Table of Contents
Preface v
1 Recurrence and Regeneration 1
2 Markov Renewal and Markov-Additive Processes 7
2.1 Introduction 7
2.2 Markov Renewal Processes: Basic Definitions 10
2.3 Elementary Properties 11
2.4 The Number of Counts 14
2.5 The Markov Renewal Equation 19
2.6 Limit Theorems 21
2.7 The Semi-Markov Process 22
2.8 Markov-Additive Processes: Basic Definitions 23
2.9 A Matrix Equation 26
2.10 The Markov-Poisson Process 28
2.11 Markov-Compound Poisson Processes; A Special Case 32
2.12 Markov-Compound Poisson Processes; General Case 34
2.13 The Use of Infinitesimal Generators 36
3 Theory of Semiregenerative Phenomena 39
3.1 Introduction 39
3.2 Basic Definitions 41
3.3 Semirecurrent Phenomena 43
3.4 A Continuous-Time Semiregenerative Phenomenon 50
3.5 Semiregenerative Phenomena With Finite Label Set 56
4 Theory of Markov Random Walks 61
4.1 Introduction 61
4.2 Definitions and Basic Properties 64
4.3 Degenerate Markov Random Walks 67
4.4 Time-Reversed MRW, Extrema, and Semirecurrent Sets 72
4.5 Fluctuation Theory for MRW 78
4.6 The Case Where Means Exist 81
4.7 Renewal Equation 85
4.8 Wiener-Hopf Factorization 87
4.9 First Exit Time From a Bounded Interval 91
5 Limit Theorems for Markov Random Walks 93
5.1 Introduction 93
5.2 A Sequence of Normalized Increments 94
5.3 Limit Theorems 97
5.4 Application 100
6 Markov-Additive Processes of Arrivals 103
6.1 Introduction 104
6.2 Univariate MAPs of Arrivals 105
6.3 MAPs of Arrivals 109
6.4 Some Properties of MAPs of Arrivals 114
6.5 Some Properties of MAPs 124
6.6 Transformations of MAPs of Arrivals 128
6.7 Markov-Bernoulli Recording of MAPs of Arrivals 134
7Markov-Modulated Single Server Queueing Systems 141
7.1 Introduction 141
7.2 Preliminary Results 143
7.3 Waiting and Idle Times 146
7.4 Markov-modulated M/M/1 Queue 151
8 A Storage Model for Data Communication Systems 155
8.1 Introduction 155
8.2 The Model 159
8.3 The Actual and the Unsatisfied Demands 163
8.4 The Inverse of the Demand 167
8.5 The Busy-Period Process 170
8.6 Some More Notations and Results 175
8.7 Laplace Transform of the Busy Period 178
8.8 The Unsatisfied Demand and Demand Rejection Rate 180
8.9 The Storage Level and Unsatisfied Demand 182
8.10 The Steady State 184
9 A Markovian Storage Model 189
9.1 Introduction 189
9.2 Preliminary Results 193
9.3 The Imbedded MRW 195
9.4 The Main Results 201
Bibliography 207
Index 215
