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Contents

Remerciements i

Table of contents iii

Introduction 2

1 Preliminaries 13

1 Functions of Matrices . . . 14

2 Matrix Equations . . . 17

3 Generalized Inverses for Markov Chains . . . 26

4 The Deviation Matrix . . . 32

5 Quasi-Birth-and-Death Process . . . 34

6 Fluid Models . . . 38

7 Phase-Type Distributions . . . 46

2 Sensitivity Analysis 50 1 Geometric Ergodicity . . . 52

2 Drift Condition for a QBD Process . . . 58

3 Perturbation of a QBD Process . . . 59

4 Perturbation of an M/PH/1 Queue . . . 67

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Table of contents

3 The Poisson Equation 73

1 Row-Block Updating for Finite Chains . . . 75

2 Makowski and Shwartz’s Poisson Solution . . . 80

3 Expected Return Times in QBDs . . . 83

4 Probabilistic Solution for QBDs . . . 88

5 The General Solution for a QBD process . . . 92

6 Sensitivity Analysis of a PH/M/1 Queue . . . 100

4 Computation of the Deviation Matrix 107 1 Recursive Equation for a Finite QBD . . . 109

2 Explicit Formula for the blocks of a QBD process . . . 114

5 Perturbation Analysis for Fluid Models 118 1 First Return Probabilities . . . 120

2 Sheep Behavior for Phases in S

0

. . . 124

3 General Case . . . 128

6 Transient Distributions of MMRWs 132 1 Erlangization . . . 134

2 Laplace-Stieltjes Transforms . . . 137

3 Erlangized Models . . . 140

4 Transient Distributions . . . 146

5 Numerical Illustration . . . 150

6 Link with Bilateral PH Distributions . . . 154

7 Illustration . . . 159

8 Computational Issue . . . 162 7 Transient Distributions of Regulated MMRWs 168

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Table of contents

1 Boundary Effect . . . 169

2 Transient distributions when Z(0) = 0 . . . 173

3 Transient distributions when Z(0) = a > 0 . . . 176

4 Joint Distributions . . . 178

Perspectives 183

Bibliography 184

v

Références

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