z,O )e- '< ZoCl- z)-JifoTCV) dV AZ(Z - l)p o R(z,O) = z - B*( l'.' - j.Z .)

234

THE QUEUE

(f)

M/G/I

Definin g R (z) ~

S;' R (z , x o) dxo, show th at I - B*(). - AZ) R(z) = R(z, 0) - --'------ ---:. A( I - z)

(g)

From the normalizati on equation of (c), now show th at

Po = I - p (h)

(p

= Ax)

Con sistent with Eq. (5.78) we now define

Q(z) = Po

+ R (z)

Sh ow th at Q(z) expressed this way is identical to the P-K transform eq uation (5.86). (See [COX 55] for additional de tails of this meth od.) 5.8.

Consider the M/G/ oo queue in which each customer always finds a free server; thus s(y)

=

bey) and T

=

x. Let Pk(l )

=

P[N (I )

=

k]

-

and assume PoCO) = !. (a) Sh ow that Pk(l ) =

[11'

(AI)n(n) Ico e-.lt-

n~k

n!

k

[1 - B(x)] d x

l o

Jk[11' -

t

B(x) dx

In-k

0

[HINT: (I /I) S~ B(x) dx is the probability th at a customer's service terminates by time I , given th at his a rrival time was uniforml y distr ibuted ove r the interval (0, I). See Eq. (2.137) also. ] (b) Sh ow th at P» ~ lim Pk(l ) as 1 ->- 00 is

r-< :«: ().X)k

-AX

-

regardless of the fo rm of B(x)! 5.9.

5.10.

Co nsider M/ E./ !. (a) F ind the po lynomial for G*(s). (b) Solve for S(y) = P[time in system

~ y].

Conside r an M/D/I system for which x = 2 sec. (a) Sh ow th at the residu al service time pdf hex) is a rectan gular distr ibuti on. (b) For p = 0.25, show that the result of Eq . (5.111) with four term s may be used as a goo d approxi matio n to the distribution of queueing time.

I

I /

EXERCISES

235

5.11.

Co nsider a n M/G/I que ue in which bul k arrivals occur at rate A and with a probability gr that r customers arrive together at an arrival instant. (a) Show that the z-t ransforrn of the n umber of customers arriving in an inte rva l of lengt h t is e- ,l '[l - Gl zl ] where G(z) = 2: g.zr. (b) Show th at the z-transform of t he random va riables Un . the number of arrivals during the service of a customer, is B * [A - i.G(z)].

5.12.

Consider the M/G/I bulk arrival system in the pre viou s problem . Usi ng the method of imbedded M a rkov chains: (a) Fi nd th e expe cted queue size. [HI NT: show th a t ij = p and

;? _

o = d2V~Z) I

z~ l

dz-

= /(C b2

+ 1) + ~(c: + 1 _ ~) (g)2 P.

g

where C. is the coefficient of va riatio n of the bulk group size and

it is t he mean group size.] (b)

Show that the generating fu nctio n for queue size is (I - p)(l - z)B*[A - AG(Z)]

Q(z)

=

B*[A _ AG(Z)] _ z

-

Using Litt le's result, find the ratio W/x of the expected wait on queue to the ave rage service time. (c) Using the same method (imbedded Markov chain) find the expected nu mb er of groups in th e qu eu e (averaged over depa rture times). [H IN TS : Show tha t D(z) = f3* (A - Az), where D(z) is the generating functi on for the number of groups arri ving during the ser vice time for an entire group and where f3 *(s) is the Laplace tra nsform o f the service-time den sity for an entire gro up. Also not e th a t f3 *(s) = G [B*(s) ], which a llows us to show that r 2 = (X) 2(g2 - g) + x 2g , where r2 is the second moment o f the group service time.] (d) U sin g Little's result, find W., the expected wa it on queue for a gr oup (measured from the arrival time of the gr oup until the start of service o f the firs t mem ber of the group) a nd show that

xII' =

• (e)

P g 2(1 - p)

C [1+ ~ + C2J 2

g

•

If the customers within a gr oup a rriving together are served in ran d om order, show that the ra tio of the mean wai ting time fo r a single customer to the average service time for a single cu stomer is W.l x from (d) increased by (1/2)g( 1 + C; ) - 1/2.

236 5.13.

TH E QUEUE

M/Gfl

Con sider an MIGII system in which service is instant aneou s bu t is only available at " service instants," the interval s between successive service instants being independently distributed with PDF F(x ). T he maximum number of custom ers that can be served at any service instant is m. Note that thi s is a bulk service system. (a) Show that if qn is the number of customer s in the system ju st before the nth service instant, then q n+t

={

qn + V n - m

vn

qn

<m

where V n is the number of arrivals in the interval between the nth and (n + I)th service instants. (b) Prove that the probability generating function of u, is P (/, - k) . Hence show that Q(z) is

m-'

Q(z) = (c)

I

I

I Pk(zm - Zk) Zm [;:~A _ AZ)r' _

1

where fk = p[ij = kl (k = 0, .. . , m - I). The {Pt} can be determined from the cond ition that within the unit disk of the z-plane, the numerator must vanish when the denominat or does. Hence show that if F(x) = I - «r', Q(z)

z - 1 = _m __ zm -

where Zm is the zero of zm [I disk.

+ A(I

Z

- z)lfll - I out side the unit

5.14 . Con sider an M/Gfl system with bulk service. Whenever the server becomes free , he accepts 11\"0 cust omer s from the queue into service simult aneou sly, or, if only one is on qu eue, he accepts that one; in either case, the service time for the group (of size I or 2) is taken from B (x ). Let qn be the number of customers remaining after th e nth service instant. Let V n be the number of arrivals during the nth service. Define B*(s), Q(z), and V(z) as transform s associated with the rand om va riables x, ij , and ii as usual. Let p = ).X12. (a) Using the meth od of imbedded Markov chains, find

E(ij) = lim E(q n) in terms of p, G b2 , and P(ij = 0) ~ Po. (b) F ind Q(z) in terms of B*('), Po, and p, ~ P(ij = I). (c) Express p, in term s of po.

1

237

EXERCISES

5.15. Consider an MIGII queueing system with the following variation. The server refuses to serve any customers unless at least two customers are ready for service, at" which time both are "taken into " service. These two customers are served individually and independently, one after the other. The instant at which the second of these two is finished is called a "critical" time and we shall use these critical times as the points in an imbedded Markov chain. Immediately following a critical time, if there are two more ready for service, they are both "taken into" service as above. If one or none are ready, then the server waits until a pair is ready, and so on. Let

q.

= number of customers left behind in the system immediately following the nth critical time

= number of customers arriving during the combined service time of the nth pair of customers

Vn

(a) (b)

Derive a relationship between q. +1> q., and v. H Find

.

eo

V(z)

= .L P[vn =

k] Zk

k -O

(c)

Derive an expression for Q( z) = lim Q.(z) as n ~ of po = P[ij = 0], where

CX)

in terms

co

Q.(z)

= .L P[qn =

k]Zk

k- O

(d) (e)

How would you solve for Po? Describe (do not calculate) two methods for finding ij.

5.16. Consider an M /G /I queueing system in which service is given as follows . Upon entry into service. a coin is tossed, which has probability p of giving Heads. If the result is Heads , then the service time for that customer is zero seconds. If Tails , his service time is drawn from the following exponential distribution : x~O

(a) Find the average service time x. (b) Find the variance of service time CJb ' . (c) Find the expected waiting time W. (d) Find W*(s). (e) From (d). find the expected waiting time W. (f) From (d), find Wet) = P[waiting time ~ t]. 5.17.

Consider an M/G{I queue. Let E be the event that Tsec have elapsed since the arrival of the last customer. We begin at a random time and

(

238

THE QUEUE

M /G /!

measure the time IV until event E next occurs. This measurement may invol ve the o bserva tion o f man y customer a rriva ls before E occurs. (a) Let A(t ) be the intera rrival-tirne distribution for th ose interva ls during which E d oes no t occur. F ind A(1) . (b) Find A *(s) = f;; e-st dA(t). (c) Find W* (s I n) = f;; e- S W dW(1V I n). where W (IV I 11) = P[time to event E :s;; IV I II arrivals occur before E). (d) F ind W *(s) = f;; e- SW dW( IV), where W (w) = P[time to event E:S;; w). (e) Find the mean time to event E .

. 5.18.

Consider a n M /G f! system in which time is di vided in to intervals of length q sec each . Assume that arrivals are B~rn oulli, th at is, P[l arrival in an y interval)

prO arrival s in a ny interval) P[ > I arrival in any interval) A ssume th at a customer's service time such th at

x is

= nq sec) = K»

11

P[service time (a) (b) (c) (d) (e)

= = =

).q

10

so me multiple o f q sec

=

0 , 1,2" . ,

Find E[number of arrivals in an interval). Find the average a rr iva l rate. Express E[ i) ;;, x a nd E[x (i - q») ~ x 2 - xq in terms of th e mom ents of the g n distribution (i.e., let gk ;;, L :"o IIkg n)' Find Ymn = P[m cu stomers a rrive in IIq sec). Let v m = P[m cu stomers a rrive during the service of a customer) and let 00

V( z)

=

L vmz m ffl = O

(f)

).q

00

and

G(z)

=

L gmzm m= O

Express V(z) in terms of G(z) and the system par ameters A a nd q. Find the mean number of a rriva ls during a customer service tim e from (e).

5.19. Suppose th at in an M /G/l queueing system the cos t of making a cu stomer wait t sec is c(t) dollars , where c(t) = rJ.eP t • Find the averag e cost of queueing for a customer. Also determine th e cond iti on s necessary to keep the average cost finite, 5.20.

We wish to find the interdeparture time p robability density fun ct ion d (t ) for an M IGII queueing system, \ (a)

Find the Laplace transform D *(s) of th is den sity conditione d first on a nonempty queue left behind , and seco nd on a n empty queue left behind by a departing customer. Co mbine these results

EXER CISES

(b)

239

to get the Laplace transform of the interdeparture time density and from this find the density itself. Give an explicit form fo r the p robability distribution D(I), or density d(l) = dD(t)fdl, of the interdeparture time when we ha ve a con stant service time, that is B(x)

o

x

I

x? T

={

=

E[lv]

W

=L -

(k - m

k- m mfl

=

But in this ran ge we know that rk W

1

K

= -.

+ l )r k

Ko" a nd so

<X>

L (k

- m

mf,l k- m

+ l )a

k

and this is easily calculated to yield Ja

-

6.6. THE QUEUE G/M/2 Let us see how far we can get with the system G/M /2. F rom Eq. (6.19) we have immediately k = 1, 2, . .. r k = K~ Conserving probability we find <X>

<Xl

L rk = 1 = ro + L K~ k- O k~l

This yields the following relationship between K and ro: K = ( I - ro)(1 - a) a

(6.46)

Our task now is to find another relation between K and roo Thi s we may do from Eq. (6.41), which states co

R1

-

R 0 -_

".L. a'-1Pn. '-1

(6.47)

POl

But R,

=

1. The denominator is given by Eq. (6.9), namely, P Ol

=

f '(D

[I -

e-"~oe-"t dA( I)

Thi s we recognize as P Ol

=

A *Cfl)

(6.48)

Regardin g the one-step transition probabilities in the numerat or sum of Eq. (6.47) we fi nd they break into two regions: the term Pu. must be calculated from Eq. (6.9) and the terms Pi: for i = 2,3 ,4 , .. . must be calculated from

6.6.

THE QUEUE

G/M/2

257

Eq. (6.13). Proceeding we have Pu

=1"'(7)[1 -

e-·'V·'dA(t)

Again we recognize this as the transform (6.49)

Pa = 2A *(Il) - 2A *(21l)

Also for i

=

2,3,4 , . . . , we have

Pi! =

f"'(2)e-.'[f' (~lly)i-2 (e-·" Jo 1 Jo (I - 2)!

e-·')21l dyJ dA(t)

(6.50)

Substituting these last equations into Eq. (6.47) we the n have Ro =

_1_[1-

2A*{fl)

A *(Il)

+ 2A*(21l) -

iIa Ip il] i-2

(6.51)

Th e summation in this equation may be ca rried out withi n the integral signs of Eq. (6.50) to give ~ i-I

""'" a i- 2

*

Pn. = 2A (21l) +

2A *(21l - 21la) - 4a A *(Il)

_....0....:_----'----'- _ _----''--'-

20' - 1

(6.52)

But from Eq. (6.21) we recognize that a = A*(21l - 21la) and so we have . L'" a,-I PiI i- 2

=

2A*(21l)

+ - 20'-

20' - I

[I - 2A*{fl) ]

Substituting back into Eq. (6.51) we find R _ 2A*(Il) - 1 o - (20' - I)A *(Il)

However from Eq. (6.39) we know that

R0 -- ..!:!. and so we may express

r« as

Ka

r - Ka[l - 2A *(Il)] 0 - (I - 2a)A*(Il)

(6 .53)

Thus Eqs. (6.46) and (6.53) give us two equations in our two unknowns K and ro, which when solved simultaneously lead to (l - 0')[1 - 2A *(Il)] ro = I - 0'- A*(Il)

K = A*(Il) (l - 0')(1 - 20') 0'[1 - a - A *(Il) ]

EXERCISES

259

Compa ring Eq. (6.56) with ou r results from Chapter 3 [Eqs. (3.37) and (3.39)] we find that they agree for m. = 2. Th is compl etes our study of the G[M [m queue . Some further results of interest may be found in [DESM 73]. In the next chapter, we view transform s as probabilities and gain considerable reduction in the ana lytic effort required to solve equ ilibrium and transient queueing problems. REFERENCES COHE 69 Cohen, J. W., The Single Ser ver Queue , Wiley (New York) 1969. DESM 73 De Smit, J. H. A., "On the Many Server Queue with Exponential Service Times," Advances in Applied Probability, 5,1 70-1 82 (1973). KEND 51 Kendall, D. G. , " Some Problems in the Theory of Queues," Journal of the Royal Statistical Society , Ser. E., 13, 151-1 85 (1951).

TAKA 62 Takacs, L., Introduction to the Theory of Queues, Oxford University Press (New York) 1962. EXERCISES 6.1. Pro ve Eq. (6.13). [HINT: condition on an interarrival time of dur at ion t and then further conditi on on the time (~ t) it will take to empty the queue.] 6.2. Cons ider E2[M[ I (with infinite qu eueing room). (a) Solve for r k in terms of G. (b) Evaluate G explicitly. 6.3.

Conside r M[M[m . (a) How do Pk and r k co mpare? (b) Co mpare Eqs. (6.22) and (3.40).

6.4.

Prove Eq. (6.31).

6.5. Show that Eq. (6.52) follows from Eq. (6.50). 6.6.

Consider an H 2 /MfI system in which ()(l

=

(a) (b) (c) (d)

}.t

=

2,

}'2

=

I, f1 = 2, and

5[8.

Find G . Find r k • Fin d Il'(Y) . Find W.

6.7. Conside r a D[MfI system with f1 = 2 and with the same p as in the previous exercise. (a) Find G (correct to two decimal places).

260

TH E QUEUE

(b) (c) (d)

G/M/m

Find r k • Find w( y). Find W.

6.8.

Consider a G/M /I queueing system with room for at most two customers (one in service plus one waiting) . Find rk (k = 0, 1,2) in terms of fl and A * (5).

6.9.

Consider a G/M/I system in which the cost of making a customer wait y sec is c(y) = aebV (3) Find the average cost of queueing for a customer. (b) Under what conditions will the average cost be finite?

J

7 The Method of Collective Marks

W hen on e stud ies stoch astic pr ocesses such as in queueing the ory , one finds th at the wo rk d ivides int o two parts. The first part typically requires a careful probabilistic argument in order to arrive a t expressions inv olvin g the random va riables of interest. * The second part is then one of a nalysis in which the f ormal manipulation of symbo ls takes place either in the o rigina l domain or in so me transformed d omain. Whereas the prob abil istic a rgu men ts typica lly must be made with great care, they nevertheless leave one with a comfort abl e feeling th at the " p hysics" of th e situ ation a re con stantly withi n one's understand ing a nd gras p. On the other hand , whereas the a na lytic manipulations that o ne carries out in the seco nd part tend to be rather stra ightfo rwa rd (albeit difficul t) formal o peratio ns, one is unfortunately left with the uneasy feelin g th a t these man ipul at ion s relate back to the origina l p roblem in no clearly understandable fashi on. This " no nphysica l" aspect to problem so lving typically is taken o n when one moves into the domain of transforms, (either Laplace o r z-t ra nsfo rms). In th is ch ap ter we de mon str at e that one ma y deal wit h tr a nsforms a nd sti ll maintain a hand le on the prob abilistic arguments taking place as the se tra nsfo rms a re manipulated. Th ere a re tw o separat e opera tions involved : the " ma rking" of custo mers ; a nd the observatio n of "catas tro phe" p roc esses. Together the se meth od s a re referred to as the meth od of collective marks. Both opera tio ns need not necessarily be used simultaneou sly , and we study them sepa ra tely bel ow. This ma terial is dra wn princip ally from [R U N N 65] ; these ideas were introduced by van Dan tzig [VA N 48] in order to expose th e probabil ist ic int erpreta tion for tr an sforms. 7.1.

THE MARKING OF CUSTOMERS

Assume that , a t the entrance to a queueing system , the re is a gremlin wh o ma rks (i.e., tags) a rrivi ng custo mers with the following p roba bilities : P [cu stomer is ma rked]

=

I - z

P[customer is not marked] = z

(7. 1) (7.2)

• As, for exa mple. the arguments leading up to Eqs. (5.31) and (6.1).

26 1

I ! I

262

THE METHOD OF COLL ECTIV E MARKS

where 0 ~ z ~ I. We assume that the gremlin marks customers with these prob ab ilities independent o f all other as pects of the queueing p roces s. As we shall see below, th is marking process allows us to cre ate genera ting fun ctions in a very natural way. It is most instructive if we illustrate the use o f th is marking p roc ess by examples:

Ex ample 1: Poisson Arrivals We first consider a Pois son a rrival pr ocess with a mean a rrival rat e of }. customers per second. Assume that customers are marked as ab ove. Let us co nsider the probability q(z, t) ~ P[no marked customers arrive in (0, t) ]

(7.3)

It is clear that k cu stomers will arrive in the int er val (0 , t ) with the pro babil ity (At)ke-At/k !. M oreover, with probability zk, none of these k cu stomers will be marked ; thi s last is true since marking takes place independently a mo ng customers. Now summing over all values o f k we have immediately that 0)

q(z, t)

= .2

(}. t)ke-.lt

k!

k- O

=

Zk

e.lt( z-Il

(7.4)

G oin g back to Eq . (2. 134) we see that Eq . (7.4) is merel y the generati ng fun ction for a Poisson arriva l proce ss. We thus conclude th at the genera ting functi on for th is arrival process may also be interpreted as the prob abili stic qu antity expressed in Eq . (7.3). This will not be the first time we may give a probabili stic in terpretati on for a generat ing fun ction!

Example 2: M /M/ ro We con sider the birth-death queueing system with a n infinite number of servers. We also assume a t time t = 0 that there a re i customers present. Th e parameters of our system as usual are). a nd f.l [i.e., A(t) = I - e- At a nd B (x ) = I - e-~X] . We are intere sted in the qu antity

Pk(t ) = P[k cu stomers in th e system a t time t ]

(7.5)

and we define its generating functi on as we did in Eq, (2. I53) to be 0)

P(z, r) =

.2 Pk(t)Zk k=O

(7.6)

I ' f

7.1.

TH E MARKI NG OF CUS TOMERS

263

Once again we mar k customer s according to Eqs. (7.1) and (7.2). In analogy with Exa mple I , we recognize that Eq . (7.6) may be interpreted as the pr oba bility that the system contains no marked customers at time t (where the term Zk again repre sents the probability that none of the k customer s present is marked). Here then is our crucial ob servati on : We may calculate P (z , t ) directly by findin g the probab ility that there are no marked customers in the system at time t, rather than calculating Pk(t) and then finding its z-transform ! We pr oceed as follows: We need merel y find the probability that none of the customers still present in the system at time t is mark ed and this we do by accounting for all customers present at time 0 as well as all customer s who arrive in the interval (0, t ). For any cust omer present at time 0 we may calculate the probability that he is still present at time t and is mar ked as (I - z)[ l - B(t )] where the first factor gives the probabil ity that our customer was marked in the first place and the second factor gives the prob abil ity that his service time is greater than t . Clearly, then , this quantity subtracted from unity is the pr obability that a customer originally present is not a marked customer present at time t ; and so we have P[custome r present initially is not a marked customer present at time t] = 1 - ( 1 - z)e-P ' Now for the new customers who enter in the interval (0, r), we have as before P[k arrivals in (0; t)] = ().t)'e- ;" /kL Given that k ha ve arrived in this interval then their arrival instants are uniformly distributed over this interval [see Eq. (2.136)]. Let us consider one such arrivin g customer and assume that he arri ves at a time or < t. Such a customer will not be a marked customer present at time t with probability P[new arrival is not a mar ked customer pre sent at time arrived at or

~

=

t]

t

given he

I - (I - z)[ l - B(t - or)]

(7.7)

However, we have that P[arrival time ~ or]

=!

for 0

t

~

or

~

t

and so P[new arrival still in syste m at t]

=f' e-p('-,) dor t

t=O

1 - e-

pt

(7.8) flt U nconditioning the arrival time from Eq. (7.7) as shown in Eq. (7.8) we have 1 - pi P[n ew arr ival is not a mark ed customer present at t] = 1 - ( 1 _ z) - e p.t

1

264

THE METHOD OF COLLECTIVE MARKS

Thus we may calculate the probability that there a re no marked c usto mers a t time t as follows: co

P( z, t)

=I

P[k arrive in (0, t)]

k= O

X {P [new arrival is not a marked customer present a t tW x {P [initial customer is not a marked customer present at t]}i

II

I r

U sing o ur established relati onships we a rrive a t P(z , t) =

co (J.t)k I - e-

k~O

At

[

1 - (I - z)

1-

k!

- PJ k e [I - ( I - z)e-p'r /it

which then gives the known result P(z, t)

=

[I - (I - z ) e - P']ie - W pll l -Z )[l - e-P' J

(7.9)

It sho uld be clear to the student that the usual method for obtaining this result would have been extremely complex. Example 3: M JGJI

In this exa mple we co nsider the FCFS M JGfI system. Re call that the random va riables w. , t . +! , t . + 2 , ••• , x., x.+1 , . . . a re all independent of each o ther. As usu al , we define B *(s) a nd W n *(s) as the Laplace tr an sform's for the service-time pdf b(x) and the waitin g-time pdf w n(Y) for C n, respe ctively. We define the event

. {no u In 1Y,

} IV .

a {no customers who arrive during the waiting} . f Cn a re mark e d trrne 0

--

(7.10)

We wish to find the probability of this event, that is, P [no M in Jr. ]. Con ditioning on the number of arriving customers and on the wait ing time lI ' n , a nd then removing the se conditions , we have P[no M

.

In

w.]

. = I<Xl 1 <Xl (i.)k ....JL. e- A·zk dWn(y ) k- O

=

0

f"

k!

e -A.(l- z)

d W.(y)

We recogni ze the inte gral as W. *(J. - Az) a nd so P[no M in w. ] = W n *(J. - }.z)

(7 .11)

Thus once again we ha ve a very simple probabili stic int erpretat ion for the (Laplace) transform of an important distribution . By identical a rguments we may arrive at (7.12) P[no M in x.] = B*(J. - },z)

7.1.

