Mathematical Methods in Queuing Theory

Mathematics and Its Applications

Managing Editor:

M. HAZEWINKEL

Centre for Mathematics and Computer Science, Amsterdam, The Netherlands

Volume 271

Mathematical Methods in Queuing Theory

by

Vladimir V. Kalashnikov Institute for System Studies, Moscow, Russia

SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.

A c.I.P. Catalogue record for this book is available from the Library of Congress

ISBN 978-90-481-4339-9 ISBN 978-94-017-2197-4 (eBook) DOI 10.1007/978-94-017-2197-4

Printed on acid-free paper

All Rights Reserved © 1994 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 1994

No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner.

TABLE OF CONTENTS

PREFACE 1 Chapter 1. QUEUEING THEORY 5

1.1. Examples of queueing models 5 1.2. Kendall's notation 10 1.3. Algebraic descriptions of queueing models 10

1.3.1. The GIGI11°o model 10 1.3.2. The GIGINloo model 13 1.3.3. The (GIGI1Ioo) -+ (GI1Ioo) -+ ... -+ (GI1Ioo) model 14 1.3.4. General remarks about the construction of queueing models 14

Problems 15 Comments 15 Chapter 2. NECESSARY FACTS FROM PROBABILITY

THEORY AND THE THEORY OF ANALYTIC FUNCTIONS 16

2.1. Probability 16 2.1.1. Introductory remarks 16 2.1.2. Events 16 2.1.3. Probability space 17 2.1.4. Independence. Conditional probabilities 19

2.2. Random variables and their distributions 21 2.2.1. Main notions. One-dimensional case 21 2.2.2. Main notions. General case 22 2.2.3. Independence of r.v. 'so Conditional probabilities 22 2.2.4. Classification of d.f.'s 24 2.2.5. Moments 26 2.2.6. Chebyshev's and Jensen's inequalities 28 2.2.7. Operations with r.v.'s 30 2.2.8. Fubini's Theorem 30

2.3. Examples of probability distributions 31 2.4. Uniformly integrable r.v.'s 37 2.5. Convergence of r.v.'s and their distributions. Probability

metrics 39 2.5.1. Different types of convergence 39 2.5.2. Probability metrics 41 2.5.3. Minimality property of probability metrics 44 2.5.4. Some useful relations in terms of probability metrics 45 2.5.5. Regularity and homogeneity 46

2.6. Analytic functions. The Laplace-Stieltjes transform. Generating functions 47 2.6.1. Analytic functions of a complex variable 47 2.6.2. The Laplace-Stieltjes and Laplace transforms 48

VI TABLE OF CONTENTS

2.6.3. Generating functions 50 Problems 52 Comments 53 Chapter 3. RANDOM FLOWS 54

3.1. General definitions. Classification 54 3.2. Poisson flow 55

3.2.1. Main characteristics 55 3.2.2. Excess and defect 57

3.3. Recurrent flows 59 3.3.1. A number of arrivals during [0, t] 59 3.3.2. Renewal function. Renewal equations 59 3.3.3. Palm's flow 60 3.3.4. Elementary renewal theorem 62 3.3.5. The excess of the Palm's flow 63

3.4. Stationarity 64 3.5. Construction of recurrent flows via Poisson ones 66 3.6. Thinning of recurrent flows 68

3.6.1. Geometric thinning 68 3.6.2. Renyi's theorem 69 3.6.3. Metric approach to Renyi's theorem 70

3.7. Superposition of random flows 72 3.7.1. Statement of the problem 72 3.7.2. Grigelionis' theorem 73 3.7.3. Example 77

Problems 78 Comments 79 Chapter 4. ELEMENTARY METHODS IN QUEUEING

THEORY 80 4.1. Preliminary remarks 80 4.2. "Conditional Poisson flow"method 80

4.2.1. The meaning of the term 80 4.2.2. Pre-stationary and stationary behaviour of the M>.IGlloo model 80 4.2.3. A model of a dam 83 4.2.4. Linear systems 86

4.3. Construction of ''restoration points" 88 4.3.1. Busy and idle periods 88 4.3.2. Busy and idle periods for the M>.IGIllloo model 89 4.3.3. The number of customers served during a busy period in the

M>.IGIllloo model 92 4.3.4. Busy and idle periods for the GIIMJLllloo model 93 4.3.5. The number of customers served during a busy period in the

