Introduction to Matrix Analytic Methods in Stochastic Modeling

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1999－Introduction to Matrix Analytic Methods in Stochastic Modeling(ASA-SIAM Series on Statistics and Applied Probability)
By G. Latouche （Universite Libre de Bruxelles Brussels, Belgium ）and V. Ramaswami （AT&T Labs Holmdel, New Jersey ）

published by the American Statistical Association and the Society for Industrial and Applied Mathematics. 



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关键词：introduction Stochastic troduction Stochast Analytic Methods Modeling matrix introduction Analytic

Contents
Preface ix

I Quasi-Birth-and-Death Processes 1
1 Examples 3
1.1 The M/M/1 Queue 3
1.2 The M/M/1 Queue in a Random Environment 5
1.3 Phase-Type Queues 8
1.4 A Queue with Two Priority Classes 18
1.5 Tandem Queues with Blocking 19
1.6 Multiprogramming Queues 21
1.7 Open Jackson Networks 25
1.8 A Retrial Queue 28


II The Method of Phases 31

2 PH Distributions 33
2.1 The Exponential Distribution 34
2.2 Simple Generalizations 36
2.3 PH Random Variables 39
2.4 Distribution and Moments 41
2.5 Discrete PH Distributions 47
2.6 Closure Properties 50
2.7 Further Properties 55
2.8 Algorithms 56

3 Markovian Point Processes 61
3.1 The PH Renewal Process 61
3.2 The Number of Renewals 65
3.3 The Stationary Process 70
3.4 Discrete Time PH Renewal Processes 71
3.5 A General Markovian Point Process 72
3.6 Analysis of the General Process 75


III The Matrix-Geometric Distribution 81

4 Birth-and-Death Processes 83
4.1 Terminating Renewal Processes 84
4.2 Discrete Time Birth-and-Death Processes 87
4.3 The Basic Nonlinear Equations 94
4.4 Transient Distributions 99
4.5 Continuous Time Birth-and-Death Processes 99
4.6 Passage and Sojourn Times 102

5 Processes Under a Taboo 107
5.1 Expected Time to Exit 108
5.2 Linking Subsets of States 109
5.3 Local Time 118
5.4 Gaussian Elimination 120
5.5 Continuous Time Processes 122

6 Homogeneous QBDs 129
6.1 Definitions 129
6.2 The Matrix-Geometric Property 130
6.3 Boundary Distribution 139
6.4 Continuous Time QBDs 141

7 Stability Condition 147
7.1 First Passage Times 147
7.2 Simple Drift Condition 151
7.3 Drift Conditions in General 159


IV Algorithms 163

8 Algorithms for the Rate Matrix 165

8.1 A Basic Algorithm 167
8.2 Analysis of the Linear Algorithm 175
8.3 An Improved Algorithm 179
8.4 Quadratically Convergent Algorithms 187
8.5 Special Structure 197
8.6 Boundary Probabilities 199
8.7 The Continuous Time Case 199

9 Spectral Analysis 203
9.1 The Caudal Characteristic 204
9.2 Examples 206
9.3 The Eigenvalues of the Matrix G 212
9.4 The Jordan Normal Form 213
9.5 Construction of the Towers 216

10 Finite QBDs 221
10.1 Linear Level Reduction 222
10.2 The Method of Folding 228
10.3 Matrix-Geometric Combination 230
10.4 Subdiagonal Matrices of Rank 1 235
11 First Passage Times 239
11.1 Generating Functions 240
11.2 Linear Level Reduction 241
11.3 Reduction by Bisection 245
11.4 Odd-Even Reduction 249


V Beyond Simple QBDs 257

12 Nonhomogeneous QBDs 259
12.1 The Stationary Distribution 260
12.2 Algorithmic Approaches

13 Processes, Skip-Free in One Direction 267
13.1 Markov Chains of M/G/1-Type 268
13.2 Markov Chains of GI/M/1-Type 275

14 Tree Processes 281
14.1 The M/PH/1 LIFO Queue 281
14.2 Tree-Like Transition Diagrams 284
14.3 Matrix-Product Form Distribution 289
14.4 Skip-Free Processes 294

15 Product Form Networks 295
15.1 Independence of Level and Phase 295
15.2 Network of Exponential Servers 299
15.3 The General Case 302

16 Nondenumerable States 305
16.1 General Irreducible Markov Chains 305
16.2 The Operator-Geometric Property 307
16.3 Computational Issues 310

Bibliography 313
Index 325

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