Introduction to matrix analytic methods in stochastic modeling.

*(English)*Zbl 0922.60001
ASA-SIAM Series on Statistics and Applied Probability, 5. Philadelphia, PA: SIAM, Society for Industrial and Applied Mathematics. xvi, 334 p. (1999).

The book deals with the basic mathematical ideas and algorithms of the matrix analytical theory. The approach uses probabilistic arguments to the fullest extent. The many new proofs given emphasize the unity of the matrix analytical approach. Necessary prerequisites for the book are undergraduate advanced calculus and linear algebra, and a basic course in stochastic processes. Thus, the book serves as a textbook for students and workers in applied probability, operations research, computer and industrial engineering.

In the first part (Chapter 1) the authors begin by describing several examples of quasi birth and death processes (QBDs), illustrating the variety of the models, which may be hidden by the general block notations. The second part deals with phase type distribution (Chapter 2) and related-point processes (Chapter 3), providing a versatile set of tractable models for applied probability, like the PH-renewal process in discrete and continuous time and a general Markovian point process. The third part is devoted to the matrix geometric distribution of QBDs in discrete and continuous time. First, in Chapter 4, birth and death processes are reviewed by using renewal theory. In Chapter 5, processes under taboo for Markov chains are considered in discrete and continuous time. They are needed later in the book. In Chapter 6 the arguments of Chapter 4 are carried over to the matrix geometric solution for homogeneous QBDs. Stability conditions (drift conditions) for QBDs are given in Chapter 7. Part IV is devoted to numerical algorithms; the material presented in Chapter 8 combines algorithmic and probabilistic reasoning most intimately. Linear and quadratically convergent algorithms for computing the rate matrix of QBDs are given. Chapter 9 shortly outlines some aspects of spectral analysis and why this works in principle. Chapter 10 deals with QBDs with a finite number of levels and Chapter 11 with the problem of the first passage times for those processes. This allows to bring together material from different authors about different problems. It is shown how these results may be interpreted in the light of the general properties of Chapter 5. The last part (Part V) of the book contains five short chapters that go beyond simple QBDs and discuss various extensions of the analyzed processes: nonhomogeneous QBDs (level-dependent transitions) in Chapter 12, processes that are skip-free in one direction in Chapter 13, processes on a tree-like state space in Chapter 14, product form networks in Chapter 15, and in Chapter 16, processes with a general state space for the phase transition.

In the first part (Chapter 1) the authors begin by describing several examples of quasi birth and death processes (QBDs), illustrating the variety of the models, which may be hidden by the general block notations. The second part deals with phase type distribution (Chapter 2) and related-point processes (Chapter 3), providing a versatile set of tractable models for applied probability, like the PH-renewal process in discrete and continuous time and a general Markovian point process. The third part is devoted to the matrix geometric distribution of QBDs in discrete and continuous time. First, in Chapter 4, birth and death processes are reviewed by using renewal theory. In Chapter 5, processes under taboo for Markov chains are considered in discrete and continuous time. They are needed later in the book. In Chapter 6 the arguments of Chapter 4 are carried over to the matrix geometric solution for homogeneous QBDs. Stability conditions (drift conditions) for QBDs are given in Chapter 7. Part IV is devoted to numerical algorithms; the material presented in Chapter 8 combines algorithmic and probabilistic reasoning most intimately. Linear and quadratically convergent algorithms for computing the rate matrix of QBDs are given. Chapter 9 shortly outlines some aspects of spectral analysis and why this works in principle. Chapter 10 deals with QBDs with a finite number of levels and Chapter 11 with the problem of the first passage times for those processes. This allows to bring together material from different authors about different problems. It is shown how these results may be interpreted in the light of the general properties of Chapter 5. The last part (Part V) of the book contains five short chapters that go beyond simple QBDs and discuss various extensions of the analyzed processes: nonhomogeneous QBDs (level-dependent transitions) in Chapter 12, processes that are skip-free in one direction in Chapter 13, processes on a tree-like state space in Chapter 14, product form networks in Chapter 15, and in Chapter 16, processes with a general state space for the phase transition.

Reviewer: A.Brandt (Berlin)

##### MSC:

60-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to probability theory |