Performance Analysis of Communications Networks and Systems

PERFORMANCE ANALYSIS OF COMMUNICATIONS NETWORKS AND SYSTEMS

PIET VAN MIEGHEM Delft University of Technology

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Waar een wil is, is een weg. to my father

to my wife Saskia and my sons Vincent, Nathan and Laurens

Contents

Preface 1

2

Introduction

1

Part I

7 9

Probability theory and set theory Discrete random variables Continuous random variables The conditional probability Several random variables and independence Conditional expectation

Basic distributions 3.1 3.2 3.3 3.4 3.5 3.6 3.7

4

Probability theory

Random variables 2.1 2.2 2.3 2.4 2.5 2.6

3

xi

37

Discrete random variables Continuous random variables Derived distributions Functions of random variables Examples of other distributions Summary tables of probability distributions Problems

Correlation 4.1 4.2 4.3

9 16 20 26 28 34

37 43 47 51 54 58 59 61

Generation of correlated Gaussian random variables Generation of correlated random variables The non-linear transformation method v

61 67 68

vi

Contents

4.4 4.5 4.6 5

Inequalities 5.1 5.2 5.3 5.4 5.5 5.6 5.7

6

Examples of the non-linear transformation method Linear combination of independent auxiliary random variables Problem

The minimum (maximum) and infimum (supremum) Continuous convex functions Inequalities deduced from the Mean Value Theorem The Markov and Chebyshev inequalities The Hölder, Minkowski and Young inequalities The Gauss inequality The dominant pole approximation and large deviations

Limit laws 6.1 6.2 6.3 6.4

7

8

A stochastic process The Poisson process Properties of the Poisson process The nonhomogeneous Poisson process The failure rate function Problems

Renewal theory 8.1 8.2 8.3 8.4 8.5

9

Stochastic processes

The Poisson process 7.1 7.2 7.3 7.4 7.5 7.6

Basic notions Limit theorems The residual waiting time The renewal reward process Problems

Discrete-time Markov chains 9.1

78 82 83 83 84 86 87 90 92 94 97

General theorems from analysis Law of Large Numbers Central Limit Theorem Extremal distributions

Part II

74

Definition

97 101 103 104 113 115 115 120 122 129 130 132 137 138 144 149 153 155 157 157

Contents

9.2 9.3 9.4 10

Continuous-time Markov chains 10.1 10.2 10.3 10.4 10.5 10.6 10.7 10.8

11

Discrete-time Markov chain The steady-state of a Markov chain Problems

Definition Properties of continuous-time Markov processes Steady-state The embedded Markov chain The transitions in a continuous-time Markov chain Example: the two-state Markov chain in continuous-time Time reversibility Problems

Applications of Markov chains 11.1 Discrete Markov chains and independent random variables 11.2 The general random walk 11.3 Birth and death process 11.4 A random walk on a graph 11.5 Slotted Aloha 11.6 Ranking of webpages 11.7 Problems

12

Branching processes 12.1 12.2 12.3 12.4 12.5

13

General queueing theory 13.1 13.2 13.3 13.4 13.5 13.6

14

The probability generating function The limit Z of the scaled random variables Zn The Probability of Extinction of a Branching Process Asymptotic behavior of Z A geometric branching processes

A queueing system The waiting process: Lindley’s approach The Bene˘s approach to the unfinished work The counting process PASTA Little’s Law

Queueing models

vii

158 168 177 179 179 180 187 188 193 195 196 199 201 201 202 208 218 219 224 228 229 231 233 237 240 243 247 247 252 256 263 266 267 271

viii

Contents

14.1 14.2 14.3 14.4 14.5 14.6 14.7 14.8 14.9

The M/M/1 queue Variants of the M/M/1 queue The M/G/1 queue The GI/D/m queue The M/D/1/K queue The N*D/D/1 queue The AMS queue The cell loss ratio Problems

Part III 15

Physics of networks

General characteristics of graphs 15.1 15.2 15.3 15.4 15.5 15.6 15.7

Introduction The number of paths with m hops The degree of a node in a graph Connectivity and robustness Graph metrics Random graphs The hopcount in a large, sparse graph with unit link weights 15.8 Problems

16

The Shortest Path Problem 16.1 The shortest path and the link weight structure 16.2 The shortest path tree in NQ with exponential link weights 16.3 The hopcount kQ in the URT 16.4 The weight of the shortest path 16.5 The flooding time WQ 16.6 The degree of a node in the URT 16.7 The minimum spanning tree 16.8 The proof of the degree Theorem 16.6.1 of the URT 16.9 Problems

17

The e!ciency of multicast 17.1 General results for jQ (p) 17.2 The random graph Js (Q ) 17.3 The n-ary tree

271 276 283 289 296 300 304 309 312 317 319 319 321 322 325 328 329 340 346 347 348 349 354 359 361 366 373 380 385 387 388 392 401

Contents

17.4 17.5 17.6 17.7 17.8 18

The Chuang—Sirbu law Stability of a multicast shortest path tree Proof of (17.16): jQ (p) for random graphs Proof of Theorem 17.3.1: jQ (p) for n-ary trees Problem

The hopcount to an anycast group 18.1 18.2 18.3 18.4 18.5 18.6

Introduction General analysis The n-ary tree The uniform recursive tree (URT) Approximate analysis The performance measure  in exponentially growing trees

ix

404 407 410 414 416 417 417 419 423 424 431 432

Appendix A

Stochastic matrices

435

Appendix B

Algebraic graph theory

471

Appendix C

Solutions of problems

493

Bibliography

523

Index

529

Preface Performance analysis belongs to the domain of applied mathematics. The major domain of application in this book concerns telecommunications systems and networks. We will mainly use stochastic analysis and probability theory to address problems in the performance evaluation of telecommunications systems and networks. The first chapter will provide a motivation and a statement of several problems. This book aims to present methods rigorously, hence mathematically, with minimal resorting to intuition. It is my belief that intuition is often gained after the result is known and rarely before the problem is solved, unless the problem is simple. Techniques and terminologies of axiomatic probability (such as definitions of probability spaces, filtration, measures, etc.) have been omitted and a more direct, less abstract approach has been adopted. In addition, most of the important formulas are interpreted in the sense of “What does this mathematical expression teach me?” This last step justifies the word “applied”, since most mathematical treatises do not interpret as it contains the risk to be imprecise and incomplete. The field of stochastic processes is much too large to be covered in a single book and only a selected number of topics has been chosen. Most of the topics are considered as classical. Perhaps the largest omission is a treatment of Brownian processes and the many related applications. A weak excuse for this omission (besides the considerable mathematical complexity) is that Brownian theory applies more to physics (analogue fields) than to system theory (discrete components). The list of omissions is rather long and only the most noteworthy are summarized: recent concepts such as martingales and the coupling theory of stochastic variables, queueing networks, scheduling rules, and the theory of long-range dependent random variables that currently governs in the Internet. The confinement to stochastic analysis also excludes the recent new framework, called Network Calculus by Le Boudec and Thiran (2001). Network calculus is based on min-plus algebra and has been applied to (Inter)network problems in a deterministic setting. As prerequisites, familiarity with elementary probability and the knowledge of the theory of functions of a complex variable are assumed. Parts in the text in small font refer to more advanced topics or to computations that can be skipped at first reading. Part I (Chapters 2—6) reviews probability theory and it is included to make the remainder self-contained. The book essentially starts with Chapter 7 (Part II) on Poisson processes. The Poisxi

xii

Preface

son process (independent increments and discontinuous sample paths) and Brownian motion (independent increments but continuous sample paths) are considered to be the most important basic stochastic processes. We briefly touch upon renewal theory to move to Markov processes. The theory of Markov processes is regarded as a fundament for many applications in telecommunications systems, in particular queueing theory. A large part of the book is consumed by Markov processes and its applications. The last chapters of Part II dive into queueing theory. Inspired by intriguing problems in telephony at the beginning of the twentieth century, Erlang has pushed queueing theory to the scene of sciences. Since his investigations, queueing theory has grown considerably. Especially during the last decade with the advent of the Asynchronous Transfer Mode (ATM) and the worldwide Internet, many early ideas have been refined (e.g. discrete-time queueing theory, large deviation theory, scheduling control of prioritized flows of packets) and new concepts (self-similar or fractal processes) have been proposed. Part III covers current research on the physics of networks. This Part III is undoubtedly the least mature and complete. In contrast to most books, I have chosen to include the solutions to the problems in an Appendix to support self-study. I am grateful to colleagues and students whose input has greatly improved this text. Fernando Kuipers and Stijn van Langen have corrected a large number of misprints. Together with Fernando, Milena Janic and Almerima Jamakovic have supplied me with exercises. Gerard Hooghiemstra has made valuable comments and was always available for discussions about my viewpoints. Bart Steyaert eagerly gave the finer details of the generating function approach to the GI/D/m queue. Jan Van Mieghem has given overall comments and suggestions beside his input with the computation of correlations. Finally, I thank David Hemsley for his scrupulous corrections in the original manuscript. Although this book is intended to be of practical use, in the course of writing it, I became more and more persuaded that mathematical rigor has ample virtues of its own. Per aspera ad astra

January 2006

Piet Van Mieghem

1 Introduction

The aim of this first chapter is to motivate why stochastic processes and probability theory are useful to solve problems in the domain of telecommunications systems and networks. In any system, or for any transmission of information, there is always a non-zero probability of failure or of error penetration. A lot of problems in quantifying the failure rate, bit error rate or the computation of redundancy to recover from hazards are successfully treated by probability theory. Often we deal in communications with a large variety of signals, calls, sourcedestination pairs, messages, the number of customers per region, and so on. And, most often, precise information at any time is not available or, if it is available, deterministic studies or simulations are simply not feasible due to the large number of dierent parameters involved. For such problems, a stochastic approach is often a powerful vehicle, as has been demonstrated in the field of physics. Perhaps the first impressing result of a stochastic approach was Boltzmann’s and Maxwell’s statistical theory. They studied the behavior of particles in an ideal gas and described how macroscopic quantities as pressure and temperature can be related to the microscopic motion of the huge amount of individual particles. Boltzmann also introduced the stochastic notion of the thermodynamic concept of entropy V, V = n log Z where Z denotes the total number of ways in which the ensembles of particles can be distributed in thermal equilibrium and where n is a proportionality factor, afterwards attributed to Boltzmann as the Boltzmann constant. The pioneering work of these early physicists such as Boltzmann, Maxwell and others was the germ of a large number of breakthroughs in science. Shortly after their introduction of stochastic theory in classical physics, the 1

2

Introduction

theory of quantum mechanics (see e.g. Cohen-Tannoudji et al., 1977) was established. This theory proposes that the elementary building blocks of nature, the atom and electrons, can only be described in a probabilistic sense. The conceptually di!cult notion of a wave function whose squared modulus expresses the probability that a set of particles is in a certain state and the Heisenberg’s uncertainty relation exclude in a dramatic way our deterministic, macroscopic view on nature at the fine atomic scale. At about the same time as the theory of quantum mechanics was being created, Erlang applied probability theory to the field of telecommunications. Erlang succeeded to determine the number of telephone input lines p of a switch in order to serve QV customers with a certain probability s. Perhaps his most used formula is the Erlang E formula (14.17), derived in Section 14.2.2, Pr [QV = p] =

p p! Pp m m=0 m!

where the load or tra!c intensity  is the ratio of the arrival rate of calls to the telephone local exchange or switch over the processing rate of the switch per line. By equating the desired blocking probability s = Pr [QV = p], say s = 1034 , the number of input lines p can be computed for each load . Due to its importance, books with tables relating s,  and p were published. Another pioneer in the field of communications that deserves to be mentioned is Shannon. Shannon explored the concept of entropy V. He introduced (see e.g. Walrand, 1998) the notion of the Shannon capacity of a channel, the maximum rate at which bits can be transmitted with arbitrary small (but non zero) probability of errors, and the concept of the entropy rate of a source which is the minimum average number of bits per symbol required to encode the output of a source. Many others have extended his basic ideas and so it is fair to say that Shannon founded the field of information theory. A recent important driver in telecommunication is the concept of quality of service (QoS). Customers can use the network to transmit dierent types of information such as pictures, files, voice, etc. by requiring a specific level of service depending on the type of transmitted information. For example, a telephone conversation requires that the voice packets arrive at the receiver G ms later, while a file transfer is mostly not time critical but requires an extremely low information loss probability. The value of the mouth-to-ear delay G is clearly related to the perceived quality of the voice conversation. As long as G ? 150 ms, the voice conversation has toll quality, which is roughly speaking, the quality that we are used to in classical

Introduction

3

telephony. When G exceeds 150 ms, rapid degradation is experienced and when G A 300 ms, most of the test persons have great di!culty in understanding the conversation. However, perceived quality may change from person to person and is di!cult to determine, even for telephony. For example, if the test person knows a priori that the conversation is transmitted over a mobile or wireless channel as in GSM, he or she is willing to tolerate a lower quality. Therefore, quality of service is both related to the nature of the information and to the individual desire and perception. In future Internetworking, it is believed that customers may request a certain QoS for each type of information. Depending on the level of stringency, the network may either allow or refuse the customer. Since customers will also pay an amount related to this QoS stringency, the network function that determines to either accept or refuse a call for service will be of crucial interest to any network operator. Let us now state the connection admission control (CAC) problem for a voice conversation to illustrate the relation to stochastic analysis: “How many customers p are allowed in order to guarantee that the ensemble of all voice packets reaches the destination within G ms with probability s?”This problem is exceptionally di!cult because it depends on the voice codecs used, the specifics of the network topology, the capacity of the individual network elements, the arrival process of calls from the customers, the duration of the conversation and other details. Therefore, we will simplify the question. Let us first assume that the delay is only caused by the waiting time of a voice packet in the queue of a router (or switch). As we will see in Chapter 13, this waiting time W of voice packets in a single queueing system depends on (a) the arrival process: the way voice packets arrive, and (b) the service process: how they are processed. Let us assume that the arrival process specified by the average arrival rate  and the service process specified by the average service rate  are known. Clearly, the arrival rate  is connected to the number of customers p. A simplified statement of the CAC problem is, “What is the maximum  allowed such that Pr [W A G] ? ?” In essence, the CAC problem consists in computing the tail probability of a quantity that depends on parameters of interest. We have elaborated on the CAC problem because it is a basic design problem that appears under several disguises. A related dimensioning problem is the determination of the buer size in a router in order not to lose more than a certain number of packets with probability s, given the arrival and service process. The above mentioned problem of Erlang is a third example. Another example treated in Chapter 18 is the server placement problem: “How many replicated servers p are needed to guarantee that any user can access the information within n hops with probability Pr [kQ (p) A n]  ”, where

4

Introduction

 is certain level of stringency and kQ (p) is the number of hops towards the most nearby of the p servers in a network with Q routers. The popularity of the Internet results in a number of new challenges. The traditional mathematical models as the Erlang B formula assume “smooth” tra!c flows (small correlation and Markovian in nature). However, TCP/IP tra!c has been shown to be “bursty” (long-range dependent, self-similar and even chaotic, non-Markovian (Veres and Boda, 2000)). As a consequence, many traditional dimensioning and control problems ask for a new solution. The self-similar and long range dependent TCP/IP tra!c is mainly caused by new complex interactions between protocols and technologies (e.g. TCP/IP/ATM/SDH) and by other information transported than voice. It is observed that the content size of information in the Internet varies considerably in size causing the “Noah eect”: although immense floods are extremely rare, their occurrence impacts significantly Internet behavior on a global scale. Unfortunately, the mathematics to cope with the self-similar and long range dependent processes turns out to be fairly complex and beyond the scope of this book. Finally, we mention the current interest in understanding and modeling complex networks such as the Internet, biological networks, social networks and utility infrastructures for water, gas, electricity and transport (cars, goods, trains). Since these networks consists of a huge number of nodes Q and links O, classical and algebraic graph theory is often not suited to produce even approximate results. The beginning of probabilistic graph theory is commonly attributed to the appearance of papers by Erdös and Rényi in the late 1940s. They investigated a particularly simple growing model for a graph: start from Q nodes and connect in each step an arbitrary random, not yet connected pair of nodes until all O links are used. After about Q@2 steps, as shown in Section 16.7.1, they observed the birth of a giant component that, in subsequent steps, swallows the smaller ones at a high rate. This phenomenon is called a phase transition and often occurs in nature. In physics it is studied in, for example, percolation theory. To some extent, the Internet’s graph bears some resemblance to the Erdös-Rényi random graph. The Internet is best regarded as a dynamic and growing network, whose graph is continuously changing. Yet, in order to deploy services over the Internet, an accurate graph model that captures the relevant structural properties is desirable. As shown in Part III, a probabilistic approach based on random graphs seems an e!cient way to learn about the Internet’s intriguing behavior. Although the Internet’s topology is not a simple ErdösRényi random graph, results such as the hopcount of the shortest path and the size of a multicast tree deduced from the simple random graphs provide

Introduction

5

a first order estimate for the Internet. Moreover, analytic formulas based on other classes of graphs than the simple random graph prove di!cult to obtain. This observation is similar to queueing theory, where, beside the M/G/x class of queues, hardly closed expressions exist. We hope that this brief overview motivates su!ciently to surmount the mathematical barriers. Skill with probability theory is deemed necessary to understand complex phenomena in telecommunications. Once mastered, the power and beauty of mathematics will be appreciated.

Part I Probability theory

2 Random variables

This chapter reviews basic concepts from probability theory. A random variable (rv) is a variable that takes certain values by chance. Throughout this book, this imprecise and intuitive definition su!ces. The precise definition involves axiomatic probability theory (Billingsley, 1995). Here, a distinction between discrete and continuous random variables is made, although a unified approach including alsoR mixed cases via the Stieltjes integral (Hardy et al., 1999, pp. 152—157), j({)gi ({), is possible. In general, the distribution I[ ({) = Pr [[  {] holds in both cases, and Z X j(n) Pr[[ = n] where [ is a discrete rv j({)gI[ ({) = n

Z =

j({)

gI[ ({) g{ g{

where [ is a continuous rv

In most practical situations, the Stieltjes integral reduces to the Riemann integral, else, Lesbesgue’s theory of integration and measure theory (Royden, 1988) is required.

2.1 Probability theory and set theory Pascal (1623—1662) is commonly regarded as one of the founders of probability theory. In his days, there was much interest in games of chance1 and the likelihood of winning a game. In most of these games, there was a finite number q of possible outcomes and each of them was equally likely. The 1

“La règle des partis”, a chapter in Pascal’s mathematical work (Pascal, 1954), consists of a series of letters to Fermat that discuss the following problem (together with a more complex question that is essentially a variant of the probability of gambler’s ruin treated in Section 11.2.1): Consider the game in which 2 dice are thrown q times. How many times q do we have to throw the 2 dice to throw double six with probability s = 12 ?

9

10

Random variables

probability of the event D of interest was defined as qD Pr [D] = q where qD is the number of favorable outcomes (samples points of D). If the number of outcomes of an experiment is not finite, this classical definition of probability does not su!ce anymore. In order to establish a coherent and precise theory, probability theory employs concepts of group or set theory. The set of all possible outcomes of an experiment is called the sample space . A possible outcome of an experiment is called a sample point $ that is an element of the sample space . An event D consists of a set of sample points. An event D is thus a subset of the sample space . The complement Df of an event D consists of all sample points of the sample space that are not in (the set) D, thus Df = \D. Clearly, (Df )f = D and the complement of the sample space is the empty set, f = > or, vice a versa, >f = . A family F of events is a set of events and thus a subset of the sample space that possesses particular events as elements. More precisely, a family F of events satisfies the three conditions that define a -field2 : (a) f > 5 F, (b) if D1 > D2 > = = = 5 F, then ^" m=1 Dm 5 F and (c) if D 5 F, then D 5 F. These conditions guarantee that F is closed under countable unions and intersections of events. Events and the probability of these events are connected by a probability measure Pr [=] that assigns to each event of the family F of events of a sample space a real number in the interval [0> 1]. As Axiom 1, we require that Pr [ ] = 1. If Pr [D] = 0, the occurrence of the event D is not possible, while Pr [D] = 1 means that the event D is certain to occur. If Pr [D] = s with 0 ? s ? 1, the event D has probability s to occur. If the events D and E have no sample points in common, D _ E = >, the events D and E are called mutually exclusive events. As an example, the event and its complement are mutually exclusive because D _ Df = >. Axiom 2 of a probability measure is that for mutually exclusive events D and E holds that Pr [D ^ E] = Pr [D]+Pr [E]. The definition of a probability measure and the two axioms are su!cient to build a consistent framework on which probability theory is founded. Since Pr [>] = 0 (which follows from 2

A field F posseses the properties: (i)  M F; (ii) if D> E M F, then D  E M F and D K E M F; (iii) if D M F, then Df M F= This definition is redundant. For, we have by (ii) and (iii) that (D  E)f M F. Further, by De Morgan’s law (D  E)f = Df K E f , which can be deduced from Figure 2.1 and again by (iii), the argument shows that the reduced statement (ii), if D> E M F, then D  E M F, is su!cient to also imply that D K E M F.

2.1 Probability theory and set theory

11

Axiom 2 because D _ > = > and D = D ^ >), for mutually exclusive events D and E holds that Pr [D _ E] = 0. As a classical example that explains the formal definitions, let us consider the experiment of throwing a fair die. The sample space consists of all possible outcomes: = {1> 2> 3> 4> 5> 6}. A particular outcome of the experiment, say $ = 3, is a sample point $ 5 . One may be interested in the event D where the outcome is even in which case D = {2> 4> 6}  and Df = {1> 3> 5}. If D and E are events, the union of these events D ^ E can be written using set theory as D ^ E = (D _ E) ^ (Df _ E) ^ (D _ E f ) because D_E, Df _E and D_E f are mutually exclusive events. The relation is immediately understood by drawing a Venn diagram as in Fig. 2.1. Taking

AˆBc

AˆB

AcˆB

A

B :

Fig. 2.1. A Venn diagram illustrating the union D ^ E.

the probability measure of the union yields Pr [D ^ E] = Pr [(D _ E) ^ (Df _ E) ^ (D _ E f )] = Pr [D _ E] + Pr [Df _ E] + Pr [D _ E f ]

(2.1)

where the last relation follows from Axiom 2. Figure 2.1 shows that D = (D _ E) ^ (D _ E f ) and E = (D _ E) ^ (Df _ E). Since the events are mutually exclusive, Axiom 2 states that Pr [D] = Pr [D _ E] + Pr [D _ E f ] Pr [E] = Pr [D _ E] + Pr [Df _ E] Substitution into (2.1) yields the important relation Pr [D ^ E] = Pr [D] + Pr [E]  Pr [D _ E]

(2.2)

Although derived for the measure Pr [=], relation (2.2) also holds for other measures, for example, the cardinality (the number of elements) of a set.

12

Random variables

2.1.1 The inclusion-exclusion formula A generalization of the relation (2.2) is the inclusion-exclusion formula, Pr [^qn=1 Dn ] =

q X

Pr [Dn1 ] 

n1 =1

+

q X

q X

q X

Pr [Dn1 _ Dn2 ]

n1 =1 n2 =n1 +1 q X

q X

Pr [Dn1 _ Dn2 _ Dn3 ]

n1 =1 n2 =n1 +1 n3 =n2 +1 q X q1

+ · · · + (1)

q X

···

n1 =1 n2 =n1 +1

q X

¤ £ Pr _qm=1 Dnm

(2.3)

nq =nq1 +1

The formula shows that the probability of the union consists of the sum of probabilities of the individual events (first term). Since sample points can belong to more than one event Dn , the first term possesses double countings. The second term removes all probabilities of samples points that belong to precisely two event sets. However, by doing so (draw a Venn diagram), we also subtract the probabilities of samples points that belong to three events sets more than needed. The third term adds these again, and so on. The inclusion-exclusion formula can be written more compactly as, q q q q i h X X X X q m31 (2.4) (1) ··· Pr _mp=1 Dnp Pr [^n=1 Dn ] = m=1

n1 =1 n2 =n1 +1

or with

h i Pr _mp=1 Dnp

X

Vm =

nm =nm31 +1

1$n1 ?n2 ?===?nm $q

as Pr [^qn=1 Dn ] =

q X (1)m31 Vm

(2.5)

m=1 Proof of the inclusion-exclusion formula 3 : Let D = q31 n=1 Dn and E = Dq such that 3

Another proof (Grimmett and Stirzacker, 2001, p. 56) uses the indicator function defined in Section 2.2.1. Useful indicator function relations are 1DKE = 1D 1E 1Df = 1 3 1D 1DX E = 1 3 1(DE)f = 1 3 1Df KE f = 1 3 1Df 1E f = 1 3 (1 3 1D )(1 3 1E ) = 1D + 1E + 1D 1E = 1D + 1E + 1DKE Generalizing the last relation yields 1q D =13 n=1 n

q \

(1 3 1Dn )

n=1

Multiplying out and taking the expectations using (2.13) leads to (2.3).

2.1 Probability theory and set theory

13

  q31 q31 D  E = q n=1 Dn and D K E = Dq K n=1 Dn = n=1 Dn K Dq by the distributive law in set theory, then application of (2.2) yields the recursion in q l l k k q31 q31 Pr [q n=1 Dn ] = Pr n=1 Dn + Pr [Dq ] 3 Pr n=1 Dn K Dq

(2.6)

By direct substitution of q < q 3 1, we have k l k l l k q32 q32 Pr q31 n=1 Dn = Pr n=1 Dn + Pr [Dq31 ] 3 Pr n=1 Dn K Dq31 while substitution in this formula of Dn < Dn K Dq gives l k l l k k q32 q32 Pr q31 n=1 Dn K Dq = Pr n=1 Dn K Dq + Pr [Dq31 K Dq ] 3 Pr n=1 Dn K Dq K Dq31 Substitution of the last two terms into (2.6) yields l k q32 Pr [q n=1 Dn ] = Pr [Dq31 ] + Pr [Dq ] 3 Pr [Dq31 K Dq ] + Pr n=1 Dn k l k l k l q32 q32 3 Pr q32 n=1 Dn K Dq31 3 Pr n=1 Dn K Dq + Pr n=1 Dn K Dq K Dq31 (2.7) Similarly, in a next iteration we use (2.6) after suitable modification in the right-hand side of (2.7) to lower the upper index in the union, k l k l l k q33 q33 Pr q32 n=1 Dn = Pr n=1 Dn + Pr [Dq32 ] 3 Pr n=1 Dn K Dq32 l k l k q33 Pr q32 n=1 Dn K Dq31 = Pr n=1 Dn K Dq31 + Pr [Dq32 K Dq31 ] k l 3 Pr q33 n=1 Dn K Dq31 K Dq32 l k l l k k q33 q33 Pr q32 n=1 Dn K Dq = Pr n=1 Dn K Dq + Pr[Dq32 K Dq ]3Pr n=1 Dn K Dq K Dq32 k l k l q33 Pr q32 n=1 Dn K Dq K Dq31 = Pr n=1 Dn K Dq K Dq31 + Pr [Dq32 K Dq K Dq31 ] l k 3 Pr q33 n=1 Dn K Dq K Dq31 K Dq32 The result is Pr [q n=1 Dn ] = Pr [Dq32 ] + Pr [Dq31 ] + Pr [Dq ] + 3 Pr [Dq32 K Dq31 ] 3 Pr [Dq32 K Dq ] l k 3 Pr [Dq31 K Dq ] + Pr [Dq32 K Dq31 K Dq ] + Pr q33 n=1 Dn k l k l k l q33 q33 3 Pr q33 n=1 Dn K Dq32 3 Pr n=1 Dn K Dq31 3 Pr n=1 Dn K Dq k l k l q33 + Pr q33 n=1 Dn K Dq31 K Dq32 + Pr n=1 Dn K Dq K Dq32 k l k l q33 + Pr q33 n=1 Dn K Dq K Dq31 3 Pr n=1 Dn K Dq K Dq31 K Dq32 which starts revealing the structure of (2.3). Rather than continuing the iterations, we prove the validity of the inclusion-exclusion formula (2.3) via induction. In case q = 2, the basic expression (2.2) is found. Assume that (2.3) holds for q, then the case for q + 1 must obey (2.6) where q < q + 1, k l q q Pr q+1 n=1 Dn = Pr [n=1 Dn ] + Pr [Dq+1 ] 3 Pr [n=1 Dn K Dq+1 ]

14

Random variables

Substitution of (2.3) into the above expression yields, after suitable grouping of the terms, q q k l [  [  Pr q+1 Pr Dn1 3 n=1 Dn = Pr[Dq+1 ] + n1 =1

+

q  [    Pr Dn1 K Dn2 3 Pr Dn1 K Dq+1

n1 =1 n2 =n1 +1

q [

q [

q [

q [

n1 =1

q  [  Pr Dn1 KDn2 K Dn3 +

n1 =1 n2 =n1 +1 n3 =n2 +1

+ · · · + (31)q31

q [

+ · · · + (31)q

q [

···

=

Pr [Dn ] 3

n1 =1

+

q+1 [

q+1 [

k l Pr Kq m=1 Dnm 3

q [

···

n1 =1 n2 =n1 +1 q+1 [

  Pr Dn1 KDn2 KDq+1

n1 =1 n2 =n1 +1 q [

n1 =1 n2 =n1 +1 q [

q [

nq =nq1 +1

l k Pr Kq m=1 Dnm K Dq+1

q [ nq =nq1 +1

  Pr Dn1 K Dn2

n1 =1 n2 =n1 +1

q+1 [

q+1 [

q+1 [

  Pr Dn1 K Dn2 K Dn3

n1 =1 n2 =n1 +1 n3 =n2 +1

+ · · · + (31)q

q+1 [

q+1 [

···

n1 =1 n2 =n1 +1

which proves (2.3).

q+1 [

l k Pr Kq m=1 Dnm K Dq+1

nq+1 =nq +1

¤

Although impressive, the inclusion-exclusion formula is useful when dealing with dependent random variables because of its general nature. In parh i m ticular, if Pr _p=1 Dnp = dm and not a function of the specific indices np , the inclusion-exclusion formula (2.4) becomes more attractive, Pr [^qn=1 Dn ]

q X = (1)m31 dm m=1

X

1

1$n1 ?n2 ?===?nm $q

µ ¶ q X m31 q = (1) dm m m=1

An application of the latter formula to multicast can be found in Chapter 17 and many others are in Feller (1970, Chapter IV). Sometimes it is useful to reason with the complement of the union (^qn=1 Dn )f = \ ^qn=1 Dn = _qn=1 Dfn . Applying Axiom 2 to (^qn=1 Dn )f ^ (^qn=1 Dn ) = , Pr [(^qn=1 Dn )f ] = Pr [ ]  Pr [^qn=1 Dn ] and using Axiom 1 and the inclusion-exclusion formula (2.5), we obtain Pr [(^qn=1 Dn )f ] = 1 

q q X X (1)m31 Vm = (1)m Vm m=1

m=0

(2.8)

2.1 Probability theory and set theory

15

with the convention that V0 = 1. The Boole’s inequalities Pr [^qn=1 Dn ] 

q X

Pr [Dn ]

(2.9)

n=1

Pr [_qn=1 Dn ]  1 

q X

Pr [Dfn ]

n=1

are derived as consequences of the inclusion-exclusion formula (2.3). Only if all events are mutually exclusive, the equality sign in (2.9) holds whilst the inequality sign follows from the fact that possible overlaps in events are, in contrast to the inclusion-exclusion formula (2.3), not subtracted. The inclusion-exclusion formula is of a more general nature and also applies to other measures on sets than Pr [=], for example to the cardinality as mentioned above. For the cardinality of a set D, which is usually denoted by |D|, the inclusion-exclusion variant of (2.8) is |(^qn=1 Dn )f | =

q X (1)m |Vm |

(2.10)

m=0

where the total number of elements in the sample space is |V0 | = Q and ¯ ¯ X ¯ m ¯ |Vm | = ¯_p=1 Dnp ¯ 1$n1 ?n2 ?===?nm $q

A nice illustration of the above formula (2.10) applies to the sieve of Eratosthenes (Hardy and Wright, 1968, p. 4), a procedure to construct the table of prime numbers4 up to Q . Consider the increasing sequence of integers

= {2> 3> 4> = = = > Q } and remove successively all multiples of 2 (even numbers starting from 4, 6, ...), all multiples of 3 (starting from 32 and not yet removed previously), all multiples of 5, all multiples of the next number larger than 5 and still in the list (which is the prime 7) and so on, up to all multiples hs i of the largest Q . Here [{] is the possible prime divisor that is equal to or smaller than largest integer smaller than or equal to {. The remaining numbers in the list are prime numbers. Let us now compute the number of primes (Q ) smaller than or equal to Q by using the inclusion-exclusion formula (2.10). 4

An integer number s is prime if s A 1 and s has no other integer divisors than 1 and itself s. The sequence of the first primes are 2, 3, 5, 7, 11, 13, etc. If I d and e are divisors of q, then q = de from which it follows that d and e cannot exceed both q. Hence, any composite I number q is divisible by a prime s that does not exceed q.

16

Random variables

The number of primes smaller than a real number { is ({) and, evidently, if sq denotes the q-th prime, then  (sq ) = q. Let Dn denote the set of the multiples of the n-th prime sn that belong to . The number of such sets Dn in the sieve of Eratosthenes is equal to³the largest prime number sq smaller hs i s ´ than or equal to Q , hence, q =  Q . If t 5 (^qn=1 Dn )f , this means that t is not divisible by each prime s number smaller than sq and that t is a prime number lying between Q ? t sQ . The cardinality of the set (^qn=1 Dn )f , the number of primes between Q ? t  Q is ³s ´ f q |(^n=1 Dn ) | = (Q )   Q On the other hand, if u 5 _mp=1 Dnp for 1  n1 ? n2 ? · · · ? nm  q, then u is a multiple of sn1 sn2 = = = snm and the number of multiples of the integer sn1 sn2 = = = snm in is ¸ ¯  ¯ Q ¯ ¯ = ¯_mp=1 Dnp ¯ sn1 sn2 = = = snm Applying ´ inclusion-exclusion formula (2.10) with | | = V0 = Q  1 and ³s the Q gives q= (Q )  

q ³s ´ X Q =Q 1 (1)m m=1

X 1$n1 ?n2 ?===?nm $q



Q sn1 sn2 = = = snm

¸

hs i Q , i.e. the The knowledge of the prime numbers smaller than or equal to ³s ´ first q =  Q primes, su!ces to compute the number of primes (Q ) smaller than s or equal to Q without explicitly knowing the primes t lying between Q ? t  Q .

2.2 Discrete random variables Discrete random variables are real functions [ defined on a discrete probability space as [ : $ R with the property that the event {$ 5 : [ ($) = {} 5 F for each { 5 R. The event {$ 5 : [ ($) = {} is further abbreviated as {[ = {}. A discrete probability density function (pdf) Pr[[ = {] has the following properties: (i) 0  Pr[[ = {]  1 for real { that are possible outcomes of an

2.2 Discrete random variables

17

experiment. The set of values { can be finite or countably infinite and constitute the discrete probability space. P (ii) { Pr[[ = {] = 1= In the classical example of throwing a die, the discrete probability space

= {1> 2> 3> 4> 5> 6} and, since each of the six edges of the (fair) die is equally possible as outcome, Pr[[ = {] = 16 for each { 5 . 2.2.1 The expectation An important operator acting on a discrete random variable [ is the expectation, defined as X { Pr [[ = {] (2.11) H [[] = {

The expectation H [[] is also called the mean or average or first moment of [. More generally, if [ is a discrete random variable and j is a function, then \ = j([) is also a discrete random variable with expectation H [\ ] equal to X H [j([)] = j({) Pr [[ = {] (2.12) {

A special and often used function in probability theory is the indicator function 1| defined as 1 if the condition | is true and otherwise it is zero. For example, X X H [1[Ad ] = 1{Ad Pr [[ = {] = Pr [[ = {] = Pr[[ A d] {

{Ad

H [1[=d ] = Pr[[ = d]

(2.13)

The higher moments of a random variable are defined as the case where j({) = {q , X H [[ q ] = {q Pr [[ = {] (2.14) {

From the definition (2.11), it follows that the expectation is a linear operator, # " q q X X dn [n = dn H [[n ] H n=1

n=1

The variance of [ is defined as

h i Var[[] = H ([  H [[])2

(2.15)

18

Random variables

The variance is always non-negative. Using the linearity of the expectation operator and  = H [[], we rewrite (2.15) as £ ¤ Var[[] = H [ 2  2 (2.16) £ ¤ Since Var[[]  0, relation (2.16) indicates that H [ 2  (H [[])2 . Often p the standard deviation, defined as  = Var [[], is used. An interesting variational principle of the variance follows, for the variable x, from i h i h H ([  x)2 = H ([  )2 + (x  )2 which is minimized at x =  = H [[] with value Var[[]. Hence, the best least square approximation of the random variable [ is the number H [[].

2.2.2 The probability generating function The probability generating function (pgf) of a discrete random variable [ is defined, for complex }, as £ ¤ X { } Pr [[ = {] (2.17) *[ (}) = H } [ = {

where the last equality follows from (2.12). If [ is integer-valued and nonnegative, then the pgf is the Taylor expansion of the complex function *[ (}). Commonly the latter restriction applies, otherwise the substitution ¡ ¢ } = hlw is used such that (2.17) expresses the Fourier series of *[ hlw . The importance of the pgf mainly lies in the fact that the theory of functions can be applied. Numerous examples of the power of analysis will be illustrated. Concentrating on non-negative integer random variables [, *[ (}) =

" X

Pr [[ = n] } n

(2.18)

n=0

and the Taylor coe!cients obey

¯ 1 gn *[ (}) ¯¯ Pr [[ = n] = n! g} n ¯}=0 Z *[ (}) 1 = g} 2l F(0) } n+1

(2.19) (2.20)

where F(0) denotes a contour around } = 0. Both are inversion formulae5 . Since the general form H[j([)] is completely defined when Pr[[ = {] is 5

A similar inversion formula for Fourier series exist (see e.g. Titchmarsh (1948)).

2.2 Discrete random variables

19

known, the knowledge of the pgf results in a complete alternative description, ¯ " X j(n) gn *[ (}) ¯¯ H [j([)] = (2.21) n! g} n ¯}=0 n=0

Sometimes it is more convenient to compute values of interest directly from (2.17) £ ¤rather than from (2.21). For example, q-fold dierentiation of *[ (}) = H } [ yields µ ¶ ¸ £ ¤ [ [3q gq *[ (}) 1 [3q } = H [([  1) · · · ([  q + 1)} = H q g} q q! such that

¯ µ ¶¸ [ 1 gq *[ (}) ¯¯ H = q q! g} q ¯}=1

(2.22)

Similarly, let } = hw , then ¤ £ gq *[ (hw ) = H [ q hw[ q gw from which the moments follow as

¯ gq *[ (hw ) ¯¯ H [[ ] = gwq ¯w=0 q

(2.23)

and, more generally,

¡ ¢¯ gq h3wd *[ (hw ) ¯¯ H [([  d) ] = ¯ ¯ gwq q

(2.24)

w=0

2.2.3 The logarithm of the probability generating function The logarithm of the probability generating function is defined as ¡ £ ¤¢ O[ (}) = log (*[ (})) = log H } [

(2.25) *0 (})

from which O[ (1) = 0 because *[ (1) = 1. The derivative O0[ (}) = *[ [ (}) ³ 0 ´2 00 (}) * * (}) shows that O0[ (1) = *0[ (1), while from O00[ (}) = *[  *[ , it follows [ (}) [ (}) that O00[ (1) = *00[ (1)  (*0[ (1))2 . These first few derivatives are interesting because they are related directly to probabilistic quantities. Indeed, from (2.23), we observe that H[[] = *0[ (1) = O0[ (1)

(2.26)

20

Random variables

and from H[[ 2 ] = *00[ (1) + *0[ (1)

¡ ¢2 Var[[] = *00[ (1) + *0[ (1)  *0[ (1) = O00[ (1) + O0[ (1)

(2.27)

2.3 Continuous random variables Although most of the concepts defined above for discrete random variables are readily transferred to continuous random variables, the calculus is in general more di!cult. Indeed, instead of reasoning on the pdf, it is more convenient to work with the probability distribution function defined for both discrete and continuous random variables as I[ ({) = Pr [[  {]

(2.28)

Clearly, we have lim{. For mutually exclusive events D _ E = >, Axiom 2 in Section 2.1 states that Pr [D ^ E] = Pr [D] + Pr [E] which proves (2.29). As a corollary of (2.29), I[ ({) is continuous at the right which follows from (2.29) by denoting d = e   for any  A 0. Less precise, it follows from the equality sign at the right, [  e, and inequality at the left, d ? [. Hence, I[ ({) is not necessarily continuous at the left which implies that I[ ({) is not necessarily continuous and that I[ ({) may possess jumps. But even if I[ ({) is continuous, the pdf is not necessary continuous6 . The pdf of a continuous random variable [ is defined as i[ ({) = 6

gI[ ({) g{

(2.30)

Weierstrass was the first to present a continuous non-dierentiable function, i ({) =

" [

eq cos (dq {)

q=0

where 0 ? e ? 1 and d is an odd positive integer. Since the series is uniformly convergent for any {, i ({) is continuous everywhere. Titchmarsh (1964, Chapter IX) demonstrates for ({) de A 1 + 3 that i ({+k)3i takes arbitrarily large values such that i 0 ({) does not exist. 2 k Another class of continuous non-dierentiable functions are the sample paths of a Brownian motion. The Cantor function which is discussed in (Berger, 1993, p. 21) and (Billingsley, 1995, p. 407) is an other classical, noteworthy function with peculiar properties.

2.3 Continuous random variables

21

Assuming that I[ ({) is dierentiable at {, from (2.29), we have for small, positive { Pr [{ ? [  { + {] = I[ ({ + {)  I[ ({) ³ ´ gI[ ({) { + R ({)2 = g{ Using the definition (2.30) indicates that, if I[ ({) is dierentiable at {, Pr [{ ? [  { + {] {{ 0) are both integrable over D. Although this restriction seems only of theoretical interest, in some applications (see the

2.3 Continuous random variables

23

\ = j([) is also a continuous random variable with expectation H [\ ] equal to Z "

H [j([)] = 3"

j({)i[ ({)g{

(2.34)

It is often useful to express the expectation H [[] of a non-negative random variable [ in tail probabilities. Upon integration by parts, ¯" Z " Z " Z " Z " ¯ {i[ ({)g{ = { i[ (x)gx¯¯ + g{ i[ (x)gx H [[] = { 0 { 0 Z0 " (1  I[ ({)) g{ (2.35) = 0

The case for a non-positive random variable [ is derived analogously, ¯0 Z { Z 0 Z { Z 0 ¯ ¯ {i[ ({)g{ = { i[ (x)gx¯  g{ i[ (x)gx H [[] = 3" Z 0

= 3"

3"

3"

3"

3"

I[ ({)g{

The general case follows by addition: Z " Z H [[] = (1  I[ ({)) g{ 

0

3"

0

I[ ({)g{

A similar expression exists for discrete random variables. In general for any discrete random variable [, we can write H [[] =

" [ n=3"

=

31 [

31 [

n Pr [[ = n] =

n Pr [[ = n] +

n=3"

31 [

n Pr [[ = n]

n=0 " [

n (Pr [[ $ n] 3 Pr [[ $ n 3 1]) +

n=3"

=

" [

n (Pr [[ D n] 3 Pr [[ D n + 1])

n=0 32 [

n Pr [[ $ n] 3

n=3"

= 3 Pr [[ $ 31] 3

(n + 1) Pr [[ $ n] +

n=3" 32 [ n=3"

Pr [[ $ n] +

" [ n=1

" [

n Pr [[ D n] 3

" [

(n 3 1) Pr [[ D n]

n=1

Pr [[ D n]

n=1

Cauchy distribution defined U in (3.38)) the Riemann integral may exists where the Lesbesgue does not. For example, 0" sin{ { g{ equals, in the Riemann sense, 2 (which is a standard excercise in contour integration), but this integral does not exists in the Lesbesgue sense. Only for improper integrals (integration interval is infinite), Riemann integration may exist where Lesbesgue does not. However, in most other cases (integration over a finite interval), U Lesbesgue integration is more general. For instance, if i ({) = 1{{ is ra tio n a l} , then 01 i (x)gx does not exist in the Riemann sense (since upper and lower sums do not converge to each U other). However, 01 i (x)gx = 0 in the Lesbesgue sense (since there is only a set of measure zero dierent from 0, namely all rational numbers in [0> 1] ). In probability theory and measure theory, Lesbesgue integration is assumed.

24

Random variables

or the mean of a discrete random variable [ expressed in tail probabilities is9 " 31 X X H [[] = Pr [[  n]  Pr [[  n] (2.36) n=1

n=3"

2.3.3 The probability generating function The probability generating function (pgf) of a continuous random variable [ is defined, for complex }, as the Laplace transform Z " ¤ £ *[ (}) = H h3}[ = h3}w i[ (w)gw (2.37) 3"

Again, in some cases, it may be more convenient to use } = lx in which case the double sided Laplace transform reduces to a Fourier transform. The strength of these transforms is based on the numerous properties, especially the inverse transform, Z f+l" 1 i[ (w) = *[ (})h}w g} (2.38) 2l f3l" where f is the smallest real variable Re(}) for which the integral in (2.37) converges. as for discrete random variables, we have h}d *[ (}) = ¤ £ 3}([3d)Similarly H h ¯ q }d ¯ q q g (h *[ (})) ¯ (2.39) H [([  d) ] = (1) ¯ g} q }=0

£ ¤ The main dierence £with¤ the discrete case lies in the definition H h3}[ (continuous) versus H } [ (discrete). Since the exponential is an entire 9

We remark that " [

H [[] =

n Pr [[ = n] =

n=3" " [

6=

" [

n (Pr [[ D n] 3 Pr [[ D n + 1])

n=3"

n Pr [[ D n] 3

n=3"

" [

n Pr [[ D n + 1] =

n=3"

" [

Pr [[ D n]

n=3"

because the series in the second line are diverging. In fact, there exists a finite integer n such that, for any real arbitrarily small  A 0 holds that Pr [[ D n ] = 1 3  and Pr [[ D n ] $ Pr [[ D n] for all n ? n . Hence, H [[] =

n [ n=3"

Pr [[ D n] +

" [ n=n

Pr [[ D n] D (1 3 )

n [

1+f y)gxgy

(2.60)

"

H [j([> \ )] = 3"

{

(2.59)

3"

j({> |)i[\ ({> |)g{g|

(2.61)

Most of the di!culties occur in the evaluation of the multiple integrals. The change of variables in multiple dimensions involves the Jacobian. Consider the transformed random variables X = j1 ([> \ ) and Y = j2 ([> \ ) and denote the inverse transform by { = k1 (x> y) and | = k2 (x> y), then iX Y (x> y) = i[\ (k1 (x> y)> k1 (x> y)) M (x> y) where the Jacobian M (x> y) is  M (x> y) = det

C{ Cx C| Cx

C{ Cy C| Cy

¸

If [ and \ are independent and ] = [ + \ , we obtain the convolution, Z " Z " i] (}) = i[ ({)i\ (}  {)g{ = i[ (}  |)i\ (|)g| (2.62) 3"

3"

which is often denoted by i] (}) = (i[  i\ )(}). If both i[ ({) = 0 and i\ ({) = 0 for { ? 0, then the definition (2.62) of the convolution reduces to Z } (i[  i\ )(}) = i[ ({)i\ (}  {)g{ 0

2.5.5 The sum of independent random variables P Let VQ = Q n=1 [n , where the random variables [n are all independent. We first concentrate on the case where Q = q is a (fixed) integer. Since VQ = VQ31 + [Q , direct application of (2.62) yields the recursion Z " iVQ (}) = iVQ 31 (}  |)i[Q (|)g| (2.63) 3"

2.5 Several random variables and independence

33

which, when written out explicitly, leads to the Q -fold integral Z iVQ (}) =

Z

4

4

i[Q (|Q )g|Q · · ·

4

4

i[1 (|1 )i[0 (}  |Q  · · ·  |1 )g|1

(2.64)

In many cases, convolutions are more e!ciently computed via generating functions. The generating function of Vq equals # " q h Sq i Y £ Vq ¤ [ [ = H } n=1 n = H *Vq (}) = H } } n n=1

Since all [n are independent, (2.55) can be applied, *Vq (}) =

q Y

¤ £ H } [n

n=1

or, in terms of generating functions, *Vq (}) =

q Y

*[n (})

(2.65)

n=1

Hence, we arrive at the important result that the generating function of a sum of independent random variables equals the product of the generating functions of the individual random variables. We also note that the condition of independence is crucial in that it allows the product and expectation operator to be reversed, leading to the useful result (2.65). Often, the random variables [n all possess the same distribution. In this case of independent identically distributed (i.i.d.) random variables with generating function *[ (}), the relation (2.65) further simplifies to *Vq (}) = (*[ (}))q

(2.66)

In the case where the number of terms Q in the sum VQ is a random variable with generating function *Q (}), independent of the [n , we use the general definition of expectation (2.54) for two random variables, " X £ V ¤ X Q = *VQ (}) = H } } { Pr [VQ = {> Q = n] n=0 {

=

" X X

} { Pr [VQ = {|Q = n] Pr [Q = n]

n=0 {

where the conditional probability (2.45) is used. Since the value of VQ

34

Random variables

depends on the number of terms Q in the sum, we have Pr [VQ = {|Q = n] = Pr [Vn = {]. Further, with X } { Pr [VQ = {|Q = n] = *Vn (}) {

we have *VQ (}) =

" X

*Vn (}) Pr [Q = n]

(2.67)

n=0

The average H [VQ ] follows from (2.26) as H [VQ ] =

" X

*0Vn (1) Pr [Q = n] =

n=0

hP n

" X

H [Vn ] Pr [Q = n]

(2.68)

n=0

i

Pn Since H [Vn ] = H m=1 [m = m=1 H [[m ] and assuming that all random variables [m have equal mean H [[m ] = H [[], we have H [VQ ] =

" X

nH [[] Pr [Q = n]

n=0

or H [VQ ] = H [[] H [Q ]

(2.69)

This relation (2.69) is commonly called Wald’s identity. Wald’s identity holds for any random sum of (possibly dependent) random variables [m provided the number Q of those random variables is independent of the [m . In the case of i.i.d. random variables, we apply (2.66) in (2.67) so that *VQ (}) =

" X

(*[ (}))n Pr [Q = n] = *Q (*[ (}))

(2.70)

n=0

This expression is a generalization of (2.66).

2.6 Conditional expectation The generating function (2.67) of a random sum of independent random variables can be derived using the conditional expectation H [\ |[ = {] of two random variables [ and \ . We will first define the conditional expectation and derive an interesting property. Suppose that we know that [ = {, the conditional density function

2.6 Conditional expectation

35

i\ |[ (||{) defined by (2.51) of the random variable \f = \ |[ can be regarded as only function of |. Using the definition of the expectation (2.33) for continuous random variables (the discrete case is analogous), we have Z " |i\ |[ (||{) g| (2.71) H [\ |[ = {] = 3"

Since this expression holds for any value of { that the random variable [ can take, we see that H [\ |[ = {] = j ({) is a function of { and, in addition since [ = {, H [\ |[ = {] = j ([) can be regarded as a random variable that is a function of the random variable [. Having identified the conditional expectation H [\ |[ = {] as a random variable, let us compute its expectation or the expectation of the slightly more general random variable k ([) j ([) with j ([) = H [\ |[ = {]. From the general definition (2.34) of the expectation, it follows that Z " Z " H [k ([) j ([)] = k ({) j ({) i[ ({) g{ = k ({) H [\ |[ = {] i[ ({) g{ 3"

3"

Substituting (2.71) yields Z "Z " H [k ([) j ([)] = k ({) |i\ |[ (||{) i[ ({) g|g{ 3" 3" Z "Z " k ({) |i[\ ({> |) g|g{ = H [k ([) \ ] = 3"

3"

where we have used (2.51) and (2.61). Thus, we find the interesting relation H [k ([) H [\ |[ = {]] = H [k ([) \ ]

(2.72)

As a special case where k({) = 1, the expectation of the conditional expectation follows as H [\ ] = H[ [H\ [\ |[ = {]] where the index in H] clarifies that the expectation is over the random P variable ]. Applying this relation to \ = } VQ where VQ = Q n=1 [n and all [n are independent yields ¤ £ £ £ ¤¤ *VQ (}) = H } VQ = HQ HV } VQ |Q = q £ ¤ Since HV } VQ |Q = q = *VQ (}) and specified in (2.65), we end up with *VQ (}) = HQ [*VQ (})] =

" X n=0

which is (2.67).

*Vn (}) Pr [Q = n]

3 Basic distributions

This chapter concentrates on the most basic probability distributions and their properties. From these basic distributions, other useful distributions are derived.

3.1 Discrete random variables 3.1.1 The Bernoulli distribution A Bernoulli random variable [ can only take two values: either 1 with probability s or 0 with probability t = 1  s. The standard example of a Bernoulli random variable is the outcome of tossing a biased coin, and, more generally, the outcome of a trial with only two possibilities, either success or failure. The sample space is = {0> 1} and Pr[[ = 1] = s, while Pr [[ = 0] = t. From this definition, the pgf follows from (2.17) as £ ¤ *[ (}) = H } [ = } 0 Pr [[ = 0] + } 1 Pr [[ = 1] or *[ (}) = t + s}

(3.1)

From (2.23) or (2.14), the q-th moment is H [[ q ] = s which shows that  = H[[] = s. From (2.24), we find H [([  d)q ] = s(1  d)q + t(d)q such that the moments centered around the mean  are ¢ ¡ H [([  )q ] = st q + (1)q tsq = st t q31 + (1)q sq31 £ ¤ Explicitly, with s + t = 1, Var[[] = st and H ([  )3 = st(t  s). 37

38

Basic distributions

3.1.2 The binomial distribution A binomial random variable [ is the sum of q independent Bernoulli random variables. The sample space is = {0> 1> · · · > q}. For example, [ may represent the number of successes in q independent Bernoulli trials such as the number of heads after q-times tossing a (biased) coin. Application of (2.66) with (3.1) gives *[ (}) = (t + s})q

(3.2)

Expanding the binomial pgf in powers of }, which justifies the name “binomial”, q µ ¶ X q n q3n n *[ (}) = } s t n n=0

and comparing to (2.18) yields µ ¶ q n q3n Pr[[ = n] = s t n

(3.3)

The alternative, probabilistic approach starts with (3.3). Indeed, the probability that [ has n successes out of q trials consists of precisely n successes (an event with probability sn ) and q  n failures (with probability equal to t q3n ). The total number ¡q¢ of ways in which n successes out of q trials can be obtained is precisely n . P The mean follows from (2.23) or from the definition [ = qm=1 [Bernoulli and the linearity of the expectation as H [[] = qs. Higher order moments around the mean can be derived from (2.24) as ¯ ¡ ¡ ¢q ¢ ¯¯ q µ ¶ gp h3wqs t + shw gp X q n q3n w(tq3n) ¯¯ ¯ p H [([  ) ] = t s = p h ¯ ¯ ¯ ¯ gwp gw n n=0 w=0 w=0 q µ ¶ X q n q3n t s = (tq  n)p n n=0

In general, this form seems di!cult to express more elegantly. It illustrates that, even for simple random variables, computations may rapidly become unattractive. For p = 2, the above dierentiation leads to Var[[] = qst. But, this result is more economically obtained from (2.27), since O[ (}) = qs qs2 and O00[ (}) =  (t+s}) q log (t + s}), O0[ (}) = t+s} 2 . Thus, Var [[] = qs2 + qs = qst

(3.4)

3.1 Discrete random variables

39

3.1.3 The geometric distribution The geometric random variable [ returns the number of independent Bernoulli trials needed to achieve the first success. Here the sample space is the infinite set of integers. The probability density function is Pr [[ = n] = st n31

(3.5)

because a first success (with probability s) obtained in the n-th trial is proceeded by n  1 failures (each having probability t = 1  s). Clearly, Pr [[ = 0] = 0. The series expansion of the probability generating function, *[ (}) = s}

" X

tn }n =

n=0

s} 1  t}

(3.6)

justifies the name “geometric”. The mean H [[] = *0[ (1) equals H [[] = s1 . The higher-order moments can be deduced from (2.24) as à !¯ " X h3wt@s ¯¯ gq q = s t n (n  t@s)q H [([  ) ] = s q ¯ gw 1  thw ¯ w=0

n=0

Similarly as for the binomial random variable, the variance most easily folt , lows from (2.27) with O[ (}) = log s+log (})log(1t}), O0[ (}) = }1 + 13t} O00[ (}) =  }12 +

t2 (13t})2 .

Thus, Var [[] =

t t2 t + = 2 2 s s s

(3.7)

P The distribution function I[ (n) = Pr [[  n] = nm=1 Pr [[ = m] is obtained as n31 X 1  tn tm = s = 1  tn Pr [[  n] = s 1t m=0

The tail probability is Pr[[ A n] = t n

(3.8)

Hence, the probability that the number of trials until the first success is larger than n decreases geometrically in n with rate t. Let us now consider an important application of the conditional probability. The probability that, given the success is not found in the first n trials, success does not occur within the next p trials, is with (2.44) Pr[[ A n + p|[ A n] =

Pr [{[ A n + p} _ {[ A n}] Pr [[ A n + p] = Pr [[ A n] Pr [[ A n]

40

Basic distributions

and with (3.8) Pr[[ A n + p|[ A n] = tp  Pr[[ A p] This conditional probability turns out to be independent of the hypothesis, the event {[ A n}, and reflects the famous memoryless property. Only because Pr[[ A n] obeys the functional equation i ({ + |) = i ({)i (|), the hypothesis or initial knowledge does not matter. It is precisely as if past failures have never occurred or are forgotten and as if, after a failure, the number of trials is reset to 0. Furthermore, the only solution to the functional equation is an exponential function. Thus, the geometric distribution is the only discrete distribution that possesses the memoryless property.

3.1.4 The Poisson distribution Often we are interested to count the number of occurrences of an event in a certain time interval, such as, for example, the number of IP packets during a time slot or the number of telephony calls that arrive at a telephone exchange per unit time. The Poisson random variable [ with probability density function n h3 (3.9) n! turns out to model many of these counting phenomena well as shown in Chapter 7. The corresponding generating function is Pr [[ = n] =

*[ (}) = h3

" X n n=0

n!

} n = h(}31)

(3.10)

and the average number of occurrences in that time interval is H[[] = 

(3.11)

This average determines the complete distribution. In applications it is convenient to replace the unit interval by an interval of arbitrary length w such that (w)n h3w Pr [[ = n] = n! equals the probability that precisely n events occur in the interval with duration w. The probability that no events occur during w time units is Pr [[ = 0] = h3w and the probability that at least one event (i.e. one or more) occurs is Pr [[ A 0] = 1  h3w . The latter is equal to the exponential distribution. We will also see later in Theorem 7.3.2 that the Poisson

3.1 Discrete random variables

41

counting process and the exponential distribution are intimately connected. The sum of q independent Poisson random variables each with mean n is P again a Poisson random variable with mean qn=1 n as follows from (2.65) and (3.10). The higher-order moments can be deduced from (2.24) as ³ ´¯ w gq h3(w3h ) ¯¯ ¯ H [([  )q ] = h3 ¯ gwq ¯ w=0

from which

£ ¤ H[[] = Var[[] = H ([  )3 = 

The Poisson tail distribution equals Pr [[ A p] = 1 

p X n h3 n=0

n!

which precisely equals the sum of p exponentially distributed variables as demonstrated below in Section 3.3.1. The Poisson density approximates the binomial density (3.3) if q $ 4 but the mean qs = . This phenomenon is often referred to as the law of rare events: in an arbitrarily large number q of independent trials each with arbitrarily small success s = q , the total number of successes will approximately be Poisson distributed. The classical argument is to consider the binomial density (3.3) with s =

 q

   q3n q! n 1 3 n!(q 3 n)! qn q   3n n31   n \ m    q 13 = 13 13 n! q q q m=1

Pr[[ = n] =

or  log (Pr[[ = n]) = log

n n!



   n31    [ m   log 1 3 3 n log 1 3 + + q log 1 3 q q q m=1

   33  { { {2 = 3q to obtain up to order For large q, we use the Taylor expansion log 1 3 q 3 2q 2 +R q  32  R q  n  n(n 3 1)       2  + n + R q32 3 log (Pr[[ = n]) = log + R q32 3  3 + R q32 n! q 2q 2q  n    1   (n 3 )2 3 n + R q32 = log 33 n! 2q With h{ = 1 + { + R({2 ), we finally obtain the approximation for large q,      1  n h3 13 (n 3 )2 3 n + R q32 Pr[[ = n] = n! 2q

42

Basic distributions

The coe!cient of

1 q

k is negative if n M  +

1 2

3

t  + 14 >  +

1 2

+

t l  + 14 . In that n-interval,

the Poisson density is a lower bound for the binomial density for large q and qs = . The reverse holds for values of n outside that interval. Since for the Poisson density

Pr[[=n] Pr[[=n31]

=

 , n

we

see that Pr[[ = n] increases as  A n and decreases as  ? n. Thus, the maximum of the Poisson density lies around n =  = H[[]. In conclusion, we can say that the Poisson density approximates the binomial density for large q and qs =  from below in the region of about the I standard deviation  around the mean H[[] =  and from above outside this region (in the tails of the distribution).

A much shorter derivation anticipates results of Chapter 6 and starts from the probability generating function (3.2) of the binomial distribution after substitution of s = q , ¶ µ (}  1) q lim *[ (}) = lim 1 + = h(}31) q \ ) =

[

|Pr [[ = n] 3 Pr [\ = n]|

n

and satisfies gW Y ([> \ ) = 2 sup |Pr [[ M D] 3 Pr [\ M D]| DaZ

3.2 Continuous random variables

43

3.2 Continuous random variables 3.2.1 The uniform distribution A uniform random variable [ has equal probability to attain any value in the interval [d> e] such Rthat the probability density function is a constant. e Since Pr[d  [  e] = d i[ ({)g{ = 1, the constant value equals i[ ({) =

1 1 e  d {M[d>e]

(3.12)

where 1| is the indicator function defined in Section 2.2.1. The distribution function then follows as {d + 1{Ae 1 Pr [d  [  {] = e  d {M[d>e] The Laplace transform (2.37) is2 Z " h3}d  h3}e *[ (}) = h3}w i[ (w)gw = }(e  d) 3"

(3.13)

while the mean  = H [[] most easily follows from Z " {g{ d+e H[[] = 1{M[d>e] = 2 3" e  d The centered moments are obtained from (2.39) as ³ } ´¯ (e3d) 3 }2 (e3d) ¯ 2 q q  h h ¯ (1) g ¯ H [([  )q ] = ¯ q e  d g} } ¯ }=0 ¯ ¯ )} 2(1)q gq sinh( e3d ¯ 2 = ¯ ¯ e  d g} q } }=0

Using the power series X ( e3d )2n+1 sinh( e3d 2 )} 2 = } 2n } (2n + 1)! "

n=0

leads to ¤ £ (e  d)2q H ([  )2q = (2q + 1)22q £ ¤ H ([  )2q+1 = 0 2

(3.14)

(}) Notice that }*[ equals the convolution i W j of two exponential densities i and j with rates de d and e, respectively.

44

Basic distributions

Let us define X as the uniform random variable in the interval [0> 1]. If Z = 1  X is a uniform random variable on [0> 1], then Z and X have the g

same distribution denoted as Z = X because Pr[Z  {] = Pr[1  X  {] = Pr[X  1  {] = 1  (1  {) = { = Pr [X  {] = The probability distribution function I[ ({) = Pr[[  {] = j({) whose inverse exists can be written as a function of IX ({) = {1{M[0>1] . Let [ = j 31 (X ). Since the distribution function is non-decreasing, this also holds for the inverse j 31 (=). Applying (2.32) yields with [ = j 31 (X ) £ ¤ I[ ({) = Pr j 31 (X )  { = Pr [X  j({)] = IX (j({)) = j({) g

=  lnX are exponentially random variFor instance, j 31 (X ) =  ln(13X)  31 ables (3.17) with parameter ; j (X ) = X 1@ are polynomially distributed random variables with distribution Pr [[  {] = { ; j 31 (X ) = cot(X ) is a Cauchy random variable defined in (3.38) below. In addition, we observe that X = j ([) = I[ ([), which means that any random variable [ is transformed into a uniform random variable X on [0> 1] by its own distribution function. The numbers dn that satisfy congruent recursions of the form dn+1 = (dn +) mod P , where P is a large prime number (e.g. P = 231 1),  and  are integers (e.g.  = 397 204 094 and  = 0) are to a good approximation dn are nearly uniformly uniformly distributed. The scaled numbers |n = P31 distributed on [0> 1]. Since these recursions with initial value or seed d0 5 [0> P  1] are easy to generate with computers (Press et al., 1992), the above property is very useful to generate arbitrary random variables [ = j 31 (X ) from the uniform random variable X .

3.2.2 The exponential distribution An exponential random variable [ satisfies the probability density function i[ ({) = h3{

> {  0

(3.15)

where  is the rate at which events occur. The corresponding Laplace transform is Z "  h3w h3}w gw = (3.16) *[ (}) =  }+ 0 and the probability distribution is, for {  0, I[ ({) = 1  h3{

(3.17)

3.2 Continuous random variables

45

The mean or average follows from (2.33) or from H [[] = *0[ (0) as  = H [[] = 1 . The centered moments are obtained from (2.39) as ³ ´¯ ¶q ¸ µ q h}@ ¯ g }+ ¯ 1 ¯ = (1)q H [ ¯  g} q ¯ }=0

h}@

around } = 0 is õ ¶ q ! " " " ³ ´n X 1 q X (1)n h}@ X 1 ³ } ´n X n } = }q (1) = }+ n!    n!

Since the Taylor expansion of

n=0

we find that

n=0

}+

q=0

n=0

¶ ¸ µ q 1 q q! X (1)n H [ = q   n!

(3.18)

n=0

For large q, the centered moments are well approximated by µ ¶ ¸ 1 q q! H [ ' q  h The exponential random variable possesses, just as its discrete counterpart, the geometric random variable, the memoryless property. Indeed, analogous to Section 3.1.3, consider Pr[[  w + W |[ A w] =

Pr [[  w + W ] Pr [{[  w + W } _ {[ A w}] = Pr [[ A w] Pr [[ A w]

and since Pr [[ A w] = h3w , the memoryless property Pr[[  w + W |[ A w] = Pr[[ A W ] is established. Since the only non-zero solution (proved in Feller (1970, p. 459)) to the functional equation i ({ + |) = i ({)i (|), which implies the memoryless property, is of the form f{ , it shows that the exponential distribution is the only continuous distribution that has the memoryless property. As we will see later, this memoryless property is a fundamental property in Markov processes. It is instructive to show the close relation between the geometric and exponential random variable (see Feller (1971, p. 1)). Consider the waiting time W (measured in integer units of w) for the first success in a sequence of W Bernoulli trials where only one trial occurs in a timeslot w. Hence, [ = {w is a (dimensionless) geometric random variable. From (3.8), Pr[W A nw] = (1  s)n and the average waiting time is H [W ] = wH [[] = {w s . The

46

Basic distributions

transition from the discrete to continuous space involves the limit process w $ 0 subject to a fixed average waiting time H [W ]. Let w = nw, then ¶ µ w w@{w lim Pr[W A w] = lim 1  = h3w@H[W ] {w  2 ), then is, for {  0, h i 2) µ s ¶ exp  ({+  { 2 2 s cosh i[ 2 ({) = 2  2{ In particular, for Q (0> 1) random variables where  = 0 and  = 1, i[ 2 ({) =

3.4 Functions of random variables

51

3{ 2

h I 2{

reduces to a Gamma distribution (3.25) with  = 12 and  = 12 . Since the sum of q independent Gamma random variables with (> ) is again a Gamma random variable (> q), we arrive at the chi-square "2 probability density function, q

{ { 2 31 i"2 ({) = q ¡ q ¢ h3 2 2 2  2

(3.31)

3.4 Functions of random variables 3.4.1 The maximum and minimum of a set of independent random variables The minimum of p i.i.d. random variables {[n }1$n$p possesses the distribution4  ¸ Pr min [n  { = Pr [at least one [n  {] = Pr [not all [n A {] 1$n$p

or

 Pr

¸ p Y min [n  { = 1  Pr[[n A {]

1$n$p

(3.32)

n=1

whereas for the maximum, ¸  p Y Pr[[n  {] Pr max [n A { = Pr [not all [n  {] = 1  1$n$p

n=1

or

 Pr

¸ Y p Pr[[n  {] max [n  { =

1$n$p

(3.33)

n=1

For example, the distribution function for the minimum of p independent exponential random variables follows from (3.17) as ! Ã ¸  p p Y X 3n { h = 1  exp { n Pr min [n  { = 1  1$n$p

n=1

n=1

or, the minimum of p independent exponential random variables each with Pp rate n is again an exponential random variable with rate n=0 n . In addition to the memoryless property, this property of the exponential distribution will determine the fundamentals of Markov chains. 4

An alternative argument for independent random variables is that the event {min1$n$p [n A {} is only possible if and only if {[n A {} for each 1 $ n $ p. Similarly, the event {max1$n$p [n $ {} is only possible if and only if all {[n $ {} for each 1 $ n $ p.

52

Basic distributions

3.4.2 Order statistics The set [(1) > [(2) > = = = > [(p) are called the order statistics of the set of random variables {[n }1$n$p if [(n) is the n-th smallest value of the set {[n }1$n$p . Clearly, [(1) = min1$n$p [n while [(p) = max1$n$p [n . If the set {[n }1$n$p consists of i.i.d. random variables with pdf i[ , the joint density function of the order statistics is, for only {1 ? {2 ? · · · ? {p , ¤ £ Cp i{[(m) } ({1 > {2 > = = = > {p ) = Pr [(1)  {1 > = = = > [(p)  {p C{1 = = = C{p p Y = p! i[ ({m ) (3.34) m=1

Indeed, confining to discrete random variables for simplicity, if {1 ? {2 ? · · · ? {p , then ¤ £ Pr [(1) = {1 > = = = > [(p) = {p = p! Pr [[1 = {1 > = = = > [p = {p ] else

¤ £ Pr [(1) = {1 > [(2) = {2 > = = = > [(p) = {p = 0

because there are precisely p! permutations of the set {[n }1$n$p onto the given ordered sequence {{1 > {2 > = = = > {p }. If the sequence is not ordered such that {n A {o for at least©one couple of ª indices n ? o, then the probability is zero because the event [(n) A [(o) is, by definition, impossible. Finally, the product in (3.34) follows by independence. If the set {[n }1$n$p is uniformly distributed over [0> w], then p! i{[(m) } ({1 > {2 > = = = > {p ) = p w =0

0  {1 ? {2 ? · · · ? {p  w elsewhere

while for exponential random variables with i[ ({) = h3{ i{[(m) } ({1 > {2 > = = = > {p ) = p!p h3 =0

Sp

m=1

{m

0  {1 ? {2 ? · · · ? {p elsewhere

The order relation between the set [(1)  [(2)  · · · ¡ [(p) ¢ is preserved ¡ ¢  j [  after a continuous, non-decreasing transform j, i.e. j [ (1) (2) ¡ ¢ · · ·  j [(p) . If the distribution function I[ is continuous (it is always non-decreasing), the argument shows that the order statistics of a general set of i.i.d. random variable {[n }1$n$p can be reduced to a study of the order statistics of the set of i.i.d. uniform random variables {Xn }1$n$p on [0,1] because X = I[ ([).

