Guy Latouche - Böcker

Intersecting two large research areas - numerical analysis and applied probability/queuing theory - this book is a self-contained introduction to the numerical solution of structured Markov chains, which have a wide applicability in queuing theory and stochastic modeling and include M/G/1 and GI/M/1-type Markov chain, quasi-birth-death processes, non-skip free queues and tree-like stochastic processes. Written for applied probabilists and numerical analysts, but accessible to engineers and scientists working on telecommunications and evaluation of computer systems performances, it provides a systematic treatment of the theory and algorithms for important families of structured Markov chains and a thorough overview of the current literature. The book, consisting of nine Chapters, is presented in three parts. Part 1 covers a basic description of the fundamental concepts related to Markov chains, a systematic treatment of the structure matrix tools, including finite Toeplitz matrices, displacement operators, FFT, and the infinite block Toeplitz matrices, their relationship with matrix power series and the fundamental problems of solving matrix equations and computing canonical factorizations. Part 2 deals with the description and analysis of structure Markov chains and includes M/G/1, quasi-birth-death processes, non-skip-free queues and tree-like processes. Part 3 covers solution algorithms where new convergence and applicability results are proved. Each chapter ends with bibliographic notes for further reading, and the book ends with an appendix collecting the main general concepts and results used in the book, a list of the main annotations and algorithms used in the book, and an extensive index.

Matrix-analytic and related methods have become recognized as an important and fundamental approach for the mathematical analysis of general classes of complex stochastic models. Research in the area of matrix-analytic and related methods seeks to discover underlying probabilistic structures intrinsic in such stochastic models, develop numerical algorithms for computing functionals (e.g., performance measures) of the underlying stochastic processes, and apply these probabilistic structures and/or computational algorithms within a wide variety of fields. This volume presents recent research results on: the theory, algorithms and methodologies concerning matrix-analytic and related methods in stochastic models; and the application of matrix-analytic and related methods in various fields, which includes but is not limited to computer science and engineering, communication networks and telephony, electrical and industrial engineering, operations research, management science, financial and risk analysis, and bio-statistics. These research studies provide deep insights and understanding of the stochastic models of interest from a mathematics and/or applications perspective, as well as identify directions for future research.

Matrix-analytic and related methods have become recognized as an important and fundamental approach for the mathematical analysis of general classes of complex stochastic models. Research in the area of matrix-analytic and related methods seeks to discover underlying probabilistic structures intrinsic in such stochastic models, develop numerical algorithms for computing functionals (e.g., performance measures) of the underlying stochastic processes, and apply these probabilistic structures and/or computational algorithms within a wide variety of fields. This volume presents recent research results on: the theory, algorithms and methodologies concerning matrix-analytic and related methods in stochastic models; and the application of matrix-analytic and related methods in various fields, which includes but is not limited to computer science and engineering, communication networks and telephony, electrical and industrial engineering, operations research, management science, financial and risk analysis, and bio-statistics. These research studies provide deep insights and understanding of the stochastic models of interest from a mathematics and/or applications perspective, as well as identify directions for future research.

This book explores regime-switching Brownian motion, a class of stochastic processes widely used in fields such as mathematical finance, risk theory, queueing theory, and epidemiological modeling. These processes are studied within the Markovian regime-switching framework, which captures dynamic environments characterized by shifts between different states or "regimes"—for example, economic cycles, seasonal environmental variations, or short-term surges in activity.The matrix-analytic approach, introduced approximately fifty years ago in the context of classical queueing theory, serves as the foundation for this analysis. This methodology emphasizes the examination of process trajectories over time, drawing insights from the interplay between analytic derivations and their physical or probabilistic interpretations. A central objective of the matrix-analytic framework is to produce solutions that are not only analytically tractable but also amenable to efficient, stable numerical algorithms—facilitating practical implementation using standard computational tools. This enables both quantitative performance evaluation and qualitative system understanding.Originally developed for telecommunication network modeling, matrix-analytic methods have since found applications across a broad spectrum of disciplines, including risk analysis, branching processes, and epidemiology.This book is the first to offer a systematic application of matrix-analytic techniques to Markov-modulated Brownian motion, filling a gap in the literature and providing a valuable resource for researchers and practitioners alike.The intended audience includes specialists in stochastic processes and their applications—such as applied probabilists, actuaries, financial analysts, systems and operations researchers, applied statisticians, and engineers in telecommunications and electrical domains. Readers are expected to have a background in advanced undergraduate calculus, linear algebra, and introductory stochastic processes.