Stochastic Mechanics

Random Media

Signal Processing

and Image Synthesis

Mathematical Economics and Finance

Stochastic Optimization

Stochastic Control

Stochastic Models in Life Sciences

Applications of Mathematics Stochastic Modelling 2 6lied Probability

Edited by B. Rozovskii M. Yor

Advisory Board D. Dawson D.Geman G. Grimmett I. Karatzas F. Kelly Y.LeJan B. 0ksendal E.Pardoux G. Papanicolaou

Springer-Verlag Berlin Heidelberg GmbH

Applications of Mathematics

1 Fleming!Rishel, Deterministic and Stochastic Optimal Control (1975) 2 Marchuk, Methods of Numerical Mathematics 1975, 2nd. ed. 1982) 3 Balakrishnan, Applied Functional Analysis (1976, 2nd. ed. 1981) 4 Borovkov, Stochastic Processes in Queueing Theory (1976) 5 Liptser/Shiryaev, Statistics of Random Processes I: General Theory (1977. 2nd. ed. 2001) 6 Liptser/Shiryaev, Statistics of Random Processes II: Applications (1978, 2nd. ed. 2001) 7 Vorob'ev, Game Theory: Lectures for Economists and Systems Scientists (1977) 8 Shiryaev, Optimal Stopping Rules (1978) 9 Ibragimov/Rozanov, Gaussian Random Processes (1978)

10 Wonham, Linear Multivariable Control: A Geometric Approach (1979, 2nd. ed. 1985) n Hida, Brownian Motion (1980) 12 Hestenes, Conjugate Direction Methods in Optimization (1980) 13 Kallianpur, Stochastic Filtering Theory (1980) 14 Krylov, Controlled Diffusion Processes (1980) 15 Prabhu, Stochastic Storage Processes: Queues, Insurance Risk, and Dams (1980) 16 Ibragimov/Has'minskii, Statistical Estimation: Asymptotic Theory (1981) 17 Cesari, Optimization: Theory and Applications (1982) 18 Elliott, Stochastic Calculus and Applications (1982) 19 Marchuk/Shaidourov, Difference Methods and Their Extrapolations (1983) 20 Hi jab, Stabilization of Control Systems (1986) 21 Protter, Stochastic Integration and Differential Equations (1990) 22 Benveniste/Metivier/Priouret, Adaptive Algorithms and Stochastic Approximations ( 1990) 23 Kloeden/Platen, Numerical Solution of Stochastic Differential Equations

(1992, corr. 3rd printing 1999) 24 Kushner/Dupuis, Numerical Methods for Stochastic Control Problems in Continuous

Time (1992) 25 Fleming!Soner, Controlled Markov Processes and Viscosity Solutions (1993) 26 Baccelli/Bremaud, Elements of Queueing Theory (1994, 2nd ed. 2003) 27 Winkler, Image Analysis, Random Fields and Dynamic Monte Carlo Methods

(1995, 2nd. ed. 2003) 28 Kalpazidou, Cycle Representations of Markov Processes (1995) 29 Elliott/Aggoun/Moore, Hidden Markov Models: Estimation and Control (1995) 30 Hermindez-Lerma/Lasserre, Discrete-Time Markov Control Processes (1995) 31 Devroye/Gyi:irfi/Lugosi, A Probabilistic Theory of Pattern Recognition (1996) 32 Maitra/Sudderth, Discrete Gambling and Stochastic Games (1996) 33 Embrechts/Kliippelberg/Mikosch, Modelling Extremal Events for Insurance and Finance

(1997, corr. 4th printing 2003) 34 Duflo, Random Iterative Models (1997) 35 Kushner/Yin, Stochastic Approximation Algorithms and Applications (1997) 36 Musiela/Rutkowski, Martingale Methods in Financial Modelling (1997) 37 Yin, Continuous-Time Markov Chains and Applications (1998) 38 Dembo/Zeitouni, Large Deviations Teclmiques and Applications (1998) 39 Karatzas, Methods of Mathematical Finance (1998) 40 Fayolle/Iasnogorodski/Malyshev, Random Walks in the Quarter-Plane (1999) 41 Aven/Jensen, Stochastic Models in Reliability (1999) 42 Hernandez-Lerma/Lasserre, Further Topics on Discrete-Time Markov Control Processes

