## Applied Probability and QueuesThis book serves as an introduction to queuing theory and provides a thorough treatment of tools like Markov processes, renewal theory, random walks, Levy processes, matrix-analytic methods and change of measure. It also treats in detail basic structures like GI/G/1 and GI/G/s queues, Markov-modulated models and queuing networks, and gives an introduction to areas such as storage, inventory, and insurance risk. Exercises are included and a survey of mathematical prerequisites is given in an appendix This much updated and expanded second edition of the 1987 original contains an extended treatment of queuing networks and matrix-analytic methods as well as additional topics like Poisson's equation, the fundamental matrix, insensitivity, rare events and extreme values for regenerative processes, Palm theory, rate conservation, Levy processes, reflection, Skorokhod problems, Loynes' lemma, Siegmund duality, light traffic, heavy tails, the Ross conjecture and ordering, and finite buffer problems. Students and researchers in statistics, probability theory, operations research, and industrial engineering will find this book useful. |

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### Contents

Markov Chains | 3 |

2 Aspects of Renewal Theory in Discrete Time | 7 |

3 Stationarity | 11 |

4 Limit Theory | 16 |

5 Harmonic Functions Martingales and Test Functions | 20 |

6 Nonnegative Matrices | 25 |

7 The Fundamental Matrix Poissons Equation and the CLT | 29 |

8 Foundations of the General Theory of Markov Processes | 32 |

4 The SpitzerBaxter Identities | 229 |

5 Explicit Examples MG1 GIM1 GIPH1 | 233 |

Lévy Processes Reflection and Duality | 244 |

2 Reflection and Loyness Lemma | 250 |

3 Martingales and Transforms for Reflected Lévy Processes | 255 |

4 A More General Duality | 260 |

Special Models and Methods | 265 |

SteadyState Properties of GIG1 | 266 |

Markov Jump Processes | 39 |

2 The Minimal Construction | 41 |

3 The Intensity Matrix | 44 |

4 Stationarity and Limit Results | 50 |

5 Time Reversibility | 56 |

Queueing Theory at the Markovian Level | 60 |

2 General BirthDeath Processes | 71 |

3 BirthDeath Processes as Queueing Models | 75 |

4 The Phase Method | 80 |

5 Renewal Theory for PhaseType Distributions | 88 |

6 Lindley Processes | 92 |

7 A First Look at Reflected Lévy Processes | 96 |

8 TimeDependent Properties of MM1 | 98 |

9 Waiting Times and Queue Disciplines in MM1 | 108 |

Queueing Networks and Insensitivity | 114 |

2 Jackson Networks | 117 |

3 Insensitivity in Erlangs Loss System | 123 |

4 QuasiReversibility and SingleNode Symmetric Queues | 125 |

5 QuasiReversibility in Networks | 131 |

6 The Arrival Theorem | 133 |

Some General Tools and Methods | 137 |

Renewal Theory | 138 |

2 Renewal Equations and the Renewal Measure | 143 |

3 Stationary Renewal Processes | 150 |

4 The Renewal Theorem in Its Equivalent Versions | 153 |

5 Proof of the Renewal Theorem | 158 |

6 SecondMoment Results | 159 |

7 Excessive and Defective Renewal Equations | 162 |

Regenerative Processes | 168 |

2 First Examples and Applications | 172 |

3 TimeAverage Properties | 177 |

4 Rare Events and Extreme Values | 179 |

Further Topics in Renewal Theory and Regenerative Processes | 186 |

2 The Coupling Method | 189 |

Regeneration and Harris Recurrence | 198 |

4 Markov Renewal Theory | 206 |

5 SemiRegenerative Processes | 211 |

6 Palm Theory Rate Conservation and PASTA | 213 |

Random Walks | 220 |

2 Ladder Processes and Classification | 223 |

3 WienerHopf Factorization | 227 |

2 The Moments of the Waiting Time | 269 |

3 The Workload | 272 |

4 Queue Length Processes | 276 |

5 MG1 and GIM1 | 279 |

6 Continuity of the Waiting Time | 284 |

7 Heavy Traffic Limit Theorems | 286 |

8 Light Traffic | 290 |

9 HeavyTailed Asymptotics | 295 |

Markov Additive Models | 302 |

2 Markov Additive Processes | 309 |

3 The Matrix Paradigms GIM1 and MG1 | 316 |

4 Solution Methods | 328 |

5 The Ross Conjecture and Other Ordering Results | 336 |

ManyServer Queues | 340 |

2 Regeneration and Existence of Limits | 344 |

3 The GIM₈ Queue | 348 |

Exponential Change of Measure | 352 |

2 Large Deviations Saddlepoints and the Relaxation Time | 355 |

General Theory | 358 |

4 First Applications | 362 |

5 CramerLundberg Theory | 365 |

6 Siegmunds Corrected Heavy Traffic Approximations | 369 |

7 Rare Events Simulation | 373 |

8 Markov Additive Processes | 376 |

Dams Inventories and Insurance Risk | 380 |

2 Some Examples | 387 |

3 Finite Buffer Capacity Models | 389 |

4 Some Simple Inventory Models | 396 |

5 Dual Insurance Risk Models | 399 |

6 The Time to Ruin | 401 |

Appendix | 407 |

A2 RightContinuity and the Space D | 408 |

A3 Point Processes | 410 |

A4 Stochastical Ordering | 411 |

A5 Heavy Tails | 412 |

A7 Semigroups of Positive Numbers | 413 |

A9 Transforms | 414 |

A11 Discrete Skeletons | 415 |

Bibliography | 416 |

431 | |

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### Common terms and phrases

aperiodic arrival Asmussen assume asymptotics bounded Brownian motion Consider continuous Corollary corresponding cycle deﬁne denote density diﬀerent discrete distribution F equivalent ergodic example exists exponential exponential distribution ﬁnite ﬁrst follows formula function given Hence implies independent inﬁnite intensity matrix interarrival distribution irreducible Jackson network jump process ladder height Laplace transform Lemma Levy process limit Markov chain Markov jump process Markov process Markov property Markovian martingale node nonlattice nonnegative notation Notes phase-type Poisson process positive recurrent probability measure Problems Proof of Theorem Proposition queue length Queueing Systems queueing theory random walk reﬂected regenerative renewal process renewal theorem resp Section server Show solution space stationary distribution stationary measure steady–state Stochastic stochastic process traﬃc transition matrix waiting workload yields

### Popular passages

Page 417 - Conditional limit theorems relating a random walk to its associate, with applications to risk reserve processes and the G//G/1 queue.