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Reviewer: Stefan Corneliu V. Stefanescu

Mitrani succeeds in making his subject easily accessible. Throughout the book, basic ideas are emphasized and intuitively justified. Generally, the book does not contain the proofs of difficult theorems. Difficult mathematics has been sacrificed to intuition and replaced with adequate examples, practical remarks, cross-references, and short, simple proofs for particular cases. The book is intended for operations research and computer science undergraduate and graduate students and for queueing practitioners. Every chapter ends with an annotated bibliography. These lists contain the classic books in the field and many new and important titles. The author mentions his own interesting contributions to the area, especially in chapters 3, 4, and 6. The book consists of six chapters: “Introduction to Probability Theory,” “Arrivals and Services,” “Queueing Systems: Average Performance,” “Queueing Networks,” “Markov Chains and Processes,” and “Queues in Markovian Environments.” In the introductory chapter, the author solves many interesting problems that will be used as examples in the remaining chapters, such as the distribution of the largest and smallest random variable, lack of memory for random variables (discrete and continuous case), multiple modular redundancy, order statistics, software reliability growth, sums, transforms, and limits of independent random variables. The second chapter treats the arrival and services problems that occur in complex systems. The renewal and Poisson processes are analyzed in detail, with applications. Chapters 3 and 4 are dedicated to queueing systems and networks. The main topics are Little's theorem, M/G/1 and M/M/1 queues, different scheduling policies and optimal policies, open networks and their traffic equations, performance measures, and closed and multiclass networks. Chapters 5 and 6 consider a Markovian environment and analyze a large class of queueing models where the arrival or service mechanisms, or both, are influenced by external processes. The author suggests possible applications of Markov chains and processes in queueing theory, and then develops methods for determining the steady-state joint distribution of a quasi-birth-and-death process. In all chapters, every section ends with examples and many carefully chosen, application-oriented exercises. Theory is not neglected; Mitrani gives the main theorems, usually with practical justifications. Much of the material presented here is known to specialists, but not in this form. For many scientists, it seems to be easier to enumerate definitions and theorems, with difficult proofs, than to clarify concepts, emphasize the basic principles of the domain, or pose practical problems and elucidate paradoxes. Someone who has a detailed vision of a whole field, embracing theory and applications, can present things from a philosophical point of view. In this approach, one suggests the proof of a difficult theorem by using a well-chosen simple example, not forgetting pertinent comments. This approach makes the material clear and understandable for a wide audience. In addition, complex theoretical and practical problems seem relatively easy. Mitrani's book has all these characteristics. He has achieved his aim of producing a concise, clear, and well-documented book.