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Searching... | 30000010138914 | QA274.8 B73 2005 | Open Access Book | Book | Searching... |
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Summary
Summary
The present textbook contains the recordsof a two-semester course on que- ing theory, including an introduction to matrix-analytic methods. This course comprises four hours oflectures and two hours of exercises per week andhas been taughtattheUniversity of Trier, Germany, for about ten years in - quence. The course is directed to last year undergraduate and?rst year gr- uate students of applied probability and computer science, who have already completed an introduction to probability theory. Its purpose is to present - terial that is close enough to concrete queueing models and their applications, while providing a sound mathematical foundation for the analysis of these. Thus the goal of the present book is two-fold. On the one hand, students who are mainly interested in applications easily feel bored by elaborate mathematical questions in the theory of stochastic processes. The presentation of the mathematical foundations in our courses is chosen to cover only the necessary results,which are needed for a solid foundation of the methods of queueing analysis. Further, students oriented - wards applications expect to have a justi?cation for their mathematical efforts in terms of immediate use in queueing analysis. This is the main reason why we have decided to introduce new mathematical concepts only when they will be used in the immediate sequel. On the other hand, students of applied probability do not want any heur- tic derivations just for the sake of yielding fast results for the model at hand.
Table of Contents
List of Figures | p. ix |
Foreword | p. xi |
1 Queues: The Art of Modelling | p. 1 |
Part I Markovian Methods | |
2 Markov Chains and Queues in Discrete Time | p. 11 |
1 Definition | p. 11 |
2 Classification of States | p. 15 |
3 Stationary Distributions | p. 20 |
4 Restricted Markov Chains | p. 27 |
5 Conditions for Positive Recurrence | p. 29 |
6 The M/M/1 queue in discrete time | p. 31 |
3 Homogeneous Markov Processes on Discrete State Spaces | p. 39 |
1 Definition | p. 39 |
2 Stationary Distribution | p. 46 |
4 Markovian Queues in Continuous Time | p. 51 |
1 The M/M/1 Queue | p. 51 |
2 Skip-Free Markov Processes | p. 54 |
3 The M/M/[infinity] Queue | p. 55 |
4 The M/M/k Queue | p. 56 |
5 The M/M/k/k Queue | p. 58 |
6 The M/M/k/k+c/N Queue | p. 59 |
5 Markovian Queueing Networks | p. 63 |
1 Balance Equations and Reversibility Properties | p. 65 |
2 Jackson and Gordon-Newell Networks | p. 80 |
3 Symmetric Service Disciplines | p. 99 |
Part II Semi-Markovian Methods | |
6 Renewal Theory | p. 113 |
1 Renewal Processes | p. 113 |
2 Renewal Function and Renewal Equations | p. 116 |
3 Renewal Theorems | p. 118 |
4 Residual Life Times and Stationary Renewal Processes | p. 124 |
5 Renewal Reward Processes | p. 130 |
7 Markov Renewal Theory | p. 135 |
1 Regenerative Processes | p. 135 |
2 Semi-Markov Processes | p. 138 |
3 Semi-regenerative Processes | p. 144 |
8 Semi-Markovian Queues | p. 147 |
1 The GI/M/1 Queue | p. 147 |
2 The M/G/1 Queue | p. 155 |
3 The GI/M/m Queue | p. 160 |
Part III Matrix-Analytic Methods | |
9 Phase-Type Distributions | p. 169 |
1 Motivation | p. 169 |
2 Definition and Examples | p. 171 |
3 Moments | p. 176 |
4 Closure Properties | p. 178 |
10 Markovian Arrival Processes | p. 185 |
1 The PH renewal process | p. 185 |
2 From PH renewal processes to MAPs | p. 187 |
3 From MAPs to BMAPs | p. 188 |
4 Distribution of the Number of Arrivals | p. 190 |
5 Expected Number of Arrivals | p. 192 |
11 The GI/PH/1 Queue | p. 197 |
1 The Embedded Markov Chain | p. 198 |
2 Stationary Distribution at Arrival Instants | p. 199 |
3 Ergodicity of the Embedded Markov Chain | p. 204 |
4 Asymptotic Distribution of the System Process | p. 208 |
12 The BMAP/G/1 Queue | p. 213 |
1 The Embedded Markov Chain | p. 214 |
2 The Matrix G | p. 215 |
3 Stationary Distribution at Service Completions | p. 216 |
4 Asymptotic Distribution of the System Process | p. 218 |
5 Stability Conditions | p. 224 |
13 Discrete Time Approaches | p. 229 |
1 Discrete Phase-Type Distributions | p. 229 |
2 BMAPs in Discrete Time | p. 232 |
3 Blockwise Skip-Free Markov Chains | p. 234 |
4 The PH/PH/1 Queue in Discrete Time | p. 236 |
14 Spatial Markovian Arrival Processes | p. 239 |
1 Arrivals in Space | p. 240 |
2 Properties of Spatial MAPs | p. 245 |
15 Appendix | p. 253 |
1 Conditional Expectations and Probabilities | p. 253 |
2 Extension Theorems | p. 256 |
3 Transforms | p. 258 |
4 Gershgorin's Circle Theorem | p. 260 |
References | p. 263 |
Index | p. 269 |