(1999) 43 Yong!Zhou, Stochastic Controls. Hamiltonian Systems and HJB Equations (1999) 44 Serfozo, Introduction to Stochastic Networks (1999) 45 Steele, Stochastic Calculus and Financial Applications (2001) 46 Chen/Yao, Fundamentals of Queuing Networks: Performance, Asymptotics,

and Optimization (2001) 47 Kushner, Heavy Traffic Analysis of Controlled Queueing and Communications Networks

(2001) 48 Fernholz, Stochastic Portfolio Theory (2002) 49 Kabanov/Pergamenshchikov, Two-Scale Stochastic Systems (2003) so Han, Information-Spectrum Methods in Information Theory (2003)

François Baccelli Pierre Brémaud

Elements of Queueing Theory Palm Martingale Calculus and Stochastic Recurrences

Second Edition

Springer

Authors

Frans:ois Baccelli INRIA-ENS Ecole Normale Superieure LIENS 45 Rue d'Ulm 75230 Paris Cedex 05, France e-mail: Francois.Baccelli@ens.fr

Managing Editors B. Rozovskii Center for Applied Mathematical Sciences University of Southern California 1042 West 36th Place, Denney Research Building 308 Los Angeles, CA 90089, USA

Pierre Bremaud

School of Computer and Communications Systems Ecole Polytechnique Federale de Lausanne CH-1015 Ecublens, Switzerland e-mail: Pierre.Bremaud@ens.fr

M. Yor Universite de Paris VI Laboratoire de Probabilites et Modeles Aleatoires 175, rue du Chevaleret 75013 Paris, France

Mathematics Subject Classification (2ooo): 60F20, 60F99, 60G07, 60Gl7, 60G35, 60G44, 60G55, 60K05, 60Kl0, 60Kl5, 60K20, 60K25, 60K30, 68Ml0, 68M20, 68U20, 90B05, 90Bl5, 90B22, 90B35

Cover pattern by courtesy of Rick Durrett (Cornell University, Ithaca)

Cataloging-in-Publication Data applied for

A catalog record for this book is available from the Library of Congress.

Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at http://dnb.ddb.de

ISSN 0172-4568 ISBN 978-3-642-08537-6 ISBN 978-3-662-11657-9 (eBook) DOI 10.1007/978-3-662-11657-9

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, reuse of illustrations, recitation, broadcasting, reproduction on microfilm 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-Verlag Berlin Heidelberg GmbH. Violations are liable for prosecution under the German Copyright Law.

http://www.springer.de

©Springer-Verlag Berlin Heidelberg 2003 Originally published by Springer-Verlag Berlin Heidelberg New York in 2003 Softcover reprint of the hardcover 2nd edition 2003 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.

Cover design: Erich Kirchner, Heidelberg Typesetting by the authors using a Springer TEX macro package Printed on acid-free paper SPI:-\: I 4l/3111DB-) ·1 3

Preface

Queueing theory is a fascinating subject in Applied Probability for two con­tradictory reasons: it sometimes requires the most sophisticated tools of stochastic processes, and it often leads to simple and explicit answers. More­over its interest has been steadily growing since the pioneering work of Erlang in 1917 on the blocking of telephone calls, to the more recent applications on the design of broadband communication networks and on the performance evaluation of computer architectures.

All this led to a huge literature, articles and books, at various levels of mathematical rigor. Concerning the mathematical approach, most of the explicit results have been obtained when specific assumptions (Markov, re­newal) are made. The aim of the present book is in no way to give a systematic account of the formulas of queueing theory and their applications, but rather to give a general framework in which these results are best understood and most easily derived.

What knowledge of this vast literature is needed to read the book? As the title of the book suggests, we believe that it can be read without prior knowledge of queueing theory at all, although the unifying nature of the proposed framework will of course be more meaningful to readers who already studied the classical Markovian approach.

