Lecture Notes in Computer Science Commenced Publication in 1973 Founding and Former Series Editors: Gerhard Goos, Juris Hartmanis, and Jan van Leeuwen
Editorial Board David Hutchison Lancaster University, UK Takeo Kanade Carnegie Mellon University, Pittsburgh, PA, USA Josef Kittler University of Surrey, Guildford, UK Jon M. Kleinberg Cornell University, Ithaca, NY, USA Alfred Kobsa University of California, Irvine, CA, USA Friedemann Mattern ETH Zurich, Switzerland John C. Mitchell Stanford University, CA, USA Moni Naor Weizmann Institute of Science, Rehovot, Israel Oscar Nierstrasz University of Bern, Switzerland C. Pandu Rangan Indian Institute of Technology, Madras, India Bernhard Steffen TU Dortmund University, Germany Madhu Sudan Microsoft Research, Cambridge, MA, USA Demetri Terzopoulos University of California, Los Angeles, CA, USA Doug Tygar University of California, Berkeley, CA, USA Gerhard Weikum Max Planck Institute for Informatics, Saarbruecken, Germany
6751
Khalid Al-Begain Simonetta Balsamo Dieter Fiems Andrea Marin (Eds.)
Analytical and Stochastic Modeling Techniques and Applications 18th International Conference, ASMTA 2011 Venice, Italy, June 20-22, 2011 Proceedings
13
Volume Editors Khalid Al-Begain University of Glamorgan, Faculty of Advanced Technology Pontypridd, CF37 1DL, UK E-mail:
[email protected] Simonetta Balsamo Andrea Marin Università Ca’ Foscari di Venezia Dipartimento di Scienze Ambientali, Informatica e Statistica Via Torino 155, 30172 Mestre - Venezia, Italy E-mail:{balsamo, marin}@dsi.unive.it Dieter Fiems Ghent University, Department TELIN Sint-Pietersnieuwstraat 41, 9000, Gent, Belgium E-mail: dieter.fi
[email protected] ISSN 0302-9743 e-ISSN 1611-3349 ISBN 978-3-642-21712-8 e-ISBN 978-3-642-21713-5 DOI 10.1007/978-3-642-21713-5 Springer Heidelberg Dordrecht London New York Library of Congress Control Number: 2011929030 CR Subject Classification (1998): D.2.4, D.2, D.2.8, D.4, C.2, C.4, H.3, F.1, G.3 LNCS Sublibrary: SL 2 – Programming and Software Engineering
© Springer-Verlag Berlin Heidelberg 2011 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Camera-ready by author, data conversion by Scientific Publishing Services, Chennai, India Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
Preface
It is our pleasure to present the proceedings of the 18th International Conference on Analytical and Stochastic Modelling and Applications (ASMTA 2011). The conference was held in the beautiful and historic city of Venice. This on its own is a great attraction but combined with the high-quality program, we hope it has been a memorable conference. ASMTA conferences have become established quality events in the calendar of analytical, numerical and even simulation experts in Europe and well beyond. In addition to regular participants from the main centers of expertise from the UK, Belgium, Germany, Russia, France, Italy, Latvia, Spain, Hungary and many other countries, we receive newcomers with interesting contributions from other countries such as Iran and the USA. The quality of this year’s program was exceptionally high. The conference committee was extremely selective, this year accepting only 24 papers. As ever, the International Program Committee reviewed the submissions critically and in detail, thereby helping in making the final decision as well as in providing the authors with useful comments to improve their papers. We would therefore like to thank every member of the International Program Committee for their time and efforts. We are very grateful for the generous support and sponsorship of the Universit´ a Ca’ Foscari Venezia for providing the magnificent venue and the administrative support. June 2011
Khalid Al-Begain Simonetta Balsamo Dieter Fiems Andrea Marin
Organization
Workshop Chairs Khalid Al-Begain Simonetta Balsamo
University of Glamorgan, UK Universit`a Ca’ Foscari Venezia, Italy
Program Chairs Dieter Fiems Andrea Marin
Ghent University, Belgium Universit` a Ca’ Foscari Venezia, Italy
Program Committee Ahmed Al-Dubai Imad Antonios Richard Boucherie Jeremy Bradley Giuliano Casale Sara Casolari Hind Castel Alexander Dudin Antonis Economou Jean-Michel Fourneau Rossano Gaeta Marco Gribaudo Andras Horvath Carlos Juiz Helen Karatza William Knottenbelt Remco Litjens Don McNickle Madjid Merabti Jos´e Ni˜ no-Mora Antonio Pacheco Balakrishna Prabhu Anne Remke Matteo Sereno Bruno Sericola Maria-Estrella Sousa-Vieira Janos Sztrik
Edinburgh Napier University, UK Southern Connecticut State University, USA University of Twente, The Netherlands Imperial College London, UK Imperial College London, UK University of Modena and Reggio Emilia, Italy Institut Telecom-Telecom SudParis, France Belarusian State University, Belarus University of Athens, Greece University of Versailles St-Quentin, France University of Turin, Italy Politecnico of Milano, Italy University of Turin, Italy Universitat de les Illes Balears, Spain Aristotle University of Thessaloniki, Greece Imperial College London, UK TNO Information and Communication Technology, The Netherlands University of Canterbury, New Zealand Liverpool John Moores University, UK Carlos III University of Madrid, Spain Instituto Superior Tecnico, Portugal Laas-CNRS, France University of Twente, The Netherlands University of Turin, Italy INRIA and University of Rennes, France University of Vigo, Spain University of Debrecen, Hungary
VIII
Organization
Mikl´ os Telek Nigel Thomas Petr Tuma Dietmar Tutsch Kurt Tutschku Benny Van Houdt Johan van Leeuwaarden Aad Van Moorsel Jean-Marc Vincent Sabine Wittevrongel Katinka Wolter Michele Zorzi
Technical University of Budapest, Hungary University of Newcastle, UK Charles University of Prague, Czech Republic University of Wuppertal, Germany University of Vienna, Germany University of Antwerp, Belgium Eindhoven University of Technology, The Netherlands Newcastle University, UK University of Grenoble, France Ghent University, Belgium Free University of Berlin, Germany University of Padova, Italy
External Referees Vlastimil Babka Leonardo Badia Nicola Bui Davide Cerotti Eline De Cuypere Thomas Demoor Michele Rossi Harald St¨ orrle Andrea Zanella
Charles University of Prague, Czech Republic University of Padova, Italy University of Padova, Italy University of Turin, Italy Ghent University, Belgium Ghent University, Belgium University of Padova, Italy Ludwig-Maximilians-Universit¨ at M¨ unchen, Germany University of Padova, Italy
Table of Contents
Queueing Theory I Departure Process in Finite-Buffer Queue with Batch Arrivals . . . . . . . . . Wojciech M. Kempa A Two-Class Continuous-Time Queueing Model with Dedicated Servers and Global FCFS Service Discipline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Willem M´elange, Herwig Bruneel, Bart Steyaert, and Joris Walraevens M/G/1 Queue with Exponential Working Vacation and Gated Service . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Zsolt Saffer and Mikl´ os Telek
1
14
28
Software and Computer Systems The Application of FSP Models in Automatic Optimization of Software Deployment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Omid Bushehrian
43
Model-Based Decision Framework for Autonomous Application Migration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Anders Nickelsen, Rasmus L. Olsen, and Hans-Peter Schwefel
55
Optimisation of Virtual Machine Garbage Collection Policies . . . . . . . . . . Simonetta Balsamo, Gian-Luca Dei Rossi, and Andrea Marin
70
