Henk C. Tijms - Stochastic Models.pdf

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Front Matter
566977996.001.png
A First Course
in Stochastic Models
Henk C. Tijms
Vrije Universiteit, Amsterdam, The Netherlands
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Copyright c
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Library of Congress Cataloging-in-Publication Data
Tijms, H. C.
A first course in stochastic models / Henk C. Tijms.
p. cm.
Includes bibliographical references and index.
ISBN 0-471-49880-7 (acid-free paper)—ISBN 0-471-49881-5 (pbk. : acid-free paper)
1. Stochastic processes. I. Title.
QA274.T46 2003
519.2 3—dc21
2002193371
British Library Cataloguing in Publication Data
A catalogue record for this book is available from the British Library
ISBN 0-471-49880-7 (Cloth)
ISBN 0-471-49881-5 (Paper)
Typeset in 10/12pt Times from L A T E X files supplied by the author, by Laserwords Private Limited,
Chennai, India
Printed and bound in Great Britain by T J International Ltd, Padstow, Cornwall
This book is printed on acid-free paper responsibly manufactured from sustainable forestry
in which at least two trees are planted for each one used for paper production.
2003
Contents
Preface
ix
1 The Poisson Process and Related Processes
1
1.0 Introduction
1
1.1 The Poisson Process
1
1.1.1 The Memoryless Property
2
1.1.2 Merging and Splitting of Poisson Processes
6
1.1.3 The M/G/ ∞ Queue
9
1.1.4 The Poisson Process and the Uniform Distribution
15
1.2 Compound Poisson Processes
18
1.3 Non-Stationary Poisson Processes
22
1.4 Markov Modulated Batch Poisson Processes
24
Exercises
28
Bibliographic Notes
32
References
32
2 Renewal-Reward Processes
33
2.0 Introduction
33
2.1 Renewal Theory
34
2.1.1 The Renewal Function
35
2.1.2 The Excess Variable
37
2.2 Renewal-Reward Processes
39
2.3 The Formula of Little
50
2.4 Poisson Arrivals See Time Averages
53
2.5 The Pollaczek–Khintchine Formula
58
2.6 A Controlled Queue with Removable Server
66
2.7 An Up- And Downcrossing Technique
69
Exercises
71
Bibliographic Notes
78
References
78
3 Discrete-Time Markov Chains
81
3.0 Introduction
81
3.1 The Model
82
vi
CONTENTS
3.2 Transient Analysis
87
3.2.1 Absorbing States
89
3.2.2 Mean First-Passage Times
92
3.2.3 Transient and Recurrent States
93
3.3 The Equilibrium Probabilities
96
3.3.1 Preliminaries
96
3.3.2 The Equilibrium Equations
98
3.3.3 The Long-run Average Reward per Time Unit
103
3.4 Computation of the Equilibrium Probabilities
106
3.4.1 Methods for a Finite-State Markov Chain
107
3.4.2 Geometric Tail Approach for an Infinite State Space
111
3.4.3 Metropolis—Hastings Algorithm
116
3.5 Theoretical Considerations
119
3.5.1 State Classification
119
Exercises
134
Bibliographic Notes
139
References
139
4 Continuous-Time Markov Chains
141
4.0 Introduction
141
4.2 The Flow Rate Equation Method
142
4.3 Ergodic Theorems
154
4.4 Markov Processes on a Semi-Infinite Strip
157
4.5 Transient State Probabilities
162
4.5.1 The Method of Linear Differential Equations
163
4.5.2 The Uniformization Method
166
4.5.3 First Passage Time Probabilities
170
4.6 Transient Distribution of Cumulative Rewards
172
4.6.1 Transient Distribution of Cumulative Sojourn Times
173
4.6.2 Transient Reward Distribution for the General Case
176
Exercises
179
Bibliographic Notes
185
References
185
5 Markov Chains and Queues
187
5.0 Introduction
187
5.1 The Erlang Delay Model
187
5.1.1 The M/M/ 1 Queue
188
5.1.2 The M/M/c Queue
190
5.1.3 The Output Process and Time Reversibility
192
5.2 Loss Models
194
5.2.1 The Erlang Loss Model
194
5.2.2 The Engset Model
196
5.3 Service-System Design
198
5.4 Insensitivity
202
5.4.1 A Closed Two-node Network with Blocking
203
5.4.2 The M/G/ 1 Queue with Processor Sharing
208
5.5 A Phase Method
209
3.5.2 Ergodic Theorems
126
4.1 The Model
147

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