GIIMJLIII°o model 99 Problems 101 Comments 102

TABLE OF CONTENTS

Chapter 5. MARKOV CHAINS 5.1. Main definitions and notations

5 .1.1. Preliminaries 5.1.2. General definitions 5.1.3. Transition functions and probabilities 5.1.4. Chapman-Kolmogorovequations 5.1.5. Generating operator 5.1.6. Markov times. Dynkin's formula. Strong Markov property

5.2. Accessibility 5.2.1. G-accessibility. Positive accessibility. Criteria 5.2.2. A criterion for a set to be nonpositive accessible 5.2.3. A criterion of unaccessibility 5.2.4. A criterion of accessibility

5.3. Examples 5.3.1. The M A IGlI11°o model 5.3.2. The GIIM~111°o model 5.3.3. The GIIGIJ1Joo model 5.3.4. The GIJGIJNloo model 5.3.5. The (GIIGlI1Ioo) --+ (GlI1Ioo) --+ ... --+ (GlI1Ioo) model

5.4. Classification of denumerable Markov chains 5.5. Classification of general Markov chains 5.6. Markov chains originated by piecewise-linear mapping

Problems Comments Chapter 6. RENEWAL PROCESSES

6.1. Main definitions. Crossing. Coupling 6.1.1. Notations 6.1.2. Some inequalities for the renewal function 6.1.3. Crossing and coupling

6.2. Estimates of crossing times 6.2.1. Preliminaries 6.2.2. Construction of crossing 6.2.3. Preliminary analytical estimates 6.2.4. Something about the condition (3) 6.2.5. Final estimates of crossing times

6.3. Blackwell's theorem 6.4. Monte-Carlo algorithm for estimating crossing times

Problems Comments Chapter 7. REGENERATIVE PROCESSES

7.1. Examples and definitions 7.1.1. The GIIGlI11°o model 7.1.2. The MA IGlI11°o model 7.1.3. Markov chain 7.1.4. Definitions

7.2. Construction of a stationary version

vii

103 103 103 103 104 105 105 106 113 113 119 120 122 123 123 127 130 132 139 147 152 158 160 161 163 163 163 163 166 167 167 167 171 181 186 188 195 199 199 201 201 201 203 203 203 206

Vlll TABLE OF CONTENTS

7.2.1. The desired properties of a stationary version 7.2.2. Construction of (z*, S*)

7.3. Ergodic theorems 7.4. Comparison of regenerative processes

7.4.1. Preliminaries 7.4.2. Uniform-in-time comparison estimates

7.5. Markov chains as regenerative processes 7.5.1. Denumerable chains 7.5.2. General chains 7.5.3. Renovative Markov chains generated by recursive equations

Problems

206 207 213 217 217 219 224 224 225 227 232

Comments 232 Chapter 8. DISCRETE TIME MARKOV QUEUEING MODELS233

8.1. Imbedded Markov chains 233 8.1.1. Introductory remarks 233 8.1.2. The M AIGllll°o model 233 8.1.3. The GIIMfLI11°o model 236

8.2. The GIIGlll/oo model 239 8.3. The GIIGIINloo model 240 8.4. The (GI/GlI1Ioo) ~ (Glllloo) ~ ... ~ (Glllloo) model 242 8.5. Finite-time continuity 244

8.5.1. Notations 244 8.5.2. Estimates of continuity 248 8.5.3. Examples 253

8.6. Uniform-in-time continuity 255 8.6.1. General remarks 255 8.6.2. The GIIGI/l/oo model 256 8.6.3. The GIIGIINloo model 257 8.6.4. The GIIGllllool -t (GII1Ioo) -t .. , -t (GIllloo) model 258

Problems 259 Comments 259 Chapter 9. MARKOV QUEUEING MODELS 261

9.1. Denumerable continuous-time Markov chains 9.1.1. Main definitions 9.1.2. Regularity 9.1.3. Backward Kolmogorov equations 9.1.4. Forward Kolmogorov equations 9.1.5. Limiting probabilities

9.2. The MAIMfLlll°o model 9.3. Birth-and-death processes

9.3.1. Queues as birth-and-death processes 9.3.2. Ergodicity criterion and limiting probabilities 9.3.3. Examples

9.4. Jackson's open queueing network 9.5. Discrete supplementary variables

261 261 263 265 267 268 271 274 274 276 281 284 287

TABLE OF CONTENTS ix

9.5.1. Erlang's phases 287 9.5.2. Hyper-Erlang distributions 291 9.5.3. PH-distributions 293

Problems 294 Comments 294 Chapter 10. METHOD OF SUPPLEMENTARY VARIABLES 296

10.1. How to describe queues with supplementary variables 296 10.2. The MAIGllll°o model 300 10.3. The GIIMJ.LIII°o model 306 10.4. Aggregative models 309