Queueing Theory II Performance Evaluation of a Single Node with General Arrivals and Service . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Alexandre Brandwajn and Thomas Begin Tandem Queueing System with Different Types of Customers . . . . . . . . . . Valentina Klimenok, Che Soong Kim, and Alexander Dudin Fast Evaluation of Appointment Schedules for Outpatients in Health Care . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S. De Vuyst, H. Bruneel, and D. Fiems
85 99
113
X
Table of Contents
Statistics and Inference Technique of Statistical Validation of Rival Models for Fatigue Crack Growth Process and Its Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nicholas Nechval, Maris Purgailis, Uldis Rozevskis, Juris Krasts, and Konstantin Nechval
132
Suitability of the M/G/∞ Process for Modeling Scalable H.264 Video Traffic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Maria-Estrella Sousa-Vieira
149
Reconstructing Model Parameters in Partially-Observable Discrete Stochastic Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Robert Buchholz, Claudia Krull, and Graham Horton
159
Queueing Theory III Performance Evaluation of a Kitting Process . . . . . . . . . . . . . . . . . . . . . . . . Eline De Cuypere and Dieter Fiems
175
Perfect Sampling of Phase-Type Servers Using Bounding Envelopes . . . . Bruno Gaujal, Ga¨el Gorgo, and Jean-Marc Vincent
189
Monotone Queuing Networks and Time Parallel Simulation . . . . . . . . . . . . Jean-Michel Fourneau and Franck Quessette
204
Telecommunication Networks I Traffic-Centric Modelling by IEEE 802.11 DCF Example . . . . . . . . . . . . . . Paolo Pileggi and Pieter Kritzinger Flexible Traffic Engineering in Full Mesh Networks: Analysis of the Gain in Throughput and Cost Savings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gerhard Hasslinger Investigation of the Reliability of Multiserver Computer Networks . . . . . . Saulius Minkeviˇcius and Genadijus Kulvietis
219
234
249
Performance and Performability Performability Modeling of Exceptions-Aware Systems in Multiformalism Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Enrico Barbierato, Marco Gribaudo, Mauro Iacono, and Stefano Marrone
257
Table of Contents
The Impact of Input Error on the Scheduling of Task Graphs with Imprecise Computations in Heterogeneous Distributed Real-Time Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Georgios L. Stavrinides and Helen D. Karatza Simple Correlated Flow and Its Application . . . . . . . . . . . . . . . . . . . . . . . . . Alexander Andronov and Jelena Revzina
XI
273 288
Telecommunication Networks II A Practical Tree Algorithm with Successive Interference Cancellation for Delay Reduction in IEEE 802.16 Networks . . . . . . . . . . . . . . . . . . . . . . . Sergey Andreev, Eugeny Pustovalov, and Andrey Turlikov
301
A Probabilistic Energy-Aware Model for Mobile Ad-Hoc Networks . . . . . Lucia Gallina, Sardaouna Hamadou, Andrea Marin, and Sabina Rossi
316
Size-Based Flow-Scheduling Using Spike-Detection . . . . . . . . . . . . . . . . . . . Dinil Mon Divakaran, Eitan Altman, and Pascale Vicat-Blanc Primet
331
Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
347
Departure Process in Finite-Buffer Queue with Batch Arrivals Wojciech M. Kempa Silesian University of Technology, Institute of Mathematics, ul. Kaszubska 23, 44-100 Gliwice, Poland, Phone/Fax: + 48 32 237 28 64
[email protected] Abstract. A finite-buffer queueing system with batch Poisson arrivals is considered. A system of integral equations for the distribution function of the number of customers h(t) served before t, conditioned by the initial state of the system, is built. A compact formula for probability generating function of the Laplace transform of distribution of h(t) is found using the potential technique. From this representation the mean of h(t) can be effectively calculated numerically using one of the inverse Laplace transform approximation algorithms. Moreover a limit behavior of departure process as the buffer size tends to infinity is investigated. Numerical examples are attached as well. Keywords: Departure process; Finite-buffer queue; Numerical inverse Laplace transform; Poisson arrivals; Potential method.
1
Introduction
Finite-buffer queueing systems are intensively investigated nowadays due to their applications in modelling of Internet routers. Of special significance is the analysis of systems in transient state i.e. at fixed moment t. We are often interested in the investigation of key system characteristics during a short period of time only since, because of permanent changing of Internet traffic, the notion of ”stationary state” in reality does not occur. Besides, especially the analysis of systems with batch arrivals is desired and expected from the practical point of view. In real Internet traffic the incoming packets have different sizes measured in bytes, thus treating a byte as a unit we can consider a stream of different-sized packets as a stream of single one-byte packets arriving in groups of random sizes. One of the essential characteristics determining the efficiency (the ”function”) of the service process with respect to the intensity of the arrival stream is departure process h(t), that at any fixed moment t takes on a random value equal to the number of packets completely served before t. In the paper we investigate this characteristic in the M X /G/1/N system with Poisson batch arrivals. The main goal of the analysis is to find an explicit, compact representation for probability generating function of the Laplace transform of the distribution of the departure counting process, conditioned on the number of packets present K. Al-Begain et al. (Eds.): ASMTA 2011, LNCS 6751, pp. 1–13, 2011. c Springer-Verlag Berlin Heidelberg 2011
2
W.M. Kempa
in the system at t = 0. According to the best knowledge of the author such a result is new in finite-buffer systems analysis. In any case the bibliography on transient departure processes is rather modest. In [6] a steady-state joint density function of k successive departure intervals is derived for the M/G/1/N system. Besides the problem of correlation of such intervals is discussed. Some problems connected with the service process in M X /G/1/N are solved in [2]. In particular, a joint distribution function of the length of the first lost series of batches and the time of the first loss is found. An interesting technique of investigation of finite-buffer systems is developed in [7]. The approach is based on resolvent sequences of the process being the difference of a compound Poisson process and a compound general-type renewal process. An overview of results related to finite-buffer systems with Markovian input streams can be found in [13] and [3]. Transient results for the departure process in the case of the system with batch arrivals and infinite queue can be found in [8] and [9], [10], where additionally multiple and single vacation policies are analyzed. In the article we use the technique of potential introduced in [11] and developed in [12]. Some details connected with this approach are stated in the next Section 2 where a precise system description is given as well. In Section 3 we obtain the main result i.e. the explicit representation for the expression ∞ ∞ zm e−st P{h(t) = m | ξ(0) = n}dt, (1) m=0
0
where 0 ≤ n ≤ N, t > 0, |z| ≤ 1, Re(s) > 0, and ξ(0) denotes the number of packets present in the system at the opening. In Section 4 we derive a limit representation for 2-fold transform of departure process as the buffer size tends to infinity. The last Section 5 contains sample numerical results for the mean of departure process obtained using one of the algorithms for numerical Laplace transform inversion.