3.4 Functions of random variables

53

© ª The event [(n)  { means that at least n among the p random variables {[m }1$m$p are smaller than {. Since each of the p random variables is chosen independently from a same distribution I[ , the probability that precisely q of the p random variables is smaller than { is binomially distributed with parameter s = Pr [[  {]. Hence, p µ ¶ ¤ X £ p (3.35) (Pr [[  {])q (1  Pr [[  {])p3q Pr [(n)  { = q q=n

The probability density function can be obtained in the usual, though cumbersome, way by dierentiation, ¤ £ g Pr [(n)  { i[(n) ({) = g{ p µ ¶ X p g = (Pr [[  {])q (1  Pr [[  {])p3q q g{ q=n µ ¶ p X p = i[ ({) q (Pr [[  {])q31 (1  Pr [[  {])p3q q q=n µ ¶ p X p  i[ ({) (p  q) (Pr [[  {])q (1  Pr [[  {])p3q31 q q=n ¡p31¢ ¡ ¢ ¡p31¢ ¡p¢ and lowering the upper index Using q q = p q31 , (p q) p q =p q in the last summation, we have ¶ p µ X p1 i[(n) ({) = pi[ ({) (Pr [[  {])q31 (1  Pr [[  {])p3q q1 q=n p31 X µp  1¶ (Pr [[  {])q (1  Pr [[  {])p3q31  pi[ ({) q q=n ¶ p µ X p1 = pi[ ({) (Pr [[  {])q31 (1  Pr [[  {])p3q q1 q=n ¶ p µ X p1  pi[ ({) (Pr [[  {])q31 (1  Pr [[  {])p3q q1 q=n+1

or, with I[ ({) = Pr [[  {], µ ¶ p1 i[(n) ({) = pi[ ({) (I[ ({))n31 (1  I[ ({))p3n n1

(3.36)

The more elegant and faster argument is as follows: in order for [(n) to be equal to {, exactly n  1 of the p random variables {[m }1$m$p must be

54

Basic distributions

less than {, one equal to { and the£ other p ¤ n must all be greater¡than¢ {. Abusing the notation i[ ({) = Pr [(n) = { and observing that p p31 n31 = p! p! 1!(n31)!(p3n)! is an instance of the multinomial coe!cient q1 !q2 !···qn ! which gives the number of ways of putting p = q1 + q2 + · · · + qn dierent objects into n dierent boxes with qm in the m-th box, leads alternatively to (3.36). 3.5 Examples of other distributions 1. The Gumbel distribution appears in the theory of extremes (see Section 6.4) and is defined by the distribution function 3d({3e)

IGumbel ({) = h3h

(3.37)

The corresponding Laplace transform is Z " ³ }´ 3d(w3e) *Gumbel (}) = h3}w h3h dh3d(w3e) gw = h3e}  1 + d 3" ¢¯ ¡ g 3e} from which the mean follows as H [[] =  g} h  1 + d} ¯}=0 = e + d , where  = 0.57721=== is the Euler constant. The variance is best computed 2 with (2.43) resulting in Var[[] = 6d 2. 2. The Cauchy distribution has the probability density function iCauchy ({) = and corresponding distribution, ICauchy ({) =

1  (1 + {2 )

´ 1 ³ + arctan {  2

The Laplace transform 1 *Cauchy (}) = 

Z

"

3"

(3.38)

h3}{ g{ 1 + {2

only converges for purely imaginary } = l$, in which case it reduces to a Fourier transform, Z 1 " h3l${ g{ *Cauchy (l$) =  3" 1 + {2 This integral is best evaluated by contour integration. If $  0, we consider a contour F consisting of the real axis and the semi-circle that encloses the negative Im({)-plane, Z 3l${ Z " 3l${ Z  3l$uh3l ¡ 3l ¢ h g uh h g{ h g{ = + lim 2 2 2 u e A 0 ³ ¡ ¢´ e exp  {d ¢ ¡ (3.39) iWeibull ({) = d 1 + 1e generalizes the exponential distribution (3.17) corresponding to e = 1 and d = 1 . It is related to the Gaussian distribution if e = 2. Let [ be a Weibull

56

Basic distributions

random variable. All higher moments can be computed from (2.34) as ¡ ¢ µ ³ ´¶ Z " h i dn  n+1 { e 1 n n e ¢ ¡ ¡ ¢ { exp  g{ = H [ = d d 1 + 1e 0  1e The generating function possesses the expansion µ ¶ " " £ 3}[ ¤ X n + 1 (}d)n (})n h n i 1 X = *[ (}) = H h  H [ = ¡1¢ n! e n!  e n=0 n=0 which cannot be summed in explicit form for general e. Sometimes an alternative definition of the Weibull distribution appears iWeibull ({) = de{e31 h3d{

e

(3.40)

3d{e

IWeibull ({) = 1  h

with the advantage of a simpler expression for the distribution function IWeibull ({). If [ possesses this probability density (3.40), the moments and variance are ¡ ¢ h i Z "  n + 1e n n { iWeibull ({)g{ = H [ = dn@e 0 ¡ ¢ ¡ ¢ Z "  1 + 2e  2 1 + 1e 2 Var[[] = ({  H [[]) iWeibull ({)g{ = d2@e 0 The interest of the Weibull distribution in the Internet stems from the self-similar and long-range dependence of observables (i.e. quantities that can be measured such as the delay, the interarrival times of packets, etc.). Especially if the shape factor e 5 (0> 1), the Weibull has a sub-exponential tail that decays more slowly than an exponential, but still faster than any power law. 4. Power law behavior is often described via the Pareto distribution with pdf for {  0 and  A 0> ³ { ´331 (3.41) 1+ iPareto ({) =   and with distribution function Z ³  {³ { ´3 { ´331 IPareto ({) = gw = 1  1 + (3.42) 1+  0   Since lim{  2 is£a Gaussian ¤ or normal random variable. From (2.32), it follows that Pr h\  { = Pr [\  log {] for {  0, and with (3.20)  ¸ Z log { 1 (w  )2 Ilognormal ({) = s exp  gw (3.43) 2 2  2 3" and, for { A 0,

h i {3)2 exp  (log2 2 s ilognormal ({) = { 2

(3.44)

The moments are  ¸ Z " h i (log {  )2 1 n n31 H [ = s { exp  g{ 2 2  2 0  ¸ Z " (x  )2 1 nx h exp  gx = s 2 2  2 3" or, explicitly,

µ 2 2¶ h i n  n H [ = exp (n) exp 2

and

(3.45)

³ 2 ´ 2 Var[[] = h2 h2  h

(3.46)

The probability generating function is by definition (2.37) 1 *[ (}; >  2 ) = s  2

Z 0

"

h3 h3}w

(log w3)2 2 2

w

1 gw = s  2

Z

"

{

h3}h h3

({3)2 2 2

g{

3"

(3.47) only exists for Re(})  0. The integral (3.47) indicates that *[ This means that *[ (}; >  2 ) is not analytic at any point } = lw on the imaginary axis because the circle with arbitrary small but non-zero radius around } = lw necessarily encircles points with Re(}) ? 0 where *[ (}; > 2 ) does not exist. Hence, the Taylor expansion (2.40) of the generating function around } = 0 does not exist, although all moments or derivatives at } = 0 (}; >  2 )

58

Basic distributions

exist. Indeed, the series £ ¤ µ 2 2¶ " " X (1)n H [ n n X (}h )n n  } = exp n! n! 2 n=0

n=0

is a divergent series (except for  = 0 or } = 0). The fact that the pgf (3.44) is not available in closed form complicates the computation of the sum of i.i.d. lognormal random variables via (2.66). This sum appears in radio communications with several transmitters and receivers. In radio communications, the received signal levels decrease with the distance between the transmitter and the receiver. This phenomenon is called pathloss. Attenuation of radio signals due to pathloss has been modeled by averaging the measured signal powers over long times and over various locations with the same distances to the transmitter. The mean value of the signal power found in this way is referred to as the area mean power Pd (in Watts) and is well-modeled as Pd (u) = f·u3 where f is a constant and  is the pathloss exponent5 . In reality the received power levels may vary significantly around the mean power Pd (u) due to irregularities in the surroundings of the receiving and transmitting antennas. Measurements have revealed that the logarithm of the mean power P (u) at dierent locations on a circle with radius u around the transmitter is approximately normally distributed with mean equal to the logarithm of the area mean power Pd (u). The lognormal shadowing model assumes that the logarithm of P(u) is precisely normally distributed around the logarithmic value of the area mean power: log10 (P(u)) = log10 (Pd (u))+[, where [ = Q (0> ) is a zero-mean normal distributed random variable (in dB) with standard deviation  (also in dB and for severe fluctuations up to 12 dB). Hence, the random variable P(u) = Pd (u)10[ has a lognormal distribution (3.43) equal to  Pr [P(u) $ {] = Pr [ $ log10

& %  ] { 1 { (log10 x 3 log10 (Pd (u)))2 gx = I exp 3 Pd (u) 22 x  2 log 10 0

3.6 Summary tables of probability distributions 3.6.1 Discrete random variables Name

Pr [[ = n]

H [[]

Var[[]

£ ¤ *[ (}) = H } [

Bernoulli Binomial Geometric

Pr [[ = 1] = s ¡q¢ n q3n n s (1  s) s (1  s)n31

s qs

s (1  s) qs (1  s)

1  s + s} ((1  s) + s})q

1 s

13s s2

Poisson

n





s} 13(13s)} h(}31)

5

n!

h3

The constant f depends on the transmitted power, the receiver and the transmitter antenna gains and the wavelength. The pathloss exponent  depends on the environment and terrain structure and can vary between 2 in free space to 6 in urban areas.

3.7 Problems

59

3.6.2 Continuous random variables Name Uniform Exponential Gaussian Gamma Gumbel Cauchy Weibull Pareto Lognormal

i[ ({)

H [[]

Var[[]

  *[ (}) = H h3}[

1d{e e3d h3{   ({)2 exp 3 22 I  2 ({)1 3{ h K() { 3{ h h h

d+e 2 1 

(e3d)2 12 1 2

h}d 3h}e }(e3d)  }+



2

 

 2 2 6

exp 

1  (1+{2 ) e

de{e31 h3d{  331  1+ {  

  (log {)2 exp 3 22 I { 2

 = 0=5772=== does not exist

does not exist

K(1+ 1 e)

K(1+ 2 3K2 (1+ 1 e) e)

d1@e  1{A1} 31

2 }2 2 

l 3 }

 }+

K (} + 1) h3| Im(})| (Re(}) = 0)

d2@e  2 1{A2}

(31)2 (32)

 exp () exp

k

 2

 2

 2  2 h2 h2 3 h

3.7 Problems (i) If *[ (}) is the probability generating function of a non-zero discrete random variable [, find an expression of H [log [] in terms of *[ (}). (ii) Compute the mean value of the n-th order statistic in an ensemble of (a) p i.i.d. exponentially distributed random variables with mean 1 and (b) p i.i.d. polynomially distributed random variables on [0,1]. (iii) Discuss how a probability density function of a continuous random variable [ can be approximated from a set {{1 > {2 > = = = > {q } of q measurements or simulations. (iv) In a circle with radius u around a sending mobile node, there are Q  1 other mobile nodes uniformly distributed over that circle. The possible interference caused by these other mobile nodes depends on their distance to the sending node at the center. Derive for large Q but constant density  of mobile nodes the pdf of the distance of the p-th nearest node to the center. (v) Let X and Y be two independent random variables. What is the probability that the one is larger than the other?

4 Correlation

In this chapter methods to compute bi-variate correlated random variables are discussed. As a measure for the correlation, the linear correlation coe!cient defined in (2.58) is used. First, the generation of q correlated Gaussian random variables is explained. The sequel is devoted to the construction of two correlated random variables with arbitrary distribution.

4.1 Generation of correlated Gaussian random variables Due to the importance of Gaussian correlated random variables as an underlying system for generating arbitrary correlated random variables, as will be demonstrated in Section 4.3, we discuss how they can be generated in multiple dimensions. With the notation of Section 3.2.3, a Gaussian (normal) random variable with average  and variance  2 is denoted by Q (>  2 ). By linearly combining Gaussian random variables, we can create a new Gaussian random variable with a desired mean  and variance 2 .

4.1.1 Generation of two independent Gaussian random variables The fact that a linear combination of Gaussian random variables is again a Gaussian random variable allows us to concentrate on normalized Gaussian random variables Q (0> 1). Let [1 and [2 be two independent normalized Gaussian random variables. Independent random variables are not correlated and the linear correlation coe!cient  = 0. The resulting joint probability distribution is i[1 [2 ({> |; ) = i[1 ({)i[2 (|) and with (3.19), i[1 [2 ({> |; 0) = 61

h3

{2 +| 2 2

2

62

Correlation

It is natural to consider a polar transformation ³ ´and the transformed random 2 2 2 variables U = [1 + [2 and  = arctan [ [1 . The inverse transform is s s { = u cos  and | = u sin , which diers slightly from the usual polar transformation in that we now define u = {2 + | 2 instead of u2 = {2 + |2 . The reason is that the Jacobian is simpler for our purposes, # " cos  s  C{ C{ ¸ I  u sin  1 2 u Cu C = M (u> ) = det C| C| = det sin  s I u cos  2 Cu C 2 u whereas the usual polar transformation has the Jacobian equal to the variable u. Using the transformation rules in Section 2.5.4, u

h3 2 iUX (u> ) = 4 which shows that iUX (u> ) does not depend on . Hence, we can write iUX (u> ) = iU (u) iX () with iX () = f, where f is a constant and iU (u) = u

h3 2 4f

. This implies that  is a uniform random variable over an interval 1@f. We also recognize from (3.15) that iU (u) is close to an exponential random variable with rate  = 12 . Therefore, it is instructive to choose the constant f such that U is precisely an exponential random variable with rate  = 12 . 3u

1 1 Thus, choosing f = 2 , we end up with iU (u) = h 2 2 and iX () = 2 . These two independent random variables U and  can each be generated separately from a uniform random variable X on [0,1], as discussed in Section 3.2.1, leading to

U = 2 ln(X1 )  = 2X2 and, finally, to the independent Gaussian random variables p p [1 = 2 ln(X1 ) cos 2X2 [2 = 2 ln(X1 ) sin 2X2 The procedure can be used to generate a single Gaussian random variable, but also more independent Gaussians by repeating the generation procedure. 4.1.2 The q-joint Gaussian probability distribution function A collection of q random variables [l is called a random vector [ = ([1 > [2 > = = = > [q )W , a matrix with dimension q × 1. The average of a random vector is a vector with components H [[l ] for 1  l  q. The variance of a random vector £ ¤ £ ¤ Var [[] = H ([  H [[])([  H [[])W = H [[ W  H [[] (H [[])W

4.1 Generation of correlated Gaussian random variables

63

is a matrix [ with elements ( [ )l>m = Cov[[l > [m ]. Since Cov[[l > [m ] = Cov[[m > [l ], the covariance matrix [ is real and symmetric, [ = W[ . The importance of real, symmetric matrices is that they have real eigenvectors (see Appendix A.2). Moreover, [ is non-negative definite because, using vector norms defined in Section A.3, £ ¤ {W [ { = H {W ([  H [[])([  H [[])W { i h¡ ¢W = H ([  H [[])W { ([  H [[])W { h° °2 i = H °([  H [[])W {°2  0 which implies that all real eigenvalues l are non-negative. Hence, there exists an orthogonal matrix X such that [ = Xdiag(l )X W

(4.1)

If all random variables [l are independent, Cov[[l > [m ] = 0 for l 6= m and Cov[[l > [l ] = Var[[l ]  0 then [ = diag(Var[[l ]). Gaussian random variables are completely determined by the mean and the variance, i.e. by the first two moments. We will now show that the existence of an orthogonal transformation for any probability distribution such that X W [ X = diag(l ) implies that a vector of joint Gaussian random variables can be transformed into a vector of independent Gaussian random variables. Also the reverse holds, which will be used below to generate q joint correlated Gaussian random variables. The multi-dimensional generating function of a q-joint Gaussian or q-joint normal random vector [ is defined for the vector } = (}1 > }2 > = = = > }q )W as µ ¶ £ 3}[ ¤ 1 W W = exp (4.2) } [ }  H [[] } *[ (}) = H h 2 Using (4.1), and the fact that X is an orthogonal matrix such that X 31 = X W and X X W = L, µ ¶ ¡ W ¢W W 1 ¡ W ¢W W *[ (}) = exp X } diag(l )X }  X H [[] X } 2 Denote the vectors z = X W } and p = X W H [[]. Then we have µ ¶ 1 W W z diag(l )z  p z *[ (}) = exp 2 3 4 q q  z2 2 X Y m m m zm C D = exp h 2 3pm zm  pm zm = 2 m=1

m=1

64

Correlation m zm2 3pm zm 2

and h = *[m (zm ) is the Laplace transform (3.22) of a Gaussian random variable [m because all m are real and non-negative. With (2.65), this shows that a vector of joint Gaussian random variables can be transformed into a vector of independent Gaussian random variables. Reversing the order of the manipulations also justifies that (4.2) indeed defines a general q-joint Gaussian probability generating function. If [1 > [2 > = = = > [q are joint normal and not correlated, then [ is a diagonal matrix, which implies that [1 > [2 > = = = > [q are independent. As discussed in Section 2.5.2, independence implies non-correlation, but the converse is generally not true. These properties make Gaussian random variables particularly suited to deal with correlations.

y

2

0 2

fXY(x,y)

0.25 0.2 0.15 0.1 0.05 0 2 0 2

x

Fig. 4.1. The joint probability density function (4.4) with [ = \ = 0 and [ = \ = 1 and  = 0=

The corresponding q-joint Gaussian probability density function of the vector [ can be derived after inverse Laplace transform for the vector { = ({1 > {2 > = = = > {q )W as µ ¶ 1 1 W 31 i[ ({) = ¡s ¢q s exp  ({  H[{]) [ ({  H[{]) (4.3) 2 2 det [ The inverse Laplace transform for q = 2 is computed in Section C.2. After computing the inverse matrix and the determinant in (4.3) explicitly, the two-dimensional (q = 2) or bi-variate Gaussian probability density function is 5 2 2 6 ({3[ )

exp 7 i[\ ({> |; ) =

2 [

2 [ \

3

({3[ )(|3\ )+ 2(132 )

2[ \

p 1  2

(|3\ ) 2 \

8 (4.4)

4.1 Generation of correlated Gaussian random variables

65

Figures 4.1—4.3 plot i[\ ({> |; ) for various correlation coe!cients . If  = 0, we observe that i[\ ({> |; 0) = i[ ({)i\ (|), which indicates that uncorrelated Gaussian random variables are also independent.

y

2

0

2

fXY(x,y)

0.25 0.2 0.15 0.1 0.05 0

2 0

x

2

Fig. 4.2. The joint probability density function (4.4) with [ = \ = 0 and [ = \ = 1 and  = 0=8=

y

2

0

2

fXY(x,y)

0.25 0.2 0.15 0.1 0.05 0

2 0

x

2

Fig. 4.3. The joint probability density function (4.4) with [ = \ = 0 and [ = \ = 1 and  = 0=8=

If we denote {W = (4.4) reduces to

({3[ ) [

and |W =

(|3\ ) , \

the bi-variate normal density

h i W 2 W | W +(| W )2 exp  ({ ) 32{ 2(132 ) p i[\ ({W > |W ; ) = 2[ \ 1  2

66

Correlation

from which we can verify the partial dierential equation Ci[\ ({W > |W ; ) Ci[\ ({W > |W ; ) = C C{W C|W

(4.5)

and the symmetry relations i[\ ({W > |W ; ) = i[\ ({W > | W ; ) = i[\ ({W > | W ; )

(4.6)

4.1.3 Generation of q correlated Gaussian random variables Let {[l }1$l$q be a set of q independent normal random variables, where each [l is distributed as Q (0> 1). The vector [ is rather easily simulated. The analysis above shows that H [[] = 0 (the null-vector) and [ = diag(Var[[l ]) = diag(1) = L, the identity matrix. We want to generate the correlated normal vector \ with a given mean vector H [\ ] and a given covariance matrix \ . Since linear combinations of normal random variables are normal random variables, we consider the linear transformation \ = D[ + E where D and E are constant matrices. We will now determine D and E. First, H [\ ] = H [D[] + H [E] = DH [[] + H [E] = H [E] Hence, the matrix E is a vector with components equal to the given components H [\l ] of the mean vector H [\ ]. Second, £ ¤ \ = H (\  H [\ ])(\  H [\ ])W £ ¤ £ ¤ £ ¤ = H D[(D[)W = H D[[ W DW = DH [[ W DW = D [ DW = DDW From the eigenvalue decomposition of \ = X diag(l )X W with real eigens ¡ s ¢W such values l  0 and the fact that diag(l ) = diag( l ) diag( l ) that p ³ p ´W DDW = X diag( l ) diag( l ) X W we obtain

p D = X diag( l )

The matrix D is also called the square root matrix of \ and can be found from the singular value decomposition of \ or from Cholesky factorization (Press et al., 1992). Example Generate a normal vector \ with H [\ ] = (300> 300)W , with standard deviations 1 = 106=066, 2 = 35=355 and correlation \ = 0=8.

4.2 Generation of correlated random variables

67

Solution: The covariance matrix of \ is obtained using the definition of the linear correlation coe!cient (2.58), ¶ µ 2 ¶ µ 1 11250 3000 \ 1 2 = \ = 3000 1250 \ 1 2 22 The square root matrix D of \ is ¶ µ 63=640 84=853 D= 0 35=355 which is readily checked by computing DDW = \ . It remains to generate p independent draws for [1 and [2 from a normal distribution with zero mean and unit variance as explained in Section 4.1.1. Each pair ([1 > [2 ) out of the p pairs is transformed as \ = D[ +H [\ ]. The result, component \2 versus \1 , is shown in Fig. 4.4. Y2

450 400 350 300 250 200 150 100 50 0 0

100

200

300

400

500

600

700 Y1

Fig. 4.4. The scatter diagram of the simulated vector \=

4.2 Generation of correlated random variables Let us consider the problem of generating two correlated random variables [ and \ with given distribution functions I[ and I\ . The correlation is expressed in terms of the linear correlation coe!cient ([> \ ) =  defined in (2.58). The need to generate correlated random variables often occurs in simulations. For example, as shown in Kuipers and Van Mieghem

68

Correlation

(2003), correlations in the link weight structure may significantly increase the computational complexity of multi-constrained routing, called in brief QoS routing. The importance of measures of dependence between quantities in risk and finance is discussed in Embrechts et al. (2001b). In general, given the distribution functions I[ and I\ , not all linear correlations from 1    1 are possible. Indeed, let [ and \ be positive real random variables with infinite range which means that I[ ({) = 1 if { $ 4 and that I[ ({) = I\ ({) = 0 for { ? 0. Consider \ = d[ + e with d ? 0 and e  0. For all finite | ? 0,  ¸  ¸ |e |e I\ (|) = Pr [\  |] = Pr [d[ + e  |] = Pr [   Pr [ A d d ¶ µ |e = 1  I[ A0 d which contradicts the fact that I\ (|) = 0 for | ? 0. Hence, positive random variables with infinite range cannot be correlated with  = 1= The requirement that the range needs to be unbounded is necessary because two uniform random variables on [0> 1], X1 and X2 , are negatively correlated with  = 1 if X1 = 1  X2 . In summary, the set of all possible correlations is a closed interval [min > max ] for which min ? 0 ? max . The precise computation of min and max is, in general, di!cult, as shown below. 4.3 The non-linear transformation method The non-linear transformation approach starts from a given set of two random variables [1 and [2 that have a correlation coe!cient [ 5 [1> 1]. If the joint distribution function Z {1 Z {2 i[1 [2 (x> y; [ )gxgy I[1 [2 ({1 > {2 ; [ ) = Pr [[1  {1 > [2  {2 ] = 3"

3"

is known, the marginal distribution follows from (2.60) as Z {1 Z " i[1 [2 (x> y; [ )gxgy Pr [[1  {1 ] = 3"

3"

Since for any random variable [ holds that I[ ([) = X where X is a uniform random variable on [0> 1], it follows that X1 = I[1 ([1 ) and X2 = I[2 ([2 ) are uniformly correlated random variables with correlation coe!cient X . As shown in Section 3.2.1, if X is a uniform random variable on [0,1], any other random variable \ with distribution function j({) can be constructed as j 31 (X ). By combining the two transforms, we can generate

4.3 The non-linear transformation method

69

\1 = j131 (I[1 ([1 )) and \2 = j231 (I[2 ([2 )) that are correlated because [1 and [2 are correlated. It may be possible to construct directly the correlated random variables \1 = W1 ([1 ) and \2 = W2 ([2 ) if the transforms W1 and W2 are known. The goal is to determine the linear correlation coe!cient \ defined in (2.58), H [\1 \2 ]  H [\1 ] H [\2 ] p \ = p Var [\1 ] Var [\2 ] as a function of [ . Using (2.61), £ ¤ H [\1 \2 ] = H j131 (I[1 ([1 )) j231 (I[2 ([2 )) Z "Z " j131 (I[1 (x)) j231 (I[2 (y)) i[1 [2 (x> y; [ )gxgy (4.7) = 3"

3"

This relation shows that \ is a continuous function in [ and that the joint distribution function of [1 and [2 is needed. The main di!culty lies now in the computation of the integral appearing in H [\1 \2 ]. For [1 and [2 , Gaussian correlated random variables are most often chosen because an exact analytic expression (4.4) exists for the joint distribution function I[2 [2 ({1 > {2 ; [ ). 4.3.1 Properties of \ as a function of [ From now on, we choose Gaussian correlated random variables for [1 and [2 . Theorem 4.3.1 The correlation coe!cient \ is a dierentiable and increasing function of [ . Proof:

From the partial dierential equation (4.5) of i[1 [2 (x> y; [ ), it follows that ] " ] "     C 2 i[1 [2 (x> y; [ ) CH [\1 \2 ] gxgy = j131 I[1 (x) j231 I[2 (y) C[ CxCy 3" 3"

Partial integration with respect to x and y yields     ] " ] " gj131 I[1 (x) gj231 I[2 (y) CH [\1 \2 ] = i[1 [2 (x> y; [ )gxgy C[ gx gx 3" 3" Applying the chain rule for dierentiation and

gj 1 ({) g{

=

1 j 0 (j 1 ({))

gives

 gj 31 ({)  i[ (x) g{ gj 31 (I[ (x)) = = 0 31  gx g{ j (j (I[ (x))) {=I[ (x) gx Since j 0 ({) and i[ (x) are probability density functions and positive,

CH[\1 \2 ] C[

we have shown that \ is a dierentiable, increasing function of [ .

= ¤

C\ C[

A 0. Hence,

70

Correlation

Since [ 5 [1> 1], \ increases from \ min at [ = 1 to \ max corresponding to [ = 1. In the sequel, we will derive expressions to compute the boundary cases [ = 1 and [ = 1. Theorem 4.3.2 (of Lancaster) For any two strictly increasing real functions W1 and W2 that transform the correlated Gaussian random variables [1 and [2 to the correlated random variables \1 = W1 ([1 ) and \2 = W2 ([2 ), it holds that |\ |  |[ | If two correlated random variables \1 and \2 can be obtained by separate transformations from a bi-variate normal distribution with correlation coefficient [ , the correlation coe!cient \ of the transformed random variables cannot in absolute value exceed [ . The interest of the proof is that it uses powerful properties of orthogonal polynomials and that \ is expanded in a power series in [ in (4.12). Proof: The proof is based on the orthogonal Hermite polynomials Kq ({) (see e.g. Rainville (1960) and Abramowitz and Stegun (1968, Chapter 22)) defined by the generating function " [   Kq ({) wq exp 2{w 3 w2 = q! q=0

(4.8)

  After expanding exp 2{w 3 w2 in a Taylor series and equating corresponding powers in w, we find that [ q2 ] [ (31)n (2{)q32n Kq ({) = q! (4.9) n! (q 3 2n)! n=0 with K0 ({) = 1. The Hermite polynomials satisfy the orthogonality relations ] " 2 h3{ Kq ({) Kp ({) g{ = 0 p 6= q 3"

]

"

3"

I 2 h3{ Kq2 ({) g{ = 2q q! 

These orthogonality relations enable us to expand functions in terms of Hermite polynomials (similar to Fourier analysis). If the expansion of a function i ({), i ({) =

" [

dn Kn ({)

n=0

converges for all {, then it follows from the orthogonality relations that ] " 2 1 h3{ i ({) Kn ({) g{ dn = n I 2 n!  3" The joint normalized Gaussian density function can be expanded (Rainville, 1960, pp. 197—198) in terms of Hermite polynomials   2 2 exp 3 { 32{|+| " 2 (13 ) 2 2 [ q s = h3{ 3| Kq ({) Kq (|) q (4.10) 2 q! 1 3 2 q=0

4.3 The non-linear transformation method

71

    In order for the covariance Cov[\1 \2 ] to exist, both H \12 and H \22 must be finite. Since \m = Wm ([m ) for m = 1> 2, the mean is & % ] " ] " 1 ({ 3 [ )2 H [\m ] = g{ Wm ({)i[m ({) g{ = I Wm ({) exp 3 2 2[ 2[ 3" 3" ] " I 2 1 = I Wm ([ + 2[ x)h3x gx  3" Let "   [ I Wm [ + 2[ x = dn;m Kn (x)

(4.11)

n=0

with dn;m =

1 I 2n n! 

]

"

3"

  I 2 h3{ Wm [ + 2[ { Kn ({) g{

then, since K0 ({) = 1, H [\m ] = d0;m k l The second moment H \m2 follows from (2.34) as   H \m2 =

]

"

3"

1 = I 

]

1

2

(Wm ({)) i[m ({) g{ = I 2[ " I 2 3x2 Wm ([ + 2[ x)h gx

]

"

3"

% Wm2 ({) exp

& ({ 3 [ )2 3 g{ 2 2[

3"

Substituting (4.11) gives ] " " [ " " [   [ 2 1 dp;m dn;m I Kn (x) Kp (x) h3x gx = d2n;m 2n n! H \m2 =  3" n=0 p=0 n=0 which is convergent. Similarly, using (4.4), ] " ] " W1 (x) W2 (y) i[1 [2 (x> y; [ )gxgy H [\1 \2 ] = 3"

]

"

3"

]

5 9 exp 73

"

W1 (x) W2 (y)

(x[ )2 2 [

2[ (y\ )2 (x3[ )(y3\ )+ 2 [ \ \ 2(132 [)

3

6 : 8

s gxgy 1 3 2   {2 32[ {|+| 2 ] " ] "    exp 3  I I (132[ ) W1 [ + 2[ { W2 \ + 2\ | g{g| = t 3" 3"  1 3 2[ % & ] " ] " " " [ [ 1 {2 3 2[ {| + | 2   = t dn;1 dp;2 Kn ({) Kp (|) exp 3 g{g| 1 3 2[ 3" 3"  1 3 2 n=0 p=0 =

3"

3"

2[ \

[

Using (4.10), H [\1 \2 ] =

] " ] " " " " [ [ q 2 2 1 [ [ dn;1 dp;2 h3{ Kn ({) Kq ({) g{ h3| Kp (|) Kq (|) g| q  n=0 2 q! 3" 3" p=0 q=0

72

Correlation

Introducing the orthogonality relations for Hermite polynomials leads to H [\1 \2 ] =

" [

dq;1 dq;2 2q q!q [

q=0

The correlation coe!cient becomes S" S" q q dq;1 dq;2 2q q!q q=0 dq;1 dq;2 2 q![ 3 d0;1 d0;2 [ \ = tS tS = tS q=1 S" " " " 2 2 2 2 2 2 2n n! n n n n! d 2 n! 3 d d 2 n! 3 d d 2 d n=0 n;1 n=1 n;2 n=1 n;1 n=1 n;2 0;1 0;2 I I S S" 2 2 Denote q = dq;1 2q q! and q = dq;2 2q q!, then Var[\1 ] = " n=1 n and Var[\2 ] = n=1 n . Since the linear correlation coe!cient  ([> \ ) equals the correlation coe!cient of the corresponding normalized random with mean zero and variance 1, as shown in Section 2.5.3, we may S S"variable 2 2 choose " n=1 n = n=1 n = 1 such that \ =

" [

q q q [

(4.12)

q=1

If 21 = 1 and 12 = 1, then |\ | = |[ | because all other n and n must then vanish. In all other cases, either 21 ? 1 or 12 ? 1 or both, such that \ = 1 1 [ +

" [

q q q [

q=2

and

y y   x" x" " " [  [ x[ x[  q q q 2 | |q q q [  $ |q q | |[ | $ w 2q |[ | w q  [   q=2

q=2

q=2

where we have used the Cauchy-Schwarz inequality partial summation, " [

2q |[ |q = (1 3 |[ |)

q=2

because

Sq

n=2

2n ?

S"

n=2

S

de $

q=2

sS

d2

sS

e2 (see Section 5.5). By

" [ q [

 |[ |2    2n |[ |q $ (1 3 |[ |) 1 3 21 = 1 3 21 |[ |2 1 3 | | [ q=2 n=2

2n = 1 3 21 . Thus

  t t |\ | $ |[ | |1 1 | + 1 3 21 1 3 12 |[ | Finally, for 21 $ 1 and 12 $ 1, the inequality |1 1 | + Lancaster’s theorem because |[ | $ 1.

t t 1 3 21 1 3 12 $ 1 holds. This proves

¤

4.3.2 Boundary cases Let us investigate some cases for special values of [ . 1. [ = 0. Since uncorrelated Gaussian random variables ([ = 0) are independent, also \1 = j131 (I[1 ([1 )) and \2 = j231 (I[2 ([2 )) are independent such that \ = 0. Hence, uncorrelated Gaussian random variables with [ = 0 lead to uncorrelated random variables \1 and \2 with \ = 0.

4.3 The non-linear transformation method

73

2. [ = 1. Perfect positively correlated Gaussian random variables [1 = [2 = [ have joint distribution  ¸ 1 (x  [ )2 i[1 [2 (x> y; 1) = s exp  (x  y) 2 2[ 2[ which follows from Pr [[1  {1 > [2  {2 ] = Pr [[  {1 > [  {2 ] = Pr [[  {] with { = max({1 > {2 ). In that case, Z " j131 (I[ (x)) j231 (I[ (x)) gI[ (x) H [\1 \2 ] = 3" 1

Z =

0

j131 ({) j231 ({) g{

(4.13)

which may lead to \ max ? 1 depending on the specifics of j1 and j2 . By transforming { = j1 (x), we obtain Z j31 (1) 1 xj231 (j1 (x)) j10 (x)gx H [\1 \2 ] = j131 (0)

which shows that, if j1 = j2 = j, Z j31 (1) £ ¤ x2 j 0 (x)gx = H \ 2 H [\1 \2 ] = j 31 (0)

Hence, if \1 and \2 have the same distribution function j as \ , the case [ = 1 leads to £ ¤ H \ 2  (H [\ ])2 =1 \ = Var [\ ] 3. [ = 1. Perfect negatively correlated Gaussian random variables [1 = [2 = [ have joint distribution  ¸ 1 (x  [ )2 i[1 [2 (x> y; 1) = s exp  (x + y) 2 2[ 2[ which follows from the symmetry relations (4.6). In that case, Z " j131 (I[ (x)) j231 (I[ (x)) gI[ (x) H [\1 \2 ] = Z3" " j131 (I[ (x)) j231 (1  I[ (x)) gI[ (x) = 3" 1

Z =

0

j131 ({) j231 (1  {) g{

(4.14)

which may lead to \ min A 1, depending on the specifics of j1 and j2 .

74

Correlation

4.4 Examples of the non-linear transformation method 4.4.1 Correlated uniform random variables Let us first focus on the relation between [ and X . Since H [X ] = 12 and 1 X2 = 12 , the definition of the linear correlation coe!cient (2.58) gives H [X1 X2 ] 

X =

1 4

1 12

where, using (2.61), H [X1 X2 ] = H [I[1 ([1 )I[2 ([2 )] Z "Z " = I[1 (x)I[2 (y)i[1 [2 (x> y; [ )gxgy 3"

3"

In the case of Gaussian correlated random variables specified by (3.20) and (4.4), we must evaluate the integral ]

]

"

H [X1 X2 ] =

gx 3"

]

"

gy

5

gw h

3"

9 exp 9 73

1

H [X1 X2 ] =

(2)2

gx0

x3[1 [1

]

3"

"

]

3"

, w0 =

gy 0

]

3"

x0

3

y

g h

( [ )2 2 22 [2

3"

(x3[1 )(y3[2 )+ 2(132 [)

2 2 (2)2 [ 1 [2

Substituting successively x0 =

"

(w[ )2 1 22 [1

 2 x[ 2[ 1 3 2  [1 [2 [1

×

]

3

x

w3[1 [1

w02 h3 2

, y0 =

gw0

]

3"

6 : : 8

t 1 3 2[

y3[2 [2

y0

 2 y[ 2 2  [2

, 0 =

 02 h3 2

3"

 3[2 [2

, we obtain

  02 0 0 02 [ x y +y exp 3 x 32 2(132 [) g 0 t 1 3 2[

We now use the partial dierential equation (4.5),    4  3 02 0 0 02 x02 32[ x0 y 0 +y02 C2 exp 3 x 32[ x 2y +y exp 3 Cx0 Cy 0 2(13[ ) 2(132 F C E [) E F= t t D C C 1 3 2[ 1 3 2[ such that CH [X1 X2 ] = C

]

"

gx0

]

3"

"

gy0

]

3"

x0

h3

w02 2

3"

gw0

]

y0

3"

h3

 02 2

g 0

C2 Cx0 Cy 0

   02 0 0 02 exp 3 x 32[ x 2y +y 2(13[ ) t 2 (2) 1 3 2[

Partial integration in the last integral y 0 &$ # % C2 x02 3 2[ x0 y0 + y 02   exp 3 L2 = gy h g Cx0 Cy 0 2 1 3 2[ 3" 3" &$ # % ] " y2 C x02 3 2[ x0 y + y2   = exp 3 gyh3 2 Cx0 2 1 3 2[ 3" ]

"

0

]

y0

02

3 2

0

4.4 Examples of the non-linear transformation method

75

yields CH [X1 X2 ] = C

]

"

y2 gyh3 2

]

3"

"

gx0

3"

]

x0

w02 h3 2

gw0

C Cx0

3"

   02 0 0 02 exp 3 x 32[ x 2y +y 2(13[ ) t (2)2 1 3 2[

and similarly in the x0 integral, %    & ] " ] " x2 2 3 2[ 3 2[ xy + y 2 2 3 2[ CH [X1 X2 ] 1   t = gy gx exp 3 C 2 1 3 2[ 3" (2)2 1 3 2[ 3"     5  (432 ) $2 6  # ] " y2 ] " exp 3 2 23[2 ( 2 3 2[ y  [) [  x3   8 = gy t gx exp 73  2 3 2[ 2 1 3 2[ 3" 3" (2)2 1 3 2[ v  v   2 2 3 2[ 2 1 3 2[ 1 1 1     = t t = 2 2 2 2 4 3  2 3  2 [ [ (2) 1 3 [ 4 3 2[ Thus, we find that 6 1 CX = t C[  4 3 2 [ or that X =

6 

]

  1 6 [ t +f + f = arcsin  2 4 3 2[

It remains to determine the constant f. We have shown in Section 4.3.2 that random variables generated from uncorrelated Gaussian random variables are also uncorrelated implying that X = 0 if [ = 0 and, hence, that the constant f = 0. This finally results in   6 [ (4.15) X = arcsin  2

In summary, two uniform correlated random variables X1 and X2 with correlation coe!cient X are found by transforming two Gaussian correlated ¡ ¢ random variables [1 and [2 with correlation coe!cient [ = 2 sin 6X . Equation (4.15) further shows that X = ±1 if [ = ±1, which indicates that the whole range of the correlation coe!cient X is possible. 4.4.2 Correlated exponential random variables In Section 3.2.1, we have seen that, if X is a uniform random variable on [0,1], j 31 (X ) =  1 log X is an exponential random variable with mean 1 . The correlation coe!cient for two exponential random variables, \1 and \2 , with mean 11 and 12 respectively, is \ =

H [\1 \2 ]  1 1 2

1 1 2

= H [1 2 \1 \2 ]  1

76

Correlation

As above, we generate \1 =  11 log I[1 ([1 ) and \2 =  12 log I[2 ([2 ), where [1 and [2 are correlated Gaussian random variables with correlation coe!cient [ . Then, 1 2 H [1 2 \1 \2 ] = H [log I[1 ([1 ) log I[2 ([2 )]   Z 1" 2Z " log I[1 (x) log I[2 (y)i[1 [2 (x> y; [ )gxgy = 3"

3"

In the general case for [ 6= 0, the previous method can be followed, which yields after substitution towards normalized variables,   2 2 ] x  ] y  exp 3 x 32[ xy+y ] " ] " 2 2 2 2 13 ( ) w  [ gx gy log h3 2 gw log h3 2 g t H [1 2 \1 \2 ] = 3" 3" 3" 3" (2)2 1 3 2[ Unfortunately, we cannot evaluate this integral analytically.

Let us compute the upper bound \ max from (4.13) with j131 ({) =  11 log { and j231 ({) =  12 log {, Z 1 log2 {g{ = 2 H [1 2 \1 \2 ; [ = 1] = 0

and thus \ max = 1. The lower boundary \ min follows from (4.14) as1 , Z 1 2 H [1 2 \1 \2 ; [ = 1] = log { log(1  {)g{ = 2  6 0 2

Here, we find \ min = 1  6 = 0.644 934===. In summary, exponential correlated random variables can be generated from Gaussian correlated random variables, but the correlation coe!cient 1

Substituting the Taylor expansion log(1 3 {) = 3 ]

1

log { log(1 3 {)g{ = 3 0

and

]

1

]

"

{n log {g{ = 3

0

S"

{n n=1 n

" [ 1 n n=1

]

1

gives {n log {g{

0

h3(n+1)x xgx = 3

0

1 (n + 1)2

Thus, ]

1

log { log(1 3 {)g{ = 0

Since ]

1 n(n+1)2

=

1 n

3

1 n+1

1

log { log(1 3{)g{ = 0

3

" [ n=1

1 n(n + 1)2

1 , (n+1)2

" " " " " [ [ [ 1 [ 1 [ 1 1 1 2 3 3 = 13 = 23 = 2 3(2) = 2 3 2 2 2 n n=2 n n=2 n n n 6 n=1 n=2 n=1

4.4 Examples of the non-linear transformation method

77

h i 2 \ is limited to the interval 1  6 > 1 . As explained in the introduction of Section 4.2, the exponential random variables are positive with infinite range for which not all negative correlations are possible. The analysis demonstrates that it is not possible to construct two exponential random 2 variables with correlation coe!cient smaller than \ min = 1  6 ' 0=645.

4.4.3 Correlated lognormal random variables Two correlated lognormal random variables \1 and \2 with distribution specified in (3.43) can be constructed directly from two correlated Gaussian random variables [1 and [2 . In particular, let \1 = hd1 [1 and \2 = hd2 [2 . The explicit scaling parameters can be used to determine the desired mean. From (4.7), Z

"

Z

"

H [\1 \2 ] = 3"

3"

hd1 x hd2 y i[1 [2 (x> y; [ )gxgy

 ¸ 12 2 22 2 = exp d1 1 + d2 2 + d1 + d1 d2 [ 1 2 + d2 2 2 where the Laplace transform (4.2) for q = 2 has been used. Invoking (3.45) and (3.46) with m $ dm m and m2 $ d2m m2 , the correlation coe!cient \ is hd1 1 d2 2 [  1 \ = r³ ´³ ´ 2 2 2 2 hd1 1  1 hd2 2  1

(4.16)

If at least one (but not all) of the quantities 1 > 2 > d1 or d2 grows large, \ tends to zero irrespective of [ . Thus even if [1 and [2 and, hence also \1 and \2 , have the strongest kind of dependence possible, i.e. [ = ±1, the correlation coe!cient \ can be made arbitrarily small. In case d1 1 = d2 2 = , (4.16) reduces to 2

\ =

h [  1 h2  1 2

We observe that \ max = 1, while \ min = h3 A 1 for  A 0; again a manifestation that for positive random variables with infinite range not all negative correlations are possible.

78

Correlation

4.5 Linear combination of independent auxiliary random variables In spite of the generality of the non-linear transformation method, the involved computational di!culty suggests us to investigate simpler methods of construction. It is instructive to consider two independent random variables Y and Z with known probability generating functions *Y (}) and *Z (}) respectively. In the discussion of the uniform random variable in Section 3.2.1, it was shown how to generate by computer an arbitrary random variable from a uniform random variable. We thus assume that Y and Z can be constructed. Let us now write [ and \ as a linear combination of Y and Z, [ = d11 Y + d12 Z + e1 \ = d21 Y + d22 Z + e2 which is specified by the matrix  D=

d11 d12 d21 d22

¸

and compute the covariance defined in (2.56), Cov [[> \ ] = H [[\ ]  [ \ h £ ¤ i = d11 d21 H Y 2  (H [Y ])2 + (d11 d22 + d12 d21 ) H [Y Z ] h £ i ¤  (d11 d22 + d12 d21 ) H[Y ] H[Z ] + d12 d22 H Z 2  (H[Z ])2 Since X and Y are independent, H [Y Z ] = H [Y ] H [Z ], and with the definition of the variance (2.16) and denoting Y2 =Var[Y ] and similarly for Z , we obtain 2 Cov [[> \ ] = d11 d21 Y2 + d12 d22 Z

In a same way, we find 2 2 [ = d211 Y2 + d212 Z 2 \2 = d221 Y2 + d222 Z

(4.17)

such that the correlation coe!cient, in general, becomes 2 d11 d21 Y2 + d12 d22 Z q = q 2 2 d211 Y2 + d212 Z d221 Y2 + d222 Z

which is independent of the constants e1 and e2 since for a centered moment H ([  H [[])2 = H ([ + e  H [[ + e])2 .

4.5 Linear combination of independent auxiliary random variables

79

In order to achieve our goal of constructing two correlated random variables [ and \ , we can choose the coe!cients of the matrix D to obtain an expression as simple as possible. If we choose [ = Y or d11 = 1, d12 = e1 = 0, the correlation coe!cient reduces to 2 d21 [ 1 = q q =r 2 2 2 + d2  2 d2 Z [ d221 [ 22 Z 1 + d22 2 2 21

[

By rewriting this relation, we obtain d21 = ± If we choose d22 =

d Z p 22 [ 1  2

p 1  2 , the random variables [ and \ are specified as [=Y \ =±

p Z Y + 1  2 Z + e2 [

2 =  2 and  2 =  2 . Finally, and the corresponding variances (4.17) are [ Y \ Z we require that H [Z ] = Z = 0, which specifies

e2 = H [\ ]

\ H [[] [

If Z is a zero mean random variable with standard deviation Z = \ , the random variables [ and \ are correlated with correlation coe!cient  p \ \ \ =± [ + 1  2 Z + \

[ (4.18) [ [ In the sequel, we take the positive sign for . Let us now investigate what happens with the distribution functions of [¤ £ 3}[ and \ . Using the pgfs for continuous random variables * (}) = H h [ ¤ £ and *\ (}) = H h3}\ , the last relation (4.18) becomes £ ¤ 3} H h3}\ = h

   \ 3  \ [ [

h i h s 2 i  3} \ [ H h [ H h3} 13 Z

because Y = [ and Z are independent, or ¶ µ   ´ ³p  \ 3} \ 3  \ [ [ *\ (}) = h *[ } *Z 1  2 } [

(4.19)

In order to produce two random variables [ and \ that are correlated with correlation coe!cient , the pgf of the zero mean random variable Z with

80

Correlation

variance \2 must obey, h

   \ 3  \ [ [ I } 132

µ

*Z (}) = *[

[

µ *\

\ s

13

¶ s} 2 13 ¶

(4.20)

} 2

which can be written in terms of the translated random variables \ 0 = \  \ and [ 0 = [  \ [ , µ ¶ } *\ 0 s 2 13 ¶ µ *Z (}) = \ s *[ 0 } 2 [

13

This form shows that, if [ 0 and \ 0 have a same distribution, Z possesses, in general, a dierent distribution. Only the pgf of a Gaussian (with zero mean) obeys the functional equation µ ¶ } i s 2 13 ¶ i (}) = µ  i s 2} 13

The joint probability generating function follows from (2.61) as Z "Z " ¤ £ h3}1 {3}2 | i[\ ({> |)g{g| *[\ (}1 > }2 ) = H h3}1 [3}2 \ = 3"

and the inverse is i[\ ({> |) =

1 (2l)2

Z

f1 +l" Z f2 3l"

f1 3l"

f2 3l"

3"

h}1 {+}2 | *[\ (}1 > }2 )g}1 g}2

(4.21)

Using (4.18), we have

   ¸ s 2 i h  3 }1 +}2  \ [ [ [ H h *[\ (}1 > }2 ) = h H h3}2 13 Z µ ¶   ³ p ´  \ 3}  3 \  *Z }2 1  2 (4.22) = h 2 \ [ [ *[ }1 + }2 [    3}2 \ 3  \ [

Introduced into the complex double integral (4.21), the joint probability density function of the two correlated random variables can be computed. The main deficiency of the linear combination method is the implicit assumption that any joint distribution function i[\ ({> |) can be constructed from two independent random variables [ and Z . The corresponding joint

4.5 Linear combination of independent auxiliary random variables

81

pgf (4.22) possesses a product form that cannot always be made compatible with the form of an arbitrary pgf *[\ (}1 > }2 ). The examples below illustrate this deficiency.

4.5.1 Correlated Gaussian random variables

¤ £ If [ and \ are Gaussian random variables with Laplace transform H h3}\ given in (3.22), the expression (4.20) for *Z (}) becomes µ 2 ¶ ¸ \ *Z (}) = exp }2 2 which shows that Z is also a Gaussian random variable with mean Z = 0 and standard deviation Z = \ . Further, the joint pgf follows from (4.22) as  ¸ 2 £ 3}1 [3}2 \ ¤ [ \2 2 2 } + }1 }2 \ [ + } = exp }2 \  [ }1 + H h 2 1 2 2 Since

¸ ¸  2 2 ¤ [ \2 2 1 £ }1 [ \ [ 2 } + }1 }2 \ [ + } = }1 }2 [ \ \2 }2 2 1 2 2 2 ¤ £ formula (4.2) indicates that H h3}1 [3}2 \ is the two dimensional pgf of a joint Gaussian with pdf (4.4). The linear combination method thus provides the exact results for correlated Gaussian random variables.

4.5.2 Correlated exponential random variables Let [ and \ be two correlated, exponential random variables with rate { and | . Recall that H [[] = [ = 1{ . Using the Laplace transform (3.16) in (4.20), we obtain *Z (}) = h |

13 I

132

}

1+ 1+

s}

132 } s | 132 |

The corresponding probability distribution function follows from (2.38) as

IZ (w) =

1 2l

Z

f+l"

f3l"

1+ 1+

s}

132 s} | 132 |

 } w+

h

 |

13 I

}

132

g}

fA0

82

Correlation

Define the normalized time W = 1 IZ (w) = 2l

Z

|

s1

f+l"

f3l"

132

, then

1 + W } h}(w+(13)W ) g} 1 + }W }

Since w + (1  ) W A 0, the contour can be closed over the negative Re(}) plane encircling the poles at } =  W1 and } = 0. By Cauchy’s residue theorem, ¢ ¡ (1 + W }) } + W1 }(w+(13)W ) (1 + W }) } }(w+(13)W ) IZ (w) = lim + lim h h } |) (5.1) {|  2 ¡s s ¢2 {  |  0, they masterly extend this which directly follows from relation to the theorem of the arithmetic and geometric mean in several real variables {n , q q Y X {tnn  tn {n (5.2) min({> |) 

Pq

n=1

n=1

where n=1 tn = 1. They further move to the inequalities of CauchySchwarz, of Hölder, of Minkowski and many more. Only a few inequalities are reviewed here and we recommend the classic treatise on inequalities by Hardy, Littlewood and Polya for those who search for more depth, elegance and insight.

5.1 The minimum (maximum) and infimum (supremum) Since these concepts will be frequently used, we explain the dierence by concentrating on the minimum and infimum (the maximum and the supremum follow analogously). Let be a non-empty subset of R. The subset

1

The arithmetic-geometric mean P({> |) is the limit for q < " of the recursion {q = I 1 ({q31 + |q31 ), which is an arithmetic mean, and |q = {q31 |q31 , which is a geometric 2 mean, with initial values {0 = { and |0 = |. Gauss’s famous discovery on intriguing properties of P({> |) (which lead e.g. to very fast converging series for computing ) is narrated in a paper by Almkvist and Berndt (1988).

83

84

Inequalities

is said to be bounded from below by P if there exists a number P such that, for all { 5 holds that {  P . The largest lower bound (largest number P ) is called the infimum and is denoted by inf ( ). Further, if there exists an element p 5 such that p  { for all { 5 , then this element p is called the minimum and is denoted by min ( ). If the minimum min ( ) exists, then min ( ) = inf ( ). However, the minimum does not always exists. The classical example is the open interval (d> e), where inf ((d> e)) = d, but the minimum does not exist because d 5 @ (d> e). On the other hand, for the closed interval [d> e], we have that inf ([d> e]) = min ([d> e]) = d. This example also illustrates that every finite non-empty subset of R has a minimum. 5.2 Continuous convex functions A continuous function i ({) that satisfies for x and y belonging to an interval L, µ ¶ x+y i (x) + i(y)  i 2 2 is called convex in that interval L. If i is convex, i is concave. Hardy et al. (1999, Section 3.6) demonstrate that this condition is fundamental from which the more general condition2 ! Ã q q X X i tn {n  tn i ({n ) (5.3) n=1

Pq

n=1

where n=1 tn = 1, can be deduced. Moreover, they show that a convex function is either very regular or very irregular and that a convex function that is not “entirely irregular” is necessarily continuous. Current textbooks, in particular the book by Boyd and Vandenberghe (2004), usually start with the definition of convexity from (5.3) in case q = 2 where t1 = 1  t2 = t and 0  t  1 as i (tx + (1  t)y)  ti (x) + (1  t)i (y)

(5.4)

where x and y can be vectors in an p-dimensional space.Fig_convex Geometrically with p = 1 as illustrated in Fig. 5.1, relation (5.4) shows that each point on the chord between (x> i (x)) and (y> i (y)) lies above the 2

The convexity concept can be generalized (Hardy et al., 1999, Section 98) to several variables in which case the condition (5.3) becomes $ # [ [ [ tn {n > tn |n $ tn i ({n > |n ) i n

n

n

5.2 Continuous convex functions

85

f(x)

f

f(v) c2 c1

f(u)

u a

a'

b' b v

x

Fig. 5.1. The function i is convex between x and y.

curve i in the interval L. The more general form (5.3) asserts that the centre of gravity of any number of arbitrarily weighted points of the curve lies above or on the curve. Figure 5.1 illustrates that for any convex function i and points d> d0 > e> e0 5 [x> y] such that d  d0  e0 and d ? e  e0 , the chord f1 over (d> e) has a smaller slope than the chord f2 over (d0 > e0 ) or, 0 (d) (d0 )  i (ee)3i = Suppose that i ({) is twice dierentiable equivalently, i (e)3i 0 3d0 e3d in the interval L, then a necessary and su!cient condition for convexity is i 00 ({)  0 for each { 5 L. This theorem is proved in Hardy et al. (1999, pp. 76—77). Moreover, they prove that the equality in (5.3) can only occur if i({) is linear. Applied to probability, relation (5.3) with tn = Pr [[ = n] and {n = n is written with (2.12) as i (H [[])  H [i ([)]

(5.5)

and is known as Jensen’s inequality. The Jensen’s inequality (5.5) also hold for continuous random variables. Indeed, if i is dierentiable and convex, then i ({)  i (|)  i 0 (|)({  |). Substitute { by the random variable [ and | = H [[], then i ([)  i(H [[])  i 0 (H [[])({  H [[]) After applying the expectation operator to both sides, we obtain (5.5). An important application of Jensen’s inequality is obtained for i ({) = h3}{ with real } as £ ¤ h3}H[[]  H h3}[ = *[ (})

86

Inequalities

Any probability generating function *[ (}) is, for real }, bounded from below by h3}H[[] . A continuous analog of (5.3) with i ({) = h{ (and similarly for i ({) =  log {)  ¸ Z y Z y 1 1 exp i ({)g{  hi ({) g{ yx x yx x can be regarded as a generalization of the inequality between arithmetic and geometric mean.

5.3 Inequalities deduced from the Mean Value Theorem The mean value theorem (Whittaker and Watson, 1996, p. 65) states that if j({) is continuous on { 5 [d> e], there exists a number  5 [d> e] such that Z e j(x)gx = (e  d)j() d

or, alternatively, if i ({) is dierentiable on [d> e], then i (e)  i (d) = (e  d)i 0 () (5.6) R{ The equivalence follows by putting i ({) = d j(x)gx= It is convenient to rewrite this relation for 0    1 as i ({ + k)  i ({) = ki 0 ({ + k) In this form, the mean value theorem is nothing else than a special case for q = 1 of Taylor’s theorem (Whittaker and Watson, 1996, p. 96), i ({ + k)  i ({) =

q31 X n=1

i (n) ({) n kq (q) k + i ({ + k) n! q!

(5.7)

An important application of Taylor’s Theorem (or of the mean value theorem) to the exponential function gives a list of inequalities. First, h{ = 1 + { +

{2 { h 2

and, since h{ A 0 for any finite {, we have for any { 6= 0, h{ A 1 + {

(5.8)

5.4 The Markov and Chebyshev inequalities

87

A direct generalization follows from Taylor’s Theorem (5.7), {

h =

q31 X

{n {q { + h n! q!

n=0

such that, for q = 2p and any {, h{ A

2p31 X n=0

{n n!

and, for q = 2p + 1, h{ A

2p n X { n=0

h{ ?

2p n X { n=0

Second, estimates of the product from (5.8) as3 q Y

{A0

n!

{?0

n! Qq

n=0 (1 + dn {)

(1 + dn {) ? exp

à q X

n=0

where dn { 6= 0 are obtained ! dn {

n=0

5.4 The Markov and Chebyshev inequalities Consider first a non-negative random variable [. The expectation reads Z d Z " Z " {i[ ({)g{ = {i[ ({)g{ + {i[ ({)g{ H [[] = 0 d Z0 " Z "  {i[ ({)g{  d i[ ({)g{ = d Pr [[  d] d

d

Hence, we obtain the Markov inequality Pr [[  d]  3

H [[] d

(5.9)

A tighter bound relation indicates   SqThe above Tqis obtained if all dn A 0 (e.g. dn is a probability). that j({) = n=0 (1 + dn {) is smaller than i ({) = exp { n=0 dn for any { 6= 0 and j(0) = i (0) = 1. Further, from (1 + dn {) ? hdn { it can be verified that, S for all Taylor q n coe!cients 1 ? n $ q holds that 0 ? jn $ in and j2 ? i2 such that j({) = n=0 jn { ? Sq S S " q n ? n for { A 0. Thus, for { = 1, we have j(1) ? i { i { i or n=0 n n=0 n n=0 n q \

(1 + dn ) ?

n=0

q [ 1 n! n=0

#

q [

n=0

$n dn

88

Inequalities

Another proof of the Markov inequality follows after taking the expectation of the inequality d1[Dd  [ for [  0. The restriction to non-negative random variables can be circumvented by considering the random variable [ = (\  H [\ ])2 and d = w2 in (5.9), h i h i H (\  H [\ ])2 Var [\ ] Pr (\  H [\ ])2  w2  = w2 w2 From this, the Chebyshev inequality follows as Pr [|[  H [[]|  w] 

2 w2

(5.10)

The Chebyshev inequality quantifies the spread of [ around the mean H [[]. The smaller , the more concentrated [ is around the mean. Further extensions of the Markov inequality use the equivalence between the events {[  d} / {j([)  j(d)} where j is a monotonously increasing function. Hence, (5.9) becomes Pr [[  d] 

H [j([)] j(d)

H [[ n ] For example, if j({) = {n , then Pr [[  d]  dn . An interesting application of this idea is based on the equivalence of the events {[  H [[]+w} / {hx[  hx(H[[]+w) } provided x  0. For x  0, h i ¤ £ Pr [[  H [[] + w] = Pr hx[  hx(H[[]+w)  h3x(H[[]+w) H hx[ (5.11)

where in the last step Markov’s inequality £ (5.9) ¤ has been used. If the generating function or Laplace transform H hx[ is known, the sharpest bound is obtained by the minimizer xW in x of the right-hand side because (5.11) holds for any x A 0. In Section 5.7, we show that this minimizer xW obeying Re x A 0 indeed exists for probability generating functions. The resulting inequality h W i W (5.12) Pr [[  H [[] + w]  h3x (H[[]+w) H hx [ is called the Cherno bound. The Cherno bound of the binomial distribution Let [ denote a binomial random variable hwith probability generating function given by i £ x[ ¤ [ x = H (h ) = (t + shx )q . Then, with H [[] = qs, (3.2) such that H h £ ¤ x h3x(H[[]+w) H hx[ = h3x(qs+w)+q log(t+sh )

5.4 The Markov and Chebyshev inequalities

£ x[ ¤¯¯ g2 3x(H[[]+w) Provided gx h H h ¯ 2

x=xW

89

A 0, the minimum xW is solution of

g 3x(H[[]+w) £ x[ ¤ =0 H h h gx Explicitly, µ ¶ qshx g 3x(H[[]+w) £ x[ ¤ x = h3x(qs+w)+q log(t+sh ) (qs + w) + H h h gx t + shx from which xW follows using t = 1  s as µ ¶ qst + tw xW = log qst  sw Hence, ³ h i 1 W W h3x (H[[]+w) H hx [ = ³ 1+

w qt w qs

´w3qt ´w+qs

£ W ¤ W For large q, but s and w fixed, we observe4 that h3x (H[[]+w) H hx [ = ¡ ¢¢ w2 ¡ w2 h3 qst 1 + R q1 . Since Var[[] = qst and by denoting | 2 = Var[[] , we find that the asymptotic regime for large q, # " |[  H [[]| 2  |  h3| Pr p (5.13) Var [[] is in agreement with the Central Limit Theorem 6.3.1. The corresponding Chebyshev inequality, # " 1 |[  H [[]| |  2 Pr p | Var [[] is considerably less tight for the binomial distribution than the Cherno bound (5.13). More advanced and sharper inequalities than that of Chebyshev are surveyed by Janson (2002). 4

Write h3x

 (H[[]+w)

     k  l w w 3 (w + qs) log 1 + H hx [ = exp (w 3 qt) log 1 3 qt qs

and use the Taylor expansion of log (1 ± {) around { = 0.

90

Inequalities

5.5 The Hölder, Minkowski and Young inequalities The Hölder inequality is ³X ´ ³X ´ ³X ´ X d } d e · · · }   e ···

+ +··· = 1

Let d = [ and e = \ , and further s = 1 A 1 and t = 1 A 1 such that 1 1 s + t = 1, then we obtain as a frequently used application, H [[\ ]  (H [[ s ])1@s (H [\ t ])1@t

(5.14)

The Hölder inequality can be deduced from the basic convexity inequality (5.4). Since  log { is a convex function for real { A 0, the basic convexity inequality (5.4) is with 0    1, log (x + (1  )y)   log(x) + (1  ) log(y) After exponentiation, we obtain for x> y A 0 a more general inequality than (5.1), which corresponds to  = 12 , x y 13  x + (1  )y s Sq|{m | s |{ m=1 m |

Substitute x = Ã

|{ |s Pq m s m=1 |{m |

! Ã

and y =

|| |t Pq m t m=1 ||m |

t Sq||m | t , || m=1 m |

!13

then

|{m |s ||m |t P   Pq + (1  ) q s t m=1 |{m | m=1 ||m |

and summing over all m yields q X

|{m |s ||m |t(13)

3 4 3 413 q q X X C |{m |s D C ||m |t D

m=1

m=1

By choosing s = 1 and t = s A 1 and 1s + 1t = 1, q X m=1

1 13 ,

we arrive at the Hölder inequality with

3 41 3 41 s t q q X X s t |{m |m |  C |{m | D C ||m | D m=1

(5.15)

m=1

(5.16)

m=1

A special important case of the Hölder inequality (5.14) for s = t = 2 is the Cauchy—Schwarz inequality, £ ¤ £ ¤ (5.17) (H [[\ ])2  H [ 2 H \ 2 It is of interest to mention that the Hölder inequality is of a general type in

5.5 The Hölder, Minkowski and Young inequalities

91

the following sense (Hardy et al., 1999, Theorem 101 (p. 82)). Suppose that i ({) is convex (such Rthat the inverse j({) =Ri 31 ({) is also convex) and that { { i (0) = 0. If I ({) = 0 i (x)gx and J({) = 0 j(x)gx, and if à q à q ! ! q X X X tn dn en  I 31 tn I (dn ) J31 tn J(en ) Pq

n=1

n=1

n=1

with n=1 tn = 1 holds for all positive dn and en , then i ({) = {u and the above inequality is Hölder’s inequality. The next inequalities are of a dierent type. For s A 1, the Minkowski inequality is (H [|[ + \ |s ])1@s  (H [|[|s ])1@s + (H [|\ |s ])1@s

(5.18)

or, written algebraically, 41 3 41 3 41 3 s s s q q q X X X s s s C |{m + |m | D  C |{m | D + C ||m | D m=1

m=1

m=1

Suppose that i ({) is continuous and strictly increasing for {  0 and i(0) = 0. Then the inverse function j({) = i 31 ({) satisfies the same conditions. The Young inequality states that for d  0 and e  0 holds that Z e Z d i (x)gx + j(x)gx (5.19) de  0

0

with equality only if e = i (d). The Young inequality follows by geometrical consideration. The first integral is the area under the curve | = i ({) from [0> d], while the second is the area under the curve { = j(|) = i 31 (|) from [0> e]. Applications of the Cauchy—Schwarz inequality ¤ will demonstrate that both the generating function *[ (}) = £1. We H h3}[ and its logarithm O[ (}) = log(*[ (})) are convex functions of }. First, the ¤ derivative is continuous and non-negative because £ second *00[ (}) = H [ 2 h3}[  0. Further, since O00[ (})

*[ (})*00[ (})  (*0[ (}))2 = *2[ (})

it remains to show that *[ (})*00[ (})(*£0[ (}))2  ¤ 0. From Cauchy—Schwarz } inequality (5.17) applied to *0[ (}) = H [h3}[ with [ $ h3 2 [ and \ = ¤¢ ¤ £ ¤ ¡ £ £ } 2 [h3 2 [ , we obtain (*0[ (}))2 = H [h3}[  H h3}[ H [ 2 h3}[ = *[ (})*00[ (}). Hence, O00[ (})  0.

92

Inequalities

2. Let \ = 1[A0 in (5.17) while [ is a non-negative random variable, then with (2.13), £ ¤ £ ¤ (H [[])2  H [ 2 H [1[A0 ] = H [ 2 (1  Pr [[ = 0]) such that an upper bound for Pr [[ = 0] is obtained, Pr [[ = 0]  1 

(H [[])2 H [[ 2 ]

(5.20)

5.6 The Gauss inequality In this section, we consider a continuous random variable [ with even probability density function, i.e. i[ ({) = i[ ({), which is not increasing for { A 0. A typical example of such random variables are measurement errors due to statistical fluctuations. In his epoch-making paper, Gauss (1821) established the method of the least squares (see e.g. Section 2.2.1 and 2.5.3). In that same paper, Gauss (1821, pp. 10-11) also stated and proved Theorem 5.6.1, which is appealing because of its generality. We define the probability p as Z  i[ (x) gx (5.21) p = Pr [  [  ] = 3

p where  = Var [[] is the standard deviation.

Theorem 5.6.1 (Gauss) If [ is a continuous random variable with even probability density function, i.e. i[ ({) = i[ ({), which is not increasing for { A 0, then s if p ? 23 then  q p 3 if p = 23 then   43 2 if p A 23 then  ? 3I13p and, conversely,

q if  ? if  A

I 3

4

then p 

4 3

then p  1 

q3

4 2

Given a bound on the probability p, Gauss’s Theorem 5.6.1 bounds the extent of the error [ around its mean zero in units of the standard deviation  or, equivalently, it provides bounds for the normalized random variable

5.6 The Gauss inequality

93

[ W = [3H[[] . The proof of this theorem only uses real function theory and  is characteristic for the genius of Gauss. Proof: Consider the inverse function { = j (|) of the integral | = I[ (3{). An interesting general property of the inverse function is ]

]

1

j 2 (x) gx =

"

3"

0

U{ 3{

i[ (x) gx = I[ ({) 3

{2 i[ ({) g{

  which is verified by the substitution { = j (x). Since H [[] = 0 and Var[[] = H [ 2 , we have ]

1

j 2 (x) gx =  2 = Var [[]

(5.22)

0

Beside j (0) = 0, the derivative j 0 (|) =

1 1 = 0 ({) 3 I 0 (3{) I[ i ({) + i[ (3{) [ [

is increasing from | = 0 until | = 1 because i[ ({) attains a maximum at { = 0 and is not 00 increasing for { A 0. Hence, j (|) D 0. From the dierential   00 g |j0 (|) = j 0 (|) g| + |j (|) g| we obtain by integration |j 0 (|) 3 j (|) =

]

|

00

xj (x) gx 0

00

Since j (|) D 0, we have that |j 0 (|) 3 j (|) D 0 and since |j 0 (|) A 0 (for | A 0) that k (|) = 1 3

j (|) |j 0 (|)

lies in the interval [0> 1]. From (5.21), it follows that  = j (p) and that k (p) = 1 3 j 0 (p) =

 pj 0 (p)

or

 p (1 3 k (p))

With this preparation, consider now the following linear function J (|) =

 (| 3 pk(p)) p (1 3 k (p))

(5.23)

 Clearly, we have that J (p) =  and that J0 (|) = p(13k(p)) = j 0 (p) is independent of |. Since j 0 (|) is non decreasing — which is the basic assumption of the theorem — the dierence g j 0 (|)3J0 (|) is negative if | ? p, but positive if | A p. Since j 0 (|)3J0 (|) = g| (j (|) 3 J (|)), the function j (|) 3 J (|) is convex with minimum at | = p for which j (p) 3 J (p) = 0. Hence, j (|) 3 J (|) D 0 for all | M [0> 1]. Further, J (|) is positive for | M (pk (p) > 1]. Especially in this interval, the inequality j (|) D J (|) is sharp because j (|) is positive in (0> 1]. Thus,

]

1

] J2 (|) g| $

pk(p)

]

1

j 2 (|) g| ?

pk(p)

1

j 2 (|) g|

0

Using (5.22) and with (5.23), we have 2 2 p2 (1 3 k (p))2

(1 3 pk(p))3 ? 2 3

94

Inequalities

from which we arrive at the inequality 3p2 (1 3 })2

2 ?