THE MAR KI N G OF CUSTO MERS

265

This last gives us another interp reta tion for a n old expre ssion we have seen in Ch apter 5. Now co mes a startling insight! It is clear that the arriva l of cu stomers during the waiting time of C n and the arrival of customers during the service time of C; must be independent events since these are non overiapping intervals and our arrival process is memoryless. Thus the events of no marked customers arriving in each of these two disjoint intervals of time mu st be independent , and so the probability that no marked customers arrive in the uni on of the se two disjoint intervals must be the product of the probabilities th at none such arrive in each of the intervals sepa ra tely. Thus we may write

P[no M in

ll ' n

+ x n] = =

P[no M in lI'n]P[no M in x n]

(7.13)

W n *(A - Az)B*(I. - Az)

(7.14)

Th is last result is pleasing in two ways . First, becau se it says th at the prob abil ity of two independe nt joint events is equal to the product of the pr obabilities of the indi vidu al events [Eq. (7.13)]. Second , becau se it says that the transform of the pdf of the sum of two independent random va riables is equal to the pr oduct of the tr an sforms of the pdf of the ind ividu al random variables [Eq. (7.14)]. Thus two familiar results (rega rding disjoint events and regarding sums of independent random variables) have led to a meaningful new insight, namely, that multiplication of tran sforms implies not only the sum of two independent random variables, but also implies the product of the probab ilities of two independent eve nts ! Not often are we privile ged to see such fundamental pri nciples related. Let us now continue with the argument. At the mom ent we have Eq, (7.14) as one mean s for expressing the pr ob abil ity that no marked cu stomers a rrive during the int erval Il' n + X n . We now proceed to calcul ate thi s prob abil ity by a seco nd a rgume nt. Of course we have

P[no M in

lI ' n

+ x n] =

P[no M in \" n + + P[no Min ll ' n

X n

+

and C n H marked] X n and C n H not marked] (7.15)

Furthermore , we have

P[no Min

I

I

I

Wn

+ X n and C n+l marked] =

0

if

Wn+l

>0

since if C n H mu st wait, then he must arrive in the interval w, + X n a nd it is impossible for him to be ma rked and stilI to ha ve the event {no M in W n + Xn} . T hus the first term on the right-hand side of Eq . (7.15) mu st be P[ wnH = 0](1 - z) where th is seco nd factor is merely the probability th at C n+l is marked. Now cons ider the second term on the right-hand side of Eq. (7. I 5) ; as sh own in F igure 7.1 it is clear th at no custo mers arrive between C n and CnH , and therefore the customers of interest (namely, those arriving after

266

TH E METHOD OF COLLECTIVE MARKS

1
1

wn + 1

C n-tl

\.

v-

-

-

-

---)

A rrivals of

interest

Figure 7.1 Arrivals of interest during lV n + X n • Cn+l does, but yet in the inte rval ll'n + x n ) must arrive in the interva l 11''' +1 since this interval will end when II' n + X n ends. Thu s this second term must be

P[no Min

IV n

+ X n and

C n+l

not mar ked) = P[no Mi n IVn+l ) X P[C n+ 1 not mar ked) = P[n o M in II' n +1) Z

From the se observations and the result of Eq. (7. \ I) we may write Eq. (7.15) as

P[no M

in + lV n

Xn)

= ( I - z)P[Cn+l arrives after lV n + x n) + z W~+l ( .l. - i.z) Now if we think of a second separate marking process in which all the customers are marked (with an additional tag) with probabi lity one, and ask that no such mar ked customers arrive du ring the interval Il'n + x'" then we are ask ing that no customers at all arrive d uring this interval (which is the same as ask ing that C n+l arrive a fter W n + x n ) ; we may calculate this using Eq. (7.14) with z = 0 (since this gua rantees that all customers be ma rked) and obtai n W n* (.l.)B*(.l.) for this pro babi lity. Th us we arrive at

P[no ,\lf in lV n + x n) = ( I - z)W.* (A)B*(A) + Z W ~+ I ( )' - ).=) (7.16) We now have two expressions for P[no M in W n + X n ), which may be equat ed to obtai n Wn*(.l. - k) B*(i. - .l.z) = ( I - z)W'* (i.)B*(i.)

+ zW:+ (i. 1

i.z) (7.17)

Th e interesting pa rt is over. Th e use of the method of collective marks has brought us to Eq. (7.17), wh ich is not easily obtaine d by other methods, but which in fact checks with the result due to other meth ods. Rathe r than dwell on the techniques required to carry this equa tion further we refer the reader to Runnen burg [RUNN 65) for additional details of the time-dependent solutio n.

7.2.

THE CAT AST RO P HE P ROCESS

Now, for p < I , we ha ve an ergodic process with W Oes ) as n -;. 00 . Equation (7.17) then reduces to 1V *(i. - i.z) B*(). - i.z) = (I - z ) W *(i.)B*()')

+ z lV*()'

=

267

lim W n *(s)

- i.z)

If we make the change variable s = X- )'z and solve for W*(s) , we obtain WOes)

=

s W*( i.) B*(i. ) s - ). ).B*(s)

+

Since W *(O) = I , we evalu ate W *(i.)B*().) = ( I - p) and arrive a t WOes)

=

s( 1 - p) s - ). i.B*(s)

(7.18)

+

which, of course, is the P-K transform equ ation for waiting time. We have demonstrated three examples where the marking of customers has allowed us to a rgue purel y with pr obabilistic reasoning to deri ve exp ressions relating transforms. What we have here traded has been straightforward but tedious analysis for deep but physical probabilistic rea sonin g. We now co nsider the cat astrophe pr ocess. 7.2.

THE CATASTROPHE PROCESS

Let us pursue the method of collective marks a bit further by observing " catas tro phe" processes. Mea suring from time 0 let us con sider th at some event occurs at time t (t ~ 0), where the pdf associa ted with the time of occurrence of this event is given by f(t). Furthermore, let there be an independent " catastrophe" process taking place simultaneously which generates cat astrophes t at a rat e y acco rding to a Poisson pr ocess. We wish to calcul ate the probability th at the event at time t takes place before the first cat astr ophe (measuring from time 0). Conditioning o n t and integra ting over all t , we get Prevent occ urs before catastrophe] =

L'" e-Y'f( t) dt

= F *(y)

I I

j

(7.19)

where, as usual , f (t )¢>F*(s) are Laplace tran sform pairs. Thu s we have a prob abili stic interp reta tio n for the Laplace transform (evalua ted at the point y) of the pdf for the time of occurrence of the event, namel y , it is the pr obability that an event with this pdf occurs before a Poisson catastr ophe at rate y occurs. t A catastrophe is merely an impressive name given to these generated times to distinguish them from the " event" of intere st at time

I.

268

T HE M ETHOD OF CO LLECTIVE MARK S

A s a second illustrati on usin g catastrophe processes, con sider a sequence o f events (tha t is, a point p rocess) on the interval (0, co). Measu ring from time 0 we would like to calculate the pdf of th e time un til the nth event, which we den ote ] by f (n,(I), and with distribution Ftn,(I), where the time between events is given as before with density f( I). That is,

We are interested in deriving an expression for the ren ewal fun ction H (I), which we rec all fr om Section 5.2 is equal to the expected number of events (renewa ls) in a n interval of len gth I . We proceed by defin ing

=

P[exactly n events occur in (0, I)]

(7.20)

The renewal functi on may therefore be calculat ed as

H(I)

=

E[number of events in (0, I)] OX>

=J,nPn(t) n -O

But from its definition we see that Pn(l ) = F(n,(I) - F(n+I)(I) and so we ha ve OX>

H(I)

= J, n(F(n,(I) -

F tn+I,(I)]

n= O OX>

=

.L F(n,(I)

(7.2 1)

n- l

If we now permit a Poisson catastrophe process (a t rate y) to devel op we may a sk for the expectat ion o f the followin g random var iable:

N;

~ number of even ts occurring before th e first ca tas tro p he

(7.22)

W ith probability ve:" dt the first catastrophe will occur in th e interval (I, I + dl ) and then H (I) will give the expected num ber of events occurr ing before this first ca ta strophe, th at is,

H (t ) = E[ N e I first ca tastrophe occurs in (I, I

+ dl )]

Summing over a ll possibilities we may then writ e

E[N , ]

= f.OX> H(t)ye- y t dl

(7.23)

In Section 5.2 we had defined H *(s) to be the Lapl ace transform of th e renewal density h(l ) defined a s h(l ) ~ dH (I)/dl , th at is ,

H*( s)

~

fO

h(I)e- " dt

E,

u tl 1

F(n)(I) = P[nth event has occu rre d by time I]

Pn(l )

If

(7.24)

t We use the subscript (n) to remind the reader of the definition in Eq . (5.110) denoting the n-fold convolution. We see that[. n,(t ) is indeed the n-fold convo lution of the lifet ime density J(t) .

REFERENCES

269

If we integra te th is last equation by parts, we see that the right-hand side of Eq. (7.24) is merely sH (t )e-'t dt a nd so from Eq . (7.23) we ha ve (ma king the substitutio n s = y) . E[N e ] = H*(y) (7.25) Let us now calcul ate E[N e ] by an alternate mean s. From Eq . (7.19) we see that the cat astrophe will occur before the first event with probability I F* (y) and in th is case N , = O. On the other hand , with probability F* (y) we will get a t least one event occurring before the catastrophe. Let Nc' be the random varia ble N; - I conditioned on at least o ne event ; then we have N; = I + Nc'. Becau se of the memoryless property of the Po isson process as well as the fac t th at the event occurrences genera te an imbedded Markov process we see th at Nc' mu st have the sa me distribution as N, itself. Forming expectati on s o n N; we may therefore write

S:

E[N e] = 0[1 - F*(y)] + {I + E[N e]}F*(y ) This gives immed iately E[N] = F*(y) e 1 _ F*(y)

(7.26)

We no w have two expressions for E[N e ] and so by equating them (and making the change of va ria ble s = y) we ha ve the final result

H*(s) _

F*(s) (7. 27) F*(s) Thi s last we recogni ze as the tr an sform expression for the integral equation of renew al the ory [see Eq. (5.2 1)]; its integral formulation is given in Eq . (5.22). It is fair to say that th e method of collective marks is a rather elegant way to get so me useful and imp ortant results in the theory of stochastic processes. On the other han d , th is method has as yet yielded no results th at were not prev iousl y known throu gh the application of other methods. Thus at present its principal use lies in providin g a n alternati ve way for viewing the fundamental relat ion ship s, thereby enhancing one's insight int o the prob abili st ic structure of the se processes. Thus end s o ur treatment of intermediate queueing the ory. In the next part, we venture into the kingdo m of the GIG II queue. 1 -

REFERENCES RUNN 65 Runnenburg. J. Th ., " On the Use of the Method of Collective Marks in Queueing Theory," Proc. Symposium O il Congest ion Theory , eds, W. L. Smith and W. E. Wilkinson, University of North Carolina Press (1965). VAN 48 van Dantzig. D., " Sur la methode des fonctions generatrices," Colloques internationaux du CN RS, 13, 29-45 (1948).

270

TH E METH OD OF COLLECTIVE MARKS

EXERCISES 7.1.

Consider the M/G /l system sho wn in the figure belo w with average arri val rate A and service-time distribution = B(x) . Customers a~e served first-come-first-served from queue A until they either leave or receive a sec of service, at which time they join an entra nce box as shown in the figure. Cu stomers continue to collect in the ent rance box forming

,J

sec of service received

)----'~ Depart wit h service

com pleted

Server

a gro up until queue A empties and the server becomes free. At this point , the entrance box "dumps" all it has collected as a bulk arrival to queue B. Queue B will receive service until a new arrival (to be referred to as a "starter") join s queue A at which time the server switche s from queue B to serve queue A and the customer who is preempted returns to the head of queue B. The entrance box then begins to fill and the process repeat s. Let g. = P[entrance box delivers bulk of size n to queue B]

G(z) =

'"

I gn .-0

zn

(a)

Give a pr obabili stic interpretat ion for G(z) using the method of collective marks. (b) Gi ven th at the " starter" reaches the entrance box, and usin g the method of collective marks find [in term s of A, a, B (' ), and G(z)] P k = P[k customers arrive to queue A during the " starter's"

service time and no mar ked customers arrive to the entrance box fr om the k sub-busy per iods creat ed in queue A by each of these customers] Gi ven that the "starter" does not reach the entrance box, find Pk as defined above. (d) From (b) and (c), give an expres sion (involving an integral) for G(z) in terms of A, a , B(') , and itself. (e) From (d) find the average bulk size ii = I ~-o ng•.

(c)

EXER CISES

271

7.7..

Consider the M/G/ oo system. We wish to find P(z, I) as defined in Eq . (7.6). Assume the system contains i = 0 customers at I = O. Let p(l) be the probability that a customer who ar rived in the interval (0, I) is still pre sen t at I. Proceed as in Example 2 of Sect ion 7.1. (a) Express p(l) in terms of B(x). (b) Find P(z, t ) in terms of A, I, z, and p (I) . (c) From (b) find Pk(l) defined in Eq . (7.5). (d) From (c), find lim Pk(t) = P» as 1->- 00 .

7.3.

Con sider a n M/G/I queue, which is idle at time O. Let p = P[no catastrophe occurs during the time the server is bu sy with those cu stom ers who a rri ved during (0, I)] and let q = P[no catastrophe occurs during (0, t + U(I» ] where U(I) is the unfinished work a t time t. Catastrophes occur at a rate y. (a) Find p. (b) Find q. (c) Interpret p - q as a probability a nd find an independent expression for it. We may then use (a) and (0) to relate the distribution of unfini shed work to 8* (s).

7.4.

Consider the G/M /m system. The root 0, 'Y+(s) is an analytic functio n of s with no zeroes in this half-pl an e.

•

For Re (s) < D , 't'_(s) is an analytic function of s with no zeroes in this half-plan e.

(8.36)

8.2.

SPECTRAL SOLUTIO N TO LI NDLE Y'S I NTEGRAL EQUATION

287

F urthe rmo re, we wish to find these functions with the additional pr operties :

•

For Re (5) For Re (5)

> 0,
D , are also ana lytic; therefore their analytic continuation contains no singularities in the entire finite s-plane, Since we have establi shed the behavior of the functi on +(5)'1"+(5) = _(s)'I'_(s) to be analytic and bounded in the finite s-plane, and since we ass ume Co ndition (8.37), we may then apply Liou ville's theor em * • Liouville's theorem states, "If I(~) is analytic and bounded for all finite values of z, then I(z) is a constant."

288

TH E QUEUE

G/G /l

[TIT C 52], which immediately establishes that this function must be a constant (say, K). We thus have . $ _(s)'Y_(s) = $ +(s)'Y+(s)

=

(8.40)

K

This immediately yields K

(8.41)

$ +( s) = - 'Y+(s)

The reader should recall that what we are seeking in this development is an expression for the distribution of queue ing time whose Laplace tran sform is exactly the function $ +(s), which is now given through Eq. (8.41). It remains for us to demonstrate a method for evaluating the constant K. Since s$+(s) = W* (s) , we have

Let us now consider the limit of this equation as s -+ 0 ; working with the right-h and side we have lim 8- 0

r ~ e-"

Jo

dW( Y)

= r~ dW(Y) =

Jo

1

We have thu s established lim s $ +(s)

.-0

=

1

(8.42)

This is nothing mor e than the final value theorem (entry 18, Table 1.3) and comes ab out since W( (0 ) = I. Fr om Eq. (8.41) and this last result we then have

and so we may write K

=

lim 'Y+(s) 8- 0

(8.43)

S

Equat ion (8.43) provides a means of calcul atin g the constant K in our solution for $ +(s) as given in Eq. (8.41). If we make a Taylor expan sion of the funct ion 'Y+(s) around s = 0 [viz., 'Y...(s) = 'Y+(O) + s'Y (O) + (s2/2 !)'Y~ I(O) + ...] and note from Eqs. (8.35) and (8.36) that 'Y +(0) = 0, we then recognize that this limit may also be written as .

d'Y+(s)

K=hm---

.-0

(8.44)

ds

J

8.2.

SPECTRAL SOLUTION TO LINDLEY'S INT EGRAL EQUATION

289

and this provides us with an alternate way for calculating the constant K. We may further explore this constant K by examining the behavior of +(5)o/+(S) anywhere in the region Re (5) > 0 [i.e., see Eq. (8.40»); we choose to examine this beh avior in the limit as 5 --->- a::J where we kn ow from Eq. (8.37) that 0/+(5) behaves as 5 does ; that is,

= lim s s .... 00

Making the change of vari able 5Y = K

=

lim

r ~ e- ' vW(y) dy

Jo X

we have

r~e-"w(~) s

s.. . oo )o

dx

As 5 --->- cc we may pull the con stant term W(O+) outside the inte gral and then obtain the value of the rema ining integral, which is unit y. We thus obtain '(8.45) This establishes that the con stant K is merely the probability that an arriving . customer need not queue]. In conclusion then , assuming that we can find the appro priate spectru m fact ori zati on in Eq . (8.35) we may immediately solve for the Lapl ace transform of the waitin g-time distribution through Eq . (8.41), where the con stant K is given in eith er of the three forms Eq . (8.43), (8.44), or (8.45). Of course it then remain s to invert the transform but the pr oblems involved in that calcul ati on have been faced before in numerous of our other solution form s. H is possible to carry out the solution of this problem by concentrating on 0/_ (5) rather than '1'+(5) , and in some cases this simplifies the calcul ation s. In such cases we may proceed from Eq, (8.35) to obtai n

= o/_(s)[A *( -

1)

(8.46)

(s) - - - - - =K-- - - [A *( -s) 8 *( s) - 1]'Y_(s)

(8.47)

'F+(s)

s)8*(s) -

From Eq . (8.4 1) we then have +

t Note t ha t W (O+) is not necessaril y equ al to I - p. which is the fracti on of lime the server is idle . (T hese two are equ al for the system M{G{I.)

290

THE QUEUE

GIGII

In order to evalua te the constant K in this ca se we di fferentiate Eq. (8.46) at s = 0, th at is , 'F~)(O) = [04 *(0 )8*(0) -

I]'¥~)(O)

+ 0/_(0) [04*(0)8 *(1)(0) -

o4 *(1\O)B*(O) ] (8.48)

From Eq. (8.44) we recognize the left-hand side of Eq . (8.48) a s the consta nt

K and we may now evaluate the right-hand side to o bta in K = giving

°+

o/_(O)[-x

+ f]

K = 0/_(0 )(1 - p)i

(8 .49)

Thus , if we wish to use 'F_(s) in our so lutio n form , we obtai n the transform of the waiting-time di st ribution fr om Eq. (8.47), where the unknown constant K is evalu ated in terms of 'L(s) through Eq. (8.49) . Summari zin g then , o nce we have ca rried out the spectru m factori zat ion as indicated in Eq . (8.35), we may proceed in one of two directions in solving fo r 0, we may collect together the two zeroes (at s = 0 and s = - po + I.) and one pole (at s = - po) and still satisfy this requ ired condition. Similarly , 0/ _(s) must be a nalytic and free from zeroes for the Re (s) < D for some D > 0; we can obtain such a cond ition if we allow this functi on to contain the rema ining pole (at s = J.) and choose D = J.. Th iswe show in Figur e 8.5. Thus we have 'F+(s)

= s(s + po

- A) s + p.

(8.53)

'L (s) = J, - s

(8.54)

No te that Co ndition (8.37) is satisfied for the limit as s ->-

00 .

Im (s)

s-plane

- - -"*--()---e>---*- - - - - Re(s) -p

Figure 8.4 Zeroes (0) and poles ( x) of'i'+(s)/,C(s) for M iM I!.

_I

292

THE QUEUE

GIG II

Im(s)

Im(s)

s-plane

s-cp lane

*

----{J--{)--

(a)

-

-

-----l-~;_- R e(s)

Re(s)

-

'!'. (s)

Figure 8.5 Factorization into '1'+{s) and 1/'Y_{s) for M IMI !. We are now faced with find ing K. F ro m Eq . (8.43) we ha ve K

=

lim o/+(s) .-0

=

.

S

s +/l-A

hm -----'-.- 0

S

+ /l

(8.55)

=I - p

Our expression for the La place tran sform of the waiting time PDF for M IMI ! is therefore fro m Eq. (8.41), +(s)

=

(I - p)(s s(s

+ /l

+

/l) - A)

(8.56)

A t this poi nt, typically, we attempt to invert the transform to get the waitingtime dist ribution. H owever , for thi s M/M /I example, we have already carr ied out th is inversion for W *(s) = s+(s) in going from Eq. (5.120) to Eq . (5.123). T he solutio n we o btai n is th e familiar form, y~O

(8.57)

Example 2: GIMllt In this case B*(s) = /l l(s giving us

+ /l)

bu t now A *( s) is completely arbitrary.

A*(- s) B*(s) _ I = A*( -s)/l _ I s +/l

t Thi s examp le forces us to locale roo ts using Rouche's theorem in a way often nccessar y for specific G/G !I problems when the spectrum facto rizatio n meth od is used. Of co urse, we have already studi ed th is system in Section 6.4 and will compare the results for both methods.

1

8.2.

SPECTRAL SOLUTION TO LI NDLEY'S INTEGRAL EQU ATION

293

and so we have o/+(s} = flA*( - s) - s - fl s +fl

'I'_(s)

(8.58)

In order to factorize we must find the roots of the numerator in this equ ation. We need not concern ourselves with the poles due to A *( - s) since they mu st lie in the region Re (s) > 0 [i.e., A (t ) = 0 for t < 0) and we are attempting to find o/+(s), which cannot include any such poles. Thus we only study the zeroes of the function

s

+ fl

- flA*(- s) = 0

(8.59)

Clearly, one root of this equation occ urs at s = O. In order to find the remaining roots , we make use of Rouche's theorem (given in Appendix I but which we repeat here) :

Rouche's Theorem Iff( s) and g(s ) are analytic functions of s inside and on a closed contour C, and also iflg(s)1 < /f(s)1 on C, thenf(s) andf (s) + g (s) have the same number of zeroes inside C. In solving fo r the roots of Eq . (8.59) we make the iden tification

f (s) = s g(s)

=

+ fl

-flA*( - s)

We have by definition A*( -s) = .Ce"dA (t )

We now ch oose C to be the contour that runs up the imaginary axis a nd then forms a n infinite-radius semicircle moving counterclockwise and surround ing the left half of the s-plane, as shown in Figure 8.6. We consider thi s contour since we are concerned abo ut all the pole s a nd zeroes in Re (s) < 0 so that we may properly include them in 'Y+(s) [recall that 0/ _(s) may contain none such); Rouche's theorem will give us information concerning the number of zeroes in Re (s) < 0, which we must consider. As usual , we assume that the real a nd imaginary parts of the complex variable s are given by a and (0), respectively, that is, for j

=

J=I s

= a + jw

294

GIGII

TH E QUEUE

Im (, )

s- plane

-

-1--

-

-

-

-

-

I-::--

a

- - Re(. Ig(s)1 in the vicin ity s = O. For thi s purpose we allow the conto ur C to make a small semicircula r excu rsion to th e left of the o rigin as show n in Figure 8.7. We note at s = 0 tha t Ig(O)1 = 1/(0)1 = fl, which doe s no t sati sfy the conditions for Rouch e's the ore m. The small semicircular excursion of radius £(£ > 0) that we take to the left of the ori gin overco mes thi s difficult y as follows. Cons ider ing a n a rbitra ry point s on thi s semicircle (see the figure) , which lies at an a ngle () with the a-axis, we may write s = a + jw = - £ cos () + j £ sin 0 and so we ha ve 2

1/ (5)1 =

Is + fl l 2 = 1-£ cos () + j £sin () + fl l 2

Formin g the product of (s

+ fl) and

its co mp lex co njugate, we get

If(sW = (fl - £ co s ()' + 0(£)

= fl '

- 2fl€ cos 0

+ 0(£)

(8.63)

N ote that the sma llest value for I/(s) I occurs for () = O. Eva lua ting g( s) on th is sa me semicircula r excursion we have

F ro m the power-series expan sion of the expon enti al inside the inte gral we have

Ig(sW =

fl21I.~ [I + (-

£

cos () + j« sin 0)1 +... 1

dA(e)12

I 296

T HE Q UEUE

GIGII

We recognize the integrals in this series as proportional to the moments of the interarrival time , and 'so Ig(sW

= !1- 2 11-

ei eos () + j dsin () + O(EW

Forming Ig(s)j2 by multiplying g(s) by its complex conju gate , we ha ve jg(s)12 = !1-2 (1 - 2d cos () + o( E» (8.64)

= xli =

II!1-i. Now since () lies in the ran ge -n/2 ~ (} ~ 0, we have as E -+-O that on the shrinking semicircle surro unding the origin

where, as usual, p

71"12 , which gives cos (}

~

2 2 2!1-E !1- - 2!1- E co s (} >!1- - - cos /}

(8.65)

p

This last is true since p < I for our stable system . The left-ha:nd side of Inequality (8.65) is merely the expression given in Eq. (8.63) for I/(s)/2 correct up to thefirst order in E , and the right-hand side is merely the expression in Eq. (8.64) for Ig(S)j2, again correct up to the first order in E. Thus we have shown that in the vicinit y s = 0, I/(s) 1 > jg(s)l. Thi s fact now having been established for all points on the contour C , we may apply Rouche's theorem and state that I (s) and I (s) + g(s) have the same number of zeroes inside the contour C. Since I (s) has only one zero (at s = - !1-) it is clear that the expression given in Eq. (8.59) [/(s) + g(s) ] ha s only one zero for Re (s) < 0 ; let this zero occur at the point s = - S1' As discussed a bove, the point 5 = 0 is also a root of Eq. (8.59). We may therefore write Eq . (8.58) as

'I"+(s) = r!1- A *( - 5) - 5 - !1-J [ 5(5 + 51)J '1'_(5) L 5(5 + 51) 5 +!1-

(8.66)

where the first bracketed term contains no poles and no zero es in Re (s) ~ 0 (we have di vided out the only two zeroes at s = 0 and s = - 51 in this halfplane). We now wish to extend the region Re (s) ~ 0 into the region Re (s) < D and we choose D (> 0) such that no new zeroes or poles of Eq. (8.59) are introduced as we extend to this new region . The first br acket qu alifies for ['I"_(S)]-1, and we see immediatel y that the second bracket qual ifies for 'I"+(s) since none of its zeroes (s = 0, s = - S1) or poles (s = -!1-) are in

-8.2. Re (5)

SPECTR AL SOLU TION TO LINDLEY 'S INT EGRAL EQUATION

> O. We may then factorize Eq.