10 .4.1. Piecewise-linear aggregate 309 10.4.2. Canonical aggregate 310 10.4.3. Aggregative models 311 10.4.4. Queues as aggregative models 312

10.5. Regeneration in continuous-time queueing processes 315 10.5.1. Preliminaries 315 10.5.2. Main constructions 316 10.5.3. Multi-server model in continuous time 319

Problems 320 Comments 321 Chapter 11. FIRST-OCCURRENCE EVENTS 322

11.1. Motivation 322 11.2. Piecewise-linear processes 322

11.2.1. Equations for d.f.'s of the first-occurrence times 322 11.2.2. Semi-Markov process 324

11.3. Estimates in terms of test functions 327 11.3.1. Main assertions 327 11.3.2. Examples 330

11.4. Regenerative processes 336 11.4.1. Statement of the problem 336 11.4.2. Metric estimates 338 11.4.3. Lower and upper bounds 348

11.5. Examples 354 Problems 363 Comments 363 REFERENCES 364 LIST OF NOTATIONS AND ABBREVIATIONS 369 SUBJECT INDEX 370

PREFACE

The material of this book is based on several courses which have been delivered for a long time at the Moscow Institute for Physics and Technology. Some parts have formed the subject of lectures given at various universities throughout the world: Freie Universitat of Berlin, Chalmers University of Technology and the University of Goteborg, University of California at Santa Barbara and others.

The subject of the book is the theory of queues. This theory, as a mathematical discipline, begins with the work of A. Erlang, who examined a model of a telephone station and obtained the famous formula for the distribution of the number of busy lines which is named after him. Queueing theory has been applied to the study of numerous models: emergency aid, road traffic, computer systems, etc. Besides, it has lead to several related disciplines such as reliability and inventory theories which deal with similar models. Nevertheless, many parts of the theory of queues were developed as a "pure science" with no practical applications.

The aim of this book is to give the reader an insight into the mathematical methods which can be used in queueing theory and to present examples of solving problems with the help of these methods. Of course, the choice of the methods is quite subjective. Thus, many prominent results have not even been mentioned. Those who need these methods have to turn to other books, some of which are referred to at the end of various chapters. The examples considered are intended to serve as illustrations of the methods described. This is the reason why only a few queueing models are applied for these illustrations. Such an approach makes it possible not to have to spend lots of efforts explaining the features of the underlying models. Each chapter concludes with comments from which the reader can learn something about other applications of the methods. Besides, the text is supple­mented by numerous problems which are intended to help the reader to understand the material.

The book presupposes a certain familiarity with the elements of probability theory and stochastic processes. Nevertheless, some facts which are of especial importance for understanding the book are contained in an auxiliary Chapter 2.

Chapter 1 briefly discusses what queueing theory is all about. Such important components as input flow, server, service discipline, etc., are introduced. Besides, it is shown how to describe the dynamics of "typical" queues by recurrent algebraic equations.

Chapter 2 collects various mathematical topics that are needed in order to study queueing theory. These are: fundamental notions of probability theory including rather new facts from the theory of probability metrics (unfortunately, the latter have not been widely used in queueing up to now); some facts from the theory of analytic functions; the Laplace-Stieltjes transform; and generating functions.

Chapter 3 examines random flows. A lot of attention is dedicated to Poisson flows and their relation to so-called order statistics. In addition, recurrent flows

1

2 PREFACE

are studied intensively. Limit theorems concerning thinning and superpositions of random flows are proved. A rather new aspect is the proof of Renyi's theorem concerning the limiting behaviour of thinning recurrent flows with the help of prolr ability metrics leading to obtaining estimates of the convergence rate. This proof is used in Chapter 11 for studying first-occurrence times for regenerative processes.

Chapter 4 attempts to show how one can study some queueing problems keeping in mind only elementary probabilistic facts. In particular, characteristics related to busy and idle periods for single server queues are examined. We emphasize the existence of so-called "restoration points" (this is a new term coined by the author meaning some mixture of Markov and regeneration times, though such points have been studied in queueing for a long time) which enable us to derive equations defining the necessary characteristics.

Chapter 5 contains useful facts about Markov chains (denumerable and gen­eral). We describe not only traditional results, but also those which are not usually mentioned in textbooks. Among these are: generating operators, Dynkin's formula, test functions, etc. They enable not only to obtain criteria of accessibility, positive recurrency, etc., but lead to corresponding quantitative estimates. These results are applied to an examination of single server, multi-servers and multi-phase queueing models.