(5.24)

(1 3 p})3

where } = k (p) M [0> 1]. The derivative of the right-hand side with respect to }, # $ g 3 (1 3 })2 2 3p2 (1 3 }) =3 p (2 3 3p + p}) g} (1 3 p})3 (1 3 p})4 3p2 (13})2 is monotonously decreasing for all } M [0> 1] if p ? 23 with maximum at (13p})3 I } = 0. Thus, if p ? 23 , evaluating (5.24) at } = 0 yields  ? 3p. On the other hand, if p A 23 , 2 2 (13}) 2 then 3p is maximal provided 2 3 3p + p} = 0 or for } = 3 3 p . With that value of }, (13p})3 2 2 2 I the inequality (5.24) yields  ? 3 13p . Both regimes p A 3 and p ? 3 tend to a same bound  ? I2 if p < 23 . The converse is similarly derived from (5.24). ¤ 3

shows that

1 If [ has a symmetric uniform distribution with i[ ({) = 2d 1{M[3d>d] , then  d  p = d and  = I3 from which p = I3 . This example shows that Gauss’s

Theorem 5.6.1 is sharpsfor p  the first condition   3p.

2 3

in the sense that equality can occur in

5.7 The dominant pole approximation and large deviations In this section, we relate asymptotic results of generating functions to the theory of large deviations. An asymptotic expansion in discrete-time is compared to established large deviations results. The first approach using the generating function *[ (}) of the random variable [ is an immediate consequence of Lemma 5.7.1. Lemma 5.7.1 If *[ (}) is meromorphic with residues un at the (simple) poles sn ordered as 0 ? |s0 |  |s1 |  |s2 |  · · · and if *[ (}) = r(} Q+1 ) as } $ 4, then holds *[ (}) = =

Q X n=0 Q X

n

Pr [[ = n] } + n

Pr [[ = n] } +

n=0

" X

sQ +1 n=0 n à " X un

n=0

un } Q+1 (}  sn )

Q X 1 }p + }  sn p=0 sp+1 n

(5.25) ! (5.26)

The normalization condition *[ (1) = 1 implies that Pr[[ A Q ] = 1 

Q X n=0

Pr [[ = n] =

" X

un Q+1 s (1  sn ) n=0 n

(5.27)

5.7 The dominant pole approximation and large deviations

95

The Lemma follows from Titchmarsh (1964, Section 3.21). Rewriting (5.26) gives, Ã" ! Q " X X X un n }m *[ (}) = Pr [[ = n] }  (5.28) m+1 s m=Q+1 n=0 n n=0 and hence, Pr [[ = m] = 

" X un n=0

sm+1 n

(m A Q )

(5.29)

The cumulative density function for N A Q follows from (5.29) as Pr[[ A N] =

" X

Pr [[ = m] =

m=N+1

" X

un N+1 s (1  sn ) n=0 n

(N A Q ) (5.30)

Lemma 5.7.1 means that, ¡if the ¢ plot Pr [[ = m] versus m exhibits a kink at m = Q , then *[ (}) = R } Q as } $ 4. Alternatively5 , the asymptotic regime does not start earlier than m  Q . For large N, only the pole with smallest modulus, s0 , will dominate. Hence, u0 Pr[[ A N]  N+1 (5.31) s0 (1  s0 ) This approximation is called the dominant pole approximation with the residue at the simple pole s0 equal to u0 = lim}<s0 *[ (})(}  s0 ). The second approach is a large deviations approximation in discrete-time. We have " X Pr [[ = m]  log Pr [[ A N] =  log   log

m=N+1 " X

{m3N31 Pr [[ = m]

m=N+1

3

  log C{3N31

" X

({ 5 R and {  1) 4

{m Pr [[ = m]D

m=0

= (N + 1) log {  log *[ ({)

(5.32)

This inequality holds for all real {  1. To get the tightest bound, we determine the maximizer {max of (5.32), thus L(N) = sup{D1 [(N +1) log { log *[ ({)]. There exists such a supremum on account of the convexity of 5

In terms of the queue occupancy in ATM, the initial Pr [[ = m]-regime for m ? Q reflects the cell scale, while the asymptotic regime m D Q refers to the burst scale.

96

Inequalities

L(N) because *[ ({) and log *[ ({) are convex for {  1 as shown in Section 5.5. Assuming that the maximum, say {max exists, then it is solution of [ ({m a x ) {max = (N + 1) * *0 ({m a x ) and the large deviations estimate becomes [

3(N+1) Pr [[ A N]  h3[(N+1) log {m a x 3log *[ ({m a x ))] = *[ ({max ) {max

(5.33)

Observe that (5.33) can be obtained directly from (5.11) with N = w+H[[]. Comparing (5.33) and (5.31) indicates, for large N, that {max = s0 because lim

N 1). The theorem then follows by virtue of the Continuity Theorem 6.1.3. ¤ 4

Feller also mentions sharper bounds.

104

Limit laws

An alternative formulation of the Central Limit Theorem is that the nfold convolution of any probability density function converges to a Gaussian h i 2 (nW) ({3) 1 probability distribution, i[ ({) $ I2 exp  22 with  = nH [[] and  2 = nVar[[]. Both the Law of Large Numbers and the Central Limit Theorem can be shown to be valid for a surprisingly large class of sequences {[n } where each random variable may have a dierent distribution. The conditions for the extension of the Central Limit Theorem are summarized in the Lindeberg conditions (Feller, 1971, p. 263). An example where the sum of independent random variables tend to a dierent limit distribution than the Gaussian appears in Section 16.5.1. If higher moments are known, the convergence to the Gaussian distribution can be bounded. Feller (1971, Chapter XVI) devotes a chapter on expansions related to the Central Limit Theorem culminating in the BerryEsseen Theorem. Theorem 6.3.2 (Berry—Esseen Theorem) Let {[n } be a sequence of independent random variables each with distribution identical to that h of the i 3 2 random variable [ with finite  = H [[],  = Var[[] and  = H |[3| . 3  Then, with F = 3, ¯  ¯ ¸ ¯ ¯ F Vq  q ¯ s  {  ({)¯¯  s sup ¯Pr  q q {

(6.5)

Proof : See e.g. Feller (1971, Section XVI.5). The constant F can be slightly improved to F  2=05. ¤ As an example of the rate of convergence towards the Gaussian distribution, the n-fold convolutions of the uniform density given by (3.30) is plotted in Fig. 6.1 together with the Gaussian approximation (3.19).

6.4 Extremal distributions 6.4.1 Scaling laws In this section, limit properties of the maximum and minimum of a set {[n } of independent random variables are discussed. For simplicity, we assume that all random variables [n have identical distribution I ({) = Pr [[  {]

6.4 Extremal distributions

105

1.0

Pdf of k-convolved uniform random variables

k=2

Exact convolution Gaussian approximation

0.8 k=3

k=4

0.6

k=8 0.4 k = 16 0.2

0.0 0

2

4

6

8

10

12

14

x

(n)

Fig. 6.1. Both the exact iX ({) with iX ({) = 10{1 and the Gaussian approximation for several values of n.

such that (3.33) and (3.32) simplify to ¸ Pr max [n  { = I p ({) 1$n$p ¸  Pr min [n A { = (1  I ({))p 

1$n$p

Consider the limit process when p $ 4. Let {{p } be a sequence of real numbers. Then, confining to the maximum first, µ  ¸¶ log Pr max [n  {p = p log I ({p ) 1$n$p

Since 0  I ({p )  1 and since the logarithm has a Taylor expansion log(1 P {n {) =  " n=1 n around { = 0 and convergent for |{| ? 1, we rewrite the right-hand side as log I ({p ) = log [1  (1  I ({p ))] and, after expansion, ¸¶ µ  = p (1  I ({p )) + r [p (1  I ({p ))] log Pr max [n  {p 1$n$p

106

Limit laws

If limp [(w2 )  [(w1 )> = = = > [(wq )  [(wq31 ) are independent. The continuous (time) stochastic process has stationary increments if [(w + v)  [(v) possesses the same distribution for all v. Hence, changes in the value of the process only dependent on the distance w between process events, not on the time point v. 6

6

4

4

2

2 t

t

Fig. 7.1. Two dierent sample paths of the experiment: roll a die and record the outcome. The total number of dierent sample paths is 6W where W is the number of times an outcome is recorded. The state space only contains 6 possible outcomes {1> 2> 3> 4> 5> 6}.

Stochastic processes are distinguished by (a) their state space, (b) the index set W and (c) by the dependence relations between random variables [(w). For example, a standard Brownian motion (or Wiener process)2 is defined as a stochastic process [(w) having continuous sample paths, stationary independent increments and [(w) has a normal distribution Q (0> w). A Poisson process, defined in more detail in Section 7.2, is a stochastic process [(w) having discontinuous sample paths, stationary independent increments and [(w) has a Poisson distribution. A generalization of the Poisson process is a counting process. A counting process is defined as a stochastic process Q (w)  0 with discontinuous sample paths, stationary independent increments, but with arbitrary distribution. A counting process Q (w) represents the total number of events that have occurred in a time interval [0> w]. Examples of a counting process are the number of telephone calls at a local exchange during an interval, the number of failures in a telecommunication network, the number of corrupted bits after transmission due to channel errors, etc. 2

Harrison (1990) shows that the converse is also true: if \ is a continuous process with stationary independent increments, then \ is a Brownian motion.

7.1 A stochastic process

117

7.1.2 Modeling a stochastic process from measurements In practice, understanding observed phenomena often asks for a stochastic model that captures the main characteristics of the studied phenomena and that enables computations of diverse quantities of interest. Examples in the field of data communications networks are the determination of the arrival process at a switch or router in order to dimension the number of buer (memory) places, the modeling of the graph of the Internet, the distribution of the duration of a telephone call or web browsing session, the number of visits to certain websites, the number of links that refer to a web page, the amount of downloaded information, the number of traversed routers by an email, etc. Accurate modeling is in general di!cult and often trades o complexity against accuracy of the model. Let us illustrate some aspects of modeling by considering Internet delay measurements. A motivation for obtaining an end-to-end delay model for (a part of) the Internet is the question whether massive service deployment of voice over IP (VoIP) can substitute classical telephony with a comparable quality. Specifically, classical telephony requires that the end-to-end delay of an arbitrary telephone conversation hardly exceeds 100 ms.

end-to-end delay D [ms]

50

Interdepature time = 12 s hopcount IP path = 13 # measurement points = 1006 E[D] = 35.03 ms V[D] = 1.36 ms min[D] = 34.18 ms max[D] = 53.95 ms

45

40

35 5:00 a.m.

6:00 a.m.

7:00 a.m.

8:00 a.m.

8:30 a.m.

Time from 5:00 a.m. until 8:30 a.m. measured on 21/11/2002

Fig. 7.2. The raw data of the end-to-end delay of IP test packets along a same path of 13 hops in the Internet measured during 3.5 hours.

118

The Poisson process

The end-to-end delay along a fixed path between source and destination measured during some interval is an example of a continuous time stochastic process. We have received data of the delay measured at RIPE-NCC as illustrated in Fig. 7.2. Figure 7.2 shows a sample path of this continuous stochastic process. The precise details of the measurement configuration are for the present purpose not relevant. It su!ces to add that Figure 7.2 shows the time dierence between the departure of an IP test packet of 100 byte at the sending box and its arrival at the destination box accurate within 10 s. 1 packets per second. Each The average sending rate of IP test packets is 12 IP test packet is assumed to follow the same path from sending to receiving box. The steadiness of the path is checked by trace-route measurements every 6 minutes. Usually, in the next step, the histogram of the raw data is made. A histogram counts the number of data points that lie in an interval of G ms, which is often called the bin size. Most graphical packages allow to choose the bin size. Figure 7.3 shows two dierent histograms with bin size G = 0=5 ms and G = 0=1 ms. In general, there is no universal rule to choose the bin size G. Clearly, the bin size is bounded below by the measurement accuracy, in our case G A 10 s. A finer bin size provides more detail, but the resulting histogram exhibits also more stochastic variations because there are fewer data points in a small bin and adjacent bins may possess a significantly dierent amount of data points. Hence, compared to one larger bin that covers a same interval, less averaging or smoothing occurs in a set of smaller bins. The normalized histogram obtained by dividing the counts per bin by the total number of data points provides a first approximation to the probability density function of G. However, it is still discrete and approximates Pr [n ? G  n + G]. A more precise description of constructing a histogram is given in Section C.1. The histogram is generally better suited to decide whether outliners in de data points may be due to measurement errors or not. Figure 7.3 suggests to either neglect the data points with G A 40 ms or to measure at a higher sending rate of IP test packets in order to have more details in the intervals exceeding 38 or 40 ms. If there existed a good3 stochastic model for the endto-end delay along fixed Internet paths, a normal procedure4 in engineering and physics would be to fit the histogram with that stochastic model to obtain the parameters of that stochastic model. The accuracy of the fit can 3 4

Which is still lacking at the time of writing. Other more di!cult methods in the realm of statistics must be invoked in case the measurement data are so precious and rare that any additional measurement point has a far larger cost than the cost of extensive additional computations.

7.1 A stochastic process

119

be expressed in terms of the correlation coe!cient  explained in Section 2.5.3. The closer  tends to 1, the better the fit, which gives confidence that the stochastic model corresponds with the real phenomenon.

Number of data points

140 120

300

100

200

80 100

60 0

40

35

40

45

50

20 0 34

35

36

37

38

39

40

41

42

43

44

D in ms

Fig. 7.3. The histogram of the end-to-end delay with a bin size of 0.1 ms (the insert has bin size of 0.5 ms).

Assuming that the presented measurement is a typical measurement along a fixed Internet path (which is true for about 80% of the investigated dierent paths), it demonstrates that there is a clear minimum at about 34 ms due to the propagation delay of electromagnetic waves. In addition, the end-toend delay lies for 99% between 34 and 38 ms. However, there is insu!cient data to pronounce claims in the tail behavior (Pr [G A {] for { A 40 ms). Just this region is of interest to compute the quality of service expressed as the probability that the end-to-end delay exceeds { ms is smaller than 103d where d specifies the stringency on the quality requirement. Toll quality in classical telephony sets { at 100 ms and d in the range of 4 to 5. The existence of a good stochastic model covering the whole possible range of the end-to-end delay G would enable us to compute tail probabilities based on the parameters that can be fitted from the measurements. The histogram is in fact a projection of the raw measurement data onto the ordinate (end-to-end delay axis). All time information (the abscissa in Fig. 7.2) is lost. Usually, the time evolution and the dependencies or correlations over time of a stochastic phenomenon are di!cult and most analyses are only tractable under certain simplifying conditions. For example, often only a steady state analysis is possible and the increments [(wn )  [(wn31 )

120

The Poisson process

of the process for all w0 ? · · · ? wn31 ? wn ? · · · ? wq are assumed to be independent or weakly dependent. The study of Markov processes (Chapters 9—11) basically tries to compute and analyze the process in steady state. Figure 7.2 is measured over a relatively long period of time and indicates that after 8.00 a.m. the background tra!c increases. The background traffic interferes with the IP test packets and causes them to queue longer in routers such that larger variations are observed. However, it is in general di!cult to ascertain that (a part of) the measurement is performed while the system operates in a certain stable regime (or steady state). We have touched upon some aspects in the art of modeling to motivate the importance of studying stochastic processes. In the sequel of this chapter, one of the most basic and simplest stochastic processes is investigated.

7.2 The Poisson process A Poisson process with parameter or rate  A 0 is an integer-valued, continuous time stochastic process {[(w)> w  0} satisfying (i) [(0) = 0 (ii) for all w0 = 0 ? w1 ? · · · ? wq , the increments [(w1 )  [(w0 )> [(w2 )  [(w1 )> = = = > [(wq )  [(wq31 ) are independent random variables (iii) for w  0, v A 0 and non-negative integers n, the increments have the Poisson distribution Pr [[(w + v)  [(v) = n] =

(w)n h3w n!

(7.1)

It is convenient to view the Poisson process [(w) as a special counting process, where the number of events in any interval of length w is specified via condition (iii). From this definition, a number of properties can be derived: (a) Condition (iii) implies that the increments are stationary because the right-hand side does not dependent on v. In other words, the increments only depend on the length of the interval w and not on the time v when the interval begins. Further, with (3.11), the mean H [[(w + v)  [(v)] = w and because the increments are stationary, this holds for any value of v. In particular with v = 0 and condition 1, the expected number of events in a time interval with length w is H [[(w)] = w

(7.2)

Relation (7.2) explains why  is called the rate of the Poisson process, namely, the derivative over time w or the number of events per time unit.

7.2 The Poisson process

121

(b) The probability that exactly one event occurs in an arbitrarily small time interval of length k follows from condition (iii) as Pr [[(k + v)  [(v) = 1] = kh3k = k + r(k) while the probability that no event occurs in an arbitrarily small time interval of length k is Pr [[(k + v)  [(v) = 0] = h3k = 1  k + r(k) Similarly, the probability that more than one event occurs in an arbitrarily small time interval of length k is Pr [[(k + v)  [(v) A 1] = r(k) Example 1 A conversation in a wireless ad-hoc network is severely disturbed by interference signals according to a Poisson process of rate  = 0=1 per minute. (a) What is the probability that no interference signals occur within the first two minutes of the conversation? (b) Given that the first two minutes are free of disturbing eects, what is the probability that in the next minute precisely 1 interfering signal disturbs the conversation? (a) Let [ (w) denote the Poisson interference process, then Pr [[(2) = 0] needs to be computed. Since [ (0) = 0 and with (7.1), we can write Pr [[(2) = 0] = Pr [[(2)  [ (0) = 0] = h32 , which equals Pr [[(2) = 0] = h30=2 = 0=8187. (b) The events during two non-overlapping intervals of a Poisson process are independent. Thus the event {[(2)  [(0) = 0} is independent from the event {[(3)  [(2) = 1} which means that the asked conditional probability Pr [[(3)  [(2) = 1|[(2)  [(0) = 0] = Pr [[(3)  [(2) = 1]. From (7.1), we obtain Pr [[(3)  [(2) = 1] = 0=1h30=1 = 0=0905= Example 2 During a certain time interval [w1 > w1 + 10 s], the number of IP packets that arrive at a router is on average 40/s. A service provider asks us to compute the probability that there arrive 20 packets in the period [w1 > w1 + 1 s] and 30 IP packets in [w1 > w1 + 3 s]. We may regard the arrival process as a Poisson process. We are asked to compute Pr [[(1) = 20> [(3) = 30] knowing that  = 40 31 s . Using the independence of increments and (7.1), we rewrite Pr [[(1) = 20> [(3) = 30] = Pr [[(1)  [(0) = 20> [(3)  [(1) = 10] = Pr [[(1)  [(0) = 20] Pr [[(3)  [(1) = 10] =

()20 h3 (2)10 h32 = 10326  0 20! 10!

122

The Poisson process

which means that the request of the service provider does not occur in practice.

7.3 Properties of the Poisson process The first theorem is the converse of the above property (b) that immediately followed from the definition. The Theorems presented here reveal the methodology of how stochastic processes are studied. Theorem 7.3.1 A counting process Q (w) that satisfies the conditions (i) Q (0) = 0, (ii) the process Q (w) has stationary and independent increments, (iii) Pr [Q (k) = 1] = k + r(k) and (iv) Pr [Q (k) A 1] = r(k) is a Poisson process with rate  A 0. Proof: We must show that conditions (iii) and (iv) are equivalent to condition (iii) in the definition of the Poisson process. Denote Sq (w) = Pr [Q (w) = q] and consider first the case q = 0, then S0 (w + k) = Pr [Q (w + k) = 0] = Pr [Q (w + k)  Q (w) = 0> Q (w) = 0] Invoking independence via (ii) S0 (w + k) = Pr [Q (w + k)  Q (w) = 0] Pr [Q (w) = 0] By definition, S0 (w) = Pr [Q (w) = 0] and from (iii), (iv) and the fact that P" n=0 Pr [Q (k) = n] = 1, it follows that Pr [Q (k) = 0] = 1  k + r(k)

(v)

Combining these with the stationarity in (ii), we obtain S0 (w + k) = S0 (w) (1  k + r(k)) or r(k) S0 (w + k)  S0 (w) = S0 (w) + k k from which, in the limit k $ 0, the dierential equation S00 (w) = S0 (w) is immediate. The solution is S0 (w) = Fh3w and the integration constant F follows from (i) and S0 (0) = Pr [Q (0) = 0] = 1 as F = 1. This establishes condition (iii) in the definition of the Poisson process for n = 0.

7.3 Properties of the Poisson process

123

The verification for q A 0 is more involved. Applying the law of total probability (2.46), Sq (w + k) = Pr [Q(w + k) = q] q X Pr [Q (w + k)  Q (w) = m|Q (w) = q  m] Pr [Q (w) = q  m] = m=0

By independence (ii), Pr [Q (w + k)  Q (w) = m|Q (w) = q  m] = Pr [Q (w + k)  Q (w) = m] and by definition Pr [Q(w) = q  m] = Sq3m (w), we have Sq (w + k) =

q X

Pr [Q (w + k)  Q (w) = m] Sq3m (w)

m=0

By the stationarity (ii) Pr [Q(w + k)  Q (w) = m] = Pr [Q (k)  Q (0) = m] we obtain using (i) Sq (w + k) =

q X

Pr [Q (k) = m] Sq3m (w)

m=0

while (v) and (iii) suggest to write the sum as Sq (w + k) = Sq (w) Pr [Q (k) = 0] + Sq31 (w) Pr [Q (k) = 1] q X Sq3m (w) Pr [Q (k) = m] + m=2

Since Sq (w)  1 and using (iv), q X

Sq3m (w) Pr [Q (k) = m] 

m=2

q X

Pr [Q(k) = m] = Pr [Q (k) A 1] = r(k)

m=2

we arrive with (v), (iii) at Sq (w + k) = Sq (w) (1  k + r(k)) + Sq31 (w) (k + r(k)) + r(k) or Sq (w + k)  Sq (w) r(k) = Sq (w) + Sq31 (w) + k k which leads, after taking the limit k $ 0, to the dierential equation Sq0 (w) = Sq (w) + Sq31 (w)

124

The Poisson process

with initial condition Sq (0) = Pr [Q(0) = q] = 1{q=0} . This dierential equation is rewritten as ´ g ³ w (7.3) h Sq (w) = hw Sq31 (w) gw In ¡case q ¢= 1, the dierential equation reduces with S0 (w) = h3w to g w w gw h S1 (w) = . The general solution is h S1 (w) = w + F and, from the initial condition S1 (0) = 0, we have F = 0 and S1 (w) = wh3w . The general q 3w solution to (7.3) is proved by induction. Assume that Sq (w) = (w)q!h holds for q, then the case q + 1 follows from (7.3) as ´ (w)q g ³ w h Sq+1 (w) =  gw q! q+1 3w

h and integrating from 0 to w using Sq+1 (0) = 0, yields Sq+1 (w) = (w)(q+1)! which establishes the induction and finalizes the proof of the theorem. ¤

The second theorem has very important applications since it relates the number of events in non-overlapping intervals to the interarrival time between these events. Theorem 7.3.2 Let {[(w); w  0} be a Poisson process with rate  A 0 and denote by w0 = 0 ? w1 ? w2 ? · · · the successive occurrence times of events. Then the interarrival times q = wq wq31 are independent identically distributed exponential random variables with mean 1 . Proof: For any v  0 and any q  1, the event {q A v} is equivalent to the event {[(wq31 +v)[(wq31 ) = 0}. Indeed, the q-th interarrival time q can only be longer than v time units if and only if the q-th event has not yet occurred v time units after the occurrence of the (q  1)-th event at wq31 . Since the Poisson process has independent increments (condition (ii) in the definition of the Poisson process), changes in the value of the process in nonoverlapping time intervals are independent. By the equivalence in events, this implies that the set of interarrival times q are independent random variables. Further, by the stationarity of the Poisson process (deduced from condition (iii) in the definition of the Poisson process), Pr [q A v] = Pr [[(wq31 + v)  [(wq31 ) = 0] = h3v which implies that any interarrival time has an identical, exponential distribution, Iq ({) = Pr [q  {] = 1  h3{

7.3 Properties of the Poisson process

This proves the theorem.

125

¤

The converse of Theorem 7.3.2 also holds: if the interarrival times {q } of a counting process {Q (w)> w  0} are i.i.d. exponential random variables with mean 1 , then {Q (w)> w  0} is a Poisson process with rate . An association to the exponential distribution is the memoryless property, Pr[q A v + w|q A v] = Pr[q A w] By the equivalence of the events, for any w> v  0, Pr[q A v + w|q A v] = Pr[[(wq31 + v + w) 3 [(wq31 ) = 0|[(wq31 + v) 3 [(wq31 ) = 0] = Pr[[(wq31 + v + w) 3 [(wq31 + v) = 0|[(wq31 + v) 3 [(wq31 ) = 0]

By the independence of increments (in non-overlapping intervals), Pr[q A v + w|q A v] = Pr[[(wq31 + v + w)  [(wq31 + v) = 0] and by the stationarity of the increments, the memoryless property is established, Pr[q A v + w|q A v] = Pr[[(wq31 + w)  [(wq31 ) = 0] = Pr[q A w] Hence, the assumption of stationary and independent increments is equivalent to asserting that, at any time v, the process probabilistically restarts again with the same distribution and is independent of occurrences in the past (before v). Thus, the process has no memory and, since the only continuous distribution that satisfies the memoryless property is the exponential distribution, exponential interarrival times q are a natural consequence. The arrival time of the q-th event or the waiting time until the q-event is P Zq = qn=1 n . In Section 3.3.1, it is shown that the probability distribution of the sum of independent exponential random variables has a Gamma distribution or Erlang distribution (3.24). Alternatively, the equivalence of the events, {Zq  w} +, {Q (w)  q}, directly leads to the Erlang distribution, IZq (w) = Pr [Zq  w] = Pr [Q (w)  q] =

" X (w)n h3w n=q

n!

The equivalence of the events, {Zq  w} +, {Q (w)  q}, is a general relation and a fundamental part of the theory of renewal processes, which we will study in the next Chapter 8. Theorem 7.3.3 Given that exactly one event of a Poisson process {[(w); w  0} has occurred during the interval [0> w], the time of occurrence of this event is uniformly distributed over [0> w].

126

The Poisson process

Proof: Immediate application of the conditional probability (2.44) yields for 0  v  w, Pr [1  v|[(w) = 1] =

Pr [{1  v} _ {[(w) = 1}] Pr [[(w) = 1]

Using the equivalence {1  v} +, {[(w0 + v)  [(w0 ) = 1} and the fact that {[(w0 + v)  [(w0 ) = 1} = {[(v) = 1} by the stationarity of the Poisson process gives {1  v} _ {[(w) = 1} = {[(v) = 1} _ {[(w) = 1} = {[(v) = 1} _ {[(w)  [(v) = 0} Applying the independence of increments over non-overlapping intervals and (7.1) yields Pr [1  v|[(w) = 1] = =

Pr [[(v) = 1] Pr [[(w)  [(v) = 0] Pr [[(w) = 1] (v) h3v h3(w3v) v = (w) h3w w ¤

which completes the proof.

Theorem 7.3.3 is immediately generalized to q events. For any set of real variables vm satisfying 0 = v0 ? v1 ? v2 ? · · · ? vq ? w and given that q events of a Poisson process {[(w); w  0} have occurred during the interval [0> w], the probability of the successive occurrence times 0 ? w1 ? w2 ? · · · ? wq ? w of these q Poisson events is Pr [w1  v1 > = = = > wq ? vq |[(w) = q] =

Pr [{w1  v1 > = = = > wq ? vq } _ {[(w) = q}] Pr [[(w) = q]

Using a similar argument as in the proof of Theorem 7.3.3, s = Pr [{w1  v1 > w2  v2 > = = = > wq ? vq } _ {[(w) = q}] = Pr [[(v1 )  [ (v0 ) = 1> = = = > [(vq )  [(vq31 ) = 1> [(w)  [(vq ) = 0] 3 4 q Y = C Pr [[(vm )  [ (vm31 ) = 1]D Pr[[(w)  [(vq ) = 0] m=1

3 =C

q Y

4

h3(vm 3vm31 )  (vm  vm31 )D h3(w3vq )

m=1

= q

q Y m=1

(vm  vm31 ) h3

Sq

m=1 (vm 3vm31 )3(w3vq )

= q

q Y m=1

(vm  vm31 ) h3w

7.3 Properties of the Poisson process

127

Thus, q Pr [w1  v1 > w2  v2 > = = = > wq ? vq |[(w) = q] =

q Y

(vm  vm31 ) h3w

m=1 (w)q h3w q!

q! Y (vm  vm31 ) wq q

=

m=1

from which the density function i{wm } (v1 > = = = > vq |[(w) = q) =

Cq Pr [w1  v1 > = = = > wq ? vq |[(w) = q] Cv1 = = = Cvq

follows as q! wq which is independent of the rate . If 0 ? w1 ? w2 ? · · · ? wq ? w are the successive occurrence times of q Poisson events in the interval [0> w], then the random variables w1 > w2 > = = = > wq are distributed as a set of order statistics, defined in Section 3.4.2, of q uniform random variables in [0> w]. In other words, if q i.i.d. uniform random variables on [0> w] are assorted in increasing order, they may represent q successive occurrence times of a Poisson process. The average spacing between these q ordered i.i.d. uniform random variables is qw as computed in Problem (ii) of Section 3.7. A related example is the conditional probability where 0 ? v ? w and 0  n  q, i{wm } (v1 > v2 > = = = > vq |[(w) = q) =

Pr [{[(v) = n} _ {[(w) = q}] Pr [[(w) = q] Pr [{[(v) = n} _ {[(w)  [(v) = q  n}] = Pr [[(w) = q] Pr [[(v) = n] Pr[[(w)  [(v) = q  n] = Pr [[(w) = q]

Pr [[(v) = n|[(w) = q] =

q!(v)n h3v ((w  v))q3n h3(w3v) n!(w)q h3w (q  n)! µ ¶ n q v = (w  v)q3n n wq

=

Hence, if s = vw , the conditional probability becomes µ ¶ q n Pr [[(v) = n|[(w) = q] = s (1  s)q3n n

128

The Poisson process

Given that a total number of q Poisson events have occurred in time interval [0> w], the chance that precisely n events have taken place in the sub-interval [0> v] is binomially distributed with parameter q and s = vw . Observe that also this conditional probability is independent of the rate . In addition, since limw v] is Poisson distributed with mean equal to the length of the interval. Application The arrival process of most real-time applications (such as telephony calls, interactive-video, ...) in a network is well approximated by a Poisson process. Suppose a measurement configuration is built to collect statistics of the arrival process of telephony calls in some region. During a period [0> W ], precisely 1 telephony call has been measured. What can be said of the time { 5 [0> W ] at which the telephony call has arrived at the measurement device? Theorem 7.3.3 tells us that any time in that interval is equally probable. Theorem 7.3.4 If [(w) and \ (w) are two independent Poisson processes with rates { and | , then is ](w) = [(w) + \ (w) also a Poisson process with rate { + | . Proof: It su!ces to demonstrate that the counting process Q] (w) = Q[ (w) + Q\ (w) has exponentially distributed interarrival times ] . Suppose that Q] (wq ) = q, it remains to compute the next arrival at time wq+1 = wq +v for which Q] (wq + v) = q + 1. Due to the memoryless property of the Poisson process, the occurrence of an event from wq on for each random variable [ and \ is again exponentially distributed with parameter { and | , respectively. In other words, it is irrelevant which process [ or \ has previously caused the arrival at time wq . Further, the event that the interarrival time of the sum processes {] A v} is equivalent to {[ A v} _ {\ A v} or Pr [] A v] = Pr [[ A v> \ A v] = Pr [[ A v] Pr [\ A v] = h3({ +| )v where the independence of [(w) and \ (w) has been used. This proves the theorem. ¤

7.4 The nonhomogeneous Poisson process

129

A direct consequence is that any sum of independent Poisson processes is also a Poisson process with aggregate rate equal to the sum of the individual rates. This theorem is in correspondence with the sum property of the Poisson distribution. 7.4 The nonhomogeneous Poisson process As will be shown later in Section 11.3.2, the Poisson process is a special case of a birth-and-death process, which is in turn a special case of a Markov process. Hence, it seems more instructive to discuss these special processes as applications of the Markov process. Therefore, only associations to the Poisson process are treated here. In many cases, the rate is a time variant function (w) and such process is termed a nonhomogeneous or nonstationary Poisson process. For example, the arrival rate of a large number p of individual IP-flows at a router is well approximated by a nonhomogeneous Poisson process, where the rate (w) varies over the day depending on the number p and the individual rate of each flow of packets. Since the sum of independent Poisson random variables is again a Poisson random variable, Pp(w) we have (w) = m=1 m (w). If [(w) is a nonhomogeneous Poisson process with rate (w), the increment [(w) [(v) reflects the number of events in an interval (v> w] and increments of non-overlapping intervals are still independent. Rw Theorem 7.4.1 If (w) = 0 (x)gx and v ? w, then [(w) [(v) is Poisson distributed with mean (w)  (v). The demonstration is analogous to the proof of Theorem 7.3.1. Proof (partly): Denote by Sq (w) = Pr [Q (w)  Q (v) = q], then S0 (w + k) = Pr [Q (w + k)  Q (v) = 0] = Pr [Q (w + k)  Q (w) = 0> Q(w)  Q (v) = 0] Invoking independence of the increments, S0 (w + k) = Pr [Q(w + k)  Q (w) = 0] Pr[Q (w)  Q(v) = 0] = S0 (w)(1  (w)k + r(k)) or S0 (w + k)  S0 (w) r(k) = (w)S0 (w) + k k from which, in the limit k $ 0, the dierential equation S00 (w) = (w)S0 (w)

130

The Poisson process

g is immediate. Rewritten as gw log S0 (w) = (w), after integration over (v> w], we find log S0 (w) =  ((w)  (v)) since S0 (v) = Pr [Q (v)  Q (v) = 0] = 1. Thus, for the case q = 0, we find S0 (w) = exp [ ((w)  (v))], which proves the theorem for q = 0. The remainder of the proof (q A 0) uses the same ingredients as the proof of Theorem 7.3.1 and is omitted. ¤

A nonhomogeneous Poisson process [(w) with rate (w) can be transformed to a homogeneous Poisson process \ (x) with rate 1 by the time transform x = (w). For, \ (x) = \ ((w)) = [(w), and \ (x + x) = \ ((w) + (w)) = [(w + w) because (w) = (w) for small w such that Pr [\ (x + x)  \ (x) = 1] = Pr [[(w + w)  [(w) = 1] = (w)w + r(w) = x + r(x) because x = (w)w + r(w). Hence, all problems concerning nonhomogeneous Poisson processes can be reduced to the homogeneous case treated above.

7.5 The failure rate function Previous sections have shown that the Poisson process is specified by a rate function (w). In this section, we consider the failure rate function of some object or system. Often it is interesting to know the probability that an object will fail in the interval [w> w + w] given that the object was still functioning well up to time w. Let [ denote the lifetime of an object5 , then this probability can be written with (2.44) as Pr [{w  [  w + w} _ {[ A w}] Pr [[ A w] Pr [w ? [  w + w] = Pr [[ A w]

Pr [w  [  w + w|[ A w] =

If i[ (w) is the probability density function of [ and I[ (w) = Pr [[  w], then for small w and assuming that i[ (w) is well behaved6 such that 5

6

In medical sciences, [ can represent in general the time for a certain event to occur. For example, the time it takes for an organism to die, the time to recover from illness, the time for a patient to respond to a therapy and so on. Recall the discussion in Section 2.3.

7.5 The failure rate function

131

Pr [w ? [  w + w] = i[ (w)w, Pr [w  [  w + w|[ A w] =

i[ (w) w 1  I[ (w)

This expression shows that u(w) =

i[ (w) 1  I[ (w)

(7.4)

can be interpreted as the intensity or rate that a w-year old object will fail. It is called the failure rate u(w) and U(w) = 1  I[ (w) = Pr [[ A w]

(7.5)

for is usually termed7 the reliability function. Since u(w) = Pr[w$[$w+{w|[Aw] {w small w, the failure rate u (w) A 0 because u (w) = 0 would imply an infinite lifetime [. Using the definition (2.30) of a probability density function, we observe that u(w) = 

gU(w) gw

U(w)

=

g ln U(w) gw

Or, since U(0) = 1, the corresponding integrated relation is  Z w ¸ U(w) = exp  u(x)gx

(7.6)

(7.7)

0

The expressions (7.6) and (7.7) are inverse relations that specify u(w) as function of U(w) and vice versa. The reliability function U(w) is non-increasing with maximum at w = 0 since it is a probability distribution function. On the other hand, the failure rate u(w) being a probability density function can take any positive real value. From (7.4) we obtain the density function of the lifetime [ in terms of failure rate u(w) as  Z w ¸ i[ (w) = u(w)U(w) = u(w) exp  u(x)gx 0

with i[ (0) = u(0). Using the tail relation (2.35) for the expectation of the lifetime [ immediately gives the mean time to failure, Z " H [[] = U(w)gw (7.8) 0

In case I[ (W ) = 1 and i[ (W ) 6= 0 for a finite time W , which is the maximum lifetime, the definition (7.4) demonstrates that u(w) has a pole at 7

In biology, medical sciences and physics, U(w) is called the survival function and u(w) is the corresponding mortality rate or hazard rate.

132

The Poisson process

w = W= In practice, the failure rate u(w) is relatively high for small w due to initial imperfections that cause a number of objects to fail early and u(w) is increasing towards the maximum life time W due to aging or wear and tear. This shape of u(w) as illustrated in Fig. 7.4 is called a “bath-tub” curve, which is convex. r(t) fX(0)

t T

0

Fig. 7.4. Example of a “bath-tub” shaped failure rate function u(w).

An often used model for the failure rate is u(w) = dwd31 with corresponding reliability function U(w) = exp [wd ] and where the lifetime [ has a Weibull distribution function I[ (w) = 1  U(w) as in (3.40). In case d = 1, the failure rate u(w) =  is constant over time, while d A 1 (d ? 1) reflects an increasing (decreasing) failure rate over time. Hence, a “bath-tub” shaped (realistic) failure function as in Fig. 7.4 can be modeled by a Weibull model for u (w) with d ? 1 in the beginning, d = 1 in the middle and d A 1 at the end of the life time. For an exponential lifetime where i[ (w) = h3w , the failure rate (7.4) equals u(w) =  and is independent of time. This means that the failure rate for a w-year-old object is the same as for a new object, which is a manifestation of the memoryless property of the exponential distribution. It also explains why  in both the exponential as Poisson process is often called a ’rate’.

7.6 Problems (i) A series of test strings each with a variable number Q of bits all equal to 1 are transmitted over a channel. Due to transmission errors, each 1-bit can be eected independently from the others and only arrives non-corrupted with probability s. The length Q of the test strings (words) is a Poisson random variable with mean length  bits. In

7.6 Problems

133

this test, the sum \ of the bits in the arriving words is investigated to determine the channel quality via s. Compute the pdf of \ . (ii) At a router, four QoS classes are supported and for each class packets arrive according to a Poisson process with rate m for m = 1> 2> 3> 4. Suppose that the router had a failure at time w1 that lasted W time units. What is the probability density function of the total number of packets of the four classes that has arrived during that period? (iii) Let Q (w) = Q1 (w) + Q2 (w) be the sum of two independent Poisson processes with rates 1 and 2 . Given that the process Q (w) had an arrival, what is the probability that that arrival came from the process Q1 (w)? (iv) Peter has been monitoring the highway for nearly his entire life and found that the cars pass his house according to a Poisson process. Moreover, he discovered that the Poisson process in one lane is independent from that in the other lanes. The rate of these independent processes diers per lane and is denoted by 1 > 2 > 3 , where m is expressed in the number of cars on lane m per hour. (a) Given that one car passed Peter, what is the probability that it passed in lane 1? (b) What is the probability that q cars pass Peter in 1 hour ? (c) What is the probability that in 1 hour q cars have passed and that they all have used lane 1? (v) In a game, audio signals arrive in the interval (0> W ) according to a Poisson process with rate , where W A 1@. The player wins only if at least one audio signal arrives in that interval, and if he or she pushes a button (only one push allowed) upon the last of the signals. The player uses the following strategy: he or she pushes the button upon the arrival of the first signal (if any) after a fixed time v  W . (a) What is the probability that the player wins? (b) Which value of v maximizes the probability of winning, and what is the probability in that case? (vi) The arrivals of voice over IP (VoIP) packets to a router is close to a Poisson process with rate  = 0=1 packets per minute. Due to an upgrade to install weighted fair queueing as priority scheduling rule, the router is switched o for 10 minutes. (a) What is the probability of receiving no VoIP packets when switched o? (b) What is the probability that more than ten VoIP packets will arrive during this upgrade?

134

The Poisson process

(c) If there was one VoIP in the meantime, what is the most probable minute of the arrival? (vii) A link of a packet network carries on average ten packets per second. The packets arrive according to a Poisson process. A packet has a probability of 30 % to be an acknowledgment (ACK) packet independent of the others. The link is monitored during an interval of 1 second. (a) What is the probability that at least one ACK packet has been observed? (b) What is the expected number of all packets given that five ACK packets have been spotted on the link? (c) Given that eight packets have been observed in total, what is the probability that two of them are ACK packets? (viii) An ADSL helpdesk treats exclusively customer requests of one of three types: (i) login-problems, (ii) ADSL hardware and (iii) ADSL software problems. The opening hours of the helpdesk are from 8:00 until 16:00. All requests are arriving at the helpdesk according to a Poisson process with dierent rates: 1 = 8 requests with login problems/hour, 2 = 6 requests with hardware problems/hour, and 3 = 6 requests with software problems/hour. The Poisson arrival processes for dierent types of requests are independent. (a) What is the expected number of requests in one day? (b) What is the probability that in 20 minutes exactly three requests arrive, and that all of them have hardware problems? (c) What is the probability that no requests will arrive in the last 15 minutes of the opening hours? (d) What is the probability that one request arrives between 10:00 and 10:12 and two requests arrive between 10:06 and 10:30? (e) If at the moment w + v there are n + p requests, what is the probability that there were n requests at the moment w? (ix) Arrival of virus attacks to a PC can be modeled by a Poisson process with rate  = 6 attacks per hour. (a) What is the probability that exactly one attack will arrive between 1 p.m. and 2 p.m.? (b) Suppose that at the moment the PC is turned on there were no attacks on PC, but at the shut-down time precisely 60 attacks have been observed. What is the expected amount of time that the PC has been on?

7.6 Problems

135

(c) Given that six attacks arrive between 1 p.m. and 2 p.m., what is the probability that the fifth attack will arrive between 1:30 p.m. and 2 p.m.? (d) What is the expected arrival time of that fifth attack? (x) Consider a system V consisting of q subsystems in series as shown in Fig. 7.5. The system V operates correctly only if all subsystems operate correctly. Assume that the probability that a failure in a subsystem Vl occurs is independent of that in subsystem Vm . Given the reliability functions Um (w) or each subsystem Vm , compute the reliability function U(w) of the system V.

S1

S3

S2

Sn

Fig. 7.5. A system consisting of q subsystems in series.

(xi) Same question as in previous exercise but applied to a system V consisting of q subsystem in parallel as shown in Fig. 7.6.

S1 S2 Sn Fig. 7.6. A system consisting of q subsystems in parallel.

8 Renewal theory

A renewal process is a counting process for which the interarrival times q are i.i.d. random variables with distribution I (w). Hence, a renewal process generalizes the exponential interarrival times in the Poisson process (see Theorem 7.3.2) to an arbitrary distribution. Since the interarrival times are i.i.d. random variables, at each event (or renewal) the process probabilistically restarts. The classical example of a renewal process is the successive replacement of light bulbs: the first bulb is installed at time Z0 , fails at time Z1 = 1 , and is immediately exchanged for a new bulb, which in turn fails at Z2 = 1 + 2 , and thereafter replaced by a third bulb, and so on. How many light bulbs are replaced in a period of w time units given the life time distribution I (w)?

N(t)

t

W1 W0

W2 W1

W3 W2

W4 W3

W5 W4

W5

Fig. 8.1. The relation between the renewal counting process Q (w), the interarrival time q and the waiting time Zq .

137

138

Renewal theory

8.1 Basic notions

P As illustrated in Fig. 8.1, the waiting time Zq = qn=1 n (for q  1, with Z0 = 0 by convention) is related to the counting process {Q (w)> w  0} by the equivalence {Q (w)  q} +, {Zq  w}: the number of events (renewals) up to time w is at least q if and only if the q-th renewal occurred on or before time w. Alternatively, the number of events by time w equals the largest value of q for which the q-th event occurs before or at time w, Q (w) = max [q : Zq  w]. The convention that Z0 = 0 implies that Q(0) = 0: the counting process starts counting from zero at time 0. The main objective of renewal theory is to deduce properties of the process {Q(w)> w  0} as a function of the interarrival distribution I (w) = Pr [  w]. 8.1.1 The distribution of the waiting time Zq If we assume that the interarrival times are i.i.d. having a Laplace transform Z " Z " 3}w * (}) = h gI (w) = h3}w i (w)gw 0

0

the waiting time Zq is the sum of q i.i.d. random variables specified by (2.66) as Z " h3}w iZq (w)gw = *q (})= (8.1) *Zq (}) = 0

By partial integration, we R w find the Laplace transform of the distribution IZq (w) = Pr [Zq  w] = 0 iZQ (x)gx Z " *Zq (}) *q (}) h3}w IZq (w)gw = =  (8.2) } } 0 The inverse Laplace transform follows1 with (2.38) as Z f+l" q * (}) }w 1 h g} Pr [Zq  w] = 2l f3l" }

(8.3)

As an alternative to the approach with probability generating functions, 1

In general, by integration of (2.38), we find ]

w

i[ (x)gx =

I[ (w) = 0

1 2l

]

f+l"

f3l"

*[ (})

h}w 3 1 g} }

U f+l" *[ (}) 1 whose form seems dierent from (8.3). However, 2l g} = 0 because the contour f3l" } can be closed over the positive Re(}) A f plane where *[ (}) is analytic and because limU w] when w $ 0 is assumed to be zero such that p(0) = 0. From (8.5), it follows at each point w for which I (w) ? 1 that p(w) 

" X n=1

(I (w))n =

1 1 1  I (w)

Hence, for finite w where I (w) ? 1, the renewal function p(w) converges at least as fast as a geometric series and is bounded. In the limit w $ 4, where limw w], which is, due to the stationarity of the

142

Renewal theory

renewal process, equal to n  1 renewals in (0> w  v]. By the law of total probability (2.46), we uncondition to find for n  1, Z " g Pr [Z1  v] Pr [Q (w) = n] = Pr [Q (w) = n|Z1 = v] gv gv 0 Z w Pr [Q (w  v) = n  1] i (v)gv (8.10) = 0

Multiplying both sides by n and summing over all n  1 gives the average at the left-hand side, H [Q (w)] =

" X

n Pr [Q (w) = n]

n=1

The sum at the right-hand side is " X

n Pr [Q (w  v) = n  1] =

n=1

" X (n + 1) Pr [Q (w  v) = n] n=0

= H [Q (w  x)] + 1 Combining both sides yields

Z

H [Q (w)] = I (w) +

w

H [Q (w  x)] gI (v) 0

which is again the renewal equation (8.9) since p(w) = H [Q (w)]. 8.1.4 A generalization of the renewal equation The renewal equation (8.9) is a special case of the more general class of integral equations Z w \ (w  x)gI (x)> w0 (8.11) \ (w) = k (w) + 0

in the unknown function \ (w), where k (w) is a known function and I (w) is a distribution function. This equation can be written using the convolution notation as \ (w) = k (w) + \  I (w) By conditioning on the first renewal as shown above, many renewal problems can be recasted into the form of the general renewal equation (8.11). An example is the derivation of the residual life or waiting time given in Section 8.3. Therefore, it is convenient to present the solution to the general renewal equation (8.11).

8.1 Basic notions

143

Lemma 8.1.1 If k (w) is bounded for all w, then the unique solution of the general renewal equation (8.11) is Z w k(w  x)gp(x) (8.12) \ (w) = k (w) + where p (w) =

P"

n=1

0

I (nW) (w)

is the renewal function.

Proof: Let us first concentrate on the formal solution. In general, convolutions are best treated in the transformed domain. After taking the Laplace transform of the general renewal equation (8.11), we obtain *\ (}) = *k (}) + *\ (}) *I (}) such that *\ (}) =

*k (}) 1  *I (})

There always exists a region in the }-domain where |*I (})| ? 1 such that the geometric series applies, *\ (}) = *k (})

" X

(*I (}))n = *k (}) + *k (})

n=0

" X

(*I (}))n

n=1

Back transforming and taking into account that (*I (}))n is the transform of a n-fold convolution yields \ (w) = k (w) + k 

" X

I (nW) (w) = k (w) + k  p (w)

n=1

This formal manipulation demonstrates2 that (8.12) is a solution of the general renewal equation (8.11). Suppose now that there are two solutions \1 (w) and \2 (w). Their dierence Y (w) = \1 (w)  \2 (w) obeys Z w Y (w) = Y (w  x)gI (x) = Y  I (w) 0 2

Alternatively, by substituting the solution into the equation, a check is \ (w) = k (w) + \ W I (w) = k (w) + k W I (w) + k W p W I (w) % & " [ (nW) = k (w) + k W I (w) + I (w) W I (w) % = k (w) + k W I (w) +

n=1 " [ n=2

= k (w) + k W p (w)

& I

(nW)

(w) = k (w) + k W

%

" [

n=1

& I

(nW)

(w)

144

Renewal theory

By convolving both sides with I and using the original equation, we deduce that Y (w) = Y  I  I (w). Continuing this process, for each n, we have that Y (w) = Y  I (nW) (w). Since I (nW) (w) $ 0 for all finite w and n $ 4 (because p (w) exists for all finite w), and if Y (w) is bounded, this implies that Y (w) = 0 for all finite w. This demonstrates the uniqueness and motivates the requirement that k (w) should be bounded. ¤

8.1.5 The renewal function for a Poisson process Before showing below that the renewal function p(w) can be specified in detail as w $ 4, we consider first the Poisson process where the interarrival times {q }qD1 are i.i.d. exponentially distributed with rate . Since * (}) =  }+ , Z f+l" }w h 1 g} fA0 p(w) = 2l f3l" } 2 The contour can be closed over the negative Re(})-plane (because w  0). The only singularity of the integrand is a double pole at } = 0 with residue ¯ gh}w ¯ p(w) =  g} ¯ = w. This result, of course, follows directly from the }=0

definition of the Poisson process given in (7.2). We see that the renewal function p(w) for the Poisson process is linear for all w. Moreover, the Poisson process is the only continuous time renewal process with a linear renewal function p(w). Indeed, if3 p (w) = w, the renewal equation is Z w Z w w = ((w  x)) gI (x) + I (w) =  I (x)gx  wI (0) + I (w) 0

0

By dierentiation with respect to w and assuming non-zero interarrival times such that I (0) = Pr [  0] = 0, we obtain a dierential equation  = I (w) +

gI (w) gw

whose solution is I (w) = 1  h3w . By Theorem 7.3.2, exponential interarrival times characterize a Poisson process with rate  = . 8.2 Limit theorems In the limit w $ 4, the equivalence relation (8.6) indicates that, for any fixed value of q, Pr [Q (w)  q] = 1, which means that the number of events 3

A linear form p(w) = w +  with  6= 0 is impossible because p (0) = 0.

8.2 Limit theorems

145

Z

Q (w) Q (w) $ 4 as w $ 4. Let us consider Q(w) , which is the sample mean of the first Q (w) interarrival times in the w]. ¤The Strong Law of £ intervalZ(0> q Large Numbers (6.3) indicates that Pr limq w] per interval length w, converges to the average interarrival time  = H [ ]. Unfortunately4 , weh cannot i simply deduce the intuQ(w) itive result that also the expectation, H tends to 1 . On the other w hand,£ the expectation of ZQ(w) is obtained from Wald’s identity (2.69) ¤ as H ZQ(w) = H [Q (w)] H [ ]. Taking the expectation in the inequality H[Q(w)] 1 ZQ(w)  w ? ZQ(w)+1 , leads to H[Q(w)]  H[ + 1w from which, after w w ] ? the limit w $ 4, the intuitive result follows. Thus, we have proved5 the following theorem:

Theorem 8.2.1 (Elementary Renewal Theorem) If  = H [ ] is the average interarrival time of events in the renewal process, then H [Q (w)] p(w) 1 = lim = w 4), then Z Z w 1 " lim j(w  x)gp(x) = j(x)gx (8.14) w v]. In this case, we have either that the residual waiting time U(w) is certainly shorter than { if v is contained in the interval [w> w + {], else the residual waiting time U(w) is surely larger than {. In summary, ; ? Pr [U(w  v) A {] if 0  v  w Pr [U(w) A {|Z1 = v] = 0 if w ? v  w + { = 1 if v A { + w Using the total law of probability (2.46), Z " g Pr [Z1  v] Pr [U(w) A {] = gv Pr [U(w) A {|Z1 = v] gv 0 Z w Z " g Pr [  v] gv = Pr [U(w  v) A {] i (v)gv + gv 0 {+w Z w = Pr [U(w  v) A {] gI (v) + 1  I ({ + w) 0

This relation is an instance of the general renewal equation (8.11). Since 1  I ({ + w) is monotonously decreasing, for all {, it holds with (2.35) that Z " Z " (1  I ({ + w)) gw  (1  I (w)) gw = H [ ] ? 4 0

0

which also implies that limw \1 > [2 > \2 > = = = where each on-period [q has a same distribution Ion and is followed by an o-period \q . Each o-period has also a same distribution Io . The o-period \q may dependent on the on-period [q , but the q-th renewal cycle with duration [q + \q is independent of any other cycle. An alternating renewal process can be used to model a data stream of packets, where the on-period reflects the time to store or process an arriving packet and the o-period a (random) delay between two packets. Another example is the modeling of the end-to-end delay from a source v to a destination g in the Internet, where the o-period describes a queueing in a router due to other interfering tra!c along that path. During the onperiod, a packet is not blocked by other packets. The on-period equals the propagation delay to travel from the output port of one router to the output port of the next-hop router. The end-to-end delay along a path with k hops equals the sum of k consecutive o-periods augmented by the propagation time from v to g. 8.4 The renewal reward process The renewal reward process associates at each renewal at time Zq a certain cost or reward Uq , which may vary over time and can be negative. For example, each time a light bulb fails, it must be replaced at a certain cost (negative reward) or each customer in a restaurant pays for his meal (positive reward). The reward Uq may depend on the interarrival time q or length of the q-th renewal interval, but it is independent of other renewal epochs (dierent from the q-th). Thus, the pairs (Uq > q ) are assumed to be independent and identically distributed. Most often one is interested in the total reward U(w) over a period w (not to be confused with the residual life time) defined as X

Q(w)

U(w) =

Uq

(8.19)

q=1

In this setting, the renewal reward process is a generalization of the counting process where Uq = 1.

154

Renewal theory

By slightly rewriting the total reward U(w) earned over an interval w as PQ(w) U(w) Uq Q (w) = q=1 w Q (w) w and taking the limit w $ 4, the first fraction tends with probability one to the average reward H [U] per renewal period by the Strong Law of Large 1 Numbers (Theorem 6.2.2), while the second fraction tends to 1 = H[ ] by the Elementary Renewal Theorem 8.2.1. Hence, with probability one holds that H [U] U(w) = (8.20) lim w m 5 F. Theorem 9.2.3 together with Corollary 9.2.2 implies that, if F is a finite, irreducible closed set, all states are recurrent. 9.3 The steady-state of a Markov chain 9.3.1 The irreducible Markov chain The steady-state vector  = limn q], the relation is equivalent to H [Qq (m)|[0 = l] = m q 0  x  w]

q [(0) = l] which describes the probability that, if a transition occurs, the process moves from state l to a dierent state m 6= l. Using the definition of conditional probability (2.44), Ylm (k) =

Slm (k) Pr [{[(k) = m} _ {[(k) 6= l} |[(0) = l] = Pr [[(k) 6= l|[(0) = l] 1  Sll (k)

In the limit k & 0, we have Slm (k) k k0 13Sll (k) k

Ylm = lim Ylm (k) = lim k0

=

tlm tl

P By (10.7), we see that m=1>m6=l Ylm = 1, demonstrating that, given a transition, it is a transition out of state l to another state m. The quantities Ylm correspond to the transition probabilities of the embedded Markov chain.

10.4 The embedded Markov chain

189

Alternatively, we can write the rate tlm in terms of the transition probabilities Ylm of the embedded Markov chain as tlm = tl Ylm

(10.21)

Since tl is the rate (i.e. the number of transitions per unit time) of the process in state l, relation (10.21) shows that the transition rate tlm from state l to state m equals the rate of transitions in state l multiplied by the probability that a transition from state l to state m occurs. By definition, Yll = 0. For, if we assume that Ylm A 0, relation (10.21) would result in tll = Yll tl A 0 which contradicts the definition tll = tl . Hence, in the embedded Markov chain specified by the transition probability matrix Y , there are no self-transitions (Yll = 0), which is equivalent to the fact that the sum of the eigenvalues of Y is zero (A.7), since trace(Y ) = 0. From the steady-state equation or balance equation (10.20), (10.21) and Yll = 0, we observe that l tl =

Q X

m tm Yml

m=1

On the other hand, the embedded Markov chain has a steady-state vector y that obeys (9.22) or (9.23) yl =

Q X

ym Yml

m=1

and kyk1 = 1. The relations between the steady-state vectors of the continuoustime Markov chain  and of its corresponding embedded discrete-time Markov chain y, are l tl (10.22) yl = PQ m=1 l tl yl @tl l = PQ m=1 ym @tm

(10.23)

The classification in the discrete-time case into transient and recurrent can be transferred via the embedded Markov chain to continuous Markov processes.

10.4.1 Uniformization The restriction Yll = 0 or tll = 0, which means that there are no selftransitions from a state into itself, can be removed. Indeed, we can rewrite

190

Continuous-time Markov chains

the basic relation (10.12) between the transition probability matrix S (w) and the infinitesimal generator T for all  as  µ ¶¸ ¶¸  µ T T = h3w exp w L + S (w) = exp Lw + w L +   Defining W () = L + T  and   maxl tl , a description, alternative to (10.15), (10.16) and (10.17), appears Slm (w) = h3w

" X (w)n n=0

n!

Wlmn ()

(10.24)

where W () is a stationary transition probability matrix and, hence, a stochastic matrix. We also observe that W () = T + L can be regarded as a rate matrix, with the property that, for each state l, Q X

Wlm () =

m=1

Q X m=1

Tlm + 

Q X

lm = 

m=1

the transition rate in any state l is precisely the same, equal to . Whereas the embedded Markov chain defined by (10.21) has no self-transitions (Yll = P tl 0), we see for any l and m, that Wll () = 1  1 Q m=1;m6=l tlm = 1    0 while t Wlm () = lm . Hence, W () can be interpreted as an embedded Markov chain that allows self-transitions. In view of (10.21), the embedded structure of W () is summarized as tlm = Wlm ()

for l 6= m

tll = 1  Wll () where the constant rate tl =  for any state l is, besides self-transitions tll 6= 0, the characterizing property. These properties also reveal that, starting from the embedded chain Y where Yll = 0, we can add self-transitions tll A 0 with the eect that, on (10.7), the transition rate tl $ tl + tll . The opposite figure illustrates an embedded Markov chain with self-transitions and the corresponding transition probability matrix, where the transition P t rates tl follow from 6m=1 Ylm = 1 with Ylm = tlml . This change in transition rate will change the steady-state vector since the balance equations (10.20) change. However, the Markov process {[(w)> w  0} is not modified because a self-transition does not change [(w) nor the distribution of the time until the next transition to a dierent state. But self-transitions clearly change the number of transitions during some period of time. When the transition

10.4 The embedded Markov chain

rate tm at each state m are the same, the embedded Markov called a uniformized chain. 5 0 0 tt121 q22 9 t22 q12 q46 0 9 0 1 t2 9 q61 q24 t32 t33 2 9 0 4 t3 t3 Y =9 t43 q51 q32 9 0 0 q65 9 t 6 q52 q43 9 t51 t52 t534 7 3 t5 t5 t5 5 q53 q56 t61 0 0 t6 q36

q33

191

chain W () is 0 t24 t2

0 0 0 0

0 0 0 0 0 t65 t6

6 0 : 0 : t36 : : t3 : t46 : t4 : t56 : t5 8 0

In addition, in a uniformized chain, the steady-state vector w() of W () is the same as the steady-state vector . Indeed, from (9.23), wm () =

Q X

Wnm ()wn ()

n=1

we have, with W () = L +

T ,

¶ Q µ X tnm wm () = wn () nm +  n=1

= wm () +

Q 1X wn ()tnm  n=1

or, wn ()tn =

Q X

wn ()tnm

n=1;n6=m

where wn () = n (independent of ) since it satisfies the balance equation (10.20) and Theorem 9.3.5 assures that the steady-state of a positive recurrent chain is unique. We will now interpret (10.24) probabilistically. Let Q(w) denote the total number of transitions in [0> w] in the uniformized (discrete) process {[n ()}. Since the transition rates tl =  are all the same, Q (w) is a Poisson process with rate  because, for any continuous-time Markov chain, the inter-transition or sojourn times are i.i.d. exponential random variables. n is recognized as the probability that the Thus, Pr [Q (w) = n] = h3w (w) n! number of transitions that occur in [0> w] in the uniformized Markov chain with rate  equals n. With (9.10), Wlmn () = Pr [[n () = m|[0 () = l] is the n-step transition probability of that discrete {[n ()} uniformized Markov

192

Continuous-time Markov chains

process. Relation (10.24) can be interpreted as Slm (w) =

" X

Pr [[n () = m|[0 () = l> Q (w) = n] Pr [Q (w) = n]

n=0

or, the probability that the continuous Markov process moves from state l to state m in a time interval of length w, can be decomposed in an infinite sum of probabilities. Each probability corresponds to a transition from state l to state m in n-steps, where the number of intermediate transitions n is a Poisson counting process with rate . 10.4.2 A sampled-time Markov chain The sampled-time Markov chain approximates the continuous Markov process in that the transition probabilities Slm (w) are expanded to first order as in (10.8) with fixed step k = w. The transition probabilities of the sampledtime Markov chain are Slm = tlm w

(l 6= m)

Sll = 1  tl w Clearly, the sampled-time Markov chain also allows self-transitions, as illustrated in Fig. 10.1. 1  q12't q12

1 q51

2 q52

5

q23

3

q53

q34 4

q45 Continuous-time Markov process

1

1  q23't q12't

2

q23't q53't

q51't

1  q34't 3 q34't

q52't

4 1  q45't q45't 1  (q51 + q52 + q53 't Sampled-time Markov chain 5

Fig. 10.1. A continuous-time Markov process and its corresponding sampled-time Markov chain.

From (10.8), we observe that the approximation lies in two facts: (a) w is fixed such that tlm w  Pr [[(w + w) = m|[(w) = l] is increasingly accurate as w $ 0 and (b) transitions occur at discrete times every w time units. The sampling step w should be chosen such that the transition probabilities obey 0  Slm  1, from which we find that w  max1l tl .

10.5 The transitions in a continuous-time Markov chain

193

Let y denote the steady-state vector of the sampled-time Markov chain with kyk1 = 1. Being a discrete Markov chain, the steady-state vector components ym satisfy (9.23) for each component m, ym =

Q X

Q X

Snm yn = w

n=1

tnm yn + (1  tm w) ym

n=1;n6=m

or tm ym =

Q X

tnm yn

n=1;n6=m

By comparing with the balance equation (10.20) and on the uniqueness of the steady-state (Theorem 9.3.5), we observe that y =  or, the steady-state of the sampled-time Markov chain is exactly (not approximately) equal to the steady-state of the continuous Markov chain for any sampling step w  max1l tl . Although we can possibly miss by sampling every w time units the smaller-scale dynamics of the continuous Markov chain, the long-run behavior or steady-state is exactly captured!

10.5 The transitions in a continuous-time Markov chain Based on the embedded Markov chain, there exists a framework that deduces all properties of the continuous Markov chain. In particular, the exponential sojourn times of a continuous-time Markov (Theorem 10.2.3) chain is postulated as a defining characteristic. Theorem 10.5.1 Let Ylm denote the transition probabilities of the embedded Markov chain and tlm the rates of the infinitesimal generator. The transition probabilities of the corresponding continuous-time Markov chain are found as Z w X Slm (w) = lm h3tl w + tl Yln h3tl x Snm (w  x)gx (10.25) n6=l

0

Proof: If l is an absorbing state (tl = 0), then, by definition, Slm (w) = lm for all w  0. For a non-absorbing state l and a process starting from state l, the event {  w> [( ) = n} _ {[(w) = m} is possible if and only if the first transition from l to n occurs at some time x 5 [0> w] and the next transition from n to m takes place in the remaining time w  x. The probability density g Pr [l  w] = tl h3tl w for w  0 and function of the sojourn time is il (w) = gw

194

Continuous-time Markov chains

for infinitesimally small , we have s = Pr [  w> [( ) = n> [(w) = m|[(0) = l] Z w gx Pr [ = x> [(x  ) = l|[(0) = l] = 0

× Pr [[(x) = n|[(x  ) = l] Pr [[(w) = m|[(x) = n] Z w gxil (x)Yln Snm (w  x) = 0 Z w tl h3tl x Snm (w  x)gx = Yln 0

Furthermore, Pr[  w and [(w) = m|[(0) = l] =

X

Pr[w  > [( ) = n> [(w) = m|[(0) = l]

n6=l

and Pr [ A w and [(w) = m|[(0) = l] = lm Pr [l A w] = lm h3tl w Finally, Slm (w) = Pr [[(w) = m|[(0) = l] = Pr[  w and [(w) = m|[(0) = l] + Pr[ A w and [(w) = m|[(0) = l] Combining all above relations into the last one proves the theorem. By a change of variable v = w  x in (10.25), we have Slm (w) = lm h3tl w + tl

X

Z Yln h3tl w

w

htl v Snm (v)gv

0

n6=l

and, after dierentiation with respect to w, we find for w  0, Z w X X tn Yln h3tl w htl v Snm (v)gv + tl Yln Snm (w) Slm0 (w) = tl lm h3tl w  tl n6=l

0

n6=l

X ¡ ¢ = tl lm h3tl w  tl Slm (w)  lm h3tl w + tl Yln Snm (w) = tl Slm (w) + tl

X n6=l

n6=l

Yln Snm (w)

¤

10.6 Example: the two-state Markov chain in continuous-time

195

Evaluated at w = 0, recalling that S 0 (0) = T and S (0) = L, X Yln Snm (0) Slm0 (0) = tl Slm (0) + tl tlm = tl lm + tl

X

n6=l

Yln nm = tl lm + tl Ylm

n6=l

which is precisely relation (10.21). With tl = tll and (10.21), we arrive at Slm0 (w)

=

Q X

tln Snm (w)

n=1

which is precisely the backward equation (10.10). Hence, (10.25) can be interpreted as an integrated form of the backward equation and thus of the entire continuous-time Markov process.

10.6 Example: the two-state Markov chain in continuous-time The continuous-time two-state Markov chain is defined by the infinitesimal generator ¸    T=   where >   0. We will solve S (w) from the forward equation (10.9), ¸  0 ¸  ¸ 0 (w) S11 (w) S12 S11 (w) S12 (w)   = 0 (w) S 0 (w)   S21 S21 (w) S22 (w) 22 which actually contains two independent transition probabilities because S12 (w) = 1  S11 (w) and S21 (w) = 1  S22 (w). The forward equation simplifies to 0 (w) = ( + )S11 (w) +  S11 0 (w) = ( + )S22 (w) +  S22

Only the first equation needs to be solved since, by symmetry, the solution of S11 (w) equals that of S22 (w) after changing the role of  $  and  $ . The linear, first-order, non-homogeneous dierential equation consists of the solution to the corresponding homogeneous dierential equation and a particular solution. The solution of the homogeneous dierential equation, 0 (w) = ( + )S (w), is S (w) = Fh3(+)w . The particular solution S11 11 11 is generally found by variation of the constant F, which proposes S11 (w) =

196

Continuous-time Markov chains

F(w)h3(+)w as general solution, where F(w) needs to satisfy the original dierential equation. Hence, F 0 (w) = h(+)w  (+)w or, after integration, F(w) = + h + f. The integration constant f follows from the initial condition (10.5), S11 (0) = 1. Finally, we arrive at

 3(+)w  + h + +   3(+)w S22 (w) = + h + +

S11 (w) =

from which the steady-state vector is immediate, i h    = + + 10.7 Time reversibility In this section, we consider only ergodic Markov chains that have a non-zero steady-state distribution . Suppose the Markov process operates already in the steady-state, or, in other words, the Markov process is stationary. We are interested in the time-reversed process defined by the sequence [q > [q31 > = = = We will show that this reversed time sequence again constitutes a Markov process. Theorem 10.7.1 The time-reversed Markov process is a Markov chain. Proof: It su!ces to demonstrate that the time-reversed process satisfies the Markov property Pr [[q = {q |[q+1 = {q+1 > = = = > [q+n = {q+n ] = Pr [[q = {q |[q+1 = {q+1 ] By definition of the conditional probability (2.44), U = Pr [[q = {q |[q+1 = {q+1 > [q+2 = {q+2 > = = = > [q+n = {q+n ] ¤ £ = Pr [q = {q | _np=1 {[q+p = {q+p } ¤ £ Pr _np=0 {[q+p = {q+p } £ ¤ = Pr _np=1 {[q+p = {q+p } Since the intersection is commutative D _ E = E _ D, the indices can be reversed, ¤ £ Pr _0p=n {[q+p = {q+p } U= Pr [_1p=n {[q+p = {q+p }] ¤ £ ¤ £ Pr [q+n = {q+n | _0p=n1 {[q+p = {q+p } Pr _0p=n1 {[q+p = {q+p } = Pr [_1p=n {[q+p = {q+p }]

10.7 Time reversibility

197

The original stationary process is a Markov process that satisfies (9.2). Using (9.2) and (9.3) we have ¤ £ Pr [q+n = {q+n | _0p=n1 {[q+p = {q+p } = Pr[[q+n = {q+n |[q+n1 = {q+n1] and n1 Y ¤ £ Pr[[q+p = {q+p |[q+p1 = {q+p1 ] Pr _0p=n1 {[q+p = {q+p } = Pr[[q = {q ] p=1

and, similarly, n Y ¤ £ Pr[[q+p = {q+p |[q+p1 = {q+p1 ] Pr _1p=n {[q+p = {q+p } = Pr[[q+1 = {q+1 ] p=2

Hence, Pr [[q+n = {q+n |[q+n1 = {q+n1 ] Pr [[q+1 = {q+1 |[q = {q ] Pr [[q = {q ] Pr [[q+n = {q+n |[q+n1 = {q+n1 ] Pr [[q+1 = {q+1 ] Pr [[q+1 = {q+1 |[q = {q ] Pr [[q = {q ] = Pr [[q+1 = {q+1 ]

U=

Applying Bayes’ rule (2.48) to the last relation finally proves the theorem.