2
System Description and Potential Approach
In this article we consider the M X /G/1/N queueing system in that the incoming batches of packets arrive according to a Poisson process with intensity λ and the size of the arriving batch is k with probability pk , ∞ k=1 pk = 1. Packets are served individually with a distribution function F (·). The system capacity is assumed to be N i.e. we have N − 1 places in the buffer queue and one place for service. To complete necessary notation let us put ∞ ∞ e−st dF (t), Re(s) > 0, p(θ) = θk pk , |θ| ≤ 1. (2) f (s) = 0
k=1
By pi∗ j we denote the jth term of the i-fold convolution of the sequence (pk ) with itself. Besides, let δi,j denotes the Kronecker delta function. The notion of potential of a random walk continuous from below was introduced in [11] (see also [12]). Here we state only selected elements of potential
Departure Process in Finite-Buffer Queue with Batch Arrivals
3
theory, which will be used later. Originally (see [11]) the potential is defined as follows. Introduce a discrete-time random walk Yn in the following way: Y0 = 0,
n
Yn =
Xk ,
n = 1, 2, ...,
(3)
k=1
where Xn are independent and identically distributed random variables with a discrete distribution rk = P{Xn = k}, k = −1, 0, 1, ..., where r−1 > 0. A sequence (Rk )∞ k=0 satisfying condition ∞
θ k Rk =
k=0
1 , P (θ) − 1
|θ| < 1,
(4)
k where P (θ) = ∞ k=−1 θ rk is called a potential of random walk Yn or, simply, a potential of the sequence (rk ). Thus, the potential (Rk ) of the sequence (rk ) is such a sequence the generating function of which can be written down using the generating function of the sequence (rk ) as in (4). In the analysis of finite-buffer queue characteristics the following property of sequence (Rk )∞ k=0 can be applied ([11]): ∞ Theorem 1. Introduce two number sequences (αn )∞ n=0 and (ψn )n=1 with the assumption α0 = 0. The general solution of the following boundary problem: n
αk+1 un−k − un = ψn ,
n≥0
(5)
n ≥ 0,
(6)
k=−1
can be written in the form un = CRn+1 +
n
Rn−k ψk ,
k=0
where C is a constant independent on n and Rk is the potential connected with sequence (αn ) i.e. ∞ k=0
θk Rk =
1 , Pα (θ) − 1
Pα (θ) =
∞
θ k αk+1 , |θ| < 1
(7)
k=−1
and, besides, Rk can be found in the following recurrent way: R0 = 0,
R1 = α−1 0 ,
Rk+1 = α−1 0 (Rk −
k
αi+1 Rk−i ),
k ≥ 1.
(8)
i=0
Theorem 1 is essential for the analysis. Its importance lies in the fact that it allows to write down in a compact form the general solution of a specific-type system of equations, which appears in the transient investigation of finite-buffer queues with Poisson arrivals. A detailed proof of this theorem can be found in [11]. Theorem 1, perhaps in a slightly different form, was applied e.g. in [2], [3] and [4].
4
3
W.M. Kempa
System of Integral Equations for Departure Process
In this section we will write the system of integral equations for conditional distributions of departure process h(t) and obtain the explicit formula for probability generating function of Laplace transform of probabilities P{h(t) = m | ξ(0) = n}. Introduce the following notation: Hn (t, m) = P{h(t) = m | ξ(0) = n},
0 ≤ n ≤ N, m ≥ 0, t > 0.
(9)
Note that the following equation is true: H0 (t, m) = λ
t N−1 0
+
∞
pk Hk (t − y, m)
k=1
pk HN (t − y, m) e−λy dy + δm,0 e−λt .
(10)
k=N
Indeed, if the system is initially empty the first summand in (10) relates to the situation in that the first group of packets enters before t. The number of packets in the system after the first arrival differs in dependence on the first group size, and for groups of size k ≥ N reaches the maximum level N. In the second summand the first arrival epoch is after t, so h(t) = m if and only if m = 0. As it is well known (see e.g. [5]) departure epochs are Markov moments in the queueing system with Poisson arrivals. Hence the formula of total probability used with respect to the first (after t = 0) departure epoch x > 0 leads to the following system of integral equations: Hn (t, m) =
t N−n−1 0
+
ak (x)Hn+k−1 (t − x, m − 1)
k=0
ak (x)HN −1 (t − x, m − 1) dF (x) + 1 − F (t) δm,0 ,
∞
(11)
k=N −n
where 1 ≤ n ≤ N, and ak (x) =
k i=0
pi∗ k
(λx)i −λx e , i!
k ≥ 0, p0∗ 0 =1
(12)
denotes the probability that exactly k customers enters to the system before time x. The interpretation of both sides of (11) is similar to that in the case of (10). The first summand in the integral on the right in (11) denotes the situation in that the first departure epoch occurs at time x < t. If exactly k packets have entered until x then, since x is a Markov moment, we can consider x as the initial moment of the system starting with n + k − 1 packets (k arrivals till x and one
Departure Process in Finite-Buffer Queue with Batch Arrivals
5
departure), and the event {h(t) = m} will become {h(t − x) = m − 1}. In the second summand in the integral on the right in (11) the number of packets which arrive before the first departure epoch x equals at least N − n, so N − n packets are enqueued only (to saturate the buffer), the remaining ones are lost. Thus, after the first departure epoch the number of packets in the system is N − 1. The last summand on the right side in (11) describes the situation in that the first departure occurs after t. Introduce the Laplace transform of Hn (t, m) with respect to argument t as follows: ∞ n (s, m) = H e−st Hn (t, m)dt, Re(s) > 0. (13) 0
Now (10)–(11) lead to 0 (s, m) = H
N −1 ∞ λ δm,0 k (s, m) + H N (s, m) pk H pk + , λ+s λ+s k=1
n (s, m) = H
N −n−1
∞
n+k−1 (s, m − 1) + H N −1 (s, m − 1) ak (s)H
k=0
+
(14)
k=N
ak (s)
k=N−n
1 − f (s) δm,0 , s
1 ≤ n ≤ N,
where
(15)
ak (s) =
∞
e−sx ak (x)dF (x).