297

(8.66) in the following form:

0/+(5) = 5(5 + 51) 5+P

(8.67)

'1" _(5) =

(8.68)

-5(5 + 51) 5 +p.-.P.A*(-5)

We have now assured that the functions given in these last two equations satisfy Conditions (8.36) and (8.37). We evaluate the unknown constant K as follows: .

0/+(5)

.

S + SI

K = hm-- = hm-,- 0

=~=

S

. - 0 S + p.

(8.69)

W(o+)

P. Thu s we have from Eq. (8.41)

+(5) = St(p. + s) P.5(5 SI)

+

The partial-fraction expansion for this last function gives us . +(5) =

! _15

sdp. S +51

(8.70)

Inverting by inspection we obtain the final solution for G{M{1 : W(y)

=

1- (1 - ; )e-

S 1Y

y

~0

(8 .71)

The reader is urged to compare this last result with that given in Eq. (6.30), also for the system G{Mjl; the comparison is clear and in both cases there is a single constant that must be solved for. In the solution given here that constant is solved as the root of Eq. (8.59) with Re (s) < 0; in the equati on given in Chap ter 6, one must solve Eq. (6.28), which is equivalent to Eq. (8.59). Example 3: The example for G{Mjl can be carried no further in the general case. We find it instructive therefore to consider a more specific G{M{I example and finish the calculations; the example we choose is the one we used in Chapter 6, for which A *(s ) is given in Eq . (6.35) and corresponds to an Ez{M{1 system, where the two arrival stages have different death rates. For that example we

298

THE QUEUE

GIG II s-plane

)( I'

Figure 8.8 Pole-zero pattern for E2 /M{1 example. note that the poles of A *( -s) occur at the points s = fl , s = 21t, which as promised lie in the region Re (s) > O. As our first step in factori zing we form 'I-'+(s) = A*(- s)B*(s) _ 1 '¥_(s) \ \

= [(fl -

S~~2~

- s(s - It

(s

- S)]

C: J-

+ fl / 2)(S -

+ fl )(fl

fl -

1

,u.j2 )

- s)(2fl - s)

(8.72)

The spectrum factorization is considerably simplified if we plot these poles and zeroes in the complex plane as shown in Figure 8.8. It is clear that the two poles and one zero in the right half-plane must be associated with 'r'_(s). Fu rthe rmore, since the strip 0 < Re (s) < fl contains no zeroes and no poles we cho ose D = fl and iden tify the remaining two zeroes and the single pole in the region Re (s) < D as being associated with 'I'"+(s) . Note well that the zero located at s = (l - .J2)fl is in fact the single root of the expre ssion flA *( -s) - s - fl located in the left half-plane, as discussed a bove, and therefo re s, = -(I - J 2),u. Of course, we need go no further to solve our problem since the solution is now given thro ugh Eq . (8.71) ; howe ver, let us co ntin ue identifying various forms in our solution to clarify the rema ining steps. With this factorization we may rewrite Eq. (8.72) as

'f'"+(s)

-- 'L(s)

- (S - fl - fl JZ)] rls(s - It + It J [ (fl - s)(2fl - s) s + ,u

2l]

•

8.3.

KI NGMAN'S ALGE BRA FOR QUEUES

299

In th is form we rec ogn ize the first bracket as II' F_(s) and the seco nd bracket as 'F+(s). Thus we have \I'"+(s)

+

=

s(s - It It J 2) s +1t

(8.73)

We may evaluate the con stant K fro m Eq . (8.69) to find

K

s

-1

+ ./2-

(8.74) It a nd thi s of co urse corresponds to W(O+) , whic h is the probability tha t a ne w arrival mu st wa it for service. Finally then we substi tute these values into Eq. (8.7 1) to find W(y) = I -

= ...! =

(2 - J 2)e- p ( v "2-

Uy

y

~0

(8 .75)

which as expected correspon ds exactly to E q . (6.37). T his method of spectru m facto rizatio n has been used successfully by Rice [R IC E 62], who con siders the busy peri od for the GfGfl system. Amon g the interesting results available, there is one corresponding to the limiting distribution of lon g waiting time s in the hea vy-traffic case (which we de velop in Section 2.1 of Volume II) ; Rice gives a similar a pp ro ximatio n for the duration of a busy peri od in the heav y tr affic ca sco

8.3.

KINGMAN'S ALGEBRA FOR QUEUES

Let us once again sta te the funda menta l relat ionships underlying the que ue. Fo r u; = X n - t n +! we have th e basic relat ion ship

GfGfl (8.76)

a nd we have a lso seen th at U,"n

=

max [0,

U n _ 17 U n _ 1

+

,• un _ 2 ,

. . . ,

U n_ 1

+ . .. + U 1 , U n_ 1 + . . . + U o + H'o]

We observed ea rlier th at {lV n } is a Markov process with sta tio nary tr an sition prob abilities ; its tot al stoc has tic structure is givcn by P[ lV m+ n :::; y 11I'm= x], which may be calcu lated as a n n-fold integral ove r the n-d imensional joint d istri b utio n of the n rando m va ria bles \I' m +!' .• . , \l'm+ n ove r that regio n o f the space which results in 11·.,+ n :::; y. T his ca lculation is mu ch too complicated an d so we look fo r alternative means to so lve this p ro blem. Pollaczek [POLL 57] used a spectra l ap proach and comp lex integrals to carry ou t the sol ution. Lind ley [LI ND 52) observed that Il' n has the sa me d istribution as defin ed ea rlier as

IV : ,

300

THE QUEU E

G/G/I

If we have the case E[u n ] < 0, which corresponds to p = xli < I, then a stable solution exists for the limiting random variable IV such that IV = sup U';

(8.77)

l1 ~ O

independent of 11'0 ' The method of spectrum factorization given in the previous section is Smith's [SMIT 53] approach to the solution of Lindley's Wiener-Hopf integral equation. Another approach due to Spitzer using combinatorial methods leads to Spitzer's identity [SPIT 57]. Many proofs for this identity exist and Wendel [WEND 58] carried it out by exposing the underlying algebraic str ucture of the problem. Keilsen [KEIL 65] demonstrated the application of Green's functions to the solution of G/G /1. Bend [BENE 63] also considered the G/G/I system by investigating the unfinished work and its variants. These many approaches, each of which is rather complicated , forces one to inquire whether or not there is a larger underlying structure, which places these solution methods in a common framework. In 1966 Kingman [KING 66] addressed this problem and introduced his algebra for queues to expose the comm on structure ; we study this algebra briefly in this section . From Eq . (8.76) we clearly could solve for the pdf of IV n +! iteratively starting with n = and with a given pdf for 11,'0; recall that the pdf for u ; [i.e., c(u)] is independent of n. Our iterative procedure would proceed as follows. Suppose we had already calculated the pdf for IV .. which we denote by IVn(Y) ~ dWn(y) /dy , where Wn(y) = P[lV n :::;: y]. To find wn+1 (y) we follow the prescription given in Eq. (8.76) and begin by formin g the pdf for the sum IV n + U n> which, due to the independence of these two random variable s, is clearly the convolution IVn(Y) 0 c(y). This convolution will result in a density funct ion that has nonne gative va lues for negative as well as positive values of its argument. However Eq. (8.76) requires that our next step in the calculation of IVn+1 (Y) is to calculate the pdf associated with (wn + u n)+; this requires that we take the total probability associated with all negative arguments for this density just found [i.e., for II'n(Y)0 c(y) ] and collect it together as an impul se of probability located at the origin for wn+1 (Y)' The value of this impulse will ju st be the integral of our former density on the negative half line. We say in this case that " we sweep the probability in the negative half line up to the origin ." The values found from the convolution on the positive half line are correct for W n +1 in that region. The algebra that describes this operation is that which Kingman introduces for stud ying the system G/G /1. Our iterative procedure continues by next forming the convolution of wn+1(Y) with c(y), sweeping the pr obability in the negative half line up to the origin to form \I' n+2(Y) and then proceed s to form ll'n+3(Y) in a like fashion , and so on.

°

7

8.3.

KINGMAN'S ALGEBRA FOR ' QUEUES

301

The elements of this algebra consist of all finite signed measures on the real line (for example, a pdf on the real line). For any two such measures, say hi and h 2 , the sum hi + h2 a nd also all scalar multiples of either belong to th is algebra. The product o peration hi 0 h 2 is defined as the convoluti on of hi with h2 • It can be shown th at this algebra is a real commutative algebra. There a lso exists an identity element denoted by e such that e 0 h = h for an y h in the algebra, and it is clear that e will merely be a unit impulse located at the origin. We are interested in operators that map real functions into other real functi ons and that are measurable. Specifically we are interested in the operato r that takes a value x and maps it into the value (z) " , where as usual we have (x)+ ~ max [0, x). Let us denote this operator by 11', which is not to be confused with the matrix of the transition probabilities used in Chapter 2 ; thus, if we let A denote some event which is measurable, and let h(A) = P{w : X(w)tA } denote the measure of this event, then 11' is defined through 11'[h(A») = P{w: X +(w)€A } We note the linearity of this operator, that is, 11'(ah) = a11'(h) and 11'(h l + h2 ) = 11'(h l ) + 11'(h2 ) . Thus we have a commutative algebra (with identity) alo ng with the line ar o pera to r 11' that maps this algebra into itself. Since [(x) +]+ = (x)+ we see that an important property of this operator 11' is th at A linear operator satisfying such a condition is referred to as a projection. Furthermore a projection whose range and null space are both subalgebras of the underlying algebra is called a Wendel projection; it can be shown th at 11' has this property, and it is this that makes the solution for G/Gfl possible. Now let us return to considerations of the queue G /G/1. Recall that the random va riable u; has pdf c(u) and that the waiting time for the nth cu stomer II' n has pdf 1\'n(Y). Ag ain since u.; and IVn are independent then IVn + U n has pdf c(y) @ lI'n(Y). Furthermore, since II' n+! = (IV n + u n)+ we have therefore

n=O,I, . ..

(8.78)

and thi s equati on gives the pdf for waiting times by induction. Now if p < I the limiting pdf 1I'(Y) exists and is independent of 11'0' That is, Ii- must hate the same pdf as (Ii' + ii)+ (a remark due to Lindley [LIND 52]). Th is gives us the ba sic equation defining the stationary pdf for waiting time in G /G/ I : ll'(Y) = 11'(c(y) @ If (Y»

_ (8.79)

The solution of this equation is of main interest in solving G/G/1. The remaining p orti on of th is secti on gives a succinct summa ry of some elegant results invol ving thi s algebra; only the courageous are encouraged to continue.

,•

302

TH E QUEUE

GIGII

The particular formalism used for constructing this algebra and car rying out the solution of Eq. (8.79) is what distinguishes the various meth ods we have menti oned above . In order to see the relationship among the various approaches we now introduce Spitzer's identity . In order to state this identit y, which involves the recurrence relation given in Eq. (8.78), we must intr oduce the following z-transform : co

X(z, y)

=I

wn(y)z'

(8.80)

n= O

Addition and scalar multiplication may be defined in the ob vious way for this power series and "multiplication" will be defined as corre sponding to convolution as is the usual case for tran sforms. Spitzer's identity is then given as (8.81) where y ~ log [e - zc(y)] (8.82) Thus wn(Y) may be found by expanding X( z, y) as a power series in z and picking out the coefficient of Z' . It is not difficult to show that X(z, y )

=

wo(Y)

+ Z1T(C(Y)®

X (z, y)

(8.83)

We may also 'form a generating function on the sequence E[e- Su: n ) ; ; W . *(s) , which permits us to find the transform of the limiting waiting time; that is, lim W n *(s) = W *(s) ;; E[e- SW) n_ ",

so long as p < I. This leads us to the following equation , which is also referred to as Spitzer's identity and is directly applicable to our queueing problem : W*(s)

= exp ( - Icc -I E[I - e- s lU n ) + ] ) n= l

n

(8.84)

1

/.

,I

We never claimed it would be simple! '] If we deal with W . *(s) it is possible to define another real commutative algebra (in which the product is defined as multiplicati on rather than con volution as one might expect). The algebraic solution to our basic equat ion (8.79) may be carried out in either of these two algebra s ; in the transformed t Fr om this identity we easily find th at A

E[ lj'l = W =

I'" -I E[ (U . )+j

n :>; 1 n

II

I

I

•

8.3.

KIN GMAN'S ALG EBRA FOR QUEUES

303

ca se o ne de als with the power series -

il

00

L Wn*(s)zn

X* (Z, S) =

(8.85)

n =O

rather than with the series given in Eq. (8.80). Pollaczek considers th is latter case and for G /G/I obtains the followin g equation which serves to define the system behavior :

*

*

X (z, s) = Wo (s)

ZS J ie+oo C*(s')X*(z, s') ds'

+ -. 21r}

ie-oo

S

"(s -

s)

(8.86)

and he then show s after considerable complexity tha t thi s solutio n mu st be of the form (8.87 ) where Yes) ~ log (I - zC *(s)) a nd S J ie+oo yes') ds' ,T(Y(s)) ~ - . , , 21r} ie-oo S (S - s) When C*( s) is simple enough then these expressions can be evaluated by contour integrals . On the other hand , the method we have described in the previous section using spectru m factorization may be phrased in terms of this algebra as follows . If we replace s 0 then II'n+l = 0 in which case Yn is merely the length of the idle period, which is terminated with the arrival of C n+!. Moreover, since either II' n +l or Yn must be 0, we have that (8.90) We adopt the convention that in order for an idle period to exist, it must have nonzero length, and so if Yn and w n + 1 are both 0, then we say that the busy period continues (an annoying triviality). From the definitions we observe the following to be true in all cases: (8.91) From this last equation we may obtain a number of important results and we proceed here as we did in Chapter 5, where we derived the expected queue size for the system M jGjl using the imbedded Markov chain approach. In particular, let us take the expectation of both sides of Eq. (8.91) to give E[wn+l1 - E[Yn] = E[wnl

+ E[un]

We assume E[unl < 0, which (except for D{D{I where ~ 0 will do) is the necessary and sufficient condi tion for there to be a stationary (and unique) waiting-time distribution independent of n ; this is the same as requiring p = xji < I. In this case we have* lim E[wn+!1

n-co

=

lim E[wnl n_ CX)

• One must be cautious in claiming that lim E[wn kl

f'l _ oo

=

lim E[w: +1l '"- co

since these are distinct random variables . We permit that step here, but refer the interested reader to Wolff [WOLF 70) for a careful treatment.

, •

8.4.

THE IDLE TIME AND DUALITY

305

and so our earlier equation gives E[Y]

=

(8.92)

-E[u]

where Yn -- Yand u; -- ii . (We note that the idle periods are independent and identicall y distributed, but the duration of an idle period does depend upon the duration of the previous busy period.) Now from Eq. (8.13) we have E[u] = i (p - I) and so (8.93) E[y] = i(l - p) Let us now square Eq. (8.91) and then take expected values as follows :

Using Eq. (8.90) and recognizing that the moments of the limiting distribution on IV n must be independent of the subscript we have E[(y)2]

=

2E[lvii]

+ E[(ii)2]

We now revert to the simpler notation for moments, " ok ~ E[(w)'], etc. Since W n and Un are independent random variables we have E[wu] = wii; using this and Eq , (8.92) we find _ .:l

u2

y2

2ii

2y

IV=W=----

- (8.94)

Recalling that the mean residual life of a random variable X is given by X2f2X, we observe that W is merely the mean residual life of -u less the mean residu al life of y! We must now evaluate the second moment of ii. Since ii = x - i ; then u2 = (x - i)2, which gives . (8.95) where a; and a/ are the variance of the interarrival-time and service-time densities, respectively. Using this expression and our previou s result for u we may thu s convert Eq. (8.94) to

y2 2y

(8.96)

We must now calculate the first two moments of y [we already know that = i(l - p) but wish to express it differently to eliminate a constant]. This we do by conditi oning these moments with respect to the occurrence of an idle peri od. Th at is, let us define

fj

Go

= P[y > 0] = P[arrival finds the system idlej

1

(8.97)

,

•

306

TH E QUEUE

GIGII

It is clear tha t we have a stable system when ao > O. Fu rthermore , since we have defined an idle period to occur only when the system remain s idle for a nonzero interva l of time, we have that

I > 0] =

P[fi ~ y y

P[idle period ~ y]

(8.98)

a nd this last is just the idle-period distribution earlier den oted by F(y ). We denote by I the random variable represent ing the idle period. Now we may calculate the following:

y = E[Y I y = O]P[Y = 0] = 0 + auE[y I y > 0]

+ E[Y I y > O]P[y > 0]

The expectation in this last equation is merely the expected value of I and so we have . (8.99) Similarly, we find (8.100) Thus, in particul ar, y2/2y = 12/21 (a o cancels!) and so we may rewrite the expre ssion for Win Eq. (8.96) as IV =

ao" + ao" + (i)"(1

- p)2

2i(1 - p)

2 1

21

- (8.101)

Unfortunately this is as far as we can go in establ ishing W for GIG /I. The calculation now invo lves the dete rmin ation of the first two moments of the idle period. In general, for G/Gfl we cannot easily solve for these rnojnents since the idle period depends upon the particul ar way in which the previous busy period terminated. However in Chapter 2, Volume II , we place bounds on the second term in this equation , thereby bounding the mean wait W. As we did for M IGII in Chapter 5 we now return to our basic equat ion ' (8.91) relat ing the import ant rand om variables and attempt to find the transform of the waiting time density W* (s) ~ E[ e- siD ] for GIG/I. As one might expect this will involve the idle-time distribution as well. Formin g the tran sform on both sides of Eq . (8.91) we have

However since

II' n

and u.; are independent, we find (8.102)

•

8.4.

THE IDL E T IME AND D UA LIT Y

307

In order to evalu a te the left-h and side o f thi s tra nsform expression we ta ke advantage of the fact that o nly o ne or the other of the ra ndom variables IV n +l and Yn may be nonzero. Accordingly, we have E [e- S1wn+.--1In) ]

= E[e- sI- On ) IYn > O]P[Yn > 0] + E [e- swn+ I s; = O]P[Yn = 0] 1

(8.103)

T o determine the right -h and side o f this last eq uatio n we may use the following simi lar expansion :

I

E[e- swB+'] = E [e- SIDn+l s; = O]P[Yn = 0] E [e- sID B+' I s; O]P[Yn 0]

+

>

> E[e- SIDft +' I Yn > 0] =

(8.104)

I. M a king use H owever , since IV n +l Y n = 0 , we have of the defin ition fo r 0 0 in Eq . (8.97) a nd allowi ng the limit as» -+ ex) we obta in the following tr an sform expression from Eq . (8.104) :

E [r S ;;; I fj = O]P[y

= 0] =

w *(s) -

00

We may then write the limiting form of Eq . (8. 103) as E [e- sl ;;;- '

)]

= 1* ( -

s)oo + W*(s) -

00

(8.105)

wher e 1*(s) is the Laplace transform of the idle-time pdf [see Eq. (8.98) for the defin ition of th is d istribution ]. Thus, fro m thi s last a nd from Eq. (8.102), we ob tai n immediatel y

W*(s)C*(s) =

0 01*( -

s) + W*(s) -

00

where as in the past C*(s) is the Laplace tr an sform for the den sity describing the random varia ble ii . Thi s last equation fina lly gives us [M ARS 68]

* 0 0 [1 - 1*(- s)] W (s) =~ 1 - C*(s)

- (8.106)

which represents the genera lization o f the Pollaczek-Khinchin tran sform equ at ion given in Chap ter 5 and which now appl ies to the system G/G/ 1. Clearly this eq ua tio n hold s a t least alo ng the imagi nary ax is of the complex splane , since in that case it become s the characteristic func tion of the various distr ibution s which a re kn own to exist . Let us now co nsider some examples. Example 1: M/M /1

For thi s system we know that the idle-period d istributi on is the same as the intera rri val-time dist ribution, nam ely , F(y) = P[l ~ y ] = I - e- 1 •

y ~ O

(8.107)

•

308

THE QUEUE

G /G/I

A nd so we ha ve th e first two moments 1 = 1/)., / 2 = 2/).2; we also have aa2 = 1/ ).2 and a b 2 = I/fl2. Using these value s in Eq. (8. 10 1) we find

+

+

).2(1/).2 l /fl2) ( I _ p)2 W = ----'--'-----'-'--'---'-- ..:....:..2).( 1 - p)

I ).

a nd so

w=

P/fl

(8. 108)

1 - p

which of course checks with o ur earlier results for M/M /!. We kn ow th at I *(s) = A/ (s + A) and C*(s) = ).fl/ (A - s)(s + fl). Moreove r , since the prob ability that a Poisson a rrival finds the system empty is the sa me as the lon g-run proportion of time the system is empty, we have that Go = I - P and so Eq . (8 . 106) yields

*

W (s) =

( I - p)[1 - A/(A - s)]

"------'-'~-'-"---'-=

I - ).fl/(A - s)(s

=

- (I - p)s(s

+ fl)

+ fl) p)(s + fl)

(A - s)(s

(1 -

+ fl) Afl

(8.109)

S + fl - A which is the sa me as Eq . (5. I20).

Example 2: M /G//

..

In thi s case the idle-time d istribution is as in M/M /I ; howe ver , we mu st lea ve the va ria nce for the ser vice-t ime distribution as a n unkn own. We obta in A2 [(I/A 2 ) + a b2] + (I - p)2 1 W = --=-'---'---'-"---.::....:..."---'-------'--'J,

2).(1 - p) =p

(1

+C

2 b )

p) which is the P-K formula . Also , C*(s) (I - pl· Equation (8 . 106) then gives

• (8 .1 10)

2fl (1 -

w*(s)

=

( I - p)[1 -

B*(S)A/ <J. - s) a nd agai n

J./U -

Go

=

s)]

= ..:....:..---'--'-'----'-----'--'

1 - [)./(A - s)]B*(s) s( I - p) s - A + AB*(s)

which is the P-K transform equation for waiting time!

(8.II I)

8.4 .

THE IDLE TIME AND DUALITY

309

Example 3: D!D!] In this case we ha ve that the length of the idle period is a constant and is given by i = i - x = t(l - p); therefore 1 = t(l - p), and j2 = (1) 2. Moreover, aa2 = a/ = O. Therefore Eq . (8.101) gives IV = 0

+ Ufo -

p)2 _ H(1 _

2t(l - p)

-

) p

and so

w=o

(8.112)

This last is of course correct since the equilibrium waiting time in the (stable) system D/D!I is always zero. Since i, I, and] are all constants, we have B*(s) = e- ii , A *(s) = e-" and I*(s) = e- ,l = r ,m -p). Also, with probabil ity one an arrival finds the system empty; thus ao = I. Then Eq. (8.106) gives

*

IV (s)

=

1[1 - e"ll- p)] ,._ - , I 1- e e (8.113)

=1

and so w(y) = uo(y ), an impulse at the origin which of course checks with the result that no waiting occurs. Considerations of the idle-time distribution naturally lead us to the study of duality in queues. This material is related to the ladder indices we had defined in Section 5.11. The random walk we are interested in is the sequence of values taken on by U'; [as given in Eq. (8.9)]. Let us denote by Un. the val ue taken on by U; at the kth ascending ladder index (instants when the function first drops below its latest maximum). Since u < 0 it is clear that lim U; = - ~ as 11 ->- 00 . Therefore, there will exist a (finite) integer K such th at K is the large st as cend ing ladder index for Un' Now from Eq. (8.11) repe ated below Ii '

It is clear that

=

sup U n n 2:: o

Ii' = U n"

Now let us define the random varia ble an idle time) as

t, (which as we shall see is related to (8 .114)

310

TH E QUEU E

GIG II

for k ::;; K. That is t, is merely the amount by which the new ascending ladder height exceeds the previous ascending ladder height. Since all of the random variables u; are independent then the rand om variables i k cond itioned on K are independent and identically distributed. If we now let I - a = P[Un ::;; Un. for all n > nk ] then we may easily calculate the distribution for K as P[K = k] = (I - a)a k

(8.115)

In exercise 8.16, we show that ( I - c) = p[iv = 0]. Also it is clear that

l+l+···+I-=u 1 2 K n l -Un o + Un! -Un l + ···+ Un g - U" K - l =Un x where no ~ 0 and Uo ~ O. Thus we see that + i K and so we may write

11 + ...

