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**Comparison methods for queues and other stochastic models. Ed. with rev. by Daryl J. Daley. (Transl. from the German).**
*(English)*
Zbl 0536.60085

Wiley Series in Probability and Mathematical Statistics. Applied Section. Chichester etc.: John Wiley & Sons. XIII, 217 p. £18.50 (Orig. Akademie-Verlag, Berlin) (1983).

This book is a revised and enlarged version of ”Qualitative Eigenschaften und Abschätzungen stochastischer Modelle” published 1977 in German (see Zbl 0356.60056 and Zbl 0395.60082) and 1979 in Russian. These first two editions have had much influence on current research in the fields of stochastic partial orderings, monotonicity properties of and bounds for stochastic models, especially queueing systems and reliability models. Furthermore the book has become a standard reference book in these fields and therefore its English edition will be welcomed very much.

The author’s main interest lies in a part of the theory and application of stochastic models which often is neglected in classical monographs in this field of research: bounds and approximations. Here bounds and approximations are not considered for numerical computations but in the following way: Consider the queue length of a G/G/m queue, which is very hard to investigate. Are there other queues the (transient or stationary) queue length process of which can serve as a bound for the original process (Chapter 5,6). To define ”bounds” for a queue length process one has to compare stochastic processes (Chapter 4) and to define (different) orderings of distributions (Chapter 1).

Another point of interest can be sketched as follows: Given a queueing system where the input intensity is controllable; if we increase the input intensity - is it true that the queue length distribution increases and in what terms have we to define an increasing sequence of stochastic processes (Chapter 2,5,6 on internal monotonicity)?

The approximation methods work in a similar way: Given a complicated system which must be investigated; is there a sequence of approximating simpler systems such that the convergence of certain characteristics of the approximating systems implies (weak) convergence of the approximating systems’ distributions to the original system’s distribution?

Each chapter ends with historical and bibliographical notes and a list of open problem.

The author’s main interest lies in a part of the theory and application of stochastic models which often is neglected in classical monographs in this field of research: bounds and approximations. Here bounds and approximations are not considered for numerical computations but in the following way: Consider the queue length of a G/G/m queue, which is very hard to investigate. Are there other queues the (transient or stationary) queue length process of which can serve as a bound for the original process (Chapter 5,6). To define ”bounds” for a queue length process one has to compare stochastic processes (Chapter 4) and to define (different) orderings of distributions (Chapter 1).

Another point of interest can be sketched as follows: Given a queueing system where the input intensity is controllable; if we increase the input intensity - is it true that the queue length distribution increases and in what terms have we to define an increasing sequence of stochastic processes (Chapter 2,5,6 on internal monotonicity)?

The approximation methods work in a similar way: Given a complicated system which must be investigated; is there a sequence of approximating simpler systems such that the convergence of certain characteristics of the approximating systems implies (weak) convergence of the approximating systems’ distributions to the original system’s distribution?

Each chapter ends with historical and bibliographical notes and a list of open problem.

Reviewer: H.Daduna

### MSC:

60Kxx | Special processes |

60-02 | Research exposition (monographs, survey articles) pertaining to probability theory |

60K25 | Queueing theory (aspects of probability theory) |

60K10 | Applications of renewal theory (reliability, demand theory, etc.) |

60K30 | Applications of queueing theory (congestion, allocation, storage, traffic, etc.) |

90B22 | Queues and service in operations research |