¤

Consider the transition probability of the time-reversed Markov process Ulm = Pr [[q = m|[q+1 = l] With Bayes’ rule (2.48), Pr [[q = m|[q+1 = l] =

Pr [[q+1 = l|[q = m] Pr [[q = m] Pr [[q+1 = l]

and, since the process is stationary, Pr [[q = m] = m > Pr [[q+1 = l] = l the transition probability of the time-reversed process is Ulm =

m Sml l

(10.26)

A Markov chain is said to be time reversible if, for all l and m, Slm = Ulm . From (10.26), the condition for time reversibility is l Slm = m Sml

(10.27)

This condition means that, for all states l and m, the rate l Slm from state l $ m equals the rate m Sml from state m $ l. An interesting property of time reversible Markov chains is that any vector { satisfying k{k1 = 1 and

198

Continuous-time Markov chains

{l Slm = {m Sml is a steady-state vector of a time-reversible Markov chain. Indeed, summing over all l, X X {l Slm = {m Sml = {m l

l

Theorem 9.3.5 indicates that the steady-state is unique and, thus, { = . As a side remark, we note that a transition matrix is only equal to its transpose S = S W if the Markov process is time reversible and doubly stochastic (i.e. l = Q1 for all l, as shown in Appendix A.5.1). The continuous-time analogon can be immediately deduced from the discrete-time embedded Markov chain defined by the transition probabilities Ylm . Let Xlm denote the transition probabilities of the time-reversed embedded Markov chain and ulm the rates of the corresponding continuous Markov chain, then by (10.21) ulm = ul Xlm

(10.28)

We will now show that the rates ul of the time-reversed continuous Markov process are indeed exponential random variables. Assume that the timereversed process is in state l at time w. The probability that the process is still in state l at reversed time w  x is, using Theorem 10.2.3, Pr [[( ) = l> w  x    w] Pr [[(w) = l] Pr [[(w  x) = l] h3tm w = h3tm w = Pr [[(w) = l]

Pr [[( ) = l> w  x    w|[(w) = l] =

because, in steady-state w $ 4, Pr [[(w  x) = l] = Pr [[(w) = l] = l for any finite x. Thus, the sojourn time in state l of the time-reversed process is exponentially distributed with precisely the same rate ul = tl as the forward time process. The steady-state vector y of the embedded Markov chain can be written in terms of the steady-state vector of the continuous Markov chain via (10.22). By (10.26), we obtain Xlm =

ym Yml m tm Yml = yl l tl

With (10.21) and (10.28) l ulm m tml = ul tl but, since ul = tl , we finally arrive at l ulm = m tml

(10.29)

10.8 Problems

199

Comparing (10.29) with the discrete case (10.26), we see that the transition probabilities Slm and Ulm are changed for the rates tlm and ulm . We know that m is the portion of time the process (both forward and reversed) spend in state m and that tlm is the rate at which the process makes transitions from state l to state m. Equation (10.29) has again a balance interpretation: m tml is the rate at which the forward process moves from state m to l, while l ulm is the rate of the time-reversed process from state l to m and both rates are equal. Intuitively, when a process jumps from state l $ m in forward time, it is plain that the process makes, in reversed time, just the opposite transition from m $ l. Similarly as above, a continuous-time Markov chain is time reversible if, for all l and m, it holds that ulm = tlm . For these processes (which occur often in practice, as demonstrated in the chapters on queueing), the rate from l $ m is equal to the rate from m $ l since l tlm = m tml .

10.8 Problems (i) Consider a computer that has two identical and independent processors. The time between failures has an exponential distribution. The mean value of this distribution is 1000 hours. The repair time for a damaged processor is exponentially distributed as well, with a mean value of 100 hours. We assume damaged processors can be repaired in parallel. There are clearly three states for this computer: (1) both processors work, (2) one processor is damaged and (3) both processors are damaged. (a) Make a continuous Markov chain presentation of these states. (b) What is the infinitesimal generator matrix T for this Markov chain? Give the relation between the state probability at time w and its derivative. (c) Calculate the steady-state of this process. (d) What is the availability of the computer if (i) both processors are required to work, or (ii) at least one processor should work. (ii) Consider two identical servers that are working in parallel. When one server fails, the other has to do the whole job alone under a higher load. The failure times of servers are exponentially distributed: H = 3 × 1034 k31 , when the servers are equally loaded and F = 7 × 1034 k31 , when one of the servers works under the “full load”. In addition, both servers may fail at the same time with a failure rate of E = 6 × 1035 k31 . As soon as one of the servers fails, the repair is initiated. The

200

Continuous-time Markov chains

average downtime of a server is 31 = 10 hours. However, if both servers are damaged, the whole system must be shut down. The average time needed to repair both damaged servers is 31 B = 20 hours. (a) (b) (c) (d) (e)

Draw the Markov chain for this system. Determine the infinitesimal generator matrix T. Determine the steady-state probabilities. Determine the average lifetime of dierent states. What is the average number of server repairs needed during a period of one year?

11 Applications of Markov chains

This chapter illustrates the theory of Markov chains with several examples. Examples of queueing problems are deferred to later chapters. Generally, Markov processes can be solved explicitly provided the transition probability matrix S or the infinitesimal generator T has a special structure. Only in a very small number of problems is the entire time dependence of the process available in analytic form.

11.1 Discrete Markov chains and independent random variables This section illustrates examples of some simple Markov chains. Consider a set {\q }qD1 of positive integer, independent random variables that are identically distributed with Pr [\ = n] = dn . The discrete-time Markov process, defined by [q = \q for q  1, possesses the (infinite) transition probability matrix, 6 5 d1 d2 d3 d4 · · · 9 d1 d2 d3 d4 · · · : : 9 : S =9 9 d1 d2 d3 d4 · · · : 7 d1 d2 d3 d4 · · · 8 ··· ··· ··· ··· ··· All rows are identical and Pr [[q+1 = m|[q = l] = dm shows that the states [q+1 and [q are independent from each other. Another, more interesting, discrete-time Markov process is defined by [q = max [\1 > \2 > \3 > = = = > \q ] = Hence, the process [q mirrors the maxima of the first q random variables. Clearly, [q+1 = max [[q > \q+1 ] reflects the Markov property: the next state is only dependent on the previous state of the process. From [q+1 = 201

202

Applications of Markov chains

max [[q > \q+1 ], we observe that Pr [[q+1 = m|[q = l] = 0 if m ? l because the maximum does not decrease by adding a new random variable in the list. If m A l, the state m is determined by \q+1 = m with probability dm . If m = l, then \q+1  m, which has probability Pr [\q+1  m] =

m X

Pr [\q+1 = n] =

n=1

m X

dn = Dm

n=1

The corresponding probability matrix 5 D1 d2 9 0 D2 9 S =9 0 9 0 7 0 0 ··· ···

is

6 d4 · · · d4 · · · : : d4 · · · : : D4 · · · 8 ··· ··· P A related discrete-time Markov chain is [q = qn=1 \n which obeys [q+1 = [q + \q+1 . Furthermore, if m  l, Pr [[q+1 = m|[q = l] = 0 because the random variables \q are non-negative such that the sum cannot decrease by adding a new member. If m A l, then d3 d3 D3 0 ···

Pr [[q+1 = m|[q = l] = Pr [[q + \q+1 = m|[q = l] = Pr [\q+1 = m  l] = dm3l The corresponding probability matrix has the same elements on diagonal lines 5 0 d1 d2 9 0 0 d1 9 9 S =9 0 0 0 7 0 0 0 ··· ··· ···

a Toeplitz structure possessing d3 d2 d1 0 ···

··· ··· ··· ··· ···

6 : : : : 8

This list can be extended by considering other integer functions of the set Q {\q }qD1 (such as [q+1 = min [[q > \q+1 ] or [q = qn=1 \n etc.). 11.2 The general random walk The general random walk is an important model that describes the motion of an item that is constrained to moving either one step forwards, stay at the position where it currently is or move one step backwards. In general, this “three-possibility motion” has transition probabilities that depend on the position m as depicted in Fig. 11.1. Figure 11.1 illustrates that, if the process is in state m, it has three possible choices: remain in state m with probability

11.2 The general random walk

203

um = Pr [[n+1 = m|[n = m], move to the next state m + 1 with probability sm = Pr [[n+1 = m + 1|[n = m] or jump back to state m  1 with probability tm = Pr [[n+1 = m  1|[n = m]. A general random walk is defined by the (Q + 1) × (Q + 1) band matrix 5 9 9 9 9 S =9 9 9 7

u0 s0 0 0 t1 u1 s1 0 0 t2 u2 s2 .. .. .. .. . . . . 0 0 0 0 0 0 0 0

··· ··· ··· .. . ··· ···

0 0 0 .. .

0 0 0 .. .

6

0 0 0 .. .

tQ31 uQ31 sQ31 0 tQ uQ

: : : : : : : 8

(11.1)

where sm A 0, tm A 0, um  0 and tm + um + sm = 1 for all 0  m  Q . The bordering states zero and Q are special: s0  0, u0  0 and s0 + u0 = 1 and tQ  0, uQ  0 and tQ + uQ = 1. rj 0

...

j1

pj j

j+1

...

N

qj

Fig. 11.1. A transition graph of the general random walk.

The general random walk serves as model for a number of practical phenomena: • The one-dimensional motion of physical particles, electrons that hop from one atom to another. In this case, the number of states Q can be very large. • The gambler’s ruin problem: a state m reflects the capital of a gambler whereas the sm is the chance that the gambler wins while tm is the probability that he looses. The gambler achieves his target when he reaches state Q , but he is ruined at state 0. In that case are both states absorbing states with u0 = uQ = 1. In games, most often the probabilities are independent of the state and simplify to sm = s, tm = t and um = 1  s  t= • The continuous-time counterpart, the birth and death process (Section 11.3), has applications to queueing processes. For a wealth of examples and applications of the random walk, we refer to the classical treatise of Feller (1970, Chapter III, XIV).

204

Applications of Markov chains

11.2.1 The probability of gambler’s ruin The probability of gambler’s ruin is defined as xm = Pr [[W = 0|[0 = m] where W = minn {[n = 0} is the hitting time to state 0, which is equivalent to xm = Pr [W0 ? 4|[0 = m]. By definition, x0 = 1 and since the gambler achieves his goal at state Q , he stops and never gets ruined, xQ = 0. The law of total probability (2.46) gives the situation after the first transition, Pr [[W = 0|[0 = m] = =

Q X n=0 Q X

Pr [[W = 0|[0 = m> [1 = n] Pr [[1 = n|[0 = m] Pr [[W = 0|[1 = n] Pr [[1 = n|[0 = m]

n=0

(Markov property) =

Q X

Smn Pr [[W = 0|[1 = n]

n=0

= tm Pr [[W = 0|[1 = m  1] + um Pr [[W = 0|[1 = m] + sm Pr [[W = 0|[1 = m + 1] After the first transition, the probability Pr [[W = 0|[1 = m] = xm remains the same as the initial Pr [[W = 0|[0 = m] because W is a random variable depending on the state and not on the discrete-time. Hence, we obtain the equations x0 = 1 xm = tm xm31 + um xm + sm xm+1

(1  m ? Q )

which are dierent from the corresponding steady-state equations (11.5). The dierence lies in left-multiplication of S , yielding x = S x instead of right-multiplication in  = S . Substituting um = 1  sm  tm gives after some modification µ ¶ tm tm (11.2) xm+1 =  xm31 + 1 + xm sm sm Iteration on m for the first few values using x0 = 1 yields µ ¶ t1 t1 x2 =  + 1 + x1 s1 s1 µ µ ¶ ¶ µ ¶ t2 t2 t1 t1 t2 t1 t2 t1 x3 = x1 + 1  + + x2 =  + 1+ x1 s2 s2 s1 s1 s2 s1 s1 s2

11.2 The general random walk

205

which suggests ! Ã m31 Y m31 Y n n X X tp tp x1 xm =  + 1+ s s p p p=1 p=1 n=1

n=1

as readily verified by substitution in (11.2). The unknown x1 is determined by the last relation, xQ = 0. Finally, the probability of gambler’s ruin is PQ31 Qn tp Pr [[W = 0|[0 = m] =

n=m

1+

p=1 sp tp n=1 p=1 sp

PQ31 Qn

(11.3)

Similarly, the mean hitting time m = H [W |[0 = m] follows by reasoning on the possible transitions. From state m, there is a transition to state m  1 with probability tm . In case of a transition from state m to state m  1, the hitting time W consists of the first transition plus the remaining time from state m  1 on which is m31 . Using the law of total probability or reading all possible transitions from the transition graph (see Fig. 11.1), we find m = 1 + tm m31 + um m + sm m+1 with the boundary equations for the state 0 and state Q (which in the gambler’s ruin problem are absorbing states), 0 = Q = 0. With um = 1  sm  tm , µ ¶ tm tm 1 m+1 =   m31 + 1 + m sm sm sm By iteration, 1 s1 1 3 =  s2 1 4 =  s3 1 = s1 2 = 

¶ µ t1 1 + 1+ s1 ¶ µ ¶ µ ¶ µ t2 t1 t2 t2 1 1 t1 t2 2 =   1+ + 1+ 1  1 + 1 + + s2 s2 s1 s2 s1 s1 s1 s2 ¶ µ t3 t3 3  2 + 1 + s3 s3 µ ¶ µ ¶ µ ¶ t2 1 t3 t1 1 t2 t3 t1 t2 t1 t2 t3 1+  1+ + 1+ 1  + + + s2 s1 s3 s2 s1 s2 s1 s1 s2 s1 s2 s3

or m31 X 1 m =  sn n=1

Ã

n31 Y q X tn3p+1 1+ sn3p q=1 p=1

!

à +

m31 Y n X tp 1+ sp

! 1

n=1 p=1

Eliminating 1 from Q = 0, finally leads to mean hitting time to ruin or

206

Applications of Markov chains

the mean duration of the game à ! m31 q n31 Y X X 1 tn3p+1 m =  1+ sn sn3p q=1 p=1 n=1 ´ Pn31 Qq ! PQ31 1 ³ à tn3p+1 m31 Y n 1 + X n=1 sn q=1 p=1 sn3p tp + 1+ PQ31 Qn s 1 + n=1 p=1 tp p=1 p

(11.4)

sp

n=1

In spite of the relatively simple dierence equations, the solution rapidly grows unattractive. The particular case of the Markov chain where tn = t and sn = s simplifies considerably. The probability of gambler’s ruin (11.3) becomes ³ ´m ³ ´Q PQ31 ³ t ´n t  st n=m s s = Pr [[W = 0|[0 = m] = ³ ´Q PQ31 ³ t ´n t 1  n=0 s s and, if s = t = 12 , via de l’Hospital’s rule, Pr [[W = 0|[0 = m] = target fortune Q (at which the game ends) is infinitely large, Pr [[W = 0|[0 = m] = 1 µ ¶m t = s

Q3m Q .

If the

if t  s if t ? s

which demonstrates that the gambler surely will loose all his money if his chances s on winning are smaller than those t on losing. Even in a fair game where s = t, he will be defeated surely. In a favorable game (s A t) ³ ´m and with start capital m, ruin is possible with probability st . Another interpretation is a game with two players d and e in which player d starts with capital m and has winning chance of s, while player e starts with capital Q  m and wins with probability t = 1  s. Similarly, the mean duration of the game (11.4) simplifies to 3 ³ ´n 4 3 ³ ´m 4 ³ ´n t t t m31 Q31 1  1  1  XE s s s FX F E m =  C D+C ³ ´Q D st st n=1 n=1 1  st or

5 H [W |[0 = m] =

3

³ ´m 4 t s

6

1 9 E1 : F 7Q C ³ ´Q D  m 8 st 1  st

11.2 The general random walk

207

11.2.2 The steady-state The steady-state equation (9.23) for the vector component m becomes, for 1  m ? Q, m = sm31 m31 + um m + tm+1 m+1

(11.5)

and, for m = 0 and m = Q, 0 = u0 0 + t1 1 Q = sQ31 Q31 + uQ Q We rewrite these equations using um = 1tm sm , u0 = 1s0 and uQ = 1tQ as s0 0 = t1 1

(11.6)

sm m = (sm31 m31  tm m ) + tm+1 m+1

(11.7)

sQ31 Q31 = tQ Q Explicitly, for a few values of m, we observe that s1 1 = (s0 0  t1 1 ) + t2 2 = t2 2 s2 2 = (s1 1  t2 2 ) + t3 3 = t3 3 ··· or, in general, for all m, sm m = tm+1 m+1 By iteration of m+1 = m+1

sm tm+1 m

starting at m = 0, we find

m Y sm sm31 sp s0 = · · · 0 = 0 tm+1 tm t1 tp+1 p=0

The normalization kk1 = 1 yields a condition for 0 , 431 3 Q m31 X Y sp D 0 = C1 + t p=0 p+1 m=1

which determines the complete steady-state vector  for the general random walk as Qm31 sp m =

p=0 tp+1

1+

PQ Qm31 m=1

sp p=0 tp+1

(11.8)

These relations remain valid even when the number of states Q tends to infinity provided the infinite sum converges.

208

Applications of Markov chains

In the simple case where sn = s and tn = t, we obtain with  = st , m =

(1  ) m 1  Q+1

(11.9)

11.3 Birth and death process A birth and death process1 is defined by the infinitesimal generator matrix 5 9 9 9 T=9 9 9 7

0 0 0 0 0  (1 + 1 ) 1 0 0 0  (2 + 2 ) 2 0 0 2  (3 + 3 ) 3 0 0 3  (4 + 4 ) 4 0 0 4 .. .. .. .. .. . . . . .

0 1 0 0 0 .. .

0 0 0 0 0 .. .

··· ··· ··· ··· ··· .. .

6 : : : : : : 8

The transition graph is shown in Fig. 11.2. Although the theory in the previous chapter was derived for finite-state Markov chains, the birth and death process is a generalization to an infinite number of states. The general random walk (Section 11.2) forms the embedded Markov chain of the birth and death process with transition probabilities specified by (10.21) resulting l l , Yl>l+1 = l+ and Yln = 0 for n 6= l  1 6= l + 1. in Yl>l31 = l+ l l The transition probability matrix is a tri-band diagonal matrix which is irreducible if all m A 0 and m A 0. O0

Oj1 ...

0

j1

P1

Oj j

Pj

j+1

...

Pj+1

Fig. 11.2. The transition graph of a birth and death process.

The basic system of dierential equations that completely describes the birth and death process follows from the general state probability equations (10.11) for vn (w) = Pr [[(w) = n] as v00 (w) = 0 v0 (w) + 1 v1 (w)

(11.10)

v0n (w) =  (n + n ) vn (w) + n31 vn31 (w) + n+1 vn+1 (w)

(11.11)

with initial condition vn (0) = Pr [[(0) = n]. Exact analytic solutions for 1

Kleinrock (1975, p. ix) mentions that William Feller was the father of the birth and death process.

11.3 Birth and death process

209

any n and n are not possible. Indeed, let us denote the Laplace transform of vn (w) by Z " h3}w vn (w)gw= (11.12) Vn (}) = 0

Since vn (w) is a continuous and bounded function (|vn (w)|  1 for all w A 0), the Laplace transform exists for Re(}) A 0. The Laplace transform of (11.10) and (11.11) becomes, (0 + }) V0 (}) = v0 (0) + 1 V1 (})

(11.13)

(n + n + }) Vn (}) = vn (0) + n31 Vn31 (}) + n+1 Vn+1 (})

(11.14)

which is a set of dierence equations more complex due to the initial condition vn (0) than the set (A.51) in Appendix A.5.2.3. That set (A.51) appears in the general random walk whose solution is shown to be intractable in general. This infinite set of dierential equations has been thoroughly studied over years under several simplifying conditions for n and n , for example, n =  and n =  for all n. As shown in Chapter 13, they form the basis for the simplest set of queueing models of the family M/M/m/K.

11.3.1 The steady-state The steady-state follows from (10.19) as solution of the set 0 0 + 1 1 = 0 m31 m31  (m + m ) m + m+1 m+1 = 0 This set is identical to (11.6) and (11.7) provided sm and tm are changed for m and m . After this modification in (11.8), the steady-state of the birth and death process is 0 =

m =

p p=0 p+1 m=1 Qm31 p p=0 p+1 P" Qm31 p + m=1 p=0 p+1

1+

1

1 P" Qm31

(11.15)

m1

(11.16)

Theorem 9.3.4 states that an irreducible Markov chain with a finite number of states is necessarily recurrent. However, it is in general di!cult to decide whether an irreducible Markov chain with an infinite number of states is recurrent or transient. In case of the birth and death process, it is possible to determine when the process is transient or recurrent. The process is transient if and only if the embedded Markov chain (determined

210

Applications of Markov chains

above) is transient. Section 9.2.3 discusses that, for a recurrent chain, ulm = Pr [Wm ? 4|[0 = l] equals 1: a finite hitting time means every state m is certainly visited starting from initial state l. Applied to the embedded Markov chain, it follows from the gambler’s ruin (11.3) that Pm31 Qn tp n=0

p=1 sp tp p=1 sp

Pr [W0 ? 4|[0 = m] = 1  PQ31 Qn n=0

Thus, for any fixed initial state m, the condition for a recurrent chain Pr [Wm ? 4|[0 = l] = 1 P Qn tp is only possible in the limit Q $ 4 if limQ }) = n=0 0

"

0

h3}w

" X

vn (w){n gw

n=0

where the reversal of summation and integration is allowed because all terms are positive. Since 0  vn (w)  1, the sum is at least convergent for |{| ? 1, which shows that *({> }) is analytic inside the unit circle |{| ? 1 for any Re(}) A 0. After multiplying (11.14) by {n and summing over all n, we obtain ( + }) V0 (}) + ( +  + })

" X n=1

Vn (}){n =

" X

vn (0){n + 

n=0

+ V1 (}) + 

" X

Vn31 (}){n

n=1 " X n=1

Vn+1 (}){n

11.3 Birth and death process

215

and, written in terms of *({> }), P n+1 +  (1  {) V (})  " 0 n=0 vn (0){ *({> }) = {2  { ( +  + }) +  Note that in the general case n and n , an expression in terms of *({> }) is not possible. Before continuing with the computations, we make the additional simplification that vn (0) = nm : we assume that the constant rate birth and death process starts in state m. With this initial condition, the generating function *({> }) =

 (1  {) V0 (})  {m+1 {2  { ( +  + }) + 

(11.25)

still depends on the unknown function V0 (}). The following derivation involving the theory of complex functions demonstrates a standard procedure that will also be useful in other queuing problems. The denominator in (11.25) has two roots, q ++} 1 ( +  + })2  4 + {1 = 2 2 q ++} 1 ( +  + })2  4  {2 = 2 2 We need the powerful theorem of Rouché (Titchmarsh, 1964, p. 116) to deduce more on the location of {1 and {2 . Theorem 11.3.1 (Rouché) If i (}) and j(}) are analytic inside and on a closed contour C, and |j(})| ? |i (})| on C, then i (}) and i (}) + j(}) have the same number of zeros inside C. Choose i ({) =   { ( +  + }) and j({) = {2 such that i({) + j({) = {2  { ( +  + }) + , the denominator in (11.25). Since both i ({) and j({) are polynomials, they are analytic everywhere in the complex {-plane. We know that *({> }) is analytic inside the unit disk. If the roots {1 or {2 lie inside the unit disk, the numerator in (11.25) must have zeros at precisely the same place in order for *({> }) to be analytic inside the unit disk. Hence, we consider as contour C in Rouché’s Theorem, the unit circle |{| = 1. Clearly,  inside the unit circle (because  A 0,  A 0 i ({) has one single zero ++} and Re(}) A 0). Furthermore, on the unit circle |{| = 1, |  { ( +  + })|  |  |{| |( +  + })|| = |  |( +  + })|| A  = |{2 | which shows that |j(})| ? |i (})| on the unit circle. Rouché’s Theorem then tells us that i ({) + j({) has precisely one zero inside the unit circle. This

216

Applications of Markov chains

implies that |{1 | A 1 and |{2 | ? 1 and that the numerator in (11.25) has a zero {2 ,  (1  {2 ) V0 (})  {m+1 =0 2 This relation determines the unknown function V0 (}) as V0 (}) =

{m+1 2  (1  {2 )

such that (11.25) becomes 13{ m+1 {m+1 2 13{2  {

(1  {)  {m+1 (1  {2 ) {m+1 = 2 *({> }) =  ({  {1 ) ({  {2 )  (1  {2 ) ({  {1 ) ({  {2 ) We know that the numerator can be divided by ({  {2 ), or explicitly, {m+1 (1  {)  {m+1 (1  {2 ) = {m+1  {m+1 + {2 {({m  {m2 ) 2 2 # " m m31 X m3n X = ({  {2 )  {2 {n + {2 { {m313n {n 2 " = ({  {2 )

n=0

{m2

n=0

+ ({2  1)

m X

#

n {m3n 2 {

n=1

Finally,

P n {m2 + ({2  1) mn=1 {m3n 2 { *({> }) =  (1  {2 ) ({  {1 )

(11.26)

By expanding the denominator in a Taylor series around { = 0, and denoting d0 = {m2 dn = ({2  1){m3n 2 dn = 0 we have 1 *({> }) =  (1  {2 ) =

1  (1  {2 )

nAm Ã" X

!Ã dn {n

n=0 Ã n " X X n=0



dn3p {p 1 p=0

" X

! p {3p 1 {

p=0

!

{n

Comparing with (11.24) and equating the corresponding powers in {, we

11.3 Birth and death process

217

find an explicit form of the Laplace transforms of the probabilities that the birth and death process is in state n, n X dn3p 1  (1  {2 ) p=0 {p 1 Ã ! n31 p n X 1 { {m3n { 2 2 1n?m = 2 +  (1  {2 ) {n1 p=0 {p 1 Ã ! m 13n 1 {  { { 1 {m2 {m3n 1 2 1 1n?m = + 2  (1  {2 ) {n1 {1  {2

Vn (}) =

This expression can be put in dierent forms by using relations among the zeros {1 and {2 , such as {1 + {2 = ++} and {1 {2 =  . This ingenuity  is required to recognize in Vn (}) a known Laplace transform. Otherwise, one has to proceed by computing the inverse Laplace transform by contour integration via (2.38). In any case, the computation needs advanced skills in complex function theory and we content ourselves here to present the result without derivation (see e.g. Cohen (1969, pp. 80—82)), i h (11.27) vn (w) = h3(+)w (n3m)@2 Ln3m (dw) + (n3m31)@2 Ln+m+1 (dw) 3(+)w

+h

n

(1  )

" X

3p@2 Lp (dw)

p=n+m+2

s where  =  , d = 2  and where Lv (}) denotes the modified Bessel function (Abramowitz and Stegun, 1968, Section 9.6.1). Using the asymptotic formulas for the modified Bessel function, the behavior of vn (w) for large w can be derived (see e.g. Cohen (1969, p. 84)), s 2 µ ¸ s ¶ s ¶µ   (nm)@2 h(1 ) w 1 m + R(w ) n vn (w) = (1  ) + s s s ¡s ¢3@2 1  1  2  w ¤ 1 £ 1 + R(w1 ) only if  = 1 =s w n

This expression demonstrates that the constant rate birth death process ¡ s ¢2 converges to the steady-state (1  )n with a relaxation rate 1   . Clearly, the higher , the lower the relaxation rate and the slower the process tends to equilibrium as illustrated in Fig. 11.3. Intuitively, two eects play a role. Since the probability that states with large n are visited increases with increasing , the built-up time for this occupation will be larger. In addition, the variability of the number of visited states (further derived for the M/M/1 queue in Section 14.1) increases with increasing , which

218

Applications of Markov chains

0.10 U = 0.8 U = 0.9 U = 0.7

400 300

0.06

U = 0.6

200

0.04 100

0.02

U = 0.4

0.0

0.00

0.2

0.4

20

0.8

0

U

U = 0.2

0

0.6

Relaxation time

s4(t,U)

0.08

40

60

80

100

t Fig. 11.3. The probability v4 (w) that the process is in state 4 given that it started from state 0 as function of time (in units of average death s time,  = 1) for various  = . The insert shows the relaxation time  = (1  )2 (in units of average death time,  = 1). The corresponding steady state probability 4 are 0.0012, 0.015, 0.051, 0.072, 0.082, 0.065 for  = 0=2> 0=4> 0=6> 0=7> 0=8> 0=9 respectively. Observe that for  = 0=9, the plotted 100 time units are smaller than the relaxiation time, which is 379 time units.

suggests that larger oscillations of the sample paths around the steady-state are likely to occur, enlarging the convergence time.

11.4 A random walk on a graph Let J(Q> O) denotes a graph with Q nodes and O links. Suppose that the link weight zlm = zml of an edge from node l $ m (or vice versa) is proportional to the transition probability Slm that a packet at node l decides to move to node m. Clearly, zll = 0. Specifically, with (9.8), zlm Slm = PQ

n=1 zln

This constraint (9.8) destroys the symmetry in link weight structure (zlm = P PQ zml ) because, in general, Slm 6= Sml since Q n=1 zln 6= n=1 zmn . The sequence of nodes (or links) visited by that packet resembles a random walk on the graph J(Q> O) and constitutes a Markov chain. Moreover, the steadystate of this Markov process is readily obtained by observing that the chain is

11.5 Slotted Aloha

219

time reversible. Indeed, the condition for time reversibility (10.27) becomes l zlm m zml = PQ PQ n=1 zln n=1 zmn or, since zlm = zml , l PQ

n=1 zln

PQ

m = PQ

n=1 zmn

This implies that l =  n=1 zln and using the normalization kk1 = 1, we obtain the steady-state probabilities for all nodes l, PQ PQ zln n=1 zln l = PQ PQ = PQ n=1 PQ 2 l=1 n=l+1 zln l=1 n=1 zln This Markov process can model an active packet that monitors the network by collecting state information (number of packets, number of lost or retransmitted packets, etc.) in each router. Of course, the link weight structure zlm for the active packet is decisive and requires additional information to be chosen e!ciently. For example, for tra!c monitoring, the distribution of the number of packets forwarded by each router must be obtained. For the collection of these data, the active packet should in steady-state visit all nodes about equally frequently or l = Q1 , implying that the Markov transition matrix S must be doubly stochastic (see Appendix A.5.1). 11.5 Slotted Aloha The Aloha protocol is a basic example of a multiple access communication scheme of which Ethernet2 is considered as the direct descendant. Aloha — which means “hello” in the Hawaiian language — was invented by Norman Abramson at the university of Hawaii in the beginning of 1970s to provide packet-switched radio communication between a central computer and various data terminals at the campus. Slotted Aloha is a discrete-time version of the pure Aloha protocol, where all transmitted packets have equal length and where each packet requires one timeslot for transmission. Consider a network consisting of Q nodes that can communicate with each other via a shared communication channel (e.g. a radio channel) using the slotted Aloha protocol. The simplest arrival process D of packets at each node is a Poisson process. We assume that these Poisson arrivals at a 2

The essential dierence with the Ethernet’s CSMA/CD (carrier sense multiple access with collision detection) is that Aloha does not use carrier sensing and does not stop transmitting when collisions are detected. Carrier sensing is only adequate if the nodes are near to each other (as in a local area network) such that collisions can be detected before the completion of transmission. Only then is a timely reaction possible.

220

Applications of Markov chains

node are independent from the Poisson arrivals at another node and that all Poisson arrivals at a node have the same rate Q where  is the overall arrival rate at the network of Q nodes. The idea of the Aloha protocol is that, upon receipt of a packet, the node transmits that newly arrived packet in the next timeslot. In case two nodes happen to transmit a packet at the same timeslot, a collision occurs, which results in a retransmission of the packets. A node with a packet that must be retransmitted is said to be backlogged. Even if new packets arrive at a backlogged node, the retransmitted packet is the first one to be transmitted and, for simplicity (to ignore queueing of packets at a node), we assume that those new packets are discarded. If backlogged nodes retransmit the packet in the next timeslot, surely a new collision would occur. Therefore, backlogged nodes wait for some random number of timeslots before retransmitting. We assume, for simplicity, that su is the probability (which is the same for all backlogged nodes) that a successful transmission occurs in the next time slot. Moreover, the probability su of retransmission is the same for each timeslot. The number of time slots between the occurrence of a collision and a successful transmission is a geometric random variable Wu (see Section 3.1.3) with parameter su such that Pr [Wu = n] = su (1  su )n31 .

11.5.1 The Markov chain The slotted Aloha protocol constitutes a discrete-time Markov chain [n 5 {0> 1> = = = > Q }, where a state m counts the number of backlogged nodes out of the Q nodes in total and the subscript n refers to the n-th timeslot. Each of the m backlogged nodes retransmits a packet in the next time slot with probability su , while each of the Q  m unbacklogged nodes will transmit surely a packet in the next time slot provided a packet arrives in the current timeslot. The latter event (at least one arrival D) occurs with probability sd = Pr [D A 0] = 1  Pr [D = ¡0]. If¢ we assume that the arrival process is Poissonean, then sd = 1  exp  Q , but the computations in this section are more generally valid. The probability that q backlogged nodes in state m retransmit in the next time slot is binomially distributed µ ¶ m q eq (m) = s (1  su )m3q q u and, similarly, the probability that q unbacklogged nodes in state m transmit

11.5 Slotted Aloha

221

in the next time slot is µ ¶ Q m q sd (1  sd )Q3m3q xq (m) = q A packet is transmitted successfully if and only if (a) one new arrival and no backlogged packet or (b) no new arrival and one backlogged packet is transmitted. The probability of successful transmission in state m and per time slot equals sv (m) = x1 (m)e0 (m) + x0 (m) e1 (m) The transition probability Sm>m+p equals ; A A ?

Sm>m+p

2pQ m xp (m) p=1 x1 (m) (1  e0 (m)) = x (m) e (m) + x (m) (1  e (m)) p=0 A 0 0 1 A = 1 p = 1 x0 (m) e1 (m)

The state with m backlogged nodes jumps to the state m  1 with one backlogged node less if no new packets are sent and there is precisely 1 successful retransmission. The state m remains in the state m if there is 1 new arrival and there are no retransmission or if there are no new retransmissions and none or more than 1 retransmission. The state m jumps to state m +1 if there is 1 new arrival from a non-backlogged node and at least 1 retransmission because then there are surely collisions and the number of backlogged nodes increases by 1. Finally, the state m jumps to state m + p if p new packets arrive from p dierent non-backlogged nodes, which always causes collisions irrespective of how many backlogged nodes also retransmit in the next time slot. The Markov chain is illustrated in Fig. 11.4, which shows that the state can only decrease by at most 1. P0i P01 0

1 P10

P00

2 P21

P11

P04

P03

P02

3 P32

P22

4

N

P44

PNN

P43 P33

Fig. 11.4. Graph of the Markov chain for slotted Aloha. Each state m counts the number of backlogged nodes.

222

Applications of Markov chains

The transition probability matrix S has the structure 5 S00 S01 S02 ··· ··· S0Q 9 S10 S11 S12 · · · · · · S1Q 9 9 0 S21 S22 ··· ··· S2Q 9 S =9 . . . . .. . .. .. .. .. 9 .. . 9 7 0 0 · · · SQ31>Q32 SQ31;Q31 SQ31;Q 0 0 ··· 0 SQ;Q31 SQQ

6 : : : : : : : 8

whose eigenstructure is computed in Appendix A.5.3. In the asymptotic regime when Q $ 4, slotted Aloha has the peculiar property that the steady-state vector  does not exist. Although for a small number of nodes Q , the steady-state equations can be solved, when the number Q grows, Slotted Aloha turns out to be instable. It seems di!cult to prove that limQ w + 1). This time interval is precisely equal to two timeslots in slotted Aloha which explains why PAloha = 12 SAloha . The same observation tells us that, in pure Aloha, sqr (m) ' h32w(m) because in the successful interval the expected number of arrivals and retransmissions is twice that in slotted Aloha. The throughput V roughly equals the total rate of transmission attempts J (which is the same as in slotted Aloha) multiplied by sqr (m) ' h32w(m) , hence, VPAloha = Jh32J .

11.6 Ranking of webpages To retrieve webpages related to a user’s query, current websearch engines first perform a search similar to that in text processors to find all webpages containing the query terms. Due to the massive size of the World Wide Web, this first action can result in a huge number of retrieved webpages. Several thousands of webpages related to a query are not uncommon. To reduce the list of webpages, many websearch engines apply a ranking criterion to sort this list. In this section, we discuss PageRank, the hyperlink-based ranking system used by the Google search engine. PageRank elegantly exploits the power of discrete Markov theory.

11.6.1 A Markov model of the web The hyperlink structure of the World Wide Web can be viewed as a directed graph with Q nodes. Each node in the webgraph represents a certain webpage and the directed edges represent hyperlinks. Let us consider a small collection of webpages as in Fig. 11.5 to illustrate the underlying idea of PageRank, invented by Brin and Page, the founders of Google. 2 3

1

P

5 4

(a)

ª0 «0 « «0 « «0 «¬ P51

P12 0 0 0 P52

0 P23

P14 P24

0 P43

0 0 0

0

P15 º P25 »» 0» » 0» 0 »¼

(b)

Fig. 11.5. A subgraph of the World Wide Web (a) and the corresponding transition probability matrix S (b).

11.6 Ranking of webpages

225

The topology of any graph is determined by an adjacency matrix (see Appendix B.1). A reasonable criterion to assess the importance of a webpage is the number of times that this webpage is visited. This criterion suggests us to consider a discrete Markov chain whose transition probability matrix S corresponds to adjacency matrix of the webgraph (shown in (b) in Fig. 11.5). The element Slm of the Markov transition probability matrix is the probability of moving from webpage l (state l) to webpage m (state m) in one time step. The components vl [n] of the corresponding state vector v[n] denotes the probability that at time n the webpage l is visited. The long run mean fraction of time that webpage l is visited equals the steadystate probability l of the Markov chain. This probability l is the ranking measure of the importance of webpage l used in Google. The basic idea is indeed simple, but we have not shown yet how to determine the elements Slm nor whether the steady-state probability vector  exists. In particular, we will demonstrate that guaranteeing that the steady-state vector  exists and that  can be computed for a Markov chain containing some billion of states — the order of magnitude of the number Q of webpages — requires a deeper knowledge of discrete Markov chains. To start determining the elements Slm , we assume that, given we are on webpage l, any hyperlink on that webpage has equal probability to be clicked on. This assumption implies that Slm = g1l where the degree gl of a node l (see also Section 15.3) equals the number of adjacent neighbors of node l in the webgraph. This number gl is thus equal the number of hyperlinks on webpage l. The transition probability matrix in Fig. 11.5 then becomes 6 5 0 13 0 13 13 9 0 0 1 1 1 : 3 3 3 : 9 : S =9 9 0 0 0 0 0 : 7 0 0 1 0 0 8 1 2

1 2

0

0

0

The uniformity assumption is in most cases the best we can make if no additional information is available. If, for example, web usage information is available showing that a random surfer accessing page 2 is twice as likely to jump to page 4 than to any¡ other neighboring ¢ webpage of 2, then the 1 1 1 second row can be replaced by 0 0 4 2 4 . When solely adopting the adjacency matrix of the webgraph as underlying structure of the Markov transition probability matrix S , we cannot assure that S is a stochastic matrix. For example, it often occurs that a node such as node 3 in our example in Fig. 11.5 does not contain outlinks. Such nodes are called dangling nodes. For example, many webpages may point

226

Applications of Markov chains

to an important document on the web, which itself does not refer to any other webpage. The corresponding row in S possesses only zero elements, which violates the basic law (9.8) of a stochastic matrix. To rectify the deviation from a stochastic matrix, each zero row must be replaced by a particular non-zero row vector3 y W that obeys (9.8), i.e. kyk1 = y W x = 1 where xW = [1 1 · · · 1]. Again, the simplest recipe is to invoke uniformity W and to replace zero row by¢y W = xQ . In our example, we replace the ¡ any third row by 15 51 15 51 51 and obtain 5

0 9 0 9 1 S¯ = 9 9 5 7 0 1 2

1 3

0 1 5

0 1 2

1 3 1 5

1 3 1 3 1 5

1 0

0 0

0

1 3 1 3 1 5

6

: : : : 0 8 0

However, this adjustment is not su!cient to insure the existence of a steady-state vector . In Section 9.3.1, we have shown that, if the Markov chain is irreducible, the steady-state vector exists. In an irreducible Markov chain any state is reachable from any other (Section 9.2.1.1). By its very nature, the World Wide Web leads almost surely to a reducible Markov chain. In order to create an irreducible matrix, Brin and Page have considered =

S = S¯ + (1  )

x=xW Q

where 0 ?  ? 1, x=xW is a Q × Q matrix with each element equal to 1 and S¯ is the previously adjusted matrix without zero-rows. The linear combination of the stochastic matrix S¯ and a stochastic perturbation matrix =

x=xW ensures that S is an irreducible stochastic matrix. Every node is now directly connected (reachable in one step) to any other (because of x=xW ), which makes the Markov chain irreducible with aperiodic, positive recurrent states (see Fig. 9.1). Slightly more general, we can replace the W W matrix x=x Q by xy , where y is a probability vector as above but where we must additionally require that each component of y is non-zero in order to guarantee reachability. Brin and Page have called yW the personalization vector which enables to deviate from non-uniformity. Hence, we arrive at the Brin and Page Markov transition probability matrix =

S = S¯ + (1  )xy W 3

(11.29)

We use the normal vector algebra convention, but remark that the stochastic vectors  and v[n] are also row vectors (without the transpose sign)!

11.6 Ranking of webpages

£

1 4 6 4 1 For y W = 16 16 16 16 16 matrix in our example becomes

5 9 = 9 S =9 9 7

1 80 1 80 1 16 1 80 33 80

¤

19 60 1 20 1 4 1 20 9 20

and  =

3 40 41 120 3 8 7 8 3 40

19 60 19 60 1 4 1 20 1 20

4 5,

227

the probability transition

67 240 67 240 1 16 1 80 1 80

6 : : : : 8

If the presented method were implemented, the initially very sparse matrix =

S would be replaced by the dense matrix S , which for the size Q of the web would increase storage dramatically. Therefore, a more eective way is to define a special vector u whose component um = 1 if row m in S is a zero-row or node m is dangling node. Then, S¯ = S + uy W is a rank-one update of S =

and so is S because = ¢ ¡ S =  S + uy W + (1  )x=y W = S + (u + (1  )x) y W 11.6.2 Computation of the PageRank steady-state vector The steady-state vector  obeys the eigenvalue equation (9.22), thus  =  S¯ . Rather than solving this equation, Brin and Page propose to compute the steady-state vector from  = limn 1). Since { = 1 obeys { = *\ ({) and *\ (0) = Pr [\ = 0] M (0> 1), the strict convex function | = *\ ({) can only intersect the line | = { in some point { M (0> 1) if *\ ({) is below that line near their intersection at { = 1 or if *0\ (1) = H [\ ] A 1. In the other case, if H [\ ] ? 1, the only intersection is at { = 1. ¤

The two possibilities are drawn in Fig. 12.2. MY (x) 1

a b

Pr[Ya = 0]

Pr[Yb = 0] x3 0

S0 x2

x1

x0

1

x

Fig. 12.2. The generating function *\ ({) along the positive real axis {. The two possible cases are shown: curve d corresponds to H [\ ] ? 1 and curve e to H [\ ] A 1. The fast convergence towards the zero 0 is exemplified by the sequence {0 A {1 = *\ ({0 ) A {2 = *\ ({1 ) A {3 = *\ ({2 ).

A root equation such as (12.13) also appears in queuing models such as the M/G/1 (Section 14.3) and GI/D/1 (Section 14.4) and reflects the asymptotic behavior as explained in Section 5.7. The extinction probability 0 can be expressed explicitly as a Lagrange series as demonstrated in Van Mieghem (1996). The branching process with infinitely many generations n $ 4 can be viewed as an infinite directed tree where each node has a finite degree a.s. The fact that 0 ? 1 if H [\ ] A 1 implies that, in infinite directed trees, there exists an infinitely long path starting from the root with probability 1  0 . Theorem 12.3.2 The limiting branching process with [0 = 1 obeys for

240

Branching processes

n$4 Pr [[n = 0] $ 0 Pr [[n = m] $ 0

for any m A 0

Proof: First, if H [\ ] ? 1, then Theorem 12.3.1 states that 0 = 1. For any probability generating function * (}), it holds that |* (}) | $ 1 for |}| $ 1. Hence, *[n ({) $ 1 for real { M [0> 1]. Moreover, tn = *[n (0) $ *[n ({). In the limit n < ", tn < 0 = 1, which implies that for all { M [0> 1] it holds that *[n ({) < 0 = 1. The fact that a probability generating function, a Taylor series around } = 0, converges to a constant 0 for 0 $ { $ 1 implies that Pr [[n = m] < 0 for any m A 0 and Pr [[n = 0] < 0 . The second case H [\ ] A 1 possesses an extinction probability 0 ? 1. For { M (0 > 1), Fig. 12.2 shows that 0 ? *\ ({) ? { ? 1. By induction using (12.5), we find that 0 ? *[n ({) ? *[n1 ({) ? · · · ? 1 or limn 1). This proves the theorem. ¤

Theorem 12.3.2 states that, regardless of the value of H [\ ], the probability that the n-th generation will consists of any finite positive number of items tends to zero if n $ 4. Theorem 12.3.2 is equivalent to the statement that, after an infinite number n $ 4 of evolutions or generations, [n $ 4 with probability 1  0 . Theorem 12.3.2 also illustrates that the Markov chain with an infinite number of states behaves dierently than a chain with a finite number of states. In particular, Theorem 12.3.2 shows that the infinite Markov chain {[n }nD0 has a single absorbing state [n = 0 while all other states m are transient (limq 1]. By Rolle’s Theorem applied to the continuous function i (}) = *\ (}) 3 }, there exists an  M (0 > 1)

12.4 Asymptotic behavior of Z

241

for which i 0 () = 0. Equivalently, *0\ () = 1 and  A 0 . Since *0\ (}) is monotonously increasing in } M [0> 1], we have that *0\ (0) = Pr [\ = 1] $ *0\ (0 ) ? 1. Since "Z (w) is continuous and    monotone decreasing, there exists an integer N0 such that *0\ "Z m w ? 1 for m A N0 and any w A 0. Hence, lim

N1  Z1 n and [n>2  Z2 n may be largely dierent for large n, because of the random nature of Z : although the prefactor Z1 and Z2 both have the same probability density function iZ ({), Z1 can dier substantially from Z2 as illustrated by the pdf iZ ({) in Fig. 12.3.

12.5 A geometric branching processes

243

0

10

P =5 0.8

P =4

-1

10

P =3

-2

10 fW(x)

0.6

increasing P

fW (x)

-3

10

P =2 0.4

increasing P -4

10

-5

10

0.2

0

2

4

6

8

10

x

increasing P

Poisson Geometric

0.0 0

2

4

6

8

10

x

Fig. 12.3. The probability density function of the limit random variable Z for both a geometric and a Poisson production process for a same set of values of the average  = H[\ ].

12.5 A geometric branching processes Consider a production generation function *\ (}) of the form i (}) = d}+e f}+g . Beside straightforward iteration of (12.5), a more elegant approach3 relies on the following property of i (}). For } = {, the dierence is i (})  i ({) =

dg  ef (}  {) (g + f}) (g + f{)

and, hence, for any two points {0 and {1 , i (})  i ({0 ) = i (})  i ({1 ) 3

µ

f{1 + g f{0 + g

¶µ

}  {0 }  {1

¶

The linear fractional transformation i (}) is an automorphism of the extended complex plane and basic in the geometric theory of a complex function for which we refer to the book of Sansone and Gerretsen (1960, vol. 2). Fixed points of an automorphism of the extended plane are solutions of } = i (}), which is a quadratic equation f} 2 + (g 3 d) } 3 e = 0 and which shows that there are at most two dierent fixed points.

244

Branching processes

Let us now confine to the two fixed points, {0 and {1 , of i (}) that are 1 +g solution of i (}) = } and let  = f{ f{0 +g , then i (})  {0 }  {0 = i (})  {1 }  {1 Now, substitute } $ i (}), then i (i (}))  {0 }  {0 i (})  {0 = = 2 i (i (}))  {1 i (})  {1 }  {1 Let us denote the iterates of i (}) by zq = iq (}) = i (iq31 (})). By iterating, we find that the iterates obey }  {0 zq  {0 = q zq  {1 }  {1 or zq =

{0 (}  {1 )  {1  q (}  {0 ) }  {1   q (}  {0 )

(12.20)

Since the probability generating function (3.6) of a geometric random variable \ is of the form i (}) = d}+e f}+g , a geometric branching process is regarded as a basic reference model in the study of branching processes. The production process in each generation obeys Pr [\ = n] = tsn for n  0 t leading to *\ (}) = 13s} , which is slightly dierent from (3.6). We know that the equation *\ ({) = { can have two real zeros in [0> 1], one at {1 = 1 since *\ (}) is a probability generating function and another at {0 = st = 1 1 H[\ ] =  = 0 such that =

s{1 + 1 t 1 = = s{0 + 1 s 

The functional equation (12.5) associates zq = *[q (}) and after substitution in (12.20) we obtain ¡ q31 ¢   1 }  q31 + 1 *[q (}) = (12.21) (q  1) }  q + 1 In the case that H [\ ] =  $ 1 or s = t, using the rule of de l’Hospital gives *[q (}) =

(q  1)}  q q}  q + 1

From (12.21), the probabilities of extinction at the n-th generation are à ! 1 n  1 Pr [[n = 0] = *[n (0) =  n  1

12.5 A geometric branching processes

245

If H [\ ] =  A 1, then limn q > q31 + q > q32 + q31 + q > = = = >

q X

253

# n + z1

(13.4)

n=1

A number of observations are in order: First, if both the interarrival times q+1 and the service times {q are i.i.d. random variables and mutually independent, then the dierences q are i.i.d. random variables. In addition, zq and q are also independent because (13.4) shows that zq only depends on n with indices n ? q. Then, the waiting time process {zq }qD1 is a discrete time Markov process with a continuous state space (the waiting times zq are positive real numbers) because the general relation (13.3) reveals that, since the random variable q is independent of zq , the waiting time for the (q + 1)-th packet is only dependent on the waiting time of the previous q-th packet. This is the Markov property. Since the state space is a continuum, it is not a Markov chain, merely a Markov process. Second, if there exists a packet p for which zp = 0 (e.g. packet p = 5 in Fig. 13.2), which means that the p-th packet finds the system empty, then all packets after the p-th packet are isolated from the eects of those before the p-th. Mathematically, this separation between two busy periods directly follows from (13.3) because zp+1 = max [0> p ] leading via iteration for q  p to # " q q X X zq+1 = max 0> q > q31 + q > = = = > n > n n=p+1

n=p

In other words, this relation is similar to (13.4) as if the system were started from n = p and zp = 0 instead of n = 1 with z1 = z. Any busy period can be regarded as a renewal of the waiting process, independent of the previous busy periods. Third, again invoking the assumption that q are i.i.d. random variables, then the order in the sequence {q }qD1 is of no importance in (13.4) and we may relabel the random variables in (13.4) as n $ q3n+1 to obtain a new random variable # " q31 q X X n > n + z1 $q+1 = max 0> 1 > 1 + 2 > = = = > n=1

n=1

which is identically distributed as zq+1 . The interest of this observation P is that, provided z1 = 0 and, hence, $q = max1$m$q q3m n=0 n where 0 = 0, the sequence {$q }qD1 can only increase with q, because the maximum

254

General queueing theory

P cannot decrease2 if an additional term q+1 n=0 n is added. Thus, if z1 = 0, the event {$q+1 ? {} is always contained in {$q ? {}. In steady-state, which is reached if q $ 4, lim {$q ? {} =

q 0) = 2l f3l" }  d to treat the complicating non-linear function max({> 0) in (13.3). Several other approaches (Kleinrock, 1975, Chapter 8) have been proposed to solve (13.3). We will only discuss the approach due to Bene˘s, because his approach does not make the confining assumption that both the interarrival times q+1 and the service times {q are i.i.d. random variables. As mentioned before, in Internet tra!c, which has been shown to be long-range dependent

256

General queueing theory

(i.e. correlated over many time units mainly due to TCP’s control loop), the interarrival times can be far from independent. D(t) [(W)

v(W)

D(u) x1+ x2+ x3 [(u)

v(u)

x1+ x2 u

x1

idle t 0 t1 t2

t3 t 4

u

t5

t 6 t7

t8 t9

W

t10

Fig. 13.3. The amount of work arriving to the queueing system (w) versus time w. At w = x, we observe that (w) = (w)  w + y(0 ) A 0. The largest value of (w)  (x) is found for x = w1 because (w1 ) = w1 , the only negative value of (w) in [0> x). Graphically, we shift the line at 45r so to intersect the point (w1 > (w1 )) to determine y(x). At w =  , ( ) ? 0 and the largest negative value of (w) in [0>  ) is attained at w = w8 . Three of the five idle periods have also been shown.

13.3 The Bene˘ s approach to the unfinished work Instead of observing the queueing system at a time w, the Bene˘s approach considers the behavior over a time interval [0> w). Let (w), (w) and e(w) denote the amount of work arriving to the queueing system in the interval [0> w), the total idle time of the server and the total busy time of the server in the interval [0> w), respectively. The amount of work arriving to the system is expressed in units of time and must be regarded as the time needed to process this work, similarly to the definition of the unfinished work. If the work to process arrives at discrete times, then (w) increases in jumps, as

13.3 The Bene˘s approach to the unfinished work

257

illustrated in Fig. 13.3, X

D(w)

(w) =

{m

m=0

In general, however, the work may arrive continuously over time with possibly jumps at certain times. The purpose is to determine the unfinished work or virtual waiting time y(w) at time instant w, and not Rover a time w interval [0> w) as the previously defined quantities and D(w) = 0 QD (x)gx. Clearly, for w A 0, the unfinished work at time w consists of the total amount of work brought in by arrivals during [0> w) plus the amount of work present just before w = 0 minus the total time the server has been active, y(w) = y(03 ) + (w)  e(w)

(13.6)

From the definitions above, (w) + e(w) = w

(13.7)

Moreover, (w), (w) and e(w) are non-decreasing and right continuous (jumps may occur) functions of time w. Since (w) and e(w) are complementary, it is convenient to eliminate e(w) from (13.6) and (13.7) and further concentrate on the total idle time (w) given as (w) = y(w) + w  y(03 )  (w)

(13.8)

If y(x) A 0 at any time x 5 [0> w), then (w) = 0. On the other hand, if y(x) = 0 at some time x 5 [0> w), then it follows from (13.8) that (x) = x  y(03 )  (x)

(13.9)

Since (w) is non-decreasing in w, the total idle time in the interval [0> w) at moments x when the buer is empty (y(x) = 0) is the largest value for  that can be reached in [0> w), ¢ ¡ (w) = sup x  y(03 )  (x) 0?x?w

and the supremum is needed because (w) can increase discontinuously (in jumps). Combining the two regimes, we obtain in general that  ¸ ¡ ¢ 3 (w) = max 0> sup x  y(0 )  (x) (13.10) 0?x?w

258

General queueing theory

Equating the two general expressions (13.8) and (13.10) for the total idle time of the server leads to an new relation for the unfinished work,  ¸ ¡ ¢ 3 3 y(w) = y(0 ) + (w)  w + max 0> sup x  y(0 )  (x) 0?x?w ¸  3 = max y(0 ) + (w)  w> sup {(w)  w  ((x)  x)} 0?x?w

The quantity (w) = (w)  w + y(03 ) is recognized as the server overload during [0> w), while (w)  w is the amount of excess work arriving during the interval [0> w). Thus, (w)  (x) is the amount of excess work during [x> w) or the overload of the server during [x> w) provided x A 0 and (0) = y(03 ). Then, ¸  y(w) = max (w)> sup {(w)  (x)} 0?x?w

and, with the convention that y(03 ) = sup0?x?03 {(0)  (x)} = (0), y(w) = sup {(w)  (x)}

(13.11)

0?x?w

The unfinished work y(w) at time w is equal to the largest value of the overload or excess work during any interval [x> w)  [0> w). The relation (13.11) is illustrated and further explained in Fig. 13.3. This general relation (13.11) shows that the unfinished work is the maximum of a stochastic process. Furthermore, if y(w) = 0, (13.11) indicates that sup0?x?w {(w)  (x)} = 0. Let xW denote the value at which sup0?x?w {(w)  (x)} = (w)  (xW ) = 0. But (xW ) is the lowest value of (w) in [0> w) and, unless an arrival occurs during the interval [w> w+ w], (w + w) = (w)  w ? (w). This argument shows that, as soon as a new idle period begins, (w) attains the minimum value so far. During the idle period as shown in Fig. 13.4, (w) further decreases linearly with slope 1 towards a new minimum (em ) in [0> em ] until the beginning of a new busy period, say the m-th at w = em . Then, for all em ? w ? em+1 , y(w) = (w)  (em ) = sup {(w)  (x)} em ?x?w

In other words, we observe that idle periods decouple the past behavior from future behavior, as deduced earlier from the waiting time analysis in Section 13.2. As illustrated in Fig. 13.4, the series {(em )}, where em denotes the start of the m-th busy period, is monotonously decreasing in em , i.e. (em ) A (em+1 ) for any m.

13.3 The Bene˘s approach to the unfinished work

259

[(t) b1

b3

b2

b4 b5

0

t 0 t1 t2

t3 t4

t5

t6

t7

t8 t9

t10

Fig. 13.4. The excess work (w) for the same process as in previous plot. The arrows with em denote the start of the m-th busy period. Observe that (em ) is the minimum so far and that a busy period ends at w A em for which (w) = (em ). The length of a busy period has been represented by a double arrow.

Let us proceed to compute the distribution of the unfinished work following an idea due to Bene˘s. Bene˘s applies the identity4 , valid for all }, Z w 3}w =1} h3} g h 0

to the total idle time of the server by putting  = (x) Z 3}w

h

 31 (w)

=1}

h3}(x) g(x)

31 (0)

where  31 (w) is the inverse function. Note that (0) = 0 and that g(x) = 1{y(x)=0} gx =  (y(x)) gx where  ({) is the Dirac impulse. Let w $ (w), then Z w 3}(w) =1} h3}(x)  (y(x)) gx h 0

Substituting (13.9) in the integral, which is only valid if y(x) = 0, and (13.8) at the left-hand side, which is generally valid, gives Z w 3 3 h3}(y(w)+w3y(0 )3(w)) = 1  } h3}(x3y(0 )3(x))  (y(x)) gx 0 4

Borovkov (1976, p. 30) proposes another but less simple approach by avoiding the use of the identity ingeniously introduced by Bene˘s=

260

General queueing theory

or, in terms of the excess work (w) = (w)  w + y(03 ), Z w 3}y(w) 3}(w) =h } h3}((w)3(x))  (y(x)) gx h 0

Taking the expectation of both sides yields, Z w h h i i h i 3}y(w) 3}(w) 3}((w)3(x)) H h H h  (y(x)) gx =H h } 0

Recall that, with (2.34), with the definition of a generating function (2.37), and further with (2.61), i h T = H h3}((w)3(x))  (y(x)) Z "Z " C2 = h3}{ (|) Pr [(w)  (x)  {> y(x)  |] g{g| C{C| 3" 3" and with (2.45), we have Z

Z

C2 Pr [(w)  (x)  {|y(x)  |] Pr [y(x)  |] g{g| h}{ (|) C{C| 4 4 Z 4 Z 4 C Pr [y(x)  |] C = g{h}{ (|) Pr [(w)  (x)  {|y(x)  |] g| C{ 4 C| 4 Z 4 g h}{ = Pr [(w)  (x)  {|y(x) = 0] Pr [y(x) = 0] g{ g{ 4 4

4

T=

Combining all leads to ]

"

3"

] h3}{ iy(w) ({)g{ =

"

3"

h3}{ i(w) ({)g{ ]

"

g g{

h3}{

3}

3"

]

w

 Pr [(w) 3 (x) $ {|y(x) = 0] Pr [y(x) = 0] gx g{

0

By partial integration, we can remove the factor } at both sides. Indeed, since Pr [y(w)  {] = 0 for { ? 0, Z " Z " 3}{ h iy(w) ({)g{ = } h3}{ Pr [y(w)  {] g{ 3"

3"

Hence, we arrive at ]

"

] h3}{ Pr [y(w) $ {] g{ =

3"

k h3}{ Pr [(w) $ {]

"

3"

3

which is equivalent to Pr [y(w)  {] = Pr [(w)  {] 

g g{

]

Z 0

 Pr [(w) 3 (x) $ {|y(x) = 0] Pr [y(x) = 0] gx g{

w

0

w

g Pr [(w)  (x)  {|y(x) = 0] Pr [y(x) = 0] gx g{ (13.12)

13.3 The Bene˘s approach to the unfinished work

261

This general relation for the distribution of the unfinished work in terms of the excess work is the Bene˘s equation. If y(x) = 0 for all x 5 [0> w), this means that during that interval no work arrives and that (w)  (x) = x  w or that w  (w)  (x)  0 for any x 5 [0> w). Thus, if we choose { 5 [w> 0) such that the event {(w)  (x) = x  w  {} is possible, the probabilities appearing in the right-hand side are not identically zero while Pr [y(w)  {] = 0. Hence, for { 5 [w> 0), the Bene˘s equation reduces to Z w g Pr [(w)  (x)  {|y(x) = 0] Pr [y(x) = 0] gx Pr [(w)  {] = g{ 0 from which the unknown probability of an empty system Pr [y(x) = 0] can be found5 for w + {  x  w. The Bene˘s equation translates the problem of finding the time-dependent virtual waiting time or unfinished work in an integral equation that, in principle, can be solved. We further note that in the derivation hardly any assumptions about the queueing system nor the arrival process are made such that the Bene˘s equation provides the most general description of the unfinished work in any queueing system. Of course, the price for generality is a considerable complexity in the integral equation to be solved. However, we will see examples6 of its use in ATM. 13.3.1 A constant service rate If the server operates deterministically as in ATM, for example, the amount of work arriving to the queueing system in the interval [0> w) simplifies to (w) = D(w), the number of ATM cells in the interval [0> w), because {m = { is the time to process one ATM cell, which we take as time unit { = 1. With this convention, we have that (w) = D(w)  w. After substitution of x = w  |, the integral L in (13.12) is Z w g Pr [(w)  (w  |)  {|y(w  |) = 0] Pr [y(w  |) = 0] g| L= g{ 0 and the event {(w)  (w  |)  {} = {D(w)  D(w  |)  { + |} 5

This relation in the unknown function i (x) = Pr [y(x) = 0] is a Volterra equation of the first kind (see e.g. Morse and Feshbach (1978, Chapter 8)) ] } j(}) = N(}|x)i (x)gx d

6

These integral equations frequently appear in physics in boundary problems, potential and Green’s function theory. Borovkov (1976) investigates the Bene˘s method in more detail. He further derives from (13.12) formulae for light and heavy tra!c, and the discrete time process.

262

General queueing theory

Since D(w)  D(w  |) is a non-negative integer n and thus a discrete random variable, the probability density function is gPr[(w)(w|) {|y(w|) = 0] = Pr[D(w)D(w|) = n|y(w|) = 0] 1{+|=n g{ which implies that only values at | = n  { contribute to the integral L. Hence, 0  |  w implies that d{e  n  b{ + wc where b}c (respectively d}e) are the integer equal to or smaller (respectively larger) than }, X

b{+wc

L=

Pr [D(w)  D(w + {  n) = n|y(w + {  n) = 0] Pr [y(w + {  n) = 0]

n=d{e

Hence, for a discrete queue with time slots equal to the constant service time, the Bene˘s equation reduces to Pr [y(w)  {] = Pr [D(w)  b{ + wc] X

(13.13)

b{+wc



Pr [D(w)D(w+{n) = n|y(w+{n) = 0]Pr[y(w+{n) = 0]

n=d{e

13.3.2 The steady-state distribution of the virtual waiting time Let us turn to the steady-state distribution Y ({) = limw w  0} interacts

13.6 Little’s Law

267

with the system process {QV (w)> w  0} because every arrival increases the system content with one, they are dependent processes. Relation (13.20) is true for Poisson arrivals and this property is called “Poisson arrivals see time averages” (PASTA). Theorem 13.5.1 (PASTA) The long-run fraction of time that a process spends in state m is equal to the long-run fraction of Poisson arrivals that find the process in state m> Pr [QV;D = m] = Pr [QV = m] Proof: See8 e.g. Wol (1982).

¤

The Poisson process has the typical property that future increments are independent of the past and, thus also of the past system history. In certain sense, Poisson arrivals perform a random sampling which is su!cient to characterize the steady-state of the system exactly. The PASTA property also applies to Markov chains. The transitions in continuous time Markov chains are Poisson processes if self-transitions are allowed (see Section 10.4.1). For any state m, the fraction of Poisson events that see the chain in state m is m , which (see Lemma 6.1.2) also equals the fraction of time the chain is in state m.

13.6 Little’s Law Little’s Law is perhaps the simplest of the general queueing formulae. Theorem 13.6.1 (Little’s Law) The average number of packets (customers) in the system H [QV ] (or in the queue H [QT ]) equals the average arrival rate  times the average time spent in the system H [W ] (or in the queue H [z]), H [QV ] = H [W ]

(13.21)

H [QT ] = H [z] 8

Although Wollf’s general proof (Wol, 1982) only contains two pages, it is based on martingales and on axiomatic probability theory.

268

General queueing theory

Little’s Law holds if two of the three limits D(w) = w

(13.22)

QV (x)gx = H [QV ]

(13.23)

lim

1 lim wwm +Wm ) gx =

D(w) Z w X m=1

0

X

D(w)

1{xM[wm >wm +Wm ) gx =

Wn

n=1

The general case where QV (w)  0 is more complicated as Fig. 13.6 shows for w =  because not all intervals [wm > wm + Wm ) for 1  m  D( ) are contained PD( ) in n=1 Wn counts too much and is an upper bound for R  [0>  ). Hence, Q (x)gx. If G(w) denotes the number of departures in [0> w], Fig. 13.6 V 0 illustrates that the area (in grey) in an interval [0> w],Rwhich equals the total w number of packets in the system in that interval 0 (D(x)  G(x))gx = Rw 0 QV (x)gx, can be bounded for any realization (sample path) and any w  0

13.6 Little’s Law

269

by Z

X

Wn 

QV (x)gx  0

n:Wn +wn $w

X

D(w)

w

Wn

n=1

where the lower bound only counts the packets that have left the system by time w. By dividing by w, we have D(w) w

X n:Wn +wn $w

1 Wn  D(w) w

Z

w

0

D(w) X Wn QV (x)gx  w D(w) D(w)

(13.25)

n=1

Since we assume that the limit (13.22) exists, we have that D(w) = R(w) for w $ 4. From the existence of the limit (13.24), we can thus write X Wn = H [W ] w un+1 ], respectively. The transition probability of the embedded Markov chain is Ylm = Pr [Sn+1 = m|Sn = l]

284

Queueing models

and the evolution over time follows from (13.16) as Sn+1 = (Sn  1)+ + An

(14.21)

Hence, since An  0, we see that Sn+1  (Sn  1)+ or that Sn+1 ? (Sn  1)+ is impossible. Hence, Ylm = 0 for m ? l and l A 1 while, for l A 0, Ylm = Pr [l  1 + An = m|Sn = l] = Pr [An = m  l + 1]. The case for l = 0 results in Y0m = Pr [An = m] = Y1m . Denoting dm = Pr [An = m], the transition probability matrix becomes2 6 5 d0 d1 d2 d3 · · · 9 d0 d1 d2 d3 · · · : : 9 : 9 Y = 9 0 d0 d1 d2 · · · : 9 0 0 d d ··· : 0 1 8 7 .. .. .. .. . . . . . . . and the corresponding transition graph is sketched in Fig. 14.2. aj  i + 1 a1

...

i2

a3

i1

a2

i

i+1

i+2

...

j

...

a0

Fig. 14.2. State transition graph for the M/G/1 embedded Markov chain.

The number of Poisson arrivals during a time slot [un > un+1 ] clearly depends on the length of the service time {n+1 = un+1  un that is distributed according to I{ (w), which is independent of a specific packet n. Furthermore, the arrival process is a Poisson process with rate  and independent of the state of the queueing process, thus Pr [An = m] = Pr [A = m]. Hence, using the law of total probability (2.46), Z " Pr [A = m] = Pr [A = m|{ = w] gI{ (w) 0

Z =

0 2

"

h3w

(w)m gI{ (w) m!

The structure of this transition probability matrix Y has been investigated in great depth by Neuts (1989). Moreover, Y belongs to the class of matrices whose eigenstructure is explicitly given in Appendix A.5.3.

14.3 The M/G/1 queue

285

If we denote the Laplace transform of the service time by Z " *{ (}) = h3}w gI{ (w) 0

then we observe that m dm = Pr [A = m] = m!

