Applied probability and queues.

*(English)*Zbl 0624.60098
Wiley Series in Probability and Mathematical Statistics. Applied Probability and Statistics. Chichester etc.: John Wiley & Sons. X, 318 p.; £34.60 (1987).

The aim of this book is to give an introduction into the mathematical methods of queueing theory and related fields. The main point is: “probabilistic” methods and proofs are presented in contrast to the more traditional analytic methods of queueing theory.

The book has three parts of nearly equal length: Part 1: Markov processes and Markovian queueing theory; Part 2: Renewal theory; Part 3: Special models and methods.

Possibly the intentions of the author become more transparent from the following examples: i) The first propositions after defining a Markov chain are stated in terms of conditional expectations (strong Markov property) and the techniques available from this are applied. ii) The renewal theorem is proved twice: Firstly the analytic proof of Feller is presented, and after that the coupling proof which goes back to Lindvall.

A sound mathematical background is required, especially in probability [the author cites the books of L. Breiman, Probability and stochastic processes: with a view toward applications (1969; Zbl 0246.60033), or K. L. Chung, A course in probability theory (1974; Zbl 0345.60003), for example]. Some experiences with Markov jump processes would make reading much easier. Working out the proofs in detail will surely be an exacting work, because the style of the book is announced to be “informal and inpretentious”. The author tries to develop the reader’s intuition and understanding of the models’ behavior by arguments of partially heuristic character, e.g. in the chapter on queueing theory there is a section “Pitfalls for intuition”.

The book may be used as a textbook for a graduate or postgraduate courses in applied probability for students having a background in stochastic processes. This book is a useful supplement to the existing literature on queueing theory and applied probability. Finally, for a rough information we give the headings of the sections of part 3:

Steady-state properties of GI/G/1; Explicit examples in the theory of random walks and single-server queues; Muldimensional methods; Many- server queues; Conjugate processes; Insurance risk, dam and storage models.

The book has three parts of nearly equal length: Part 1: Markov processes and Markovian queueing theory; Part 2: Renewal theory; Part 3: Special models and methods.

Possibly the intentions of the author become more transparent from the following examples: i) The first propositions after defining a Markov chain are stated in terms of conditional expectations (strong Markov property) and the techniques available from this are applied. ii) The renewal theorem is proved twice: Firstly the analytic proof of Feller is presented, and after that the coupling proof which goes back to Lindvall.

A sound mathematical background is required, especially in probability [the author cites the books of L. Breiman, Probability and stochastic processes: with a view toward applications (1969; Zbl 0246.60033), or K. L. Chung, A course in probability theory (1974; Zbl 0345.60003), for example]. Some experiences with Markov jump processes would make reading much easier. Working out the proofs in detail will surely be an exacting work, because the style of the book is announced to be “informal and inpretentious”. The author tries to develop the reader’s intuition and understanding of the models’ behavior by arguments of partially heuristic character, e.g. in the chapter on queueing theory there is a section “Pitfalls for intuition”.

The book may be used as a textbook for a graduate or postgraduate courses in applied probability for students having a background in stochastic processes. This book is a useful supplement to the existing literature on queueing theory and applied probability. Finally, for a rough information we give the headings of the sections of part 3:

Steady-state properties of GI/G/1; Explicit examples in the theory of random walks and single-server queues; Muldimensional methods; Many- server queues; Conjugate processes; Insurance risk, dam and storage models.

Reviewer: H.Daduna

##### MSC:

60Kxx | Special processes |

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

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

60-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to probability theory |

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