zbMATH — the first resource for mathematics

Stochastic models in operations research. Vol. 1. Stochastic processes and operating characteristics. Unabridged reprint of the 1982 original. (English) Zbl 1072.90001
Mineola, NY: Dover Publications (ISBN 0-486-43259-9/pbk). xii, 547 p. (2004).
During the last decades the use of stochastic models in optimisation has been revolutionised with the development of informatics. This fact is a motivation for the republication of this book from the 1982’s edition made by McGraw Hill (for a review see Zbl 0503.90031). To study stochastic models arising in the practice of Operations Research [OR] is a challenge that this book overcomes. Commonly in OR it is necessary to model processes and systems which involve multiple uncertainty sources and the distortions may be described, often, by models related with Stochastic Processes [SP]. Then the modeller needs to have a knowledge going beyond the general theory, as he must be able of dealing with particular models such as Markov Chains [MC] and Processes, Queueing etc. These stochastic models are used for modelling, summarising and/or capturing the uncertainty present in the data. Different books deal with optimisation in a stochastic frame work, looking for an approach to modelling OR problems, see for example G. Ch. Pflug [Optimization of stochastic models. Kluwer (1996; Zbl 0909.90220)]. The book under review deals with the general conceptualisation of the theory of stochastic processes [SP], operating characteristics and stochastic systems focusing in their role in modelling usual problems appearing in OR’s practices. It is divided into two parts and the introduction is the first chapter.
Part A deals with SP and models through 9 chapters, numbered from 2 to 10. In Chapter 2 different models of queues are presented as well as the ideas of posterior analysis. Some usually encountered OR’s problems are developed (computer maintenance, congestion, occupancy, cash management, reservoir regulation, inventory, replenishment , harvesting, etc.). Chapter 3 addresses the basic models and ideas of SP; chapter 4 provides an introduction to Birth and Death processes through a reanalysis of the problems discussed in the previous chapter. The use of Kalman equations for transition functions, the steady state problems and other issues on \(M/M/1\) queues are treated. It contains an extensive study of queues with Poisson arrivals and exponential service times. Chapters 5 and 6 are devoted to present Renewal Theory and how to deal with its concepts and models at large. Poisson SP and the distribution of the sum of random variables are studied. A discussion on the use of approximations to the analysis of simulation experiments in \(M/G/1\) queues is developed. The main features of the existent theory on infinitely many servers queue are given. Sufficient conditions for the existence of steady state distribution for the different Operating Characteristics of \(GI/G/c\) queues are derived. The next three chapters deal with Markov Models. Chapter 7 is lengthy and develops the main issues related with Discrete MC. They play a key role in the analysis of \(M/G/1\) queues as well as in other stochastic models. The representation through digraphs is presented and six examples are discussed for establishing how this approach works in OR’s modelling. The fact that renewal processes may be embedded in MC is discussed. Chapter 8 provides a similar discussion for the Continuous time MC and the next one on Semi- Markov and Markov Renewal. The first part closes with a short but rather complete discussion on Ergodicity for Stationary and Regenerative SP.
Part B develops a set of issues related with qualitative aspects of queuing, Chapter 11 considers models of storage systems and the generalities of some particular queuing models [work-in-systems, multi-server, queuing disciplines and priorities etc.]. Chapter 12 develops a similar exposition with important problems in a family of network of queues. SP’s are used for describing how networks evolution and enhance to model more efficiently. Jackson Networks are presented and its modelling role in the study of the somewhat paradigmatic Multiprogram Computer System problem is developed. Chapter 13 points key results in Bounds and approximations to the OCs of important models as the delay in \(G1/G/1\) and its applications, diffusion processes , steady state exponential delay, \(M/G/c\) queues etc.
The book contains 62 figures, 8 tables and an Appendix which provides the basics on Probability Theory and Mathematical Calculus. Each chapter includes bibliographical notes, a list of references and recommended further lecture.
For Vol. II Part C see Zbl 0531.90062 and the reprint Zbl 1076.90001.

90-02 Research exposition (monographs, survey articles) pertaining to operations research and mathematical programming
62L20 Stochastic approximation
65C40 Numerical analysis or methods applied to Markov chains
90-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to operations research and mathematical programming
60-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to probability theory
60Kxx Special processes