Z

"

h 0

¯ ()m gm *{ (}) ¯¯ w gI{ (w) = m! g} m ¯}=

3w m

(14.22)

so that the transition probability matrix Y is specified. Since all dm A 0 for all m A 0, Fig. 14.2 indicates that all states are reachable from an arbitrary state l as the Markov process evolves over time in at least l steps. These l steps occur in the transition from state l to state 0. This implies that the Markov process is irreducible and the steady-state stability requirement  ? 1 makes the Markov process ergodic. The steady-state vector with components m = Pr [QV;G = m] where limqpn+m

n=0

where the Kronecker delta n>p = 1 if n = p else n>p = 0. The sum " X

" X

q>pn+m =

n=0

q3m>pn = 1p|q3m

n=3"

is one if p divides q  m else it is zero. Such expression can be written as " X

q3m>pn

n=3"

p31 X 1  h2l(q3m) ¢= = ¡ h2ln(q3m)@p p 1  h2l(q3m)@p n=0

Using the latter summation yields W (} p ) =

p31 " p31 X } p X 3m X } Pr [R = q] } q h2ln(q3m)@p p q=0 m=0

=

=

n=0

p31 p31 }p X X ³

p

}h2ln@p

n=0 m=0

p31 }p X

p

n=0

´3m

" X

³ ´q Pr [R = q] }h2ln@p

q=0

³ ´ 1  } 3p 2ln@p ¢31 U }h ¡ 1  }h2ln@p

where we have introduced the generating function U(}) = Thus, we arrive at W (} p ) =

P"

q=0 Pr [R

= q] } q .

p31 ³ ´ }p  1 1 X 2ln@p U }h ¢ ¡ 2ln@p 31 p n=0 1  }h

The generating function U(}) can be specified further since h i i £ ¤ h + + U(}) = H } (S3p) +F = H } F H } (S3p) where independence of the arrival process and queueing process (GI) has

14.4 The GI/D/m queue

295

been used. From (14.31), the corresponding steady-state relation is h i i h i £ ¤ h + + + V(}) = H } (S3p) +A = H } A H } (S3p) = D(})H } (S3p) h i + while from V(}) = D(})T(}), we observe that T(}) = H } (S3p) . Hence, U(}) = I (})T(}) where T(}) is given by (14.34). We now turn our attention to the determination of I (}), the generating function of the number of packets in front of the “test” packet. The “test” packet has been uniformly chosen out of the total flow of arriving packets at the system. Let us denote by AW the number of arriving packets in the same time slot as the “test” packet. The random variable AW is not the same as the number of arriving cells A per time slot. For example, we know that there is at least one arrival in the time slot of the “test” packet, namely, the “test” packet itself, hence, Pr [AW = 0] = 0. Furthermore, the larger the number of arriving packets in a time slot, the higher the probability that the “test” packet is chosen out of those packets in this time slot. Hence, Pr [AW = m] is proportional to the number m of arriving packets in a time slot. In addition, Pr [AW = m] is also proportional to Pr [A = m], which describes how likely a number m of arriving packets is. Combining both shows that Pr [AW = m] = m Pr [A = m]

P 1 W with the proportionality factor equal to  = H[A] because " m=0 Pr [A = m] = 1. The “test” packet is uniformly distributed among the arriving packets AW in the time slot of the “test” packet (in steady-state). The probability of having precisely n packets in front of the “test” packet given AW = m  1 equals 1n?m Pr [F = n|AW = m] = m Indeed, the “test” packet has equal probability 1m of occupying any of the m possible positions. The occupation of a position n + 1 implies precisely n cells in front of the “test” packet in a FIFO discipline. Using the law of total probability (2.46), Pr [F = n] = =

" X

Pr [F = n|AW = m] Pr [AW = m]

m=1 " X m=n+1

" X 1 m Pr [A = m] 1 Pr [A = m] = m H [A] H [A] m=n+1

296

Queueing models

The generating function I (}) becomes I (}) =

" X n=0

" " 1 X X Pr [F = n] } = Pr [A = m] } n H [A] n

n=0 m=n+1

m X

" " 1 X 1 X 1  }m Pr [A = m] } n31 = Pr [A = m] H [A] H [A] 1} m=1 m=1 n=1 3 4 " " X X 1 C Pr [A = m] } m  Pr [A = m]D = H [A] (}  1)

=

m=1

=

m=1

} (D(})  Pr [A = 0]  1 + Pr [A = 0]) H [A] (}  1)

Since H [A] = D0 (1) = , we finally arrive at I (}) =

D (})  1 (}  1) D0 (1)

Combining all involved expressions leads to the expression of the generating function of the total time spent in the GI/D/m queue ¡ 2ln@p ¢ p31 p31 p1 X  1 Y }h2ln@p  q D }h } p   p ¢ ¡ W (} ) = ¢31 p ¡ p 1  q }  D }h2ln@p q=1 1  }h2ln@p n=0

For the single-server case (p = 1), the generating function of the system time (queueing time plus service time) considerably simplifies to ¶ µ D (})  1 1 } W (}) =  }  D(}) from which H [W ] and Var[W ] readily follow. The computation of the pdf given the arrival process D(}) is more complex, as illustrated for the M/D/1/K queue in the next section.

14.5 The M/D/1/K queue Suppose we have a buer of N cells and an aggregate arrival stream consisting of a large number of sources with none of them dominating the others. This input process is well modeled by a Poisson process with arrival rate . Both the input process as well as the buer content and the output process have been simulated in Fig. 14.4. Observe the eect of variations and maximum number of cells in the queue and input process!

14.5 The M/D/1/K queue

297

6 Poisson(0.8)

number of cells in queue

2

number of served cells

8

4 number of arriving cells

1

10

6

4

2 0 0

0 0

200

400

600

800

1000

0

timeslot

200

400

600

800

1000

0

200

400

600

800

1000

timeslot

timeslot

Fig. 14.4. On the left, the Poisson input process with  = 0=8 in terms of the number of cells versus the timeslot. In the middle, the buer occupancy for a buer with N = 20 as function of time. On the right, the M/D/1/K output process in cells served per timeslot.

14.5.1 The pdf of the buer occupancy in the M/D/1 queue n

For a Poisson process, Pr[A = n] = n! h3 and D(}) = h(}31) . The pgf T(}) of the buer content immediately follows from (14.36) as

T(}) = (1  )

(1  }) h(13}) 1  } h(13})

The average queue length is obtained from (14.37) as ¤ £ H QT;P@G@1 = H [Q] =

2 2 (1  )

and Little’s law (13.21) provides the average waiting time in the queue ¤ £ £ ¤ H QT;P@G@1  = H zP@G@1 =  2 (1  )

(14.38)

Since } h(13}) is analytic everywhere, there always exists a neighborhood (depending on ) around } = 0 for which |} h(13}) |  1. Hence, we can use the series expansion for the geometric series to obtain " " X X T(}) } n hn(13}) = (1  }) } n h(n+1)(13}) = (1  }) h(13}) 1 n=0

n=0

298

Queueing models

Integrating with respect to  removes the factor (1  }), Z " X }n T(}) g = h(n+1) h3(n+1)} 1 n+1 n=0

" " X X }n (1)q (n + 1)q q q h(n+1) } = n+1 q! q=0

n=0

" X " X (1)q (n+1) = (n + 1)q31 q } q+n h q! q=0 n=0

After a change in the variable p = q + n with p  0, which implies that n  p since q = p  n  0, we have Ãp ! Z " X X (1)p3n T(}) g = h(n+1) (n + 1)p3n31 p3n } p 1 (p  n)! p=0 n=0 Ãp+1 ! " X X (1)p+13n = hn n p3n p+13n } p (p + 1  n)! p=0 n=1

Dierentiation with respect to , gives Ãp+1 ¸!  " p+13n p3n X X (n) (n) T(}) = (1  ) (1)p+13n hn + }p (p + 1  n)! (p  n)! p=0 n=1

from which we finally deduce the probability t[p] = Pr [Q = p] that the p-th position in the buer is occupied6 ¸  p+1 X (n)p3n (n)p+13n p+13n n + t[p] = (1  ) (1) h (14.39) (p + 1  n)! (p  n)! n=1

One readily observes from the derivation above that the probability v[p] = Pr [S = p] that p positions in the system are occupied, is v[p] = t[p  1] 6

Explicitely, we have for the queue content probabilities t[0] = h (1 3 ) t[1] = h (1 3 ) (h 3  3 1) t[2] = h (1 3 ) (

2 3 h2 +  3 h 3 2 h ) 2

while the system content probabilities are v[0] = (1 3 ) v[1] = (h 3 1) (1 3 ) v[p] = t[p 3 1]

(p D 2)

14.5 The M/D/1/K queue

299

for p  2 because the }-transform is V(}) = (1  ) 13}(13}) . This result is h(13}) a characteristic property of a deterministic server. Next, we rewrite (14.39) as # "p+1 p p+13n p3n X X n (n) n (n)  h h t[p] = (1  ) (p + 1  n)! (p  n)! n=1 n=1 h i = (1  ) h(p+1) j(h3 ; p + 1)  hp j(h3 ; p) (14.40) where j({; p) =

p X {n (p  n)n n!

(14.41)

n=0

Due to the nature of the dierences, we immediately find the cumulative distribution, N X

i h t[p] = (1  ) h(N+1) j(h ; N + 1)  h

p=1

and since t[0] = (1  ) h , we arrive at Pr[Q  N] =

N X

t[p] = (1  ) h(N+1) j(h ; N + 1)

(14.42)

p=0

The expressions in (14.40) are numerically only useful for small p because the series is alternating. This problem may be solved by considering a famous result due to Lagrange (Markushevich, 1985, Vol. 2, Chapter 3, Section 14) he} = 1 + e

" X (e + qd)q31 ¡ 3d} ¢q }h q!

(14.43)

q=1

that converges for |}|  d31 . Dierentiation of (14.43) with respect to z = }h3d} leads to X (e + d + qd)q h(e+d)} = } q h3qd} 1  d} q! q=0 "

g e} g e} g} because gz h = g} h gz = } = , we obtain

ehe} (13d})h3d} .

Choosing d = 1, e + d = p and

" " X X h3p (p  q)q (q  p)q = (h3 )q = j(h3 ; p) + (h3 )q 1   q=0 q! q! q=p+1

300

Queueing models

and thus 3

j(h

" X (q  p)q h3p  (h3 )q ; p) = 1   q=p+1 q!

where the infinite series consists of merely positive terms. Substituted in (14.42), this finally yields N+1

Pr[Q A N] = (1  ) 

" X

qq+N+1 (h3 )q (q + N + 1)! q=1

(14.44)

In the heavy tra!c limit  =  $ 1, the dominant zero (5.35) of 1  and the resulting tail } h(13}) is approximately equal to   1 + 2 (13)  asymptotic (5.31) for the buer occupancy pdf is Pr [Q A N]  

(13) 1   3N31  h3N log   h32N   1  

(14.45)

14.6 The N*D/D/1 queue The N*D/D/1 queue is a basic model for constant bit rate sources in ATM, as shown in Fig. 14.5. The input process consists of a superposition of Q independent periodic sources with the same period G but randomly phased, i.e. arbitrarily shifted in time with respect to each other. The server operates deterministically and serves one ATM cell per timeslot. The buer size is assumed to be infinitely long mainly to enable an exact analytic solution. During a time period G measured in time slots or server time units, precisely Q cells arrive such that the tra!c intensity or load (13.2) equals  = Q G. 1 2 3

K D

. . .

N Fig. 14.5. Sketch of an ATM concentrator where Q input lines are multiplexed onto a single output line. The N*D/D/1 queue models this ATM basic switching unit accurately.

Whereas the arrivals in the M/D/1 queue are uncorrelated, the successive

14.6 The N*D/D/1 queue

301

interarrival times in the N*D/D/1 queue are negatively correlated. For the same average arrival rate, this more regular arrival process results in shorter queues than in the M/D/1 queue, where the higher variability in the arrival process causes longer queues. Due to the dependence of the arrivals over many timeslots, the solution method is based on the Bene˘s approach and starts from the complementary distribution (13.15) for the virtual waiting time or unfinished work in steady-state, Pr [y(w" ) A {] = limw G). The computations are most conveniently performed if we choose the steady-state observation point w" = G. Each of the Q ATM cells arrives uniformly in [0> G] due to the random phasing of each cell train and the probability that it arrives in [G +{n> G] is s = n3{ G . Hence, the number of arrivals in [G + {  n> G] is a sum of Bernoulli random variables, which is binomially distributed, Z G ¸ µ ¶µ ¶ µ ¶ Q n{ n n  { Q3n Pr QD (x) gx = n = 1 n G G G+{3n The conditional probability is obtained as follows. The unfinished work at time G + {  n only depends on past arrivals in the interval [0> G + {  n]. Given that the number of arrivals in [G + {  n> G] equals n while there are always precisely Q in [0> G], the number of arrivals in [0> G + {  n] Q3n ? 1 since equals Q  n and the corresponding tra!c intensity 0 = G+{3n Q ? G and, thus, Q  n ? G  n for any n. From Section 13.3.2, we use the local stationary result: for any stationary single-server queueing system with tra!c intensity , the probability of an empty system at an arbitrary time is 1  . If we take a random point in w 5 [0> G + {  n], then stationarity 0 implies that Pr [y (w) = 0] = 1 0 = G+{3Q G+{3n . Since  ? 1, the system

302

Queueing models

is necessarily empty at some instant wW in [0> G + {  n). As explained in Section 13.2 and Section 13.3, we may consider that the system restarts from wW on ignoring the past. But the probability Pr [y (w) = 0] at a random time in the interval [0> wW ] and in [wW > G + {  n] is the same, which means that Pr [y(G + {  n) = 0] = G+{3Q G+{3n because we can periodically repeat the system’s process in [0> wW ] while omitting any activity in [wW > G + {  n]. With respect to this newly constructed periodic arrival pattern, the point w = G + {  n is arbitrary such that the local stationary result is applicable. In summary, we arrive at the overflow probability in a N*D/D/1 system, Pr [QV A {] =

µ ¶µ ¶ µ ¶ Q X G+{Q Q n{ n n  { Q3n (14.46) 1 G+{n n G G

n=d{e

Observe that Pr [QV A Q ] = 0. Rewriting (14.46) yields Q µ ¶ G+{Q X Q (n  {)n (G + {  n)Q3n31 Pr [QV A {] = GQ n n=d{e

Q3d{e µ ¶ G+{Q X Q = (Q  m  {)Q3m (G + {  Q + m)m31 Q m G m=0

G+{Q = GQ

Q µ X m=0

G+{Q  GQ

Q m

¶ (G + {  Q + m)m31 (Q  {  m)Q3m

Q X

µ ¶ Q (G+{Q +m)m31 (Q m {)Q3m m

m=Q3d{e+1

Applying Abel’s identity (Comtet, 1974, p. 128), valid for all x> |> }, q µ ¶ X q (x  n})n31 (| + n})q3n (x + |) = x n q

(14.47)

n=0

with q = Q , x = G + {  Q , | = Q  { and } = 1 gives Pr[QV  {] =

G+{Q GQ

Q X

µ ¶ Q (G + {  Q + m)m31 (Q  m  {)Q3m m

m=Q3d{e+1

(14.48) demonstrating that, indeed, Pr [QV  0] = 0. For small {, relation (14.48) is convenient, while (14.46) is more suited for large { $ Q . For example, ¡ 1 ¢Q ¡ ¢ 1 Q , while Pr [QV A Q  1] = G . 1+ G Pr [QV  1] = G+13Q G+1

14.6 The N*D/D/1 queue

In the heavy tra!c regime for  = mation (Roberts, 1991) is

Q G

303

$ 1, a Brownian motion approxi-

  { 32{ Q + 13 

Pr [QV A {] ' h

(14.49)

Figure 14.6 compares the exact (14.46) overflow probability and the Brownian approximation (14.49) for  = 0=95. Observe from (14.45) that Pr [QV A {] ' h3

2{2 Q

£ ¤ Pr QM/D/1 A {

which shows that, for su!ciently high Q , the overflow probability of the N*D/D/1 queue tends to that of the M/D/1 queue. Thus, an arrival process consisting of a superposition of a large number of periodic processes tends to 2{2

a Poisson arrival process. The decaying factor h3 Q reflects the eect of the negative correlations in the arrival process and shows that a Poisson process overestimates the tail probability in heavy tra!c. Comparison of (14.46) and (14.44) for lower loads  = Q G illustrates that the Poisson approximation becomes more accurate. 0

10

-1

Exact Brownian Approximation

10

-2

10

-3

10

M/D/1: N of

-4

10

-5

10

-6

Pr[NS > x]

10

N = 5000

-7

10

-8

10

-9

10

N = 1000

-10

10

N = 500

-11

10

N = 200

-12

10

N = 100

-13

N = 50

10

U = 0.95

-14

10

-15

10

0

20

40

60

80

100

x

Fig. 14.6. The overflow probability Pr [QV A {] in the N*D/D/1 queue for  = 0=95 and various number of sources Q .

304

Queueing models

14.7 The AMS queue Many arrival patterns in telecommunication exhibit active or “on” periods succeeded by silent inactive or “o” periods. At the burst or flow level phenomena of the order of time of an on-o period are dominant and the finer details of the individual packet arrivals within an on-period can be ignored. The stream of packets can be regarded as a continuous fluid characterized by the flow arrival rate. The AMS queue is perhaps the simplest exact solvable queueing model that describes the queueing behavior at the burst or flow level. The AMS queue named after Anick, Mitra and Sondhi (Anick et al., 1982; Mitra, 1988) considers Q homogeneous, independent on-o sources in a continuous fluid flow approach. For each source, both the on- and o-period are exponentially distributed, which makes the model Markovian by Theorem 10.2.3. In the on-period each source emits a unit amount of information. Hence, at each moment in time when u sources are in the on-period, u packets (units of information) arrive at the buer. The service time is constant and equal to f ? Q packets per unit time. If f A Q , then the buer is always empty. The time unit is chosen as the average time of an on-period while the average time of an o-period is denoted by 1 . The buer size is infinitely long. The Q tra!c intensity then equals  = f(1+) and stability requires that  ? 1.  The ratio 1+ is the long term “on” time fraction of the sources. Suppose the number of on-sources at time w is l. During the next time interval w only two elementary actions can take place: a new source can start with probability (Q  l)w or a source can turn o with probability lw. Compound events have probabilities R(w2 ). The probability of no change in the arrival process is 1  [(Q  l) + l]w during which l sources are active and the queue empties at rate f  l. The AMS queueing process is a birth-death process where state l describes the number of on-sources and where the birth rate l = (Q  l)  and the death rate l = l. Let Sl (w> {) where 0  l  Q , w  0, {  0 be the probability that at time w, l sources are on and the buer content does not exceed {. Then, we have

Sl (w + w> {) = [Q  (l  1)]  w Sl31 (w> {) + (l + 1) w Sl+1 (w> {) + [1  {(Q  l)  + l} w] Sl (w> {  (l  f) w) + R(w2 ) Passing to the limit w $ 0 yields CSl (w> {) CSl (w> {) + (l  f) = (Q  l + 1)Sl31 (w> {)  [(Q  l) + l]Sl (w> {) Cw C{ + (l + 1) Sl+1 (w> {)

14.7 The AMS queue

305

The time-independent equilibrium probabilities, l ({) = limw Q [f]1]. In addition, m = Q that corresponds to the eigenvalue Q = 0 and the (4) eigenvector. The general bounded solution of (14.51) is X

Q3[f]31

({) = (4) +

dm hm { {m

(14.53)

m=0

where the scalar coe!cients dm = |mW (0) still need to be determined. Rather than determining |mW (0) as in Appendix A.5.2.2, a more elegant and physical method is used. The eigenvalue solution in Appendix A.5.2.2 has scaled the eigenvectors by setting the Q component equal to 1, hence, ({m )Q = 1. Writing the Q -th component in (14.53) gives with (14.52) Q + Q ({) = (1 + )Q

X

Q3[f]31

dm hm {

(14.54)

m=0

The most convenient choice of { is { = 0. If the number of on-source m at any time exceeds the service rate f, then the buer builds-up and cannot be empty, m (0) = 0

for [f] + 1  m  Q

This observation provides one equation in (14.54) for the coe!cients dn , X

Q3[f]31 m=0

dm = 

Q (1 + )Q

and shows that Q  [f]  1 additional equations are needed to determine all coe!cients dm . By dierentiating (14.54) p-times and evaluating at { = 0, we find these additional equations ¯ Q3[f]31 X gp Q ({) ¯¯ = dm mp ¯ p g{ {=0 m=0

which will be determined with the help of the dierential equation (14.51). Indeed, for p = 1, the dierential equation (14.51) gives ¯ gQ ({) ¯¯ = G31 T(0) g{ ¯{=0 31 The important observation is that the eect of multiplication by ¯ G T gm ({) ¯ decreases the number of zero components in (0) by 1, i.e. g{ ¯ =0 {=0

for [f] + 2  m  Q . Any additional multiplication by G31 T has the same

14.7 The AMS queue

eect. Since that

gp ({) g{p

307

¡ ¢p = G31 T ({), we thus find, for 0  p  Q  [f]  1 ¯ gp Q ({) ¯¯ =0 g{p ¯{=0

We write these Q  [f] equation in the unknown dn in matrix form, 5

1 1 12 .. .

9 9 9 9 9 9 9 Q [f]2 9  7 1 Q [f]1 1

1 2 22 .. .

1 3 32 .. .

Q [f]2

3

Q [f]1

3

2 2

Q[f]2 Q[f]1

6

··· ··· ··· .. . ···

Q [f]1 2 Q [f]1 .. . Q [f]2 Q [f]1

···

Q [f]1

1

5

: d0 : d1 : 9 : 9 d2 :=9 : 9 .. : 7 . : 8 dQ [f]1

Q [f]1

6

5

9 : 9 : 9 :=9 : 9 8 9 9 7

Q (1+)Q

0 0 .. . 0 0

6 : : : : : : : 8

and recognize the matrix, denoted by Y , as a Vandermonde matrix (Section A.1, art. 5) with Y

Y

l=0

m=l+1

Q3[f]31Q3[f]31

det (Y ) =

(m  l )

Since all eigenvalues appearing in the Vandermonde matrix are distinct (Appendix A.5.2.2) det (Y ) 6= 0 and a unique solution follows for all 0  m  Q  [f]  1 from Cramer’s theorem as µ dm = 

 1+

¶Q

Y

Q3[f]31 l=0;l6=m

m l  m

(14.55)

Together with the exact determination of the eigenvalues m and corresponding right-eigenvector {m explicitly given in Appendix A.5.2.2, the coe!cients dm completely solve the AMS queue. P The buer overflow probability Pr [QV A {] = 1  Q m=0 m ({) becomes PQ with m=0 m (4) = 1, X

Q3[f]31

Pr [QV A {] = 

m=0

m {

dm h

Q X

({m )o

o=0

Using the explicit form of the generating function (A.44) where the roots u1 and u2 belonging to eigenvalue n are specified in (A.42) and the residue n = nm = f1 in (A.43), the buer overflow probability is X

Q 3[f]31

Pr [QV A {] = 

m=0

dm hm { (1  u1 )nm (1  u2 )Q 3nm

(14.56)

308

Queueing models

For large {, Pr [QV A {] will be dominated by the exponential with the largest negative eigenvalue 0 (for which n0 = Q ), Pr [QV A {]  d0 h0 {

Q X

({Q )m

m=0

Writing that largest negative eigenvalue (A.47) in terms of the tra!c intensity , gives (1 + ) (1  ) 0 =  1  Qf ¡ Q ¢Q P From (A.49), we have Q . Combined with (14.55), the m=0 ({Q )m = f asymptotic formula for the buer overflow probability becomes Y

Q3[f]31 Q 0 {

Pr [QV A {]   h

l=1

l l  0

(14.57)

0

10

-1

10

-2

10

-3

U = 0.9

10

-4

10

-5

10

-6

Pr[NS > x]

10

N = 40

-7

10

-8

10

U = 0.7

-9

10

N = 100

-10

10

-11

10

-12

10

U = 0.5

-13

10

-14

10

-15

10

0

5

10

15

20

x

Fig. 14.7. The overflow probability (14.56) in the AMS queue versus the buer level { for fixed  = 12 . For each tra!c intensity  = 0=5, 0=7 and 0.9, the upper curve corresponds to Q = 40 and the lower to Q = 100. The asymptotic formula (14.57) is shown in dotted line.

Figure 14.7 shows both the exact (14.56) and asymptotic (14.57) overflow

14.8 The cell loss ratio

309

probability as function of { for various tra!c intensities  and two size of Q . The average o-period in Fig. 14.7 is two times the average on-period. For large values of { and large tra!c intensities , the asymptotic formula is adequate. For smaller values, clear dierences are observed. As mentioned in Section 5.7, the asymptotic regime that nearly coincides with (14.57) refers to the burst scale phenomena while the non-asymptotic regime reflects the smaller scale variations. The AMS queue allows us to analyze the eect of the burstiness of a source by varying . 14.8 The cell loss ratio Due to its importance in ATM and in future time-critical communication services, the QoS loss-performance measure, the cell loss ratio, deserves some attention. In designing a switch for time-critical services with strict delay requirements smaller than GW , the buer size N is dimensioned as follows. The order of magnitude of GW is about 10 ms, the maximum end-to-end delay for high-quality telephony (world wide) advised in ITU-T standards. Let H[K] be the average number of hops of a path in an ATM network that rarely exceeds 10 hops. The buer size N is determined such that the GW GW , thus, N maximum waiting time of a cell never exceeds H[K]   10 . For example, for STM-1 links where F = 155 Mb/s, we have that  ' 366 800 W ATM cell/s such that N  G 10 ' 366 ATM cell buer positions. This firstorder estimate shows that the ATM buer for time-critical tra!c consists of a few hundreds of ATM cell positions. That small number for N indeed assures that the delay constraints are met, but introduces the probability to loose cells. Hence, the QoS parameter to be controlled for time-critical services is the cell loss ratio. The cell loss ratio fou is defined as the ratio of the long-time average number of lost cells because of buer overflow to the long-time average number of cells that arrive in steady-state. There are typically two dierent views to describe the cell loss ratio: a conservation-based and a combinatorial one. The conservation law simply states that cells entering the system also must leave it. The average number of entering cells are all those oered per time slot minus the ones that have been rejected, thus (1  fou). On the other side, the average number of cells that leave the system are related to the server activity as (1  t [0]), where  is the service rate and t[m] = Pr [QT = m]. Hence, we have (1  fou)  = (1  t [0]) 

(14.58)

In the combinatorial view, only the arrival process is viewed from a position

310

Queueing models

in the buer and the number of ways in which cells are lost are counted leading to " N 1 X X q t [N  m] d [m + q] (14.59) fou = 0 D (1) q=0 m=0

with D0 (1) =  and d[m] = Pr [A = m]. Although equation (14.58) is simple, its practical use is limited since the quantities involved are to be known with extremely high accuracy if fou is of the order of 10310 , which in practice means a virtually loss-free service. Therefore, we confine ourselves to the combinatorial result and express (14.59) in terms of a generating function as ¯ gY (}) ¯¯ 0 fou D (1) = (14.60) g} ¯}=1 where Y (}) =

" X

}q

q=0

=

N X

N X

t [N  m] d [m + q] =

m=0

Ã

t [N  m] } 3m

" X q=0

" X

N X

}q !

t [N  m] } 3m d [m + q] } m

m=0

d [m + q] } m+q

q=0

m=0

Rearranging in terms of the generating function for the arrivals D(}) and P m for the buer occupancy T(}) = N m=0 t [m] } , where t [m] = 0 for m A N, yields à ! m31 N X X t [N  m] } 3m D(})  d [q] } q Y (}) = q=0

m=0

= D(}) }

3N

N X

t [N  m] }

N3m

m=0

= } 3N D(})T(})  } 3N



N X

t [N  m] }

3m

N X m=0

t [m] } m

d [q] } q

q=0

m=0 N3m31 X

m31 X

d [q] } q

(14.61)

q=0

In order to express the cell loss ratio entirely in terms of the generating functions D(}) and T(}), we employ (2.20), Ã !  Z Z q q ³ } ´q+1 ¸ X X 1 \ ($) \ ($) 1 m m |[m]} = } g$ = 1 g$ 2l F(0) $ m+1 2l F $  } $ m=0 m=0 Z \ ($) ³ } ´q+1 1 = \ (})  g$ (14.62) 2l F $  } $

14.8 The cell loss ratio

311

where F is a contour enclosing the origin and the point } and lying within the convergence region of \ (}). Combining (14.61) and (14.62), we rewrite Y (}) as Z D($)T($) 1 g$ Y (}) = } 3N D(})T(})  } 3N T(})D(}) + 2l F ($  }) $ N Z 1 D($)T($) = g$ 2l F ($  }) $ N Finally, our expression for the cell loss ratio in a GI/G/1/K system reads Z D($)T($) 1 g$ (14.63) fou = 2lD0 (1) F ($  1)2 $N where the contour F encloses both the origin and the point } = 1 and lies in the convergence region of D(}). Usually, D(}) is known while T(}) proves to be more complicated to obtain. The product T(})D(}) = V(}) is the pgf of the system content. If T(}) and D(}) are meromorphic functions7 and if ¯ ¯ ¯ D(}) T(}) ¯ ¯ ¯ = 0> lim } O) equals the number of its neighboring nodes and 0  gm  Q  1. Clearly, the node m is disconnected from the rest of the graph if gm = 0. Hence, in connected graphs, 1  gm  Q  1. The basic law for the degree (see also Appendix (B.2)) is Q X

gm = 2O>

m=1

since each link belongs to precisely two nodes and, hence, is counted twice. In directed graphs, the in(out)-degree is defined as the number of the in(out)going links at a node, while the sum of in- and out-degree equals the degree. The minimum nodal degree in the graph J is denoted by gmin = minmMJ gm . P 2O The average degree of a graph is defined as gd = Q1 Q m=1 gm = Q which is, for a connected graph, bounded by 2  Q2  gd  Q  1. The lower bound is obtained for any spanning tree, a graph that connects all nodes and that contains no cycles and where O = Omin = Q  1. The upper bound is reached in the complete graph NQ with Omax = Q(Q31) . Graphs where 2 gmin = gd such as NQ and the ring topology in Fig. 15.1 are called regular graphs since any node has precisely gd links. Sometimes networks are classified either as dense if gd is high or as sparse

15.3 The degree of a node in a graph

323

Fig. 15.2. Degree graph with  = 2=4 and Q = 300. All nodes are drawn on a circle.

if gd is small. For instance, the Internet is sparse with average degree gd  3, although some backbone routers may have a much higher degree, even exceeding 100. The distribution of the degree GInternet of an arbitrary node in the Internet is shown to be approximately polynomial (Siganos et al., 2003), Pr [GInternet = n] 

n3  ( )

(15.4)

P 3v for Re(v) A 1 is the Riemann with3  5 (2=2> 2=5) and  (v) = " n=1 n Zeta function (Titchmarsh and Heath-Brown, 1986). A graph of this class is called a degree graph. Figures 15.2 and 15.3 show two instances of a degree graph. Also the web graph consisting of websites and hyperlinks features a power law for the in-degree. David Aldous has given the following argument why a power law of the in-degree of the web graph is natural. To a good approximation, the number of websites is growing exponentially at rate  A 0. This means that the lifetime W of a random website satisfies Pr [W A w]  h3w . 3

A more general expression than (15.4) is Pr [gm = n] = fn j(n), where f is a normalization constant and where j(n) is a slowly varying function (Feller, 1971, pp. 275-284) with basic property that limw O). Natural metrics of a graph are the degree distribution and the hopcount distribution of an arbitrary path. Beside quantities such as the diameter and the complexity of a graph defined in algebraic graph theory (Appendix B), some other metrics are the clustering coe!cient, the expansion, the resilience, the distortion and the betweenness. The clustering coe!cient fJ (y) characterizes the density of connections in the environment of a node y and is defined as the ratio of the number of links | connecting the gy neighbors of y over the total possible gy (g2y 31) , fJ (y) =

2| gy (gy  1)

(15.6)

The expansion hJ (k) of a graph reflects the number of nodes that can be

15.6 Random graphs

329

reached in k hops from a node y, hJ (k) =

1 X |F (k)| Q2

(15.7)

yMN

where F (k) is the set of nodes that can be reached in k hops from a node y and |D| represents the number of elements in the set D. We can interpret F (k) geometrically as a ball centered at node y with radius k. The resilience uJ (p) measures the connectivity or robustness of a graph. Let p = |F (k)| denote the number of nodes in a ball centered at node y and with radius k, and define o (y> p) as the number of links that needs to be removed to split F (k) into two sets with roughly equal numbers of nodes (around p@2). The resilience uJ (p) of a graph is 1 X o(y> p) (15.8) uJ (p) = O yMN

The distortion wJ (p) measures how closely the graph resembles a tree and is defined as 1 X z (F (k)) (15.9) wJ (p) = Q yMN

where z (J) is the value of the minimum spanning tree in J with unit link weight z (l $ m) = 1 for each link of J= Consider a flow with a unit amount of tra!c between each pair of nodes in the graph J. Each flow between a node pair follows the shortest path between that node pair. The betweenness E of a link (node) is defined as the number of shortest paths between all possible pairs of nodes in J that traverse the link (node). If Kl<m denotes the number of hops in the shortest path from l $ m, then the total number of hops KJ in all shortest paths in P PQ PO J is KJ = Q l=1 m=l+1 Kl<m . This number is also equal to KJ = o=1 Eo , where Eo is the betweenness of a link o in J. Taking the expectation of both relations gives the average betweenness of a link in terms of the average hopcount ¡Q ¢ 2 H[KQ ]  H[KQ ] O with equality only for the complete graph.

H [E] =

15.6 Random graphs Besides the regular topologies in Fig. 15.1, the class of random graphs constitutes an attractive set of topologies to analyze network performance. The

330

General characteristics of graphs

theory of random graphs originated from a series of papers by Erdös and Rényi in the late 1940s. There exists an astonishingly large amount of literature on random graphs. The standard work on random graphs is the book by Bollobas (2001). We also mention the work of Janson et al. (1993) on evolutionary processes in random graphs. The two most frequently occurring models for random graphs are the Erdös-Rényi random graphs Js (Q ) and Ju (Q> O). The class of random graphs denoted by Js (Q ) consists of all graphs with Q nodes in which the links are chosen independently and with probability s. In the class Js (Q ) the total number of links is not deterministic, but on average equal ¡ ¢ to H [O] = s Omax where Omax = Q2 . Since s = OH[O] we also call s the max link density of Js (Q ). An instance of the class J0=013 (300) is drawn in Fig. 15.6. Related to Js (Q ) are the geometric random graphs J{slm } (Q ) where the links are still chosen independently but where the probability of l $ m being an edge is slm . An example of J{slm } (Q ) is the Waxman graph ul  um |) and where (Waxman, 1998; Van Mieghem, 2001) with slm = exp(d| the vector ul represents the position of node l and d a real, non-negative number. Geometric random graphs are good models for ad-hoc wireless networks where the probability slm = i (ulm ) that there is a wireless link between node l and m is specified by the radio propagation that is briefly explained at the end of Section 3.5. random graphs with Q nodes and O links. The class Ju (Q> O) is the¡ set of Om a x ¢ In total, we can construct O dierent graphs, which corresponds to the number of ways we can distribute a set of O ones in the Omax possible places in the upper triangular part above the diagonal of the adjacency matrix D. Each of the possible Omax links has equal probability to belong to a random graph of the class Ju (Q> O). The probability that an element in the adjacency matrix D is dlm = 1 equals s = OmOa x . As opposed to the class Js (Q ), the number of non-zero elements in D in each random graph of Ju (Q> O) is precisely 2O (see Appendix B.1, art. 2), which induces weak dependence between links in Ju (Q> O). The latter also explains why more computations are easier in Js (Q ) than in Ju (Q> O). The average number of paths with m hops between two arbitrary nodes in Js (Q ) follows from (15.1) and (2.13) as

H[[m ] =

(Q  2)! m s (Q  m  1)!

(15.10)

for 1  m  Q 1. The average total number of paths between two arbitrary

15.6 Random graphs

331

Fig. 15.6. A connected random graph Js (Q ) with Q = 300 and s = 0=013 drawn on a circle.

nodes D and E equals 6 5 Q31 Q32 X X s3o 1 H7 [m 8 = (Q  2)!sQ 31  (Q  2)!sQ31 h s o! m=1

o=0

where the latter bound is closely approached for large Q. Moreover, when the random graph reduces to the complete graph (s = 1), we again obtain (15.3). Since the degree gm of a node m is the number of links incident with that node, it follows directly from the definition of Js (Q ) that the probability density function of the degree Grg of an arbitrary node in Js (Q ) equals µ ¶ Q 1 n Pr [Grg = n] = s (1  s)Q313n (15.11) n The interest in random graphs is fueled by the fact that the topology of the Internet is inaccurately known and also that good models7 are lacking. In some sense, the Internet can be regarded as a growing and changing organism. Such complex networks also arise in other fields. Increased interest 7

A detailed discussion on di!culties in modeling or simulating the Internet is presented by Floyd and Paxson (2001).

332

General characteristics of graphs

from dierent disciplines to understand network behavior resulted in a new wave of science, which may be termed “the physics of networks” and which was recently reviewed by Strogatz (2001). Random graphs are an elegant vehicle to thoroughly analyze the performance of, for example, routing algorithms. Some constructed overlay networks such as Gnutella and mobile wireless ad-hoc networks seem reasonably well modeled by Js (Q ). However, the class Js (Q) does not describe the Internet topology well, and the degree distribution especially deviates significantly. The degree distribution (15.11) in Js (Q ) is binomially distributed, while that of the Internet is close to a power law (15.4). Hence, there is a discrepancy between Internet measurements and properties of the random graph Js (Q ).

15.6.1 The number F(Q> O) of connected random graphs in the class Ju (Q> O) From the point of view of telecommunications networks, by far the most interesting graphs are those with connected topology. This limitation restricts the value of the link density s from below by a critical threshold sf . For large Q, the critical threshold is sf  logQQ , as shown in Section 15.6.3. In the theory of random graphs, the problem to determine the number of connected random graphs F(Q> O) in the class Ju (Q> O) has been intensively studied. Gilbert (1956) has presented an exact recursion formula for F(Q> O) via the technique of enumeration. Erdös and Rényi (1959, 1960) have determined the asymptotic behavior of random graphs via the probabilistic method, largely introduced by Erdös himself. Since the analysis8 of Gilbert is both exact and simple, we will review his results here and those of Erdös and Rényi in the next section. Consider a particular random graph M of the class of random graphs Ju (Q + 1> O), which is constructed from the class Ju (Q> O) by adding one node labelled Q + 1. Suppose that the node labelled Q + 1 in the random graph M belongs to a connected component N that possesses y other nodes and some number  of links. ¡ ¢ The remaining part of M has Q  y nodes and O   links. There are Qy ways in which the y nodes of N can be chosen out of the Q nodes in Ju (Q> O). On the other hand, there are F(y + 1> ) ¡ Q 3y ¢ ways of picking a connected random graph N while there are ( 2 ) ways O3

of constructing the remaining part of M. Hence, since the number of ways ¡ Q +1 ¢ we can construct a graph M equals ( O2 ) , we obtain Gilbert’s recursion 8

A dierent, less admissible approach is found in Goulden and Jackson (1983).

15.6 Random graphs

333

formula, µ¡Q+1¢¶ 2

=

O

y+1 Q µ ¶ (X 2 ) X Q

y=0

y

µ¡Q3y¢¶ 2

F(y + 1> )

O

=y

(15.12)

Gilbert (1956) further derives the generating function for F(Q> O) as ! Ã n " X " " X X F(Q> O) Q O (1 + |)(2) {n (15.13) { | = log 1 + Q! n! Q=1 O=1

n=1

which converges for 2  |  0 and all {. So far, no other explicit formulae for F(Q> O) exist. In 1889, Caley (see Appendix B.1, art. 3) proved that, in the special case where O = Q  1, there holds F(Q> Q  1) = Q Q32 . In the other extreme where O = Omax¡ corresponding with a full mesh, we have F(Q> Omax ) = 1. ¢ Actually, when Q2  Q  1 ? O  Omax , the graph is always connected because the adjacency matrix D has necessarily at least one non-zero element ¡ ¢ ¡Q ¢ per row. This means that F(Q> O) = ( 2 ) for Q  Q  1 ? O  O . In max

2

O

all cases where O ? Q  1, the random graphs are necessarily Disconnected, leading to F(Q> O) = 0. For computational purposes, we rewrite (15.12) as µ¡Q+1¢¶ 2

=

O ¡

¢

Q +1 (X 2 )

=Q

y+1 µ ¡0¢ ¶ Q31 µ¡Q3y¢¶ 2 ) X µQ ¶ (X 2 2 F(Q +1> ) F(y+1> ) + y =y O O

y=0

0 = >O , we arrive after a substitution of Q $ Q  1 at the Since O3 recursion formula,

µ¡Q ¢¶ F(Q> O) =

2

O



µ¡Q 313y¢¶ ¶ (y+1 2 ) Q 1 X 2 F(y + 1> ) y O   =y

Q32 Xµ y=0

(15.14)

Below we list a few values: F(2> 1) = 1 F(3> 2) = 3 F(4> 3) = 16 F(5> 4) = 125 F(5> 8) = 45 F(6> 5) = 1296 F(6> 9) = 4945 F(6> 13) = 105 F(7> 6) = 16807 F(7> 10) = 331506 F(7> 14) = 116175 F(7> 18) = 1330

F(3> 3) = 1 F(4> 4) = 15 F(5> 5) = 222 F(5> 9) = 10 F(6> 6) = 3660 F(6> 10) = 2997 F(6> 14) = 15 F(7> 7) = 68295 F(7> 11) = 343140 F(7> 15) = 54257 F(7> 19) = 210

F(4> 5) = 6 F(5> 6) = 205 F(5> 10) = 1 F(6> 7) = 5700 F(6> 11) = 1365 F(6> 15) = 1 F(7> 8) = 156555 F(7> 12) = 290745 F(7> 16) = 20349 F(7> 20) = 21

F(4> 6) = 1 F(5> 7) = 120 F(6> 8) = 6165 F(6> 12) = 455 F(7> 9) = 258125 F(7> 13) = 202755 F(7> 17) = 5985 F(7> 21) = 1

334

General characteristics of graphs

15.6.2 The Erdös and Rényi asymptotic analysis In a classical paper, Erdös and Rényi (1959) proved that ¸ ¶ ¸  µ  1 32{ is connected = h3h lim Pr Ju Q> Q log Q + {Q Q O{ ) with n  1 isolated nodes equals ¡Q ¢¡(Q 3n)¢ 2 , the number of ways in which n isolated nodes can be chosen n

O{

out of the total of Q nodes multiplied by the number of graphs that can be constructed with Q  n nodes and O{ links. Observe that this total number also includes those graphs where not all the Q  n nodes are necessarily connected. In other words, this total number includes the graphs that do not possess property Dn . The total number of graphs W0 without isolated node follows from the inclusion-exclusion formula (2.10) as µ ¶µ¡Q3n¢¶ Q X n Q 2 W0 (Q> O{ ) = (1) n O{ n=0

where the index n = 0 equals the total number of graphs with Q nodes and O{ links, i.e. the total number of elements in the sample space. Evi-

336

General characteristics of graphs

dently, the total number F(Q> O{ ) of connected random graphs of the class Ju (Q> O{ ) is smaller than W0 (Q> O{ ) because all of them must obey property D0 as well. Since9 ¡Q ¢¡(Q 3n)¢ 2 (h32{ )n n O{ lim = Q ¢ ¡ (2) Q

 A 0>

(16.1)

The corresponding density is iz ({) = {31 1{M[0>1] . The exponent  = lim {0

log Iz ({) log {

is called the extreme value index of the probability distribution of z and  = 1 for regular distributions. By varying the exponent  over all nonnegative real values, any extreme value index can be attained and a large class of corresponding SPTs, in short -trees, can be generated. Fw(x) 1

D D 

0

H

D!

larger scale

1

x

Fig. 16.1. A schematic drawing of the distribution of the link weights for the three dierent -regimes. The shortest path problem is mainly sensitive to the small region around zero. The scaling invariant property of the shortest path allows us to divide all link weights by the largest possible such that Iz (1) = 1 for all link weight distributions.

16.2 The shortest path tree in NQ with exponential link weights

349

Figure 16.1 illustrates schematically the probability distribution of the link weights around zero (0> ], where  A 0 is an arbitrarily small, positive real number. The larger link weights in the network will hardly appear in a shortest path provided the network possesses enough links. These larger link weights are drawn in Fig. 16.1 from the double dotted line to the right. The nice advantage that only small link weights dominantly influence the property of the resulting shortest path tree implies that the remainder of the link weight distribution (denoted by the arrow with larger scale in Fig. 16.1) only plays a second order role. To some extent, it also explains the success of the simple SPT model based on the complete graph NQ with i.i.d. exponential link weights, which we derive in Section 16.2. A link weight structure eectively thins the complete graph NQ — any other graph is a subgraph of NQ — to the extent that a specific shortest path tree can be constructed. Finally, we assume the independence of link weights, which we deem a reasonable assumption in large networks, such as the Internet with its many independent autonomous systems (ASs). Apart from the Section 16.7, we will mainly consider the case for  = 1, which allows an exact analysis.

16.2 The shortest path tree in NQ with exponential link weights 16.2.1 The Markov discovery process Let us consider the shortest path problem in the complete graph NQ , where each node in the graph is connected to each other node. The problem of finding the shortest path between two nodes D and E in NQ with exponentially distributed link weights with mean 1 can be rephrased in terms of a Markov discovery process. The discovery process evolves as a function of time and stops at a random time W when node E is found. The process is shown in Fig. 16.2. The evolution of the discovery process can be described by a continuoustime Markov chain [(w), where [(w) denotes the number of discovered nodes at time w, because the characteristics of a Markov chain (Theorem 10.2.3) are based on the exponential distribution and the memoryless property. Of particular interest here is the property (see Section 3.4.1) that the minimum of q independent exponential variables each with parameter l is again an P exponential variable with parameter ql=1 l . The discovery process starts at time w = W0 with the source node D and for the initial distribution of the Markov chain, we have Pr[[(W0 ) = 1] = 1. The state space of the continuous Markov chain is the set VQ consisting of all positive integers (nodes) q with q  Q . For the complete graph NQ , the

350

The Shortest Path Problem

transition rates are given by q = q(Q  q)>

q 5 VQ

(16.2)

Indeed, initially there is only the source node D with label2 0, hence q = 1. From this first node D precisely Q  1 new nodes can be reached in the complete graph NQ . Alternatively one can say that Q  1 nodes are competing with each other each with exponentially distributed strength to be discovered and the winner amongst them, say F with label 1, is the one reached in shortest time which corresponds to an exponential variable with rate Q  1. v8 v7

corresponding URT

Markov discovery process

v6 v5 v4 v3

2 5

v2

h=0

0

6

3

4

1

5

h=1 h=2

6

v1

1 7

0

8

h=3

time 2 4 3

8

W6 7

Fig. 16.2. On the left, the Markov discovery process as function of time in a graph with Q = 9 nodes. The circles centered at the discovering node D with label 0 present equi-time lines and yn is the discovering time of the n-th node, while n = yn  yn1 is the n-th interattachment time. The set of discovered nodes redrawn per level are shown on the right, where a level gives the number of hops k from the source node D. The tree is a uniform recursive tree (URT). 2

When continuous measures such as time and weight of a path are computed, the source node is most conveniently labeled by zero, whereas in counting processes, such as the number of hops of a path, the source node is labeled by one.

16.2 The shortest path tree in NQ with exponential link weights

351

After having reached F from D at hitting time y1 , two nodes q = 2 are found and the discovery process restarts from both D and F. Although at time y1 we were already progressed a certain distance towards each of the Q  2 other, not yet discovered, nodes, the memoryless property of the exponential distribution tells us that the remaining distance to these Q  2 nodes is again exponentially distributed with the same parameter 1. Hence, this allows us to restart the process from D and F by erasing the previously partial distance to any other not yet discovered node as if we ignore that it were ever travelled. From the discovery time y1 of the first node on, the discovery process has double strength to reach precisely Q  2 new nodes. Hence, the next winner, say G labeled by 2, is reached at y2 in the minimum time out of 2(Q  2) traveling times. This node G has equal probability to be attached to D or F because of symmetry. When G is attached to D (the argument below holds similarly for attachment to F), symmetry appears to be broken, because G and F have only one link used, whereas D has already two links used. However, since we are interested in the shortest path problem and since the direct link from D to G is shorter than the path D $ F $ G, we exclude the latter in the discovery process, hereby establishing again the full symmetry in the Markov chain. This exclusion also means that the Markov chain maintains single paths from D to each newly discovered node and this path is also the shortest path. Hence, there are no cycles possible. Furthermore, similar to Dijkstra’s shortest path algorithm, each newly reached node is withdrawn from the next competition round, which guarantees that the Markov chain eventually terminates. Besides terminating by extinction of all available nodes, after each transition when a new node is discovered, the Markov chain stops with 1 , since each of the q already discovered nodes has probability equal to Q3q precisely 1 possibility out of the remaining Q  q to reach E and only one of them is the discoverer. The stopping time W is defined as the infimum for w  0 at which the destination node E is discovered. In summary, the described Markov discovery process, a pure birth process with birth rate q = q(Q  q), models exactly the shortest path for all values of Q . 16.2.2 The uniform recursive tree A uniform recursive tree (URT) of size Q is a random tree rooted at D. At each stage a new node is attached uniformly to one of the existing nodes until the total number of nodes is equal to Q . The hopcount kQ (equivalent to the depth or distance) is the smallest number of links between the root D and a destination chosen uniformly from all nodes {1> 2> = = = > Q }.

352

The Shortest Path Problem

n o (n) Denote by [Q the n-th level set of a tree W , which is the set of nodes in the tree W at hopcount n from the root nD in aograph with Q nodes, and (n) (n) (0) by [Q the number of elements in the set [Q . Then, we have [Q = 1 because the zeroth level can only contain the root node D itself. For all (n) n A 0, it holds that 0  [Q  Q  1 and that Q31 X

(n)

[Q = Q

(16.3)

n=0 (q)

Another consequence of the definition is that, if [Q = 0 for some level (m) q ? Q  1, then all [Q = 0 for levels m A q. In such a case, the longest possible shortest path in the tree has a hopcount of q. The level set o n (1) (2) (Q31) OQ = 1> [Q > [Q > = = = > [Q (n)

of a tree W is defined as the set containing the number of nodes [Q at each level n. An example of a URT organized per level n is drawn on the right in Fig. 16.2 and in Fig. 16.3. A basic theorem for URTs proved in van der Hofstad et al. (2002b), is the following: (n)

(n)

Theorem 16.2.1 Let {\Q }n>QD0 and {]Q }n>QD0 be two independent copies of the vector of level sets of two sequences of independent URTs. Then (n)

g

(n31)

{[Q }nD0 = {\Q1

(n)

+ ]Q3Q1 }nD0 >

(16.4)

where on the right-hand side the random variable Q1 is uniformly distributed over the set {1> 2> = = = > Q  1}. Theorem 16.2.1 also implies that a subtree rooted at a direct child of the root is a URT. For example, in Fig. 16.3, the tree rooted at node 5 is a URT of size 13 as well as the original tree without the tree rooted at node 5. By applying Theorem 16.2.1 to the URT subtree, any subtree rooted at a member of a URT is also a URT. An arbitrary URT X consisting of Q nodes and with the root labeled by 1 can be represented as X = (q2 # 2) (q3 # 3) = = = (qQ # Q )

(16.5)

where (qm # m) means that the m-th node is attached to node qm 5 [1> m  1] and q2 = 1. Hence, qm is the predecessor of m and the predecessor relation is indicated by the arrow “#”. Moreover, qm is a discrete uniform random variable on [1> m  1] and all q2 > q3 > = = = > qQ are independent.

16.2 The shortest path tree in NQ with exponential link weights

Root 1

6

12

2

18

3

4

22

24

5

26

9

7

8

10

14

21

13

16

20

23

25

19

11

15

17

353

X N( 0)

1

X N(1)

5

X N( 2)

9

X N( 3)

7

X N( 4)

4

Fig. 16.3. An instance of a uniform recursive tree with Q = 26 nodes organized per level 0  n  4. The node number (inside the circle) indicates the order in which the nodes were attached to the tree.

Theorem 16.2.2 The total number of URTs with Q nodes is (Q  1)! Proof: (a) Let the nodes be labeled in the order of attachment to the URT and assign label 1 for the root. The URT growth law indicates that node 2 can only be attached in one way, node 3 in two ways, namely to node 1 and node 2 with equal probability. The n-th node can be attached in n  1 possible nodes. Each of these possible constructions leads to a URT. (b) By summing over all allowable configurations in (16.5), we obtain 1 X 2 X q2 =1 q3 =1

and this proves the theorem.

===

Q31 X

1 = (Q  1)!

qQ =1

¤

In general, Cayley’s Theorem (Appendix B.1 art. 3) states that there are Q Q32 labeled trees possible. The URT is a subset of the set of all possible labeled trees. Not all labeled trees are URTs, because the nodes that are further away from the root must have larger labels. The shortest path tree from the source or root D to other nodes in the complete graph is the tree associated with the Markov discovery process, where the number of nodes [(w) at time w is constructed as follows. Just as the discovery process, the associated tree starts at the root D. We now investigate the embedded Markov chain (Section 10.4) of the continuous-time discovery process. After each transition in the continuous-time Markov chain, [(w) $

354

The Shortest Path Problem

[(w)+1, an edge of unit length is attached randomly to one of the q already discovered nodes in the associated tree because a new edge is equally likely to be attached to any of the q discovering nodes. Hence, the construction of the tree associated with the Markov discovery process and illustrated in Fig. 16.2 on the right demonstrates that the shortest path tree in the complete graph NQ with exponential link weights is an uniform recursive tree. This property of the shortest path tree in NQ with exponential link weights is an important motivation to study the URT. More generally, in van der Hofstad et al. (2001) we have proved that, for a fixed link density s and su!ciently large Q , the shortest path tree in the class RGU, the class of random graphs Js (Q ) with exponential or uniformly distributed link weights, is a URT. Smythe and Mahmoud (1995) have reviewed a number of results on recursive trees that have appeared in the literature from the late 1960s up to 1995.

16.3 The hopcount kQ in the URT 16.3.1 Theory The hopcount kQ from the root to an arbitrary chosen node in the URT equals the number of links or hops from the root to that node. We allow the arbitrary node to coincide with the root in which case kQ = 0. Theorem 16.3.1 The probability generating function of the hopcount in the URT with Q nodes is h i (Q + }) *kQ (}) = H } kQ = (16.6) (Q + 1)(} + 1) Proof: Since the number of nodes at hopcount n from the root (or at (n) level n) is [Qk , al node uniformly chosen out of Q nodes in the URT has (n)

probability

H [Q Q

of having hopcount n, Pr[kQ = n] =

h i (n) H [Q Q

(16.7)

If the size of the URT grows from q to q + 1 nodes, each node at hopcount n  1 from the root can generate a node at hopcount n with probability 1@q. Hence, for n  1, h i (n31) i Q31 h X H [q (n) H [Q = q q=n

16.3 The hopcount kQ in the URT

355

With (16.7), a recursion for Pr[kQ = n] follows for n  1 as Pr[kQ

Q 31 1 X = n] = Pr[kq = n  1] Q q=n

The generating function of kQ equals Q31 h i X kQ Pr [kQ = n] } n = Pr[kQ = 0] + *kQ (}) = H } n=1

=

1 1 + Q Q

1 1 = + Q Q

Q 31 Q31 X X n=1 q=n q Q 31 X X q=1 n=1

Pr[kq = n  1]} n

Q31 1 } X Pr[kq = n  1]} = *k (}) + Q Q q=1 q n

Taking the dierence between (Q + 1)*kQ +1 (}) and Q *kQ (}) results in the recursion (Q + 1)*kQ +1 (}) = (Q + })*kQ (}) £ ¤ £ ¤ Iterating this recursion starting from *k1 (}) = H } k1 = H } 0 = 1 leads to (16.6). ¤ Corollary 16.3.2 The probability density function of the hopcount in the URT with Q nodes is (n+1)

(1)Q 3(n+1) VQ (16.8) Q! Proof: The probability generating function *kQ (}) in (16.6) is also the (n) generating function of the Stirling numbers VQ of the first kind (Abramowitz and Stegun, 1968, 24.1.3) such that the probability that a uniformly chosen node in the URT has hopcount n equals (16.8). ¤ Pr[kQ = n] =

The explicit form of the generating function shows that the average hopcount kQ in a URT of size Q equals ¯ Q X ¯ 1 g 0 H[kQ ] = *kQ (1) = = log *kQ (})¯¯ (16.9) g} o }=1 o=2

= #(Q + 1) +   1 0

(}) is the digamma function (Abramowitz and Stegun, where #(}) = KK(}) 1968, Section 6.3) and the Euler constant is  = 0=57721 = = =. Similarly,

356

The Shortest Path Problem

the variance (2.27) follows from the logarithm of the generating function OkQ (}) = log (Q + })  log (Q + 1)  log (} + 1) as Var[kQ ] = # 0 (Q + 1)  # 0 (2) + #(Q + 1) +   1 2 + # 0 (Q + 1) 6 Using the asymptotic formulae for the digamma function leads to µ ¶ 1 H[kQ ] = log Q +   1 + R Q µ ¶ 2  1 +R Var[kQ ] = log Q +   6 Q = #(Q + 1) +  

(16.10) (16.11)

For large Q , we apply an asymptotic formula of the Gamma function (Abramowitz and Stegun, 1968, Section 6.1.47) to the generating function of the hopcount (16.6), µ µ ¶¶ Q }31 1 1+R *kQ (}) = (} + 1) Q P 1 n = " Introducing the Taylor series of K(}) n=1 fn } where the coe!cients fn are listed in Abramowitz and Stegun (1968, Section 6.1.34), we obtain with Q } = h} log Q , µ µ ¶¶ " " X 1 logn Q n 1 X n31 } 1+R fn } *kQ (}) = Q n! Q n=1 n=0 ¡1¢ " n 1+R Q X X logn3p Q n } fp+1 = Q (n  p)! n=0 p=0

With the definition (2.18) of the probability generating function, we conclude that the asymptotic form of the probability density function (16.8) of the hopcount in the URT is ¡ ¢ n 1 + R Q1 X logn3p Q fp+1 Pr[kQ = n] = (16.12) Q (n  p)! p=0 Since the coe!cients fn are rapidly decreasing, approximating the sum in (16.12) by its first term (p = 0) yields to first order in Q , (log Q )n (16.13) Pr[kQ = n]  Q n! which is recognized as a Poisson distribution (3.9) with mean log Q . Hence, for large Q and to first order, the average and variance of the hopcount in

16.3 The hopcount kQ in the URT

357

the URT are approximately H[kQ ]  Var[kQ ]  log Q . The accuracy of the Poisson approximation can be estimated by comparison with the average (16.10) and the variance (16.11) found above up to second order in Q . For example, if the URT has Q = 104 nodes, the Poisson approximation yields H[kQ ] = Var[kQ ] = 9=21034, while the average (16.10) is H[kQ ] = 8=78756 accurate up to 1034 and the variance (16.11) is Var[kQ ] = 8=14262. The exact results are H[kQ ] = 8=78761 and Var[kQ ] = 8=14277.

16.3.2 Application of the URT to the hopcount in the Internet In trace-route measurements explained in Van Mieghem (2004a), we are interested in the hopcount KQ denoted with capital K, which equals kQ in the URT excluding the event kQ = 0. In other words, the source and the destination are dierent nodes in the graph. Since from (16.8) Pr[kQ = 0] = (1)

(31)Q 31 VQ Q!

=

1 Q

we obtain, for 1  n  Q  1,

Pr[KQ = n] = Pr[kQ = n|kQ 6= 0] = =

Pr[kQ = n> kQ 6= 0] Pr[kQ 6= 0]

Q Pr[kQ = n] Q 1

Using (16.8), we find (n+1)

Pr[KQ = n] =

Q (1)Q3(n+1) VQ Q 1 Q!

(16.14)

with corresponding generating function, *KQ (}) =

Q31 X

Pr[KQ = n] } n

n=1 Q31 Q X Q Pr[kQ = n] } n  Pr[kQ = 0] Q 1 Q 1 n=0 µ ¶ Q 1 = *kQ (})  Q 1 Q

=

The average hopcount H[KQ ] = H[kQ |kQ 6= 0] is H[KQ ] =

Q 31 Q X1 Q 1 o o=2

(16.15)

358

The Shortest Path Problem

Hence, for large Q and in practice, we find that

µ

Pr[KQ = n] = Pr[kQ = n] + R

1 Q

¶

which allows us to use the previously derived expressions (16.12), (16.10) and (16.11). The histogram of the number of traversed routers in the Internet measured between two arbitrary communicating parties seems reasonably well modeled by the pdf (16.12). Figure 16.4 shows both the histogram of the hopcount deduced from paths in the Internet measured via the trace-route utility and the fit with (16.12). From the fit, we find a rather high number of nodes

Asia Europe USA fit with log(NAsia) = 13.5 fit with log(NEurope) = 12.6 fit with log(NUSA) = 12.9

0.10

Pr[H = k]

0.08

0.06

0.04

0.02

0.00 0

5

10

15

20

25

30

hop k

Fig. 16.4. The histograms of the hopcount derived from the trace-route measurement in three continents from CAIDA in 2004 are fitted by the pdf (16.12) of the hopcount in the URT.

h12=6  3 105  Q  h13=5  7 105 , which points to the approximate nature of modeling the Internet hopcount by that deduced from a URT. The relation between Internet measurements and the properties of the URT is further analyzed in a series of articles (Van Mieghem et al., 2000; van der Hofstad et al., 2001; Van Mieghem et al., 2001b; Janic et al., 2002; van der Hofstad et al., 2002b). At the time of writing, an accurate model of the hopcount in the Internet is not available.

16.4 The weight of the shortest path

359

16.4 The weight of the shortest path The weight — sometimes also called the length — of the shortest path is defined as the sum of the link weights that constitute the shortest path. In Section 16.2.1, the shortest path tree in the complete graph with exponential link weights was shown to be a URT. In this section, we confine ourselves to the same type of graph and require that the source node D (or root) is dierent from the destination node E. By Theorem 10.2.3 of a continuous-time Markov chain, the discovery time Pn of the n-th node from node D equals yn = q=1 q , where 1 > 2 > = = = > n are independent, exponentially distributed random variables with parameter q = q(Q q) with 1  q  n. We call m the interattachement time between the discovery or the attachment to the URT of the m  1-th and m-th node in the graph. The Laplace transform of yn is Z " £ 3}y ¤ g n = h3}w Pr [yn  w] H h gw 0 For a sum of independent exponential random variables, using the probability generating function (3.16), we have !# " Ã n n n X Y £ 3}y ¤ £ ¤ Y q(Q  q) H h n = H exp } = q H h3}q = } + q(Q  q) q=1 q=1 q=1 (16.16) ¤ £ The probability generating function3 *ZQ (}) = H h3}ZQ of the weight ZQ of the shortest path equals *ZQ (}) =

Q31 X

¤ £ H h3}yn Pr [E is n-th attached node in URT]

n=1

=

Q31 n 1 X Y q(Q  q) Q 1 } + q(Q  q)

(16.17)

n=1 q=1

because any node apart from the root D but including the destination node E has equal probability to be the n-th attached node. The average weight is ¯ ¯ Q 31 n g*ZQ (}) ¯¯ 1 X g Y q(Q  q) ¯¯ = H [ZQ ] =  ¯ ¯ g} Q 1 g} q=1 } + q(Q  q) ¯ }=0 n=1

3

1 d

}=0

If the link weights have mean (instead of 1), then ZQ is multiplied by d as explained in Sections 16.2.1 and 3.4.1. The weight of the scaled shortest path ZQ>d has pgf l k *ZQ>d (}) = H h3}dZQ = *ZQ (d})

360

The Shortest Path Problem

Using the logarithmic derivative of the product, ¯ n g Y q(Q  q) ¯¯ ¯ g} q=1 } + q(Q  q) ¯

n Y

}=0

q(Q  q) g = } + q(Q  q) g} q=1 =

Ã

n X

q(Q  q) log } + q(Q  q) q=1

!¯ ¯ ¯ ¯ ¯

}=0

n X

1 q(Q  q) q=1

gives Q31 n Q31 Q 31 X 1 1 1 XX 1 X H [ZQ ] = 1 = Q 1 q(Q  q) Q  1 q=1 q(Q  q) q=1

=

1 Q 1

n=1 Q31 X q=1

n=q

Q q q(Q  q)

The average weight is H [ZQ ] =

Q31 #(Q ) +  1 X 1 = Q 1 q Q 1

(16.18)

q=1

For large Q , log Q +  H [ZQ ] = +R Q

µ

1 Q2

¶

Similarly, the variance is computed (see problem (ii) in Section 16.9) as, ³P ´2 Q 31 1 Q31 X q=1 q 1 3 Var [ZQ ] =  (16.19) 2 Q (Q  1) q (Q  1)2 Q q=1 and for large Q, 2 +R Var [ZQ ] = 2Q 2

µ

log2 Q Q3

¶

By inverse Laplace transform of (16.17), the distribution Pr [ZQ  w] can be computed. The asymptotic distribution for the weight of the shortest path is (see problem (iii) in Section 16.9) 3{

lim Pr [Q ZQ  log Q  {] = h3h

Q for Q large. We rewrite (16.21) as [(Q  1)!]2 h *WQ ({) = Q ¡ Q31 Q2 q=1 { + 4  q  For Q = 2P , using

K(}+p) K(}+1)

=

à *W2P ({) =

Qp31 q=1

¢ Q 2 2

i

(16.24)

(q + }), we deduce that

!2 s (2P )(1 + { + P 2  P ) s (P + { + P 2 )

(16.25)

s { For large P , there holds { + P 2  P + 2P , provided |{| ? 2P . After substitution of { = 2P | in (16.25), with ||| ? 1, we obtain *W2P (2P |)  2 (1 + |)

2 (2P )  2 (1 + |)(2P )32| 2 (2P + |)

from which follows the asymptotic relation lim Q 2| *WQ (Q |) = 2 (1 + |)>

Q 1> 2> ===> we find that Q 2| *WQ (Q|) is analytic whenever the real part of | is non-negative. Evaluation along the line Re(|) = f = 0 then gives ] " 1 lim jQ (w) = lim hlwx Q 2lx *WQ (lQx)gx Q where (w) = h3h is the Gumbel distribution (3.37). sity function gw Furthermore, the two-fold convolution is given by Z " g ³ (2W) ´ 3x 3(w3x) h3h h3h gx  (w) = h3w gw 3"  µ ¶¸ Z " w 3w 3w@2 =h exp 2h cosh  x gx 2 3" Z " ³ ´ h i exp 2h3w@2 cosh (x) gx = 2h3w N0 2h3w@2 = 2h3w 0

where N ({) denotes the modified Bessel function (Abramowitz and Stegun, 1968, Section 9.6) of order . In summary, ³ ´ g ³ (2W) ´  (w) = 2h3w N0 2h3w@2 (16.29) lim jQ (w) = j(w) = QGQ @ GQ >GQ Pr GQ +1 = m = Pr GQ = m|qQ +1 5 n n oi h oi h (n) (n) (n) Pr qQ+1 5 GQ + Pr GQ = m + 1|qQ+1 5 GQ n n h oi h oi (n) (n31) (n31) + Pr GQ = m  1|qQ+1 5 GQ Pr qQ+1 5 GQ If the process of attaching a new node Q + 1 does not depend on the way thehQ previous nodes i arehattached ibut rather on their number, there holds (n) (n) Pr GQ = m|qQ+1 = Pr GQ = m . This property holds for the URT. We obtain a three point recursion for n A 1, h i h i h n oi (n) (n) (n) (n31) Pr GQ+1 = m = Pr GQ = m Pr qQ+1 5 @ GQ > GQ oi i h n h (n) (n) + Pr GQ = m + 1 Pr qQ +1 5 GQ i h n oi h (n) (n31) + Pr GQ = m  1 Pr qQ +1 5 GQ The probability generating function " h (n) i X h i (n) Pr GQ = m } m *G (}> Q ; n) = H } GQ = m=0

is obtained after multiplication by } m and summing over all m, h n oi (n) (n31) @ GQ > GQ *G (}> Q ; n) *G (}> Q + 1; n) = Pr qQ+1 5 h n oi * (}> Q ; n)  * (0> Q; n) (n) G G + Pr qQ+1 5 GQ } n oi h (n31) }*G (}> Q ; n) + Pr qQ+1 5 GQ h i (n) Now *G (0> Q ; n) = Pr GQ = 0 is the probability of the event that there are no nodes with degree n for 1  n  nmax  Q  1. Since the normalization of the generating function requires that *G (1> Q ; n) = 1 and since n n n oi h oi h oi h (n) (n31) (n) (n31) @ GQ >GQ +Pr qQ+1 5 GQ +Pr qQ+1 5 GQ =1 Pr qQ+1 5 h i h n oi (n) (n) it follows that Pr GQ = 0 Pr qQ+1 5 GQ = 0. Further, h n oi (n) Pr qQ+1 5 GQ 6= 0

368

The Shortest Path Problem

for any n 5 [1> nmax ] because the attachment of the node qQ+1 is possible to any non-empty set, this means that the absence of nodes with degree h i (n) n 5 [1> nmax ] cannot occur in URTs, thus Pr GQ = 0 = 0. A consequence is that the probability generating function *G (}> Q ; n) is at least R (}) as h } $ 0 n(for Q A 1).oiAfter using *G (0> Q ; n) = 0 and eliminat(n) (n31) @ GQ > GQ ing Pr qQ +1 5 , the recursion relation for the probability generating function becomes5 n oi 3 h 4 (n) n h oi Pr q 5 G Q+1 Q *G (}> Q + 1; n) (n31) D = 1+C  Pr qQ+1 5 GQ (1  }) *G (}> Q ; n) } (16.33) The special case for n = 1 and Q A 1 is h n oi 4 3 (1) Pr qQ+1 5 GQ (1  }) D *G (}> Q ; 1) *G (}> Q + 1; 1) = C1 + }

16.6.2 The Average Number of Degree n Nodes in the URT In the URT, a new node qQ+1 is attached uniformly to any of Q previously attached nodes such that h i (n) n oi H GQ h (n) = Pr qQ+1 5 GQ Q Also, the probability that an arbitrary node in a URT of size Q has degree k l (n)

n equals

H GQ Q

. We obtain from (16.33) 3

3

*G (}> Q + 1; n) = C1 + C

i h (n) H GQ Q}

h i4 4 (n31) H GQ D (1  })D *G (}> Q ; n)  Q (16.34)

5

k (n) l (n) = 1 for all p $ n because Gp = 0 if With the initialization *G (}> p; n) = H } Gp 1 ? p $ n, after iterating (16.33) we arrive at

*G (}> Q ; n) =

Q 31  \ p=n

  k k rl  rl q q 1 (n) (n31) 1 3 Pr qp+1 M Gp 3 Pr qp+1 M Gp 13 (1 3 }) }

16.6 The degree of a node in the URT

369

By taking the derivative of both sides in (16.34) with respect to } and evaluating at } = 1, a recursion for the average is found, h i (n31) i Q 1 h i H GQ h (n) (n) H GQ + (16.35) H GQ+1 = Q Q h i (n) (n) Let uQ = (Q  1)H GQ , then the recursion valid for 1 ? n  Q  2 becomes un31 (n) (n) uQ+1 = uQ + Q (16.36) Q 1 Theorem 16.6.1 In the URT, the average number of degree n nodes is given by (n) n31 i Q h (1)Q+n31 VQ31 (1)Q X (m) (n) VQ31 (2)m (16.37) + n H GQ = n + 2 (Q  1)! 2 (Q  1)! m=1

Proof: See Section 16.8.

