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E-grāmata: Switching Processes in Queueing Models

(GlaxoSmithKline, UK)
  • Formāts: PDF+DRM
  • Izdošanas datums: 05-Jan-2010
  • Izdevniecība: ISTE Ltd and John Wiley & Sons Inc
  • Valoda: eng
  • ISBN-13: 9780470393956
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Switching Processes in Queueing ModelsPalielināt
  • Formāts: PDF+DRM
  • Izdošanas datums: 05-Jan-2010
  • Izdevniecība: ISTE Ltd and John Wiley & Sons Inc
  • Valoda: eng
  • ISBN-13: 9780470393956
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While at Kiev University and Bilkent University in Turkey, statistician Anisimov developed switching processes for describing the operation of stochastic systems with the property that their development in time varies spontaneously at some random points of time that may depend on the previous system trajectory. He looks at the limit theorems of averaging principle and diffusion approximation type in the case of fast switching, limit theorems for switching processes with slow switching, and the asymptotic aggregation of switching processes in different time scales. Annotation ©2008 Book News, Inc., Portland, OR (booknews.com)

Switching processes, invented by the author in 1977, is the main tool used in the investigation of traffic problems from automotive to telecommunications. The title provides a new approach to low traffic problems based on the analysis of flows of rare events and queuing models. In the case of fast switching, averaging principle and diffusion approximation results are proved and applied to the investigation of transient phenomena for wide classes of overloading queuing networks. The book is devoted to developing the asymptotic theory for the class of switching queuing models which covers models in a Markov or semi-Markov environment, models under the influence of flows of external or internal perturbations, unreliable and hierarchic networks, etc.
Preface 13
Definitions 17
Chapter
1. Switching Stochastic Models		 19
1.1. Random processes with discrete component
19
1.1.1. Markov and semi-Markov processes
21
1.1.2. Processes with independent increments and Markov switching
21
1.1.3. Processes with independent increments and semi-Markov switching
23
1.2. Switching processes
24
1.2.1. Definition of switching processes
24
1.2.2. Recurrent processes of semi-Markov type (simple case)
26
1.2.3. RPSM with Markov switching
26
1.2.4. General case of RPSM
27
1.2.5. Processes with Markov or semi-Markov switching
27
1.3. Switching stochastic models
28
1.3.1. Sums of random variables
29
1.3.2. Random movements
29
1.3.3. Dynamic systems in a random environment
30
1.3.4. Stochastic differential equations in a random environment
30
1.3.5. Branching processes
31
1.3.6. State-dependent flows
32
1.3.7. Two-level Markov systems with feedback
32
1.4. Bibliography
33
Chapter
2. Switching Queueing Models		 37
2.1. Introduction
37
2.2. Queueing systems
38
2.2.1. Markov queueing models
38
2.2.1.1. A state-dependent system MQ/MQ/1/infinity
39
2.2.1.2. Queueing system MM,Q/MM,Q/1/m
40
2.2.1.3. System MQB/MQB/1/infinity
41
2.2.2. Non-Markov systems
42
2.2.2.1. Semi-Markov system SM/MSM,Q/1
42
2.2.2.2. System MSM,Q/MSM,Q/1/infinity
43
2.2.2.3. System MSM,Q/MSM,Q/1/V
44
2.2.3. Models with dependent arrival flows
45
2.2.4. Polling systems
46
2.2.5. Retrial queueing systems
47
2.3. Queueing networks
48
2.3.1. Markov state-dependent networks
49
2.3.1.1. Markov network (MQ/MQ/m/oo)infinity
49
2.3.1.2. Markov networks (MQ,B/MQ,Bm/infinity)r with batches
50
2.3.2. Non-Markov networks
50
2.3.2.1. State-dependent semi-Markov networks
50
2.3.2.2. Semi-Markov networks with random batches
52
2.3.2.3. Networks with state-dependent input
53
2.4. Bibliography
54
Chapter
3. Processes of Sums of Weakly-dependent Variables		 57
3.1. Limit theorems for processes of sums of conditionally independent random variables
57