(16)
0
To eliminate dependence on m let us now introduce a generating function of n (s, m) of the form H ∞
n (s, z) = H
n (s, m), z mH
|z| ≤ 1.
(17)
m=0
Equations (14)–(15) will take now the form
0 (s, z) = H
N −1 ∞ λ 1
k (s, z) + H
N (s, z) pk H pk + , λ+s λ+s k=1
n (s, z) = H
N −n−1
n+k−1 (s, z) + H
N −1 (s, z) αk (s, z)H
k=0
+
1 − f (s) , s
(18)
k=N
∞
αk (s, z)
k=N −n
1 ≤ n ≤ N,
(19)
where αk (s, z) = z ak (s). Now let us make the following substitution:
N −n (s, z), Dn (s, z) = H
0 ≤ n ≤ N.
(20)
6
W.M. Kempa
Introducing (20) into (18)–(19) gives DN (s, z) = n
N −1 ∞ λ 1 pk DN −k (s, z) + D0 (s, z) pk + , λ+s λ+s k=1
(21)
k=N
αk+1 (s, z)Dn−k (s, z) − Dn (s, z) = ψn (s, z),
0 ≤ n ≤ N − 1,
(22)
1 − f (s) . s
(23)
k=−1
where ∞
ψn (s, z) = αn+1 (s, z)D0 (s, z) − D1 (s, z)
αk (s, z) −
k=n+1
Note that the system (22) has the form stated in (5). The only difference is in that functions depend on arguments s and z. The general solution of (22) can be written in the form (compare (6)) Dn (s, z) = C(s, z)Rn+1 (s, z) +
n
n ≥ 0,
Rn−k (s, z)ψk (s, z),
(24)
k=0
where (see (8)) R0 (s, z) = 0,
R1 (s, z) = α0 (s, z)−1 ,
Rk+1 (s, z) = α0 (s, z)−1 (Rk (s, z) −
k
αi+1 (s, z)Rk−i (s, z)),
k ≥ 1.
(25)
i=0
Of course, taking into consideration that ∞ ∞ ∞ θk αk+1 (s, z) = z θk e−sx ak+1 (x)dF (x) k=−1
0
k=−1
=z
∞
θk
0
k=−1
=
= =
z θ z θ z θ
∞
k+1
pi∗ k+1
i=0
∞
(λx)i −λx e dF (x) i!
∞ ∞ (λx)i i=0
∞
0
e−sx
e−(λ+s)x dF (x)
0
∞
i!
k+1 pi∗ k+1 θ
k=i−1
i ∞ λxp(θ) −(λ+s)x e dF (x) i! i=0 e−(λ+s)x eλxp(θ) dF (x) =
0
z f λ 1 − p(θ) + s , θ
we get from (7) R(θ, s, z) =
∞ k=0
θ k Rk (s, z) =
θ , zf λ 1 − p(θ) + s − θ
|θ| < 1.
(26)
Departure Process in Finite-Buffer Queue with Batch Arrivals
7
Now we must find the representation for C(s, z). Substituting n = 0 to (24) leads to ⇐⇒
D0 (s, z) = C(s, z)R1 (s, z)
C(s, z) = D0 (s, z)α0 (s, z).
(27)
Taking n = 0 in (22) we obtain α0 (s, z)D1 (s, z) + α1 (s, z)D0 (s, z) − D0 (s, z) = ψ0 (s, z), that, since ∞ k=0 αk (s, z) = zf (s), gives D1 (s, z) =
1 1 − f (s) D0 (s, z) − . zf (s) s
(28)
Introduce the following functionals: bn (s, z) =
n k 1 Rn−k (s, z) αi (s, z) zf (s) i=0
(29)
k=0
and χn (s, z) =
n
Rn−k (s, z)αk+1 (s, z) −
k=−1
n
Rn−k (s, z) + bn (s, z).
(30)
k=0
Now, substituting n = N to (24) and using (23), (27) and (28) we get N N
DN (s, z) = D0 (s, z) RN−k (s, z)αk+1 (s, z) − RN −k (s, z) + bN (s, z) k=−1
−
k=0
1 − f (s) 1 − f (s) bN (s, z) = D0 (s, z)χN (s, z) − bN (s, z). s s
(31)
Similarly, introducing (23), (24), (27) and (28) into (21), after a little laborious calculations we obtain DN (s, z) =
−1 ∞ N λ D0 (s, z) pN−k χk (s, z) + pk + GN −1 (s, z), λ+s k=1
(32)
k=N
where N −1 k i λ 1 − f (s) 1 GN−1 (s, z) = − pN −k Rk−i (s, z) αj (s, z). (33) λ + s (λ + s)szf (s) i=0 j=0 k=1
Comparing the right sides of (31) and (32) we eliminate D0 (s, z) as follows:
N (s, z) D0 (s, z) = H
GN −1 (s, z) + 1 − f (s) s−1 bN (s, z) = N −1 . ∞ χN (s, z) − λ(λ + s)−1 k=1 pN −k χk (s, z) + k=N pk
(34)
8
W.M. Kempa
Now from (24) (applying (29) and (30)) we get for 1 ≤ n ≤ N
N −n (s, z) = D0 (s, z)χn (s, z) − 1 − f (s) bn (s, z) Dn (s, z) = H s
(35)
and hence, after operation on indexes, we obtain
n (s, z) = H
N (s, z)χN−n (s, z) − 1 − f (s) bN −n (s, z), H s
0 ≤ n ≤ N − 1. (36)
Collecting formulae (34) and (36) we can state the following main theorem: Theorem 2. For any |z| ≤ 1 and Re(s) > 0 the following representations hold true: χN−n (s, z) GN −1 (s, z) + 1 − f (s) s−1 bN (s, z)
n (s, z) = H N −1 ∞ χN (s, z) − λ(λ + s)−1 k=1 pN −k χk (s, z) + k=N pk −
1 − f (s) bN −n (s, z), s
0 ≤ n ≤ N − 1,
(37)
−1 G (s, z) + 1 − f (s) s bN (s, z) N −1
N (s, z) = H N −1 . ∞ χN (s, z) − λ(λ + s)−1 k=1 pN−k χk (s, z) + k=N pk
4
(38)
Limit Theorem
This section is devoted to the limit behavior of 2-fold transform of departure process in the case of infinitely increasing buffer size i.e. as N → ∞. We start with the formula (24). Taking into consideration (27) we have Dn (s, z) = D0 (s, z)α0 (s, z)Rn+1 (s, z) +
n
Rn−k (s, z)ψk (s, z),
n ≥ 0. (39)
k=0
Multiplying (39) by θn and, under the assumption N → ∞, taking a sum over n from 0 to ∞ we obtain, along the way applying identity (26) ∞
θn Dn (s, z) = D0 (s, z)α0 (s, z)θ−1
n=0 ∞
+
k=0
θ k ψk (s, z)
∞
θ n+1 Rn+1 (s, z)
n=0
θn−k Rn−k (s, z)
n=k
= D0 (s, z)α0 (s, z)θ
∞
−1
∞ R(θ, s, z) − R0 (s, z) + R(θ, s, z) θk ψk (s, z), k=0
where |θ| < 1. Using in (40) representations (23) and (28) we derive
(40)
Departure Process in Finite-Buffer Queue with Batch Arrivals ∞
9
θ n Dn (s, z)
n=0 ∞
= R(θ, s, z) D0 (s, z) θk αk+1 (s, z) − k=−1
−
1 − f (s) szf (s)
∞ k=0
θk
k
∞ k 1 1 k + θ αi (s, z) 1 − θ zf (s) i=0 k=0
αi (s, z) − D0 (s, z)θ−1 ,
|θ| < 1.