¤ (n)

For large Q and using the asymptotics of the Stirling numbers VQ of the first kind (Abramowitz and Stegun, 1968, Section 24.1.3.III), the asymptotic law is i h à ! (n) H GQ logn31 Q 1 Pr [GURT = n] = (16.38) = n +R Q 2 Q2 The ratio of the average number of nodes with degree n over the total number of nodes, which equals the probability that an arbitrary node in a URT of size Q has degree n, exponentially fast with rate ln 2. h decreases i (n) The variance Var GQ is most conveniently computed from the logarithm of the probability generating function with (2.27). By taking the logarithm of both sides in (16.34) and dierentiating twice and adding (16.35), we obtain i h i h (n) (n) Var GQ+1 = i (Q ; n) + Var GQ where 3 i(Q ; n) =  C

h i (n) H GQ Q

h h i 42 3 h i i4 (n31) (n) (n31) H GQ H GQ H GQ D +C D + + Q Q Q

370

The Shortest Path Problem

h i (n) Since Var Gp = 0 for p  n, the general solution is Q i X h (n) i (m; n) Var GQ = m=n

For large Q , using (16.38), we observe that h i à ! (n) µ ¶ Var GQ log2n32 Q 1 3  = +R Q 2n 22n Q2

(16.39)

(n)

G (n) In practice, if we use the estimator wˆQ = QQ for the probability that the degree of a node equals n, then (a) the estimator is unbiased because the l k (n) h i H G (n) Q mean of the estimator H wˆQ equals the correct mean and (b) the Q l k ¸  (n) h i (n) ¡ ¢ Var GQ G (n) variance Var wˆQ = Var QQ = $ 0 as R Q1 for large Q . 2 Q

10

0

RIPE data (May-June 2003) N = 2574, L = 3992 fit: ln(Pr[D U = k]) = 0.44 - 0.67 k with U = 0.99

Pr[DU = k]

10

10

10

10

RIPE data (Jan.-Feb. 2004) N = 3850, L = 6743 fit: ln(Pr[D U = k]) = -0.49 - 0.41 k with U = 0.95

-1

-2

-3

-4

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

32

34

k

Fig. 16.6. The histogram of the degree GX derived from the graph JX formed by the union of paths measured via trace-route in the Internet. Both measurements in 2003 and 2004 are fitted on a log-lin plot and the correlation coe!cient  quantifies the quality of the fit.

The law (16.38) is observed in Fig. 16.6, which plots the histogram of the degree GX in the graph JX . The graph JX is obtained from the union of trace-routes from each RIPE measurement box to any other box positioned

16.6 The degree of a node in the URT

371

mainly in the European part of the Internet. For about 50 measurement boxes in 2003, the correspondence is striking because the slope of the fit on a log-lin scale equals 0.668 while the law (16.38) gives  ln 2 = 0=693. Ignoring in Fig. 16.6 the leave nodes with n = 1 suggests that the graph JX is URT-like. For 72 measurement boxes in 2004 which obviously results in a larger graph JX , deviations from the URT law (16.38) are observed. If measurements between a larger full mesh of boxes were possible and if the measurement boxes were more homogeneously spread over the Internet, a power law behavior is likely to be expected as mentioned in Section 15.3. However, these earlier reported trace-route measurements that lead to power law degrees have been performed from a relatively small number of sources to a large number of destinations. These results question the observability of the Internet: how accurate are Internet properties such as hopcount and degree that are derived from incomplete measurements, i.e. from a selected small subset of nodes at which measurement boxes are placed?

16.6.3 The degree of the shortest path tree in the complete graph with i.i.d. exponential link weights In the complete graph NQ with i.i.d. exponential link weights, any node q possesses equal properties in probability because of symmetry. If we denote by gq the degree of node q in the shortest path tree rooted at that node q, the symmetry implies that Pr [gq = n] = Pr [gl = n] for any node q and l. In fact, we consider here the degree of a URT as an overlay tree in a complete graph. Concentrating on a node with label 1, we obtain from (16.32) h i i h (n) (1) H GQ = Q Pr [g1 = n] = Q Pr [Q = n The latter follows from the fact that the degree of a node is equal to the (1) number of its direct neighbors, the nodes at level 1, [Q . By definition of the URT, the second node surely belongs to the level set 1, while node 3 has equal probability to be attached to the root or to node 2. In general, when attaching a node m to a URT of size m  1, the probability that node m is 1 . Thus, the number of nodes at level 1 in the attached to the root equals m31 URT (constructed upon the complete graph) is in distribution equal to the sum of Q  1 independent Bernoulli random variables each with dierent 1 , mean m31 (1) g [Q =

Q X m=2

µ Bernoulli

1 m1

¶ =

Q31 X m=1

µ ¶ 1 Bernoulli m

372

The Shortest Path Problem

because each node in the complete graph is connected to Q  1 neighbors. The generating function is  SQ 31  ¸ Q31 h (1) i Y  Bernoulli 1  ¸ 1 [Q m=1 Bernoulli m m =H } H } = H } m=1

Using the generating function (3.1) of a Bernoulli random vari probability  ¸ Bernoulli 1m = 1  1m + }m , yields able, H } h (1) i Q31 Y µ } + m  1 ¶ (} + Q  1) [Q H } = = m (})(Q ) m=1

Compared to the generating function (16.6) of the hopcount kQ , we recognize that Q32 Q31 h (1) i X X Pr[kQ31 = n]} n+1 = Pr[kQ31 = n  1]} n H } [Q = }*Q31 (}) = n=0

n=1

from which we deduce, for 1  n  Q  1, h i (1) Pr [Q = n = Pr[kQ31 = n  1] Using (16.7), we arrive at the curious result h i (n31) h i H [Q 31 (1) Pr [Q = n = for 1  n  Q  1= Q 1 The probability that the number of level 1 nodes in the shortest path tree in the complete graph with i.i.d. exponential link weights is n equals the average number of nodes on level n  1 in a URT of size Q  1 divided by that size Q  1. In other words, the “horizontal” distribution at level 1 is related to the “vertical” distribution of the size of the level sets. In summary6 , in the complete graph with i.i.d. exponential link weights, the probability that an arbitrary node q as root of a shortest path tree has degree n is (n)

Pr [gq = n] = Pr[kQ31

(1)Q313n VQ31 = n  1] = (Q  1)!

(16.40)

The degree of an arbitrary node in the union of all shortest paths trees in the complete graph NQ with i.i.d. exponential link weights is also given by (16.40) because in that union each node q is once a root and further plays, 6

This result is due to Remco van der Hofstad (private communication).

16.7 The minimum spanning tree

373

by symmetry, the role of the m-th attached node in the URT rooted at any other node in NQ . 16.7 The minimum spanning tree From an algorithmic point of view, the shortest path problem is closely related to the computation of the minimum spanning tree (MST). The Dijkstra shortest path algorithm is similar to Prim’s minimum spanning tree algorithm (Cormen et al., 1991). In this section, we compute the average weight of the MST in a graph with a general link weight structure.

16.7.1 The Kruskal growth process of the MST Since the link weights in the underlying complete graph are chosen independently and assigned randomly to links in the complete graph, the resulting graph is probabilistically the same if we first order the set of link weights and assign them in increasing order randomly to links in the complete graph. In the latter construction process, only the order statistics or the ranking of the link weights su!ce to construct the graph because the precise link weight can be unambiguously associated to the rank of a link. This observation immediately favors the Kruskal algorithm for the MST over Prim’s algorithm. Although the Prim algorithm leads to the same MST, it gives a more complicated, long-memory growth process, where the attachment of each new node depends stochastically on the whole growth history so far. Pietronero and Schneider (1990) illustrate that in our approach Prim, in contrast with Kruskal, leads to a very complicated stochastic process for the construction of the MST. The Kruskal growth process described here is closely related to a growth process of the random graph Ju (Q> O) with Q nodes and O links. The construction or growth of Ju (Q> O) starts from Q individual nodes and in each step an arbitrary, not yet connected random pairs is connected. The only dierence with Kruskal’s algorithm for the MST is that, in Kruskal, links generating loops are forbidden. Those forbidden links are the links that connect nodes within the same connected component or “cluster”. As a result, the internal wiring of the clusters diers, but the cluster size statistics (counted in nodes, not links) is exactly the same as in the corresponding random graph. The metacode of the Kruskal growth process for the construction of the MST is shown in Fig. 16.7. The growth process of the random graph Js (Q ), which is asymptotically equal to that of Ju (Q> O), is quantified in Section 15.6.4 for large Q . The

374

The Shortest Path Problem KruskalGrowthMST 1. start with Q disconnected nodes 2. repeat until all nodes are connected 3. randomly select a node pair (l> m) 4. if a path Pl$m does not exist 5. then connect l to m Fig. 16.7. Kruskal growth process

fraction of nodes V in the giant component of Js (Q ) is related to the average degree or to the link density s because rg = s(Q  1) in Js (Q ) by (15.20). For large Q , the size of the giant cluster in the forest is thus determined as a function of the number of added links that increase rg .

a e c

d f b

Fig. 16.8. Component structure during the Kruskal growth process.

We will now transform the mean degree rg in the random graph Js (Q ) to the mean degree MST in the corresponding stage in Kruskal growth process of the MST. In early stages of the growth each selected link will be added with high probability such that MST = rg almost surely. After some time the probability that a selected link is forbidden increases, and thus rg exceeds MST . In the end, when connectivity of all Q nodes is reached, MST = 2 (since it is a tree) while rg = R(log Q ), as follows from (15.19) and the critical threshold sf  logQQ . Consider now an intermediate stage of the growth as illustrated in Fig. 16.8. Assume there is a giant component of average size Q V and qo = Q(1 V)@vo small components of average size vo each. Then we can distinguish six types of links labelled d-i in Fig. 16.8. Types d and e are links that have been

16.7 The minimum spanning tree

375

chosen earlier in the giant component (d) and in the small components (e) respectively. Types f and g are eligible links between the giant component and a small component (f) and between small components (g) respectively. Types h and i are forbidden links connecting nodes within the giant component (h), respectively within a small component (i ). For large Q , we can enumerate the average number of links O{ of each type {: Od + Oe = 12 PVW Q Of = VQ · (1  V)Q Og = 12 q2o · v2o

Oh = 12 (VQ )2  VQ Oi = 12 qo vo (vo  1)  qo (vo  1)

To highest order in R(Q 2 ), we have Of = Q 2 V(1  V)>

1 Og = Q 2 (1  V)2 > 2

1 Oh = Q 2 V 2 2

The probability that a randomly selected link is eligible is t = ¡ ¢ or, to order R Q 2 ,

Of +Og Of +Og +Oh +Oi

t = 1  V2

(16.41)

In contrast with the growth of the random graph Js (Q ) where at each stage a link is added with probability s, in the Kruskal growth of the MST we are only successful to add one link (with probability 1) per 1t stages on average. Thus the average number of links added in the random graph corresponding 1 to one link in the MST is 1t = 13V 2 . This provides an asymptotic mapping between rg and MST in the form of a dierential equation, grg 1 = gMST 1  V2 By using (15.22), we find gMST grg (1 + V) (V + (1  V) log(1  V)) gMST = = gV grg gV V2 Integration with the initial condition MST = 2 at V = 1, finally gives the average degree MST in the MST as function of the fraction V of nodes in the giant component MST (V) = 2V 

(1  V)2 log(1  V) V

(16.42)

As shown in Fig. 16.9, the asymptotic result (16.42) agrees well with the simulation (even for a single sample), except in a small region around the transition MST = 1 and for relatively small Q . The key observation is that all transition probabilities in the Kruskal

376

The Shortest Path Problem

Fraction S of nodes in the giant component

1.0

0.8

N = 1000 N = 10000 N = 25000 Theory

0.6

0.4

0.2

0.0 0.0

0.5

1.0

1.5

2.0

Mean degree PMST

Fig. 16.9. Size of the giant component (divided by Q ) as a function of the mean degree M ST . Each simulation for a dierent number of nodes Q consists of one MST sample.

growth process asymptotically depend on merely one parameter V, the fraction of nodes in the giant component, and V is called an order parameter in statistical physics. In general, the expectation of an order parameter distinguishes the qualitatively dierent regimes (states) below and above the phase transition. In higher dimensions, fluctuations of the order parameter around the mean can be neglected and the mean value can be computed from a selfconsistent mean-field theory. In our problem, the underlying complete (or random) graph topology makes the problem eectively infinite-dimensional. The argument leading to (15.20) is essentially a mean-field argument.

16.7.2 The average weight of the minimum spanning tree By definition, the weight of the MST is ZMST =

O X

z(m) 1mMMST

(16.43)

m=1

where z(m) is the m-th smallest link weight. The average MST weight is H [ZMST ] =

O X m=1

¤ £ H z(m) 1mMMST

16.7 The minimum spanning tree

377

The random variables z(m) and 1mMMST are independent because the m-th smallest link weight z(m) only depends on the link weight distribution and the number of links O, while the appearance of the m-th link in the MST only depends on the graph’s topology, as shown in Section 16.7.1. Hence, ¤ £ ¤ ¤ £ £ H z(m) 1mMMST = H z(m) H [1mMMST ] = H z(m) Pr [m 5 MST] such that the average weight of the MST is H [ZMST ] =

O X

£ ¤ H z(m) Pr [m 5 MST]

(16.44)

m=1

In general for independent link weights with probability density function iz ({) and distribution function Iz ({) = Pr [z  {], the probability density function of the m-th order statistic follows from (3.36) as µ ¶ miz ({) O (16.45) (Iz ({))m (1  Iz ({))O3m iz(m) ({) = Iz ({) m ¡ ¢ The factor Om (Iz ({))m (1  Iz ({))O3m is a binomial distribution with mean  = Iz ({) O and variance  2 = OIz ({) (1  Iz ({)) that, by the Central Limit Theory 6.3.1, tends for large O to a Gaussian m O,

I1 h3  2 £

(m3)2 2 2

¤

, which

peaks at m = . For large Q and fixed we {m = H z(m) ' Iz31 ( Om ). We found before in (16.41) that the link ranked m appears in the MST with probability have7

Pr [m 5 MST] = 1  Vm2 where Vm is the fraction of nodes in the giant component during the construction process of the random graph at the stage where the number of links precisely equals m. Since links are added independently, that stage in fact establishes the random graph Ju (Q> O = m). Our graph under ¡Q ¢consideration is the complete graph NQ such that we add in total O = 2 links. 7

31 In general, it holds that z(n) = Iz (X(n) ) and

   31    31 H z(n) = H Iz (X(n) ) 6= Iz (H X(n) ) but, for a large number of order statistics O, the Central Limit Theorem 6.3.1 leads to       m 31 31 H z(n) ' Iz (H X(n) ) ' Iz O because for a uniform random variable X on [0,1] the average weight of the m-th smallest link is exactly   m m H z(l) = ' O+1 O

378

The Shortest Path Problem

With (15.22) and rg =

2O Q,

it follows that log(1  Vm ) 2m = Q Vm

Hence, H [ZMST ] '

O X m=1

Iz31

(16.46)

µ ¶ ¢ m ¡ 1  Vm2 O

We now approximate the sum by an integral, Z O ³x´¡ ¢ H [ZMST ] ' Iz31 1  Vx2 gx O 1 Substituting { = 2x Q (which is the average degree in any graph J (Q> x)) 2 yields for large Q where O ' Q2 , Z Z ´ ¢ Q Q 3131 ³ { ´³ Q Q 31 ³ { ´¡ 2 Iz Iz H [ZMST ] ' 1  V Q { g{ ' 1  V 2 ({) g{ 2 2 Q 2 0 Q 2 Q

It is known (Janson et al., 1993) that, if the number of links in the growth process of the random graph is below Q2 , with high probability (and ignoring a small onset region just below Q2 ), there is no giant component such that V ({) = 0 for { 5 [0> 1]. Thus, we arrive at the general formula valid for large Q , Z Z ¢ Q 1 31 ³ { ´ Q Q 31 ³ { ´ ¡ Iz Iz H [ZMST ] ' g{ + 1  V 2 ({) g{ 2 0 Q 2 1 Q (16.47) The first term is the contribution from the smallest Q@2 links in the graph, which are included in the MST almost surely. The remaining part comes from the more expensive links in the graph, which are included with diminishing probability since 1  V 2 ({) decreases exponentially for large { as can be deduced from (15.21). The rapid decrease of 1  V 2 ({) makes only ¡ ¢ relatively small values of the argument Iz31 Q{ contribute to the second integral. At this point, the specifics of the link weight ¡ ¢ distribution needs to be introduced. The Taylor expansion of Q2 Iz31 Q{ for large Q to first order is µ ¶ µ ¶ 1 1 Q 31 ³ { ´ Q 31 { { Iz +R +R = Iz (0) + = 2 Q 2 2iz (0) Q 2iz (0) Q since we require that link weights are positive such that Iz31 (0) = 0. This expansion is only useful provided iz is regular, i.e. iz (0) is neither zero nor

16.7 The minimum spanning tree

379

infinity. These cases occur, for example, for polynomial link weights with iz ({) = {31 and  6= 1. For polynomial link weights, however, holds ¡ ¢ 13 1 1 that Q2 Iz31 Q{ = Q 2  {  . Formally, this latter expression reduces to the first order Taylor approach for  = 1, apart from the constant factor iz1(0) . Therefore, we will first compute H [ZMST ] for polynomial link weights and then return to the case in which the Taylor expansion is useful. 16.7.2.1 Polynomial link weights The average weight of the MST for polynomial link weights follows8 from (16.47) as à ! 1 Z Q ¢ 1 ¡ 1 Q 13  {  1  V 2 ({) g{ + H [ZMST ()] ' 1 2 + 1 1  and g{ = Let | = V ({) and use (15.22), then { = V 31 (|) =  log(13|) | ³ ´ log(13|) g g| while | = V (1) = 0 and | = V (Q ) = 1, such that  g| | Z L= 1

Z =

0

Q

¢ 1 ¡ {  1  V 2 ({) g{

1µ

log(1  |)  |

¶1



After partial integration, we have L =1 

1 + +1

Finally, we end up with H [ZMST ()] ' Q 8

1 

2 +1

1 

1 +1

à 1 13 

µ ¶ log(1  |)  g| |

¡ ¢ g 1  |2 g| Z

"

0

Z

h3{

1

{  +1

"

{

1

(1  h3{ ) 

0

!

h3{

1 +1 

g{

1

(1  h3{ ) 

g{

(16.48)

Since the average of the n-th smallest link weight can be computed from (3.36) as   1  K n+   H!  H z(n) =  1 K (n) K H+1+  the exact formula (16.44) reduces to H [ZM S T

H! ()] =  K H+1+

1 



 H [ K m+ m=1

1 

K (m)



  1 3 Vm2

Analogously to the above manipulations, after convertion to an integral, substituting { = K(}+ 1 ) 2x and using (Abramowitz and Stegun, 1968, Section 6.1.47), for large }, that K(}) = Q   1  (})  1 + R }1 , we arrive at the same formula.

380

The Shortest Path Problem

If  ? 1, then H[ZMST ()] $ 0 for Q $ 4, while for  A 1, H[ZMST ()] $ 4. In particular, lim" , and we can simply replace iz (0) by siz (0) in the expression (16.50) to obtain the average weight of the MST in the random graph Js (Q ). 16.8 The proof of the degree Theorem 16.6.1 of the URT h i (Q31) 16.8.1 The case n = Q: H GQ (Q )

If n = Q, the recursion (16.35) becomes with GQ k l (Q ) H GQ +1 = With initial value

(1) G2

= 0, l k (Q 31) H GQ Q

= 2, the solution l k (Q 31) = H GQ

2 (16.51) (Q 3 1)! l k (n) is readily verified. Since for any URT, it holds that Pr GQ = m = 0 for m A Q 3 n, we have k l k l (Q 31) (Q 31) that H GQ = Pr GQ = 1 . Since there exists in total (Q 3 1)! dierent URTs of size Q, this result (16.51) means that there are precisely two possible URTs with a node of degree Q 3 1. Indeed, one is the root with Q 3 1 children and the other is the root with one child of degree Q 3 1 that in turn possesses Q 3 2 children. Also, (Q 31)

uQ

l k (Q 31) = = (Q 3 1)H GQ

2 (Q 3 2)!

(16.52)

16.8 The proof of the degree Theorem 16.6.1 of the URT

h i (1) 16.8.2 The case n = 1: H GQ

381

If n = 1 and Q D 3, the recursion (16.35) is slightly dierent because the newly attached node nQ +1 necessarily belongs to the set of degree 1 nodes in the URT of size Q + 1 such that

(0)

(1)

with G1 = 1 and G2 for n = 1 becomes

k l Q 3 1 k (1) l (1) H GQ +1 = H GQ + 1 Q l l k k (1) (1) (1) = 2. With uQ = (Q 3 1)H GQ , the recursion = 2. Hence, H G3 (1)

(1)

uQ +1 = uQ + Q (1)

(1)

The particular solution is uQ ;s = dQ 2 + eQ + f. Substitution of uQ = dQ 2 + eQ + f into the dierence equation yields dQ 2 + (e + 2d) Q + d + e + f = dQ 2 + (e + 1) Q + f or, by equating corresponding power in Q, we find the conditions e + 2d = e + 1 and d + e = 0 from which d = 12 , e = 3 12 . Thus, l k Q (1) (1) uQ = (Q 3 1)H GQ = (Q 3 1) + f 2 and

k l Q f (1) H GQ = + 2 Q 31

k l (1) Using H G3 = 2 shows that f = 1 such that, for Q A 2, l k 1 Q (1) + H GQ = 2 Q 31

(16.53)

h i (n) 16.8.3 The general case: H GQ Let us denote U({> |) =

31 " Q [ [

(n)

uQ {n |Q

(16.54)

Q =3 n=2

then the recursion (16.36) is transformed into 31 31 31 " Q " Q " Q [ [ [ [ [ [ (n) (n) (n31) n Q (Q 3 1)uQ +1 {n | Q = (Q 3 1)uQ {n | Q + uQ { | Q =3 n=2

Q =3 n=2

Q =3 n=2

Now, the left-hand side is 31 " Q " Q 32 " Q 32 [ [ 1 [ [ 1 [ [ (n) (n) (n) (Q 3 1)uQ +1 {n | Q = (Q 3 2)uQ {n | Q = (Q 3 2)uQ {n | Q | | Q =3 n=2 Q =4 n=2 Q =3 n=2

=

" Q 31 " 1 [ [ 1 [ (n) (Q 31) Q 31 Q QuQ {n |Q 3 QuQ { | | Q =3 n=2 | Q =3

3

" Q 31 " 2 [ [ (n) n Q 2 [ (Q 31) Q 31 Q uQ { | + u { | | Q =3 n=2 | Q =3 Q

382

The Shortest Path Problem

Using (16.54) yields 31 " Q [ [ 2 CU({> |) (n) 3 U({> |) (Q 3 1)uQ +1 {n | Q = C| | Q =3 n=2

3

" " 1 [ 2 [ (Q 31) Q 31 Q (Q 31) Q 31 Q QuQ { | + u { | | Q =3 | Q =3 Q

Invoking (16.52) yields " " " [ 2 [ (Q 31) Q 31 Q ({|)Q 4 [ ({|)Q = 4{| = 4{| (h{| 3 1) uQ { | = | Q =3 {| Q =3 (Q 3 2)! Q! Q =1 " " " [ 2 [ Q ({|)Q (Q + 2) ({|)Q 1 [ (Q 31) Q 31 Q QuQ { | = = 2{| | Q =3 {| Q =3 (Q 3 2)! Q! Q =1

= 2 ({|)2

" " [ [ ({|)Q ({|)Q + 4{| = 2 ({|)2 h{| + 4{| (h{| 3 1) Q! Q! Q =0 Q =1

such that 31 " Q [ [ 2 CU({> |) (n) 3 U({> |) 3 2 ({|)2 h{| (Q 3 1)uQ +1 {n | Q = C| | Q =3 n=2

Similarly, 31 " Q [ [

(n31) n Q

uQ

{ |

={

Q =3 n=2

32 " Q [ [

(n)

uQ {n |Q

Q =3 n=1

={

31 " Q [ [

(n)

uQ {n |Q + {2

Q =3 n=2 " [

(1)

uQ | Q 3 {

Q =3 (1)

Q 2

{2

" [

(1)

uQ |Q = {2

Q =3

{

" [

(Q 31) Q 31 Q uQ { |

=2

Q =3

(1)

uQ |Q 3 {

Q =3

= {U({> |) + {2

Using both (16.52) and uQ =

" [

" [

(Q 31) Q 31 Q

uQ

{

|

Q =3 " [

(Q 31) Q 31 Q

uQ

{

|

Q =3

(Q 3 1) + 1 leads to

 3  " " [ [ ({|)2 g2 {2 |3 Q | |Q = (Q 3 1)|Q + {2 + 2 2 2 g| 13| 13| Q =3 Q =3

" [ Q =3

" [ ({|)Q ({|)Q = 2 ({|)2 = 2 ({|)2 (h{| 3 1) (Q 3 2)! Q! Q =1

such that 31 " Q [ [

(n31) n Q

uQ

{ |

Q =3 n=2

= {U({> |) +

({|)2 g2 2 g| 2



|3 13|

 +

{2 | 3 3 2 ({|)2 (h{| 3 1) 13|

Combining all transforms the recursion (16.36) to a first order linear partial dierential equation (1 3 |)

CU({> |) + C|

  {2 |3 2 3 3 3| + | 2 + 13{3 U({> |) = {2 | 3 + 2 ({|)2 | 13| (1 3 |)3   1 1 = {2 | 2 + 13| (1 3 |)3

16.8 The proof of the degree Theorem 16.6.1 of the URT

383

with boundary equations U({> 0) = U(0> |) = 0. Further, ]

(n) 31 31 " Q " Q l k [ [ [ [ uQ U({> |) (n) n Q 31 { g| = | = H GQ {n | Q 31 |2 Q 3 1 Q =3 n=2 Q =3 n=2

Hence, if { = 1, ]

31 " Q "  l k l k [ [ [ U(1> |) (n) (1) Q 3 H GQ | Q 31 g| = H GQ |Q 31 = 2 | Q =3 n=2 Q =3 " " " l k [ [ [ Q Q 31 |Q 31 (1) H GQ | Q 31 = Q|Q 31 3 3 | 2 Q 31 Q =3 Q =3 Q =3 Q =3 Q =3 ] 1 1 1 | = 3 (1 + 2|) 3 g| 2 (1 3 |)2 2 13|

=

" [

Q| Q 31 3

" [

or U(1> |) =

|2 (1 3 |)3

3

|2 (1 3 |)

(16.55)

It is more convenient to consider the dierential equation as an ordinary dierential equation in | and to regard the variable { as a parameter. The homogeneous dierential equation,   CUk ({> |) 2 (1 3 |) = { + 3 1 Uk ({> |) C| | is solved after integration with respect to |,   ] {+ 2 31 ] ] g| g| | ln Uk ({> |) = g| = ({ 3 1) +2 13| 13| | (1 3 |) l k = (1 3 {) ln(1 3 |) + 2 (ln | 3 ln(1 3 |)) = ln (1 3 |)3{31 |2 or Uk ({> |) = (1 3 |)3{31 | 2 . The particular solution is of the form U ({> |) = F (|) Uk ({> |) where F (|) obeys   CF (|) 1 1 = {2 (1 3 |){ + C| 13| (1 3 |)3 or F (|) = {2 =3

]   (1 3 |){33 + (1 3 |){31 g| + f ({)

{2 (1 3 |){32 {2 (1 3 |){ 3 + f ({) {32 {

where f ({) is a function of {, independent of |, to be determined later. The solution is U ({> |) = 3

({|)2 3

({ 3 2) (1 3 |)

3

{| 2 + f ({) (1 3 |)3{31 | 2 (1 3 |)

The initial condition U (0> |) = 0 shows that f (0) = 0, while the boundary condition (16.55) implies that f(1) = 0. Expanding this solution in a power series around { = 0 and | = 0 yields Uk ({> |) = (1 3 |)3{31 |2 =

"  [ 3{ 3 1 (31)Q |Q +2 Q Q =0

384

The Shortest Path Problem

From the generating function of the Stirling numbers of the first kind (Abramowitz and Stegun, 1968, Section 24.1.3), q [ K({ + 1) (m) = Vq {m K({ + 1 3 q) m=0

(16.56)

we observe that (n+1) Q 3{ 3 1 [ VQ +1 (31)n n K(3{) = { = Q !K(3{ 3 Q) Q! Q n=0

such that Uk ({> |) =

(n+1) " [ Q [ VQ +1 Q =0 n=0

Q!

(31)Q +n {n | Q +2 =

(n+1) 32 " Q [ [ VQ 31 Q =2 n=0

(Q 3 2)!

(31)Q +n {n |Q

Hence,

U ({> |) =

" [ " [ Q =2 n=2

1 2n31

(n+1) 32 " " Q Q  [ [ [ VQ 31 {n |Q 3 { |Q + f ({) (31)Q +n {n | Q 2 (Q 3 2)! Q =2 Q =2 n=0

It remains to determine f ({) by equating the corresponding powers in { and | at both sides. With the definition (16.54), equating the second power (Q = 2) in | yields 0=

" [ n=2

1 {n 3 { + f ({) 2n31

which indicates that f ({) = { 3

{2 23{

agreeing with f (0) = f (1) = 0. The Taylor series around { = 0 is f ({) = 1 f1 = 1 and fn = 3 2n1 for n A 1. Equating the power Q A 2 in |, Q 31 [

(n)

uQ {n =

n=2

" [ n=2

S"

n n=0 fn {

with f0 = 0,

(n+1) Q 32 Q  [ VQ 31 n (31)Q +n {n { 3 { + f ({) 2n31 2 (Q 3 2)! n=0

1

(n+1) " " Q  [ [ VQ 31 n n (31)Q +n {n { 3 { + f { n 2n31 2 (Q 3 2)! n=2 n=0 n=0 3 4 (m+1) n31 " " [ [ [ VQ 31 1 Q  n Q +m D n C = fn3m (31) { 3{+ { 2n31 2 (Q 3 2)! m=0 n=2 n=1 3 3 4 4 (1) (m+1) n31 " " [ [ [ (31)Q VQ 31 (31)Q +m VQ 31 1 Q  n C C D D {n { 3 { + f1 = fn3m {+ 2n31 2 (Q 3 2)! (Q 3 2)! m=0 n=2 n=2

=

" [

1

(1)

which, by using VQ 31 = (31)Q (Q 3 2)! and f1 = 1, equals Q 31 [ n=2

(n) uQ {n

=

" [ n=2

3 4 (m+1) n31 " Q  [ [ VQ 31 Q +m D n n C { + fn3m (31) { 2n31 2 (Q 3 2)! m=0 n=2 1

16.9 Problems

385

Finally, by equating the corresponding powers in {, leads to (n)

uQ =

=

(m+1) Q  n31 [ VQ 31 f (31)Q +m + n3m 2n31 2 (Q 3 2)! m=0

1

1 2n31

n31 Q  (31)Q +n31 V (n) [ (m) (31)Q Q 31 + n + V (32)m (Q 3 2)! 2 (Q 3 2)! m=1 Q 31 2

or to (16.37). As a check, using (16.56) the generating function reveals that l k (Q 31) = H GQ

Q 2Q 31

(31)q K(q3{) K(3{)

=

Q! 1 2 1 Q 3 Q 31 + = + 2Q 31 (Q 3 1)! 2 (Q 3 1)! (Q 3 1)! (Q 3 1)! +

m=0

(m)

Vq {m ,

Q 32 [ (31)Q 1 (m) + Q 31 V (32)m (Q 3 1)! 2 (Q 3 1)! m=1 Q 31 % & (31)Q (31)Q 31 K(Q 3 1 + 2) 1 (Q 31) Q 31 + Q 31 3 VQ 31 (32) + (Q 3 1)! 2 (Q 3 1)! K(2)

Q 2Q 31

Q 2

Sq

+

=

 1 = Also H GQ

=

1 Q 31

is readily verified.

16.9 Problems (i) Comparison of simulations with exact results. Many of the theoretical results are easily verified by simulations. Consider the following standard simulation: (a) Construct a graph of a certain class, e.g. an instance of the random graphs Js (Q ) with exponentially distributed link weights (b) Determine in that graph a desired property, e.g. the hopcount of the shortest path between two dierent arbitrary nodes, (c) Store the hopcount in a histogram and (d) repeat the sequence (a)-(c) q times with each time a dierent graph instance in (a). Estimate the relative error of the simulated hopcount in Js (Q ) with s = 1 for q = 104 > 105 and 106 . (ii) Given the probability generating function (16.17) of the weight of the shortest path in a complete graph with independent exponential link weights, compute the variance of ZQ . (iii) Prove the asymptotic law (16.20) of the weight of the shortest path in a complete graph with i.i.d. exponential link weights. (iv) In a communication network often two paths are computed for each important flow to guarantee su!cient reliability. Apart from the shortest path between a source D and a destination E, a second path between D and E is chosen that does not travel over any intermediate router of the shortest path. We call such a path node-disjoint to the shortest path. Derive a good approximation for the distribution of

386

The Shortest Path Problem

the hopcount of the shortest node-disjoint path to the shortest path in the complete graph with exponential link weights with mean 1.

17 The e!ciency of multicast

The e!ciency or gain of multicast in terms of network resources is compared to unicast. Specifically, we concentrate on a one-to-many communication, where a source sends a same message to p dierent, uniformly distributed destinations along the shortest path. In unicast, this message is sent p times from the source to each destination. Hence, unicast uses on average iQ (p) = pH [KQ ] link-traversals or hops, where H [KQ ] is the average number of hops to a uniform location in the graph with Q nodes. One of the main properties of multicast is that it economizes on the number of linktraversals: the message is only copied at each branch point of the multicast tree to the p destinations. Let us denote by KQ (p) the number of links in the shortest path tree (SPT) to p uniformly chosen nodes. If we define the multicast gain jQ (p) = H [KQ (p)] as the average number of hops in the SPT rooted at a source to p randomly chosen distinct destinations, then jQ (p)  iQ (p). The purpose here is to quantify the multicast gain jQ (p). We present general results valid for all graphs and more explicit results valid for the random graph Js (Q ) and for the n-ary tree. The analysis presented here may be valuable to derive a business model for multicast: “How many customers p are needed to make the use of multicast for a service provider profitable?” Two modeling assumptions are made. First, the multicast process is assumed1 to deliver packets along the shortest path from a source to each of the p destinations. As most of the current Internet protocols forward packets based on the (reverse) shortest path, the assumption of SPT delivery is quite realistic. The second assumption is that the p multicast group member nodes are uniformly chosen out of the total number of nodes Q . This assumption has been discussed by Phillips et al. (1999). They concluded 1

The assumption ignores shared tree multicast forwarding such as core-based tree (CBT, see RFC2201).

387

388

The e!ciency of multicast

that, if p and Q are large, deviations from the uniformity assumption are negligibly small. Also the Internet measurements of Chalmers and Almeroth (2001) seem to confirm the validity of the uniformity assumption. 17.1 General results for jQ (p) Theorem 17.1.1 For any connected graph with Q nodes, Qp (17.1) p+1 Proof: We need at least one edge for each dierent user; therefore jQ (p)  p and the lower bound is attained in a star topology with the source at the center. We will next show that an upper bound is obtained in a line topology. It is su!cient to consider trees, because multicast only uses shortest paths without cycles. If the tree has not a line topology, then at least one node has degree 3 or the root has degree 2. Take the node closest to the root with this property and cut one of the branches at this node; we paste that branch to a node at the deepest level. Through this procedure the multicast function jQ (p) stays unaltered or increases. Continuing in this fashion until we reach a line topology demonstrates the claim. For the line topology we place the source at the origin and the other nodes at the integers 1> 2> = = = > Q  1. The links of the graph are given by (l> l + 1)> l = 0> 1> = = = > Q  2. The multicast gain jQ (p) equals H [P ], where P is the maximum of a sample of size p, without replacement, from the integers 1> 2> = = = > Q  1. Thus, ¡n¢ p  jQ (p) 

p Pr [P  n] = ¡Q31 ¢>

pn Q 1

p

from which jQ (p) = H [P ] is ¡ n ¢ ¡n31¢ Q31 ¡ n31 ¢ Q31 X X p31  jQ (p) = n p ¡Q31¢p = n ¡Q31¢ n=p Q31 X

p

¡n¢

n=p Q 31 X

p

¡n¢

pQ pQ p ¡ Q ¢= p+1 p+1 p n=p n=p p+1 PQ31 ¡ n ¢ ¡ Q ¢ where we have used that n=p p @ p+1 = 1, because it is a sum of probabilities over all possible disjoint outcomes. ¤ =p

p ¡Q31 ¢=

Figure 17.1 shows the allowable space for jQ (p).

17.1 General results for jQ (p) gN(m)

389

Nm/(m + 1)

N1

N/2

clog(N) 1

m 1

N1

Fig. 17.1. The allowable region (in white) of jQ (p). For exponentially growing graphs, H[KQ ] = f log Q , implying that the allowable region for these graphs is smaller and bounded at the left (in dotted line) by the straight line p(f log Q ).

Theorem 17.1.2 For any connected graph with Q nodes, the map p 7$ (p) jQ (p) is concave and the map p 7$ ijQ is decreasing. Q (p) Proof: Define \p to be the random variable giving the additional number of hops necessary to reach the p-th user when the first p1 users are already connected. Then we have that H [\p ] = jQ (p)  jQ (p  1) Moreover, let \p0 be the random number of additional hops necessary to reach the p-th multicast group member, when we discard all extra hops of the (p  1)-st group member. An example is illustrated in Fig. 17.2. The random variable \p0 has the same distribution as \p31 , because both the (p  1)-st and the p-th group member are chosen uniformly from the remaining Q  p  1 nodes. In general, \p0 6= \p31 > but, for each n, Pr[\p0 = n] = Pr[\p31 = n] and, hence, £ ¤ (17.2) H \p0 = H [\p31 ] Furthermore, we have by construction that \p  \p0 with probability 1, implying that £ ¤ (17.3) H [\p ]  H \p0 Indeed, attaching the p-th group member to the reduced tree takes at least as many hops as attaching that same group member to the non-reduced tree because the former is contained in the latter and the extra hops added by

390

The e!ciency of multicast

the p  1 group member can only help us. Combining (17.2) and (17.3) immediately gives £ ¤ jQ (p)  jQ (p  1) = H [\p ]  H \p0 = jQ (p  1)  jQ (p  2) (17.4) This is equivalent to the concavity of the map p 7$ jQ (p). Root A

C

B

D

3

1

4

5

2

Fig. 17.2. A multicast session with p = 5 group members where \5 = 1 (namely link C-5). To construct \50 the three dotted lines must be removed and we observe that \50 = 2 (A-C-5), which is referred to as the reduced tree. In this example, \50 = \4 = 2 because A-C-4 and A-C-5 both consist of 2 hops. In general, they are equal in distribution because the role of group member 4 and 5 are identical in the reduced tree.

In order to show that jQ (p) p

jQ (p) iQ (p)

is decreasing it su!ces to show that p 7$

is decreasing, since iQ (p) is proportional to p. Defining jQ (0) = 0, we can write jQ (p) as a telescoping sum p p X X {jQ (n)  jQ (n  1)} = {n jQ (p) = n=1

n=1

where {n = jQ (n)  jQ (n  1)> n = 1> = = = > p. Then, jQ (p) 1 X {n = p p p

n=1

is the mean of a sequence of p positive numbers {n . By (17.4) the sequence {n  {n31 is decreasing and, hence, p p31 jQ (p) 1 X jQ (p  1) 1 X {n  {n = = p p p1 p1 n=1

n=1

17.1 General results for jQ (p)

This proves that p 7$ jQ (p)@p is decreasing.

391

¤

Next, we will give a representation for jQ (p) that is valid for all graphs. Let [l be the number of joint hops that all l uniformly chosen and dierent group members have in common, then the following general theorem holds, Theorem 17.1.3 For any connected graph with Q nodes, p µ ¶ X p jQ (p) = (1)l31 H [[l ] l

(17.5)

l=1

Note that jQ (1) = iQ (1) = H [[1 ] = H [KQ ] so that the decrease in average hops or the “gain” by using multicast over unicast is precisely p µ ¶ X p (1)l31 H [[l ] jQ (p)  iQ (p) = l l=2

However, computing H [[l ] for general graphs is di!cult. Proof of Theorem 17.1.3: Let D1 > D2 > = = = > Dp be sets where Dl consists of all links that constitute the shortest path from the source to multicast group member l. Denote by |Dl | the number of elements in the set Dl . The multicast group members are chosen uniformly from the set of all nodes except for the root. Hence, H [[1 ] = H [|Dl |] >

for 1  l  Q

and H [[2 ] = H [|Dl _ Dm |] >

for 1  l ? m  Q

¡ ¢ etc.. Now, jQ (p) = H [|D1 ^ D2 ^ · · · ^ Dp |]. Since T(D) = H [|D|] @ Q2 is a probability measure on the set of all links, we obtain from ¡Q ¢ the inclusionexclusion formula (2.3) applied to T and multiplied with 2 afterwards, H [|D1 ^ D2 ^ · · · ^ Dp |] =

p X

H [|Dl |] 

l=1

X

H [|Dl _ Dm |] + · · ·

l?m

+ (1)p31 H [|D1 _ D2 _ · · · _ Dp |] µ ¶ p H [[2 ] + · · · + (1)p31 H [[p ] = pH [[1 ]  2 This proves Theorem 17.1.3.

¤

392

The e!ciency of multicast

Corollary 17.1.4 For any connected graph with Q nodes, p µ ¶ X p H [[p ] = (1)l31 jQ (l) l

(17.6)

l=1

The corollary is a direct consequence of the inversion formula for the binomial (Riordan, 1968, Chapter 2). Alternatively, in view of the GregoryNewton interpolation formula (Lanczos, 1988, Chapter 4, Section 2) for P ¡p¢ l l31 l j (0) where  j jQ (p) = " Q (0), we can write H [[l ] = (1) Q l=1 l  is the dierence operator, i (0) = i(1)  i (0). Corollary 17.1.5 For any connected graph, the multicast e!ciency jQ (p) is bounded by iQ (p)  H [KQ ] (17.7) jQ (p) where H [KQ ] is the average number of hops in unicast. Proof: We give two demonstrations. (a) From jQ (Q  1) = Q  1 (all nodes, source plus Q  1 destinations, of the graph are spanned by a tree (p) (see Theorem consisting of Q  1 links) and the monotonicity of p 7$ jiQ Q (p) 17.1.2) we obtain: jQ (Q  1) Q 1 1 jQ (p)  = = iQ (p) iQ (Q  1) (Q  1)H [KQ ] H [KQ ] (b) Alternatively, Theorem 17.1.1 indicates that jQ (p)  p, which, with the identity iQ (p) = pH [KQ ], immediately leads to (17.7). ¤ Corollary 17.1.5 means that for any connected graph, including the graph describing the Internet, the ratio of the unicast over multicast e!ciency is bounded by the expected hopcount in unicast. In order words, the maximum savings in resources an operator can gain by using multicast (over unicast) never exceeds H [KQ ], which is roughly about 15 in the current Internet. 17.2 The random graph Js (Q ) In this section, we confine to the class RGU, the random graphs of the class Js (Q ) with independent identically and exponentially distributed link weights z with mean H [z] = 1 and where Pr[z  {] = 1  h3{ , { A 0. In Section 16.2, we have shown that the corresponding SPT is, asymptotically, a URT. The analysis below is exact for the complete graph NQ while asymptotically correct for connected random graphs Js (Q ).

17.2 The random graph Js (Q )

393

17.2.1 The hopcount of the shortest path tree Based on properties of the URT, the complete probability density function of the number of links KQ (p) in the SPT to p uniformly chosen nodes can be determined. We first derive £ K (p)a¤recursion for the probability generating of the number of links KQ (p) in the function *KQ (p) (}) = H } Q SPT to p uniformly chosen nodes in the complete graph NQ . Lemma 17.2.1 For Q A 1 and all 1  p  Q  1, *KQ (p) (}) =

(Q  p  1)(Q  1 + p}) p2 } * (}) + K (p) Q 1 2 *KQ 1 (p1) (}) (Q  1)2 (Q  1) (17.8)

Proof: To prove (17.8), we use the recursive growth of URTs: a URT of size Q is a URT of size Q  1, where we add an additional link to a uniformly chosen node. 1

N

N

2 N N

Case A

Case B

Case C and D

Fig. 17.3. The several possible cases in which the Q -th node can be attached uniformly to the URT of size Q  1. The root is dark shaded while the p multicast member nodes are lightly shaded.

In order to obtain a recursion for KQ (p) we distinguish between the p uniformly chosen nodes all being in the URT of size Q  1 or not. The p probability that they all belong to the tree of size Q  1 is equal to 1  Q31 (case A in Fig. 17.3). If they all belong to the URT of size Q  1, then we have that KQ (p) = KQ31 (p). Thus, we obtain µ ¶ h i p p *KQ (p) (}) = 1  *KQ 31 (p) (}) + H } 1+OQ 31 (p) (17.9) Q 1 Q 1 where OQ31 (p) is the number of links in the subtree of the URT of size Q  1 spanned by p  1 uniform nodes and the “one” refers to the link from

394

The e!ciency of multicast

the added Q -th node to its ancestor in the URT of size Q  1. We complete the proof by investigating the generating function of OQ31 (p). Again, there are two cases. In the first case (B in Fig. 17.3), the ancestor of the added Q -th node is one of the p  1 previous nodes (which can only happen if it is unequal to the root), else we get one of the cases C and D in Fig. 17.3. The probability of the first event equals p31 Q31 , the probability of the latter equals p31 1  Q31 . If the ancestor of the added Q-th node is one of the p  1 previous nodes, then the number of links OQ31 (p) equals KQ31 (p  1), otherwise the generating function of the number of additional links equals ¶ µ 1 1 (}) *KQ 31 (p) (}) + * 1 Q p Q  p KQ 31 (p31) The first contribution comes from the case where the ancestor of the added Q -th node is not the root, and the second from where it is equal to the root, 1 1 = Q3p . Therefore, which has probability Q313(p31) i p1 h (}) * H } OQ 31 (p) = Q  1 KQ 31 (p31) µ ¶ *KQ 31 (p31) (}) Q p Q p1 *KQ 31 (p) (}) + + Q 1 Q p Q p p Q p1 = *KQ 31 (p31) (}) + *KQ 31 (p) (}) (17.10) Q 1 Q 1 Substitution of (17.10) into (17.9) leads to (17.8).

¤

Since jQ (p) = H[KQ (p)] = *0KQ (p) (1), we obtain the recursion for jQ (p), µ ¶ p2 p2 p (p) + jQ31 (p  1) + jQ (p) = 1  j Q31 (Q  1)2 (Q  1)2 Q 1 (17.11) Theorem 17.2.2 For all Q  1 and 1  p  Q  1, p µ ¶ h i p!(Q  1  p)! X (Q + n}) p KQ (p) *KQ (p) (}) = H } (1)p3n = 2 (1 + n}) n ((Q  1)!) n=0 (17.12) Consequently, (m+1) (p)

p!(1)Q3(m+1) VQ Sm Pr [KQ (p) = m] = ¢ ¡ (Q  1)! Q31 p

(17.13)

17.2 The random graph Js (Q ) (m+1)

395

(p)

where VQ and Sm denote the Stirling numbers of first and second kind (Abramowitz and Stegun, 1968, Section 24.1). Proof: By iterating the recursion (17.8) for small values of p, the computations given in van der Hofstad et al. (2006a, Appendix) suggest the solution (17.12) for (17.8). One can verify that (17.12) satisfies (17.8). This proves (17.12) of Theorem 17.2.2. Using (Abramowitz and Stegun, 1968, Section 24.1.3.B), the Taylor expansion around } = 0 equals *KQ (p) (}) =

  p 1 p!Q(Q 3 1 3 p)! [ p K(Q + n}) 3 (31)p3n (Q 3 1)! Q!K(1 + n}) Q n n=0

(m+1) Q 31 p [ (31)Q 3(m+1) VQ p!Q(Q 3 1 3 p)! [ p nm } m (31)p3n (Q 3 1)! Q! n m=1 n=0 # p $ (m+1) Q 31 [ p p!Q(Q 3 1 3 p)! [ (31)Q 3(m+1) VQ (31)p3n nm } m = n (Q 3 1)! Q! m=1 n=0

=

Using the definition of Stirling numbers of the second kind (Abramowitz and Stegun, 1968, 24.1.4.C), p   [ p (p) p!Sm (31)p3n nm = n n=0 (p)

for which Sm

= 0 if m ? p, gives *KQ (p) (}) =

Q 31 (p!)2 (Q 3 1 3 p)! [ (m+1) (p) m (31)Q 3(m+1) VQ Sm } 2 ((Q 3 1)!) m=1

This proves (17.13) and completes the proof of Theorem 17.2.2.

¤

Figure 17.4 plots the probability density function of K50 (p) for dierent values of p. Corollary 17.2.3 For all Q  1 and 1  p  Q  1, jQ (p) = H [KQ (p)] =

Q X 1 pQ Q p n

(17.14)

n=p+1

and

P 1 2 (p) p2 Q 2 Q jQ Q 1+p n=p+1 n2 jQ (p)   Var [KQ (p)] = Q +1p (Q + 1  p) (Q  p)(Q + 1  p) (17.15)

The formula (17.14) is proved in two dierent ways. The earlier proof presented in Section 17.6 below does not rely on the recursion in Lemma 17.2.1 nor on Theorem 17.2.2. The shorter proof is presented here. Formula (17.14) can be expressed in terms of the digamma function #({) as µ ¶ #(Q )  #(p) 1 (17.16) jQ (p) = pQ Q p

396

The e!ciency of multicast 0.5

Pr[H50(m) = j]

0.4

0.3

0.2

0.1

0.0 0

10

20

30

40

50

j hops

Fig. 17.4. The pdf of K50 (p) for p = 1, 2, 3, 4, 5, 10, 15, 20, 25, 30, 35, 40, 45, 47. Proof of Corollary 17.2.3: The expectation and variance of KQ (p) will not be obtained using the explicit probabilities (17.13), but by rewriting (17.12) as p k l K(p + 1)K(Q 3 p) [ p (31)p3n CwQ 31 wQ 31+n} 2 w=1 n K (Q) n=0 k l K(p + 1)K(Q 3 p) (31)p CwQ 31 wQ 31 (1 3 w} )p = w=1 K2 (Q)

*KQ (p) (}) =

(17.17)

Indeed, k l K(p + 1)K(Q 3 p) Q 31 Q 31 p } p (31) w C C (1 3 w ) } w w=}=1 K2 (Q) k l K(p + 1)K(Q 3 p) p(31)p31 CwQ 31 wQ log w(1 3 w)p31 = > 2 w=1 K (Q) k l K(p + 1)K(Q 3 p) H[KQ (p) (KQ (p) 3 1)] = (31)p C}2 CwQ 31 wQ 31 (1 3 w} )p 2 w=}=1 K (Q) K(p + 1)K(Q 3 p) p31 = p(31) K2 (Q) k l × CwQ 31 wQ log2 w(1 3 w)p32 [3(p 3 1)w + (1 3 w)] H[KQ (p)] =

w=1

We will start with the former. Using H[KQ (p)] =

Cwl (1

3

w)m |w=1

=

m!(31)m 

l>m

and Leibniz’ rule, we find

l K(p + 1)K(Q 3 p) Q 3 1 Q 3p k Q p! C w log w w=1 K2 (Q) p31 w

17.2 The random graph Js (Q )

397

Since Cwn [wq log w]w=1 =

q! (q 3 n)!

q [ m=q3n+1

1 m

we obtain expression (17.14) for H[KQ (p)]. We now extend the above computation to H[KQ (p)(KQ (p) 3 1)] that we write as H[KQ (p) (KQ (p) 3 1)] =

K(p + 1)K(Q 3 p) (U1 + U2 ) K2 (Q)

(17.18)

where k l U1 = p(p 3 1)(31)p32 CwQ 31 wQ +1 log2 w(1 3 w)p32 w=1 k l U2 = p(31)p31 CwQ 31 wQ log2 w(1 3 w)p31 w=1

Using Cwn [wq

2

log w]w=1

q! =2 (q 3 n)!

q [

q [

l=q3n+1 m=l+1

53 q [ q! 1 7C = lm (q 3 n)! l=q3n+1

6 42 q [ 1D 18 3 l l2 l=q3n+1

we obtain, k Q 3 1 l p(p 3 1)(p 3 2)!CwQ 3p+1 wQ +1 log2 w w=1 p32 6 53 42 Q +1 Q +1 Q 3 1 [ 1 [ 1 8 7C D 3 = (Q + 1)! p32 n n2 n=p+1 n=p+1

U1 =

Similarly, Q 3 1 l k p(p 3 1)!CwQ 3p wQ log2 w w=1 p31 6 42 53 Q Q Q 3 1 [ [ 1 1 8 D 3 7C = Q! n n2 p31 n=p+1 n=p+1

U2 =

Substitution into (17.18) leads to 6 53 42 Q Q [ [ 1 1 p2 Q 2 8 7C D 3 H[KQ (p)(KQ (p) 3 1)] = (Q + 1 3 p)(Q 3 p) n n2 n=p+1 n=p+1 +

2p(p 3 1)Q (Q + 1 3 p)(Q 3 p)

Q [ n=p+1

1 n

2 (p), we From jQ (p) = H[KQ (p)] and Var [KQ (p)] = H[KQ (p)(KQ (p) 3 1)] + jQ (p) 3 jQ

obtain (17.15). This completes the proof of Corollary 17.2.3.

¤

For Q = 1000, Fig. 17.5 illustrates the typical behaviorp for large Q of the expectation jQ (p) and the standard deviation Q (p) = Var [KQ (p)] for all values of p. For any spanning tree, the number of links KQ (Q  1) is precisely Q  1, so that Var[KQ (Q  1)] = 0.

398

The e!ciency of multicast 1000

12

10

8 600 6 400 4

g1000 (m) V1000 (m)

Standard deviation VN (m)

average hopcount gN (m)

800

200 2

0

200

400

600

0 1000

800

m

Fig. 17.5. The average number of hops jQ (p) (left axis) in the SPT and the corresponding standard deviation Q (p) (right axis) as a function of the number p of multicast group members in the complete graph with Q = 1000.

Figure 17.5 also indicates that the standard deviation Q (p) of KQ (p) is much smaller than the average, even for Q = 1000. In fact, we obtain from (17.15) that Var [KQ (p)] 

2 (p) 2jQ Q 1+p 2Q jQ (p) jQ (p)  = PQ Q +1p Q p p n=p+1

1 n

2 = r(jQ (p))

This bound implies that with probability converging to 1 for every p = 1> = = = > Q  1, ¯ ¯ ¯ KQ (p) ¯ ¯ ¯%  1 ¯ ¯ jQ (p) KQ (p)3jQ (p) s jQ (p) g KQ (p)3jQ (p) s $ jQ (p)

In van der Hofstad et al. (2006a), the scaled random variable

is proved to tend to a Gaussian random variable, i.e. s Q (0> 1), for all p = r( Q ). For large graphs of the size of the Internet and larger, this observation implies that the mean jQ (p) = H [KQ (p)] is a good approximation for the random variable KQ (p) itself because the variations of KQ (p) around the mean are small. Consequently, it underlines the importance of jQ (p) as a significant measure for multicast.

17.2 The random graph Js (Q )

399

17.2.2 The weight of the shortest path tree In this section, we summarize results on the weight ZQ (p) of the SPT and omit derivations, but refer to van der Hofstad et al. (2006a). For all 1  p  Q  1, the average weight of the SPT is H [ZQ (p)] =

p X m=1

Q31 1 X 1 Q m n

(17.19)

n=m

In particular, if the shortest path tree spans the whole graph, then for all Q  2, H [ZQ (Q  1)] =

Q31 X q=1

from which H [ZQ (Q  1)] ?  (2) =

2 6

1 q2

(17.20)

for any finite Q . The variance is

m Q31 Q31 Q31 X 1 X X 1 1 4 X 1  5 Var [ZQ (Q  1)] = + 4 Q n3 m3 n m4 n=1

m=1

or asymptotically, for large Q , 4 (3) +R Var [ZQ (Q  1)] = Q

µ

(17.21)

m=1

n=1

log Q Q2

¶ (17.22)

Asymptotically for large Q , the average weight of a shortest path tree is (2) = 1=645 = = =, while the average weight of the minimum spanning tree, given by (16.49), is (3) ? (2). This result has an interesting implication. The Steiner tree is the minimum weight tree that connects a set of p members out of Q nodes in the graph. The Steiner tree problem is NP-complete, which means that it is unfeasible to compute for large Q . If p = 2, the weight of the Steiner tree equals that of the shortest path, ZSteiner>Q (2) = ZQ , while for p = Q , we have ZSteiner>Q (Q ) = ZMST . Hence, for any p ? Q and Q , H[ZSteiner>Q (p)]  (3) because the weight of the Steiner tree does not decrease if the number of members p increases. The ratio (2) (3) = 1=368 indicates that the use of the SPT (computationally easy) never performs on average more than 37% worse than the optimal Steiner tree (computationally unfeasible). In a broader context and referring to the concept of the “Prize of Anarchy”, which is broadly explained in Robinson (2004), the SPT used in a communications network is related to the Nash equilibrium, while the Steiner tree gives the hardly achievable global optimum. Simulations — even for small Q, which allow us to cover the entire p-range

400

The e!ciency of multicast 10

fW* (m)(x) N

10

10

0

m m m m m m m m m m m m

-1

-2

=1 =5 = 10 = 20 = 30 = 40 = 50 = 60 = 70 = 80 = 90 = 95

N = 100 10

10

-3

Normalized Gumbel Normalized Gaussian N(0,1)

-4

-4

-2

0

2

4

6

x

Fig. 17.6. The pdf of the normalized random variable ZQ (p) for Q = 100.

as illustrated in Fig. 17.6 — indicate that the normalized random variable ZQW (p) = ZQs(p)3H[ZQ (p)] lies between a normalized Gaussian Q (0> 1) and ydu[ZQ (p)]

a normalized Gumbel (see Theorem 6.4.1). Fig. 17.6 may suggest that, for all p ? Q ,  {3 3I 6

lim Pr [ZQW (p)  {]  h3h

Qn (p) = Q  1  n 1 G



G31 X

p Q 1

nm+1 31 31 n31

n G3m

m=2

Y

¶ n

G31

n µ Y 1 t=0

µ 1

t=0

p Q 1t

t=

n31

n

p Q313t

´ 1 ? n

à 1

(17.26)

Q31 n .

Moreover,

!nm

p Q 1

¶

¶

which shows that jQ>n (p) is a polynomial in p of degree  the terms in the m-sum rapidly decrease; their ratio equals ³ 31 Q nm+1 31 n31 1 m n 31

p Q 1t

nm 31 n31

?? 1

17.3 The n-ary tree

403

Figure 17.8 indicates that formula j(17.25), althoughk derived subject to (n31)]  1 , where b{c is the (17.24), also seems valid when G = log[1+Q log n largest integer smaller than or equal to {. This suggests that the deepest level G need not be filled completely to count nG nodes and that (17.25) may extend to “incomplete” n-ary trees. As further observed from Fig. 17.8, jQ>n (p) is monotonously decreasing in n. Hence, it is quite likely that he map n 7$ jQ>n (p) is decreasing in n 5 [1> Q  1]. Intuitively, this conjecture can be understood from Fig. 17.7. Both the n = 2 and n = 5 trees have an equal number of nodes. We observe that the deeper G (or the smaller n), the more overlap is possible, hence, the larger jQ>n (p). Theorem 17.1.1 can also be deduced from (17.25). The lower bound is attained in a star topology where n = Q  1, G = 1 and H[KQ ] = 1. The upper bound is attained in a line topology where n = 1, G = Q  1 and H[KQ ] = Q2 . Furthermore, for real values of n 5 [1> Q  1], the set of curves specified by (17.25) covers the total allowable space of jQ>n (p), as shown in Fig. 17.1. This suggests to consider (17.25) for estimating n in real topologies. Since jQ (1) = H[KQ ], the average hopcount in a n-ary tree follows from (17.25) as H[KQ ] = Q  1 

G31 X

G31 Q  1  n n3131 1 X G3m nm+1  1 = n Q 1 Q 1 n1 m+1

n

m=0

G3m

m=0

QG G 1 = +  Q  1 (Q  1)(n  1) n  1

(17.27)

For large Q , we find with ¹

º log[1 + Q (n  1)] G=  1  logn Q + logn (1  1@n) + R(1@Q ) log n that 1 +R H[KQ ] = logn Q + logn (1  1@n)  n1

µ

logn Q Q

¶ (17.28)

Comparing (17.28) with the average hopcount in the random graph (16.10) shows equality to first order if nrg = h. Moreover, both the second order 1 terms   1 = 0=42 and log(1  1@h) h31 = 1=04 are R(1) and independent of Q . This shows that the multicast gain in the random graph is well approximated by jQ>h (p).

404

The e!ciency of multicast

17.4 The Chuang—Sirbu law We discuss the empirical Chuang—Sirbu scaling law, which states that jQ (p)  H [KQ ] p0=8 for the Internet. Based on Internet measurements, Chuang and Sirbu (1998) observed that jQ (p)  H [KQ ] p0=8 . Subsequently, Phillips et al. (1999) dubbed this observation the Chuang—Sirbu law. Corollary 17.1.5 implies that the empirical law of Chuang—Sirbu cannot hold true for all p  Q . Indeed, if jQ (p) = H [KQ ] p0=8 > we obtain from the inequality (17.7) and the identity iQ (p) = pH [KQ ], that p0=2  H [KQ ]. Write p = {Q for a fixed 0 ? { ? 1 and { independent of Q . Hence, we have shown that Q] $ 0> Corollary 17.4.1 For all graphs satisfying the condition that H[K Q 0=2 for large Q , the empirical Chuang—Sirbu law does not hold in the region p = {Q with 0 ? {  1 and su!ciently large Q .

The most realistic graph models for the Internet assume that H [KQ ]  f log Q , since this implies that the number of routers that can be reached from any starting destination grows exponentially with the number of hops. For these realistic graphs, Corollary 17.4.1 states that empirical Chuang— Sirbu law does not hold for all p. On the other hand, there are more regular graphs (such as a g-lattice, where H[KQ ] ' g3 Q 1@g ) with H [KQ ]  Q 0=2+ (and  A 0) for which the mathematical condition p0=2  H [KQ ] is satisfied for all p and Q . As shown in Van Mieghem et al. (2000), however, these classes of graphs, in contrast to random graphs, are not leading to good models for SPTs in the Internet.

17.4.1 Validity range of the Chuang—Sirbu law For the random graph Js (Q ), the SPT is very close to a URT for large Q and with (16.10), we obtain iQ (p)  p(log Q +   1) From the exact jQ (p) formula (17.16) for the random graph Js (Q ), the asymptotic for large Q and p follows as µ ¶ pQ Q 1 jQ (p)  log  (17.29) Q p p 2 The above scaling explains the empirical Chuang—Sirbu law for Js (Q ): ¡for¢p pQ log Q small with respect to Q, the graphs of (log Q +1)p0=8 and Q3p p  1 2 look very alike in a log-log plot, as illustrated in Fig. 17.9.

17.4 The Chuang—Sirbu law

405

Using the asymptotic properties of the digamma function #, we obtain (17.29) as an excellent approximation for large Q (and all p) or, in normalized form with p = {Q and 0 ? { ? 1, jQ ({Q ) + 0=5 { log {  Q {1

(17.30)

Q ] 0=8 = H[K The normalized Chuang—Sirbu law is jQ ({Q) Q Q 0=2 { . It is interesting Q] = 1, since then both to note that the Chuang—Sirbu law is “best” if H[K Q 0=2 endpoints { = 0 and { = 1 coincide with (17.30). This optimum is achieved when Q  250 000, which is of the order of magnitude of the estimated number of routers in the current Internet. This observation may explain the fairly good correspondence on a less sensitive log-log scale with Internet measurements. At the same time, it shows that for a growing Internet, the fit of the Chuang—Sirbu law will deteriorate. For Q  106 , the Chuang—Sirbu law underestimates jQ (p) for all p. 7

10

0.8

m  law random graph 6

10

5

10

gN (m)

1.00 4

10

0.95 Effective Power Exponent

3

10

2

10

0.90

0.85

0.80 Number of Nodes N 0.75

1

1

10

10 0

10

1

10

2

10

2

3

10

10 3

4

5

10

10 4

10

10

6

7

10

10 5

10

8

9

10

10 6

10

10

10

7

10

m

Fig. 17.9. The multicast e!ciency for Q = 10m with m = 3> 4> ===> 7. The endpoint of each curve jQ (Q  1) = Q  1 determines Q . The insert shows the eective power exponent versus Q .

17.4.2 The eective power exponent (Q ) For small to moderate values of p, jQ (p) is very close to a straight line in a log-log plot. This “power law behavior” implies that log jQ (p) 

406

The e!ciency of multicast

log H(KQ )+(Q ) log p, which is a first order Taylor expansion of log jQ (p) in log p. This observation suggests the computation4 of the eective power exponent (Q ) as ¯ g log jQ (p) ¯¯ (17.31) (Q ) = g log p ¯ p=1

Only for a straight line, the dierential operator can be replaced by the dierence operator such that (Q)   W (Q )> where W

 (Q) =

jQ (2) log H[K Q]

log 2

(17.32)

In general, for small p, the eective power exponent (17.31) is not a constant 0.8 as in the Chuang—Sirbu law, but dependent on Q . Since jQ (p) is concave jQ (p) by Theorem 17.1.2, (Q ) is the maximum possible value for g log at any g log p p  1. A direct consequence of Theorem 17.1.1 is that the eective power exponent (Q ) 5 [ 12 > 1]. From recent Internet measurements, Chalmers and Almeroth (2001) found that 0=66  (Q)  0=7. The eective power exponent (Q) as defined in (17.31) for the random graph is ³ ´ 2 2 Q #(Q ) +   6 + 6Q ¢ ¡ (Q ) = (Q  1) #(Q ) + (  1) + Q1 while, according to the definition (17.32), W

 (Q ) =

jQ (2) log H[K Q]

log 2

 = 1 + log2

(Q  1) (#(Q ) +   3@2 + 1@Q ) (Q  2) (#(Q ) +   1 + 1@Q )

¸

The dierence (Q ) W (Q ) monotonously decreases and is largest, 0.048 at Q = 3 while 0.0083 at Q = 105 and 0.0037 at Q = 1010 . This eective power exponent (Q ) is drawn in the insert of Fig. 17.9, which shows that (Q ) is increasing and not a constant close to 0.8. More interestingly, for Q] large Q , we find with (16.10) and (16.11) that (Q )  Var[K H[KQ ] and that Q] limQ Q 3 p; Dm 3 p + 1; }) = }n K(Dm 3 p + 1) K(D m 3 p + 1 + n) n=0

Hence, Vm =

1 K(Q 3 p) gp [(1 3 })p I (1> Q 3 p; Dm 3 p + 1; })]|}=0 p! K(Dm 3 p + 1) g} p

Invoking the dierentiation formula (Abramowitz and Stegun, 1968, Section 15.2.7),  (31)p K(d + p)K(f 3 e + p)K(f) gp  (1 3 })d+p31 I (d> e; f; }) = (13})d31 I (d+p> e; f+p; }) g} p K(d)K(f 3 e)K(f + p) we have, since d = 1 and I (d> e; f; 0) = 1, Vm =

(31)p K(Q 3 p)K(Dm + 1 3 Q + p) K(Dm + 1 3 Q)K(Dm + 1)

Thus, jQ>n (p) =

=

  G31 [ Dm ! (31)p (Q 3 p 3 1)!(Dm 3 Q + p)! pnG (Q 3 1)! nG3m 3Q 3 Q 31 Q! (Dm 3 Q)!Dm ! Dm ! m=1 G31 G31 [ (31)p31 (Q 3 p 3 1)! [ G3m (Dm 3 Q + p)! pnG + nG3m + n Q 31 (Q 3 1)! (Dm 3 Q)! m=1 m=1

from which (17.25) is immediate.

¤

17.8 Problem (i) Compute the eective power exponent  W (Q) for the n-ary tree.

18 The hopcount to an anycast group

In this chapter, the probability density function of the number of hops to the most nearby member of the anycast group consisting of p members (e.g. servers) is analyzed. The results are applied to compute a performance measure  of the e!ciency of anycast over unicast and to the server placement problem. The server placement problem asks for the number of (replicated) servers p needed such that any user in the network is not more than m hops away from a server of the anycast group with a certain prescribed probability. As in Chapter 17 on multicast, two types of shortest path trees are investigated: the regular n-ary tree and the irregular uniform recursive tree treated in Chapter 16. Since these two extreme cases of trees indicate that the performance measure   1  d log p where the real number d depends on the details of the tree, it is believed that for trees in real networks (as the Internet) a same logarithmic law applies. An order calculus on exponentially growing trees further supplies evidence for the conjecture that   1  d log p for small p.