The general framework in question concerns what can be said when only stationarity and ergodicity assumptions are made on the input point process to a queueing system. What is the interest of such a framework, beyond its generality? In most cases, the approaches based on specific computational assumptions fail to find properties of point processes which are kept invariant by a queue: for instance, if the input point process to a stationary queue is a renewal process, then its output point process is not renewal in general; since queues should primarily be seen as network elements that transform input point processes into output point processes within a larger network, this is a major drawback as it shows that such computational frameworks cannot be the basis of a constructive queueing network theory. It turns out that stationarity and ergodicity are preserved by queues. In this sense, the proposed framework brings the key ingredient to build the network theory alluded to above.

VI Preface

Typical of the kind of results that can be obtained within this stationary­ergodic framework are the construction of the stationary regime via Loynes' method, Little's formula L = ,\ W and stochastic comparison results. Each of these topics will actually be the central theme of a chapter of the book.

As indicated in the subtitle of the book, the framework in question has two main mathematical foundations: Palm calculus and stochastic recurrences. By stochastic recurrences, we mean sequences obtained by iterating stationary random maps, like for instance the sequence of waiting times of successive customers in a queue. The stability of such sequences, which constitutes one of the central problems of queueing theory, is covered in Chapter 2. In many situations of practical interest, one can prove or disprove stability only for special models, mostly Markovian or renewal. However, in his pioneering article of 1962 concerning the recurrence equation satisfied by waiting times, Loynes was able to prove stability for a single server, infinite capacity queue, under the natural condition that the rate of work brought into the system be smaller than the processing speed of the server. Loynes' proof did not require the usual independence assumptions. It only required the input to be stationary and ergodic. The generality of his proof provided impetus to the stability theory of stationary and ergodic systems which is still flourishing.

Loynes' theory of stability of the GIG I 1 I oo queue has since been refined and extended. We shall present various extensions of the basic theory to the case when the critical monotonicity assumption is not verified (a typi­cal example is the GIGI1IO queue). Borovkov's theory of renovating events provides a general method to analyze systems of this type. This theory gen­eralizes the notion of construction points and allows one to prove the con­vergence in variation of non-stationary sequences to stationary limits, via coupling arguments. Another basic tool is Kingman's sub-additive ergodic theorem, which provides the stability region for open separable queueing sys­tems, like blocking queues or generalized Jackson networks for instance.

In a queueing context, the stochastic recurrences arise by sampling a con­tinuous time stochastic process at events such as the arrival or the departure of a customer, or its transfer from one queue to another. They are also called embedded sequences, and are often used as a way to go back to the Markovian framework, when the original continuous time model is not Markovian. The pioneering work in this respect is that of Kendall who studied the MIG I 11 I oo queue via embedding at departure times. However, as is well known in con­tinuous time Markov chain theory, the stationarity of an embedded sequence does not imply the stationarity of the original continuous time processes and vice-versa. Also, in the ergodic context, empirical averages of the continuous time process and of the embedded sequence need not coincide.

The theory of Palm probabilities provides the natural framework for the study of relations between time averages and event averages, such as Little's theorem, Brumelle's H = ,\G formula and the PASTA (Poisson Arrivals See Time Averages) property. The first chapter of the book is devoted to the

Preface VII

aspects of Palm probabilities which are relevant to queueing theory, and in particular of the so-called Palm calculus based on Matthes' definition and Mecke's formula. In this book, the treatment is in terms of abstract shifts. This is the so-called Bt-formalism which makes the basic rules clear and easy to use, by allowing one to remain in the same measurable space. The connec­tion is also made between the theories of Palm probabilities and of stochastic intensities of point processes via Papangelou's Radon-Nikodym theorem.

Once the Palm calculus and the stability theory of stochastic recurrences are available, the harvesting of the classical results of queueing theory is easy; this is done in Chapter 3, where Little's L = >. W formula, Brumelle's H = >.G formula, Kleinrock's conservation law, the PASTA principle, and their applications are given.

The constructive theory of stationary states in terms of stochastic re­currences is also a very useful tool in the theory of stochastic compari­son of queues, together with the sample path approach to stochastic orders (Strassen's theorems). The corresponding theory is covered in Chapter 4, where it is applied to provide non-analytical proofs of classical monotonic­ity results in queueing theory, and various optimality properties, such as the optimality of the SRPT (shortest remaining processing time) discipline and the extremal properties of the FIFO (first-in-first-out) discipline. Finally, the relationship between the stochastic monotonicity of point processes and that of stochastic recurrences based on these point processes is studied, together with extensions to continuous time processes.