(41)
i=0
Taking into consideration the formula (26) we get from (41) for |θ| < 1 ∞
θn Dn (s, z)
n=0
=
D0 (s, z)
U (λ,s,z) z(1−θ)f (s)+θ −θf (s) θ(1−θ)f (s)
−
U(λ,s,z)θ 1−f (s) sf (s)(1−θ)
zU (λ, s, z) − θ where we denote U (λ, s, z) = f λ 1 − p(θ) + s . Let us take into consideration the following equation: zf λ 1 − p(θ) + s − θ = 0.
− D0 (s, z)θ−1 , (42)
(43)
Treat the left side as a function Q of variable θ, for fixed 0 < z < 1 and s such that Im(s) = 0, Re(s) > 0. Substituting θ = 0 we have Q(0) = zf (λ + s) > 0, and similarly taking θ = 1 we get Q(1) = zf (s) − 1 < z − 1 < 0. Since Q (θ) = −zλp (θ)f λ 1 − p(θ) + s < 0, thus there exists a unique root θ0 = θ0 (λ, z, s) of (43) between 0 and 1. The series on the left side of (42) is convergent in particular for 0 < θ < 1, hence the numerator of the expression on the right side of (42) should be 0 at θ0 . Substituting θ0 to the numerator of the expression on the right side of (42) we eliminate D0 (s, z) as follows: θ0 1 − f (s) . D0 (s, z) = (44) s zf (s)(1 − θ0 ) + θ0 − f (s)
N (s, z) we can now formulate Since for N → ∞ we have D0 (s, z) = limN →∞ H the following limit theorem: Theorem 3. For conditional 2-fold transform of departure process as the initial number of packets tends to infinity the following limit representation holds: ∞ ∞
N (s, z) = lim H zm e−st P{h(t) = m | ξ(0) = ∞}dt N →∞
m=0
0
θ0 1 − f (s) , = s zf (s)(1 − θ0 ) + θ0 − f (s)
(45)
10
W.M. Kempa
where 0 < z < 1, s ∈ R+ and θ0 = θ0 (λ, s, z) is a unique root of equation zf λ 1 − p(θ) + s − θ = 0 in (0, 1).
5
Numerical Results
In this section we present sample numerical results for conditional mean of departures before fixed t i.e. for the expression En h(t), where the symbol En stands for the mean on condition ξ(0) = n. In computations we use firstly the following obvious identity: ∞
n (s, z) e−st En h(t)dt = H (46) z=1
0
and next the algorithm of approximate numerical Laplace transform inversion introduced in [1]. The base of the algorithm is the Bromwich inversion integral that finds value of the function g at fixed t > 0 from its transform g as δ+i∞ 1 g(t) = est g(s)ds, (47) 2πi δ−i∞ where δ ∈ R is located on the right to all singularities of g . Using trapezoidal method with step h to estimate the integral in (47) we obtain the approximation gh (t) of g(t) in the form gh (t) =
∞ heδt heδt ikht g g(δ) + Re e g (δ + ikht) . 2π π
(48)
k=1
π Setting h = Lt and δ = series representation:
A , 2Lt
after algebraic calculations, we reach the following
gh (t) = gA,L (t) =
∞
(−1)k qk (t),
(49)
k=0
where qk (t) =
eA/2L ωk (t), 2Lt
k ≥ 0,
(50)
L
A A ijπ ijπ/L ω0 (t) = g +2 Re g + e 2Lt 2Lt Lt
(51)
j=1
and ωk (t) = 2
A ijπ ikπ ijπ/L Re g + + e , 2Lt Lt t j=1
L
k ≥ 1.
(52)
Departure Process in Finite-Buffer Queue with Batch Arrivals
11
Let us take into consideration the following Euler formula for approximate summation of alternating series: ∞ m n+k m 1 k (−1) ck ≈ (−1)j cj , (53) k 2m j=0 k=0
k=0
where parameter m is usually of order of dozen and parameter n - of order of several dozen ([3]). Applying (53) in (48) we finally obtain g(t) ≈
m n+k m 1 (−1)j qj (t), k 2m j=0
(54)
k=0
where typical values of parameters are following (see [1]): m = 38, n = 11, A = 19, L = 1. A precise evaluation of the estimation error in (54) is not possible, first of all, due to the lack of such an estimation for the Euler summation formula (53). The practice has shown that a good evaluation of the error one can get by executing the calculation twice, and changing by 1 one of the parameters. Then the difference between the results, e.g. for A = 19 and A = 20 provides a good evaluation of the error estimation. More details can be found in [1] and [3]. Let us consider the system of size N = 4 in that individual service times have Erlang distributions with parameters n = 2 (shape) and μ = 1 (rate) i.e. F (t) = 1 − e−t (t + 1),
t > 0,
thus the mean service time is 2. Let us take into consideration two different simple batch size distributions: p2 = 1 (each arriving batch consists of 2 packets) and p1 = p2 = 12 . The system with the second possibility we denote by M 1,2 /G/1/N. Below we present a numerical comparison of values E0 h(t) and EN h(t), computed for t = 20, 50, 100, 200, 500 and 1000 for these two systems. Possibilities of underloaded (ρ < 1) and overloaded system (ρ ≥ 1) are presented separately. Each time values of the arrival rate parameter λ are selected in such a way to fix the same value of ρ for both systems. In the table below (Table 1) we present results obtained for λ1 = 0.15 (for M 2 /G/1/N queue) and λ1,2 = 0.20 (for M 1,2 /G/1/N queue) that gives = 0.6 < 1 in both systems. Table 1. Conditional means of departures before t for ρ = 0.6 (underloaded systems) t 20 50 100 200 500 1000
M 2 /G/1/N E0 {h(t)} EN {h(t)} 4.299 7.133 12.204 15.059 25.384 28.240 51.745 54.600 130.827 133.682 262.630 265.486
M 1,2 /G/1/N E0 {h(t)} EN {h(t)} 4.479 7.334 12.632 15.523 26.231 29.121 53.427 56.318 135.016 137.907 270.998 273.889
12
W.M. Kempa
Table 2. Conditional means of departures before t for ρ = 2.4 (overloaded systems) t 20 50 100 200 500 1000
M 2 /G/1/N E0 {h(t)} EN {h(t)} 8.559 9.495 22.997 23.933 47.061 47.996 95.187 96.123 239.568 240.503 480.202 481.138
M 1,2 /G/1/N E0 {h(t)} EN {h(t)} 8.798 9.601 23.470 24.273 47.922 48.726 96.828 97.631 243.545 244.348 488.073 488.876
The next table present similar results for λ1 = 0.6 and λ1,2 = 0.8 (so for ρ = 2.4 > 1 in both systems). Considering the above results one can observe that the average number of departures is greater for the M 1,2 /G/1/N system (for the same ρ). But the difference is the smaller, the greater is traffic load ρ. Similarly, of course, the mean of h(t) is greater in the case of the opening with maximum number of customers present than in the case of the system empty at t = 0. The difference decreases as t increases.
