Introduction to Stochastic Networks

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Juli 1999



In a stochastic network, such as those in computer/telecommunications and manufacturing, discrete units move among a network of stations where they are processed or served. Randomness may occur in the servicing and routing of units, and there may be queueing for services. This book describes several basic stochastic network processes, beginning with Jackson networks and ending with spatial queueing systems in which units, such as cellular phones, move in a space or region where they are served. The focus is on network processes that have tractable (closed-form) expressions for the equilibrium probability distribution of the numbers of units at the stations. These distributions yield network performance parameters such as expectations of throughputs, delays, costs, and travel times. The book is 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. Assuming a graduate course in stochastic processes without measure theory, the emphasis is on multi-dimensional Markov processes. There is also some self-contained material on point processes involving real analysis. The book also contains rather complete introductions to reversible Markov processes, Palm probabilities for stationary systems, Little laws for queueing systems and space-time Poisson processes. This material is used in describing reversible networks, waiting times at stations, travel times and space-time flows in networks.


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 Lévy 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.


From the reviews:
"...integral to today's development of analytical methods for queuing networks...[collects] an interesting summary of what one might view as the heart of network theory."
"The book provides self-contained introductions to point processes and Palm probabilities, a point that I emphasize because these parts are extremely well-presented and might also be valuable for readers not particularly interested in stochastic networks...Serfozo has certainly written a book that is unique in its choice of topics and captures the most recent developments in research in stochastic networks. I recommend it to all mathematically interested readers in this area. I found the book extremely well written, with many insightful explanations that make it a pleasure to read. The notations used are very elegant and intuitive...thoroughly edited with very few typographical errors."
EAN: 9780387987736
ISBN: 0387987738
Untertitel: 'Stochastic Modelling and Applied Probability'. 1999. Auflage. Book. Sprache: Englisch.
Verlag: Springer
Erscheinungsdatum: Juli 1999
Seitenanzahl: 320 Seiten
Format: gebunden
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