Chapter 6 is devoted to a very important topic - renewal theory. We discuss here traditional assertions (renewal theorems, Blackwell's theorem) but use for their proof rather modern "coupling arguments" leading to both an effective way of obtaining the proof and to quantitative estimates of convergence rates.

The material of Chapter 7 is based on the results of Chapter 6 and provides a detailed study of regenerative processes. Here, not only regenerative processes in the sense of W. Smith but wide sense regenerative processes (S. Asmussen, H. Thorisson and others) which arise naturally in queueing theory are studied. Fol­lowing H. Thorisson, we present the construction of a so-called stationary version of a regenerative process. Some ergodic theorems with convergence rate estimates are proved. We develop methods for a uniform-in-time comparison of regenerative processes and give estimates displaying the difference between pre-stationary distri­butions of these processes which are uniform in time. We discuss relations existing between Markov chains and regenerative processes, introduce a useful notion of ren­ovation (A. Borovkov) and consider its relation to splitting.

Chapter 8 contains some topics on queueing models described by Markov chains. One of these is the well-known method of imbedded Markov chains. Another is a continuity analysis of queues based on the results of Chapter 7. In addition, we analyse ergodic properties of the underlying models.

In Chapter 9, we examine Markov queueing models with the help of continuous time Markov chains. This is a rather traditional topic in queueing theory. The central role here is performed by the birth-and-death process. Jackson's queueing network serves as an example of a Markov chain which is not a birth-and-death pro­cess. We also discuss Erlang-phase and PH-distribution methods in order to enlarge the scope of Markov models. Some estimates of the accuracy of these methods are given.

PREFACE 3

Chapter 10 is devoted to an important method of supplementary variables when a non-Markovian process is imbedded into a more complicated Markov one. Tradi­tional analytic results concerning limiting and pre-limiting characteristics of single server queues are presented. We suggest a rather general scheme (aggregative model) for a description of non-Markovian queues which can be used in simulation. A use­ful construction of regeneration times for continuous-time queueing models is also proposed.

In Chapter 11, we discuss a problem which is of importance not only in queueing, but in storage, reliability, etc. This is an estimation of the distribution function of the first occurrence time (break-down time in reliability, overflow time in queueing, depletion time in storage, etc.). We consider here various approaches for solving this problem: with the help of linear equations, test functions, probability metrics, and renewal theory.

In referring to various monographs or papers it has not been the intention to mention all the available literature. This is far too extensive. Note that many thousands of papers and books have been published on queueing theory and related fields. Nevertheless, we hope, that the reader can find necessary results starting from our references.

Every chapter consists of sections and most sections are divided into subsections. Each section has its own enumeration of formulas, theorems, lemmas, etc. If we mention formula (2), this refers to the formula from the current Section. If we refer to formula (4.2), then we mean formula (2) from Section 4 in the current Chapter. The reference (5.4.2) means formula (2) from Section 4 in Chapter 5. The same is true for theorems, etc. There is only one exception: the Figure numbering; namely, Figure 10.4 refers to Figure 4 from Chapter 10.

I hope that some fragments of the book seem to be rather "fresh", and that one can use them (together with the suggested problems) in special courses on applied probability or operations research.

I am deeply indebted to those people who taught me, had an influence on my interests and career. In this connection, I would like to mention first of all my alma mater - Moscow Institute for Physics and Technology. Up to now I remember the enthusiasm with which the Professors of this Institute (F. Gantmacher, M. Naimark, L. Kudryavtzev, V. Lidsky and others) delivered their lectures. They persuaded us (without any resistance from our side) to fall in love with mathematics. Later on, N. Buslenko and B. Gnedenko showed me the beauty of the probabilistic world. I am happy to consider them my teachers. Having worked in this area, I was lucky to meet many nice people and very qualified specialists, some of whom are now my friends. It is almost impossible to list all of these. But it is quite necessary to single out S. Asmussen, A. Borovkov, S. Foss, J. Gani, P. Jagers, I. Kovalenko, E. Nummelin, N. Prabhu, A. Soloviev, D. Stoyan, H. Thorisson, G. Tsitsiashvili, R. Tweedie, V. Zolotarev. Their help in this work is significant: some of their results are included in the book, they discussed various topics, provided valuable comments, they promoted my studies in one way or another and so on. At last (or, maybe, at first?) I ought to thank Mrs. E. Orlova and my children Ira and Slava for their help in typesetting and drawing. Unfortunately, their help does not relieve me of any

4 PREFACE

responsibility for possible mistakes, misleads, errors which are present (a.s.) in the book.

Moscow, June 1993 Vladimir Kalashnikov