References 1. Abate, J., Choudhury, G.L., Whitt, W.: An introduction to numerical transform inversion and its application to probability models. In: Grassmann, W. (ed.) Computational Probability, pp. 257–323. Kluwer, Boston (2000) 2. Bratiichuk, M.S., Chydzinski, A.: On the loss process in a batch arrival queue. App. Math. Model. 33(9), 3565–3577 (2009) 3. Chydzinski, A.: Queueing characteristics for Markovian traffic models in packetoriented networks. Silesian University of Technology Press, Gliwice (2007) 4. Chydzinski, A.: On the distribution of consecutive losses in a finite capacity queue. WSEAS Transactions on Circuits and Systems 4(3), 117–124 (2005) 5. Cohen, J.W.: The single server queue. North-Holand Publishing Company, Amsterdam (1982) 6. Ishikawa, A.: On the joint distribution of the departure intervals in an ”M/G/1/N” queue. J. Oper. Res. Soc. Jpn. 34(4), 422–435 (1991) 7. Kadankov, V., Kadankova, T.: Busy period, virtual waiting time and number of the customers in Gδ |M κ |1|B system. Queueing Syst 65(2), 175–209 (2010) 8. Kempa, W.M., et al.: The departure process for queueing systems with batch arrival of customers. Stoch. Models 24(2), 246–263 (2008) 9. Kempa, W.M.: On one approach to study of departure process in batch arrival queues with multiple vacations. In: Applied stochastic models and data analysis. ASMDA-2009. The XIIIth International Conference pp. 203–207. Vilnius Gediminas Technical University, Vilnius (2009)
Departure Process in Finite-Buffer Queue with Batch Arrivals
13
10. Kempa, W.M.: Some new results for departure process in the M X /G/1 queueing system with a single vacation and exhaustive service. Stoch. Anal. Appl. 28(1), 26–43 (2010) 11. Korolyuk, V.S.: Boundary-value problems for complicated Poisson processes. Naukova Dumka, Kiev (1975) (in Russian) 12. Korolyuk, V.S., Bratiichuk, M.S., Pirdzhanov, B.: Boundary-value problems for random walks. Ylym, Ashkhabad, (1987) (in Russian) 13. Takagi, H.: Queueing Analysis, vol. 2. North-Holland, Amsterdam (1993)
A Two-Class Continuous-Time Queueing Model with Dedicated Servers and Global FCFS Service Discipline Willem M´elange, Herwig Bruneel, Bart Steyaert, and Joris Walraevens Department of Telecommunications and Information Processing, Ghent University - UGent {wmelange,hb,bs,jw}@telin.Ugent.be http://telin.ugent.be/
Abstract. This paper considers a continuous-time queueing model with two types (classes) of customers each having their own dedicated server. The system adopts a “global FCFS” service discipline, i.e., all arriving customers are accommodated in one single FCFS queue, regardless of their types. As a consequence of the “global FCFS” rule, customers of one type may be blocked by customers of the other type, in that they may be unable to reach their dedicated server even at times when this server is idle, i.e., the system is basically non-workconserving. One major aim of the paper is to estimate the negative impact of this phenomenon on the (mean) system occupancy and mean system delay. For this reason, the systems with and without “global FCFS” are studied and compared. The motivation of our work are systems where this kind of blocking is encountered, such as input-queueing network switches or road splits. Keywords: queueing; blocking; global FCFS; Markov; non-workconserving.
1
Introduction
In general, queueing phenomena occur when some kind of customers, desiring to receive some kind of service, compete for the use of a service facility (containing one or multiple servers) able to deliver the required service. Most queueing models assume that a service facility delivers exactly one type of service and that all customers requiring this type of service are accommodated in one common queue. If more than one service is needed, multiple different service facilities can be provided, i.e., one service facility for each type of service, and individual separate queues are formed in front of these service facilities. In all such models, customers are only hindered by customers that require exactly the same kind of service, i.e., that compete for the same resources. In some applications, it may not be physically feasible or desirable to provide separate queues for each type of service that customers may require, and it may be necessary or desirable to accommodate different types of customers (i.e., customers requiring different types of service) in the same queue. In such K. Al-Begain et al. (Eds.): ASMTA 2011, LNCS 6751, pp. 14–27, 2011. c Springer-Verlag Berlin Heidelberg 2011
A Two-Class Continuous-Time Queueing Model
15
cases, customers of one type (i.e., requiring a given type of service) may also be hindered by customers of other types. For instance, if a road or a highway is split in two or more subroads leading to different destinations, cars on that road heading for destination A may be hindered or even blocked by cars heading for destination B, even when the subroad leading to destination A is free, simply because cars that go to B are in front of them. In other words, there is a first-come-first-served (FCFS) order on the main road. This blocking also takes place in weaving sections on highways albeit to a lesser extent [1,2]. We refer to [3,4] for a general overview and validation of the modelling of traffic flows with queueing models. Similarly, in switching nodes of telecommunication networks, information packets with a given destination of node A may have to wait for the transmission of packets destined to node B that arrived earlier, even when the link to node A is free, if the arriving packets are accommodated in so-called input queues according to the source from which they originate (the well-known HOL-blocking effect, see [5,6,7,8,9]). In order to gain insight in the impact of this kind of phenomenon on the performance of the involved systems, we analyse and compare in this paper two systems. The first system is a system where different types of customers are accommodated in the same queue and global FCFS is assumed. The second system is a system where different types of customers are still accommodated in the same queue, but without global FCFS.
