Applied Probability and Queues

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Springer Science & Business Media, May 15, 2003 - Business & Economics - 438 pages
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This 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
LÚvy Processes Reflection and Duality
244
2 Reflection and Loyness Lemma
250
3 Martingales and Transforms for Reflected LÚvy 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 LÚvy 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
Index
431
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Page 417 - Conditional limit theorems relating a random walk to its associate, with applications to risk reserve processes and the G//G/1 queue.

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