Markov-Modulated Processes & Semiregenerative Phenomena

Contents......Page 10
Preface......Page 6
1. Recurrence and Regeneration......Page 14
2.1 Introduction......Page 20
2.2 Markov Renewal Processes: Basic Definitions......Page 23
2.3 Elementary Properties......Page 24
2.4 The Number of Counts......Page 27
2.5 The Markov Renewal Equation......Page 32
2.6 Limit Theorems......Page 34
2.7 The Semi-Markov Process......Page 35
2.8 Markov-Additive Processes: Basic Definitions......Page 36
2.9 A Matrix Equation......Page 39
2.10 The Markov-Poisson Process......Page 41
2.11 Markov-Compound Poisson Processes; A Special Case......Page 45
2.12 Markov-Compound Poisson Processes; General Case......Page 47
2.13 The Use of Infinitesimal Generators......Page 49
3.1 Introduction......Page 52
3.2 Basic Definitions......Page 54
3.3 Semirecurrent Phenomena......Page 56
3.4 A Continuous-Time Semiregenerative Phenomenon......Page 63
3.5 Semiregenerative Phenomena With Finite Label Set......Page 69
4.1 Introduction......Page 74
4.2 Definitions and Basic Properties......Page 77
4.3 Degenerate Markov Random Walks......Page 80
4.4 Time-Reversed MRW, Extrema, and Semirecurrent Sets......Page 85
4.5 Fluctuation Theory for MRW......Page 91
4.6 The Case Where Means Exist......Page 94
4.7 Renewal Equation......Page 98
4.8 Wiener-Hopf Factorization......Page 100
4.9 First Exit Time From a Bounded Interval......Page 104
5.1 Introduction......Page 106
5.2 A Sequence of Normalized Increments......Page 107
5.3 Limit Theorems......Page 110
5.4 Application......Page 113
6. Markov-Additive Processes of Arrivals......Page 116
6.1 Introduction......Page 117
6.2 Univariate MAPs of Arrivals......Page 118
6.3 MAPs of Arrivals......Page 122
6.4 Some Properties of MAPs of Arrivals......Page 127
6.5 Some Properties of MAPs......Page 137
6.6 Transformations of MAPs of Arrivals......Page 141
6.7 Markov-Bernoulli Recording of MAPs of Arrivals......Page 147
7.1 Introduction......Page 154
7.2 Preliminary Results......Page 156
7.3 Waiting and Idle Times......Page 159
7.4 Markov-modulated M/M/1 Queue......Page 164
8.1 Introduction......Page 168
8.2 The Model......Page 172
8.3 The Actual and the Unsatisfied Demands......Page 176
8.4 The Inverse of the Demand......Page 180
8.5 The Busy-Period Process......Page 183
8.6 Some More Notations and Results......Page 188
8.7 Laplace Transform of the Busy Period......Page 191
8.8 The Unsatis ed Demand and Demand Rejection Rate......Page 193
8.9 The Storage Level and Unsatis ed Demand .......Page 195
8.10 The Steady State......Page 197
9.1 Introduction......Page 202
9.2 Preliminary Results......Page 206
9.3 The Imbedded MRW......Page 208
9.4 The Main Results......Page 214
Bibliography......Page 220
Index......Page 228