Probabilistic Modelling

Probabilistic modelling is the most cost-effective means of performance and reliability evaluation of complex dynamic systems. This self-contained text will be welcomed by students and teachers for its no-nonsense treatment of the basic results and examples of their application. The only mathematical background that is assumed is basic calculus. The necessary fundamentals of probability theory are included, as well as an introduction to renewal, Poisson and Markov processes. Models arising in the fields of manufacturing, computing and communications, involving single or multiple service stations and one or more customer classes, are examined in some detail. Both exact and approximate solution methods are discussed, including recent techniques such as spectral expansion. Special attention is devoted to models of systems subject to breakdowns and repairs. Throughout the book, strong emphasis is placed on explaining the ideas behind the results and helping the reader to use them, making the book ideal for students in computer science, engineering or operations research taking courses in modern system design.

Preface

ix

1 Introduction to probability theory

1

(46)

1.1 Sample points, events and probabilities

1

(9)

1.1.1 An algebra of events

3

(1)

1.1.2 Probabilities

4

(3)

1.1.3 Conditional probability

7

(3)

1.2 Random variables

10

(10)

1.2.1 Distribution functions

11

(2)

1.2.2 Probability density functions

13

(2)

1.2.3 Joint distributions

15

(3)

1.2.4 Largest and smallest

18

(2)

1.3 Expectation and other moments

20

(8)

1.3.1 Properties of expectation

23

(1)

1.3.2 Variance and covariance

24

(4)

1.4 Bernoulli trials and related random variables

28

(8)

1.4.1 The geometric distribution

28

(3)

1.4.2 The binomial distribution

31

(5)

1.5 Sums, transforms and limits

36

(10)

1.5.1 Generating functions

37

(2)

1.5.2 Laplace transforms and characteristic functions

39

(2)

1.5.3 The normal distribution and central limit theorem

41

(5)

1.6 Literature

46

(1)

2 Arrivals and services

47

(27)

2.1 Renewal processes

47

(8)

2.1.1 Forward and backward renewal times

50

(3)

2.1.2 The `paradox' of residual life

53

(2)

2.2 The exponential distribution and its properties

55

(4)

2.2.1 First and last

56

(3)

2.3 The Poisson process

59

(10)

2.3.1 Properties of the Poisson process

61

(6)

2.3.2 Arrivals during a random interval

67

(2)

2.4 Application: ALOHA and CSMA

69

(4)

2.5 Literature

73

(1)

3 Queueing systems: average performance

74

(48)

3.1 Little's theorem and applications

75

(6)

3.1.1 Utilization and response time laws

77

(4)

3.2 The single-class M/G/1 queue

81

(11)

3.2.1 Busy periods

86

(2)

3.2.2 The M/M/1 queue

88

(4)

3.3 Different scheduling policies

92

(7)

3.3.1 The Processor-Sharing policy

94

(2)

3.3.2 Symmetric policies and multiclass queues

96

(2)

3.3.3 Which policy is better?

98

(1)

3.4 Priority scheduling

99

(8)

3.4.1 Preemptive-resume priorities

104

(1)

3.4.2 Averaging and lumping job types

105

(2)

3.5 Optimal scheduling policies

107

(12)

3.5.1 A conservation law for waiting times

108

(2)

3.5.2 The c/p rule

110

(2)

3.5.3 Optimization with preemptions

112

(4)

3.5.4 Characterization of achievable performance

116

(3)

3.6 Literature

119

(3)

4 Queueing networks

122

(34)

4.1 Open networks

124

(12)

4.1.1 Traffic equations

125

(3)

4.1.2 Performance measures

128

(8)

4.2 Closed networks

136

(10)

4.2.1 Mean value analysis

138

(8)

4.3 Multiclass networks

146

(8)

4.3.1 Closed multiclass networks

150

(4)

4.4 Literature

154

(2)

5 Markov chains and processes

156

(45)

5.1 Markov chains

157

(11)

5.1.1 Steady state

161

(7)

5.2 A Markov chain embedded in the M/G/1 model

168

(4)

5.3 Markov processes

172

(12)

5.3.1 Transient behaviour

178

(2)

5.3.2 Steady state

180

(4)

5.4 First passages and absorptions

184

(2)

5.5 Birth-and-Death queueing models

186

(13)

5.5.1 The M/M/n model

188

(4)

5.5.2 The M/M/n/n and M/M/XXX systems

192

(2)

5.5.3 The M/M/1/N queue

194

(1)

5.5.4 The M/M/n/./K model

195

(4)

5.6 Literature

199

(2)

6 Queues in Markovian environments

201

(20)

6.1 Quasi-Birth-and-Death models

202

(8)

6.1.1 The MMPP/M/1 queue

204

(1)

6.1.2 A multiserver queue with breakdowns and repairs

205

(2)

6.1.3 Manufacturing blocking

207

(1)

6.1.4 Phase-type distributions

208

(2)

6.2 Solution methods

210

(6)

6.2.1 Spectral expansion method

211

(4)

6.2.2 Matrix-geometric solution

215

(1)

6.3 Generalizations

216

(3)

6.4 Literature

219

(2)

Index

221

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