According to the preface, this textbook is designed for a two-semester graduate course in stochastic models for students of operations research, computer science, business and public policy analysis, and engineering. Although it is not a mathematics book, it is surely mathematical, and to include in its target audience students majoring in business or public policy seems a little optimistic to me. As the back-cover blurb states, the treatment is “user-friendly, yet rigorous.” Kulkarni’s viewpoint is encapsulated in two pieces of text, each enclosed in a box, on the page facing the table of contents:

Jack and Harry were lost over a vast farmland while on their balloon ride. When they spotted a bicyclist on a trail going through the farmland, they lowered their balloon and yelled, “Good day, Sir! Could you tell us where we are?” The bicyclist looked up, and said in a surprised voice: “Sure! You are in a balloon!” Jack turned to Harry and said, “Oh great! Just our luck that we should run into a mathematician!” “What makes you think he is a mathematician?” asked Harry. “Well, his answer was correct, but totally useless!”

and

The author sincerely hopes that a student mastering this book will be able to use stochastic models to obtain correct as well as useful answers.

I include these excerpts to emphasize my observation that while this is a careful, mathematically correct treatment of stochastic models, Kulkarni always keeps in mind that his objective is to explain the intricacies of a difficult subject to students. His strategy is to concentrate on the modeling of “real life” situations with stochastic elements, and analysis of the resulting stochastic models. There are nine chapters:

• Introduction

• Discrete-Time Markov Chains: Transient Behavior

• Discrete-Time Markov Chains: Limiting Behavior

• Discrete-Time Markov Chains: First Passage Times

• Poisson Processes

• Continuous-Time Markov Chains

• Applications of Markov Chains to Queueing Theory

• Renewal Processes

• Markov Renewal Processes

The book contains a substantial set of appendices (including coverage of elementary probability theory, transforms, modes of convergence, and stochastic ordering), answers to selected problems, a bibliography of books, and an index (with separate indexes for examples, theorems, corollaries, lemmas, propositions, and definitions--a nice touch in support of user friendliness). Each chapter concludes with “Reference Notes,” which direct readers to the appropriate books listed in the bibliography, and a set of exercises. The exercises are separated into three classes--modeling, computational, and conceptual--according to subject, not level of relative difficulty. Use of the computer for numerical calculations is encouraged. Answers are provided for selected exercises of each class.

Although Kulkarni emphasizes the theory of stochastic models, a full chapter is devoted to queueing models (primarily to illustrate Markov chains, rather than as a treatment of queueing theory per se), and queueing models are used as examples throughout the book.

I liked this carefully and thoughtfully written book. At the time of the writing of this review, the publisher was Chapman & Hall, which was then in the process of being acquired by Kluwer. During this (long) handover period, I saw no advertising. Indeed, it took some detective work to locate the book and to receive a review copy. Furthermore, the book is priced at about twice the going rate for a textbook. It would be a shame if this book were to perish because of poor exposure and marketing.