Probabilistic Modelling

International Journal of Electrical Engineering Education, Jan 1999 by Bagley, Jonathan H

Probabilistic Modelling: I. MITRANI

(Cambridge University Press, 1997, 223 pp., 45 hardback, ;16.95 paperback)

This book, apart from the first chapter, is devoted to various aspects of the probabilistic modelling of queues. The book is intended to be read by operations research undergraduates and postgraduates, and practitioners in the field.

Chapter 1 is a self-contained introduction to probability theory. It is comprehensive and clearly written although, from necessity, it covers a large amount of material fairly quickly. The reader without a previous knowledge of probability might wish to work through it in conjunction with an elementary probability text.

Chapter 2 covers renewal processes, including the Poisson process, and contains clear explanations of the residual life "paradox" and the P.A.S.T.A. (Poisson Arrivals See Time Averages) property.

Queuing systems are introduced in chapter 3. The average performances of the M/M/1 and M/G/1 queues are investigated via Little's theorem. The concept of the busy period is introduced. Finally, a substantial section is devoted to job scheduling policies.

Queuing networks, both open and closed, are covered in chapter 4. The Jackson, and Gordon-Newell theorems, together with applications of Little's theorem, are used to derive various performance measures. The final section of the chapter introduces multiclass networks.

Chapter 5 introduces both discrete and continuous time Markov chains, enabling the investigation of the equilibrium distributions of various birthdeath type queuing models. The telephone exchange model, the multi-server queue and queues with finite waiting room are all discussed.

The application of Markov chains is continued in chapter 6, in which the use of two dimensional continuous time Markov chains allows the formulation and analysis of more realistic queuing models. Included in this general approach is the method of phases. The final part of the chapter is devoted to methods of solution of the (often complicated) steady-state equations arising from this approach.

Overall I was impressed by this book. In view of the intended readership, the author chooses a sensible point on the line between the rigorous proving of every result, and the opposite extreme of plugging numbers into formulae. The excellent lists of references following each chapter will enable the interested reader with a more mathematical background to investigate further any topic which captures his or her imagination.

JONATHAN H. BAGLEY Department of Mathematics, UMIST