E[e-'W]

=

whas the same distribution as

= E[(i*(s))K] \

\

I

E[E[e- ,l1,+·· ·+Jx ) K]]

(8.116)

where I *(s) is the Lapl ace transform for the pdf of each of the t, (each of which we now denote simply by I). We may now evaluate the expectation in Eq . (8.116) by using the distribution for Kin Eq. (8.115) finally to yield

\

W*( ) 1- a s = I _ ai*(s)

I

- (8.117)

Here then , is yet another expre ssion for W*(s) in the GIGII system. We now wish to interpret the random variable 1 by considering -a ."dual" queue (whose varia bles we will distinguish by the use of the symbol "), The dual queue for the GIGII system considered above is the queue in which the service times x n in the original system become the interarrival times i n+1 in the dual queue and also the interarrival time s I n + 1 from the origin al queue become the service time s xn in the dual queue.] It is clear then that the random variable Un for the dual queue will merely be Un = xn '- i n+! = I n+! - X n = -Un and defining O n = Uo + ... + Un_ 1 for the dual queu e we have (8.118) t Clearly , if the origina l queue is sta ble, the du al must be unstabl e, a nd conversely (except that both may be unstable if p = I) .

8.4.

THE IDLE TIME AND D UALITY

311

as the relationship am ong the dual and the original queues. It is then clear from our discussion in Section 5.11 that the ascending and descending ladder indice s are interchanged for the original and the dual queue (the same is true of the ladder heights). Therefore the first ascending ladder index n) in the original queue will correspond to the first descending ladder index in the dual queue ; however, we recall that descending ladder indices correspond to the arrival of a customer who terminates an idle period. We denote this customer by en,. Clearly the length of the idle period that he terminates in the dual queue is the difference between the accum ulated interarrival times and the accumulated service times for all customers up to his arrival (these services must have taken place in the first busy period), that is, for the dual queue,

Length of first idle period} {following first busy period

n 1-1.

= n~o t n+l n l-l

=

L

A

n~o X n

n l -l

Xn -

n ""O

=

n , -1

-

L

I n+J

n= O

.U n,

(8.119)

where we have used Eq. (8.114) at the last step. Thus we see that th e random variable j is merely th e idle period in th e dual queue and so our Eq . (8.117) relate s the transform of the waiting time in the original queue to the transform of the idle time pdf in the du al queue [contrast this with Eq . (8.106), which relates this waiting-time transform to the tran sform of the idle time in its own queue] . Thi s d uality observation permits some rather powerful conclusions to be d rawn in simp le fashion (and these are discussed at length in [FELL 66], especially Sections VI.9 and XII.5). Let us discuss two of these . Example 4: GIMII

If we have a sta ble G IM II queue (with i = II), and x = IIp.) then the du al is an unstable queue of the type M IGI I (with i = IIp. and if = II), and so 1 (the distribution of idle time in the dual queue) will be of exp onential form; therefore 1*(s ) = p.1(s + p.), which gives from Eq. (8.11 7) the follo wing

--312

TIlE QU EUE

G/GfI

result for the original G{M /I queue : W *(s) = (1 - O")(s s + f-L -

+ /-l ) 0"1-'

In vertin g this a nd form ing the PDF for waiting time we have W(y) = I

_· O"e- p ll- a lY

y ~ O

(8.120)

which correspond s exactly to Eq. (6.30)_

Example 5: M IGI}

As a second example let the original queue be of the form M /G fl and therefore the dual is of the form G/M /I. Since 0" = P[ w > OJ it must be that 0" = P for M IG/I . Now in the du al system, since a busy period end s at a random point in time (and since the service time in this dual queue is memoryless), an idle per iod will have a durati on equ al to the residual life of an interarrival time ; therefore from Eq. (5.1I) we see that

1*(s) = I - B*(s) sx

(8.121)

and when these calc ulatio ns are applied to Eq. (8.117) we have

W *(s)

= 1-

I - P p{[l - B*(s)]/sx }

(8.122)

which is the P-K transform equ ati on for wailing time rewritten as in Eq. (5.106). Th is conclu des our study of G/G /1. Sad to say, we have been unable 10 give analytic expressions for the waiting-time distribution explicitly in terms of known qu antities. In fact, we have not even succeeded for the mean wait W! Nevert heless, we have given a method for handling the rational case by spectrum fact orizati on , which is quite effective. In Chapter 2, Volume II, we return to G/G{I and succeed in extracting many of its important properties throu gh the use of bounds , inequ alities, and approximations.

REFERENCES

313

REFERENCES ANDE 53a ANDE 53b ANDE 54 BENE 63 FELL 66 KEIL 65

KING 66 LIND 52 MARS 68

POLL 57

RICE 62

SMIT 53 SPIT 56

SPIT 57 SPIT 60

SYSK 62 TITC 52 WEND 58

Andersen , S. E., "On Sum s of Symmetrically Dependent Random Vari ables," Skan . A k tuar., 36, 123-138 (1953). Andersen, S. E., "On the Fluctuations of Sum s of Random Variables I," Math . Scand., 1,263-285 (1953). Andersen , S. E., "On the Fluctuati on s o f Sums o f Random Varia bles II," Math . Scand., 2, 195-223 (1954). Benes, V. E., General St ochastic Processes in the Theory of Queues , Addison-Wesley (Reading, Mass.) , 1963. Feller, W. , Probability Theory and its Applications, Vol. II, Wiley (New Yo rk), 1966. Keilson, J ., " The Role of Green's Function s in Congestion The ory," Proc. Symp, on Congestion Theory (edited by W. L. Smith and W. E. Wilkinson) Uni v. of North Carolina Press (Chapel Hill) , 43-71 (1965). Kingman, J. F. C,; "On the Algebra of Queues," Journal of Applied Probability , 3, 285-326 (1966). Lindley , D. V., "The Theory of Queues with a Single Server," Proc. Cambridge Philosophical Society , 48, 277-289 (1952). Marshall, K. T., "Some Relationships between the Distributions of Waiting Time , Idle Time, and Interoutput Time in the GI/G /I Queue," SIAM Journal Applied Math ., 16,324-327 (1968). Pollaczek, F ., Problemes St ochastiques Poses par le Phenomene de Formation dune' Queue d'Attente a un Guichet et par de Phenomenes Apparentes, Gauthiers Villars (Paris), 1957. R ice, S. o., " Single Server System s," Bell Sys tem Technical Journal, 41, Part I : "Relations Between Some Averages," 269- 278 a nd Part II: "Busy Period s," 279-310 (1962). Smith, W. L., " O n the Di stribution of Queueing Times," Proc. Cambridg e Philosophical Society , 49, 449-461 (1953). Spitzer, F . "A Combinatorial Lemma and its Application to Probability Theory," Transactions of the American Math ematical Society , 82, 323-339 ( 1956). Spitzer, E , "The Wiener-Hopf Equ ation whose Kernel is a Probability Density," Duke Math ematics Journal, 24, 327-344 (1957). Spit zer , F . "A Tauberian Theorem a nd its Probability Interpretatio n," Transactions ofthe American M athematical Society, 94,150-1 60 (1960). Syski, R. , Introduction to Cong estion Theory in Telephone Systems , Oliver and Boyd (London), 1962. Titchmarsh, E. C, ; Theory of Functions, Oxford Uni v, Press (London) , 1952. Wendel, F. G ., "Spitzer'S Formula; a Short Proof," Proc. American Math ematical Society, 9, 905-908 (1958).

3 14

THE QUEUE

WOLF 70

GIGII

Wolff, R. W., " Bounds and Inequalities in Queueing," unpublished notes, Department of Industrial Engineering and Operations Research, University of California (Berkeley), 1970.

EXERCISES 8.1.

From Eq. (8.18) show th at C*(s)

=

A *( - s)B *(s).

8.2. Find ceu) for MIM I!. 8.3.

Consider the system MIDI I with a fixed service time of x sec. (3) Find ceu) = P[u n .::;; II] a nd sketch its shape. (b) Find E[u n ].

8.4 .

For the seq uence o f random variables given below, generate the figure corresponding to Figure 8.3 and complete the ta ble. 0

II

\

t n -t 1

\

x.

5

6

2 I I 5 7 2 3 4 2 3 3 4

2 2

2

3 4

7

8

9 6 3

/In

lV .

measured

w. calculated

8.5.

Co nside r the case where p = I W (y - II) in Eq . (8.23) as W(y -

E

u) = W(y) - u W(l)(y)

for 0

+ -u

<E«

I . Let us expa nd

W(21(y)

+ R(u, y)

2

2

where w (nl(y) is the nth derivat ive of W(y) and R (II, y) is such th at ~'" R (u, y) dC(~) is negligible due to the slow varia tion of W( y) when p = I - E. Let Il k denote the kth mom ent of ii. (3) Und er these conditio ns con vert Lindley's integral eq uation to a seco nd-orde r linear d ifferential eq uation involving Ii" and ii. (b) Wit h the boundary cond ition W(O) = 0, solve the equation foun d in (a) and expre ss the mean wait W in term s of the first two moments of i and x. 8.6.

Consider the DIErl1 queueing system, with a con stant intera rrival time (of [sec) and a service-time pdf given as in Eq. (4.16). (3) Fi nd ceu).

-315

EXERC ISES

(b)

Sh ow th at Lindley' s integral equation yields W (y - i) for y < { and

W( y - i) =

(c)

f

for y

W( y - w) dB(w)

=0

~ {

Assume the following solution for W (y) : r

lV(y)

= 1 + L G,e'"

y~O

i- I

where a, a nd cx, may both be complex, but where Re (cx,) < 0 for i = 1, 2, . . . , r . Usin g thi s ass umed so lutio n , show th at the following equations mu st hold:

_, .,

e '

i (rp.. +a,cx

i_ 0

. i )' +!

=

(

= 0

rp. rp.

+ cx,

)r

i

=

1,2, . . . , r

j = 0, 1, . . . , r -

1

where ao = 1 and CX o = O. Note th at {cx,} ma y be found fr om the first set of (tra nscendenta l) equation s, a nd th en the seco nd set gives {a i } . It ca n be shown th at the CX i a re di stinct. See [SYSK 62]. 8.7 .

C onsider the following queueing systems in wh ich no queue is permitted . Cu st omers wh o a rrive to find the system bu sy mu st lea ve witho ut service. (a) M/M /l : Solve for P» = P[k in system]. (b) M/H ./I: A s in Figure 4.10 with cx , = cx , cx. = 1 - CX , P., = 2p.cx a nd p.. = 2p.(1 - «). (i) Find the mean ser vice time x. (ii) Solve for Po (a n empty system) , P« (a cu st om er in th e 2p.:t. box) and P'- a (a cu stomer in the 2p.(1 - Gt) box). (c) H./M fl : Where A(t) is hyp ere xp onential as in (b), but with par am et er s tI, = 2,1,cx an d p.. = 2}.( 1 - Gt) instead . Draw the sta te-tra nsition diagram (with labels on br a nche s) for th e foll ow ing four states: E iJ is state with " a rri ving" cu stomer in a rriva l stage i and j cu stomers in service i = I , 2 and; = 0 , I.

316

TH E QUEUE

GIG fl

= P[j stages of service left to go l.

(d)

M/Er/l: Solve for Pi

(e)

M/D fl: With all service times equal to x (i) Find the probability of an empty system. (ii) Find the fracti on of lost customers.

(0

EJM /I : De fine the four states as Eii where i is the number o f " arrival" stages left to go and j is the number of cust omers in service. Draw the labeled state-tran sition diagram.

8.8.

Consider a single-server queueing system in which the interarri val time is chosen with probability cz from an exp onential distribution of mean II). and with probability I - cz from an expo nential distribution with mean 11ft . Service is exponential with mean 11ft. (a) Find A *(s) and B* (s) . (b) Find the expression for 'I'+(s)/'I" _(s) and show the pole- zero plot in the s-pla ne. (c) Find 'I' +(s) and 'L(s) . (d) Find
8.9.

Con sider a G/G/I system in which

A*(s) - -

(s

2 :::....-_

_

-

+ 1)(s + 2) 1

8 *( s) = - -

s

+

1

Find the expression for 'I'+(s)/'I"_ (s) and show the pole- zero plot in the s-plane. (b) Use spectrum factorization to find 'I' +(s) and 'F _(s). (c) Find +(s). (d) Find W(y) . (e) Find the average waiting time W. (0 We solved for W(y) by the method of spectrum fact or izati on . Can you describe another way to find W( y ) ? (a)

8.10.

Consider the system M/G/1. Using the spectral so lution meth od for Lindley' s integral equation, find (a) . 'Y+ (s). {HINT: Interpret [l - B * (s)l/sx. } (b) 'I'_(s). (c) s+(s).

,

J

EXERCISES

8.11.

317

Consider the queue Eo/E r /1. (a) Show th at

'¥+(5) '1" _(5)

=

F(5) 1 - F(5)

where F(5) = I - ( 1 - 5/I..Q)o(l + 5/W Y . For p < 1, show th at F(5) has one zero at the o rigin, zeroes 5 \ ,52 , . . • . s, in Re (.I) < 0, and zeroes .1' + 1> Sr +2, .• • , 5 r +o- 1 in Re (5) > O. (c) Expre ss 'Y+(.1) and q.'_ (s) in term s of S i ' (d) Expre ss W*(s) in te rms of s, (i = 1,2, .. . , r + q - I). (b)

8.12.

Show that Eq. (8.71) is equivalent to Eq. (6.30).

8,13.

Consider a 0 /0/1 queue with p < 1. Assume W o = 4f(1 - p). (a) Calculate lI'n(Y) using the procedure defined by Eq. (8.78) for n = 0, 1,2, . . .. (b) Show th at the known solution for w(y)

= lim wn(y)

satisfies Eq. (8.79).

8.14.

Con sider an M/M /I queue with p < 1. Assume W o = O. (a) Calcul ate 11'\ (y) using the procedure defined by Eq. (8.78). (b) Repeat for 1t'2(Y)' (c) Show th at o ur known solution for W(y )

(d)

= lim wn(y)

satisfies Eq . (8.79). Compare 1t'2(Y) with I\·(Y).

8.15.

By first cub ing Eq. (8.91) and then forming expectations, expre ss a;;; (the variance of the waiting time) in terms of the first three moments of i, i , and I .

8.16.

Show th at P [w = 0] = I - a from Eq . (8.117) by find ing the cons tant term in a power-series expansion of W*( s).

8.17.

Consider a G/GfI system. (a) Express 1*(.1) in terms of the transform of the pdf of idle time in the given system. (b) Using (a) find 1*(.1 ) when the original system is the ordinary M/M /1.

318

THE QUEUE

(C)

G/G /!

Using (a), show that the transform of the idle-time pdf in a G/M/I queue is given by 1*(s)

=

1 - ~ *(s) sl

(d)

thereby reaffirming Eq. (8.121). Since either the original or the dual queue must be unstable (except for D/DfI), discuss the existence of the transform of the idle-time pdf for the unstable queue.

-Epilogue We have invested eight chapters (and two a ppendices!) in studying the theory of queuein g systems. Occasionally we have been overjoyed at the beaut y and generality of the results, but more often we have been overco me (with frustration) at the lack of real pr ogress in the theory. (No, we never promi sed you a rose garden.) However, we did seduce you into believing that this study would pro vide worthwhile methods for pract ical application to man y of today' s pressing congestion problems, We confirm that belief in Volume II. In the next volume, after a brief review of th is one, we begin by tak ing a more relaxed view of GIG II. In Chapter 2, we enter a new world leaving behind the rigor (and pain) of exact soluti ons to exact probl ems. Here we are willing to accep t the raw facts of life, which state that our models are not perfect pictures of the systems we wish to ana lyze so we should be willing to accept approximations and bounds in o ur problem solution. Upper and lower bound s are found for the average delay in GIG II a nd we find that these are related to a very useful heavy traffic approximati on for such qu eues. This approximation, in fact , predicts that the long waiting times are exponentially distributed. A new class of models is then introduced whereby the discrete a rrival and departure processes of queueing systems are replaced first by a fluid approximation (in which these stoc hastic pr ocesses are replaced by their mean values as a function of time), and then secondly by a diffusion ap proximation (in which we permit a variation about these means). We happil y find that these approximations give quite reasonable result s for rather general queueing systems. In fact th ey even permi t us to study the transien t behavio r not only of stab le queues but also of saturated queues, a nd this is the material in the final section of Chapter 2 whereby we give Newell' s tr eatm ent of the ru sh-hour approxi mation-an effective method indeed. Cha pter 3 points the way to our application s in time-shared computer systems by presenting some of the prin cipal results for pr iority queueing systems. We study general methods and appl y them to a number of import ant queue ing disciplines. The con servati on law for priority systems is establi shed , preventing the useless search for non realizable disciplines. In the remainder , we cho ose applications pr incipally from the computer field, since these application s are perh aps the most recent and successful for the theory of queue s. In fact, the queueing an alysis of allocation of resources and job flow through computer systems is perh aps the only tool available 319

320

EP ILOGUE

to computer scientists in understanding the behavior of the complex interaction of users, programs , processes, and resour ces. In Chapter 4 we emphasize multi-access computer systems in isolati on , handling demands of a large collection of competing users. We look for throughput and response time as well as utilization of resources . The major portion of this chapter is devoted to a particular class of algorithms known as pr ocessor-shar ing algorithms, since they are singularly suited to queueing analysis and capture the essence of more difficult and more complex algorithms seen in real scheduling problems. Chapter 5 addresses itself to computers in network s, a field that is perhaps the fastest growing in the young computer indu stry itself (most of the references there are drawn from the last three year s-a tell-tale indicator indeed) . The chapter is devoted to developing method s of anal ysis and design for computer-communication networks and identifies many unsolved important problems. A specific existing network , the ARPANET, is used th roughout as an example to guide the reader through the motivati on and evaluati on of the various techn iques developed. Now it remain s for you , the reader, to sharpen and appl y your new set of tools . The world awaits and you must serve !

I

L

A P PEN DIX

I

Transform Theory Re fresher: z - Transform and Laplace Transform

In this appendix we develop some of the properties and expressio ns for the z-transform and the Laplace transform as they apply to our studies in queueing theory. We begin with the z-transforrn since it is easier to visualize its operation. The forms and properties of both transforms are very similar, and we compare them later under the discussion of Laplace transforms. 1.1.

WHY TRANSF O RMS ?

So often as we progress through the study of interesting physical systems we find that transforms appear in one form or another. These transforms occur in many varieties (e.g., z-tra nsform, Laplace tr ansform, Fourier transform, Mellin transform, Hankel transform, Abel transform) and with a variety of names (e.g., transform, characteristic function , generating function) . Why is it that they appear so ofte n? The answe r has two parts ; first, because they arise naturally in the form ulation and the solution of systems problems ; and second , because when we observe or introduce them into our solution method , they greatly simplify the calculations. Moreover, oftentimes they are the only tools we have available for proceeding with the solution at all. Since transforms do appear naturally, we should inquire as to what gives rise to their appearance. The answer lies in tbe consideration of linear timeinvariant systems. A system, in the sense that we use it here, is merely a transformation, or mapping, or input-output relationship between two functions. Let us represent a general system as a "black" box with an input f and an out put g, as shown in f igure 1.1. Thus the system operates on the function f to prod uce tbe function g . In what follows we will assume that these functions depend upon an independent time parameter t ; this arbitrary choice results in no loss of generality but is convenient so that we may discuss certain notions mor e explicitly. Thus we assum e th at f = J(t} . In order to represent the input-output relationship between the functions J(t} and get}

321

322

APP ENDI X I

fa

~

Figure I.l

A general system.

we use the not ati on

f (t ) -.. get)

(I.!)

to den ote the fac t that get) is the output of our system when f (t) is applied as input. A system is sa id to be linear if, when

and

then a lso the following is true : (1.2)

where a a nd b are independent of the time va riable t. Further, a system is said to be time-invariant if, when Eq. (Ll ) holds, then the following is a lso true :

f(t

+ T} -.. get + T)

(1.3)

for a ny T. If the a bove two properties both hold , then ou r system is sai d to be a linear time-invariant system, a nd it is these with which we con cern ourselves fo r the momen t. . Whenever o ne studies such systems , o ne find s th a t complex exponent ial fun ctions of time a ppea r throu ghout the solution . Further, as we sha ll see, the tr an sforms o f interes t merely represent ways of dec omposing functi on s of time int o sums (o r int egrals) of complex exp onentials. Th at is, co mp lex expo nentials form the building blocks of ou r tr an sforms, a nd so, we m ust inq uire further to d iscover why the se complex exponentials pervade o ur thin kin g with such systems. Let us now pose the fundam ent al qu estio n, namel y , which fun ction s of tim e f( t) may pass through o u r linear ti meinvariant systems with no change in form ; th at is, for whichf(t) will g( l) = Hf (t }, where H is so me sca la r multiplier (with respect to I) ? If we can discover such functi on s f( t} we will then have found the "eigenfunctions," or "characteri stic functi on s," o r " inva ria nts" of ou r system. Denotin g th ese eigenfun ction s by f ,(I) it will be show n th at th ey mu st be of the following form (to within a n a rbitrary sca la r multiplier) : !e(t} = e st

(IA)

L

1.1.

W HY TRAN SFOR," tS?

323

where s is, in general, a complex varia ble. Th at is, the compl ex exponentials given in (1.4) form the set of eigenfunctions for all linear time-invariant systems. Thi s result is so fund amental that it is worthwhile devoting a few lines to its derivation. Thu s let us assume when we appl y[. (t) that the output is of the form g.(t), tha t is, f . (t ) = e' t -+ g.(t) But , by the linearity property we have e"f.(t)

=

e'(t+ T) -+ e"g.( t)

where T and therefore e" are both constants. Moreover , from the timeinvariance property we must have f .( t

+ T) = e,(t+T)

-+

git

+ T)

From the se last two , it must be that eSTg.(t)

=

g.(t

+ T)

Th e uniq ue solution to this equati on is given by g.(t)

=

He' t

which confirms our earlier hypothesis that the co mplex exponentials pass through our linear time-invariant systems unch anged except for the scalar mult iplier H. H is independent of t but may certa inly be a funct ion of s and so we choose to write it as H = H(s) . Therefore, we have the final conclu sion th at (r.5) an d this funda menta l result exposes the eigenfun ctions of our systems. In this way the complex expon enti als are seen to be the basic functions in the study of linea r time-invaria nt systems. Moreover , if it is tru e th at the in put to such a system is a complex expon ential , then it is a tr ivial computation to evaluat e the output of that system from Eq. (1.5) if we are given the function H (s ). Thus it is natural to ask th at for a ny inputf(t ) we would hope to be ab le to decomp ose f( t) into a sum (o r integral) of com plex expo nentials, each of which contri butes to t he overa ll outp ut g(t ) thr ough a com putation of the form given in Eq. (1.5). Then the overall output may be foun d by summin g (integrating) these individual comp onents of th e output. (The fact tha t the sum of the individual out puts is the same as the output of the sum of the individua l inpu ts-that is, the complex expo nentia l decomposition-is due to the linear ity of o ur system.) The process of decomposing our input into sums of exponentials, computing the respon se to each from Eq. (r.5), and then reconstitutin g the outpu t from sums of expo nentials is

324

APPENDIX I

referred to as the transform method of analysis. This approach, as we can see, arises very naturally from our foregoing statements. In this sense, transform s arise in a perfectly natural way. Moreover, we know that such system s are described by constant-coefficient linear differential equations , and so the common use of transforms in the solution of such equations is not surprising. We still have not given a precise definition of the transform itself; be patient , for we are attempting to answer the question " why transform s ?" If we were to pursue the line of reasoning that follows from Eq. (1.5), we would quickly encounter Laplace transforms. However, it is convenient at this point to con sider only functions of discrete time rather than function s of continuous time, as we have so far been discussing. This change in directi on brings us to a con sideration of a-transforms and we will return to Laplace transforms later in this appendix. The reason for this switch is that it is easier to visualize operations on a discrete-time axis as compared to a continuoustime axis (and it also delays the introduction of the unit impulse function temporarily). Thus we consider functions I that are defined only at discrete instants in time , which, let us say , are multiples of some basic time unit T. That is, l(t) = /(t = n T) , where n= . . . ,-2,-I,O,I ,2, .. .. In order to inco rporate this discrete -time axis into our notation we will denote the function I(t = nT) by In. We assume further that our systems are also discrete in time. Thus we are led to consider linear time-invariant systems with inputs I and outputs g (also functions of discrete time) for which we obtain the following three equations corresponding to Eqs. (Ll)-(I.3): ( 1.6)

i n ->- gn ai~ll

+ bi~;'

->-

a g~ll

f n+ m - . g n+ m

+ bg ~2 )

(1.7)

( 1.8)

where m is some integer constant. Here Eq. (I.7) is the expressi on of linearit y wherea s Eq. (I.8) is the expression of time -invariance for our discrete systems. We may ask the same fundamental question for the se discrete systems and , of course, the an swer will be essentially the same, namel y, that the eigenfunction s are given by On ce again the complex exponentials are the eigenfunctions. At this point it is convenient to introduce the definition (1.9)

and so the eigenfunctions I~' ) take the form

1.1.

WH Y TRANSFORMS ?

