Introduction to Stochastic Networks

ISBN-13: 9781461271604 / Angielski / Miękka / 2012 / 301 str.

Richard Serfozo

Opis

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.

Recenzje

From the reviews:

JOURNAL OF APPLIED MATHEMATICS AND STOCHASTIC ANALYSIS

"...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."

JOURNAL OF THE AMERICAN STATISTICAL ASSOCIATION

"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."

Spis treści

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.

O autorze

Stochastic networks describe the movements of units among nodes in a network. There are applications to telecommunication systems, warehousing, logistics, and computer systems. The audience includes researchers and graduate students in applied probability, operations research, industrial engineering, computer science, mathematics, and statistics.