The general approach taken in this book has already proved fruitful in two respects: it gives simple and rigorous proofs of most classical results, and it has taken the theory one step beyond the renewal assumptions in a number of situations. We have illustrated the theory by a few examples which should be familiar to a reader already acquainted with the classical theory of queues.

For the reader wishing to obtain more information on topics of queue­ing theory that are not covered here, Stochastic Networks and Reversibility, by Frank Kelly, Applied Probability and Queues, by S0ren Asmussen and Stochastic Modeling and the Theory of Queues, by Ronald Wolff are highly recommended classical references.

Acknowledgments

We wish to acknowledge our intellectual debt to the book Queues and Point Processes, by Peter Franken, Dieter Konig, Volker Schmidt and Ursula Arndt, which introduced us to the Palm calculus approach to queues.

The present book grew out of the authors' Lecture Notes Palm Probabil­ities and Stationary Queueing Systems (1987), Springer-Verlag, and from a graduate course taught at the Ecole Polytechnique in the DEA Modelisation aleatoire et statistique, where it greatly benefited from the interaction with the students.

VIII Preface

Our colleagues Venkat Anantharam of the University of California, Berke­ley, Serguei Foss of the Institute of Mathematics of Novosibirsk, Armand Makowski of the University of Maryland, College Park, Masakiyo Miyazawa of the Science University of Tokyo and Volker Schmidt of the University of Ulm deserve our sincere thanks for reading and checking the manuscript at various steps of completion.

The typing of the first edition was diligently accomplished by Rivo Rako­tosafi; Ephie Deriche also greatly contributed to the material realization of the first edition in many ways.

Preface IX

Foreword to the Second Edition

The second edition has benefited from the experience acquired by giving a doctoral course based on the material presented in the book. The material of the first edition has also been thoroughly reorganized and revised, and in general, every effort has been made to transform the monograph into an advanced textbook. A number of sections have been augmented; some sections have been added, mainly in Chapter 2 on stationarity and coupling.

One of the main new features is the addition of exercises and problems, with complete solutions. These are organized in three types: fundamental exercises (F), which the reader should solve before proceeding further; appli­cation exercises (A) and problems (P). Most of the exercises of the last two types stem from research papers relevant to practical issues in engineering and which have been published since the first edition.

We thank our colleagues Takis Konstantopoulos of the University of Pa­tras, Jean Mairesse of Universite Paris 7, Laurent Massoulie of Microsoft Research in Cambridge, and Augustin Chaintreau and Marc Lelarge of Ecole Normale Superieure, Paris, for their help in reading and correcting the new manuscript, and for their useful suggestions to improve the presentation. We also thank Jeffrey Boys for his careful editing of the manuscript.

Paris, November 2002, F. Baccelli and P. Bremaud.

Contents

Preface....................................................... V

1. The Palm Calculus of Point Processes . . . . . . . . . . . . . . . . . . . . 1 1.1 Stationary Marked Point Process . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.1 The Canonical Space of a Point Process . . . . . . . . . . . . . 2 1.1.2 Stationary Point Process . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1.3 Marks of a Point Process . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.1.4 Two Properties of Stationary Point Processes . . . . . . . . 11 1.1.5 Intensity of a Stationary Point Process. . . . . . . . . . . . . . 12 1.1.6 The Campbell Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.2 Palm Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.2.1 The Matthes Definition in Terms of Counting........ 14 1.2.2 Invariance of Palm Probability . . . . . . . . . . . . . . . . . . . . . 16 1.2.3 Mecke's Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.2.4 The Inversion Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

1.3 Basic Formulas of Palm Calculus . . . . . . . . . . . . . . . . . . . . . . . . . 21 1.3.1 The Mean-Value Formulas......................... 21 1.3.2 The Neveu Exchange Formula. . . . . . . . . . . . . . . . . . . . . . 21 1.3.3 The Miyazawa Conservation Principle............... 23 1.3.4 Forward and Backward Recurrence Time . . . . . . . . . . . . 24 1.3.5 The Slivnyak Inverse Construction. . . . . . . . . . . . . . . . . . 25 1.3.6 Other Inversion Formulas. . . . . . . . . . . . . . . . . . . . . . . . . . 27 1.3. 7 The Swiss Army Formula . . . . . . . . . . . . . . . . . . . . . . . . . 28