325

Sin ce s is a complex variable, so, to o , is z. Following through steps essentially identical to those which led from Eq. (104) to Eq. (I.5) we find that ( 1.10) where H(z) is a function independent of n. This merely expresses the fact that the set of functi on s {z:"} for any value of z form the set of eigenfunction s for discrete line ar time-invariant sys tems. Moreover, the functi on (co ns ta nt) H either in Eq . (1.5) o r (Ll O) tells us precisely how much of a given complex exp onential we get o ut of our linear system when we ins ert a unit a mo unt of that exp onential at the input. Th at is, H really describes the effect of the system on these exponenti als; for this reason it is usually referred to a s the

system (or transfer)function. Let us pursue this line of reasoning somewhat further. As we all know , a common way to discover what is ins ide a system is to kick it-hard and quickly. F or our sys tems this corresponds to providing an input only at time t = 0 a nd then ob serving the subsequent output. Thus let us define the Kronecker delta fun ction (also known as the unit fun ction) as

n=O (I. 11) n~O

When we apply u.; to our system it is common to refer to the output as the unit response, a nd th is is usually denoted by h; That is,

F ro m Eq . (1.8) we may therefore also write U n+ m

---+

hn+ m

From the linearity p roperty in Eq . (1.7) we have therefore

Certa inly we may multiply both expressions by unity , and so z- nz nzmu n+m

---+

z -nznzmhn+m

( 1. 12)

Furthermore , if we con sider a set of inputs {J~I}, and if we define the output for each of these by then by the linearity of our system we must h ave 2J~ I _Lg ~) i

(1.l3 )

326

APPENDIX t

If we now apply this last observation to Eq. (1.12) we have .,.- n "" II

""

.£.

n+ m

z n+ 1n ----+- .,.-n OJ

m

v hn+ m'" _ n+ m

Lm

where the sum ranges over all integer values of m . Fr om the definition in Eq. (I. I I) it is clear that the sum on the left-hand side of this equation has only one nonzero term , namely, for m = -n, and this term is equal to unity; moreover, let us make a change of variable for the sum on the right-hand side of this expression , giving z - !1

---+- z-n

L hkz k k

This last equation is now in the same form as Eq. (1.10); it is obvious then that we have the relationship (l.I 4)

This last equ ation relates the system functi on H( z) to the unit respon se hk • Recall that our linear time-invariant system was completely* specified by knowledge of H( z), since we could then determine the output for any of our eigenfunctions ; similarly, knowledge of the unit response also completely* determines the operation of our linear time-invariant system. Thus it is no surprise that some explicit relationship must exist between the two , a nd, of course, this is given in Eq. (I.I4). Finally, we are in a position to answer the question-why tran sforms ? The key lies in the expression (1.14), which is, itself, a transform (in this case a z-transform), which con verts'[ the time function hk int o a function of a complex variable H(z). This transform aro se naturally in our study of linear time-invariant systems and was not intr oduced into the anal ysis in an artificial way. We shall see later that a similar relationship exists for continuoustime systems, as well, and this gives rise to the Laplace transform . Recalling that continuous-time systems may be described by constant-coefficient linear differential equations and that the use of tr ansform s greatly simplifies the solution of these equations, we are not surprised that discrete-time systems lead to sets of constant-coefficient linear difference equations whose solution is simplified by the use of e-transforms. Lastly, we comment that the input s f • Completely specified in the sense that the only additional requ ired information is the initial sta te of the system (e.g., the initial condit ions of all the ener gy sto rage elements). Usually, the system is assumed to be in the zero-energy state, in which case we truly have a complete specification. t Transforms not only change the form in which the informati on describing a given function is presented , but they also present this inform ation in a simplified form which is con venient for. mathem atical man ipulation.

1.2.

TilE Z- T RAN SFORM

327

and the outputs g are easily decomposed into weighted sums of complex exponentials by means of transforms, and of course, once this is done, then results such as (1.5) or (1.10) immediately give us the component-by-component output of our system for each of these inputs; the total output is then formed by summing the output components as in Eq. (1.13). The fact that these transforms arise naturally in our system studies is really only a partial answer to our basic question regarding their use in analysis. The other and more pragmatic reason is that they greatly simplify the analysis itself ; most often , in fact, the analysis can only proceed with the use of transforms leading us to a partial solution from which properties of the system behavior may be derived. The remainder of this appendix is devoted to giving examples and properties of these two principal transforms which are so useful in queueing theory. 1.2. THE z-TRANSFORM [JURY 64, CADZ.73] Let us consider a function of discrete time In' which takes on nonzero values only for the nonnegative integers, that is, for n = 0, 1,2, ... (i.e., for convenience we assume that/n = 0 for n < 0). We now wish to compress this semi-infinite sequence into a single function in a way such that we can expand the compressed form back into the original sequence when we so desire. In order to do this, we must place a "tag" on each of the terms in the sequence/no We choose to tag the term j', by multiplying it by z" ; since n is then unique for each term in the sequence, each tag is also unique. z will be chosen as some complex variable whose permitted range of values will be discussed shortly. Once we tag each term, we may then sum over all tagged terms to form our compressed function, which represents the original sequence . Thus we define the z-transform (also known as the generating function or geometric transform) for /n as follows: F(z)

A

<Xl

= 2. fnz'"

(1.15)

n= O

F(z) is clearly only a function of our complex variable z since we have summed over the index n; the notation we adopt for the c-transforrn is to use a capital letter that corresponds to the lower-case letter describing the sequence , as in Eq. (LIS). We recognize that Eq. (Ll4) is, of course, in exactly this form . The z-transform for a sequence will exist so long as the terms in that sequence grow no faster than geometrically, that is, so long as there is some a > 0 such that

Furthermore, given a sequence / n its e-transform F(z) is unique.

328

APPENDIX I

If the sum over all term s in the sequence f n is finite , th en certainly th e unit disk [z] ~ 1 represents a range of analyticity for F(z). * In such a ca se we have a>

F(l) =

Lfn

(I.16)

n= O

We now consider some important examples of z-transforms. It is convenient to den ote the relati onship between a sequence and its transform by mean s of a double-barred, double-headed arrowr ; thu s Eq. (US) may be written as

fn F(z)

(I.1 7)

For our first example, let us consider the unit function as defined in Eq . (1.1I). For this function and from the definition given in Eq . (1.15) we see that . exactly one term in the infinite summation is nonzero, and so we immediately have the transform pair (U8) For a related example, let us consider the unit function shifted to the right by k units, that is, Il=k

n,pk From Eq . (US) again, exactly one term will be non zero , giving U n_ k Zk

As a third example, let us consider the unit step fun ction defined by for (recall that all functions are zero for n series, that is,

n

=

0, 1,2 , . ..

< 0). In thi s case we have a geometric

a>

I

n- O

I - z

15 n L l e" = - -

(I.19)

We note in thi s case that Izl < 1 in order for the z-transform to exist. An extremely important sequence often encountered is th e geo metric series

n=0,1,2, .. .

* A functi on of a comple x varia ble is said to be analytic at a point in the complex plane if that function has a unique derivative at that point. The Ca uchy- Rieman n necessary and sufficient condition for analyticity of such functions may be found in any text on functions of a complex variable [AHLF 66). t Th e do uble bar denotes the tran sform relati on ship whereas the doubl e heads on the arrow indicate that the journe y may be made in either direction , f => F a nd F => f

1.2.

TH E Z-T RANSFORM

329

Its z-tra nsform may be calculated as co

F(z)

= L Aocnzn n=O

A 1 - ocz

And so n A A oc - - 1 - ocz

(1.20)

where, of course, the region of analyticity for th is function is [z] < i ]«; note that oc may be greater or less than unity. Linear transformations such as the z-transform enjoy a number of important properties. Many of these are listed in Table 1.1. However , it is instructive for us to derive the convolution property which is most important in queue ing systems. Let us consider two functions of discrete time I n and gn, which may take on nonzero values only for the no nnegative integers . Their respective z-tra nsforms are , of course, F(z) and G(z). Let 0 denote the convolution oper ator, which is defined for I n and g n as follows:

We are intere sted in deriving the z-transform of the convoluti on for I n and gn, a nd this we do as follows: co

f n 0 gn L U n 0 gn)zn n- O 00

n

= L L fn _kgk?n-kzk n-=O k=O

However, since

co

n

n=O k=O

we have

00

co

L L=L L 00

1;- 0 n= k 00

fn ® gnL g~k L f n_kZn- k k""'O

=

n=k

G(z)F(z)

-330

APPE NDIX I

Table 1.1 Some Properties of the z-Transf orm z-TRA NSFORM

SEQUENCE

co

I. f n

n

2. af;

+ bg n

=

0, 1,2, . . .

aF(: )

n =0,k,2k, ...

k

>0

F(zk)

f ol

-F(zkz) - iL=sl z·.-k-'j;

l- J

zF (z)

7. f n-1 8. f n-k

+ bG (z)

I - [F(z) z

5. f n+! 6. fn+k

L: f nzn n_ O

F(a z)

3. a"fn 4.fnlk

F(z) =

k>O

zkF(z)

d

9. nfn

z dz F(z) dm

10. n(n - I)(n - 2), . . . , (Il - m + I )fn

z" ' - F (z)

11. f n @ g n

F(z)G(z)

12. fn - f n-1

(I - z )F(z )

n

13.

L: f k

F C: ) n = 0, 1, 2, . ..

k~ O

a oa [«

14. -

dz'"

(a is a parameter off n)

I - z

o oa "

- F (z) co

L: f n n= O

15. Series sum property

F(I ) =

16. AlIem ating sum property

F( -I) =

17. In itial value theorem

F (O) =

18. Intermediate value theorem

--

19. Final value theorem

,_1

co

L: ( -

I) nfn

n= O

fo

I d nF(: )

n! dz"

I %=0

= j,

n

lim ( I - : )F(z ) = f oo

1.2.

THE Z-TRANSFORM

331

"he

2 Transform Pairs SEQUENCE

z-TRA NSFORM

=

F (z) =

co

n

,

(~

333

0, 1, 2, . . .

2: I nzn n= O

n = 0

rm "he ion the itly vay sed

1/ ~ 0

zk , 1

1/ = 0 ,1 , 2 , .. .

1 - z Zk

her ) is

1 - z

~i a l

A

uo r

1 - «z

wer hen

ctZ

(I - ctZ)2 Z

(I - z )' ctZ(I + ctZ) (I - ctZ )3 z(1

+ z)

(I - z)" . 1

1) ,,_

(I - ctz)2 I)

l

+ m)(1/ + m

(I - z )"-

I) . . . (1/

+ 1)ctn

(I _ ctz)m+l

' to ress In S-

.ing lOW

for h is IS a 5 in rms I to out .eed

eZ

len , we have that the a-transform of the convolutio n of two equal to the produ ct of the z-transforrn of eac h o f the sequences .1 we list a number of important pro perties of the z-tra nsfor m, ,g t hat in Table 1.2 we provide a list of importa nt common

.ach UIL

332

APPENDIX I

Some comments regarding these tables are in order. First, in the propertv table we note that Property 2 is a statement of linearity, and Properties 3 and 4 are statements regarding scale change in the transform and time domain, respectively . Properties 5-8 regard translation in time and are most useful. In particular, note from Property 7 that the unit delay (delay by one unit of time) results in multiplication of the transform by the factor z whereas Property 5 states that a unit advance involves division by the factor z, Properties 9 and 10 show multiplication of the sequence by terms of the form n(n - I) . . . (n - m). Combinations of these may be used in order to find, for example, the transform of n2jn; this may be done by recognizing that n2 = n(n - I) + n, and so the transform of n2jn is merely Z2 d 2F(z)/dz2 + zdF(z)/dz. This shows the simple differentiation technique of obtaining more complex transforms. Perhaps the most impor~ant, however, is Property I I showing that the convolution of two time sequences has a transform that is the product of the transform of each time sequence separately. Properties 12 and 13 refer to the difference and summation of various terms in the sequence. Property 14 shows if a is an independent parameter of In' differentiating the sequence with respect to this parameter is equivalenttodifferentiating the transform. Property 15 is also important and shows that the transform expression may be evaluated at z = I directly to give the sum of all term s in the sequence. Property 16 merely shows how to calculate the alternating sum. From the definition of the z-tra nsform, the initial value theorem given in Property 17 is obvious and shows how to calculate the initial term of the sequence directly from the transform. Property 18, on the other hand, shows how to calculate any term in the original sequence directly from its z-transform by successive differentiation; this then corresponds to one method for calculating the sequence given its transform. It can be seen from Property 18 that the sequence In forms the coefficients in the Taylor-series expansion of F(z) about the point o. Since this power- series expansion is unique, then it is clear that the inversion process is also unique. Property 19 gives a direct method for calculating the final value of a sequence from its z-transform. Table 1.2 lists some useful transform pairs . This table can be extended considerably by making use of the properties listed in Table 1.1 ; in some cases this has already been done. For example, Pair 5 is derived from Pair 4 by use of the delay theorem given as entry 8 in Table 1.1. One of the more useful relationships is given in Pair 6 considered earlier. Thus we see the effect of compressing a time scquence In into a single function of the complex variable z. Recall that the use of the variable z was to tag the terms in the sequence Inso that they could be recovered from the compressed function; that is,ln was tagged with the factor z", We have

see! pre F(z

firs F (:

sec ex wr

a,

I.2.

THE Z-TRANSFORM

333

seen how to form the z-transforrn of the sequence [through Eq . (U5)]. Th e problem confronting us now is to find the sequence f n given the z-tra nsform F(z). Th ere a re basically three meth ods for carrying out this inversio n. The fi rst is th e powe r-series method, which attempts to ta ke the given func tion F(z) a nd express it as a power series in z; once thi s is done the term s in the sequence f n may be picked off by inspection since the tagging is now explicitly expo sed . T he powe r series may be obtained in one of two ways : the first way we have already seen through our intermediate value theorem expressed as Item 18 in Table I.l , tha t is,

f = 1- d nF(z) I n

It!

dz"

%= 0

(t his meth od is useful if one is only interested in a few term s but is rather tedi ou s if man y term s are required) ; the second way is useful if F(z) is expressible as a rationa l fun ction of z (that is , as the rati o of a polyn omial in z over a polynomial in z) and in thi s ca se one may divide the den omin ator int o the numerator to pick off the sequence of leadin g term s in the power series directly. The power-series expan sion meth od is usually difficult when man y term s are req uired. Th e second a nd most useful meth od for inverting z-tra nsforms [that is, to calcul ate j', from F(z)] is the inspection method. That is, one att empts to express F(::.) in a fas hion such that it co nsists of term s that are recognizable as tran sform pairs, for example , fr om Table I.2. The sta nda rd approach for placing F(z) in this for m is to carry out a par tial-fraction expansion, * which we now discuss. Th e partial-fr act ion expansion is merely an algebraic techn iqu e for expre ssing rat ional fun ction s of z as sums of simple term s, each of which is easily inverte d. In pa rticular , we will attempt to express a rati onal F(z) as a sum of terms , each of which looks either like a simple pole (see entry 6 in Ta ble I.2) or as a multi ple pole (see entry 13). Since the su m of the tran sform s equals the tra nsform of the sum we may apply Property 2 from Tabl e I.l to inve rt each of these now recognizable forms sepa rately, th ereby carrying out the req uired inversion. To carry out the parti al-fraction expan sion we proceed as follows. We ass ume that F(z) is in rati on al for m, that is F(z)

=

N( z) D(z)

where both the nu merat or N (z) and the den ominat or D (z) are each • T his procedure is related to the La u rent expa nsion o f F( z) around each pole [G U IL 49 ].

334

APP ENDI X I

polyn omials in z. * F urthermo re we will assume that D (z) factored form , that is,

IS

a lready

in

k

D(z)

= II ( I -

,,;z)m;

( 1.21)

i= l

The pr oduct notation used in this last equ ati on is defined as

Equati on (1.21) implies th at the ith root at z =

II"; occurs with

multiplicity

m.. [We note here th at in most problems of interest, the difficult part of the so lution is to take a n arbitr ary polynomial such as D(z) and to find its roots so that it ma y be put in the factored form given in Eq. (1.21). A t this point we ass ume th at that difficult ta sk has been accomplished. ] If F(z) is in thi s form then it is possible to express it as follows [G UlL 49]:

IX;

This last form is exactl y wha t we were looking for, since eac h term in this sum may be found in o ur table o f transform pa irs ; in particul ar it is Pair I3 (a nd in the simplest case it is Pa ir 6). Thus if we succeed in ca rrying ou t the partial-fraction expa nsion, then by inspection we ha ve o ur time seq uence In. It rem ain s now to descr ibe the meth od for ca lculating th e coefficient s A i;' The genera l expression for such a term is given by

)J I

1 1 / - 1 d l-l [ "( A ·· = - ( I - IX .Z)m, ~ ) ( I 1 " (j - I )! «, dZ D(z) •

:~ I/..

(1.23)

This rather formida ble procedure is, in fact , rather stra ightforwa rd as long as the function F(z) is not terribly complex. • We no te here tha t a partial-frac tion ex pansion ma y be ca rrie d ou t o nly if the degree of the numerato r po lynomial is strictly less th an th e degr ee of the den omi na to r polyno mia l : if thi s is not thc case, then it is nece ssary to d ivide the den omina to r into the numerator until the remaind er is o f lower degree th an the de nom inat or. This remainder divided by th e origina l den omi na tor may then be exp anded in partial frac tions by the method show n ; the terms ge nerated from the division al so may be invert ed by inspectio n mak ing use of tr an sform pa ir 3 in T a ble 1.2. An alterna tive way of sa tisfying the de gree co ndit ion is to attempt to factor o ut e no ugh pow ers o f z from the numera tor if possi ble.

l.2.

TH E Z-TRANSFORM

335

worthwhile at th is point to ca rry o utan example in order to demonstrate etho d. Let us ass ume F(z) is given by

F(z)

=

2

4z (1 - 8z) (1 - 4z)(1 - 2z)"

(1.24)

0

; exampl e th e numerator a nd denominator both ha ve the sa me degree ) it is necessary to bring the expressio n into pr oper form (numerato r : less th an den omin ator degree). In this case our task is simp le since ly factor o ut two power s of z (we a re req uired to fact or o ut onl y one of z in o rder to brin g the numerator degree below that of the denorninalit o bviously in this case we may as well facto r out both a nd simp lify .lculati ons). Thus we have

F(z)

=

Z2 [

4(l - 8z) ] (1 - 4z)(I - 2Z)2

; define the term in sq ua re br ackets as G(z). We note in this example .ie den ominator ha s three poles: one at z = 1/4 ; a nd two (that is a ~ pole) at z = 1(2. Thus in ter ms of the variab les defined in Eq . (1.21) ve k = 2, 0(1 = 4, 1111 = I , 0(2 = 2, 111. = 2. From Eq . (1.22) we are ore seekin g the following expan sion : G(z)

t>.

=

4(1 - 8z) 0

(1 - 4z)(1 - 2z)' All 1 -4z

= ---

+

A. I A22 + - ----"' '-( 1 -2z)" ( 1-2z) 0

such as All (that is, coefficients of simple pole s) are easily obtained :q. (1.23) by mult iplying the ori ginal functi on by the factor correspond the pole and t hen evalu ating the result at the po le itself (that is, whe n ; o n a value that d rives the facto r to 0). Thus in o ur example we ha ve A , = ( 1 - 4z)G(z) l. .1

. -1/ 1

= 4[1 - (8/4)] = - I

[I _ (2(4)]2

6

ly be evaluat ed in a similar way from Eq. (l.23 ) as follows : 2

A'I = (1 - 2z) G(z)lz_l/o =

.

•

4[ 1 - (8/2) ] [I - (4/2)]

=

12

-------------
0) go es to zero; further , we often denote this lower limit merel y by 0 with the understanding th at it is meant as 0(usually thi s will cau se no confusion). There also exists what is known as the "one-sided" Lapl ace transform in which the lower limit is repl aced by 0+, which is defined as th e limit of 0 + e as €(> 0) goes to zero; th is o ne-sided tr an sform has application in the so lution of tran sient problems in linear systems . It is impo rtant th at th e reader distingu ish bet ween th ese two transfo rms with zero as th eir lower limit since in th e former case (the bilat eral tr ansform) an y accumulation at the origin (as, for example , the unit impulse defined below) will be included in the tr an sform , wherea s in the la tte r case (t he o ne-sided transform) it will be o mitted . For o ur assumed case in which f(l) = 0 for t < 0 we may write o ur t ran sform as

F*(s)

= f '! (t)e- dt S

'

(1.29)

where, we repeat , the lo wer limit is to be int erpreted as 0- . Thi s Lap lace transform will exist so lon g asf(l) gro ws no fa ster than an exponential , th at is, so lon g as there is so me real number u. such that

340

APPENDIX I

The smallest possible value for " « is referred to as the abscissa of absol ute con vergence. Aga in we stat e tha t the Laplace transfo rm F*(s) for a given functio n j (r) is unique. If the integral of f(l) is finite, then certainly the right-ha lf plane Re (s) ~ 0 represents a region of analyticity for F*(s) ; the notati on Re ( ) reads as " the rea l part of the complex function withi n the parentheses." In such a case we have, corresponding to Eq. (1.16), F*(O) = 1"'j (l) dt

(I.30)

From our earlier definition in Eq. (1.9) we see th at prope rt ies for the ztran sform when z = I will corres po nd to properties for the Lapl ace transform when s = 0 as, for example , in Eqs. (1.16) a nd (I.30). Let us now co nsider so me importa nt examples of Laplace tr an sfor ms. We use notati on here ident ical to that used in Eq. (1.17) for z-t ransforms, namely, we use a dou ble-ba rred , dou ble-headed arrow to denote the relat ion ship bet ween a functio n a nd its transform; thu s, Eq. (1.29) may be written as j(t)

t~O

= {I

0

1 0)

F* (s) =

5.:

f(t)e- ' t dt

+ bG*(s)

aF*(s)

aF*(as)

4. f (t - a)

e- "'F*(s)

5. e-a'f(t)

F*(s + a) dF *(s)

- - -

6. tf(t )

ds d nF*(s)

( _ I )n ~

7. t nf (t )

1:,

8. f (t )

F *(s, ) ds,

I

r'"

9. f( l) In

10. f (l ) ® g(l)

I I. 12.

ds;

J 51""" 3

r'" J 8: = 81

ds 2 ' "

I"

dsnF*(sn)

) Sn= 5 n _ l

F*(s)G*(s)

t df(l )

sF*(s)

---;[(

t d"f( t)

snF*(s)

di"

F* (s)

B .t f ",f(t )dl

f"",

14. t

s F*(s) sn

f J (t )(dr)n

~

a

n times

[a is a parameter)

15 . aa f(t)

16. Int egra l property 17. Ini tial value theorem

a aa F(s) F*(O) =

f(t) dt

lim sF*(s) = lim f(t) $ _00

18. Fin al va lue theo rem

fo:

t_ O

lim sF*(s) = lim f (t) 8_ 0

l _co

if sF*(s) is ana lytic for Re (s)

~

0

t To be co mplete. we wish to sh ow the form of the tran sform for entries 11 -14 in the cas e when f( l) may ha ve nonzero va lues for I < 0 a lso :

d nf (t ) snF*(s) _ sn-'[(o-) _ sn-2[(1 )(0-) - . .. - [ 0'+. Similarl y, F_ *(s) will have some abs cissa of abso lute convergence, which we will denote by 0'_ , which implies that F_ *(s) converges for Re (s ) > 0'_ , It then follows directly that F_ *( - s) will have th e same convergence ab scissa (0'_) bu t will converge for Re (s) < 0'_ , Thus we ha ve a situation where F *(s) converges for 0'+ < Re (s) < 0'_ and therefor e we will have a " convergence st rip" if and only if 0'+ < 0'_ ; if such is not the case, then it is not useful to define F*(s). Of course, a similar a rgument ca n be made in the case of z-transforrns for funct ion s that ta ke on non zero values for negati ve time indices. So far we have seen the effect of taggin g our time funct ion f( t) with the compl ex exponential e- s t a nd then compressing (integrating) over all such ta gged functi ons to form a new functi on, namel y, the tran sform F*(s). The purpose of the tagging was so that we could later " untransfo rrn" or, if you will, "unwind" the transform in order to obtainf(t) once again. In princ iple we know this is possible since a tr an sform and its time functi on are uniquely related. So far , we have specified how to go in the ~Jn e direction from f( t) to F*(s). Let us now discuss the problem of inverting the Laplace tr an sform F *(s) to recover f( t) . There are basically two meth ods for conducting this inversio n : The insp ection method a nd th e f ormal inversion int egral m ethod. These two meth od s are very similar. First let us discuss the inspection method , which is perh ap s the most useful scheme for invertin g transforms. Here , as with z-transforms, the approac h is to rewrite F*(s ) as a sum of term s. each of which ca n be recognized from the table of Laplace transform pa irs . Then , makin g use of the linear ity pr operty, we may invert the transform term by term , and then sum the result to recover f( t) . Once again , the basic method for writing F*(s) as a sum of recognizable term s is that of the partial-fraction expan sion . Our description of that meth od will be so mewhat sho rtened here since we have discussed it at so me length in the z-tra nsform sectio n. First. we will ass ume th at F*(s) is a rat ion al functio n of s , nam ely, F*(s)

=

N( s) D(s)

where bot h the numerator N(s) and den omin at or D (s ) are each polynomials in s. Again , we ass ume that the degree of N(s) is less than the degree of D (s ) ; if this is not t he case. N (s) must be divided by D(s) until the remainder is of degree less than the degree of D(s), and then the partial-fract ion expan sion is ca rried out for this remainder, whereas the terms of the qu otient resultin g from the division will be simple powers of s, which may be inverted by appealing to Tr an sform 4 in Table 1.4. In additi on , we will assume that the

350

AP PEN DIX [

" hard" part of the problem ha s been done , namely, that D (s) has been put in facto red form k

II (s + ai)m, i :::lt l

D(s) = .