3.2. Limit theorems for sums with Markov switching
65
3.2.1. Flows of rare events
67
3.2.1.1. Discrete time
67
3.2.1.2. Continuous time
69
3.3. Quasi-ergodic Markov processes
70
3.4. Limit theorems for non-homogenous Markov processes
73
3.4.1. Convergence to Gaussian processes
74
3.4.2. Convergence to processes with independent increments
78
3.5. Bibliography
81
Chapter
4. Averaging Principle and Diffusion Approximation for Switching Processes		 83
4.1. Introduction
83
4.2. Averaging principle for switching recurrent sequences
84
4.3. Averaging principle and diffusion approximation for RPSMs
88
4.4. Averaging principle and diffusion approximation for recurrent processes of semi-Markov type (Markov case)
95
4.4.1. Averaging principle and diffusion approximation for SMP
105
4.5. Averaging principle for RPSM with feedback
106
4.6. Averaging principle and diffusion approximation for switching processes
108
4.6.1. Averaging principle and diffusion approximation for processes with semi-Markov switching
112
4.7. Bibliography
113
Chapter
5. Averaging and Diffusion Approximation in Overloaded Switching Queueing Systems and Networks		 117
5.1. Introduction
117
5.2. Markov queueing models
120
5.2.1. System MQB/MQB/1/infinity
121
5.2.2. System MQ/MQ/1/infinity
124
5.2.3. Analysis of the waiting time
129
5.2.4. An output process
131
5.2.5. Time-dependent system MC,t/MQ,t/1/infinity
132
5.2.6. A system with impatient calls
134
5.3. Non-Markov queueing models
135
5.3.1. System GI/MQ/1/infinity
135
5.3.2. Semi-Markov system SM/MSM,Q/1/infinity
136
5.3.3. System MSM,Q/MSM,Q/1/infinity
138
5.3.4. System SMQ/MSM,Q/1/infinity
139
5.3.5. System GQ/MQ/1/infinity
142
5.3.6. A system with unreliable servers
143
5.3.7. Polling systems
145
5.4. Retrial queueing systems
146
5.4.1. Retrial system MQ/G/1/2.r
147
5.4.2. System M/G/1/w.r
150
5.4.3. Retrial system M/M/m/w.r
154
5.5. Queueing networks
159
5.5.1. State-dependent Markov network (MQ/MQ/1/infinity)r
159
5.5.2. Markov state-dependent networks with batches
161
5.6. Non-Markov queueing networks
164
5.6.1. A network (MSM,Q/MSM,Q/1/infinity)r with semi-Markov switching
164
5.6.2. State-dependent network with recurrent input
169
5.7. Bibliography
172
Chapter
6. Systems in Low Traffic Conditions		 175
6.1. Introduction
175
6.2. Analysis of the first exit time from the subset of states
176
6.2.1. Definition of S-set
176
6.2.2. An asymptotic behavior of the first exit time
177
6.2.3. State space forming a monotone structure
180
6.2.4. Exit time as the time of first jump of the process of sums with Markov switching
182
6.3. Markov queueing systems with fast service
183
6.3.1. M/M/s/m systems
183
6.3.1.1. System MM/M/l/m in a Markov environment
185
6.3.2. Semi-Markov queueing systems with fast service
188
6.4. Single-server retrial queueing model
190
6.4.1. Case 1: fast service
191
6.4.1.1. State-dependent case
194
6.4.2. Case 2: fast service and large retrial rate
195
6.4.3. State-dependent model in a Markov environment
197
6.5. Multiserver retrial queueing models
201
6.6. Bibliography
204
Chapter
7. Flows of Rare Events in Low and Heavy Traffic Conditions		 207
7.1. Introduction
207
7.2. Flows of rare events in systems with mixing
208
7.3. Asymptotically connected sets (Vn,-S-sets)
211
7.3.1. Homogenous case
211
7.3.2. Non-homogenous case
214
7.4. Heavy traffic conditions
215
7.5. Flows of rare events in queueing models
216
7.5.1. Light traffic analysis in models with finite capacity
216
7.5.2. Heavy traffic analysis
218
7.6. Bibliography
219
Chapter
8. Asymptotic Aggregation of State Space		 221
8.1. Introduction
221
8.2. Aggregation of finite Markov processes (stationary behavior)
223
8.2.1. Discrete time
223
8.2.2. Hierarchic asymptotic aggregation
225
8.2.3. Continuous time
227
8.3. Convergence of switching processes
228
8.4. Aggregation of states in Markov models
231
8.4.1. Convergence of the aggregated process to a Markov process (finite state space)