1.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 1.4.1 Renewal Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 1.4.2 Superposition of Independent Point Processes . . . . . . . . 33 1.4.3 Selected Marks and Conditioning . . . . . . . . . . . . . . . . . . . 35 1.4.4 Selected Transitions of a Stationary Markov Chain . . . 38 1.4.5 Stationary Semi-Markov Process . . . . . . . . . . . . . . . . . . . 39 1.4.6 Delayed Marked Point Process . . . . . . . . . . . . . . . . . . . . . 42

1.5 Local Aspect of Palm Probability. . . . . . . . . . . . . . . . . . . . . . . . . 44 1.5.1 The Korolyuk and Dobrushin Infinitesimal Estimates . 44 1.5.2 Conditioning at a Point . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

1.6 Ergodicity of a Point Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

XII Contents

1.6.1 Ergodicity of a Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 1.6.2 Invariant Event . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 1.6.3 Ergodicity Under the Stationary Probability and

Under the Palm Probability . . . . . . . . . . . . . . . . . . . . . . . 52 1.6.4 The Cross-Ergodic Theorems . . . . . . . . . . . . . . . . . . . . . . 52

1. 7 Palm Theory in Discrete Time . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 1.8 Stochastic Intensity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

1.8.1 Predictable Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 1.8.2 Stochastic Intensity Kernel . . . . . . . . . . . . . . . . . . . . . . . . 58 1.8.3 Stochastic Intensity Integration Formula . . . . . . . . . . . . 60 1.8.4 Watanabe's Characterization of Poisson Processes . . . . 61

1.9 Palm Probability and Stochastic Intensity . . . . . . . . . . . . . . . . . 64 1.9.1 Invariance of Stochastic Intensity................... 64 1.9.2 The Papangelou Formula . . . . . . . . . . . . . . . . . . . . . . . . . 65

1.10 Solutions to Exercises................................... 69 1.11 Bibliographical Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

2. Stationarity and Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 2.1 Stability of the Single Server Queue. . . . . . . . . . . . . . . . . . . . . . . 76

2.1.1 The Single Server Queue . . . . . . . . . . . . . . . . . . . . . . . . . . 76 2.1.2 The Loynes Stability Theorem . . . . . . . . . . . . . . . . . . . . . 78 2.1.3 Construction Points and Cycles . . . . . . . . . . . . . . . . . . . . 80

2.2 Proof of Loynes' Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 2.2.1 Reduction to the Palm Setting . . . . . . . . . . . . . . . . . . . . . 83 2.2.2 Construction of the Workload Sequence . . . . . . . . . . . . . 84 2.2.3 Uniqueness of the Stationary Workload . . . . . . . . . . . . . 87 2.2.4 Construction Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 2.2.5 Queueing Proof of the Ergodic Theorem............. 89

2.3 The Multiserver Queue. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 2.3.1 Ordered Workload Vector . . . . . . . . . . . . . . . . . . . . . . . . . 91 2.3.2 Existence of a Finite Stationary Workload . . . . . . . . . . . 92 2.3.3 The Maximal Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

2.4 Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 2.4.1 Coupling and Convergence in Variation . . . . . . . . . . . . . 98 2.4.2 Coupling in the Single Server Queue ................ 100

2.5 Stochastic Recurrences and Their Stationary Regimes ....... 104 2.5.1 Stochastic Recurrences ............................ 104 2.5.2 The Loynes Theorem for Stochastic Recurrences ...... 107 2.5.3 Exact Sampling of Markov Chains .................. 109 2.5.4 The Borovkov Theory of Renovating Events ......... 114

2.6 Stability of the G/G/1/0 Queue .......................... 121 2.6.1 Counter-Examples ................................ 121 2.6.2 Coupling in the G/G/1/0 Queue ................... 123 2.6.3 Enriched Probability Space ........................ 124

2.7 The Fluid Queue ....................................... 128

Contents XIII

2.7.1 Departure and Workload Processes in Fluid Queues ... 128 2.7.2 The Loynes Theorem for Fluid Queues .............. 131