(1.39)

Once F "'(s) is in this form we may th en expre ss it as the following sum:

,

+

Bkl (s

+ ak)m.

+

Bk 2 (s

+ ak)'n.-,

+ ... +

Bk m • (s

+ ak)

(1.40)

Once we ha ve expressed F* (s) as ab ove we are then ina position to invert each term in this sum by inspection from Table 1.4. In particular, Pair s 8 (for simple poles) a nd 10 (for multiple poles) give us the answer dire ctly. As before, the method for calculating th e coefficients B i; is given in general by .

Bi ;

H

= (j

I d [ _ I)! ds i- ' (s

N(S)]

+ ai)m, D(s)

I

s--a,

(1.41)

Thus we have a complete pre scription for findingf (t) from £* (s) by inspectien in those cases where F*(s) is rat ional and where D (s ) has been facto red a s in Eq. (1.39). This method works very well in those cases where F *(s ) is not overly complex. To elucidate some of these principles Jet us carry out a simple exam ple. Assume that F *(s) is given by 8(S2 + 3s

*

+

l)

F (s) - - ' - -- --'- (s + 3)(s + 1)3

(1.42)

We ha ve already written the den ominator in factored form, and so we may proceed directly to expand F* (s) as in Eq. (1.40). Not e th at we have k = 2, a, = 3, m, = I , a2 = I , m 2 = 3. Since the denominat or degree (4) is greater than the numerator degree (2), we may immediately expand F*(s) as a parti al fraction as given by Eq. (1.40), namel y,

F*(S)=~+ S+ 3

B 2,

(s

+ 1)3

+

B 22

(s

+ 1)2

+~ (s

+ 1)

1.3.

TH E LAPLA CE TRANSFORM

351

Evaluat ion o f the coe fficients Bij pr oceeds as foll ows. Bl l is especi ally simple since no differentia tions are requi red , and we obtain

+ 3)F

B l l = (s

*

(S)I'~_3 = 8

(9 - 9 + I) (_2)3 =-1

B2 1 is a lso easy to evaluate:

+ I) F *(s)I.~-1 = 3

B2 1 = (s

8

(I - 3 2

+ I)

= - 4

For B 22 we mu st d ifferentiate once, na mely ,

B = .!!...[8(S2 + 3s 22

cis

s

= 8 (s

+

+ I)J +3

3)(2s

+ 3) (s

S2

=8

+

6s

+ 81

(s + 3)"

I ,--I (S2 + 3s

+

1)(1)

+ 3)2

I .--1

I -

6

+

8

=8 --~~

.--1

(2)2

=6 Lastly , the calculati on of B23 in volve s two differentiati ons ; however , we have a lrea dy carried o ut the first diffe rentiation , a nd so we take adva ntage o f the fo rm we ha ve derived in B22 j ust pr ior to eva luation at s = -I ; furthermore , we note th at since j = 3, we have for the first time an effect due to the term (j - I)! fr om Eq . (1.4 1). Thus

1 2 [8(S2 +s +3s 3+ OJ I.-- 1

B' 3 = ci 2! ds 2

_ 1 (8) .{ [S2 + 6s + 8J

- 2

cis

= 4 (s

+

(s

3)2(2s

+ 3)2 +

6) - (S2

(s = 4

(2)2(4) - ( I - 6

+

6s

+ 3)4

+ 8)(2)(2)

'---'-~---'---'----'-:"':""':

(2)4 =1

I.-- 1 +

8)(2)(s

+

3)

I s~- I

352

APP ENDI X I

This completes the evaluation of the con stants B ii to give the parti al-fracti o n expansion F*(s) =

---=.!.- + . s

+3

(s

-4

+

+

1)3

6

(s

+ 1)2

+ _I_ (s + I)

(1.43)

This last form lend s itself to inversion 'by inspecti on as we had promi sed . In particular, we o bserve that the first a nd last terms invert directly accordin g to transform pair 8 from Table lA, whereas the seco nd a nd third term s in vert directly from Pair 10 of that table; thus we have for I ~ the following :

°

(1.44)

°

and of course, JU) = for I < 0. In the course of carrying out a n inversi on by partial-fracti on expansion s there a re two natural points at which o ne can conduct a test to see if an y errors ha ve been made : first , once we ha ve the partial-fraction expa nsion [as in o ur example, the result give n in Eq. (1.43)], then one ca n co mbine thi s sum of terms int o a single term over a co mmo n den omin at or a nd check th at thi s single term corresp onds to the o rigina l given F *(s) ; the other check is to take th e fina l form for J (I) a nd carry o ut the forw ard transform ati on a nd confirm th at it gives the o riginal F*(s) [of course, one then gets F*(s) expanded directly as a partial fraction]. The second meth od for findin g j'(r) from F *(s) is to use the incersion int egral

J( t)

= - I . J"'+iOO F*(s)es t ds 2-rr}

°

( 1.45)

", - ;00

for t ~ and (Je > (J a ' The integration in the complex s-plane is tak en to be a straight-line inte grati on parallel to the imaginary axis a nd lying to the right of (J a ' the ab scissa of ab solute con vergence for F *(s). The usu al means for carrying out this integra tion is to make use of the Cauchy residue th eorem as applied to the integral in the complex domain around a closed contour. T he closed con tour we ch oose for this purpose is a semicircle of infinite radiu s as shown in Figure IA. In thi s figure we see the path of int egrati on required for Eq . (lAS) is S3 - S1 and the semicircle of infin ite radius closing thi s co nto ur is given as S1 - S 2 - S3 ' If the integral al on g the path S 1 - S2 - S3 is 0, then the integral along the entire closed contour will in fact give us J(I) from Eq. (lAS) . To establish that this contribution is 0, we need

1.3.

TH E LAPLA CE TR ANSFORM

353

W = Im (s)

s- plane

--:-1-----;;+---':-""+------ 0 =

Re(s)

Figure 1.4 Closed contour for inversion integral.

Jordan's Lemma

If IF*(s)l- 0 as R -

00

on s, - s. for

t

S 3,

then

>0

Thus, in orde r to carry out the complex in version int egral show n in Eq . (1.45) , we mu st first express F*(s) in a form for which J o rdan' s lemm a applies. Ha ving done thi s we may then evaluate the integral a ro und the clos ed co nto u r C by ca lculating residues and using Cauchy' s residu e the orem . Thi s is most easily ca rried o ut if F*(s) is in rational form with a fact or ed den ominat or as in Eq . (1.39). In order for Jordan' s lemma to apply, we will require, as we did before , th at th e degree o f the numerator be strictly less than the degree of the den om ina tor, a nd if thi s is not so , we mu stdividetheration al . functi on until the remainder has th is property. That is all there is to the meth od . Let us carry th is o ut o n our previous example, namely th at given in Eq . (1.42). We note this is alread y in a form for which Jordan's lemm a applies, and so we ma y proceed directly with Cauchy's residue theorem. Our poles a re loca ted .at s = -3 and s = -I. We begin by ca lculating the residu e a t s = -3 , thu s

r-

3

= (s

+ 3)F*(s)eSlf'~_3

+ 3s + l )e', (s + 1)3 8(9 - 9 + 1)e-

= 8(s'

31

=

=_

( _2)3

e-3'

I s_ _ 3

354

APPENDIX I

Similarly , we mu st calculate the re sidue a t s = -I , which requires the differentiation s indicated in our residue formula Eq. (1.27) :

I

2

I d (5 + 1)3F*(s)e" r_ 1=:;-;----; . _. ds ,~ - I 2 = I d 8(S2 + 3s + l)e"/ 2 2 d5 (5 + 3) ,-- 1

= !~[(s + 2 ds

2

3)[8(2s + 3)e' t + 8(5 + 3s + 1) te")- 8(S2 + 3s + (s + 3)2

Oe"JI .•~_I

=!

I {(s + 3)"[8(25 + 3)e" + 8(5 2+ 3s + 1)le" 2 (s + 3)4

+ (s + 3)8[2e" + (25 + 3)le") + (5 + 3)8[(2s + 3)lest + (S2 + 3s + 1)1 2e") - 8(2s + 3)e" - 8(S2 + 35 + 1)le"] - [(s + 3)8[(2s + 3)e' t + (52 + 3s + ljre"] - 8(S2 + 3s + l )e"]2(s + 3)} = e-' + 61e- ' - 21 2e-'

1,--1

Combining the se residues we have

f(l )

=

_ e- 3 '

+ e:' + 6le- t

-21 2e- '

t

~

0

Thus we see that o ur solution here is the same as in Eq . (1.44), as it must be: we ha ve once again that f( l) = 0 for t < O. In o ur earlier discussion of the (bila teral) Laplace transform we discussed . functi on s of time 1_(1 ) and f +(I) defined for the negat ive a nd positive realtime a xis, respect ively. We also obse rved th at the tr an sform for each of th ese functi on s was a na lytic in a left half-plane and a right half-pl ane , respecti vely, as mea sured from their appropriate a bscissas of a bsolute con vergenc e. More over , in o ur last inversio n method [th e a pplication of Eq . (1.45) ] we observed that closing th e co nto ur by a semicircle of infinit e rad ius in a co unterclockwise directi on gave a result fo r t > O. We comment now th at had we closed the co nto ur in a clockwi se fashion to the right , we would have ob ta ined the result th at would have been a pplica ble for I < O. ass umi ng tha t the contribu tion of thi s contour could be sh own to be 0 by Jordan ' s lemma . In o rde r to invert a bilateral tr ansform, we pr oceed by o btaining first f( l) for positive values of I a nd then for negati ve values of I. For the first we tak e a path of inte gra tion within the con vergence strip de fined by G_ < G c < G... and th en closing the contour with a counterclockwise semicircle; for I < 0, we ta ke th e same vertical contour but close it with a semicircle to t he righ t.

- - - - - - - - - - - - --

-

-

1.4.

TRA NSFORMS IN DIFF ERENC E AND DIFFERE NTIAL EQUATIONS

355

A s may be anticipated from our contour integration methods, it is some times necessary to determine exactly how many singularities of a function exist wit hin a closed region. A very powerful and convenient theorem which a ids us in thi s determination is given as follows :

Rouche's Theorem [G UIL 49] If f(s) and g(s ) are analytic fun ctions of s inside and on a closed contour C, and also if /g(s)! < If(s)1on C, then f (s) and f (s) + g (s) have the same number of zeroes inside C.

1.4.

USE OF TRANSFORMS IN THE SOLUTION OF DIFFERENCE AND DIFFERENTIAL EQUATIONS

As we have already mentioned, transforms are extremely useful in the solution o f both differential and difference equations with constant coefficients. In th is sectio n we illustrate that technique ; we begin with difference equati on s using z-tra nsfo rms and then move on to differential equations usin g Laplace transforms, preparing us for the more complicated differentialdifference equations encountered in the text for which we need both methods simultaneously. Let us con sider the following general Nth-order linear difference equation with constant coefficients (1.46) where the a, ar e known con stant coefficients, g, are unknown functions to be found, and e. is a giveri function o f n. In addition , we assume we are given N boundary equations (e.g., initial condition s). As always with such equ at ion s, the solutio n which we are seeking consi sts of both a homogeneous a nd a particular solution , namely, ju st as wit h differential equations. We know that the homogeneous solution must sati sfy the homogene ous equation Gs g. - s

+ ... + Gog.

= 0

(1.47 )

The genera l form of solution to Eq. (1.47) is

where A a nd « are yet to be determined. If we substitute the proposed so lutio n int o Eq. (1.47), we find

as A rx n-

S

+ G S _ 1Arx. -.\"+ 1 + . . . + aoACI. n = 0

(1.48)

356

APPENDI X I

This Nth·order polyn omial clearly ha s N so lutio ns, which we will de note by ai ' <X 2 , • •• , <X,v , assuming fo r the moment th at a ll the <Xi a re distinct. As soci. a ted wit h each suc h so lutio n is an a r bitra ry co nstant Ai whic h will be de ter mi ned fro m the initial co nditio ns for the differen ce equati on (o f which there must be N). By ca ncelling th e com mon te rm A<xn- .Y from Eq . (1.48) we fina lly arrive at the characteristic equation which determines the va lues <X i aN

+ as_l<x + as_2<x - + .. . + a o<X' = 0 C)

, ..

( 1.49)

Thu s the sea rc h for th e homogeneous so lutio n is now reduced to find ing th e N roots o f o ur characteristic equati on (1.49) . If a ll N o f the <X i a re disti nct, th en the homogene ou s so lutio n is g

G(: )

=I

gn:n

(1.52)

n= O

Fro m o ur earlier d iscussio n we know that once we have found G(:) we may then apply o u r inversion techniques to find th e desired solution gn' Our next step is to multiply the nth equation from Eq. (1.4 6) by : n and then form the sum of a ll such multiplied equation s from k to infinity ; that is. we form ec

S

00

.2 .2 Q ig n_ i z n = .2 e nz n Tl = h: ; = 0

n=h:

We then carry out the summa tions and a ttempt to recogni ze G(z) in this sin gle equ ati on. N ext we solve for G(:) algebraica lly and then pr oceed with o ur inversion techniques to obtai n the so lution. Th is meth od does not requ ire that we guess at the particula r solution , and so in th at sense is simpler th an the di rect method : however, as we sha ll see, it still has the basic difficulty that we must solve the char acteristic equation [Eq . (1.49)] and in general this is the difficult part of the solution . However, even if we cannot so lve for the ro ot s (Xi it is possible to o btai n meaningful properties of the solution g n from the perhap s unfa ctored form for G(: ).

358

APPENDIX I

Let us solve our earlier example using the method of z-tra nsforrns. Accordingly we begin with Eq. (1.50), multiply by z" and then sum; the sum will go from 2 to infinity since this is the applicable range for that equation . Thus

We now factor out enough powers of z from each sum so that these powers match the subscript on g thusly:

Focusing on the first summation we see that it is almost of the form G(z) except that it is missing the terms for n = 0 and /I = I [see Eq. (1.52)]; applying this observation to each of the sums on the left-hand side and carrying out the summation on the right-hand side directly, we find • 6[G(z) - go - g,z] - 5z[G(z) - go] + z-G(z)

6(1/5)2z2

= --'--'---'--

1 - (1/5) z

Observe how the first term in this last equation reflects the fact that our summation was missing the first two terms for G(z). Solving for G(z) algebraically we find "

G(z)

=

6g o + 6g, z - 5goz + (6/25)Z2/[1 - (l /5)z] 6-5z+ z2

If we now use our given values for go and g" we have

G(z)

- z) (I)[I _ (1/3) z][1z(6- (1/2) z][1 -

= "5

(I /5)z]

Proceeding with a partial-fraction expansion of this last form we obtain

G(z) =

-9 1 - (1/3) z

+

8 1 - (1/2) z

+

1 1 - (1/5)z

which by our usual inversion methods yields the final solution n

=

0, 1,2, .. .

Note that this is exactly the same as Eq . (LSI) and so our method checks. We comment here that even were we not able to invert the given form for G(z) we could still have found certain of its properties; for example, we could"find

l

IA.

TRANSFORMS IN DIFFE REN CE AN D DIFFERENT IAL EQUATIONS

359

that the sum of all term s is given immed iately by G(I ), that is,

Let us now consider the application of the Laplace transform to the solution of constant-c oefficient linear differential equations. Consider an Nth- order equation of the following form: d Nf (t) aN--,-. dt"

+ aN-l

dN- 'f(t) v

dt : -

1

df(t )

+ ... + a, -dt- + aof( t) =

e(t)

( 1.53)

Here the coefficients a; are given constants, and e(t ) is a given driving function. Along with this equation we must a lso be given N initial co nditions in order to carry out a complete solution; these conditions typically a re the values of the first N derivat ives at some instant, usually at time zero. It is required to find the function f( t). As usual, we will have a homogeneous solut ion fihl( t), which solves the homogeneous equation [when e{t ) = OJ as well as a particular solut ion f (OI(t) that corresponds to the nonhomogeneous equation. The form for the homogeneous solution will be

If we substitute this into Eq. (1.53) we obt ain

Thi s equation will have N solutions characteristic equation

<x" <x" ••• , <x n ,

which must solve the

which is equivalent to Eq. (1.49) with a change in subscripts. If all of the are distinct , then the general form for our homogeneous solution will be

<X i

The evalu ation of th e coefficients A ; is carried out makin g use of the initial condition s. In the case of multiple root s we have the following modificat ion. Let us assume that <x, is' a repeated root of order k ; this multiple roo t will contri bute to the hom ogeneous solution in th e following way:

360

APPENDIX I

and in the case of more than one multiple root the modification is obviou s. As usual , one must guess in order to find the particular solution J< p)(t). The complete solution then is, of course, the sum of the homogeneous and particular solutions, namely,

+f

Jet) = j (hl(t)

p'(t)

Let us apply this method to the solution of the following differential equation for illustrative purposes: d](t) _ 6 df(t) dt 2 dt

with the two initial conditions f(O -) characteristic equation cx' - 6cx

+ 9f(t) = 2t

=

(1.54)

0 and df(O-)/dt

= O.

Forming the

+9=0

(1.55)

we find the following multiple root:

cx, = cx, = 3 and so the hom ogeneou s solution must be of the form jlh l(t)

= (Ant + A'2)e3t

Making an appropriate guess for the particular solution we try j lp'(t) = B,

+ B,t

Substituting this back into the basic equation (1.54) we find that B, and B. = 2/9. Thu s our complete solution takes the form

j (t) = ( Ant

=

4/27

+ A , 2)e + -4 + -2 t 3/

27

9

Since our initial conditions state that bothf(t) and its first derivative must be zero at t '= 0- , we find that Au = 2/9 and A" = -4/27 , which gives for our final and complete solution Je t) =

~(t - ~) e3t +~(t +~)

t~O

(1.56)

The Laplace transform provides an alternative meth od for solving constantcoefficient linear differential equations. The method is based upon Properties II and 12 of Table 1.3, which relate the derivative of a time funct ion to its Laplace transform. The approach is to make use of these properties to tr an sform both sides of the given differential equation into an equation involving the Laplace transform of the unknown functionf (t) itself, which we denote as usual by F*(s) . This algebraic equation is then solved for F*(s), and is then

IA.

TRANSFO/t.\ lS IN DIFF ERENCE AND DIFFERENTIAL EQUATIO NS

361

inverted by any of our methods in order to immediately yield the complete solution forfU ). No guess is required in order to find the particular solution, since it comes out of the inversion procedure directly. Let us apply thi s technique to our pre viou s example . We begin by transformin g both sides of Eq . (1. 54), which will require that we take adv antage of our initial conditions as follows: s2F*(s) - sJ(O-)

- l'\o-) -

6sF*(s)

+ 6J(0-) + 9F*(s) = ~

s-

In carrying out this last operation we have taken advantage of Laplace transform pair 7 from Table IA. Since our initial conditions are both zero, we may eliminate certain terms in this last equation and proceed dir ectly to solve for £*(s) thusly: 2

F*( s)

2/s = -.---'--s: - 6s

+9

We must now factor this last equation, which is the same problem we faced in findin g the roots of Eq. (1.55) in the direct method , and as usual forms the basically difficult part of all direct a nd indirect methods. Carrying this out we have F*(s) _ -

2 S2( S -

W

We are now in the position to make a partial-fract ion expansion yielding

* 2/9 4/27 F (s) = - + - + s

S2

Inverting as usuaLwe then obtai n, for J ( I)

2 9

=-

I

2/9 -4/27 + -(s - 3)2 S- 3

I ~

0,

+ -4 + -2 le 27

9

3t

-

4 3t e 27

-

which is identical to our former solution given in Eq . (1.56). In our study of queueing systems we often encounter not only difference equ ati on s and differential equations but also the co mbinatio n in the form of differential-difference equations. That is, if we refer back to Eq . (1.53) and replace the time functi on s by time functions that depend upon an inde x, say /1 , and if we then displa y a set of differenti al equations for vari ou s values of /1, th en we have an infinite set of differential-difference equations. The solution to such equations often requ ires that we take both the z-transform on the discrete index 11 and the Laplace tran sform on the continuous time parameter I. Examples of this type of analysis are to be found in the text itself.

362

APPENDIX t

REFEREN CES A hlfors, L. V., Complex Analysis, 2nd Ed ition, McGraw-Hili (New York), 1966. CA D Z 73 Ca dzow , J . A. , Discrete-Time Sys tems, Pre ntice-Ha ll (E nglewood Cliffs, N.J.), 1973. D OET 61 Doetsch, G. , Guide to the Applications of Laplace Transf orms, Van Nostrand (Princeton), 1961. GU[L 49 Guillemin, E. A., The Mathematics of Circuit Analy sis, Wiley (New Yo rk) , 1949. J URY 64 Jury, E. 1., Theory and Application of the z-Transfo rm Me /hod, Wiley (New Yor k), 1964. SC HW 59 Schwartz, L., Theorie des Distributions, 2nd printing , Ac tualities scientifiques et industrielles Nos. [245 and I [ 22, Hermann et Cie. (Paris), Vol. 1 (1957), Vol. 2 (1959). WIDD 46 Widder , D . V., The Laplace Transf orm, Princeton Uni versity Press (Princeto n), 1946. AHLF 66

APPENDIX

II

Probability Theory Refresher

In this appendix we review selected topics from probability theory , which are relevant to o ur discussion of queueing systems. Mostly , we merel y list the imp ortant definitions and results with an occasional example. The reader is expected to be famili ar with this material , which corresp onds to a good first course in probability theory. Such a course would typ ically use o ne of the follo wing texts th at conta in additional details and derivat ion s: Feller, Volume I [FELL 68]; Papouli s [PAPO 65] ; Parzen [PARZ 60]; o r Daven. port [DAVE 70]. Probability theory concerns itself with describing random events. A typical dictio nary definition of a ra ndom event is an event lacking aim, pu rpose, or regularity. N othing could be further from the truth ! In fact , it is the ex treme regularity that manifests itself in collections of random events, that makes probability the or y interesting and usefu l. The notion of statistical regularity is central to our studies. F or exa mple, if one were to toss a fair coin four times , o ne expect s o n the average two head s and tw o tails . Of course , there is one chance in sixteen that no head s will occu r. A s a con sequence, if a n unusual seq uence ca me up (tha t is, no head s) , we would not be terribly surp rised nor would we suspect the coin was unfair. On the other hand , if we tossed the coin a million times, then once again we expect a ppro ximately half head s a nd half tails, but in thi s case, if no heads occ urred, we would be more than surp rised, we would be indi gnant and with overwhelming as su rance could state that this coin was clearly unfa ir. In fact , the odds are better t ha n 1088 to I th at at least 490,000 heads will occu r! This is what we mean by statistical regularity , namely, that we can make some very precise statements about large collect ion s of random events . ILL

RULES OF THE GAME

We now descri be the rule s of the game for cre ating a mathematical model for probabilistic situatio ns, which is to corresp ond to real-world experiments. Typic ally one exa mines three features 'of such experiments: 1.

A set of possible experimental outcomes. 363

364 2. 3.

APP ENDIX II

A gro uping of the se outcomes into classes called results. The relative frequency of these classes in many independent tr ials of the experiment.

The relative frequency f e of a class c is merely the number of time s the experimental outcome falls int o that clas s, divided by the number of time s the experiment is performed; as the number of experimental trial s increases, we expect [, to reach a limit due to our noti on of statistical regularity. The mathematical model we create also has three quantities of interest that are in one-to-one relation with the three quantities listed a bove in the experimental world. They are , respectively: A sample space which is a collection of objects which we den ote by S . S co rrespo nds to the se t of mutually exclusive exhaustive outcome s of the model of an experiment. Each object (i.e., possible o utco me) w in the set S is referred to as a sample po int. 2. . A famil y of events <S' denoted {A , B, C, . . .} in which each event is a set of sa mple points {w} . An event corresponds to a class or result in the real wo rld. 3. A probability measure P which is an assignment (ma pping) of the events defined on S into the set of real numbers. P corresponds to the relative frequency in the experimental situati on. The notation P[A] is used to denote the real number associated with the event A . Th is assignment must satisfy the followin g prop erti es (axioms):

1.