232
8.4.2. Convergence of the aggregated process with a general state space
236
8.4.3. Accumulating processes in aggregation scheme
237
8.4.4. MP aggregation in continuous time
238
8.5. Asymptotic behavior of the first exit time from the subset of states (non-homogenous in time case)
240
8.6. Aggregation of states of non-homogenous Markov processes
243
8.7. Averaging principle for RPSM in the asymptotically aggregated Markov environment
246
8.7.1. Switching MP with a finite state space
247
8.7.2. Switching MP with a general state space
250
8.7.3. Averaging principle for accumulating processes in the asymptotically aggregated semi-Markov environment
251
8.8. Diffusion approximation for RPSM in the asymptotically aggregated Markov environment
252
8.9. Aggregation of states in Markov queueing models
255
8.9.1. System MQ/MQ/r/infinity with unreliable servers in heavy traffic
255
8.9.2. System MM,Q/MM,Q/1/infinity in heavy traffic
256
8.10. Aggregation of states in semi-Markov queueing models
258
8.10.1. System SM/MSM,Q/1/infinity
258
8.10.2. System MSM,Q/MSM,Q/1/infinity
259
8.11. Analysis of flows of lost calls
260
8.12. Bibliography
263
Chapter
9. Aggregation in Markov Models with Fast Markov Switching		 267
9.1. Introduction
267
9.2. Markov models with fast Markov switching
269
9.2.1. Markov processes with Markov switching
269
9.2.2. Markov queueing systems with Markov type switching
271
9.2.3. Averaging in the fast Markov type environment
272
9.2.4. Approximation of a stationary distribution
274
9.3. Proofs of theorems
275
9.3.1. Proof of Theorem 9.1
275
9.3.2. Proof of Theorem 9.2
277
9.3.3. Proof of Theorem 9.3
279
9.4. Queueing systems with fast Markov type switching
279
9.4.1. System MM,Q/MM,Q/1/N
279
9.4.1.1. Averaging of states of the environment
279
9.4.1.2. The approximation of a stationary distribution
280
9.4.2. Batch system BMM,Q/BMM,Q/1/N
281
9.4.3. System M/M/s/m with unreliable servers
282
9.4.4. Priority model MQ/MQ/m/s, N
283
9.5. Non-homogenous in time queueing models
285
9.5.1. System MM,Q,t/MM,Q,t/s/m with fast switching – averaging of states
286
9.5.2. System MM,Q/MM,Q/s/m with fast switching – aggregation of states
287
9.6. Numerical examples
288
9.7. Bibliography
289
Chapter
10. Aggregation in Markov Models with Fast Semi-Markov Switching		 291
10.1. Markov processes with fast semi-Markov switches
292
10.1.1. Averaging of a semi-Markov environment
292
10.1.2. Asymptotic aggregation of a semi-Markov environment
300
10.1.3. Approximation of a stationary distribution
305
10.2. Averaging and aggregation in Markov queueing systems with semi-Markov switching
309
10.2.1. Averaging of states of the environment
309
10.2.2. Asymptotic aggregation of states of the environment
310
10.2.3. The approximation of a stationary distribution
311
10.3. Bibliography
313
Chapter
11. Other Applications of Switching Processes		 315
11.1. Self-organization in multicomponent interacting Markov systems
315
11.2. Averaging principle and diffusion approximation for dynamic systems with stochastic perturbations
319
11.2.1. Recurrent perturbations
319
11.2.2. Semi-Markov perturbations
321
11.3. Random movements
324
11.3.1. Ergodic case
324
11.3.2. Case of the asymptotic aggregation of state space
325
11.4. Bibliography
326
Chapter
12. Simulation Examples		 329
12.1. Simulation of recurrent sequences
329
12.2. Simulation of recurrent point processes
331
12.3. Simulation of RPSM
332
12.4. Simulation of state-dependent queueing models
334
12.5. Simulation of the exit time from a subset of states of a Markov chain
337
12.6. Aggregation of states in Markov models
340
Index 343
Vladimir V. Anisimov is currently Director of the Research Statistics Unit at GlaxoSmithKline, UK. He has written about 200 papers, nine books and manuals in this area.
Biežāk uzdotie jautājumi par e-grāmatām Biežāk uzdotie jautājumi par e-grāmatām
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