Introduction to Stochastic Networks

By (author) Serfozo, Richard

Beginning with Jackson networks and ending with spatial queuing systems, this book describes several basic stochastic network processes, with the focus on network processes that have tractable expressions for the equilibrium probability distribution of the numbers of units at the stations. Intended for graduate students and researchers in engineering, science and mathematics interested in the basics of stochastic networks that have been developed over the last twenty years, the text assumes a graduate course in stochastic processes without measure theory, emphasising multi-dimensional Markov processes. Alongside self-contained material on point processes involving real analysis, the book also contains complete introductions to reversible Markov processes, Palm probabilities for stationary systems, Little laws for queuing systems and space-time Poisson processes.

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[目次]

  • 1 Jackson and Whittle Networks.- 1.1 Preliminaries on Networks and Markov Processes.- 1.2 Tandem Network.- 1.3 Definitions of Jackson and Whittle Processes.- 1.4 Properties of Service and Routing Rates.- 1.5 Equilibrium behavior.- 1.6 Production-Maintenance Network.- 1.7 Networks with Special Structures.- 1.8 Properties of Jackson Equilibrium Distributions.- 1.9 Convolutions for Single-Server Nodes.- 1.10 Throughputs and Expected Sojourn Times.- 1.11 Algorithms for Performance Parameters.- 1.12 Monte Carlo Estimation of Network Parameters.- 1.13 Properties of Whittle Networks.- 1.14 Exercises.- 1.15 Bibliographical Notes.- 2 Reversible Processes.- 2.1 Reversibility.- 2.2 Time Reversal.- 2.3 Invariant Measures.- 2.4 Construction of Reversible Processes.- 2.5 More Birth-Death Processes.- 2.6 Reversible Network Processes.- 2.7 Examples of Reversible Networks.- 2.8 Partition-Reversible Processes.- 2.9 Examples of Partition-Reversible Processes.- 2.10 Exercises.- 2.11 Bibliographical Notes.- 3 Miscellaneous Networks.- 3.1 Networks with Multiple Types of Units.- 3.2 Kelly Networks: Route-dependent Services.- 3.3 BCMP Networks: Class-Node Service Dependencies.- 3.4 Networks with Cox and General Service Times.- 3.5 Networks with Constraints.- 3.6 Networks with Blocking and Rerouting.- 3.7 Bottlenecks in Closed Jackson Networks.- 3.8 Modeling Whittle Networks by Locations of the Units.- 3.9 Partially Balanced Networks.- 3.10 Exercises.- 3.11 Bibliographical Notes.- 4 Network Flows and Travel Times.- 4.1 Point Process Notation.- 4.2 Extended Levy Formula for Markov Processes.- 4.3 Poisson Functionals of Markov Processes.- 4.4 Multivariate Compound Poisson Processes.- 4.5 Poisson Flows in Jackson and Whittle Networks.- 4.6 Palm Probabilities for Markov Processes.- 4.7 Sojourn and Travel Times of Markov Processes.- 4.8 Palm Probabilities of Jackson and Whittle Networks.- 4.9 Travel Times on Overtake-Free Routes.- 4.10 Exercises.- 4.11 Bibliographical Notes.- 5 Little Laws.- 5.1 Little Laws for Markovian Systems.- 5.2 Little Laws for General Queueing Systems.- 5.3 Preliminary Laws of Large Numbers.- 5.4 Utility Processes.- 5.5 Omnibus Little Laws.- 5.6 Little Laws for Regenerative Systems.- 5.7 Exercises.- 5.8 Bibliographical Notes.- 6 Stationary Systems.- 6.1 Preliminaries on Stationary Processes.- 6.2 Palm Probabilities.- 6.3 Campbell-Mecke Formulas for Palm Probabilities.- 6.4 Little Laws for Stationary Systems.- 6.5 Sojourn Times and Related Functionals.- 6.6 Travel Times for Stochastic Processes.- 6.7 Sojourn and Travel Times in Networks.- 6.8 Exercises.- 6.9 Bibliographical Notes.- 7 Networks with String Transitions.- 7.1 Definition of a String-Net.- 7.2 Invariant Measures of String-Nets.- 7.3 Traffic Equations, Partial Balance, and Throughputs.- 7.4 String-Nets with Unit-Vector Transitions.- 7.5 Networks with One-Stage Batch Transitions.- 7.6 Networks with Compound-Rate String Transitions.- 7.7 Networks with Multiple, Compound-Rate String Transitions.- 7.8 String-Nets with Two-Node Batch Transitions.- 7.9 Single Service Station With String Transitions.- 7.10 Bibliographical Notes.- 8 Quasi-Reversible Networks and Product Form Distributions.- 8.1 Quasi-Reversibility.- 8.2 Network to be Studied.- 8.3 Characterization of Product Form Distributions.- 8.4 Quasi-Reversibility and Biased Local Balance.- 8.5 Networks with Reversible Routing.- 8.6 Queueing Networks.- 8.7 Time-Reversals and Departure-Arrival Reversals.- 8.8 Networks with Multiclass Transitions.- 8.9 Exercises.- 8.10 Bibliographical Notes.- 9 Space-Time Poisson Models.- 9.1 Introductory Examples.- 9.2 Laplace Functionals of Point Processes.- 9.3 Transformations of Poisson Processes.- 9.4 Translations, Partitions, and Clusters.- 9.5 Service Systems with No Queueing.- 9.6 Network of M/G/? Service Stations.- 9.7 Particle Systems.- 9.8 Poisson Convergence of Space-Time Processes.- 9.9 Transformations into Large Spaces.- 9.10 Particle Flows in Large Spaces.- 9.11 Exercises.- 9.12 Bibliographical Notes.- 10 Spatial Queueing Systems.- 10.1 Preliminaries.- 10.2 Stationary Distributions and Ergodicity.- 10.3 Properties of Stationary Distributions and Examples.- 10.4 Throughputs and Expected Sojourn Times.- 10.5 Poisson Flows in Open Systems.- 10.6 Systems with Multiclass Units.- 10.7 Bibliographical Notes.- References.

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この本の情報

書名 Introduction to Stochastic Networks
著作者等 Serfozo, Richard
シリーズ名 Stochastic Modelling and Applied Probability 44
出版元 Springer-Verlag New York Inc.
刊行年月 2012.09.27
版表示 Softcover reprint of the original 1st ed. 1999
ページ数 315p
大きさ H235 x W155
ISBN 9781461271604
言語 英語
出版国 アメリカ合衆国
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