(a)

For an y event A , 0 ::::; P[A]::::; 1.

(b) P[S] (c)

=

I.

If A and B are "mutually exclusive" event s [see (11.4) below], then P[A U B] = P[A] + P[B].

(ILl) (11.2) (11.3 )

It is appropriate at thi s point to define some set the or etic not ati on [for example , the use of the symbo l U in property (c)]. Typically, we descr ibe a n event A as follows : A = {w : w sa tisfies the membership pr operty for the event A } ; thi s is read as "A is the set of sa mple points w such th at w satisfies the memb er ship property for the event A ." We further de fine

A e = {w: w not in A } = complement of A

A U B

= {w : w

A II B = AB

=

in A or B or both} = union of A a nd B

{w : w in A and B } = intersection of A and B

'P = S" = null event (contains no sample points since S contains all the points)

11.1.

RUL ES OF THE GAME

365

If A B = cp , then A and B are said to be mutually ex clusive (or disjoint). A set of event s whose union forms the sample space S is sa id to be a n exhaustive set of event s. We a re therefore led to the definiti on of a set of mutually exclusive exhaustive events {Ai> A 2 , • • . , An}, which have the properties AiA; = cp for all i '" j A , V A 2 V . . . v An = S

(1104)

We note further that A v Ac = S , AA c = cp , AS = A , Alp = tp, A V S = S, A V cp = A, S " = cp, and cpc = S. Also , we comment that the uni on and intersection opera to rs a re commutative , associative, a nd distributive. The triplet (S, tff, P ) along with Axioms (11.1)- (11.3) form a p robability system . These three axioms are all that one needs in order to develop an axiomatic theory of probability whenever the number of events th at can be defined on the sa mple space S is finite. [When the number of such events is infinite it is necessar y to include an additional axiom which extends Axiom (11.3) to include the infinite union of disjoint event s. Thi s leads us to the noti on of a Borel field a nd of infinite additiv ity of probab ility measures. We do not discuss the det ails further in this refr esher.] La stly, we comment that Axiom (11. 2) is nothing more than a normalizat ion statement and the ch oice of unity for th is normalizati on is quite arbitrary (but also very natural). Two other definition s are now in order. The first is that of conditional probability . The cond itional probability of the event A given that the event B occurr ed (denot ed as P[A I BD is defined as

I

P[ A B]

~ P[ AB] P[B]

when ever P[B] '" O. The introduction of the conditional event B force s us to restrict attention from the original sample spa ce S to a new sa mple space defined by the event B; since this new constrained sample space mu st now ha ve a total pr obability of unity, we magnify the probabilities associated with co nditio nal event s by dividing by the term P[B] as given above. The seco nd additiona l notion we need is that of statistical independence of events. Two events A, B are said to be statistically indep endent if a nd only if (11.5) P[ AB] = P[A]P[B] Fo r three events A, B , C we require that each pair of event s satisfies Eq . (11.5) a nd in additi on P[ABC] = P[ A]P[B]P[C]

Th is definition extend s of course to n event s requiring the n-fold fact oring of the probab ility expression as well as a ll the (n - I)-fold factorings all the way

366

AP PEN DIX II

down to all the pairwi se factorings. It is easy to see for two ind ependen t events A, B that PtA I B} = Pt A], which merely says that knowledge of the occurrence of the event B in no way affects the probability of the Occurrence of the independent event A . The theorem of total probability is especially simple and important. It relates the pr obab ility of an event B and a set of mutually exclusive exhau stive events {Ai} as defined in Eq . (11.4). The the orem is n

= L P[ AiB]

P[B]

i= l

which merely says that if the event B is to occur it must occur in conjunction with exactly one of the mutually exclusive exhaustive events Ai ' Howe ver from the definition of conditional probability we may always write P[AiB]

=

I

P[A i B] P[B]

=

P[B

I Ai ]

P[A i]

Thus we have the second important form of the theorem of total pro bability, namel y , n

P[B) =

L P[B I Ai] i= l

pr All

Thi s last equ ation is perhaps one of the most useful for us in st udying qu eueing theory. It suggests the following approach for finding the pr obability of some complex event B , namely, first to condition the .event B on some event Ai in such a way that the calculation of the occurrence of event B given this cond ition is less comple x, and then of course to multipl y by the pr obabil ity of the conditio nal event A , to yield the j oint probability P[ A,B) ; this having been done for a set of mutu ally exclusive exhau stive events {A ,} we may then sum these probabilities to find the probability of the event B. Of course, this approach can be extended and we may wish to condition the event B on more than one event then unc ondition each of these events suita bly (by multipl ying by the probabilit y of the appropriate condition) and then sum all possible form s of all conditions. We will use this approach man y times in the text. We now come to the well-know n Bay es' theorem. Once aga in we consi der a set of events {A ,}, which are mutually exclusive and exhaustive. Th e theo rem says P[B Ai) P[A i ) P [A ; B) = - n --'--'--.:..:....---'---''-=--

I

I

L P[B I Ai )

P[A i)

j= l

Th is theorem permits us to calculate the probability of one event conditioned on a second by calculatin g the probability of the seco nd conditioned on the first a nd other similar terms.

I J

ILl .

RUL ES OF TH E GAME

367

imple example is in order here to illustrate some of these ideas. Con sider 'ou have just entered a ga mbling casino in Las Vegas. You approach a

: who is known to have an ident ical twin brother ; the twins cannot tinguished. It is further kn own that one of the twin s is an honest dealer .a s the second twin is a cheating dealer in the sense that when you play he honest dealer yo u lose with probability one-h alf, whereas when you vith the cheating dealer you lose with probability p (if P is greater than alf, he is cheating again st you whereas if p is less than one-half he is ng for you). Further more, it is equally likely that upon entering the ) you will find one or the other of these two dealers. Con sider that you ilay one game with the particular twin who m you encounter and further -ou lose. Of course yo u are disappointed and you would now like to ate the probability that the dealer you faced was in fact the cheat, for if an establish that thi s probability is close to unity, you have a case for the casino. Let D II be the event that yo u pla y with the ho nest dealer :t D c be the event that yo u play with the cheating dealer ; further let L : event that you lose. What we are then asking for is P[D c L]. It is no t diat ely ob vious how to make th is calculation ; howe ver , if we appl y , the orem the calculation itself is trivial , for

I

I = P[L I Dc]

P[D c L]

I

P[L Del P[DC] P[Del P[L D II ] P[D ll ]

+

I

s applica tion of Bayes' theorem the collection of mutually exclusive stive events is the set { D H, D c }, for one of these two event s must occur 'oth cannot occur simulta neously. Our problem is now tr ivial since .errn on the right-hand side is easily calculated and lead s us to P D

IL

[ c

_

]-

pm __2p_ pm+ mm 2p + I

s the answer we were seeking and we find that the probability of having a cheating dealer, given that we lost in one play , ranges from 0 (p = 0) (p = I). Thus, even if we know that the cheating dealer is completel y Jest (p = I), we can only say that with probab ility 2/3 we faced this "given th at we lost one play. a final word on elementary topics, let us remind the reader that the er of perm utations of N objects taken K at a time is

N'

-----'-'-'.=--

(N - K )!

=

N(N - I) . .. (N - K

+ 1)

368

APPENDIX II

whereas the number of combinations of N things taken K at a time is denoted by

(~)

and is given by

(~)

N! K!(N - K)!

11.2. RANDOM VARIABLES So far we have described a probability system which consists of the triplet (S, Iff, P), that is, a sample space, a set of events, and a probability assignment to the events of that sample space. We are now in a position to define the important concept of a random variable. A random variable is a variable whose value depends upon the outcome of a random experiment. Since the outcomes of our random experiments are represented as points w E S then to each such outc ome w , we associate a real number X (w), which is in fact the value the random variable takes on when the experimental outcome is w . Thus our (real) random variable X (w) is nothing more than a function defined o n the sample space , or if you will, a mapping from the points of the sample space into the (real) line. As an example , let us consider the random experiment which consists of one play of a game of blackjack in Las Vegas. The sample space consists of all possible pairs of scores that can be obt ained by the dealer and the player. Let us assume th at we have grouped all such sample points into three (mutually exclusive) events of interest : lose (L), draw (D) , or win ( W). In order to complete the probability system we must assign probabilities to each of these events as follows *: P[L] = 3{8, P[D] = 1{4, P[W] = 3{8. Thus our probabilit y system may be represented as in the Venn diagram of Figure 11.1. The numbers in parentheses are of course the probabiliti es. Now for the random variable X (w). Let us assume that if we win the game we win S5, if we draw we win SO, and if we lose we win - S5 (that is, we lose S5). Let our winnings on this single play of blackjack be the random variable X(w ). We may therefore define this variable as follows :

+5 X(w)

=

{

wEW

0

W ED

-5

wE L

Similarly, we may represent this random variable as the mapping shown in Figure 11.2. • This is the most difficult step in practice , that is, determining appro priate numbers to use in o ur model of the real world.

11.2.

369

R ANDOM VARIABLES

Figure JI.I The probability system for the blackjack example. T he domain of the random variable X(w) is the set of event s e' and the values it ta kes on for m its range. We note in passing that the probability assignment P may itself be thought.of as a random var iable since it satisfies th e definition; this particular assignment P , however, has further restricti ons on it , namely, th ose given in Axioms (II.l}-(II.3). We are mainly interested in describing the probability that the rand om vari able X(w) takes on certain values. To this end we define the followin g sho rtha nd notati on for events: [X = x] ~ {w : X (w) = x } (11 .6) We may discuss the probability of this event which we define as P[ X = x ] = probability that X(w) is equal to x

which is merely the sum of the probabilities associa ted with each point w for which X (w) = x . For our example we have P[X = -5] = 3/8 P[X =0] = 1/4

(11.7)

P[ X = 5] = 3/8

An other convenient form for expressing the pr obabilities associated with the rand om vari able is the probability distribution fun ction (PDF) also known

x u»

---'----------L.--------L.----- R ~a l

~

0

Figure 11.2 The random variable X(w).

~

line

370

APPEN DIX II

as the cumulati ve distributi on function . For th is purpose we define notation similar to that given in Eq . (II .6), namely,

[X

~

x] = {w: X(w)

~

x}

We then ha ve that the PDF is defined as Fx(x) ~ P[X ~ x )

which expresses the probability that the random varia ble X ta kes on a valu e less than o r equal to x . The important properties of this functi on are Fx(x) ~ 0

(II .S)

Fx ( oo) = I Fx( - oo) = 0 Fx(b) - Fx (a) = P[a

<X

Fx (b) ~ Fx (a)

~

b)

for

a

for

O.

e-A X

O~X

x a) may be calculated in eith er of the two following ways: P[a

0) it sh ould yield a probability equal to 3/8. Such a function has alread y been studied in Appendix I and is, of course , the impulse functi on (or Dirac delta funct ion ). Recall th at such a function uo(x) is given by lI

o(x )

=

oo 0

x = 0

r

x ;= O

C'

uo(x ) d x

. - a::

=

I

a nd a lso th at it is merel y the derivati ve o f the unit step functi on as can be seen from x

r

lI

L"

o(Y) d y

= {O I

x< O x :2: 0

Using the graphica l notati on in Fi gure 1.3, we may pr operly descr ibe the pd f for o ur blackjack example as in Figure 11.4. We note immediately th at this represen tat ion gives exactly the information we had in Eq. (11.7), a nd therefore the use of impulse functions is overl y cumbersom e for such problems. In particular if we de fine a discrete random va riab le as o ne that take s on va lues over a discrete set (finite o r countable) then th e use of the pdf* is a bit heavy a nd unnecessary a lt ho ugh it does fit int o o u r genera l definition in the o bvio us way. On the o ther hand , in the case o f a random va riable that ta kes o n va lues over a continuum it is perfectly natural to use the pdf and in the • In the discrete case, the function P[X = xd is often referr ed to as the probability 1Il0SS [unction. The genera lization to the pd f lead s one to the noti on of a mass density func tion .

[1.2.

_ _ -'

-5

RANDOM VARIABLES

J

J

8

8

-'

o

--'

373

x

+5

Figure 11.4 The pdf for the blackjack examp le.

case where the re is also a non zero probability th at th e ra nd om variable ta kes on a specific value (i.e., that the PDF con tains j um ps) then th e use of the pd f is necessar y as well as is the introduction of the impulse functio n to acco unt for the se points of accumula tion. We a re thus led to distin gu ish between a discrete rando m variable, a purely continuous random va ria ble (o ne whos e PDF is co ntinuous a nd ' everywhere differenti a ble), a nd the th ird case of a mixed ran dom variable which contains so me discrete as well as co nti nuo us portion s. * So , for exa mple, let us con sider a random va ria ble that represent s the lifetime of an a uto mo bile. We will assume th at there is a finite pr oba bility , say of val ue p , t hat the autom obil e will be inoperable imme d iately upon de livery, a nd th erefor e will ha ve a lifet ime o f length zero. On th e othe r ha nd , if the a utomobi le is ope ra ble upo n delivery then we will ass ume tha t th e rem ainder of its lifetim e is exp onentially distributed a s given in Eqs. (11.I 2) a nd ([1.13). Thus for thi s a uto mo bile lifetime we have a PD F a nd a pdf as given in Figure 11.5. Thus we clearly see the need fo r impulse fu nct ions in describing interesting ran dom va riables . We have now disc ussed the notion of a pro ba bility system (5, s, P) and the no tio n of a rand om va ria ble X(w) defined upon the sa mple space 5 . Th ere is, of co urse, no reason why we cann ot define mallY rand om vari ables o n the sa me sa mple space. Let us co nsider the ca se of two ran dom va riables X a nd Y defined for so me probability system (5, 8, P ) . In th is case we have • It can be shown that an y PD F may be decomposed into a sum of th ree part s, na mely, a pure jump function (cont aining on ly discont inu ou s ju mps), a pure ly cont inuo us porti o n, a nd a singula r port ion (which ra rcly occu rs in distribution functions of interest and which will be con sidered no fu rther in thi s text) .

374

AP PEN DIX II

p

1 - - --======--x o

o (a) PD F

Figure U.s

.

(h) pdf

PD F and pdf for automobile lifetime.

the natural exte nsion of the PD F for two rand om varia bles, namely, 6

F Xy(X , y) = P[X ~ x, Y ~ y ]

which is mer ely th e prob ability tha t X ta kes on a value less tha n or eq ua l to x a t the sa me tim e Y ta kes on a va lue less th an o r equal to y; th at is, it is th e su m of th e p robabilities associated with all sample p oint s in the intersecti o n of the two events {w : X(w) ~ z }, ·{w : Y(w) ~ y }. FXY(x, y) is referred to a s t he j oint PDF. Of co urse, associa ted with thi s funct ion is a joint probability d ensity funct iondefined as A

f.yy( x, y) =

d

2F

XI'(X' y)

_-"--'.--'-~

dx dy Gi ven a joint pdf, o ne naturally inq uires as to th e "marginal" density functi on for one of th e varia bles and thi s is clearl y given by integra ting over a ll possible val ues of the second variable, thus

fx( X) = i : _J xr (X, y) dy

(11.14)

We are now in a posi tion to de fine the noti on of independence between ra nd om variables. T wo rando m variables X an d Yare sai d to be inde pendent if a nd only if f xy(x, y) = fx(x)!I .(y) th at is, if the ir joint pdf fac tors in to th e product of th e o ne-dime nsional pdf's. Thi s is very much like th e definition fo r two independent events a s given in Eq . (11. 5). H owever , for th ree o r mo re random va riables, th e definitio n is esse nt ia lly th e same as for two , na mely, X" X 2 , • • • , X n are sai d to be independent ra ndom variables if a nd o nly if

IL 2.

RANDOM VARIABLES

375

This last is a much simpler test th an th at required for multiple events to be independe nt. With more th an one random variab le, we can now define co nditiona l distr ibut ions an d den sities as follows. For example, we could ask for the PDF of the rand om variable X co nditio ned on some given value of the rand om variab le Y, which would be expre ssed as P[X ::;; x Y = y). Similarly, the cond itional pdf on X , given Y, is defined as

I

a d fXI y(x y) = - P[X ::;; x dx

I

Iy

= y) =

fxy (x, y) :...:=-'--'--'-

fy(y)

much as the definition for conditional prob abil ity of events. To review again, we see that a random variable is defined as a mapping from the sample space for so me probability system into the real line and from this mappin g the PDF may easily be determined. Usually, however, a random var iable is not given in term s of its sample space and the mapping, but rather directly in terms of its PDF or pdf. It is possible to define one random variable Y in terms of a second random variable X, in which case Y would be referred to as a function of the random variable X . In its most general form we then have

(11.15)

Y= g(X)

where g(' ) is some given function of its argument. Thus, once the value for X is determined, then the value for Y may be computed ; however, the value for X depends upon the sample point w, a nd therefore so does the value of Y which we may therefore write as Y = Y(w) = g( X(w». Gi ven the random va riable X and its PDF , one sho uld be able to calculate the PDF for the random variable Y, once the functi on gO is known. In principle, the co mputat ion take s the following form: Fy (Y)

=

P[ Y::;; y ] = P[ {w : g(X(w» ::;; y} ]

In general, this computati on is rather complex. On e random variable may be a funct ion of many other random varia bles rather than ju st one. A particularly important form which often a rises is in fact the sum of a collection of independent random va riables {Xi}' namely, n

( 11.16)

Y = L: X i i- I

Let us derive the distribution function of the sum of two independent random variables (n = 2). It is clear that this distribution is given by Fy(y)

=

P[Y::;; y]

=

P[X 1

+X

2 ::;;

y]

376

APPENDIX 11 X,

Figure I/.6 The integration region for Y = X,

+ X,

::::; y .

We have the situ ati on shown in Figure I1.6. Inte gratin g over the indic ated region we have

Due to the independence of X, and X, we then obta in the PD F for Y as F y(Y )

=

f [f"-X' OO - <Xl

-00

]

!x ,(x ,) dx, ! x ,(x,) dx,

=foo Fx,(Y -

xJ!x,(x,) dx,

- 00

Fi nally, forming the pdf from thi s PDF , we have f y(Y) = L : ! x ,(y - x,) !x,(x,) dx,

Th is last equ atio n is merely the convolut ion of the density functions for X, a nd X, a nd, as in Eq. (1.36), we denote this con volut ion opera to r (which is both associative an d com mutative) by a n asterisk enclosed within a circle. Thus

II. 3.

EXPECTATIO N

377

In a similar fash ion , o ne easily shows for the case of a rbitra ry n th at the pdf fo r Yas de fined in Eq. (11.16) is given by the conv olu tion of the pd f's fo r the X;'s, th at is, (11.17)

II.3. EXPECTATION In thi s sectio n we discuss certain measures associated with the PDF and the pdf for a random variabl e. These measures will in genera l be ca lled expectations and they deal with inte grals of the pdf. As we saw in the last section , th e pdf involves certain difficulties in its definiti on , and the se difficultie s were handily resolved by the use of impulse functions. However, in much of the literature on pr obability the or y a nd in most of the literature o n queueing the or y th e use of impulses is either not accepted, not und er stood or not kn own ; as a result, special care and not ation has been built up to get a ro und the problem of differentiating discontinuous functions. The result is that many of the int egrals encountered are Stieltjes integrals rather than the usual Riemann inte grals with which we are most fa miliar. Let us take a moment to define the Stieltjes integral. A Stieltjes inte gral is defined in term s of a nondecreasing function F(x) and a continuous function 'r ex); in additi on, two sets of points {f.} a nd { ~.} such that 1. _ 1 < ~. ~ f. a re defined and a limit is considered where max If• .:... 1. _,1 ~ O. From the se definiti on s, con sider the sum

L 'I'(~.)[F(I.)

•

- F(I . _1) ]

This sum tends to a limit as the interva ls shrink to zero indepe ndent of the sets {f. } a nd { ~.} a nd the limit is referred to as the Stieltje s integral of cp with respect to F. This Stieltjes integral is written as

J

qc(x) dF(x )

Of co urse, we rec ogn ize th at the PDF may be identified with the functi on F in thi s definitio n a nd th at dF(x) may be identified with th e pdf [say , f (x)] through

dF (x )

= f( x) d x

by defini tion . With out the use of impulses the pdf may not exist ; however, the Stieltjes integral will always exist and therefore it avo ids the issue of impulses. Howe ver , in thi s text we will feel free to incorporate impul se functi on s and therefore will work with both the Riem ann and Stieltje s integra ls ; when impulses a re pe rmitted in the fun ct ion f (x) we then have the

378

APP ENDI X II

following identity:

J'r ex ) dF(x) = Jg;(x)f(x) dx We will use both notations throughout the text in o rder to fam iliari ze the student with the more common Stieltjes integral for queueing theory, a s well a s with the more ea sily manipulated Riemann integral with impulse funct ion s. Having sa id all th is we may now introduce the definition of expectation. The expectation of a real random variable X (w) denoted by E[X] and al so by X is given by the following:

E[X]

&

=

-

X

.1

=

'1"

-X> X

(I 1.1 8)

dF_,(x)

This last is given in the form of a Stieltjes integral ; in the form of a Riemann integral we have, of course,

E[X]

= .¥ =

L:

xfx(x) dx

The expectation of X is als o referred to a s the mean or average calue of X. We may also wr ite

E[X] =

f'

(I - Fx(x)] d x -

f/ x(X) dx

which, up on int egrating by parts, is easily shown to be equal to Eq . (l1.l 8) so long a s E[X] < 00. Similarly, for X a nonnega tive random variable , thi s form bec omes

E[X] =

i"

[I - Fx (x)] dx

X

zo

In general , the expectation of a random variable is equal to the product of the va lue the random va ria ble may take on and the probability it takes o n th is value, summed (integra ted) over all po ssible values. N ow let us con sider o nce aga in a new random va riable Y, which is a function of our first rand om variable X, namely, a s in Eq. (IU5) Y

=

g(X )

We may define the expecta tio n E r [Y] for Yin terms of its PDF just as we did for X ; the subscript Yon the expectation is there to distingui sh expectation with respect to Yas opposed to any other random va ria bles (in th is ca se X ). Thus we have

II.3.

EXP ECTAT ION

379

This last computation requi res that we find either Fy(Y) o r f y(y) , which as mentioned in the previous section , may be a ra ther complex computation. However, thefu/ldamentaltheorem ofexpectation gives a much more straightfor ward calculatio n for th is expectation in ter ms of distributio n of the underlying random variable X, na mely,

Edy ] = Ex[g(X)]

= roo g(x)fx(X) dx .i_oo

We may define the expectation of the sum of two random variables given by the followi ng obvious general izatio n of th e o ne-dimensional case :

E[X

+

Y] = L : L : (x

+ Y)fxy( x, y) d x dy

= L : L : Xf x y (x , y) d x dy

=L :

xfx (x) d x

= E[X ]

+i

+L :

L :Yfxy(x, y) d x dy

:y!y(y) dy

+ E[Y ]

( 11 .19)

Th is may also be written as (X + y = X + Y). In goi ng from the seco nd line to the thi rd line we have taken advan tage of Eq. (I I. 14) of the previous section in whic h the marginal density was define d from the jo int density. We have show n the very imp ortant result, that the expectation of the sum of tlro random cariables is always equal to the sum of the exp ectations of eachthis is true whether or not these random variables are independent. This very nice prope rty comes from the fact th at th e expectation operato r is a linea r ope rator. The mor e genera l sta tement of th is prop ert y for any number of random variab les, independent o r not , is that the expectation of the sum is alway s equal to the sum of the exp ectations, tha t is,

E[X l

+ X o + . .. + X n ] = E [X tl + E[X o] + ... + E[X n ]

A similar que stion may be asked a bo ut th e product of two rando m var iab les, that is,

E[XY] = L : L : xyfXl '( x, y) d x dy In the special case where the two ran do m variables X and Ya re independent , we may write the pdf for th is joint ra ndom variab le as the product of the pdf's for the individual rando m variab les, thus obtaining

E[XY]

=

L:L:

xyf s (x )f d Y) dx dy

=

E[X]E[Y ]

( 11.20)

380

APP ENDI X II

This last equati on (which may also be written as X Y = X Y) states th at the ·expecta tion of the product is equal to the product of the expectations if the random va ria bles are indepe ndent. A result simi la r to th at expre ssed in Eq. (11.20) applies a lso to fun ctions of independent random va riab les. That is, if we have two independent random variables X and Y and fun cti ons of each denoted by g(X) a nd h( Y), then by a rguments exactl y the sa me as th ose leading to Eq. (11.20) we may sho w E [g(X) lJ(Y)]

=

E[ g(X )]E[lJ(Y )]

(l1.21 )

Often we are interested in the expectation of the pOlrer of a random variable. In fact, this is so common that a s pecia l name has been coined so that the expected value of the 11th power of a random va riable is referred to a s its nth mom ent . Thus, by definition (rea lly this follo ws from th e fundamental theorem of expectation) , the 11th moment of X is given by E[X n]

~ xn~

L:

x nfx(x) dx

Furthermore , the ntli central moment of thi s random va riable is given as follows: OC

A

( X - x )n =

(x -

[

-

X)"fx( x) d x

· - oc

The nth central moment may be expressed in terms of the first II mo ments themselves ; to show thi s we first write down the foll owin g ident ity making use of the bino mial the orem

(X -

X )"

=I(n)X'-C- Xl"-k ,·~O

k

Taking expectations o n both sides we then have

(X -

X)" =i(II) Xk(- X)"-k k~O

=

i k -O

k

(/~) Xk( - Xj',-k

( 11.22 )

k

In going fr om the first to th e seco nd line in thi s last eq ua tion we ha ve taken a dva ntag e of th e fact th at th e expectat ion o f a su m is equal to the sum of the expectation s a nd th at the expectat ion of a con stant is mere ly th e con st ant itself. Now for a few o bserva tio ns. Fir st we note th at the Oth moment o f a random va ria ble is j ust unit y. Al so . the Ot h central moment mu st be ~ n e. The first central mom en t mu st be 0 since (X - X) = X - X = 0

var

In (I fil

tc

o q d

11.4 .