18.1 Introduction IPv6 possesses a new address type, anycast, that is not supported in IPv4. The anycast address is syntactically identical to a unicast address. However, when a set of interfaces is specified by the same unicast address, that unicast address is called an anycast address. The advantage of anycast is that a group of interfaces at dierent locations is treated as one single address. For example, the information on servers is often duplicated over several secondary servers at dierent locations for reasons of robustness and accessibility. Changes are only performed on the primary servers, which are then copied onto all secondary servers to maintain consistency. If both the primary and all secondary servers have a same anycast address, a query 417

418

The hopcount to an anycast group

from some source towards that anycast address is routed towards the closest server of the group. Hence, instead of routing the packet to the root server (primary server) anycast is more e!cient. Suppose there are p (primary plus all secondary) servers and that these p servers are uniformly distributed over the Internet. The number of hops from the querying device D to the closest server is the minimum number of hops, denoted by kQ (p), of the set of shortest paths from D to these p servers in a network with Q nodes. In order to solve the problem, the shortest path tree rooted at node D, the querying device, needs to be investigated. We assume in the sequel that one of the p uniformly distributed servers can possibly coincide with the same router to which the querying machine D is attached. In that case, kQ (p) = 0. This assumption is also reflected in the notation, small k, according to the convention made in Section 16.3.2 that capital K for the hopcount excludes the event that the hopcount can be zero. Clearly, if p = 1, the problem reduces to the hopcount of the shortest path from D to one uniformly chosen node in the network and we have that kQ (1) = kQ > where kQ is the hopcount of the shortest path in a graph with Q nodes. The other extreme for p = Q leads to kQ (Q ) = 0 because all nodes in the network are servers. In between these extremes, it holds that kQ (p)  kQ (p  1) since one additional anycast group member (server) can never increase the minimum number of hops from an arbitrary node to that larger group. The hopcount to an anycast group is a stochastic problem. Even if the network graph is exactly known, an arbitrary node D views the network along a tree. Most often it is a shortest path tree. Although the sequel emphasizes “shortest path trees”, the presented theory is equally valid for any type of tree. The node D’s perception of the network is very likely dierent from the view of another node D0 . Nevertheless, shortest path trees in the same graph possess to some extent related structural properties that allow us to treat the problem by considering certain types or classes of shortest path trees. Hence, instead of varying the arbitrary node D over all possible nodes in the graph and computing the shortest path tree at each dierent node, we vary the structure of the shortest path tree rooted

18.2 General analysis

419

at D over all possible shortest path trees of a certain type. Of course, the confinement of the analysis then lies in the type of tree that is investigated. We will only consider the regular n-ary tree and the irregular URT . It seems reasonable to assume that “real” shortest path trees in the Internet possess a structure somewhere in between these extremes and that scaling laws observed in both the two extreme cases may also apply to the Internet. The presented analysis allows us to address at least two dierent issues. First, for a same class of trees, the e!ciency of anycast over unicast defined in terms of a performance measure , =

H [kQ (p)] 1 H [kQ (1)]

is quantified. The performance measure  indicates how much hops (or link traversals or bandwidth consumption) can be saved, on average, by anycast. Alternatively,  also reflects the gain in end-to-end delay or how much faster than unicast, anycast finds the desired information. Second, the so-called server placement problem can be treated. More precisely, the question “How many servers p are needed to guarantee that any user request can access the information within m hops with probability Pr [kQ (p) A m]  , where  is certain level of stringency,” can be answered. The server placement problem is expected to gain increased interest especially for real-time services where end-to-end QoS (e.g. delay) requirements are desirable. In the most general setting of this server placement problem, all nodes are assumed to be equally important in the sense that users’ requests are generated equally likely at any router in the network with Q nodes. As mentioned in Chapter 17, the validity of this assumption has been justified by Phillips et al. (1999). In the case of uniform user requests, the best strategy is to place servers also uniformly over the network. Computations of Pr [kQ (p) A m] ?  for given stringency  and hop m, allow the determination of the minimum number p of servers. The solution of this server placement problem may be regarded as an instance of the general quality of service (QoS) portfolio of an network operator. When the number of servers for a major application oered by the service provider are properly computed, the service provider may announce levels  of QoS (e.g. via Pr [kQ (p) A m] ? ) and accordingly price the use of the application. 18.2 General analysis Let us consider a particular n o shortest path tree W rooted at node D with (n) as defined in Section 16.2.2. Suppose the level set OQ = [Q 1$n$Q31

420

The hopcount to an anycast group

that the result of uniformly distributing p anycast group members over the graph leads to a number p(n) of those anycast group member nodes that (n) are n hops away n from o the root. These p distinct nodes all belong to the (n) (n) n-th level set [Q . Similarly as for [Q , some relations are immediate. First, p(0) = 0 means that none of the p anycast group members coincides with the root node D or p(0) = 1 means that one of them (and at most one) is attached to the same router D as the querying device. Also, for all n A 0, (n) it holds that 0  p(n)  [Q and that Q31 X

p(n) = p

(18.1)

n=0

Given the tree W specified the level set OQªand the anycast group members © (0) by (1) specified by the set p > p > = = = > p(Q31) , we will derive the lowest nonempty level p(m) , which is equivalent to kQ (p). Let us denote by hm the event that all first m + 1 levels are not occupied by an anycast group member, n o n o n o hm = p(0) = 0 _ p(1) = 0 _ · · · _ p(m) = 0 The probability distribution of the minimum hopcount, [kQ (p) © Pr ª = m|OQ ], (m) is then equal to the probability of the event hm31 _ p A 0 . Since the ª © ªf © event p(m) A 0 = p(m) = 0 , using the conditional probability yields hn o¯ i ¯ Pr [kQ (p) = m|OQ ] = Pr p(m) A 0 ¯ hm31 Pr [hm31 ] i´ o¯ ³ hn ¯ = 1  Pr p(m) = 0 ¯ hm31 Pr [hm31 ]

(18.2)

© ª Since hm = hm31 _ p(m) = 0 , the probability of the event hm can be decomposed as hn o¯ i ¯ Pr [hm ] = Pr p(m) = 0 ¯ hm31 Pr [hm31 ] (18.3) The assumption that all£p group members are uniformly distributed ¤ © anycast ª¯ enables to compute Pr p(m) = 0 ¯ hm31 exactly. Indeed, by the uniform assumption, the probability equals the ratio of the favorable possibilities over the total possible. The total number of ways to distribute p items over P (n) Q  m31 — the latter constraints follows from the condition n=0 [Q positions ¡Q3Sm31 [ (n) ¢ n=0 Q hm31 — equals . Likewise, the favorable number of ways to p

18.2 General analysis

421

distribute p items over the remaining levels higher than m, leads to S o¯ hn i ¡Q 3 mn=0 [Q(n) ¢ ¯ (m) p (18.4) Pr p = 0 ¯ hm31 = ¡ Sm31 (n) Q3 [ ¢ Q

n=0

p

The recursion (18.3) needs an initialization, given by h i p Pr [h0 ] = Pr p(0) = 0 = 1  Q Q 31 ¤ ¤ ª £ (0) £© ( ) which follows from Pr p = 0 = p and equals Pr p(0) = 0 |h31 Q (p) £ ¤ p (although the event h31 is meaningless). Observe that Pr p(0) = 1 = Q holds for any tree such that p Pr [kQ (p) = 0] = Q By iteration of (18.3), we obtain Pr [hm ] =

m Y v=0

¡Q3Sv

(n)

n=0

[Q

n=0

Q

¢

¡Q 3Sm

p ¡Q3Sv31 [ (n) ¢ = p

(n)

n=0

p ¡Q ¢

[Q

¢ (18.5)

p

P where the convention in summation is that en=d in = 0 if d A e. Finally, combining (18.2) with (18.4) and (18.5), we arrive at the general conditional expression for the minimum hopcount to the anycast group, ¡Q 3Sm31 [ (n) ¢ ¡Q3Sm [ (n) ¢ n=0 Q n=0 Q  p p (18.6) Pr [kQ (p) = m|OQ ] = ¡Q ¢ p

Clearly, while Pr [kQ (0) = m|OQ ] = 0 since there is no path, we have for p = 1, (m)

[ Pr [kQ (1) = m|OQ ] = Q Q It directly follows from (18.6) that

¡Q 3Sq

n=0

Pr [kQ (p)  q|OQ ] = 1 

p ¡Q ¢ p

(n)

[Q

¢ (18.7)

P 31 P (n) (n) If Q  qn=0 [Q ? p or, equivalently, Q n=q+1 [Q ? p, then equation (18.7) shows that Pr [kQ (p) A q|OQ ] = 0. The maximum possible hopcount of a shortest path to an anycast group strongly depends on the specifics of the shortest path tree or the level set OQ . A general result is worth mentioning:

422

The hopcount to an anycast group

Theorem 18.2.1 For any graph, it holds that Pr[kQ (p) A Q  p] = 0 In words, the longest shortest path to an anycast group with p members can never possess more than Q  p hops. Proof: This general theorem follows from the fact that the line topology is the tree with the longest hopcount Q  1 and only in the case that all p last positions (with respect to the source or root) are occupied by the p anycast group members, is the maximum hopcount Q  p. ¤ For the URT , Pr[kQ (p) = Q  p] is computed exactly in (18.12). Corollary 18.2.2 For any graph, it holds that 1 Q Proof: This corollary follows from Theorem 18.2.1 and the law of total probability. Alternatively, if there are Q  1 anycast members in a network with Q nodes, the shortest path can only consist of one hop if none of the anycast members coincides with the root node. This probability is precisely 1 ¤ Q. Pr[kQ (Q  1) = 1] =

Using the tail probability formula (2.36) for the average, it follows from (18.7) that P Q32 µ (n) ¶ 1 X Q  qn=0 [Q (18.8) H [kQ (p)|OQ ] = ¡Q ¢ p p q=0 from which we find, Q31 1 X (n) n[Q H [kQ (1)|OQ ] = Q n=1

Thus, given OQ , a performance measure  for anycast over unicast can be quantified as H [kQ (p)|OQ ] 1 = H [kQ (1)|OQ ] Using the law of total probability, the distribution of the minimum hopcount to the anycast group is X Pr [kQ (p) = m|OQ ] Pr [OQ ] (18.9) Pr [kQ (p) = m] = all OQ

18.3 The n-ary tree

423

or explicitly, Pr[kQ (p) = m] =

X

SQ31

¡SQ 31 {n ¢ ¡SQ 31 {n ¢ h i n=m  n=m+1 (1) (Q31) p = {Q31 Pr [Q = {1 >= = = >[Q ¡Q ¢ p

n=1 {n =Q31

p

where the integers {n  0 for all n. This expression explicitly shows the importance of the level structure OQ of the shortest path tree W . The level set OQ entirely determines the shape of the tree W . Unfortunately, a general form for Pr [OQ ] or Pr [kQ (p) = m] is di!cult to obtain. In principle, via extensive trace-route measurements from several roots, the shortest path tree and Pr [OQ ] can be constructed such that a (rough) estimate of the level set OQ in the Internet can be obtained. 18.3 The n-ary tree For regular trees, explicit expressions are possible because the summation in (18.9) simplifies considerably. For example, for the n-ary tree defined in Section 17.3, (m)

[Q = n m (m)

Provided the set OQ only contains these values of [Q for each m, we have that Pr [OQ ] = 1, else it is zero (because then OQ is not consistent with a G+1 n-ary tree). Summarizing, for the n-ary tree with Q = n n3131 and G levels, the distribution of the minimum hopcount to the anycast group is ¡Q3 nm 31 ¢ ¡Q3 nm+1 31 ¢ n31 n31  p (18.10) Pr [kQ (p) = m] = ¡Q ¢ p p

Extension of the integer n to real numbers in the formula (18.10) is expected to be of value as suggested in Section 17.3. When a n-ary tree was used to fit corresponding Internet multicast measurements (Van Mieghem et al., 2001a), a remarkably accurate agreement was found for the value n  3=2, which is about the average degree of the Internet graph. Hence, if we were to use the n-ary tree as model for the hopcount to an anycast group, we expect that n  3=2 is the best value for Internet shortest path trees. However, we feel we ought to mention that the hopcount distribution of the shortest path between two arbitrary nodes is definitely not a n-ary tree, because Pr [kQ (1) = m] increases with the hopcount m, which is in conflict with Internet trace-route measurements (see, for example, the bell-shape curve in Fig. 16.4). Figure 18.1 displays Pr [k(p)  m] for a n-ary with outdegree n = 3 and

424

The hopcount to an anycast group 1.0 N = 500 k=3

0.8

Pr[hN (m) d j]

m = 50 m = 10

0.6

m=5

0.4

m=2 m=1

0.2

0

1

2

3

4

5

j

Fig. 18.1. The distribution function of k500 (p) versus the hops m for various sizes of the anycast group in a n-ary tree with n = 3 and Q = 500

Q = 500. This type of plot allows us to solve the “server placement problem”. For example, assuming that the n-ary tree is a good model and the network consists of Q = 500 nodes, Fig. 18.1 shows that at least p = 10 servers are needed to assure that any user is not more than four hops separated from a server with a probability of 93%. More precisely, the equation Pr[k500 (p) A 4] ? 0=07 is obeyed if p  10. Figure 18.2 gives an idea how the performance measure  decreases with the size of the anycast group in n-ary trees (all with outdegree n = 3), but with dierent size Q . For values of p up to around 20% of Q , we observe that  decreases logarithmically in p.

18.4 The uniform recursive tree (URT) Chapter 16 motivates the interest in the URT. The URT is believed to provide a reasonable, first order estimate for the hopcount problem to an anycast group in the Internet.

18.4.1 Recursion for Pr [k(p) = m] Usually, a combinatorial approach such as (18.9) is seldom successful for URTs while structural properties often lead to results. The basic Theo-

18.4 The uniform recursive tree (URT)

425

1.0

k=3

0.8

0.6

K

N = 100 N = 500

0.4 N = 5000 5

N = 10

0.2 6

N = 10

0.0 2

3

4

5

6

7 8 9

2

3

4

5

6

7 8 9

0.1

1

m/N

Fig. 18.2. The performance measure  for several sizes of n-ary trees (with n = 3) as a function of the ratio of anycast nodes over the total number of nodes.

rem 16.2.1 of the URT, applied to the anycast minimum hop problem, is illustrated in Fig. 18.3. Root

i anycast members

m  i anycast members

R1

N  k nodes

T2

k nodes T1

Fig. 18.3. A uniform recursive tree consisting of two subtrees W1 and W2 with n and Q  n nodes respectively. The first cluster contains l anycast members while the cluster with Q  n nodes contains p  l anycast members.

Figure 18.3 shows that any URT can be separated into two subtrees W1 and W2 with n and Q  n nodes respectively. Moreover, Theorem 16.2.1 states

426

The hopcount to an anycast group

that each subtree is independent of the other and again a URT. Consider now a specific separation of a URT W into W1 = w1 and W2 = w2 , where the tree w1 contains n nodes and l of the p anycast members and w2 possesses Q  n nodes and the remaining p  l anycast members. The event {kW (p) = m} equals the union of all possible sizes Q1 = n and subgroups p1 = l of the event {kw1 (l) = m  1} _ {kw2 (p  l)  m} and the event {kw1 (l) A m  1} _ {kw2 (p  l) = m}, {kW (p) = m} = ^n ^l {{kw1 (l) = m  1} _ {kw2 (p  l)  m}} ^ {{kw1 (l) A m  1} _ {kw2 (p  l) = m}} Because kQ (0) is meaningless, the relation must be modified for the case l = 0 to {kW (p) = m} = {kw2 (p) = m} and for the case l = p to {kW (p) = m} = {kw1 (p) = m  1} This decomposition holds for any URT W1 and W2 , not only for the specific ones w1 and w2 . The transition towards probabilities becomes X (Pr [kw1 (l) = m  1] Pr [kw2 (p  l)  m] Pr [kW (p) = m] = all w1 >w2 >n>l

+ Pr [kw1 (l)  m  1] Pr [kw2 (p  l) = m]) × Pr [W1 = w1 > W2 = w2 > Q1 = n> p1 = l] Since W1 and W2 and also p1 are independent given Q1 , the last probability o simplifies to o = Pr [W1 = w1 > W2 = w2 > Q1 = n> p1 = l] = Pr [W1 = w1 |Q1 = n] Pr [W2 = w2 |Q1 = n] Pr [p1 = l|Q1 = n] Pr [Q1 = n] Theorem 16.2.1 states that Q1 is uniformly distributed over the set with 1 . The fact that l out of the p Q  1 nodes such that Pr [Q1 = n] = Q31 anycast members, uniformly chosen out of Q nodes, belong to the recursive subtree W1 implies that p  l remaining anycast members belong to W2 . Hence, analogous to a combinatorial problem outlined by Feller (1970, p. 43) that leads to the hypergeometric distribution, we have ¡n¢¡Q 3n¢ Pr [p1 = l|Q1 = n] =

l

¡Qp3l ¢ p

18.4 The uniform recursive tree (URT)

427

¡n¢

because all favorable combinations are those l to distribute l anycast mem¡ 3n¢ bers in W1 with n nodes multiplied by all favorable Q p3l to distribute the remaining p  l in W2 containing Q ¡ n¢nodes. The total way to distribute p anycast members over Q nodes is Q p . Finally, we remark that the hopcount of the shortest path to p anycast members in a URT only depends on its size. This means that the sum over all w1 of Pr [W1 = w1 |Q1 = n], which equals 1, disappears and likewise also the sum over all w2 . Combining the above leads to Pr [kQ (p) = m] =

Q31 X p31 X

(Pr [kn (l) = m  1] Pr [kQ3n (p  l)  m]

n=1 l=1

¡n¢¡Q3n¢

+ Pr [kn (l) A m  1] Pr [kQ3n (p  l) = m]) +

¡ ¢ Q31 X Q3n Pr [kQ 3n (p) p

= m] +

¡n¢

p ¢ ¡Q (Q  1) p

n=1

l p3l ¡ ¢ (Q  1) Q p

Pr [kn (p) = m  1]

By substitution of n 0 = Q  n and p0 = p  l, we obtain the recursion, ¡ ¢¡Q3n¢

Pr [kQ (p) = m] =

Q31 X p31 X nl n=1 l=1 Q3n31 X

×

p3l

(Pr [kn (l) = m  1] + Pr [kn (l) = m]) ¡ ¢ (Q  1) Q p

Pr [kQ3n (p  l) = t]

t=m

+

Q31 X n=1

¡n¢ p

(Pr [kn (p) = m] + Pr [kn (p) = m  1]) ¡ ¢ (Q  1) Q p (18.11)

This recursion (18.11) is solved numerically for Q = 20. The result is shown in Fig. 18.4, which demonstrates that Pr [k(p) A Q  p] = 0 or that the path with the longest hopcount to an anycast group of p members consists of Q  p links. Since there are (Q  1)! possible recursive trees (Theorem 16.2.2) and there is only one line tree with Q  1 hops where each node has precisely one child node, the probability to have precisely Q  1 hops from the root is 1 (Q31)! (which also is Pr [kQ = Q  1] given in (16.8)). The longest possible hopcount from a root to p anycast members occurs in the line tree where all p anycast members occupy the last p positions. Hence, the probability

428

The hopcount to an anycast group

for the longest possible hopcount equals Pr [kQ (p) = Q  p] =

p! ¡ ¢ (Q  1)! Q p

(18.12)

because there are p! possible ways to distribute the p¡ anycast members Q¢ at the p last positions in the line tree while there are p possibilities to distribute p anycast members at arbitrary places in the line tree. -1

10

N = 20

-3

10

-5

10

Pr[hN (m) = j]

-7

10

-9

10

-11

10

-13

10

-15

10

-17

10

-19

10

0

2

4

6

8

10

12

14

16

18

20

j

Fig. 18.4. The pdf of kQ (p) in a URT with Q = 20 nodes for all possible p. Observe that Pr[kQ (p) A Q  p] = 0= This relation connects the various curves to the value for p.

Figure 18.4 allows us to solve the “server placement problem”. For example, consider the scenario in which a network operator announces that any user request will reach a server of the anycast group in no more than m = 4 hops in 99.9% of the cases. Assuming his network has Q = 20 routers and the shortest path tree is a URT, the network operator has to compute the number of anycast servers p he has to place uniformly spread over the Q = 20 routers by solving Pr [k20 (p) A 4] ? 1033 . Figure 18.4 shows that the intersection of the line m = 4 and the line Pr [k20 (p) = 4] = 1033 is the curve for p = 7. Since the curves for p  7 are exponentially decreasing, Pr [k20 (p) A 4] is safely1 approximated by Pr [k20 (p) = 4], which leads to the placing of p = 7 servers. When following the line m = 4, we also observe that the curves for p = 5> 6> 7> 8 lie near to that of p = 7. This means that 1

More precisely, since Pr [k20 (4) A 4] = 0.001 06 and Pr [k20 (5) A 4] = 0.000 32, only p = 5 servers are su!cient.

18.4 The uniform recursive tree (URT)

429

placing a server more does not considerably change the situation. It is a manifestation of the law   1  d log p, which tells us that by placing p servers the gain measured in hops with respect to the single server case is slowly, more precisely logarithmically, increasing. The performance measure  for the URT is drawn for several sizes Q in Fig. 18.5.

18.4.2 Analysis of the recursion relation The product of two probabilities in the double sum in (18.11) seriously complicates a possible analytic treatment. A relation for a generating function of Pr [kQ (p) = m] and other mathematical results are derived in Van Mieghem (2004b). Here, we summarize the main results. p . Using Pr [kn (l)  1] = 1, the (a) Let us check Pr [kQ (p) = 0] = Q p3l convention that Pr [kn (l) = 1] = 0 and Pr [kQ3n (p  l) = 0] = Q 3n , the right hand side of (18.11), denoted by u, simplifies to µ ¶µ ¶ Q31 p XX pl n Q n 1 u= ¡ ¢ Q n l pl (Q  1) Q p n=1 l=0 Q31 X p31 X µn ¶µQ  1  n¶ 1 = ¡ ¢ l p1l (Q  1) Q p n=1 l=0 Q31 X µQ  1¶ p 1 = = ¡ ¢ Q p1 (Q  1) Q p n=1 (b) Observe that Pr [kQ (Q ) = m] = 0 for m A 0. (c) For p = 1,

Pr [kQ

Q 31 1 X n = m] = (Pr[kn = m] + Pr [kn = m  1]) Q 1 Q n=1

Multiplying both sides by } m , summing over all m leads to the recursion for the generating function (16.6) (Q + 1)*Q+1 (}) = (} + Q )*Q (})

430

The hopcount to an anycast group

(d) The case p = 2 is solved in Van Mieghem (2004b, Appendix) as µ ¶ m 2(1)Q313m (m+1) 2(1)Q3m X (n+m+1) n 2n1 Pr [kQ (2) = m] = + (1) VQ VQ Q! Q !(Q  1) n n=1 ¶ µ ¶¸ m31 µ 2n + 1 2(1)Q3m X m + n + 1 n+m+1  (1)n VQ + Q !(Q  1) m n n=0

(18.13) In van der Hofstad et al. (2002b) we have demonstrated that the covariance between the number of nodes at level u and m for u  m in the URT is µ ¶ u h i (1)Q31 X (u) (m) (n+m+1) n+m 2n + m  u (1) H [Q [Q = VQ (Q  1)! n n=0

For m  u = 1, the last term in (18.13) is recognized as ¡2n31¢ 1 ¡2n¢ = 2 n , the first sum in (18.13) is n µ ¶ m 2(1)Q3m X (n+m+1) n 2n1 (1) VQ Q !(Q  1) n n=1

With

2(31)Q 313m (m+1) VQ Q!

l k (m31) (m) H [Q [Q

(Q2 )

. Since

³ ´ ¸ (m) 2 H [Q (m+1) 2(1)Q3m31 VQ =  ¡ ¢ (Q  1) Q! 2 Q2

= 2 Pr [kQ = m], we obtain

³ ´ ¸ i h (m) 2 (m31) (m) H [Q H [Q [Q 2Q Pr [kQ = m] + Pr [kQ (2) = m] =  ¡Q ¢ ¡ ¢ Q 1 2 Q2 2 ¶ m µ 2(1)Q31 X m + n n+m + (1)n+m VQ Q !(Q  1) n n=1

It would be of interest to find an interpretation for the last sum. Without proof2 , we mention the following exact results: Q µ ¶ X Q p=1

p

Pr [kQ (p) = Q  2] =

Q1 Xµ p=1

¶ Q  1 Pr [kQ 1 (p) = Q  3] 1 + Q 1 (Q  2)! p

For p  Q  3, it holds that p! Pr [kQ (p) = Q  p  1] = ¡ ¢ (Q  1)! Q p 2

"µ ¶ # p X Q p+1 + (p  1)(p@2 + 1) + 2 n

By substitution into the recursion (18.11), one may verify these relations.

n=2

18.5 Approximate analysis

431

18.5 Approximate analysis Since the general solution (18.9) is in many cases di!cult to compute as shown for the URT in Section 18.4, we consider a simplified version of the p above problem where each node in the tree has equal probability s = Q to be a server. Instead of having precisely p servers, the simplified version considers on average and the probability that there are precisely ¡Q ¢ p p servers Q3p . In the simplified version, the associated p servers is p s (1  s) equations to (18.4) and (18.3) are o¯ oi i hn hn (m) ¯ Pr p(m) = 0 ¯ hm31 = Pr p(m) = 0 = (1  s)[Q Pr [hm ] =

m Y

Pr

hn oi Sm31 (o) p(m) = 0 = (1  s) o=0 [Q

o=0

which implies that the probability that there are no servers in the tree is (1  s)Q . Since in that case, the hopcount is meaningless, we consider the conditional probability (18.2) of the hopcount given that the level set contains at least one server (which is denoted by e kQ (p)) is ³ ´ Sm31 (o) (m) i h 1  (1  s)[Q (1  s) o=0 [Q kQ (p) = m|OQ = Pr e 1  (1  s)Q Thus, h i 1  (1  s)Sqo=0 [Q(o) e Pr kQ (p)  q|OQ = 1  (1  s)Q

h i (o) Finally, to avoid the knowledge of the entire level set OQ , we use H [Q = (o)

Q Pr [kQ (1) = o] from (16.7) as the best estimate for each [Q and obtain the approximate formula µ l¶ k Sm31 k (o) l (m) H [Q H [Q (1  s) o=0 1  (1  s) h i kQ (p) = m = Pr e (18.14) 1  (1  s)Q In the dotted lines in Fig. 18.5, we have added the approximate result for the URT where H [kQ (p)] is computed based on (18.14), but where H[kQ (1)] is computed exactly. For p = 1, the approximate analysis (18.14) is not well i Fig. 18.5 illustrates this deviation in the fact that appr (1) = h suited: kQ (1) @H [kQ (1)] ? 1. For higher values of p we observe a fairly good H e correspondence. We found that the probability (18.14) reasonably approximates the exact result plotted on a linear scale. Only the tail behavior (on

432

The hopcount to an anycast group

log-scale) and the case for p = 1 deviate significantly. In summary for the URT, the approximation (18.14) for Pr [kQ (p) = m] is much faster to compute than the exact recursion and it seems appropriate for the computation of  for p A 1. However, it is less adequate to solve the server placement problem that requires the tail values Pr [kQ (p) A m]. 1.0 N = 10 : K N = 20 : K N = 30 : K N = 50 : K

0.8

 0.404 ln(m/N) 0.295 ln(m/N)  0.252 ln(m/N)  0.210 ln(m/N)

K

0.6

0.4

0.2

0.0

2

3

4

5

6

7 8 9

2

0.1

3

4

5

6

7 8 9

1 m/N

Fig. 18.5. The performance measure  for several sizes Q of URTs as a function of the ratio p@Q

18.6 The performance measure  in exponentially growing trees In this section, we investigate the observed law   1  d log p for a much larger class of trees, namely the class of exponentially growing trees to which both the n-ary tree and the URT belong. Also most trees in the Internet are exponentially growing trees. A tree is said to grow exponentially in the ³ ´ (m) 1@m number of nodes Q with degree  if limm n=1

which, because all eigenvalues are distinct, implies that there is a smaller set of v  1 linearly depending eigenvectors. This contradicts the initial hypothesis. This important property has a number of consequences. First, it applies to left- as well as to right-eigenvectors. Relation (A.10) then shows that the sets

438

Stochastic matrices

of left- and right-eigenvectors form a bi-orthogonal system with |nW {n 6= 0. For, if {n were orthogonal to |n (or |nW {n = 0), (A.10) demonstrates that {n would be orthogonal to all left-eigenvectors |m . Since the set of lefteigenvectors span the q dimensional vector space, it would mean that the q dimensional vector {n would be orthogonal to the whole q-space, which is impossible because {n is not the null vector. Second, any q dimensional vector can be written in terms of either the left- or right-eigenvectors. 4. Let us denote by [ the matrix with in column m the right-eigenvector {m and by \ W the matrix with in row n the left-eigenvector | W . If the rightand left-eigenvectors are scaled such that, for all 1  n  q, |nW {n = 1, then \ W[ = L

(A.13)

or, the matrix \ W is the inverse of the matrix [. Furthermore, for any right-eigenvector, (A.1) holds, rewritten in matrix form, that D[ = [ diag(n )

(A.14)

Left-multiplying by [ 31 = \ W yields the similarity transform of matrix D, [ 31 D[ = \ W D[ = diag(n )

(A.15)

Thus, when the eigenvalues of D are distinct, there exists a similarity transform K 31 DK that reduces D to diagonal form. In many applications, similarity transforms are applied to simplify matrix problems. Observe that a similarity transform preserves the eigenvalues, because, if D{ = {, then K 31 { = K 31 D{ = (K 31 DK)K 31 {. The eigenvectors are transformed to K 31 {. When D has multiple eigenvalues, it may be impossible to reduce D to a diagonal form by similarity transforms. Instead of a diagonal form, the most compact form when D has u distinct eigenvalues each with multiplicity P pm such that um=1 pm = q is the Jordan canonical form F, 5 9 9 9 F=9 9 7

6

Fp1 3d (1 ) Fd (1 )

: : : : : 8

.. . Fpu31 (u31 ) Fpu (u )

A.1 Eigenvalues and eigenvectors

439

where Fp () is a p × p submatrix of the form, 5

 0 .. .

1  .. .

9 9 9 Fp () = 9 9 7 0 ··· 0 ···

0 ··· 1 0 .. .. . . 0  0 0

0 ··· .. .

6

: : : : : 1 8 

The number of independent eigenvectors is equal to the number of submatrices. If an eigenvalue  has multiplicity p, there can be one large submatrix Fp (), but also a number n of smaller submatrices Fem () such P that nm=1 em = p. This illustrates, as mentioned in art. 1, the much higher complexity of the eigenproblem in case of multiple eigenvalues. For more details we refer to Wilkinson (1965). 5. The companion matrix of the characteristic polynomial (A.3) of D is defined as 5 9 9 9 F=9 9 7

(1)q31 fq31 (1)q31 fq32 1 0 0 1 .. .. . . 0 0

··· ··· ··· .. . ···

(1)q31 f1 (1)q31 f0 0 0 0 0 .. .. . . 1 0

6 : : : : : 8

Expanding det (F  L) in cofactors of the first row yields det (F  L) = f (). If D has distinct eigenvalues, D as well as F are similar to diag(l ). It has been shown that the similarity transform K for D equals K = [. The similarity transform for F is the Vandermonde matrix Y (), where 5 9 9 9 Y ({) = 9 9 9 7

{q31 {q31 ··· 1 2 q32 {1 {q32 ··· 2 .. .. . . ··· .. { { . 1

2

1

1

···

q31 {q31 q31 {q q32 {q31 {q32 q .. .. . .

{q31 1

{q 1

6 : : : : : : 8

The Vandermonde matrix Y () is clearly non-singular if all eigenvalues are

440

Stochastic matrices

distinct. Furthermore, 5 9 9 9 Y ()diag (l ) = 9 9 9 7 while

5

9 9 9 FY () = 9 9 9 7

q1 q2 ··· q31 1 q31 ··· 2 .. .. . . ··· .. 2 . 2 1

2

1

2

···

(1)q1 f (1 ) + q1 q1 1 .. .

(1)q1 f (2 ) + q2 q1 2 .. .

21 1

22 2

qq31 qq q31 q31 q31 q .. .. . . 2q31 q31 ··· ··· ··· .. . ···

2q q

6 : : : : : : 8

(1)q1 f (q ) + qq q1 q .. . 2q q

6 : : : : : : 8

Since f (m ) = 0, it follows that FY () = Y ()diag(l ), which demonstrates the claim. Hence, the eigenvector {n of F belonging to eigenvalue n is ¤ £ {Wn = q31 q32 · · · n 1 n n 6. When left-multiplying (A.1), we obtain D2 { = D{ = 2 { or, in general for any integer n  0, Dn { = n {

(A.16)

Since any eigenvalue  satisfies its characteristic polynomial f () = 0, we directly find from (A.16) that the matrix D satisfies its own characteristic equation, f(D) = 0

(A.17)

This result is the Caley—Hamilton theorem. There exist several other proofs of the Caley—Hamilton theorem. 7. Consider an arbitrary matrix polynomial in , I () =

p X

In n

n=0

where all In are q × q matrices and Ip 6= R. Any matrix polynomial I () can be right and left divided by another (non-zero) matrix polynomial E() in a unique way as proved in Gantmacher (1959a, Chapter IV). Hence the

A.1 Eigenvalues and eigenvectors

441

left-quotient and left-remainder I () = E()TO () + O() and the rightquotient and right-remainder I () = TU ()E() + U() are unique. Let us concentrate on the right-remainder in the case where E() = L  D is a linear polynomial in . Using Euclid’s division scheme for polynomials, p31

I () = Ip 

p31

(L  D) + (Ip D + Ip31 ) 

+

£ ¤ = Ip p31 + (Ip D + Ip31 ) p32 (L  D)

p32 X

In n

n=0

X ¢ p32 p33 ¡ 2 + In n + Ip D + Ip31 D + Ip32  n=0

and continued, we arrive at 5 I () = 7Ip p31 + · · · + n31

p X

Im Dm3n + · · · +

p X

6 Im Dm31 8 (L  D)

m=1

m=n

+

p X

Im Dm

m=0

In summary, I () = TU () (L  D) + U() (and similarly for the leftquotient and left-remainder) with ³P ´ ³P ´ P Pp p p n31 m3n n31 m3n I  I D () =  D T TU () = p m m O m=n m=n P n=1 m Ppn=1 m U() = p I D = I (D) O() = D I m m=0 m m=0 (A.18) and where the right-remainder is independent of . The Generalized Bézout Theorem states that the polynomial I () is divisible by (L  D) on the right (left) if and only if I (D) = R (O() = R). By the Generalized Bézout Theorem, the polynomial I () = j()L j(D) is divisible by (L  D) because I (D) = j(D)L  j(D) = R. If I () is an ordinary polynomial, the right- and left-quotient and remainder are equal. The Caley—Hamilton Theorem (A.17) states that f(D) = 0, which indicates that f()L = T() (L  D) and also f()L = (L  D) T(). The matrix T() = (L  D)31 f() is called the adjoint matrix of D. Explicitly, from (A.18), 3 4 q q X X T() = n31 C fm Dm3n D n=1

m=n

Pq m3n . The main theand, with (A.6), T(0) =  (D)31 det D = m=1 fm D oretical interest of the adjoint matrix stems from its definition f()L =

442

Stochastic matrices

T() (L  D) = (L  D) T() in case  = n is an eigenvalue of D. Then, (n L  D) T(n ) = 0, which indicates by (A.1) that every non-zero column of the adjoint matrix T(n ) is an eigenvector belonging to the eigenvalue n . In addition, by dierentiation with respect to , we obtain f0 ()L = (L  D) T0 () + T() This demonstrates that, if T(n ) 6= R, the eigenvalue n is a simple root of f() and, conversely, if T(n ) = R, the eigenvalue n has higher multiplicity. The adjoint matrix T() = (L  D)31 f() is computed by observing that, on the Generalized Bézout Theorem, f()3f() is divisible without re3 mainder. By replacing in this polynomial  and  by L and D respectively, T() readily follows as illustrated in Section A.4.2. 8. Consider the arbitrary polynomial of degree o, j({) = j0

o Y

({  m )

m=1

Substitute { by D, then j(D) = j0

o Y

(D  m L)

m=1

Since det (DE) = det D det E and det(nD) = n q det D, we have det(j(D)) = j0q

o Y

det(D  m L) = j0q

m=1

o Y

f(m )

m=1

With (A.5), det(j(D)) = j0q

o Y q Y

(n  m ) =

m=1 n=1

=

q Y

q Y n=1

j0

o Y

(n  m )

m=1

j (n )

n=1

If k({) = j({)  , we arrive at the general result: For any polynomial j({), the eigenvalues values of j(D) are j (1 ) > = = = > j (q ) and the characteristic polynomial is q Y det(j(D)  L) = (j (n )  ) (A.19) n=1

which is a polynomial in  of degree at most q. Since the result holds for an

A.2 Hermitian and real symmetric matrices

443

arbitrary polynomial, it should not surprise that, under appropriate conditions of convergence, it can be extended to infinite polynomials, in particular to the Taylor series of a complex function. As proved in Gantmacher (1959a, Chapter V), if the power series of a function i (}) around } = }0 i (}) =

" X

im (}0 )(}  }0 )m

(A.20)

m=1

P m converges for all } in the disc |}}0 | ? U, then i (D) = " m=1 im (}0 )(D}0 L) provided all eigenvalues of D lie with the region of convergence of (A.20), i.e. |  }0 | ? U. For example, hD} = log D =

" X } n Dn n=0 " X n=1

n!

for all D

(1)n31 (D  L)n for |n  1| ? 1, all 1  n  q n

and, from (A.19), the eigenvalues of hD} are h}1 > = = = > h}1 . Hence, the knowledge of the eigenstructure of a matrix D allows us to compute any function of D (under the same convergence restrictions as complex numbers }).

A.2 Hermitian and real symmetric matrices

¡ ¢W A Hermitian matrix D is a complex matrix that obeys DK = DW = D, where dK = (dlm )W is the complex conjugate of dlm = Hermitian matrices possess a number of attractive properties. A particularly interesting subclass of Hermitian matrices are real, symmetric matrices that obey DW ¡= D.¢The W inner-product of vector | and { is defined as | K { and obeys | K { = ¡ K ¢K P | { = {K |. The inner-product {K { = qm=1 |{m |2 is real and positive for all vectors except for the null vector. 9. The eigenvalues of a Hermitian matrix are all real. Indeed, leftmultiplying (A.1) by {K yields {K D{ = {K { ¡ ¢K and, since {K D{ = {K DK { = {K D{, it follows that {K { = K {K { or  = K because {K { is a positive real number. Furthermore, since D = DK , we have DK { = {

444

Stochastic matrices

Taking the complex conjugate, yields DW {W = {W In general, the eigenvectors of a Hermitian matrix are complex, but real for a real symmetric matrix since DK = DW . Moreover, the left-eigenvector | W is the complex conjugate of the right-eigenvector {. Hence, the orthogonality relation (A.10) reduces, after normalization, to an inner-product {K n {m = nm

(A.21)

where nm is the Kronecker delta, which is zero if n 6= m and else nn = 1. Consequently, (A.13) reduces to [K [ = L which implies that the matrix [ formed by the eigenvectors is an unitary matrix ([ 31 = [ K ). For a real symmetric matrix D, the corresponding relation [ W [ = L implies that [ is an orthogonal matrix ([ 31 = [ W ). Although the arguments so far (see Section A.1) have assumed that the eigenvalues of D are distinct, the theorem applies in general (as proved in Wilkinson (1965, Section 47)): For any Hermitian matrix D, there exists a unitary matrix X such that X K DX = diag (m )

real m

and for any real symmetric matrix D, there exists an orthogonal matrix X such that X W DX = diag (m )

real m

10. To a real symmetric matrix D, a bilinear form {W D| is associated, which is a scalar defined as q q X X W W dlm {l |m { D| = {D| = l=1 m=1

We call a bilinear form a quadratic form if | = {. A necessary and su!cient condition for a quadratic form to be positive definite, i.e. {W D{ A 0 for all { 6= 0, is that all eigenvalues of D should be positive. Indeed, art. 9 shows the existence of an orthogonal matrix X that transforms D to a diagonal form. Let { = X }, then {W D{ = } W X W DX } =

q X n=1

n }n2

(A.22)

A.3 Vector and matrix norms

445

which is only positive for all }n provided n A 0 for all n. From (A.6), a positive definite quadratic form {W D{ possesses a positive determinant, det D A 0. This analysis shows that the problem of determining an orthogonal matrix X (or the eigenvectors of D) is equivalent to the geometrical problem of determining the principal axes of the hyper-ellipsoid q q X X

dlm {l |m = 1

l=1 m=1

Relation (A.22) illustrates that the eigenvalues n are the squares of the principal axis. A multiple eigenvalue refers to an indeterminacy of the principal axes. For example if q = 3, an ellipsoid with two equal principal axis means that any section along the third axis is a circle. Any two perpendicular diameters of the largest circle orthogonal to the third axis are principal axis of that ellipsoid.

A.3 Vector and matrix norms Vector and matrix norms, denoted by k{k and kDk respectively, provide a single number reflecting a “size” of the vector or matrix and may be regarded as an extension of the concept of the modulus of a complex number. A norm is a certain function of the vector components or matrix elements. All norms, vector as well as matrix norms, satisfy the three distance relations (i) k{k A 0 unless { = 0 (ii) k{k = || k{k for any complex number  (iii) k{ + |k  k{k + k|k In general, the Hölder t-norm of a vector { is defined as 3 41@t q X k{k = C |{m |t D t

(A.23)

m=1

For example, the well-known Euclidean norm or length of the vector { is found for t = 2 and k{k22 = {K {. In probability theory where { denotes P a discrete pdf, the law of total probability states that k{k1 = qm=1 {m = 1 and we will write k{k1 = k{k. Finally, max |{m | = limt t A 1, ¯ K ¯ ¯{ | ¯  k{k k|k s t

1 s

+ 1t = 1 and (A.24)

A special case of the Hölder inequality where s = t = 2 is the CauchySchwarz inequality ¯ K ¯ ¯{ | ¯  k{k k|k (A.25) 2 2 The t = 2 norm is invariant under an unitary (hence also orthogonal) transformation X , where X K X = L, because kX {k22 = {K X K X { = {K { = k{k2 . An p other example s of a non-homogeneous vector norm is the quadratic form k{kD = {W D{ provided D is positive definite. Relation (A.22) shows that, if not all eigenvalues m of D are the same, not all p components of the vector { are weighted similarly and, thus, in general, k{kD is a non-homogeneous norm. The quadratic form k{kL equals the homogeneous Euclidean norm k{k22 .

A.3.1 Properties of norms All norms are equivalent in the sense that there exist positive real numbers f1 and f2 such that, for all {, f1 k{ks  k{kt  f2 k{ks For example, k{k2  k{k1 

s q k{k2

k{k"  k{k1  q k{k" s k{k"  k{k2  q k{k" By choosing in the Hölder inequality (5.15) s = t = 1, {m $ m {vm for real v A 0 and |m $ m A 0, we obtain with 0 ?  ? 1 an inequality for the weighted t-norm à Pq

m=1 m |{m | Pq m=1 m

v

à Pq

!1

v



v m=1 m |{m | P q m=1 m

!1 v

For m = 1, the weights m disappear such that the inequality for the Hölder t-norm becomes 1 1

k{kv  k{kv q v (  31)

A.3 Vector and matrix norms

where q

1 1 ( 31) v 

447

 1. On the other hand, with 0 ?  ? 1 and for real v A 0,

³P q

´ 1v

4 1v 3 41 3 ¶ 1 v q q µ v v X X k{kv |{ |{ | | m m D =C D C Pq = P 1 = 1 Pq v q k{kv |{ | v )  n n=1 ( n=1 |{n |v ) v ( |{ | m=1 m=1 n n=1 v m=1 |{m |

Since | =

|{ |v Sq m v n=1 |{n |

 1 and

1 

1

A 1, it holds that |   | and

3 41 3 4 1v à P !1 ¶ 1 v q q µ q v v X X |{m |v v | |{ | |{ m m m=1 C D C D = Pq Pq Pq =1 v v v n=1 |{n | n=1 |{n | n=1 |{n | m=1 m=1 1 1

which leads to the opposite inequality (without normalization as q v (  31) ), k{kv  k{kv In summary, if s A t A 0, then the general inequality for Hölder t-norm is 1

1

k{ks  k{kt  k{ks q t 3 s

(A.26)

For p × q matrices D, the most frequently used norms are the Euclidean or Frobenius norm 41@2 3 q p X X |dlm |2 D (A.27) kDkI = C l=1 m=1

and the t-norm kDkt = sup

kD{kt

(A.28) k{kt ° ° ° { ° = °D k{k ° , which shows that

{6=0

On the second distance relation,

kD{kt k{kt

t

t

kDkt = sup kD{kt

(A.29)

k{kt =1

Furthermore, the matrix t-norm (A.28) implies that kD{kt  kDkt k{kt

(A.30)

Since the vector norm is a continuous function of the vector components and since the domain k{kt = 1 is closed, there must exist a vector { for which equality kD{kt = kDkt k{kt holds. Since the n-th vector component of D{ P is (D{)l = qm=1 dlm {m , it follows from (A.23) that ¯ ¯t 41@t 3 ¯ p ¯X X ¯ q ¯ ¯ ¯ D C d { kD{kt = lm m ¯ ¯ ¯ ¯ l=1 m=1

448

Stochastic matrices

For example, for all { with k{k1 = 1, we have that ¯ ¯ ¯ X q p ¯X p X q p X X X ¯ q ¯ ¯ ¯  d { |d | |{ | = |{ | |dlm | kD{k1 = lm m ¯ lm m m ¯ ¯ ¯ l=1 m=1 l=1 m=1 m=1 l=1 Ã ! q p p X X X  |{m | max |dlm | = max |dlm | m=1

m

m

l=1

l=1

Clearly, there exists a vector { for which equality holds, namely, if n is the column in D with maximum absolute sum, then { = hn , the n-th basis vector with all components zero, except for the n-th one, which is 1. Similarly, for all { with k{k" = 1, ¯ ¯ ¯X ¯ q q X X ¯ q ¯ ¯ ¯ dlm {m ¯  max |dlm | |{m |  max |dlm | kD{k" = max ¯ l ¯ l l ¯ m=1 m=1 m=1 Again, if u is the row with maximum absolute sum and {m = 1.sign(dum ) P P such that k{k" = 1, then (D{)u = qm=1 |dum | = maxl qm=1 |dlm | = kD{k" . Hence, we have proved that kDk" = max l

kDk1 = max m

from which

q X m=1 p X

|dlm |

(A.31)

|dlm |

(A.32)

l=1

° K° °D ° = kDk 1 "

The t = 2 matrix norm, kD{k2 > is obtained dierently. Consider kD{k22 = (D{)K D{ = {K DK D{ Since DK D is a Hermitian matrix, art. 9 shows that all eigenvalues are real and non-negative because a norm kD{k22  0. These ordered eigenvalues are denoted as 12  22  · · ·  q2  0. Applying the theorem in art. 9, there exists a unitary matrix X such that { = X } yields ¡ ¢ {K DK D{ = } K X K DK DX } = } K diag m2 }  12 } K } = 12 k}k22 Since the t = 2 norm is invariant under a unitary (orthogonal) transform k{k2 = k}k2 , by the definition (A.28), kDk2 = sup {6=0

kD{k2 = 1 k{k2

(A.33)

A.3 Vector and matrix norms

449

where the supremum is achieved if { is the eigenvector of DK D belonging to 12 . Meyer (2000, p. 279) proves the corresponding result for the minimum eigenvalue provided that D is non-singular, ° 31 ° °D ° = 2

1 = q31 min kD{k2

k{k2 =1

The non-negative quantity m is called the m-th singular value and 1 is the largest singular value of D. The importance of this result lies in an extension of the eigenvalue problem to non-square matrices which is called the singular value decomposition. A detailed discussion is found in Golub and Loan (1983). If D has real eigenvalues 1  2  · · ·  q , the above can be simplified and we obtain {W D{ {W {

(A.34)

{W D{ {6=0 {W {

(A.35)

1 = sup {6=0

q = inf

because, for any {, it holds that q {W¡{  {W¢ D{  1 {W {. The Frobenius norm kDk2I = trace DK D . With (A.7) and the analysis of DK D above, kDk2I =

q X

n2

(A.36)

n=1

In view of (A.33), the bounds kDk2  kDkI 

s q kDk2 may be attained.

A.3.2 Applications of norms ° ° ° n ° ° n31 ° (a) Since °D ° = °DD °  kDk °Dn31 °, by induction, we have for any integer n, that ° ° ° n° n °D °  kDk and lim Dn = 0 if kDk ? 1

n }1 A 0

which is always possible by suitable renumbering of the states and  ¸ S11 S12 S = S21 S22

A.4 Stochastic matrices

451

The relation } = (S + L)y is written as  ¸  ¸ ¸  }1 S11 y1 y1 = + 0 S21 y1 0 Since S is irreducible, S21 6= R, such that y1 A 0 implies that S21 y1 6= 0, which proves the lemma. ¤ Observe, in addition, that all components of } are never smaller than those of y. Also, transposing does not alter the result. Theorem A.4.2 (Frobenius) The modulus of all eigenvalues  of an irreducible stochastic matrix S are less than or equal to 1. There is only one real eigenvalue  = 1 and the corresponding eigenvector has positive components. Proof: The t = 4 norm (A.31) of a probability matrix S with q states defined by (9.7) subject to (9.8) precisely equals kS k" = 1. From (A.37), it follows that all eigenvalues are, in absolute value, smaller than or equal to 1. Since all elements Slm 5 [0> 1] and because an irreducible matrix has no zero element rows, y W S has positive components if yW has positive components. (yW S ) Thus, there always exists a scalar 0 ? y = min1$n$q (yW ) n , such that n

y y W  y W S . By Lemma A.4.1, we can always transform the vector y to a vector } by right-multiplying both sides with (L + S ) such that y y W (L + S )  yW S (L + S ) y } W  } W S and, by definition of y , y  } since the components of } are never smaller than those of y. Hence, for any arbitrary vector y with positive components, the transform in Lemma A.4.1 leads to an increasing set y  }  · · · , which is bounded by 1 because no eigenvalue can exceed 1. This shows that  = 1 is the largest eigenvalue and the corresponding eigenvector | W has positive components. This eigenvector | W is unique. For, if there were another linearly independent eigenvector zW corresponding to the eigenvalue  = 1, any linear combination } W = | W + zW is also an eigenvector belonging to  = 1. But  and  can always be chosen to produce a zero component which the transform method shows to be impossible. The fact that the eigenvector | W is the only eigenvector belonging  = 1, implies that the eigenvalue  = 1 is a single zero of the characteristic polynomial of S . ¤ The theorem proved for stochastic matrices is a special case of the famous Frobenius theorem for non-negative matrices (see for a proof, e.g. Gant-

452

Stochastic matrices

macher (1959b, Chapter XIII)). We note that, in the theory of Markov chains, the interest lies in the determination of the left-eigenvector | W =  belonging to  = 1, because the right-eigenvector { of S belonging to  = 1 equals xW = [1 1 · · · 1], where  is a scalar, because of the constraints (9.8). Recall (A.10) and (A.13), the proper normalization, | W x = 1, precisely corresponds to the total law of probability. Using the interpretation of Markov chains, an alternative argument is possible. If all eigenvalues were || ? 1, application (c) in Section A.3.2 indicates that the steady-state would be non-existent because S n $ 0 for n $ 4. Since this is impossible, there must be at least one eigenvalue with || = 1. Furthermore, (9.22) shows that at least one eigenvalue corresponding to the steady-state is real and precisely 1. Corollary A.4.3 An irreducible probability matrix S cannot have two linearly independent eigenvectors with positive components. Proof: Consider, apart from |W =  belonging to  = 1, another eigenvector zW belonging to the eigenvalue $ 6= 1. On art. 3, zW | = 0, which is ¤ only possible if not all components of zW are positive.

The corollary is important because no other eigenvector of S than | W =  can represent a (discrete) probability density. Since the null vector is never an eigenvector, the corollary implies that at least one component in the other eigenvectors must be negative. Since the characteristic polynomial of S has real coe!cients (because Slm is real), the eigenvalues occur in complex conjugate pairs. Since  = 1 is an eigenvalue, for an even number of state q, there must be at least another real eigenvalue obeying 1   ? 1. It has been proved that the boundary of the locations of the eigenvalues inside the unit disc consists of a finite number of points on the unit circle joined by certain curvilinear arcs. There exist an interesting property of a rank-one update S¯ of a stochastic matrix S . The lemma is of a general nature and also applies to reducible Markov chains with several eigenvalues m = 1 for 1 ? m  n. Lemma A.4.4 If {1> 2 > 3 > = = = > q } are the eigenvalues of the stochastic matrix S , then the eigenvalues of S¯ = S + (1  )xy W , where y W is any probability vector, are {1> 2 > 3 > = = = > q }.

A.4 Stochastic matrices

453

Proof: We start from the eigenvalues equation (A.2) ¢ ¡ ¢ ¡ det S¯  L = det S  L + (1  )xy W ³ ³ ´´ = det (S  L) L + (S  L)31 (1  )xyW ´ ³ = det (S  L) det L + (1  ) (S  L)31 xy W Applying the formula ¢ ¡ det L + fgW = 1 + gW f

(A.38)

which follows, after taking the determinant, from the matrix identity ¶µ ¶µ ¶ µ ¶ µ L 0 L + fgW f L f L 0 = gW 1 0 1 + gW f gW 1 0 1 gives

´ ³ ¡ ¢ det S¯  L = det (S  L) 1 + y W (1  ) (S  L)31 x

Since the row sum of a stochastic matrix S is 1, we have that S x = x and, thus, (S  L) x = (  ) x from which (S  L)31 x = (  )31 x. Using this result leads to 1 + y W (1  ) (S  L)31 x = 1 +

1 1 W 1 y x=1+ =   

because a probability vector is normalized to 1, i.e. y W x = 1. Hence, we end up with ¡ ¢ 1 det S¯  L = det (S  L)  Invoking (A.19) yields q q Y ¢ Y 1 ¯ det S  L = = (1  ) (n  ) (n  ) 

¡

n=1

n=2

which shows that the eigenvalues of S¯ are {1> 2 > 3 > = = = > Q }.

¤

A similar property may occur in a special case where a Markov chain is supplemented by an additional state q + 1 which connects to every other state and to which every other state is connected (such that S¯ is irreducible). Then, ¶ µ S (1  )x ¯ S = 0 yW

454

Stochastic matrices

with corresponding eigenvalues {1> 2 > 3 > = = = > q > 0}. This result is similarly proved as Lemma A.4.4 using (Meyer, 2000, p. 475) ¶ µ ¡ ¢ D E = det D det G  FD31 E det (A.39) F G provided D31 exists unless F = 0. A.4.2 Example: the two-state Markov chain The two-state Markov chain is defined by ¸  1s s S = t 1t Observe that det S = 1st. The eigenvalues of S satisfy the characteristic polynomial f() = 2  (2  s  t) + det S = 0, from which 1 = 1 and 2 = 1  s  t = det S . The adjoint matrix T () is computed (art. 7) via the polynomial f()3f() 3 , f()  f() =  +   (2  s  t)  and after  $ L and  $ S T () = L + S  (2  s  t)L ¸  1+t s = t 1+s The (unscaled) right- (left-) eigenvectors of S follow as the non-zero columns (rows) of T (). For 1 = 1, we find {1 = (1> 1) and |1W = (t> s). For 2 = 1st, the eigenvector {2 = (s> t) and |2W = (1> 1). Normalization 1 1 (1> 1) and {2 = s+t (s> t). If the (art. 4) requires that |nW {n = 1 or {1 = s+t eigenvalues are distinct (s + t 6= 0), the matrix S can be written as (art. 4) S = [diag(n )\ W ,  ¸ ¸ ¸ 1 1 s 1 0 t s S = 0 1st 1 1 s + t 1 t from which any power S n is immediate as ¸ ¸  ¸ 1 1 0 1 s t s n S = 1 1 0 (1  s  t)n s + t 1 t  ¸  ¸ n 1 (1  s  t) t s s s = + t t s+t t s s+t

(A.40)

A.4 Stochastic matrices

The steady-state matrix S " = limn Q ] to is transformed with {m = 0 if m 5 ¢ ¡ ¢ ¡ Q } J (})  {Q } Q  } 2 J0 (})  Q {Q } Q31  ( + 1  ) }J0 (}) [Q  f]J (}) + J0 (}) = 0 from which the logarithmic derivative is Q } + f  Q  J0 (}) = 2 J (}) } + ( + 1  ) }  1 The integration of the right-hand side requires a partial faction decomposition, } 2

Q } + f  Q  f2 f1 + = + ( + 1  ) }  1 }  u1 }  u2

where u1 and u2 are the roots of the quadratic polynomial } 2 +( + 1  ) } 1 and f1 and f2 are the residues computed for n = 1> 2 as fn = lim

} 2 (n)}0$n$Q of roots to precisely Q + 1 and confines the analysis to 0  n  Q@2. We will show that all roots are real and distinct (except for n = Q@2).  (n) = E 2 (n)  4D (n) F (n) is with | = (Q@2  n)2 5 £ The discriminant ¤ 0> (Q@2)2 , ¢ ¡  (n) = 16| 2 + 4 (1 + ) f2 (1 + )  2fQ + Q 2 | which shows that  (n) is concave in | because

g2 {(n) = 32 ? 0, for g| 2 2 Q (f(1 + )  Q )2 A 0

| = 0,  (Q@2) = 0 and, for | = (Q@2)2 ,  (0) = and, hence,  (n)  0 for n 5 [0> Q@2]. This means that, for 0  n ? Q@2, the roots 1 (n) and 2 (n) are real and distinct and, for n = Q@2 (only if Q is even) where  (Q@2) = 0, 1 (Q@2) = 2 (Q@2) =

E (Q@2) 1+ = 2D (Q@2) 1  2 Qf

For  ? n  Q@2, the roots {1 ()> 2 ()} are dierent from the roots {1 (n)> 2 (n)} because D(n) } 2 + E(n) } + F(n) ? D() } 2 + E() } + F() for all }. Indeed, D()  D(n) = (Q@2  )2  (Q@2  n)2 A 0 and the discriminant (E()  E (n))2  4 (D()  D(n)) (F()  F(n)) ? 0 shows that there are no real solutions. Thus, an extreme eigenvalue occurs for n = 0 for which F (0) = 0 such that 1 (0) = 0 and 2 (0) = 

1 +   Q E (0) f = D (0) 1  Qf

(A.47)

Q ? 1 and f ? Q shows that 2 (0) ? 0, The stability requirement  = f(1+) and thus 2 (0) is the largest negative eigenvalue. The eigenvalues for other 0 ? n  Q@2 are either larger than 0 or smaller than 2 (0). We need to consider two dierent cases (a) f ? Q@2 and (b) f A Q@2 while F (n) ? 0 for all n 5 [0> Q).

(a) If f ? Q@2 and if 0  n ? f and , then D (n) A 0. Hence, the product

A.5 Special types of stochastic matrices

463

1 (n)2 (n) = F(n) D(n) ? 0 which means that 1 (n) A 0 A 2 (n) and that there are precisely [f] positive eigenvalues. Similarly, D (n) ? 0 for f ? n ? Q@2, such that 1 (n)2 (n) A 0 while 1 (n) + 2 (n) =  E(n) D(n) ? 0 shows that both eigenvalues are negative because E (n) ? 0. Indeed, if   1 and f ? Q@2, the above expression immediately leads to E (n) ? 0 while if  ? 1 and f ? Q@2, the expression "µ µ ¶2 µ ¶2 # ¶ ¸ Q Q Q Q  2f n  f f +1 E(n) = 2(1  ) 2 2 2 f shows that both terms are negative. (b) If f A Q@2, we see that D (n) A 0 for 0 ? n ? Q  f leading to 1 (n) A 0 A 2 (n). For Q  f ? n ? Q@2, we have D (n) ? 0 and thus 1 (n)2 (n) = F(n) D(n) A 0 while their same sign follows from 1 (n) + 2 (n) =  E(n) D(n) requires us to consider the sign of E (n). If   1, then E (n) A 0. If  A 1, then µ ¶ Q E(n) = Q (1 + ) f  + 2(1  ) (Q@2  n)2 2 µ ¶ µ ¶ Q Q 2 ? Q (1 + ) f  + 2(1  ) f  2 2 µ ¶µ ¶ Q  (Q  f) = 2f f  +1 A0 2 f which shows that 0 ? 2 (n) ? 1 (n). Hence, there are Q  [f] + 2(Q@2  Q + [f]) = [f] positive eigenvalues. In summary, there are [f] positive eigenvalues, one 1 (0) = 0 and Q  [f] negative eigenvalues. Relabel the eigenvalues as (n > Q3n ) = (1 (n)> 2 (n)) in increasing order Q3[f]31 ? · · · ? 1 ? 0 ? Q = 0 ? Q31 ? · · · ? Q3[f] .This way of writing distinguishes between underload and overload eigenvalues. In terms of the discriminant by  (n) = E 2 (n)  4D (n) F (n), the non-positive eigenvalues are (a) If f ? Q@2, s

3E(n)3 {(n) 2D(n) s 3E(n)~ {(n) = 2D(n)

1 (n) = 1>2 (n)

0  n  [f] [f] + 1  n 

Q 2

(b) If f A Q@2, s

1 (n) =

3E(n)3 {(n) 2D(n)

0  n  Q  [f]  1

464

Stochastic matrices

The eigenvector belonging to m follows from (A.45) where u1 and u2 are given in (A.42) and n is determined from (A.43) since n = f1 . The eigenvectors for 1 (n) and 2 (n) belonging to a same quadratic n must be dierent. Especially in this case, the corresponding n = f1 values can be determined from (A.43). For example, for Q = 0, we find u1 = 1, u2 =  1 and n = 0 and the eigenvector belonging to Q is with (A.45), µ ¶ m µ ¶ Q  Q3m Q 3m Q 3m {m (0) = (1) = (A.48) u1 u2 m Q m After renormalization such that k{(0)k1 = 1, i.e. by dividing each com¡Q ¢ m P (1+)Q 1 PQ , the steady-state vector ponent by Q m=0 {m (0) = Q m=0 m  = Q (14.52) is obtained. Similarly, for the largest negative eigenvalue 0 in (A.47), we find with u1 = 1  Qf , u2 =  Q131 and n = f1 = Q such that (f ) µ ¶µ ¶Q3m µ ¶ Q Q Q3m Q Q 3m 0 u2 = (A.49) 1 u1 {m (Q ) = (1) m f m The left-eigenvectors | satisfy (A.9): | W G31 T = |W . The above approach is applicable. However, there is a more elegant method based on the observation that there exists a diagonal matrix q Z = gldj (Z0 > = = = > ZQ ) for ¡Q ¢ ¡ 31 ¢W 31 31 TZ is m which Z TZ = Z TZ , namely Zm = m  . Since Z symmetric, the left- and right-eigenvectors corresponding to the same eigenvalue are the same (Section A.2, art. 9). Now |W G31 T = | W is equivalent to ¡ ¢31 ¡ 31 ¢ Z TZ | W Z = |W Z Z 31 G31 Z Z 31 TZ = | W Z Z 31 GZ W = |W Z , G 31 GZ and T 31 TZ = TW , we obtain With |Z Z = Z Z = Z Z 31 31 W W |Z GZ TZ = |Z . The transpose |Z = TZ GZ |Z is

Z 2 | = TG31 Z 2 | which shows compared to G31 T{ = { that { = Z 2 | or, the vector components are, for 0  m  Q , µ ¶ Q m {m =  |m (A.50) m A.5.2.3 General tri-diagonal matrices Since tri-diagonal matrices of the form (11.1) frequently occur in Markov theory, we devote this section to illustrate how far the eigen-analysis can be

A.5 Special types of stochastic matrices

465

pushed. For an eigenpair (the right-eigenvector { belonging to eigenvalue ), the components in (S  L){ = 0 satisfy (u0  ) {0 + s0 {1 = 0 tm {m31 + (um  ) {m + sm {m+1 = 0

1m?Q

tQ {Q31 + (uQ  ) {Q = 0 If sm = s and tm = t, the matrix S reduces to a Toeplitz form for which the eigenvalues and eigenvectors can be explicitly written, as shown in Appendix A.5.2.1. Here, we consider the general case and show how orthogonal polynomials enter the scene. Using um = 1  tm  sm , u0 = 1  s0 and uQ = 1  tQ , the set becomes, with  =   1, s0 +  {0 () s0 sm + tm +  tm {m ()  {m31 () {m+1 () = sm sm tQ {Q 31 () {Q () = tQ +  {1 () =

1m?Q

(A.51)

The dependence on the eigenvalue  is made explicit. Solving (A.51) iteratively for m ? Q , ¢ {0 () ¡ 2  + (t1 + s1 + s0 )  + s1 s0 s0 s1 ¢ {0 () ¡ 3  + (t1 + t2 + s2 + s1 + s0 )  2 {3 () = s2 s1 s0 + (t2 t1 + t2 s0 + s2 t1 + s2 s1 + s2 s0 + s1 s0 )  + s2 s1 s0 {2 () =

reveals a polynomial of degree m in the eigenvalue  =   1. By inspection, the general form of {m () for m ? Q is m {0 () X {m () = Qm31 fn (m) n p=0 sp n=0

(A.52)

466

Stochastic matrices

with fm (m) = 1 fm31 (m) = f0 (m) =

m31 X p=0 m31 Y

(sp + tp ) sp

p=0

where t0 = sQ = 0. By substituting (A.52) into (A.51), m31 X

fn (m + 1) n =

n=1

m31 X

[(tm + sm ) fn (m)  tm sm31 fn (m  1) + fn31 (m)]  n

n=1

and equating the corresponding powers in , a recursion relation for the coe!cients fn (m) (0  n ? Q ) is obtained with fm (m) = 1, fn (m + 1) = (tm + sm ) fn (m)  tm sm31 fn (m  1) + fn31 (m) from which all coe!cients can be determined. Finally, for m = Q , the explicit form of {Q () follows from (A.51) as Q 31 tQ tQ {0 () X fn (Q  1) n {Q31 () = {Q () = Q 32 tQ +  tQ +  Q s p p=0 n=0

We can always scale an eigenvector without eecting the corresponding eigenvalue. If we require a normalization of the eigenvector k{()k1 = 1, then {0 () is uniquely determined, {0 () =

¯ PQ 31 ¯¯Pm 1 + m=1 ¯ n=0

1 ¯ ¯ fn (m) Tm31  n ¯¯ + p=0 sp

tQ |tQ +|

¯P ¯ Q31 ¯ n=0

¯

f (Q31) n ¯ TnQ 32  ¯ p=0 sp

Another scaling consists of choosing {0 () = 1. Hence, apart from the eigenvalue , all eigenvector components {m () are explicitly determined. If  = 1 or  = 0, the solution is {m () = {0 (0). If k{()k1 = 1, then {m () 1 , which is, after proper scaling by Q + 1 (art. 4 in Section A.1), the = Q+1 £ ¤ right-eigenvector x = 1 1 · · · 1 belonging to the left-eigenvector  (see also Section A.4.1). If {0 () = 1, we immediate obtain x. Eigenvectors belonging to dierent eigenvalues  0 6=  are linearly independent (art. 3 in Section A.1), but only orthogonal if S = S W , i.e. if sm = tm+1 . Only in the latter case (art. 9 in Section A.2), where also all eigenvalues are real, we

A.5 Special types of stochastic matrices

467

have Q X

¡ ¢ {m () {m  0 = k{()k22 0

m=0

This orthogonality requirement determines the dierent eigenvalues . Since  0 = 0 is an eigenvalue, each other real eigenvalue  6= 0 must obey Q X

{m () = 0

m=0

P while the normalization enforces k{()k1 = Q m=0 |{m ()| = 1. The scaling PQ {0 () = 1 leads to the polynomial m=0 en  n of degree Q whose Q zeros equal the eigenvalues  6= 0 and whose coe!cients are, with sm = tm+1 and for 2  n  Q  2, e0 = (Q + 1) tQ e1 = Q + tQ

Q 32 X m=1

en =

Q32 X m=n

f1 (m) f1 (Q  1) + 2tQ QQ32 Qm31 p=0 sp p=0 sp

Q31 X fn31 (m) tQ fn (m) 2tQ fn (Q  1) + + QQ32 Qm31 Qm31 p=0 sp p=0 sp p=0 sp m=n31

sQ32 + 2tQ + fQ32 (Q  1) QQ32 p=0 sp 1 = QQ31 p=0 sp

eQ31 = eQ

The Newton identities (B.9) relate these coe!cients to the sum of integer powers of the real zeros  6= 0. Proceeding much further in the case that S is not symmetric is di!cult. A similarity transform is needed to transform the linearly independent set of vectors { () for dierent  to an orthogonal set from which the eigenvalues then follow, as in the symmetric case above. Karlin and McGregor (see Schoutens (2000, Chapter 3)) have shown the existence of a set of orthogonal polynomials (similar to our set {m ()) that obey an integral orthogonality condition (similar to Legendre or Chebyshev polynomials) instead of our summation orthogonality condition. Only in particular cases, however, were they able to specify this orthogonal set explicitly.

468

Stochastic matrices

A.5.3 A triangular matrix complemented with one subdiagonal The transition probability matrix S has the structure of a triangular matrix complemented with one subdiagonal, 5 9 9 9 9 S =9 9 9 7

S00 S01 S02 ··· ··· S0Q S10 S11 S12 ··· ··· S1Q 0 S21 S22 ··· ··· S2Q .. .. .. .. .. . . . ··· . . 0 0 · · · SQ31>Q32 SQ31;Q31 SQ31;Q 0 0 ··· 0 SQ;Q31 SQQ

6 : : : : : : : 8

Besides the normalization kk1 = 1, the steady-state vector  obeys the relation  = =S , or per vector component (9.23),

m =

m+1 X

Snm n

n=0

because Snm = 0 if n A m + 1. Immediately we obtain an iterative equation that expresses m+1 (for m ? Q ) in terms of the n for 0  n  m as µ m+1 =

1  Smm Sm+1;m

¶

m31 X Snm n m  Sm+1;m n=0

Let us consider the eigenvalue equation (A.1) that is written for stochastic matrices as (S  L)W {W = 0. The matrix (S  L)W is a (Q + 1) × (Q + 1) matrix of rank Q because det(S L)W = 0 (else all eigenvectors { are zero). When writing this set of equations in terms of {0 , we produce the following set of Q equations, 5

S10 9 S11 3  9 9 9 S12 9 9 9 9 ··· 9 9 .. 7 .

S22 3 

S32

··· .. .

··· .. .

S1;Q 31

S2;Q 31

S3;Q 31

0 S21

0 0

··· ··· ··· .. .

0 0 .. .

0 0 .. .

···

···

··· ···

SQ 31;Q 32 SQ 31;Q 31 3 

0 SQ ;Q 31

6 5 { : 1 : : 9 {2 : 9 {3 :9 :=9 .. :9 :9 . :7 : {Q 31 8 {Q

6

5 3S 00 3S01 3S02 .. . 3S0;Q 32 3S0;Q 31

: 9 : 9 : 9 :=9 : 9 : 9 8 7

6 : : : : {0 : : 8

Since the right hand side matrix is a triangular Q matrix, the determinant equals the product of the diagonal elements or Q31 n=0 Sn+1;n . By Cramer’s

A.5 Special types of stochastic matrices

469

rule, we find that 5

S10 9 S11 3  9 9 9 S12 9 9 .. 9 9 . 9 det 9 .. 9 . 9 9 9 9 S1m 9 9 .. 7 . {m = {0

S1;Q 31

0 S21 S22 3  .. . .. . S2m .. . S2;Q 31

··· ··· .. .

0 0 .. .

 3 S00 3S01 .. .

···

Sm31;m32

3S0>m32

0 0 .. . .. .

···

Sm31;m31 3 

3S0;m31

0

···

··· .. . ···

Sm31;m .. .

3S0m .. . 3S0;Q 31

Sm+1;m .. .

··· .. . ···

Sm31;Q 31 TQ 31 n=0

Sm+1;Q 31

··· ··· ··· ···

0 0 .. . .. . .. . .. . 0 SQ ;Q 31

6 : : : : : : : : : : : : : : : : : 8

Sn+1;n

The above determinant is of the form (Meyer, 2000, p. 467)  ¸ Dm×m Rm×Q3m det = det F det D EQ3m×m FQ 3m×Q3m Q 31 where det F = Q n=m Sn+1;n . In the determinant det D, we can change the m-th column with the (m  1)-th, and subsequently, the (m  1) th with the (m  2)-th and so on until the last column is permuted to the first column, in total m  1 permutations. After changing the sign of that first column, the result is that det D = (1)m det (Sm×m  Lm×m ) where Sm×m is the original transition probability matrix limited to m states (instead of Q + 1). Hence, for 1  m  Q , {0 (1)m det (Sm×m  Lm×m ) {m = Qm31 n=0 Sn+1;n and the normalization of eigenvectors k{k1 = 1 determines {0 as {0 =

1+

PQ

1

(31)m det(Sm×m 3Lm×m ) Tm31 m=1 n=0 Sn+1;n

If the Q + 1 eigenvalues  are known, we observe that all eigenvectors can be expressed in terms of the original matrix S in a same way.

