Contents......Page 8
Preface......Page 6
Notation and Conventions......Page 12
Part A: Simple Markovian Models......Page 14
1 Preliminaries......Page 16
2 Aspects of Renewal Theory in Discrete Time......Page 20
3 Stationarity......Page 24
4 Limit Theory......Page 29
5 Harmonic Functions, Martingales and Test Functions......Page 33
6 Nonnegative Matrices......Page 38
7 The Fundamental Matrix, Poisson’s Equation and the CLT......Page 42
8 Foundations of the General Theory of Markov Processes......Page 45
1 Basic Structure......Page 52
2 The Minimal Construction......Page 54
3 The Intensity Matrix......Page 57
4 Stationarity and Limit Results......Page 63
5 Time Reversibility......Page 69
1 Generalities......Page 73
2 General Birth–Death Processes......Page 84
3 Birth–Death Processes as Queueing Models......Page 88
4 The Phase Method......Page 93
5 Renewal Theory for Phase–Type Distributions......Page 101
6 Lindley Processes......Page 105
7 A First Look at Reflected Lévy Processes......Page 109
8 Time–Dependent Properties of M/M/1......Page 111
9 Waiting Times and Queue Disciplines in M/M/1......Page 121
1 Poisson Departure Processes and Series of Queues......Page 127
2 Jackson Networks......Page 130
3 Insensitivity in Erlang’s Loss System......Page 136
4 Quasi–Reversibility and Single–Node Symmetric Queues......Page 138
5 Quasi–Reversibility in Networks......Page 144
6 The Arrival Theorem......Page 146
Part B: Some General Tools and Methods......Page 150
1 Renewal Processes......Page 151
2 Renewal Equations and the Renewal Measure......Page 156
3 Stationary Renewal Processes......Page 163
4 The Renewal Theorem in Its Equivalent Versions......Page 166
5 Proof of the Renewal Theorem......Page 171
6 Second–Moment Results......Page 172
7 Excessive and Defective Renewal Equations......Page 175
1 Basic Limit Theory......Page 181
2 First Examples and Applications......Page 185
3 Time–Average Properties......Page 190
4 Rare Events and Extreme Values......Page 192
1 Spread–Out Distributions......Page 199
2 The Coupling Method......Page 202
3 Markov Processes: Regeneration and Harris Recurrence......Page 211
4 Markov Renewal Theory......Page 219
5 Semi–Regenerative Processes......Page 224
6 Palm Theory, Rate Conservation and PASTA......Page 226
1 Basic Definitions......Page 233
2 Ladder Processes and Classification......Page 236
3 Wiener–Hopf Factorization......Page 240
4 The Spitzer–Baxter Identities......Page 242
5 Explicit Examples. M/G/1, GI/M/1, GI/ PH/1......Page 246
1 Lévy Processes......Page 257
2 Reflection and Loynes’s Lemma......Page 263
3 Martingales and Transforms for Reflected Lévy Processes......Page 268
4 A More General Duality......Page 273
Part C: Special Models and Methods......Page 278
1 Notation. The Actual Waiting Time......Page 279
2 The Moments of the Waiting Time......Page 282
3 The Workload......Page 285
4 Queue Length Processes......Page 289
5 M/G/1 and GI/M/1......Page 292
6 Continuity of the Waiting Time......Page 297
7 Heavy Traffic Limit Theorems......Page 299
8 Light Traffic......Page 303
9 Heavy–Tailed Asymptotics......Page 308
1 Some Basic Examples......Page 315
2 Markov Additive Processes......Page 322
3 The Matrix Paradigms GI/M/1 and M/G/1......Page 329
4 Solution Methods......Page 341
5 The Ross Conjecture and Other Ordering Results......Page 349
1 Comparisons with GI/G/1......Page 353
2 Regeneration and Existence of Limits......Page 357
3 The GI/M/s Queue......Page 361
1 Exponential Families......Page 365
2 Large Deviations, Saddlepoints and the Relaxation Time......Page 368
3 Change of Measure: General Theory......Page 371
4 First Applications......Page 375
5 Cramér–Lundberg Theory......Page 378
6 Siegmund’s Corrected Heavy Traffic Approximations......Page 382
7 Rare Events Simulation......Page 386
8 Markov Additive Processes......Page 389
1 Compound Poisson Dams with General Release Rule......Page 393
2 Some Examples......Page 400
3 Finite Buffer Capacity Models......Page 402
4 Some Simple Inventory Models......Page 409
5 Dual Insurance Risk Models......Page 412
6 The Time to Ruin......Page 414
A1 Polish Spaces and Weak Convergence......Page 420
A2 Right–Continuity and the Space D......Page 421
A3 Point Processes......Page 423
A4 Stochastical Ordering......Page 424
A6 Geometric Trials......Page 425
A8 Total Variation Convergence......Page 426
A10 Stopping Times and Wald’s Identity......Page 427
A11 Discrete Skeletons......Page 428
Bibliography......Page 429
C......Page 444
H......Page 445
L......Page 446
P......Page 447
R......Page 448
S......Page 449
V......Page 450
Z......Page 451