2
Mathematical Model
We consider a continuous-time queueing model with infinite waiting room. There are two servers which both have exponential service times with mean service rate μ. Each of the two servers is dedicated to a given class of customers. In this case, server A always serves customers of one type (say type 1) and server B always serves customers of a second type (type 2). The customers enter the system according to a Poisson arrival process with mean arrival rate λ. The types of consecutive customers are independent, i.e., every time a customer arrives, the customer is of type 1 with probability σ and of type 2 with probability (1 − σ). In the first (and most important) model, we assume that customers all queue together and are served in the order of arrival, regardless of the class they belong to. In our paper, we will call this service discipline “global FCFS”. It can be seen that the two-server system described above is non-workconserving, for two different (orthogonal) reasons. First, the fact that the two servers A and B are dedicated to only one type of customers each, may result in situations where only one of the servers is active even though the system contains more than one customer (of the same type, in such a case). This implies that we cannot expect the system to perform as well as a regular two-server queue with two equivalent servers, i.e., servers able to serve all customers. In this paper, we consider this form of inefficiency as an intrinsic feature of our system, simply caused by the fact that the customers as well as the servers are non-identical.
16
W. M´elange et al.
The second reason why the system is non-workconserving lies in the use of the global FCFS service discipline. This rule may result in situations where only one server is active although the system contains customers of both classes. Such situations occur whenever the two customers at the front of the queue are of the same type: only one of them can then be served (by its own dedicated server) and the other “blocks” the access to the second server for customers of the opposite type further in the queue. This second form of inefficiency is not an intrinsic feature of two-class systems with dedicated servers, but rather it is due to the accidental order in which customers of both types happen to arrive (and receive service) in the system. It is this second mechanism that we want to emphasize in the paper. Therefore, we also study a system without this second mechanism in order to quantify the impact of the global FCFS rule. The rest of this paper can be split up into three parts. In section 3, we will first discuss the system with global FCFS with a focus on the number of customers in the system and the stability of the system. This system is modelled by a continuous-time Markov chain and is solved using generating functions. Next in section 4, the system without the restriction of global FCFS will be discussed, again with a focus on the number of customers and stability of the system. Section 5 is devoted to the comparison between both systems. Some conclusions and directions for future work are given in section 6.
3 3.1
System with a Global FCFS Service Discipline System State Diagram and Balance Equations
The system can be described by a continuous-time Markov chain (see Fig. 1). The states of the Markov chain contain the number of customers in the system and the types of the two customers at the front of the system. For example, the state AB(k), means that one of the customers is of type 1 (is a customer for server A) and the other customer is of type 2 (is a customer for server B), and there are a total of k customers in the system. Because, at each time, only the first two customers can be served (if they are of different types) we are only interested in the type of the customer when the customer becomes the second customer at the front of the system. This type is independent of the type of previous customers, by assumption, so the customer is of type 1 with probability σ and of type 2 with probability (1 − σ). The balance equations of the state diagram of the system in Fig. 1, are γ · p(0) = pA (1) + pB (1)
(1)
(γ + 1) · pA (1) = pAA (2) + pAB (2) + γ · σ · p(0) (γ + 1) · pB (1) = pAB (2) + pBB (2) + γ · (1 − σ) · p(0)
(2) (3)
(γ + 1) · pAA (2) = σ · (pAA (3) + pAB (3)) + γ · σ · pA (1)
(4)
(γ + 2) · pAB (2) = (1 − σ) · pAA (3) + pAB (3) + σ · pBB (3) + γ · ((1 − σ) · pA (1) + σ · pB (1))
(5)
(γ + 1) · pBB (2) = (1 − σ) · (pAB (3) + pBB (3)) + γ · (1 − σ) · pB (1)
(6)
A Two-Class Continuous-Time Queueing Model (1)
λ(1-σ)
0
λσ μ
μ (3)
B 1
A 1 λσ
λ(1-σ)
μ
μ
AB 2
μ
λ μσ
AA 2
(5)
μσ
λ μσ
μ(1-σ)
μ(1-σ)
AB 3
AA 3
BB k-1
AB k-1
AA k-1
μσ
μσ
μ(1-σ) AB k
BB k
μ
λ
μ(1-σ)
μσ BB k+1
μσ
λ
μ
λ
μ(1-σ)
AA k
μσ μσ
AB k+1
λ
μ(1-σ)
(8)
λ
(4)
λ
BB 3
μ(1-σ)
(9)
λ(1-σ)
BB 2
μ(1-σ)
λσ
μ μ
(6)
(2)
(7)
λ
μ(1-σ) AA k+1
Fig. 1. State Diagram of the system with global FCFS
17
18
W. M´elange et al.
(γ + 1) · pAA (k) = σ · (pAA (k + 1) + pAB (k + 1)) + γ · pAA (k − 1)
(7) k3
(γ + 2) · pAB (k) = (1 − σ) · pAA (k + 1) + pAB (k + 1) + σ · pBB (k + 1) (8) + γ · pAB (k − 1) k3 (γ + 1) · pBB (k) = (1 − σ) · (pAB (k + 1) + pBB (k + 1)) + γ · pBB (k − 1) where γ 3.2
(9) k3
λ . μ
Analysis of the System
Stability Condition. Intuitively, stability means that the expected direction (drift) of the process for states in the repeating portion of the chain is towards lower valued states (lower number of customers in the system). If the chain tends to move towards higher valued states (higher number of customers in the system), then the chain is unstable. To calculate the average drift the probabilities are needed that the process is in an interlevel state j (here AA, AB or BB) of the repeating portion of the process for a level k far from the boundary, i.e., for k >> 0 (here called qj ). Intuitively speaking since these levels are far from the boundary, the values qj are independent of the level k. To determine these values a derived Markov chain is considered which identifies transitions only in terms of their interlevels. Specifically, if the original process has a transition from state (k, j) tot state (k − i, l) (in this case i = −1 or 1), then in the derived process it is only considered that a transition occurs from interlevel state j to interlevel state l. [10,11] Translated to the system in this paper, it is easily seen in Fig. 1 that the drift towards higher values and the drift towards lower values are respectively drif thigher = λ · (qAA + qAB + qBB ) , drif tlower = μ · (qAA + 2 · qAB + qBB ).
(10) (11)
The following equations are found by making equations (7) to (9) independent of the level k by substituting pAA (k) by qAA , pAB (k) by qAB and so forth, ⎧ ⎪ ⎨(γ + 1) · qAA = σ · (qAA + qAB ) + γ · qAA (12) (γ + 2) · qAB = (1 − σ) · qAA + qAB + σ · qBB + γ · qAB ⎪ ⎩ (γ + 1) · qBB = (1 − σ) · (qAB + qBB ) + γ · qBB This equation system lets us calculate all the probabilities (qj ) to be in an interlevel state j of the repeating portion of the chain. Inserting these qj in (10) and (11) give us the stability condition,
⇔
drif thigher < drif tlower 1 γ< . 1 − σ · (1 − σ)
(13)
A Two-Class Continuous-Time Queueing Model
19
Distribution and Moments of System Occupancy. To tackle this problem, generating functions are used. We first introduce the three following partial probability generating functions (pgf’s) PAA (z) PAB (z) PBB (z)
∞ k=2 ∞ k=2 ∞
pAA (k) · z k ,
(14)
pAB (k) · z k ,
(15)
pBB (k) · z k .
