Markov Chains

Theory and Applications

Inbunden, Engelska, 2013

2 289 kr

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Markov Chains: Theory and ApplicationsMarkov chains are a fundamental class of stochastic processes. They are widely used to solve problems in a large number of domains such as operational research, computer science, communication networks and manufacturing systems. The success of Markov chains is mainly due to their simplicity of use, the large number of available theoretical results and the quality of algorithms developed for the numerical evaluation of many metrics of interest.The author presents the theory of both discrete-time and continuous-time homogeneous Markov chains. He carefully examines the explosion phenomenon, the Kolmogorov equations, the convergence to equilibrium and the passage time distributions to a state and to a subset of states. These results are applied to birth-and-death processes. He then proposes a detailed study of the uniformization technique by means of Banach algebra. This technique is used for the transient analysis of several queuing systems.

Produktinformation

Utgivningsdatum2013-07-26
Mått163 x 241 x 28 mm
Vikt748 g
FormatInbunden
SpråkEngelska
Antal sidor410
FörlagISTE Ltd and John Wiley & Sons Inc
ISBN9781848214934

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Preface ixChapter 1. Discrete-Time Markov Chains 11.1. Definitions and properties 11.2. Strong Markov property 51.3. Recurrent and transient states 81.4. State classification 121.5. Visits to a state 141.6. State space decomposition 181.7. Irreducible and recurrent Markov chains 221.8. Aperiodic Markov chains 301.9. Convergence to equilibrium 341.10. Ergodic theorem 411.11. First passage times and number of visits 531.12. Finite Markov chains 681.13. Absorbing Markov chains 701.14. Examples 761.15. Bibliographical notes 87Chapter 2. Continuous-Time Markov Chains 892.1. Definitions and properties 922.2. Transition functions and infinitesimal generator 932.3. Kolmogorov’s backward equation 1082.4. Kolmogorov’s forward equation 1142.5. Existence and uniqueness of the solutions 1272.6. Recurrent and transient states 1302.7. State classification 1372.8. Explosion 1412.9. Irreducible and recurrent Markov chains 1482.10. Convergence to equilibrium 1622.11. Ergodic theorem 1662.12. First passage times 1722.13. Absorbing Markov chains 1842.14. Bibliographical notes 190Chapter 3. Birth-and-Death Processes 1913.1. Discrete-time birth-and-death processes 1913.2. Absorbing discrete-time birth-and-death processes 2003.3. Periodic discrete-time birth-and-death processes 2083.4. Continuous-time pure birth processes 2093.5. Continuous-time birth-and-death processes 2133.6. Absorbing continuous-time birth-and-death processes 2283.7. Bibliographical notes 233Chapter 4. Uniformization 2354.1. Introduction 2354.2. Banach spaces and algebra 2374.3. Infinite matrices and vectors 2434.4. Poisson process 2494.5. Uniformizable Markov chains 2634.6. First passage time to a subset of states 2734.7. Finite Markov chains 2754.8. Transient regime 2764.9. Bibliographical notes 286Chapter 5. Queues 2875.1. The M/M/1 queue 2885.2. The M/M/c queue 3155.3. The M/M/∞ queue 3185.4. Phase-type distributions 3235.5. Markovian arrival processes 3265.6. Batch Markovian arrival process 3425.7. Block-structured Markov chains 3525.8. Applications 3705.9. Bibliographical notes 380Appendix 1 Basic Results 381Bibliography 387Index 395