2.8 Other Queueing Systems ................................ 133 2.8.1 The Pure Delay Queue ............................ 133 2.8.2 Service Disciplines and Residual Services ............ 135 2.8.3 Single Server Queues with Vacations ................ 138 2.8.4 Single Server Queues with Mutual Service ........... 140

2.9 Stability of Queueing Networks via Coupling ............... 141 2.9.1 Single Server Queues in Tandem ................... 141 2.9.2 Kelly-Type Networks ............................. 141

2.10 Queueing Network Stability via Recurrence Equations ....... 145 2.10.1 Finite Capacity Queues in Tandem with Blocking .... 145 2.10.2 Existence of a Stationary Solution .................. 147 2.10.3 Uniqueness of the Stationary Solutions .............. 151 2.10.4 Tandem Queues .................................. 153

2.11 Non-Expansive Stochastic Recurrences .................... 154 2.11.1 The Crandall-Tartar Theorem ..................... 154 2.11.2 Monotone-Homogeneous Stochastic Recurrences ...... 158 2.11.3 The Monotone-Homogeneous-Separable Framework ... 161 2.11.4 The Saturation Rule .............................. 165

2.12 Solutions to Exercises ................................... 171 2.13 Bibliographical Comments ............................... 179

3. Formulas ................................................. 181 3.1 The Little Formula ..................................... 182

3.1.1 The Bt-Framework for Stationary Queueing Systems .. 182 3.1.2 L = -\W ......................................... 185 3.1.3 The Little Formula for Fluid Queues ................ 193

3.2 Other Applications of Campbell's Formula ................. 195 3.2.1 The Function Space Campbell-Little-Mecke Formula . 195 3.2.2 The Pollaczek-Khinchin Mean-Value Formulas ....... 195 3.2.3 Extension of the H = ,\G Formula .................. 197 3.2.4 The Kleinrock Conservation Law ................... 201 3.2.5 The Keilson Asymptotic Equivalence for Rare Events . 205

3.3 Event and Time Averages ............................... 211 3.3.1 Poisson Arrivals See Time Averages ................ 211 3.3.2 Applications of Papangelou's Formula ............... 213 3.3.3 Mean-Value Analysis ............................. 218

3.4 Formulas Derived from Conservation Equations ............ 221 3.4.1 First Order Equivalence ........................... 221 3.4.2 The Brill and Posner Formula ...................... 228 3.4.3 The Takacs Formula and Queues with Vacations ..... 231 3.4.4 Backward and Forward Recurrence Times ........... 236

3.5 Applications of the Stochastic Intensity Integration Formula . 237 3.5.1 Reminder ....................................... 237

XIV Contents

3.5.2 Priorities in M/GI/1/oo .......................... 238 3.5.3 Mean-Value Formula for Fluid Queues .............. 244

3.6 Solutions to Exercises ................................... 248 3. 7 Bibliographical Comments ............................... 257

4. Stochastic Ordering of Queues ............................ 259 4.1 Comparison of Service Disciplines ........................ 261

4.1.1 Partial Orderings on JRn ........................... 261 4.1.2 Optimality of SRPT for Single Server Queues ........ 264 4.1.3 Optimality of FIFO ............................... 266

4.2 Comparison of Queues .................................. 272 4.2.1 Integral Stochastic Orderings ...................... 272 4.2.2 Analytical Characterizations ....................... 274 4.2.3 The Strassen Pointwise Representation Theorems ..... 277 4.2.4 Comparison of Stochastic Recurrences .............. 278 4.2.5 Bounds Based on Integral Orderings ................ 279 4.2.6 Stability of Stochastic Orders by Limits ............. 281 4.2.7 Comparison of Basic Queues ....................... 282 4.2.8 Other Queueing Systems .......................... 285

4.3 Association Properties of Queues ......................... 288 4.3.1 Association of Random Variables ................... 288 4.3.2 Bounds by Association ............................ 292

4.4 Stochastic Comparison of Time-Stationary Queues .......... 294 4.4.1 Comparison of Point Processes ..................... 294 4.4.2 Comparison Under Time-Stationary Probabilities ..... 300 4.4.3 Comparison of Continuous Time Characteristics ...... 304

4.5 Solutions to Exercises ................................... 308 4.6 Bibliographical Comments ............................... 314

References .................................................... 317

Index ......................................................... 329