Contents
Acknowledgments v
Introduction 1
1 Fluid Queues with Infinite Buffers 13
1.1 Background . . . 14
1.2 Differential Equations . . . 16
1.3 Stationary Density . . . 20
1.4 First Return Probabilities . . . 26
1.5 Expected Number of Crossings . . . 31
1.6 Boundary Probability Vector . . . 33
1.7 General Fluid Input Rates . . . 35
1.8 General Fluid Input Rates by a Probabilistic Approach . . 38
1.9 Performance Measures . . . 43
1.10 A Closely Related Expression . . . 45
1.11 Phase-type Representation . . . 52
1.12 Wiener-Hopf Factorization . . . 57
2 Algorithms 63 2.1 Discrete-Time Homogeneous QBDs . . . 64
2.2 Uniformization and Interpretation . . . 67
2.3 Uniformization Using Different Parameters . . . 72
2.4 Numerical Illustration . . . 74
2.5 Convergence Analysis . . . 77
iv Contents
2.6 Other Algorithms . . . 80
2.6.1 First-Exit Algorithm . . . 81
2.6.2 Newton’s Method . . . 81
3 Fluid Queues with Finite Buffers 83 3.1 Finite QBDs . . . 84
3.2 Finite Buffer Fluid Queues: Background . . . 86
3.3 Stationary Density . . . 88
3.4 Expected Number of Crossings . . . 90
3.5 Boundary Probability Vectors . . . 96
3.6 General Fluid Input Rates . . . 103
3.7 Performance Measures . . . 104
3.8 Numerical Illustration . . . 110
4 Feedback Fluid Queues 113 4.1 Fluid Model of TCP . . . 114
4.2 Infinite Buffer and Feedback Control . . . 116
4.3 Finite Buffer and Feedback Control . . . 118
4.4 General Fluid Input Rates . . . 120
4.5 Numerical Illustration . . . 122
4.6 Fluid Queues with Thresholds . . . 125
4.7 Sticky and Repulsive Thresholds . . . 134
5 Level-Phase Independence 143 5.1 Introduction . . . 143
5.2 Asymptotic Independence . . . 145
5.3 Censoring Out Level Zero . . . 146
5.4 Exact Independence . . . 147
5.5 Construction . . . 150
A Markov Processes and PH Distributions 157 A.1 Markov Processes . . . 157
A.2 Poisson Processes and the M/M/1 Queue . . . 161
A.3 Censored Markov Processes . . . 162
A.4 Phase-Type Distributions . . . 163
Conclusion and Perspectives 165
Notations 167
Bibliography 169