(16)
k=2
Equations (7) to (9) are multiplied by z k and summed over all k 3. We find (γ + 1) · (PAA (z) − z 2 · pAA (2)) =
σ · [(PAA (z) − z 3 · pAA (3) − z 2 · pAA (2)) z + (PAB (z) − z 3 · pAB (3) − z 2 · pAB (2))]
+ γ · z · PAA (z), (17) 1 (γ + 2) · (PAB (z) − z 2 · pAB (2)) = · [(1 − σ) · (PAA (z) − z 3 · pAA (3) z − z 2 · pAA (2)) + (PAB (z) − z 3 · pAB (3) − z 2 · pAB (2)) + σ · (PBB (z) − z 3 · pBB (3) − z 2 · pBB (2))] + γ · z · PAB (z) , (18) 1 − σ (γ + 1) · (PBB (z) − z 2 · pBB (2)) = · [(PAB (z) − z 3 · pAB (3) z − z 2 · pAB (2)) + (PBB (z) − z 3 · pBB (3) − z 2 · pBB (2))] + γ · z · PBB (z).
(19)
The common probability generating function of the (total) number of customers in the system is given by P (z) = p(0) + z · (pA (1) + pB (1)) + PAA (z) + PAB (z) + PBB (z).
(20)
If we solve equations (1) to (6) and (17) to (19) and insert the solutions in (20) this equation translates into a linear equation that only contains known quantities, except for two unknown probabilities in the numerator. These can be determined, in general, by invoking the well-known property that pgf’s such as P (z) are bounded inside the closed unit disk {z : |z| ≤ 1} of the complex z-plane, at least when the stability condition (13) of the queueing system is met (only in such a case our analysis was justified and P (z) can be viewed as a legitimate pgf). In our case, the denominator of P (z) is of degree 5. The Abel-Ruffini theorem tells us that for a function of degree 5 or higher, it is not possible to find the
20
W. M´elange et al.
zeroes explicitly (see page 586 of [14]). But in this case, we already know one of the five zeroes (z = 1) of the denominator so we only have to find the zeroes of a function of fourth degree which is possible by means of the method of Ferrari [15]. The denominator of P (z) is given by D(z) = (z − 1) · (γ · (1 − σ · (1 − σ)) + 1) · (γ 3 · z 4 −2 · γ 2 · (γ + 2) · z 3 + γ · (γ + 5) · (γ + 1) · z 2 −(2 · γ + 1) · (γ + 2) · z + γ · (1 − σ · (1 − σ)) + 1) and has four zeroes (besides z = 1) of the form 2 + γ ±s 2 + γ 2 ±t 2 · 1 + 4 · γ 2 · σ · (1 − σ) z= 2·γ
(21)
where the signs ±s or ±t can be plus or minus (so four options for four zeroes). We can prove that the only zero of these four zeroes, named z0 in the rest of the paper, that is inside the closed unit disk when the stability condition is met, is the zero where ±s = − and ±t = +. It is clear that this first zero (z0 ) should also be a zero of the numerator of (20), as P (z) must remain bounded in this point (inside the closed unit disk). The requirement that the numerator should vanish yields a linear equation for the two unknowns. For the zero z = 1, this condition is fulfilled regardless of the values of the unknowns, since the numerator of (20) contains a factor z − 1. A second linear equation can however be obtained by invoking the normalizing condition of the pgf P (z), i.e., the condition P (1) = 1. In general, the two unknown probabilities can be found as the solutions of the two established linear equations. Substitution of the obtained values in (20) then leads to a fully determined and explicit expression for the steady-state pgf P (z) of the system occupancy. From this result, various performance measures of practical importance can then be derived. For instance, the mean system occupancy can be found as N = P (1). The mean system delay can then be calculated using Little’s Law [16].
4 4.1
System without Global FCFS Network Model
When the global FCFS restriction is dropped and replaced by the service discipline FCFS for each of the types separately (i.e., a customer can be served if no customers of his own type are in front of him), our system is equivalent to a network of two parallel, separate queues (see Fig. 2) where each customer upon arrival goes immediately to the queue (waiting room) for its own server. The analysis of this model turns out to be much easier. 4.2
Analysis of the System
The random split property of the Poisson process (stating that if a Poisson process with intensity λ is randomly split into two subprocesses with probabilities
A Two-Class Continuous-Time Queueing Model
λ
σ
21
serverA μ
1−σ
μ serverB
Fig. 2. Network model of the system without global FCFS
σ and (1 − σ), then the resulting processes are independent Poisson processes with intensities σ · λ and (1 − σ) · λ) makes the network meet the requirements of a Jackson network [17]. The fact that the system is a Jackson network, greatly simplifies the analysis of this network. The following traffic equations of the system in this paper are very simple λ1 = σ · λ,
(22)
λ2 = (1 − σ) · λ
(23)
and the mean service rates of the servers are given by μi = μ (i = 1, 2)
(24)
The Jackson’s theorem states (for our case) that if (and only if) the unique solution of the traffic equations (λi ) satisfies λi < μi , 1 i 2,
(25)
then the network possesses stochastic equilibrium and a steady-state distribution of the state of the Jackson network is given by a product form p(k1 , k2 ) = p1 (k1 ) · p2 (k2 ),
(26)
where the left side of the equation is the joint distribution to have k1 customers in node (queue) 1 and k2 in node (queue) 2. The right side of the equation is the product of the marginal distributions to have ki customers in node (queue) i of the network of queues. So the network can be split in two simple M/M/1-queues and the marginal distributions of those queues can be multiplied to get the joint distribution. The solution of a simple M/M/1-queue is well-known from many books about queueing theory e.g. [18]. The solutions of the two queues (one for server A and one for server B) are given in (27) and (28). The notation γ is again introduced and is defined as in section 3.
22
W. M´elange et al.
pA (k1 ) = (1 − γ · σ) · (γ · σ)k1
(27) k2
pB (k2 ) = (1 − γ · (1 − σ)) · (γ · (1 − σ)) p(k1 , k2 ) = pA (k1 ) · pB (k2 )
(28) (29)
The stability condition for the system exists out of merging the stability conditions of the two queues separately ((30) and (31)). Those stability conditions are also the conditions that must be met to be able to use Jackson’s Theorem. σ · λ < μ ⇐⇒ γ < (1 − σ) · λ < μ ⇐⇒ γ
