This book is a non-measure-theoretic introduction to stochastic processes for students knowing some calculus and probability theory. Illustrative examples used in the book are slanted toward operations research and management science, and the main text concentrates on topics relevant to these areas, viz., renewal theory, Markov chains and processes, random walk, martingales and stochastic order relationships. Whenever possible the author uses probabilistic, rather than analytic, methods of proof, and he adopts some other measures which will help the student gain a better understanding of the text. Many theorems are preceded by informal heuristic discussion of the main steps in their proof and often extra regularity conditions are assumed in proofs in order to minimise technical detail and highlight the principal ideas. Proofs are usually followed by remarks explaining the limitations of the techniques used, simple consequences of them, and so on. Each chapter is followed by about 30 problems, and solutions to some are collected at the end of the book. Annotated references are also given at the end of each chapter, many being to recent papers from whence illustrative examples are taken. The text is well presented and has very few misprints. A more detailed description of the contents follows. Chapter 1 reviews some probability and distribution theory, particularly the exponential distribution. The second chapter discusses the Poisson process and related combinatorial theorems, and applies them to the M/G/1 busy period. Nonhomogeneous, compound and conditional Poisson processes are discussed and illustrated. Chapter 3 contains a fairly complete discussion of elementary renewal theory including a heuristic discussion of Blackwell's theorem and the key renewal theorem. Versions of the former are proved later in the book, and the latter is used to discuss alternating and delayed renewal processes, the mean size of an age-dependent branching process, renewal reward processes, Little's formula and regenerative processes. The chapter ends with an introduction to stationary point processes, including Korolyuk's theorem. The main classes of Markov process are covered in Chapters 4–6. Thus, Chapter 4 treats the classification and ergodic behaviour of discrete state Markov chains. Illustrative examples include the embedded Markov chains of queuing theory, branching processes, algorithmic efficiency, runs and list ordering rules. Time reversibility is used to derive limiting distributions. An introduction to semi-Markov processes ends the chapter. Chapter 5 discusses discrete-state Markov processes, in particular, birth and death processes. Time reversibility is again effectively used to obtain the equilibrium properties of some vector-valued processes, for example, queue lengths in networks and the M/G/1 shared processor system. This chapter ends with a brief account of uniformization. Some distributional aspects of Brownian motion are discussed in Chapter 6. The exposition includes related Gaussian processes, for example, the Brownian bridge and the empirical distribution function, and absorbed and reflected Brownian motions with applications to stock options and optimal betting. The chapter ends with brief discussions of somewhat unrelated topics, viz., shot noise and stationary processes. Chapter 7 contains a variety of topics connected with random walks and martingales. Versions of the optional sampling and convergence theorems are derived. Duality arguments and the optional sampling theorem are used to obtain an exponential bound for the limiting waiting time distribution of a stable GI/G/1 queue. Other examples include hitting times and places of random walk and occurrences of patterns. The final chapter covers stochastic order relations, a subject not found in similar texts. Concepts discussed include stochastic largeness, monotone failure rates, hazard rate ordering, likelihood ratio ordering and stochastic variability. They are used to obtain comparison results for renewal and queueing processes and elegant proofs of these, and some other results, are achieved by using coupling methods. Minor detractions include the following. Chapter 1 begins by defining the basic concepts of probability theory but, surprisingly, not independence. Some definitions are repeated, for example, convexity (pp. 243, 270) and the GI/G/1 queue (pp. 82, 225, 276). The proof of Jensen's inequality is set as an exercise in both Chapters 7 and 8. More importantly, the definition given for the inverse of a distribution function in Problem 1.2 is inadequate. Consequently, constructions using this notion (pp. 189, 255) need further discussion. However, these are minor blemishes of an otherwise attractive introductory text.
Reviewer: Pakes, Anthony G. [form MathSciNet]