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Introduction to Matrix Analytic Methods in Stochastic Modeling

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by G. Latouche and V. Ramaswami


1999 / xiv + 334 pages / Softcover / ISBN: 978-0-898714-25-8 / List Price $90.50 / ASA-SIAM Member Price $63.35 / Order Code SA05

Matrix analytic methods are popular as modeling tools because they give one the ability to construct and analyze a wide class of queuing models in a unified and algorithmically tractable way. The authors present the basic mathematical ideas and algorithms of the matrix analytic theory in a readable, up-to-date, and comprehensive manner. In the current literature, a mixed bag of techniques is used-some probabilistic, some from linear algebra, and some from transform methods. Here, many new proofs that emphasize the unity of the matrix analytic approach are included.

The authors begin by describing several examples of quasi-birth-and-death (QBD) processes. These examples give the reader an idea of the variety of models which are hidden by the general block notation as well as reinforce some of the terminology and notation used throughout the text. These same examples are used as illustrations later. The second part of the book deals with phase-type distributions and related-point processes, which provide a versatile set of tractable models for applied probability. Part three reviews birth-and-death processes, and points out that the arguments for these processes carry over to more general processes in a parallel manner and are based on Markov renewal theory.

Part four covers material where algorithmic and probabilistic reasoning are most intimately connected. In three steps, the authors take you from one of the simplest iterative procedures to the fastest, relating the successive approximations to the dynamic behavior of the stochastic process itself. The final part goes beyond simple QBDs with a sequence of short chapters where the authors discuss various extensions to the analyzed processes. Their intention is to show that the fundamental ideas extend beyond simple homogeneous QBD.

Applied probabilists, systems analysts, operations research analysts, applied statisticians, and communication and computer engineers as well as electrical engineers interested in modeling and industrial engineers interested in manufacturing systems will find this book essential. Undergraduate advanced calculus and linear algebra and a course in stochastic processes are necessary prerequisites for understanding the book.

Preface; Part I: Quasi-Birth-and-Death Processes. Chapter 1: Examples; Part II: The Method of Phases. Chapter 2: PH Distributions; Chapter 3: Markovian Point Processes; Part III: The Matrix-Geometric Distribution. Chapter 4: Birth-and-Death Processes; Chapter 5: Processes Under a Taboo; Chapter 6: Homogeneous QBDs; Chapter 7: Stability Condition; Part IV: Algorithms. Chapter 8: Algorithms for the Rate Matrix; Chapter 9: Spectral Analysis; Chapter 10: Finite QBDs; Chapter 11: First Passage Times; Part V: Beyond Simple QBDs. Chapter 12: Nonhomogeneous QBDs; Chapter 13: Processes, Skip-Free in One Direction; Chapter 14: Tree Processes; Chapter 15: Product Form Networks; Chapter 16: Nondenumerable States; Bibliography; Index.

ISBN: 9780898714258

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