T RAN SFORMS AN D CHA RACTE RISTIC FUNC TIONS

38 1

The second central moment is extremel y importa nt and is referred to as the variance; a special not ation has been ado pted for the varia nce and is given by

ax 2 ~ (X - X)2 ~ X2 - (1')2

In the second line of thi s last equation we have taken ad vantage of Eq. (II.22) and have expre ssed the variance (a central moment) in term s of the first two momen ts themselves. The square root of the variance ax is referred to as the standard deviation. The ratio of the sta nda rd deviati on to the mean of a random va riab le is a most important qu ant ity in sta tistics and a lso in queueing the or y; th is ratio is referred to as the coefficient of variation and is den oted by (11 .23)

11.4. TRANSFORMS, GENERATING FUNCTIONS, AND CHARACTERISTIC FUNCTIONS In probability the or y one encounters a variety of functi ons (in part icular , expect ations) all of which are close relative s of each other. Included in this class is the characteristicf unction of a random vari able, its moment generating fun ction, the Laplace transf orm of its probability density fu nction, and its probability generating fun ction. In this secti on we wish to define and distinguish these vario us forms a nd to indicate a common centra l propert y tha t they share. The characteristic fun ction of a rand om variable X, den oted by c/> .d u), is given by c/> x( u) ~ E[ei "x ] =

L:

eiuz;-x(x) dx

where j = J~ a nd where u is a n arbitrar y real vari able . (Note that except for the sign of the exponent , the characteristic function is the Fo urier tran sform of the pd f for X). Clearly.

Ic/>x(u)I :s;; L : a nd since

lei""1=

which shows tha t

I . we have

lei "xI IJx (X)1dx

382

APPENDIX II

An important property of the char acteri stic function may be seen by expanding the exponential in the integrand in terms of its power series and then integrating each term separately as follows: epx(u)

= Lf'"/ = 1

[ + jux + ~ (jU X)2 + ...]

x(x) 1

dx

(jU)2~

. _

+ lUX + - - X " + ,.. 2!

From this expansion, we see that the characteristic function is expressed in terms of all the moments of X. Now, if we set u = 0 we find that epx(O) = I. Similarly, if we first form depx(u)/du and then set u = 0, we obtain j X Thus, in general , we have (11.24)

Thi s last important result gives a rather simple way for calculating a constant times the nth moment of the random variable X . Since this property is frequently used, we find it con venient to adopt the following simplified notation (consistent with that in Eq. · 1.37) for the nth der ivative of an arbitrary function g (x), evaluated at some fixed value x = xo: gln )(x ) o

~ d ng( x)

I

d x"

. ( 11.25)

%=:1: 0

Thus the result in Eq. (II.24) may be rewritten as ep:~)(O)

= rx».

The mom ent generating function denoted by M x(v) is given below along with the appropriate differential relationship that yields the nth moment of X directly . Mx (v) ~ E[e- x]

=L : M:~)(O) = X

e-Z! x(x) d x n

where v is a real variable. From this last property it is easy to see where the name "moment generating function" come s from. The deriv ation of this moment relationship is the same as that for the characteristic function . Another important and useful function is the Laplace transform of the pdf of a random variable X. We find it expedient to use a notation now in which the PDF for a rand om va riable is labeled in a way th at identifies the rand om variable without the use of subscripts. Thus, for example. if we have

a

IIA . TRANSFOIU\1S AN D CHARACTERISTIC FUNCTIONS

383

rand om var iable X , which represents , say, the interarrival time between adjacent customers to a system, then we define A( x) to be the PD F for X; A(x) = P[ X

~

x]

where the symbol A is keyed to the word "A rrival." Further , the pdf for th is example would be denoted a(x). Finally, then , we den ote the Laplace tr an sform of a(x) by A *(s) and it is given by the following: A *(s) ~ E[e- ' x]

~

L:

e-' '''a(x) d x

where s is a complex variable. Here we are using the "two-sided" transform ; however , as mentioned in Section 1.3, since most of the random variables we deal with are nonnegative, we often write

Th e reader should take special note that the lower limit 0 is defined as 0- ; that is, the limit comes in from the left so that we specifically mean to include an y impulse functions at the origin. In the fashion identical to that for the moment generating funct ion and for the characteristic function , we may find the moments of X through the following formula: A*(nl(o) = ( _ l)nx n

(11.26)

For nonnegative random variab les IA*(s)1

~ f ',e-'''''IG(X)' d x

But the complex variable s consists of a real part Re (s) = a and an imaginary par t 1m (s) = w such that s = a + j w. Th en we have

le-s"'l = le- ' ' 'e- ;w'' l ~ le-''' I le- ;w "'l = le-a"'l M oreover , for Re (s) ;:::: 0, le-a"'l ~ I and so we have from these last two equations a nd from

JO' a (x ) dx = IA*(s)1

~

I,

1

Re (s) ;:::: 0

It is clear tha t the three functions 1>x(u), M x (v), A *(s) are all close relatives of each other. In particular, we ha ve the following relationship :

c/>x(sj)

=

M x (- s)

=

A*(s)

384

APPENDIX II

Thus we are not surprised that the moment generating pr operties (by differentiati on) are so simila r for each; this property is the central pr operty th at we will take ad vantage of in o ur studies. Thus the nth mom ent of X is calculable from an y of the following expression s: X

X

n

= rn 4>~~)(O)

n

=;

Xn

.M~~)(o)

= (_l )nA*

=I

E[e- , x l ] .. . E [e-,xn]P[N

=

II]

n= O

: since {X ,} is a set of identically distribu ted ra ndo m va riables, we have OX>

Y *(s)

=I

[X *(s)rP[N

=

II]

( 11.33)

n= O

ere we have den oted the Lap lace transfo rm of th e pd f for each of the X i X ' (s) . The final expression given in Eq . (II .33) is immedia tely recognized

388

AP PENDI X II

as the z-transform for the random variable N, which we choose to denote by N(z) as defined in Eq. (11.27) ; in Eq , (II.33), z has been replaced by X*(s) .

Thus we finally conclude Y *( s)

= N (X*(s»

(11.34)

Thus a random sum of identically d istributed independent random varia bles has a tran sform that is related to the transform s of the sum's random varia bles and of the number of term s in the sum, as given above. Let us now find an expre ssion similar to that in Eq. (11.32); in that equation for the case of identically distributed Xi we had Y = nX, where n was a given con stant. Now, however, the number of term s in the sum is a random quantity and we mu st find the new mean Y. We proceed by takin g advantage of the momen t generating properties of our transforms [Eq. (11.26)]. Thus different iating Eq . (11.34), setting s = 0, and then takin g the negative of the result we find Y= N X

which is a perfectly reasonable result. Similarly, one can find the variance of this rand om sum by differenti at ing twice an d then subtracting off the mean squared to obta in a y 2 = Nax 2 + (X)2 all· 2 Thi s last result perhaps is not so intuitive. 11.5. INEQUALITIES AND LIMIT THEOREMS In this section we present some of the classical inequalities and limit theorem s in pro bability the ory. Let us first consider bou nding the probability that a random variab le exceeds some value. If we know on ly the mean value of the random variable, then the following Ma rkov inequality can be established for a nonnegative ran dom variable X: P[X

~

z]

x

~-

x

Since on ly the mean value of the rand om var iable is utilized, this inequ ality is rather weak. The Cheby shev inequality makes use of the mean and variance a nd is somewhat tighter ; it states that for any x > 0, a .2 P[IX - X/ ~ xl ~ ~ xOth er simple inequal ities invol ve momen ts of two ran dom varia bles. as follows: First we have the Cauchy-Schwarz inequality , which make s a statement ab out the expectation of a product of rand om varia bles in term s of

11.5.

389

INEQU ALITIES AND LIMIT TH EOREMS

the second moments of each. (I 1.35)

A gene rali zati on of this last is Holder's inequality , which states for C1. I, C1.- 1 + f3- 1 = I, and X> 0, Y> 0 that

f3 >

>

I,

XY ~ (X ")l /'(yP/IP

whenever the indicated expectations exist. Note that the Cauchy-Schwartz inequality is the (impor ta nt) special case in which C1. = f3 = 2. The triangle inequality relates the expectation of the absolute value of a sum to the sum of the expect at ion s of the absolute values, namel y,

IX + YI ~ IXI

+ IYI

A generalizat ion of the tri an gle inequality , which is kn own as the C.-inequality, is where I

C= • { 2.- 1

O- O.OIJ -< 100.25106 •

4

=

25 x 10-4

Thi s result is twice as large as it should be for our calculation since we have effectively calculated both tails (namely, the probability that more than 510 ,000 or less than 490 ,000 heads would occur); thus the appropriate an swer for the Chebyshev inequ ality would be that the probability of exceeding 5 10 ,000 heads is less than or equal to 12.5 x IQ-4. Note what a poor result this inequ ality gives comp ared to the central limit theorem approximat ion , which in this case is comp arabl e to the Chernoff bound. 11.6. STOCHASTIC PROCESSES It is often said that queueing theory is part of the theory of applied stochastic processes. As such, the main port ion of this text is really the proper sequel to this section on stochastic processes; here we merely state some of the fundamental definitions and concepts. We begin by considering a probability system (S, Iff, P) , which consists of a sample space S, a set of events {A , E, .. .} , and a probability measure P. In addition, we have already introduced the notion of a rand om variable

394

AP PEN DIX II

X (w). A stochastic process may be defined as follows : For each sample point w E S we assign a time functi on X (t, w). Thi s family of functions forms a stochastic pro cess; altern ativel y, we may say that for each t included in some appropriate parameter set, we choose a random variable X (t , w). Thi s is a collection of rand om variables depending upon t , Thus a stochastic process (or random function) is a function * X (t) whose values are rand om variables. An exampl e of a random process is the sequence of closing prices for a given security on the New York Stock Exchan ge; an other exampl e is the temperature at a given point on the earth as a function of time. We are immed iately confronted with the problem of completely specifying a random pr ocess X (t ). For this purpose we define, for each allo wed t, a PDF, which we denote by Fx(x , t) and which is given by

Fx(x , t) = P[X(t )

~

xl

Further we define for each of n allo wable t , {t l , t 2 , given by FX

1""-Y 1, ·· ·X ,,,( X 1 , X 2, · · · ,Xn ;

t h 12 ,

.. .

• • • ,

t n} a j oint PD F,

,t,J

.l.

= P[X(tl)

~ Xl ' X (t 2 ) ~

X 2, • • • ,

X(t n ) ~ x n ]

and we use the vector notation Fx(x ; t) to den ote this function. A stochastic process X(t) is said to be stat ionary if all Fx (x , t) are invariant to shifts in time ; that is, for a ny given constant T the followin g holds:

Fx(x ; t

+ T) = Fx(x; t)

where the notation t + T implies the vector (11 + T , t 2 + T, .• • , t n + T ). Of most interest in the theory of stoch astic processes are these stationary random functions. In order to completely specify a stochastic process, then , one must give Fx(x ; t) for all possible subsets of {Xi}, {I,}, and all n. Th is is a monstrou s task in general! Fortunately, for man y of the interesting stochastic pr ocesses, it is possible to pro vide this specificat ion in very simple term s. Some other definiti ons are in order. The first is the definition of the pdf . for a stochastic process, and this is defined by

Second, we often discuss the mean value of a stochastic process given by X (I)

=

E[X(t )]

=

L:

xix(x; I) d x

• Usually we denote X(I, w) by X (I ) for simp licity.

Il .6.

STOCH ASTIC PROCESSES

395

Next, we introduce the autocorrelation of X (t) given by

R x x (t" t2 ) = £ [X(t ,)X( t2 ) ]

J-: L:

=

x ,xdx,x,(x " x 2 ; I" 12) dx, d X2

A large the or y of sto chas tic pr ocess has been de veloped , kn own as secondorder theory, in which these pr ocesses a re classified a nd distingui shed o nly o n the basis of th eir mean X U) a nd autocorrelati on R x x (t" t 2) . In the case of stat iona ry rand om processes, we ha ve X(I) = X

( 11.43)

and

R x .\:(t"

12 )

= R x x (12

-

I,)

(11.44)

th at is, R x x is a functi on onl y of the time difference -r = t 2 - t,. In the sta tio nary case, then , random processes are characterized in the seco ndorder the or y only by a con stant (their mean X) and a one-d imensiona l func tion Rx x (-r). A random pr ocess is sa id to be wide-sense stationary if Eqs. (11.43) and (11.44) hold . No te that all sta tiona ry p rocesses are wide- sen se sta tio nary, but not con versely. REFERENCES DAVE 70 Davenport, W. B. Jr. , Probability and Random Processes, McGraw-Hill (New York), 1970. FE LL 68 Feller, W., An Introduction to Probability Theory and Its Applications, 3rd Edition, Vol. I , Wiley (New York), 1968. PAPO 65 Papoulis, A., Probability , Random Variables, and St ochastic Processes, McGraw-Hill (New York), 1965. PARZ 60 Parzen, E., Modern Probability Theory and Its Applications, Wiley (New York), 1960.

f

c

G G

s.

Glossary of Notation*

g }-

Ii

I,

(Only the notat ion used ofte n in this book is included below.) NOTATI ONt

A .(t) = A(t) An*(s) = A *(s) ak a. (t) = aCt) Bn(x) = B (x) B ;*(s) = B*(s) bk bn(x) = b(x)

C2 b C. Cn(u) = CCu) C. *(s) = C*(s) c.(lI) = c(u)

D dk E [X ] = Ei Er FC FS Fx(x)

X

DEFI NITION

I'

TYPICAL PAGE REFER ENCE

P[t . ~ t] = P[i ~ t] Lapl ace transform of aCt ) k th mome nt of aCt ) dA n(t) /dt = dA( t) /dt P[x . ~ x ] = P[x ~ x ] Laplace tra nsform of b(x) kth moment of b(x) dBn(x)/dx = dB(x)/dx Coefficient of variati on for service time nt h customer to enter the system P[u. ~ u] Lapl ace tra nsfor m of cn(lI) = c(u) dC . (lI)/du = dCCu)/du Deno tes determin istic distribution P[ij = k] Expectation of t he ran dom variab le X System sta te i Den otes r-stage Erlan gian distribution Fir st-come-first-served P [X ~ x]

r

13 14 14 14 14 14 14 14 187 II

281 285 281

K L

tv tv III

N

N

O( 01

p

P P

VIII

176 378 27 124 8 370

PI p< P,

Pi Pi fk

Q • In those few cases where a symbo l has more than one meaning. the context (or a specific statement) resolves the a m biguity. t The use of the notation Y n - Y is mean t to indicate tha t y = lim Yn' as /I - co wherea s y( t) - y indicates that Y = lim y(r) as t - "'J .

396

q" q"

GLOSSARY OF NOTATION

397

I,(x)

dFx(x){dx

371

G G(y) G*(s)

Denot es genera l di stri bution Busy-peri od di stribution Laplace tr an sform o f g(y)

VIII

gk

kth moment of bu sy-peri od duration

g(y)

dG(y) {dy

HIl

Den ote s R-stage hyperexponential d istri bution

Im (s) In - > I I *(s ) I*(s)

208 211 213 215

Im aginary p art of the complex varia ble s Durat ion of the (nth) idle peri od Laplace tr ansform of idle-period den sity

141 293 206 307

Laplace tran sform o f idle-time density in the

310

LC FS

dual system Size of finite sto rag e La st-c ome-first-ser ved

M

Den ote s exponential distribution

VIII

M 111

Size o f finite population Number of servers

viii

N.(I) ->- N. N(I) ->- N

Number of cu st omers in system a t time

K

o(x)

8 Vlll

N u mber 'of cu st omers in queue a t time lim O(x){x z-o

lim o(x) {x

o(x)

viii

z-o

=K< =0

48

M atri x of transition probabil itie s Pr obability of the event A Pr ob ab ility of the even t A conditioned o n the event B Pro babil ity distribution fun cti on Pr obabil ity den sity functi on

pdf

P[ N(I ) = k ] P[next sta te is

Pk(l )

Pu

I

II

284

P

PDF

t

co

Pt A] P tA I B]

17

I

t; I current

sta te is E,]

31 364

365 369 371 55 27 46 90 192 48

Pk

P[ X (t ) = j Xes) = i ] P[k custo mers in system]

Q(=)

c-tra nsfo rrn of P[ij = k]

quC l )

T ra nsi tio n rate s a t tim e

qn

N umbe r left behi nd by dep artu re (o f e n)

177

Nu mbe r found by arrival (of e n)

242

Pij(S, I)

ij

q,/ ~ ij'

t

398

GLOSSARY OF NOTAT ION

r ij

Real part of the complex variable s P[next node is j current node is i]

rk

P [q' = k ]

Rc(s)

I

Sn(Y) -> S(y)

P[sn ~ y] -> p es ~ y ]

S; *(s) s n~s

Laplace tran sform of sn(Y) ---+ sty) Lapl ace tra nsform variable T ime in system (for e n)

sn(Y) -> sty)

dS n(y) /dy

sn ---+ s = T

Average time in system (for e n)

snk

kth moment of sn(Y) Average time in system Interarrival time (between e n_1 a nd e n) Avera ge interarriva l time

S * (s)

---+

s

-Joost

T I n -> i i n = (= IIA

fi' U(/) Un ~U

V(z ) V

W IV o W_(y ) Wn(y)

W(y)

->

W n*(s)

---+

->

~vn -+-

W= W

w(y)

w nk - lvk

X(I) 53

xk

y z

---+

P[I;:' ~ y ]

Waiting time (for e n) in queue

wn(y)

Xn

P[w n ~ y ]

W*(s) Lapl ace transform of wn(y)

"'n-+- lV

X n -+-

dS(y)/dy

kth moment of a(/ ) Unfinished wor k in system at time I Unit impul se functi on Un = Xn f n+ 1 ----+ ii = 53 - i z-tra nsforrn of P[v = k] Number of arrivals du ring service time (of e n) Average time in qu eue Average rema ining service time Co mplementary waiting time

!lo(/)

Un ----+

---+

= x = 1/fl-

dWn(y) /dy

---+

dW(Y) ldy

Average waiting time (for en) kth moment of wn(y) State of stochastic pr ocess X(I) at time I Service time (of e n) kt h moment of b(x) Average service time Busy-period du ration z-tra nsform va riable

340 149 176 14 14 339 14 14 14 14 14 14 14 14 206 341 277 184 177 14 190 284 14 14 14 14 14 14 19 14 14 14 206 327

GLOSSARY OF NOTATION

(X(t)

Number of arrivals in (0 , t)

Yi bet)

(E xternal) input rate to node i Number of departures in (0, t)

},

Average arrival rate

z,

Birth (arrival) rate when N = k A verage service rate

fl fl k 1t ( n ) -)-

Cnl

1Tk

1t

---+ 1Tk

399 15 149 16

Death (service) rate when N = k Vector of sta te probabilities 7Tla )

14 53 14 54 31

P[sy stem sta te (at nth step) is Ek ]

29

k

II a,

at a2 '

p

Utilization factor

"

ak (Product notation)

334

i= l

18 249

Root for G /M/m Variance of interarrival time

305 305

Variance of service time Arrival time of e n

12

285 285

Laplace transform of W(y)

(0, t)

Laplace transform of W_(y) Equals by definition The interval from 0 to t

X= E[X]

Expectation of the random variable X

(y) +

max [0, y]

=

(:) A/B /m /K/M

FC nl(a) fCkl(x)

o f ---+ g A
I' ,m, p(,"1 tI

=

p lm )

£ [1'1;"1]

k

p on- q'p lq)

PiI(n , n+ I ) £ P[X" +l = j

I X" = i]

= "

+ 1)]

1'(11)

.

= J I X" = /)

101 2: pt'''-0Ip ik k;

=

.

= /]

.

A

= p [X" t",

HOMOG ENEO US

NON HOMOGENEOUS

[Pi /n, n

Pij(III , n) A P[ X"

= j I x'" =

£ P[ X(t

Po (1) £ P[X (s

H (III, n) £ )1'/1(11I, II)]

p o (m , n) = p ik(m , q)pk;(q , n) k 11 (11I , n) = H (m , q)H(q, n

2:

. Po U , I + ~ I ) + ~I ) = j I X( I) = i] A P[X(I + ~I ) = jl X (I ) = i]

Po

£

I'

i]

=

H (t )

1'(1)

[PiI)

+ I) = j I Xes) = i]

H (t )

PiI(1)

NO NHOMOGEN EOUS

A [P iI (I )]

2:k P,k(1 =

s)Pk,(s)

H (I - s)H (s)

£

[pi;(I, I

Po C' , I ) £ P[x(t )

=

H (s , I)

= j I Xes) =

i)

=. " [ p i; (S,I )J

H (s , I)

Piles, I)

+ ~t)]

2:k PikeS, lI)pk; (II , I)

=

H (s , II)H (II, I)

Table (continued) Forwa rd

P IUIl

=

p tm- l lp

P I11IJ

=

pp lm - l )

equa tio n

Backward equation Soluti on

p llll l =

Transition-rate matrix. State probabi lity Matrix of state probabi lities For ward equation solution

,, ~,,) ~ P [X"

./>0

o

VJ

= jl

,,~n ) ~ P[X"

n: (tll

=

n ln )

n ' n,

n 1n- 11P

= n lO'p n

[I -

= nP

%p l- 1

= n ,n- 1,p (1I _

-¢> p n

= QH (I)

H (I) = e o t

I)

d n(I) ldl

all(s, t)l as

H (s , I)

= -Q (s)lI(s, t)

= exp [f: Q (II) dll ] .

" j(t) ~ P[X (I )

= n (I )Q

=0

[sl - Ql- l -¢>H (r)

= jl

n (t ) ,1\, [" jU )l dn(I)ldl n (l )

= n (I )Q (I )

= n CO) exp WQ (II) dll ]

I) nQ

p et) - I

Q U) = lim - - t>t-O CJ. r

=j l

n(l) = n (O)e o t

= H (s, I)Q (I )

aH (s, t)lal

n (l ) ~ [" j U)l

n:11ll

-

H (I)Q

" j(t) ~ P[X(I)

=j l

= n 'OIP(O)p (1) . . . P en -

n

d H(t)ldl

,d

t> t_O CJ.r

n '" ' ~ [" I" )l

n "lt ,1\, [ 1T ~ " )l

d H(I)ldl

P - I Q =l im - -

-

-

Equilibrium so lution Tr an sform relatio nships

pm

H (III , II) = H (III , II - 1)1' (11 - I) H (III , II) = P (m)H (1II + I, II) H (III , II) = 1'(111)1'(111 + I) ' . . 1'(11 - I)

-

404

SU MMARY OF IMPORTA NT RESULTS

P [in ter departure tim e ~ tj

g(y)

~

in =

t~ O

I - e-;"

l(2n - 1

2)p n_l(1

n

215

0

n -

+ p)I- 2n

218

( .A.)k

~ - CJ.1.u)K+I .u

(

104 otherwise

Pk =

M! (M - k)!

I.

M!

,-oeMP(z)

=

148

( 1)11 e- U+P)"I 1[2 y(A.u)1/2j -

y p

l - I,I.u Pk =

=

(/.)k .u

107

(M/ M/ IIIM)

( ~)'

i) ! .u

.uO - p)(l - z) .u( I - c) - 1.z[1 - G(z)]

Pk = ( I -

(M/M/ I/ K)

~)(~J

k

=

(M/M/l bulk a rr ival )

0, 1,2, . . .

(MIMI I bulk service)

136

139

M/M/rn

102

Po= ['II (mpt + (IIl )m)(_1 )]-1 k! 1P

k- O

.

(7)(2-)

P[qu eue m gj =

['II(mp)k + (m p)m)P(_ l _ ) ] k _O

k!

III!

CJ.1.ulj m

Queueing Systems: Theory

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Advances in queueing theory and network applications

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Algebraic Systems Theory 001

Queueing Systems: Theory