Reversibility and stochastic networks. (English) Zbl 0422.60001

Kelly, F. P.

Wiley Series in Probability and Mathematical Statistics. Chichester etc.: John Wiley & Sons. VIII, 230 p. £14.50 (1979).

The stationary distribution of an ergodic reversible Markov chain can usually be obtained quite easily from the determining system of linear equations, the so-called global balance equations. This is due to the fact that, in the presence of reversibility, these equations break up into further, so-called local or partial balance equations which are also satisfied by the stationary law. This book is devoted to the study of mostly-continuous-time-Markov chains being reversible or satisfying some type of partial balance. Essentially, the book constitutes an account – the first of its kind – of the many uses of reversibility and partial balance. Already in Chapter 1, which presents the basic facts about reversible Markov chains, many illustrative examples are given. The remaining chapters, 2–9, are devoted to the presentation and investigation of certain classes of examples from numerous fields or application such as population dynamics and genetics (Chapters 2, 5–9), chemistry (Chapter 8), queuing theory (Chapters 2–4), and others. A common aspect of these examples is the fact that they can all be viewed as some kind of stochastic network, a point of view which allows a fairly unified statement of the subject matter. The book is recommended to anyone interested in Markov chain models. It is a rich source of examples for students and teachers, of ideas for research workers. The prerequisite is an understanding of Markov processes at about the level of Feller’s Introduction to Probability Theory and its Applications, Vol. I.

Reviewer: R. Schassberger (Braunschweig)

MSC:

60-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to probability theory
60K20	Applications of Markov renewal processes (reliability, queueing networks, etc.)
60J27	Continuous-time Markov processes on discrete state spaces
92D10	Genetics and epigenetics
92D25	Population dynamics (general)
92Exx	Chemistry
60K99	Special processes

Keywords:

reversibility; stochastic networks; ergodic reversible Markov chain; global balance equations; population dynamics; genetics; chemistry; queuing theory