Appendix B Algebraic graph theory

This appendix reviews the elementary basics of the matrix theory for graphs J (Q> O). The book by Cvetkovic et al. (1995) is the current standard work on algebraic graph theory.

B.1 The adjacency and incidence matrix 1. The adjacency matrix D of a graph J with Q nodes is an Q × Q matrix with elements dlm = 1 only if (l> m) is a link of J, otherwise dlm = 0. Because the existence of a link implies that dlm = dml , the adjacency matrix D = DW is a real symmetric matrix. It is assumed further that the graph J does not contain self-loops (dll = 0) nor multiple links between two nodes. The complement Jf of the graph J consists of the same set of nodes but with a link between (l> m) if there is no link (l> m) in J and vice versa. Thus, (Jf )f = J and the adjacency matrix Df of the complement Jf is Df = M  L  D where M is the all-one matrix ((M)lm = 1). Information about the direction 1

3 4

2 6

5

Fig. B.1. A graph with Q = 6 and O = 9. The links are lexicographically ordered, h1 = 1 $ 2> h2 = 1 $ 3> h3 = 1 # 6 etc.

of the links is specified by the incidence matrix E, an Q × O matrix with 471

472

Algebraic graph theory

elements

; ? 1 if link hm = l $ m elm = 1 if link hm = l # m = 0 otherwise

Figure B.1 exemplifies the definition of D and E: 5

9 9 9 D=9 9 7

0 1 1 0 0 1

1 0 1 0 1 1

1 1 0 1 0 0

0 0 1 0 1 0

0 1 0 1 0 1

1 1 0 0 1 0

6

5

: 9 : 9 : 9 : E =9 : 9 8 7

1 1 1 0 0 0 0 0 0 1 0 0 1 1 1 0 0 0 0 1 0 1 0 0 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 1 0 0 1 1 0 0 1 0 0 1 0 0 1

6 : : : : : 8

2. The relation between adjacency and incidence matrix is given by the admittance matrix or Laplacian T, T = EE W =   D where  = diag(g1 > g2 > = = = > gQ ) is the degree matrix. Indeed, if l 6= m and noting that each column has only two non-zero elements at a dierent row, O X ¡ ¢ tlm= EE W lm = eln emn = 1 n=1

PO

2 If l = m, then n=1 eln = gl , the number of links that have node l in common. Also, by the definition of D, the row sum l of D equals the degree gl of node l, Q X gl = dln (B.1)

PQ

n=1

Consequently, each row sum n=1 tln = 0 which shows that T is singular implying that det T = 0. The Laplacian is symmetric T = TW because D and  are both symmetric and the quadratic form defined in Section A.2 art. 10, {W T{ = {W TW { = {W E W E{ = kE{k22  0 is positive semidefinite, which implies that all eigenvalues of T are nonnegative and at least one is zero because det T = 0. P PQ Since Q l=1 n=1 dln = 2O, the basic law for the degree follows as Q X l=1

gl = 2O

(B.2)

B.1 The adjacency and incidence matrix

473

Notice that S = 31 D is a stochastic matrix because all elements of S lie in the interval [0> 1] and each row sum is 1. 3. Let M denote the all-one matrix with (M)lm = 1 and  (J) the total number of spanning trees in the graph J, also called the complexity of J, then adjT =  (J) M

(B.3)

adjT where T31 = det T . We omit the proof, but apply the relation (B.3) to the complete graph NQ where T = Q L  M. Equation (B.3) demonstrates that all elements of adjT are equal to  (J). Hence, it su!ces to compute one suitable element of adjT, for example (adjT)11 that is equal to the determinant of the (Q  1) × (Q  1) principal submatrix of T obtained by deleting the first row and column in T, 5 6 Q 1 1 === 1 9 1 Q  1 === 1 : 9 : (adjT)11 = det 9 : .. .. .. 7 8 . . .

1

1

=== Q  1

Adding all rows to the first and subsequently adding this new first row to all other rows gives 6 5 1 1 === 1 1 : 9 9 1 Q  1 = = = : = det 9 = det 9 .. .. 8 7 7 ... . . 1 1 === Q  1 5

(adjT)11

1 1 0 Q .. . 0 0

6 === 1 === 0 : . : = Q Q 2 .. . .. 8 === Q

Hence, the total number of spanning trees in the complete graph NQ which is also the total number of possible spanning trees in any graph with Q nodes equals Q Q32 . This is a famous theorem of Cayley of which many proofs exist (van Lint and Wilson, 1996, Chapter 2). 4. The complexity of J is also given by  (J) =

det (M + T) Q2

Indeed, observe that MT = (ME) E W = 0 since ME = 0. Hence, (Q L  M) (M + T) = Q M + QT  M 2  MT = Q T and adj ((Q L  M) (M + T)) = adj (M + T) adj (Q L  M) = adj (QT)

(B.4)

474

Algebraic graph theory

Since TNQ = Q L  M and as shown in art. 3, adj(Q L  M) = Q Q32 M and since adj(Q T) = Q Q31 adjT = Q Q 31  (J) M where we have used (B.3), adj (M + T) M = Q  (J) M Left-multiplication with M+T taking into account that MT = 0 and M 2 = Q M finally gives (M + T) adj (M + T) M = det (M + T) M = Q 2  (J) M which proves (B.4). 5. A walk of length n from node l to node m is a succession of n arcs of the form (q0 $ q1 )(q1 $ q2 ) · · · (qn31 $ qn ) where q0 = l and qn = m. A path is a walk in which all vertices are dierent, i.e. qo 6= qp for all 0  o 6= p  n. Lemma B.1.1 The ¡number of walks of length n from node l to node m is ¢ equal to the element Dn lm . Proof (by induction): For n = 1, the number of walks of length 1 between state l and m equals the number of direct links between l and m, which is by definition the element dlm in the adjacency matrix D. Suppose the lemma holds for n  1. A walk of length n consists of a walk of length n  1 from l to some vertex u which is adjacent to m. By the induction ¢ ¡ n31 and hypothesis, the number of walks of length n  1 from l to u is D lu total number the number of walks with length 1 from u to m equals d¡um . The ¢ ¡ n¢ P n31 of walks from l to m with length n then equals Q d = D lm um u=1 D lu (by the rules of matrix multiplication). ¤ Explicitly, Q Q X Q ³ ´ X X = ··· dlu1 du1 u2 · · · dun32 un31 dun31 m Dn lm

u1 =1 u2 =1

un31 =1

As shown in Section 15.2, the number of paths with n hops between node l and node m is X X X [n (l $ m; Q ) = ··· dlu1 du1 u2 · · · dun31 m u1 6={l>m} u2 6={l>u1 >m}

un31 6={l>u1 >===>un32 >m}

The definition of a path restricts the first index u1 to Q  2 possible values, the second u2 to Q  3, etc.. such that the total possible number of paths is n31 Y o=1

(Q  1  o) =

(Q  2)! (Q  n  1)!

B.2 The eigenvalues of the adjacency matrix

475

whereas the total possible number of walks clearly is Q n31 . A graph is connected if, for each pair of nodes, there a walk or, ¢ ¡ exists n equivalently, if there exists some integer n A 0 for which D lm 6= 0 for each ¡ ¢ l> m. The lowest integer n for which Dn lm 6= 0 for each pair of nodes l> m is called the diameter of the graph J. Lemma B.1.1 demonstrates that the diameter equals the length of the longest shortest hop path in J.

B.2 The eigenvalues of the adjacency matrix In this section, only general results of the eigenvalue spectrum of a graph J are treated. For special types of graphs, there exists a wealth of additional, but specific properties of the eigenvalues. 1. Since D is a real symmetric matrix, it has Q real eigenvalues (Section A.2), which we order as 1  2  · · ·  Q . Section A.1, art. 4 shows that, apart from a similarity transform, the set of eigenvalues with corresponding eigenvectors is unique. A similarity transform consists of a relabeling of the nodes in the graph that obviously does not alter the structure of the graph but merely expresses the eigenvectors in a dierent base. The classical Perron-Frobenius Theorem for non-negative square matrices (of which Theorem A.4.2 is a special case) states that 1 is a simple and positive root of the characteristic polynomial in (A.3) possessing the only eigenvector of D with non-zero components. Moreover, it follows from (A.34) that PQ PQ {W D{ l=1 m=1 dlm {l {m 1 = sup W = max PQ 2 {6=0 {6=0 { { l=1 {l The maximum is attained if and only if { is the eigenvector of D belonging W as shown in Section A.3. to 1 and for any other vector | 6= {, 1  {{WD{ { P By choosing the vector | = x = (1> 1> = = = > 1), we have, with Q m=1 dlm = gl and (B.2), 1 

Q Q Q 1 XX 1 X 2O dlm = gl = Q Q Q l=1 m=1

(B.5)

l=1

The stochastic matrix S = 31 D where  = diag(g¡1 > g2 > = = = > gQ¢) is the degree matrix has the characteristic polynomial det 31 D  L = Q Y det(D3{) where det  = gm . Since the largest eigenvalue of a stochastic det { m=1

matrix equals 1 = 1 (Theorem A.4.2), for a regular graph where gm = u, the largest eigenvalue equals 1 = u.

476

Algebraic graph theory

2. Since dll = 0, we have that trace(D) = 0. From (A.7), Q 31

(1)

fQ31 =

Q X

n = 0

(B.6)

n=1

3. The Newton identities for polynomials. Let sq (}) denote a polynomial of order q defined as sq (}) =

q X

n

dn (q) } = dq (q)

n=0

q Y

(}  }n (q))

(B.7)

n=1

where {}n (q)} are the q zeros. It follows from (B.7) that sq (0) = d0 (q) = Q dq (q) qn=1 (}n (q)). The logarithmic derivative of (B.7) is s0q (})

= sq (})

q X n=1

1 }  }n (q)

For } A maxn }n (q) (which is always possible for polynomials, but not for functions), we have that s0q (}) = sq (})

q X " X (}n (q))m

} m+1

n=1 m=0

= sq (})

" X ]m (q) m=0

} m+1

where ]m (q) =

q X

(}n (q))m

n=1

Thus q X

ndn (q) } n =

n=1

q X

dn (q) } n

" X

]m (q)} 3m =

m=0

n=0

q " X X

dn (q) ]m (q)} n3m (B.8)

m=0 n=0

Let o = n  m, then 4  o  q. Also m = n  o  0 such that n  o. Combined with 0  n  q, we have max(0> o)  n  q. Thus, q " X X

dn (q) ]m (q)} n3m =

m=0 n=0

q X

q X

dn (q) ]n3o (q)} o

o=3" n=max(o>0)

=

31 X q X o=3" n=0

dn (q) ]n3o (q)} o +

q X q X o=0 n=o

dn (q) ]n3o (q)} o

B.2 The eigenvalues of the adjacency matrix

477

Equating the corresponding powers of } in (B.8) yields d0 (q)]3o (q) +

q X

n=1 q X

dn (q) ]n3o (q) = 0

o0

dn (q) ]n3o (q) = (o  q)do (q)

n=o+1

The last set of equations for 0  o ? q, do (q) = 

q 1 X dn (q) ]n3o (q) qo

(B.9)

n=o+1

are the Newton identities that relate the coe!cients of a polynomial to the sum of the positive powers of the zeros. Applied to the characteristic polynomials (A.3) and (A.5) of the adjacency matrix with }n (q) = n , dn (q) = (1)Q fn and fQ 31 = 0 (from (B.6)) yields for the first few values, fQ 32 = 

Q 1 X 2 n 2 n=1

3n 3 n=1 3Ã 4 !2 Q Q X 1C X 2 = n  2 4n D 8

fQ 33 = 

fQ 34

Q 1X

n=1

n=1

4. From (A.4), the coe!cient of the characteristic polynomial fQ 32 = P P . Each principal minor P2 has a principal submatrix of the form  doo 2¸ 0 { with {> | 5 [0> 1]. A minor P2 is non-zero if and only if { = | = 1 | 0 in which case P2 = 1. For each set of adjacent nodes, there exists such non-zero minor, which implies that fQ32 = O From art. 3, it follows that the number of links O equals O=

Q 1 X 2 n 2 n=1

(B.10)

478

Algebraic graph theory

5. Each principal submatrix P3×3 is 5 0 P3×3 = 7 { }

of the form 6 { } 0 | 8 | 0

with determinant P3 = det P3×3 = 2{|}, which is only non-zero for { = | = }. That form of P3×3 corresponds with a subgraph of 3 nodes that are fully connected. Hence, fQ33 = 2× the number of triangles in J. From art. 3, it follows that 1X 3 the number of triangles in J = n 6 Q

(B.11)

n=1

6. In general, from (A.4) and by identifying the structure of a minor Pn , any coe!cient fQ3n can be expressed in terms of graph characteristics, X (1)f|fohv(G) (B.12) (1)Q fQ3n = GMJn

where Jn is the set of all subgraphs of J with exactly n nodes and f|fohv (G) is the number of cycles in a subgraph G 5 Jn . The minor Pn is a determinant of the Pn×n submatrix of D and defined as X (1)(s) d1s1 d2s2 · · · dnsn det Pn = s

where the sum is over all n! permutations s = (s1 > s2 > = = = > sn ) of (1> 2> = = = > n) and (s) is the parity of s, i.e. the number of interchanges of (1> 2> = = = > n) to obtain (s1 > s2 > = = = > sn ). Only if all the links (1> s1 ) > (2> s2 ) > = = = (n> sn ) are contained in J, d1s1 d2s2 = = = dnsn is non-zero. Since dmm = 0, the sequence of contributing links (1> s1 ) > (2> s2 ) > = = = (n> sn ) is a set of disjoint cycles and (s) depends on the number of those disjoint cycles. Now, det Pn is constructed ¡ ¢ from a specific set G 5 Jn of n out of Q nodes and in total there are Qn such sets in Jn . Combining all contributions leads to the expression (B.12). 7. Since D is a symmetric 0-1 matrix, we observe that using (B.1), Q Q Q X X X ¡ 2¢ 2 dln dnl = dln = dln = gl D ll = n=1

n=1

n=1

Hence, with (A.16) or (B.10), (A.7) and basic law for the degree (B.2) is expressed as Q Q X X 2n = gn = 2O (B.13) trace(D2 ) = n=1

n=1

B.2 The eigenvalues of the adjacency matrix

479

Furthermore, Q Q X X ¡

D2

¢ lm

=

l=1 m=1;m6=l

=

Q Q Q X X X

dln dnm =

Q X Q X

l=1 m=1;m6=l n=1

n=1 l=1

Q X Q X

Q X

dnl (gn  dnl ) =

n=1 l=1

Q X

dnl

Ã

gn

dnm

m=1;m6=l Q X

dnl 

Q X

l=1

n=1

! dnl

l=1

or Q Q X X ¡

D2

¢ lm

=

l=1 m=1;m6=l

Q X

gn (gn  1)

(B.14)

n=1

¡ 2¢ P PQ Lemma B.1.1 states that Q l=1 m=1;m6=l D lm equals twice the total number of two-hop walks with dierent source and destination nodes. In other words, the total number of connected triplets of nodes in J equals half (B.14). 8. The total number Qn of walks of length n in a graph follows from Lemma B.1.1 as Q Q X X Qn = (Dn )lm l=1 m=1

Since any real symmetric matrix (Section A.2, art. 9) can be written as D = X diag(m )X W where X is an orthogonal matrix of the (normalized) eigenvecP n tors of D, we have that Dn = X diag(nm )X W and (Dn )lm = Q q=1 xlq xmq q . Hence, ÃQ !2 Q X Q Q X Q X X X Qn = xlq xmq nq = xlq nq l=1 m=1 q=1

q=1

l=1

9. Applying the Hadamard inequality for the determinant of any matrix Fq×q , Ã q !1 q 2 Y X |det F|  |flm |2 m=1

l=1

yields, with dlm = dml and (B.1), |det D| 

ÃQ Q Y X m=1

l=1

! 12 d2ml

=

ÃQ Q Y X m=1

l=1

! 12 dml

=

Q Y p gm m=1

480

Algebraic graph theory

Hence, with (A.6), (det D)2 =

Q Y

2n 

Q Y

gm

(B.15)

m=1

n=1

10. Applying the Cauchy—Schwarz inequality (5.17) Ã q !2 q q X X X dn en  d2n e2n n=1

n=1

n=1

to the vector (2 > = = = > Q ) and the 1 vector (1> 1> = = = > 1) gives !2 ÃQ Q X X n  (Q  1) 2n n=2

n=2

Introducing (B.6) and (B.13) ¡ ¢ 21  (Q  1) 2O  21 leads to the bound for the largest (and positive) eigenvalue 1 , r 2O (Q  1) (B.16) 1  Q P 2O Alternatively, in terms of the average degree gd = Q1 Q m=1 gm = Q , the largest eigenvalue 1 is bounded by the geometric mean of the average degree p and the maximum possible degree, 1  gd (Q  1). Combining the lower bound (B.5) and upper bound (B.16) yields r 2O 2O (Q  1)  1  (B.17) Q Q 11. From the inequality (A.26) for Hölder t-norms, we find that, if Q X

|n |t ? t

n=1

then Q X

PQ

|n |s ? s

n=1

for s ¯A t A 0. ¯Since n=1 n = 0, not all n can ¯be positive ¯ and combined PQ ¯PQ ¯PQ s¯ s¯ s with ¯ n=1 n ¯  n=1 |n | , we also have that ¯ n=1 n ¯ ? s . Applied to the case where t = 2 and s = 3 gives the following implication: if

B.2 The eigenvalues of the adjacency matrix

481

¯ ¯P 2 ? 2 then ¯ Q 3 ¯ ? 3 . In that case, the number of triangles  ¯ n=1 n ¯ 1 1 n=2 n given in (B.11) is ¯Q ¯ Q 1 3 1 ¯¯X 3 ¯¯ 1 3 1X 3 n  1  ¯ n ¯ A 0 the number of triangles in J = 1 + ¯ 6 6 6 6¯

PQ

n=2

n=2

P 2 2 Hence, if Q n=2 n ? 1 , then the number s of triangles in J is at least one. Equivalently, in view of (B.10), if 1 A O then the graph J contains at least one triangle. 12. A Theorem of Turan states that h Theorem B.2.1 A graph J with Q nodes and more than tains at least one triangle.

Q2 4

i links con-

h 2i 2 This theorem is a consequence of art. 7 and 11. For, using O A Q4  Q4 s which is equivalent to Q ? 2 O in the bound on the largest eigenvalue (B.5), 1  and 1 A triangle.

s 2O 2O A s = O Q 2 O

s O is precisely the condition in art. 11 to have at least one

13. The eigenvalues of the complete graph NQ are 1 = Q  1 and 2 = = = = = Q = 1. This follows by computing the determinant in (A.2) in the same way as in Section B.1, art. 3. Alternatively, the adjacency matrix of the complete graph is M  L and, if xW = [1 1 · · · 1] is the all-one vector, then M = x=xW . A direct computation yields µ ¶ ¢ ¡ W x=xW Q det (M  L  L) = det x=x  ( + 1) L = ( ( + 1)) det L  +1 Using (A.38) and xW x = Q , µ det (M  L  L) = ( ( + 1)) 1 Q

Q +1

¶

= (1)Q31 ( + 1)Q31 ( + 1  Q ) , gives the eigenvalues of NQ . Since the number of links in NQ is O = Q(Q31) 2 Q(Q31) we observe that the equality sign in (B.16) can occur. Since O  for 2 any graph, the upper bound (B.16) shows that 1  Q  1 for any graph.

482

Algebraic graph theory

14. The dierence between the largest eigenvalue 1 and second largest 2 is never larger than Q , i.e. 1  2  Q

(B.18)

Since 1 A 0 as indicated by (B.17), it follows from (B.6) that 0=

Q X n=1

n  1 +

Q X

|n |  1 + (Q  1) |2 |

n=2

such that 2  

1 Q 1

Hence, 1  2  1 +

1 Q 1 = Q 1 Q 1

Art. 13 states that the largest possible eigenvalue is 1 = Q  1 of the complete graph which proves (B.18). Again, the equality sign in (B.18) occurs in case of the complete graph. 15. Regular graphs. Every node m in a regular graph has the same degree gm = u and relation (B.1) indicates that each row sum of D equals u. Theorem B.2.2 The maximum degree gmax = max1$m$Q gm is an eigenvalue of the adjacency matrix D of a connected graph J if and only if the corresponding graph is regular (i.e. gm = gmax for all m). Proof: If { is an eigenvector of D belonging to eigenvalue  = gmax so is each vector n{ for each complex n (Section A.1, art. 1). Thus, we can scale the eigenvector { such that the maximum component, say {p = 1, and {n  1 for all n. The eigenvalue equation D{ = gmax { for that maximum component {p is gmax {p = gmax =

Q X

dpm {m

m=1

which implies that all {m = 1 whenever dpm = 1, i.e. when the node m is adjacent to node p. Hence, the degree of node p is gp = gmax . For any node m adjacent to p for which the component {m = 1, a same eigenvalue relation holds and thus gm = gmax . Proceeding this process shows that every node n 5 J has same degree gn = gmax because J is connected. Hence, { = x where xW = [1 1 · · · 1]. Conversely, if J is connected and regular, P then Q m=1 dpm = gmax for each p such that x is the eigenvector belonging

B.2 The eigenvalues of the adjacency matrix

483

to eigenvalue  = gmax , and the only possible eigenvector (as follows from ¤ art. 1). Hence, there is only one eigenvalue gmax . 16. The characteristic polynomial of the complement Jf is det (Df  L) = det (M  D  ( + 1) L) ³ ³ ´´ = (1)Q det (D + ( + 1) L) L  (D + ( + 1) L)31 M ³ ´ = (1)Q det ((D + ( + 1) L)) det L  (D + ( + 1) L)31 x=xW where we have used that M = x=xW and x is the all-one vector. Similar to the proof of Lemma A.4.4, we find det (Df  L) = (1)Q j () det (D + ( + 1) L)

(B.19)

where j () = 1  xW (D + ( + 1) L)31 x In general, j () is not a simple function of  although a little more is ° °2 1 ° ° known. For example, j () = 1  °(D + ( + 1) L)3 2 x° which shows that 2 j () 5 (4> 1]. Unlike in the proof of Lemma A.4.4, x is generally not an eigenvector of D and we can write (Section A.1, art. 8) 1 X (1)n Dn +1 ( + 1)n "

(D + ( + 1) L)31 =

n=0

P n n where the last sum " n=0 D } can be interpreted as the matrix generating function of the number of walks ³ ´ of length n (see Section B.1, art. 5 n and art. 8). Since D = X diag nm X W (Section A.2, art. 9) where the orthogonal s £ xm of D, the matrix product¤ £ matrix X consists of eigenvectors ¤ W x X = x=x1 x=x2 · · · x=xQ = Q cos 1 cos 2 · · · cos Q where m is the angle between ³the´eigenvector xm and the all-one vector P n 2 x. Hence, xW Dn x = xW X diag nm X W x = Q Q m=1 m cos m and, with P" (3m )n +1 n=0 (+1)n = +1+m , we can write j () = 1  Q

Q X m=1

cos2 m  + 1 + m

With (A.5), we have f (  1) = det (D + ( + 1) L) =

QQ

n=1 (n

+ 1 + )

484

Algebraic graph theory

and, hence, det (Df  L) =

Q ¢ (1)Q X ¡  + 1 + m  Q 2 cos2 m Q m=1

Q Y

(n + 1 + )

n=1;n6=m

(B.20) which shows that the poles of j () are precisely compensated by the zeros of the polynomial f (  1). Thus, the eigenvalues of Df are generally dierent from {m  1}1$m$Q where m is an eigenvalue of D. Only if x n 3Q and is an eigenvector of D corresponding with n , then j () = +1+ +1+n all eigenvalues of Df belong to the set {m  1}1$m6=n$Q ^ {Q  1  n }. According to art. 15, x is only an eigenvector when the graph is regular.

B.3 The stochastic matrix S = 31 D The stochastic matrix S = 31 D, introduced in Section B.2, art. 1, characterizes a random walk on a graph. A random walk is described by a finite Markov chain that is time-reversible. Alternatively, a time-reversible Markov chain can be viewed as random walk on an undirected graph. Random walks on graphs have many applications in dierent fields (see e.g. the survey by Lovász (1993)); perhaps, the most important application is randomly searching or sampling. The combination of Markov theory and algebra leads to interesting properties of S = 31 D. Section 9.3.1 and A.4.1 show that the left-eigenvector of S belonging to eigenvalue  = 1 is the steady-state vector  (which is a 1×Q row vector) and that the corresponding right-eigenvector is the all-one vector x, which essentially follows from (9.8) and which indicates that, at each discrete time step, precisely one transition occurs. These eigenvectors obey the eigenvalue equations S W  W =  W and S x = x and the orthogonality relation x = 1 (Section A.1, art. 3). If g = (g1 > g2 > = = = > gQ ) is the degree vector, then the basic law for the degree (B.2) is written in vector form as ¡ g ¢W x = 1. Theorem 9.3.5 states that the steady-state gW x = 2O, or, 2O ¡ g ¢W x = 1 eigenvector  is unique such that the equations x = 1 and 2O imply that the steady-state vector is µ ¶W g = 2O or m =

gm 2O

(B.21)

B.3 The stochastic matrix S = 1 D

485

In general, the matrix S is not symmetric, but, after a similarity transform K = 1@2 , a symmetric matrix U = 1@2 S 31@2 = 31@2 D31@2 is obtained whose eigenvalues are the same as those of S (Section A.1, art. 4). The powerful property (Section A.2, art. 9) of symmetric matrices shows that all eigenvalues are real and that U = X W diag(U ) X , where the columns of the orthogonal matrix X consist of the normalized eigenvectors yn that obey ymW yn = mn . Explicitly written in terms of these eigenvectors gives U=

Q X

n yn ynW

n=1

where, with Frobenius Theorem A.4.2, the real eigenvalues are ordered as 1 = 1  2  · · ·  Q  1. If we exclude bipartite graphs (where the set of nodes is N = N1 ^ N2 with N1 _ N2 = B and where each link connects a node in N1 and in N2 ) or reducible Markov chains (Section A.4), then |n | ? 1, for n A 1. Section A.1, art. 4 shows that the similarity transform K = 1@2 maps the steady state vector  into y1 = K 31  W and, with (B.21), 31@2  W ° y1 = ° °31@2  W ° 2 or

s

r

gm 2O

y1m = s µ s ¶2 = PQ gm m=1

gm s = m 2O

2O

Finally, since S = 31@2 U1@2 , the spectral decomposition of the transition probability matrix of a random walk on a graph with adjacency matrix D is S =

Q X

31@2

n 

yn ynW 1@2

= x +

n=1

Q X

n 31@2 yn ynW 1@2

n=2

¢ ¡ The q-step transition probability (9.10) is, with yn ynW lm = ynl ynm and (B.21), s Q gm gm X q q + n ynl ynm Slm = 2O gl n=2

The convergence towards the steady state m can be estimated from s s Q Q X ¯ ¯ q gm X q ¯Slm  m ¯  gm |qn | |ynl | |ynm | ? |n | gl gl n=2

n=2

486

Algebraic graph theory

Denoting by  = max (|2 | > |Q |) and by 0 the largest element of the reduced set {|n |} \ {} with 2  n  Q , we obtain s ¯ ¯ q ¯Slm  m ¯ ? gm  q + R (0q ) gl

B.4 Eigenvalues and connectivity A graph J has n components (or clusters) if there exists a relabeling of the nodes such that the adjacency matrix has the structure 6 5 D1 R = = = R 9 .. : 9 R D2 . : : D=9 9 .. : .. 7 . 8 . R = = = Dn where the square submatrix Dp is the adjacency matrix of the connected component p. Disconnectivity is a special case of reducibility of a stochastic matrix defined in Section A.4 and expresses that no communication is possible between two states in a dierent component or cluster. Using (A.39) indicates that det (D  L) =

n Y

det (Dp  p L)

(B.22)

p=1

If D is a regular graph with degree u, so is each submatrix Dp . Since Dp is connected, Section B.2, art. 15 states that the largest eigenvalue of any Dp equals u. Hence, by (B.22), the multiplicity of the largest eigenvalue of D equals the number of components in the regular graph. As shown in Section B.1, art. 2, the Laplacian T has non-negative eigenvalues of which at least one equals zero. In addition, the matrix (Q  1)L  T = (Q  1)L   + D is non-negative with constant row sums all equal to Q  1. Although the matrix (Q  1)L  T is not an adjacency matrix and does not represent a regular graph, the main argument in the proof of Theorem B.2.2 is the property of constant row sums and non-negative matrix elements. Hence, the multiplicity of the largest eigenvalue of (Q  1)L  T is equal to the number of components of J. But the largest eigenvalue of (Q  1)L  T is the smallest of T  (Q  1)L and also of T. Hence, we have proved

B.5 Random matrix theory

487

Theorem B.4.1 The multiplicity of the smallest eigenvalue  = 0 of the Laplacian T is equal to the number of components in the graph J If T has only 1 zero eigenvalue with corresponding eigenvector x (because PQ n=1 tln = 0 for each 1  l  Q is, in vector notation, Tx = 0), then the graph is connected; it has only 1 component. Theorem B.4.1 also implies (T) that, if the second smallest eigenvalue T = Q31 of T is zero, the graph J is disconnected. Since all eigenvectors of a matrix are linearly independent, the eigenvector {T of T must satisfy {WT x = 0 since x is the eigenvector belonging to  = 0. By requiring this additional constraint and choosing the scaling of the eigenvector such that {W { = 1, we obtain similar to (A.35) that T =

min

k{k22 =1 and {W x=0

{W T{

The second smallest eigenvalue T has many interesting properties that characterize how strongly a graph J is connected. It is interesting to mention the inequality (Cvetkovic et al., 1995, p. 265) ³ ´  (J)  T  2 (J) 1  cos Q

(B.23)

where  (J) and  (J) are the vertex and edge connectivity respectively.

B.5 Random matrix theory Random matrix theory investigates the eigenvalues of an Q × Q matrix D whose elements dlm are random variables with a given joint distribution. Even in case all elements dlm are independent, there does not exist a general expression for the distribution of the eigenvalues. However, in some particular cases (such as Gaussian elements dlm ), there exist nice results. Moreover, if the elements dlm are properly scaled, in various cases the spectrum in the limit Q $ 4 seems to converge rapidly to a deterministic limit distribution. The fascinating results of random matrix theory and applications from nuclear physics to the distributions of the non-trivial zeros of the Riemann Zeta function are discussed by Mehta (1991). Random matrix theory immediately applies to the adjacency matrix of the random graph Js (Q ) where each element dlm is 1 with probability s and zero with probability 1  s.

488

Algebraic graph theory

B.5.1 The spectrum of the random graph Js (Q ) Let  denote an arbitrary eigenvalue of the adjacency matrix of the random graph Js (Q ). Clearly,  is a random variable with mean H [] = Q1 £ ¤ PQ n = 0 because of (B.6). In addition, the variance Var[] = H 2 = n=1 1 PQ 2 n=1 n and from (B.10) Q 2O = s(Q  1) Q This results implies ³s ´ that, for fixed s and large Q , the eigenvalues of Js (Q ) grow as R Q , with the exception1 of the largest eigenvalue 1 . The number of links O in Js (Q ) is binomially distributed with mean H [O] = s Q (Q231) . Taking the expectation of the bounds (B.17) on the largest eigenvalue gives r 2 2 (Q  1) hs i O H [O]  H [1 ]  H Q Q Using (2.12) yields Var [] =

(Q2 ) (Q2 ) µ¡Q ¢¶ hs i X X s s n Q 2 O = n Pr [O = n] = ns (1  s)( 2 )3n H n n=0

n=0

Unfortunately, the sum cannot be expressed in closed form, but (Q2 ) µ¡ ¢¶ Q 1 X Q2 s n s q¡ ¢ ns (1  s)( 2 )3n  s n Q 2

n=0

with equality for Q $ 4. In summary, for any Q and s, s s(Q  1)  H [1 ]  s (Q  1)

(B.24)

The degree distribution (15.11) of the random graph is a binomial distribution with mean H [Grg ] = s(Q  1) and Var[Grg ] = (Q  1)s(1  s). The inequality (5.13) indicates that the degree Grg converges exponentially fast to zero the mean H [Guj ] for fixed s and large Q , which means that the random graphs tends to a regular graph with high probability. Section B.2, art. 1 states that 1 $ s (Q  1) with high probability. Comparison with the bounds (B.24) indicates that the upper bound is less tight than the lower bound and that the upper bound is only sharp when s $ 1, i.e. for the complete graph. Section B.2, art. 13 shows that only for the complete graph the upper bound is indeed exactly attained. 1

 1  It is known that, for large Q, the second largest eigenvalue of Js (Q) grows as R Q 2 + .

B.5 Random matrix theory

489

B.5.2 Wigner’s Semicircle Law Wigner’s Semicircle Law is the fundamental result in the spectral theory of large random matrices. Theorem B.5.1 (Wigner’s Semicircle Law) Let D be a random Q × Q real symmetric matrix with independent and identically distributed elements dlm with  2 = Var[dlm ] and denote by (DQ ) an eigenvalue of the set of the Q real eigenvalues of the scaled matrix DQ = IDQ . The probability density function i(DQ ) ({) of (DQ ) tends for Q $ 4 to lim i(DQ ) ({) =

Q 1] into p + 1 subintervals. The length O of each subinterval has a same distribution, which more easily follows by symmetry if the line segment is replaced by a circle of unit perimeter. Since the length O of each subinterval is equal in distribution, one can consider the first subinterval [0> [(1) ] whose length O exceeds a value { M (0> 1) if and only if all p uniform random variables belong to [{> 1]. The latter event has probability equal to (1 3 {)p such that Pr [O A {] = (1 3 {)p 1 and, with (2.35), H [O] = p+1 . (iii) If [ were a discrete random variable, then Pr [[ = n] E qqn , where qn is the number of values in the set {{1 > {2 > = = = > {q } that is equal to n. For a continuous random variable [, the values are generally real numbers ranging from {min = min1$m$q {m until {max = max1$m$q {m . We first construct a histogram K from the set {{1 > {2 > = = = > {q } by choosing 3{min a bin size {{ = {maxp , where p is the number of bins (abscissa points). The choice of 1 ? p ? q is in general di!cult to determine. However, most computer packages allow us to experiment with p and the human eye proves sensitive enough to make a good choice

C.1 Probability theory (Chapter 2)

495

of p: if p is too small, we loose details, while a high p may lead to high irregularities due to the stochastic nature of [. Once p is chosen, the histogram consists of the set {k0 > k1 > = = = > kp31 } where km equals the number of [ values in the set {{1 > {2 > = = = > {q } that lies in the interval [{min + m{{> {min + (m + 1){{] for 0 $ m $ p 3 1. By construction, Sp31 m=0 km = q. The histogram K approximates the probability density function i[ ({) after dividing each value km by q{{ because ]

{max {min 31 { [

{max

1= {min

i[ ({)g{ = lim

{{ {min + (m + 1){{]. Alternatively from (2.31) we obtain i[ (m ) = lim

{{ |; ) =

1 (2l)2

]

f1 +l"

f1 3l"

]

f2 +l"

h}1 ({3[ )+}2 (|3\ )

f2 3l"

2 2 2 \ (1 ) 2 1 }2 2 × h 2 ([ }1 +}2 \ ) + g}1 g}2 ] f2 +l" 2 (12 ) 2 \ 1 }2 }2 (|3\ ) 2 = h h g}2 2l f2 3l" ] f1 +l" 2 1 1 × h}1 ({3[ ) h 2 ([ }1 +}2 \ ) g}1 2l f1 3l"

Evaluating the last integral, denoted by O, yields   ] f1 +l" 1 1 h}1 ({3[ ) exp ([ }1 + }2 \ )2 g}1 2l f1 3l" 2   ] " 1 1 = h(f1 +lw)({3[ ) exp ([ (f1 + lw) + }2 \ )2 gw 2 3" 2 % 2 &  ] " 2  3[ }2 \ 1 f1 ({3[ ) h w 3 l f1 + gw hlw({3[ ) exp = 2 2 [ 3"

O=

C.3 Poisson process (Chapter 7)

497

Since the integrand is an entire function, thekcontour canlbe shifted, which allows substitution as in real analysis. Thus, let x = w 3 l f1 + }2\ , then [

=

1 3 }2\ [ h 2

=

1 3 }2\ [ h 2

%

& 2 [ 2 [ h exp 3 x gx 2 3" % # $& ] " 2 ({ 3 [ ) ({3[ ) exp 3 [ x2 + 2lx gx 2 2 [ 3" 5 # &] $2 6 % 2 " [ ({ 3  ) ({ 3 [ )2 ({3[ ) [ 8 gx x+l exp 3 exp 73 2 2 2[ 2 [ 3"

1 f1 ({3[ ) O= h 2

]

"

By substituting w = x + l

 k l }  l x+l f1 + 2 \ ({3[ )

({3[ ) , 2 [

the integral becomes

% & & I ] " ] " 2 2 [ [ 2 2 2 exp 3 exp 3 h3z z31@2 gz w gw = 2 w gw = 2 2  [ 0 3" 0 I I   2 2 1 = K = [ 2 [

]

"

%

where we have used the Gamma function (Abramowitz and Stegun, 1968, Chapter 6). Hence, }  3 2 \ ({3[ )

O=

h

[

I [ 2

%

({ 3 [ )2 exp 3 2 2[

&

and   ({3 )2 ] f2 3l" \2 (12 ) 2  exp 3 22[  1 }2 3 \ +  \ [ 2 [ i[\ ({> |; ) = I h 2l f2 3l" [ 2

 ({3[ ) }2 }2 |

h

g}2

The last integral is recognized  with (3.22) as the inverse Laplace transform of a Gaussian 2 1 3 2 and mean  =  + \ ({ 3  ). Thus with variance  2 = \ \ [  [

%  2 &    |3\ 3  \ ({3[ ) [ ({3[ )2 exp 3 2 exp 3 22 2\ (132 ) [ I s i[\ ({> |; ) = I [ 2 \ 1 3 2 2 which finally leads to the joint Gaussian density function (4.4). Hence, the linear combination method leads to exact results for Gaussian random variables.

C.3 Poisson process (Chapter 7) (i) Let \ be a binomial random variable with parameters Q and s, where Q is a Poisson random variable with parameter . The probability density function of \ is obtained by applying the law of total probability (2.46), Pr [\ = n] =

" [ q=0

Pr[\ = n|Q = q] Pr [Q = q]

498

Solutions of problems With (3.3) and (3.9), we have Pr [\ = n] =

=

" "   [ sn 3 [ t (q3n) q q n (q3n) q h3 = h s t q! n! (q 3 n)! n q=0 q=n " (s)n 3+t sn n 3 [ t q q h = h n! q! n! q=0 n

Since t = 1 3 s, we arrive at Pr [\ = n] = (s) h3s , which means that \ is a Poisson n! random variable with mean s. If a su!cient sample of test strings defined above is sent and received, the average number of “one bits” at receiver divided by the average number of bits at the sender gives the probability s (if errors occur indeed independently). (ii) Since the counting process of a sum of a Poisson process is again a Poisson counting process S with rate equal to 4m=1 m , the average number of packets of the four classes in the router’s S buers during interval W is  = W 4m=1 m . Hence, the probability density function for the q

total number Q of arrivals is Pr [Q = q] = q! h3 . (iii) Theorem 7.3.4 states the Q(w) is a Poisson counting process with rate 1 + 2 . Then, Pr [{[1 (w) = 1} K {[(w) = 1}] Pr [[(w) = 1] Pr [{[1 (w) = 1} K {[2 (w) = 0}] = Pr [[(w) = 1] 1 Pr [[1 (w) = 1] Pr [[2 (w) = 0] = = Pr [[(w) = 1] 1 + 2

Pr [[1 (w) = 1|[(w) = 1] =

since the Poisson random variables [1 and [2 are independent. As an application we can consider a Poissonean arrival flow of packets at a router with rate . If the packets are marked randomly with probability s = 1 , the resulting flow consists of two types, those marked and those not. Each of these flows is again a Poisson flow, the marked flow with rate 1 = s and the non-marked flow with 2 = (1 3 s). Actually, this procedure leads to a decomposition of the Poisson process into two independent Poisson processes and leads to the reverse of Theorem 7.3.4. 1 (iv) (a) Applying the solution of previous exercise immediately gives  + 1 2 +3 (b) Since the three Poisson processes are independent, the total number of cars on the three lanes, denoted by [, is also a Poisson process (Theorem 7.3.4) with rate  = 1 + 2 + 3 . q Hence, Pr [[ = q] = q! h3 . (c) Let us denote the Poisson process in lane m by [m . Then, using the independence between the [m , Pr [[1 = q> [2 = 0> [3 = 0] = Pr [[1 = q] Pr [[2 = 0] Pr [[3 = 0] =

q q 1 31 32 33 h h = 1 h3 h q! q!

(v) (a) The player relies on the fact that during the time there is exactly one arrival. Since the game rules mention that he should identify the last signal in (0> W ), signals arriving during (0> v) do not influence his chance to win because of the memoryless property of the Poisson process. The number of arrivals in the interval (v> W ) obeys a Poisson distribution with parameter  (W 3 v). The probability that precisely one signal arrives in the interval (v> W ) is Pr [Q (W ) 3 Q (v) = 1] =  (W 3 v) h3(W 3v) . (b) Maximizing this winning probability with respect to v (by equating the first derivative to zero) yields g Pr [Q (W ) 3 Q (v) = 1] = 3h3(W 3v) + 2 (W 3 v) h3(W 3v) = 0 gv with solution  (W 3 v) = 1 or v = W 3 1@. This maximum (which is readily verified by g2 checking that gv 2 Pr [Q (W ) 3 Q (v) = 1] ? 0) lies inside the allowed interval (0> W ). The maximum probability of winning is Pr [Q (W ) 3 Q (W 3 1@) = 1] = 1@h.

C.3 Poisson process (Chapter 7)

499

(vi) (a) We apply the general formula (7.1) for the pdf of a Poisson process with mean H [[(w)] = w = 1. Then, Pr [[ (w + v) 3 [ (v) = 0] = h3w = 1h . S 1 (b) Pr [[ (w + v) 3 [ (v) A 10] = 1 3 Pr [[ (w + v) 3 [ (v) $ 10] = 1 3 1h 10 n=0 n! . (c) Each minute is equally probable as follows from Theorem 7.3.3. (vii) This exercise is an application of randomly marking in a Poisson flow as explained in solution (iii) above. The total flow of packets can be split up into an ACK stream, a Poisson process Q1 with rate s = 3v31 and a data flow, an independent Poisson process Q2 with rate (1 3 s)  = 7v31 . Then, (a) Pr [Q1 A 1] = 1 3 Pr [Q1 = 0] = 1 3 h33 (b) The average number is H [Q1 + Q2 |Q1 = 5] = H [Q1 |Q1 = 5] + H [Q2 |Q1 = 5] = 5 + H [Q2 ] = 5 + 7 = 12 packets. (c) Pr [Q1 = 2|Q1 + Q2 = 8] =

Pr[Q1 =2>Q1 +Q2 =8] Pr[Q1 +Q2 =8]

=

32 h3 76 h7 2! 6! 108 h10 8!

E 29=65%

(viii) (a) Since the three Poisson arrival processes are independent, the total number of requests will also be a Poisson process with the parameter  = 1 + 2 + 3 = 20 requests/hour (Theorem 7.3.4). The expected number of requests during an 8-hour working day is H [Q] = w = 20 × 8 = 160 requests. (b) If we denote arrival processes of requests with dierent ADSL problems each with a random variable [l for l = 1> 2> and 3, then due to their mutual independence Pr [[1 = 0> [2 = n> [3 = 0] = Pr [[1 = 0] Pr [[2 = n] Pr [[3 = 0] = h31 w 8

from which Pr [[1 = 0> [2 = 3> [3 = 0] = h3 3

(2 w)n h32 w 33 w h n!

 6 3

6

h3 3

6

h3 3 = 1=7 × 1033 . 3! 20 (c) If we denote the total number of requests by [ then Pr [[ = 0] = h3w = h3 4 = 33 6=7 × 10 . (d) The precise time is irrelevant for Poisson processes, only the duration of the interval matters. Here intervals are overlapping and we need to compute the probability 3

s = Pr [{[ (0=2) = 1} K {[ (0=5) 3 [ (0=1) = 2}] =

1 [

Pr [{[ (0=1) = n} K {[ (0=2) 3 [ (0=1) = 1 3 n} K {[ (0=5) 3 [ (0=2)} = 1 + n]

n=0

=

1 [

Pr [[ (0=1) = n] Pr [[ (0=2) 3 [ (0=1) = 1 3 n] Pr [[ (0=5) 3 [ (0=2) = 1 + n]

n=0

=

1 [ n=0

h32

(2)n 32 (2)13n 36 (6)1+n h h = 48h310 = 2=18 × 1033 n! (1 3 n)! (1 + n)!

(e) Given that at the moment w + v there are n + p requests, the probability that there were n requests at the moment w is Pr [{[ (w) = n} K {[ (w + v) = n + p}] Pr [[ (w + v) = n + p] Pr [[ (w) = n] Pr [[ (w + v) 3 [ (w) = p] = Pr [[ (w + v) = n + p]

Pr [[ (w) = n|[ (w + v) = n + p] =

(w)n h3w (v)p h3v n! p! = ( (w + v))n+p h3(w+v) (n + p)! n + p  w n  v p = n w+v w+v

500

Solutions of problems

(ix) (a) The number of attacks that are arriving to the PC is a Poisson random variable [ (w) with rate  = 6. The probability that exactly one (n = 1) attack during one (w = 1) hour follows from (7.1) as Pr [[(1) = 1] = 6h36 . (b) Applying (7.2), the expected amount of time that the PC has been on is w = H[[(w)] =  60 = 10 hours. 6 (c) The arrival time of the fifth attack is denoted by W . Given that there are six attacks in one hour (w = 1), we compute the probability Pr[W ? w|[(1) = 6] that either five attacks arrive in the interval (0> w) and one arrives in (w> 1) or all six attacks arrive in (0> w) and none arrives in the interval (w> 1). Hence, for 0 $ w ? 1, IW (w) = Pr[W ? w|[(1) = 6] Pr[{[(w) = 5} K {[(1) = 6}] + Pr[{[(w) = 6} K {[(1) = 6}] Pr[[(1) = 6] Pr[[(w) = 5] Pr[[(1) 3 [(w) = 1] + Pr[[(w) = 6] Pr[[(1) 3 [(w) = 0] = Pr[[(1) = 6] =

=

((w)5 @5!)h3w (1 3 w)h3(13w) + ((w)6 @6!)h3w h3(13w) = 6w5 3 5w6 (6 @6!)h3

The probability that the fifth attack will arrive between 1:30 p.m. and 2 p.m. is IW (1) 3  7 IW 12 = 1 3 64 = 57 . 64 U (d) The expectation of W given [(1) = 6 follows from (2.33) as H [W |[(1)] = 01 {iW ({)g{

gIW (w) derived in (c). Alternatively, the expectation can be computed from where iW (w) = gw U   (2.35), H [W |[(1)] = 01 1 3 (6{5 3 5{6 ) g{ = 57 . Hence the expected arrival time of the fifth attack between 1 p.m. and 2 p.m. is about 1:43 p.m. (x) Let [ and [m denote the lifetime of system and subsystem m respectively. For a series {[ A w} and of subsystems with independent lifetimes [m is the event {[ A w} = Kq  m=1 m  T Pr [[ A w]. Recall with (3.32) that Pr [[ A w] = Pr min Pr [[ A w] = q m 1$m$q [m A w . m=1 Using the definition of the reliability function (7.5) then yields

Userie s (w) =

q \

Um (w)

m=1

(xi) The probability that the system V shown in Fig. 7.6 fails is determined by the subsystem with longest lifetime or [ = max1$m$q [m . Invoking relation (3.33) combined with the definition of the reliability function (7.5) leads to

Up a ra llel (w) = 1 3

q \

(1 3 Um (w))

m=1

C.4 Renewal theory (Chapter 8) (i) The equivalence {Q (w) A q} Ui {Zq $ w} indicates " [   Pr ZQ (w) $ { = Pr [{Zq $ {} K {Zq+1 A w}] q=0

= Pr [Z0 $ {> Z1 A w] +

" [ q=1

Pr [Zq $ {> Zq+1 A w]

C.4 Renewal theory (Chapter 8)

501

The convention Z0 = 0 reduces Pr [Z0 $ {> Z1 A w] = Pr [Z1 A w] = Pr [1 A w] = 1 3 I (w). Furthermore, by the law of total probability, ]

"

Pr [Zq $ {> Zq+1 A w] =

Pr [Zq $ {> Zq+1 A w|Zq = x] ]

0

g Pr [Zq $ x] gx gx

{

Pr [Zq+1 A w|Zq = x] g Pr [Zq $ x]

= 0

A renewal process restarts after each renewal from scratch (due to the stationarity and the independent increments of the renewal process). This implies that Pr [Zq+1 A w|Zq = x] = Pr [q+1 A w 3 x] = 1 3 I (w 3 x) because the interarrival times are i.i.d. random variables. Combined, " ] [   Pr ZQ (w) $ { = Pr [ A w] + q=1

]

{

Pr [ A w 3 x] g Pr [Zq $ x]

0

#

{

Pr [ A w 3 x] g

= Pr [ A w] + 0

" [

$ Pr [Zq $ x]

q=1

With the basic equivalence (8.6) and the definition (8.7) of the renewal function p(w), we arrive at ] {   Pr ZQ (w) $ { = Pr [ A w] + Pr [ A w 3 x] gp (x) 0

This equation holds for all {. If { = w, we can use the renewal equation, ]

]

w

w

Pr [ A w 3 x] gp (x) = p(w) 3 0

Pr [ $ w 3 x] gp (x) 0

= p(w) 3 p(w) + I (w)   which indeed confirms Pr ZQ (w) $ w = 1. (ii) The generating function of the number of renewals in the interval [0> w] is with (8.10) " l [ k Pr [Q(w) = n] } n *Q (w) (}) = H } Q (w) = n=0

]

w

#

= Pr [Q(w) = 0] + 0

] = Pr [Q(w) = 0] + } ] 0

$ Pr [Q(w 3 v) = n 3 1] } n

n=1 #" w [

0

= Pr [Q(w) = 0] + }

" [

i (v)gv

$ Pr [Q(w 3 v) = n] } n

i (v)gv

n=0 w

*Q (w3v) (}) gI (v)

From (8.6), we have that Pr [Q(w) = 0] = 1 3 I (w) and ] *Q (w) (}) = 1 3 I (w) + }

w

0

*Q (w3v) (}) gI (v)

By derivation with respect to }, we arrive at the dierential-integral equation for the derivative of the generating function, *0Q (w) (}) = =

]

]

w 0

*Q (w3v) (}) gI (v) + }

*Q (w) (}) 3 1 + I (w) }

w

0 ] w

+} 0

*0Q (w3v) (}) gI (v) *0Q (w3v) (}) gI (v)

502

Solutions of problems which reduces to the renewal equation (8.9) for } = 1 since *0Q (w) (1) = p(w). The second derivative ] w ] w 0 (}) = 2 * (}) gI (v) + } *00 *00  Q (w) Q (w3v) Q (w3v) (}) gI (v) 0

2 2 = *0Q (w) (}) 3 } }

0

]

]

w

*Q (w3v) (}) gI (v) + }

0

w

0

*00 Q (w3v) (}) gI (v)

evaluated at } = 1, is *00 Q (w) (1) = 2p(w) 3 2I (w) +

] 0

w

*00 Q (w3v) (1) gI (v)

The variance Var[Q(w)] follows from (2.27) as  2 0 0 Var[Q(w)] = *00 Q (w) (1) + *Q (w) (1) 3 *Q (w) (1) ] w = 3p(w) 3 p2 (w) 3 2I (w) + *00 Q (w3v) (1) gI (v) 0

(iii) Every time an IP packet is launched by TCP, a renewal occurs and the reward is that 2000 km are travelled, in each renewal, thus Uq = 2000 km. The speed in a trip that suers from congestion is, on average, 40 000 km/s, while the speed without congestion experience is 120 000 km/s. Since congestion only occurs in 1/5 cases, the average length (in s) of a renewal period is H [ ] =

2000 4 2000 1 7 × + × = 120000 5 40000 5 300

The average speed of an IP packet (in km/s) then follows from (8.20) as lim

w 105 and 106 iterations.  [uq ] of the relative error for q iterations versus the hops n are H [u104 ] = 0=12 H [u105 ] = 0=047 H [u106 ] = 0=017

 [u104 ] = 0=17  [u105 ] = 0=073  [u106 ] = 0=02

where the range of n values has been limited for q = 104 to 10 hops, for q = 105 to 11 hops, and for q = 106 to 12 hops. For larger hops, the simulations return zeros because the tail probability Pr [KQ A n] decreases as R (1@n!) and simulating such a rare event requires on average at least as many simulations as (Pr [KQ = n])31 . The table roughly shows

516

Solutions of problems

  that the average error over the non-zero returned values decreases as R I1q , which is in agreement with the Central Limit Theorem 6.3.1. Each iteration of the simulation can be regarded as an independent trial and the histogram sums in a particular way the number of these trials. (ii) Using (2.43), we have  2 2 0 Var [ZQ ] = *00 = *00 ZQ (0) 3 *ZQ (0) ZQ (0) 3 (H [ZQ ])  $2 # Q 31 Q 31 n [ [ g2 \ q(Q 3 q)  1 1 1 = 3  Q 3 1 n=1 g} 2 q=1 } + q(Q 3 q)  Q 3 1 q=1 q }=0

where H [ZQ ] is given in (16.18). The derivatives of the product j (}) =

n \

q(Q 3 q) } + q(Q 3 q) q=1

are elegantly computed via the logarithmic derivative gj(}) = j (}) g}   g2 j(}) g log j(}) 2 g2 log j(}) + j (}) . With derivative is g}2 = j (}) g} g} 2

g log j(}) . g}

The second

n n [ g [ 1 q(Q 3 q) g log j (}) = =3 log g} g} q=1 } + q(Q 3 q) } + q(Q 3 q) q=1 n [ g2 log j (}) 1 = 2 g} 2 q=1 (} + q(Q 3 q))

we obtain since j(0) = 1, Q 31 [ 1 Var [ZQ ] = Q 3 1 n=1

#

n [

q=1

$2 1 q(Q 3 q)

Q 31 [ n [ 1 1 + 3 Q 3 1 n=1 q=1 q2 (Q 3 q)2

# SQ 31

1 q=1 q

$2

Q 31 (C.5)

The first sum is Q 31 [ n=1

#

n [

q=1

and, with Q 31 [ n=1

#

$2 1 q(Q 3 q)

=

Q 31 [

n=1 q=1

SQ 31 Sn

n [

q=1

n=q

1 m=1 m(Q 3m)

$2 1 q(Q 3 q)

n [

=

Q 31 n [ [ 1 1 = q(Q 3 q) m=1 m(Q 3 m) q=1

SQ 31 Sn n=1

1 m=1 m(Q 3m)

3

SQ 31 Sn n=q

1 m=1 m(Q 3m)

q(Q 3 q)

Sq31 Sn n=1

1 m=1 m(Q 3m) ,

3 4 Q 31 [ q31 n n [ [[ 1 1 1 C D = 3 q(Q 3 q) n=1 m=1 m(Q 3 m) n=1 m=1 m(Q 3 m) q=1 4 3 Q 31 Q 31 Q 31 q31 q31 [ [ [ [ [ 1 1 1 C 13 1D = q(Q 3 q) m=1 m(Q 3 m) n=m m(Q 3 m) n=m q=1 m=1 Q 31 [

=

Q 31 [ q=1

+

Q 31 Q 31 q31 [ [ [ 1 1 1 1 3 q(Q 3 q) m=1 m (Q 3 q) m(Q 3 m) q=1 m=1

Q 31 [ q=1

q31 [ 1 1 q(Q 3 q) m=1 (Q 3 m)

C.10 The uniform recursive tree (Chapter 16) Furthermore, since Q 31 [ q=1

Sn

1 q=1 q(Q 3q)

Sn

1 q=1 q

1 Q

=

+

1 Q

SQ 31

1 q=Q 3n q ,

517

we have

q31 Q 31 q31 Q 31 [ [ 1 1 [ 1 [ 1 1 1 1 = + (Q 3 q) m=1 m(Q 3 m) Q q=1 (Q 3 q) n=1 n Q q=1 (Q 3 q)

=

1 Q

Q 31 [

1 m

m=1

Q[ 3m31 n=1

1 1 + n Q

Q 31 [

1 m

m=1

Q 31 [ n=m+1

Q 31 [ n=Q 3q+1

1 n

1 n

and Q 31 [ q=1

 q31 q31 Q 31  [ [ 1 [ 1 1 1 1 1 = + q(Q 3 q) m=1 (Q 3 m) Q q=1 q Q 3 q m=1 (Q 3 m) =

Q 31 1 [ 1 Q m=1 m

Q 31 [ n=Q 3m+1

Q 31 Q 31 1 [ 1 [ 1 1 + n Q m=1 m n=m+1 n

Hence, Q 31 [ n=1

#

n [

q=1

$2

#Q 31 $2 Q 31 [ 1 1 [ 1 3 q Q m=1 m q=1

Q[ 3m31

Q 31 Q 31 [ 1 [ 1 1 1 + n Q m n m=1 n=1 n=Q 3m+1 3 4 #Q 31 $2 Q 31 Q 31 Q 31 [ 1 [ 1 1 [ 1 C[ 1 2 D 3 = 3 Q q=1 q Q m=1 m n=1 n n=Q 3m n

1 q(Q 3 q)

2 = Q

+

Q 31 1 [ 1 Q m=1 m

Q 31 [ n=Q 3m+1

1 n

#Q 31 $2 Q 31 Q 31 [ 1 1 [ 1 1 1 2 [ 1 + = + Q q=1 q Q m=1 m Q 3 m Q m=1 m #Q 31 $2 Q 31 Q 31 [ 1 2 [ 1 2 [ 1 1 + + 2 = Q q=1 q Q m=1 m Q m=1 m

Q 31 [ n=Q 3m+1

Q 31 [ n=Q 3m+1

1 n

Substituted into (C.5) yields S Q 31 Var [ZQ ] = 3

1 q=1 q 2

2 2

(Q 3 1) Q SQ 31 Sn

SQ 31

1 m 1) Q 2

m=1

(Q 3

2 +

SQ 31 m=1

1 m

SQ 31

1 n=Q 3m+1 n

Q (Q 3 1)

1 q=1 q2 (Q 3q)2

n=1

+

+

Q 31

Further, Q 31 [

n [

n=1 q=1

q2 (Q

Q 31 Q 31 Q 31 [ [ [ 1 1 1 = 1= 2 2 2 2 (Q 3 q) 3 q) q (Q 3 q) q q=1 q=1 n=q

The partial fraction expansion of Q 31 [

1 q2 (Q 3q)

n [

n=1 q=1

q2 (Q

=

1 Q 2q

+

1 1 + Q 2 (Q Q q2 3q)

Q 31 Q 31 2 [ 1 1 [ 1 1 = 2 + 2 3 q) Q q=1 q Q q=1 q2

such that

1 n

518

Solutions of problems Combined, S Q 31 Var [ZQ ] = 3

1 q=1 q 2

2 4

+

(Q 3 1) Q

SQ 31

(Q

1 q=1 q 3 1) Q 2

+

Q 31 [ 2 1 Q (Q 3 1) m=1 m

Q 31 [ n=Q 3m+1

SQ 31 1 1 q=1 q2 + n Q(Q 3 1)

Invoking the identity Q 31 [ m=1

Q 31 Q 31 [ [ 1 1 1 = Q 3 m n=m n q2 q=1

(which can be verified by induction) yields Q 31 [ m=1

1 m

Q 31 [ n=Q 3m+1

Q 31 [ 1 1 = n Q 3m m=1

=

Q 31 [ m=1

Q 31 [ n=m+1

3 4 Q 31 Q 31 [ 1D 1 1 C[ 1 = 3 n Q 3 m n=m n m m=1

Q 31 Q 31 Q 31 Q 31 [ [ [ 2 [ 1 1 1 1 1 3 3 = 2 Q 3 m n=m n m (Q 3 m) q Q q=1 q q=1 m=1

Finally, we arrive at (16.19). (iii) The limit for Q < " of the probability generating function (16.17) of the weight ZQ of the shortest path l k *ZQ (}) = H h3}ZQ =

Q 31 \ n [ q(Q 3 q) 1 Q 3 1 n=1 q=1 } + q(Q 3 q)

will be derived from which the distribution then follows by taking the inverse Laplace transform. Since 3v 4 3v 4  2  2   Q Q Q Q C D C } + q(Q 3 q) = +}+ +}3 3q 3q D 2 2 2 2 u we have with | =

Q 2

2

+ },

n \

n n \ n!(Q 3 1)! \ q(Q 3 q) 1 1     = Q Q } + q(Q 3 q) (Q 3 n 3 1)! q=1 q=1 | + 2 3 q q=1 | 3 2 + q

The products can be written in terms of the Gamma function,   3n K |+ Q 2    =  Q K |+ Q q=1 | + 2 3 q 2   n +1 K |3 Q \ 1 2  =    Q K |3 Q +n+1 q=1 | 3 2 + q 2 n \

1

Thus,

*ZQ

    Q 31 K |3 Q + 1 Q[ K (n + 1) K | + 2 3 n 2     (}) = (Q 3 2)! K(Q 3 n) K | 3 Q + n + 1 K |+ Q n=1 2 2

C.10 The uniform recursive tree (Chapter 16) I Let the number of nodes be even Q = 2P such that | = P 2 + } ; P + |}| ? 2P). The sum, denoted by V, can be split as V=

=

519 } 2P

(provided

2P 31 P [ [ K (| + P 3 n) K (| + P 3 n) K (n + 1) K (n + 1) + K(2P 3 n) K (| 3 P + n + 1) K(2P 3 n) K (| 3 P + n + 1) n=1 n=P+1 P 31 [ m=0

P 31 [ K (P 3 m + 1) K (| + m) K (P + n + 1) K (| 3 n) + K(P + m) K (| 3 m + 1) K(P 3 n) K (| + n + 1) n=1

P 31 [

=

m=3(P 31)

K (| + m) K (P 3 m + 1) K(P + m) K (| 3 m + 1)

and *Z2P (}) = (2P 3 1)!

K (| 3 P + 1) 1 K (| + P) 2P 3 1

P 31 [ m=3(P 31)

K (| + m) K (P 3 m + 1) K(P + m) K (| 3 m + 1)

For large P, (2P 3 1)!

   }  K } +1 } K (| 3 P + 1)  ; (2P)3 2P K ; K (2P)  2P +1 } K (| + P) 2P K 2P + 2P

which suggests that we consider } < 2P} since then, using (Abramowitz and Stegun, 1968, Section 6.1.47), *Z2P (2P}) ; (2P)3} K (} + 1) 3}

; (2P)

1 2P 3 1

P 31 [ m=3(P 31)

      1 1 1+R 1+R P P

K (} + 1)

Hence, lim Q } *ZQ (Q}) = K (} + 1)

Q 100> and 200. Each simulation consisted of q = 106 iterations. The corresponding weight of the shortest and node-disjoint shortest path are drawn in Fig. C.10. The weight of the node-disjoint shortest path is evidently always larger than that of the shortest path in the same graph. Nevertheless, for large Q , the simulations suggest that both pdfs tend to each other.

522

Solutions of problems

shortest path node-disjoint shortest path

10

N = 50

N

fW (x)

1

0.1

N = 100

N = 200

0.01

0.001

0.0

0.1

0.2

0.3

0.4

0.5

x

Fig. C.10. Pdf of the weight of the shortest path (thin line) and the node-disjoint shortest path (bold line).

C.11 The e!ciency of multicast (Chapter 17) (i) Using (17.25), we obtain

jQ>n (2) = Q 3 1 3

G31 [

n

G3m

 Q 313

m=0

=

nm+1 31 n31

 Q 323

nm+1 31 n31



(Q 3 1)(Q 3 2)

(2Q 3 3)G (2Q 3 3) (2Q 3 3)GQ + 3 + (Q 3 1)(Q 3 2) (Q 3 1)(Q 3 2)(n 3 1) (Q 3 2)(n 3 1) Q(Q 3 1 3 2G) 1 (Q 3 1 3 2G) 3 3 3 (Q 3 1)(Q 3 2)(n 3 1) (Q 3 1)(Q 3 2)(n 3 1)2 (Q 3 2)(n 3 1)2

or, for large Q, jQ>n (2) ; 2G 3

3 +R n31



logn Q Q



the eective power exponent  W (Q) as defined in (17.32), equals for the n-ary tree and large Q ,   3 2G3 n1 log 1 G3 n1  W (Q) ; log 2 5 6 1 8  = 1 + log2 71 3 1 2(n 3 1) logn Q + logn (1 3 1@n) 3 n31   1 1 ;13 = 1 + log2 1 3 2(n 3 1)H[KQ ] (log 4)(n 3 1)H[KQ ] which shows, for large Q, that  W (Q) ? 1, but that  W (Q) < 1 if n < ".

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Index

n-ary tree, 387, 401, 414, 417, 423, 432, 522

Pareto, 56 Poisson, 40, 116, 129, 335 polynomial, 44, 348, 494 regular, 348, 362 uniform, 43, 74 Weibull, 55, 107, 132

adjacency matrix, 320, 471, 488 eigenvalues, 475 Bayes’ rule, 28, 197 Bene˘s’ equation, 261, 301 cell loss ratio (clr), 309, 512 Central Limit Theorem, 104, 148, 366, 377, 516 Cherno bound, 88 Chuang—Sirbu scaling law, 404—407 complete graph, 319, 321, 327, 347, 349, 359, 371, 373, 380, 392, 473, 481, 482, 488, 520 conditional distribution function, 28 conditional expectation, 34, 233, 341 conditional probability, 26 conditional probability density function, 28 correlation coe!cient, 30, 61, 67, 69, 74, 119 covariance, 29, 71, 78 matrix, 63, 66 degree graph, 323 degree of a node, 225, 322, 472 disjoint paths, 520 Merger’s Theorem, 327 distribution n-th order statistics, 53, 494 Bernoulli, 37 binomial, 38, 332, 488 Cauchy, 54 chi-square, 50 Erlang, 48, 125, 274, 278 exponential, 44, 75 extremal, 106 Fréchet, 107 Gamma, 48, 51, 125 Gaussian, 46, 103, 400 geometric, 39, 272 Gumbel, 54, 107, 400 joint Gaussian, 64 lognormal, 57, 77

Engset formula, 314, 511 Erlang B formula, 2, 280 Erlang C formula, 277 event, 10, 53 mutually exclusive, 10 failure rate, 131 flooding time, 362 giant component, 335, 337, 339 Google, 224 graph connectivity, 325, 486 edge connectivity, 326, 487 vertex connectivity, 326, 487 graph metrics betweenness, 329 clustering coe!cient, 328, 346, 513 diameter, 475 distortion, 329 expansion, 328 hopcount, 329 resilience, 329 histogram, 118, 494 hopcount, 340, 347, 354, 357, 387, 392, 403, 409, 418, 420, 423, 431, 513, 520 incidence matrix, 471 inclusion-exclusion formula, 12, 335, 391 sieve of Eratosthenes, 15 indicator function, 12, 17, 43, 321 inequality Boole, 15 Cauchy-Schwarz, 90, 91, 480 Chebyshev, 88 Gauss, 92

529

530

Index

Hölder, 90, 480 Jensen, 85, 342 Markov, 88 Minkowsky, 91 infinitesimal generator, 181 Laplacian (admittance matrix), 472, 486 law of rare events, 41, 128, 495 law of total probability, 27, 123, 142, 159, 204, 205, 238, 255, 274, 278, 284, 295, 324, 341, 367, 410, 422, 445, 495 level set of a tree, 352 Lindley’s equation, 255 link weight, 320, 340, 341, 347, 349, 359, 362, 373, 392, 406, 408 Little’s law, 267, 273, 275, 281, 287, 297, 508, 510 Markov chain absorbing states, 164 communicating states, 162 conservative, 181 continuous-time, 179 discrete-time, 158 embedded, 186, 188 hitting time, 163 irreducible Markov chain, 161, 226 periodic and aperiodic, 162 transient and recurrent states, 165 mean time to failure, 131 memoryless property, 27, 40, 45, 125, 132, 185, 351 Metcalfe’s law, 320 minimum spanning tree (MST), 373, 399 modes of convergence, 99 Newton identities for polynomials, 477 order statistics, 52, 127 PageRank (Google), 224 phase transition, 335, 376 Poisson arrivals see time averages (PASTA), 267, 274, 275, 283, 288, 312, 509, 511 Pollaczek-Khinchin equation, 286 power law, 325 probability density function (pdf), 16, 20, 22 joint, 28, 32 probability generating function (pgf), 18 logarithm of, 19, 25 moment generating function, 25, 235 process arrival, 248, 270 birth and death, 208, 304, 351 branching, 229, 342 geometric, 244 Poisson, 246, 345 counting, 263 Markov, 180, 253, 349 balance equation, 187 Chapman-Kolmogorov equation, 180

forward and backward equation, 182, 195 time reversibility, 196 nonhomogeneous Poisson, 129 Poisson, 120, 210 queueing, 250 renewal, 137 service, 249, 270 stochastic, 115 modeling, 117 Yule, 212, 230 quality of service (QoS), 2, 249, 283, 309, 340, 419 random graph, 330, 332, 337, 339, 346, 354, 362, 373, 374, 377, 387, 392, 403, 404, 406, 408, 410, 488, 513 random variable continuous, 20, 59 discrete, 16, 58 expectation, 17, 22 indepedent, 97, 104 independent, 28, 29, 32, 34, 47, 49, 51, 78 normalized, 31, 93, 400 random vector, 62 random walk, 202, 484 redundancy level, 325 regular graphs, 322, 328, 475, 482, 486, 488 reliability function, 131 renewal alternating renewal process, 153 Blackwell’s Renewal Theorem, 146 Elementary Renewal Theorem, 145, 170 inspection paradox, 152 Key Renewal Theorem, 146, 151 renewal equation, 141 renewal function, 140 renewal process, 137 renewal theory, 165 server placement problem, 419, 424, 429 shortest path, 340, 347 tree (SPT), 387, 392, 399, 407, 419, 428 slotted Aloha, 219 stochastic matrix, 159, 450, 451, 455, 473 total variation distance, 42 transition probability matrix, 159, 190, 201 spectral decomposition, 184 unfinished work, 256, 261, 263, 301 uniform recursive tree (URT), 354, 380, 392, 404, 407, 411, 417, 419, 422, 424 uniformization, 189 Wald’s identity, 34, 145, 154 web graph, 224, 323 Wigner’s Semicircle Law, 489