A single-server processor-sharing system with M job classes is analyzed in the steady state. The scheduling strategy considered divides the total processor capacity in unequal fractions among the different job classes. More precisely, if there are N~jobs of classj in the system, j = 1, 2 ..... M, each class k job receives a fraction gh/(~M.~ giN~) of the processor capacity.Earlier analyses of this system are shown to be incorrect and new expressions for the conditional expected response times Wk(t) of class k jobs with required service time t are obtained (for general required service time distributions). These yield the asymptotic behavior of W~(t) as t ~ oo and rather simple formulas in the exponential case. The unconditional average response times are also obtained.

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In this paper we study the problem of designing scheduling strategies when the demand on the system is known and waiting time requirements are pre-specified. This important synthesis problem has received little attention in the literature, and contrasts with the common analytical approach to the study of service systems. This latter approach contributes only indirectly to the problem of finding satisfactory scheduling rules when the desired (or required) response-time performance is known in advance. Briefly, the model studied assumes a Markov queueing system with M (priority) classes of jobs. For each class, a desired mean waiting time is given in advance. Making use of a well known conservation law, we prove a necessary and sufficient condition for the existence of a scheduling strategy that achieves the desired performance. We also give a constructive procedure for checking the condition and, if a solution exists, a procedure for finding one such strategy. Our assumptions are discussed and the possibility of relaxing them is explored.

Probabilistic modelling is the most cost-effective means of performance and reliability evaluation of complex dynamic systems. This self-contained text will be welcomed by students and teachers for its no-nonsense treatment of the basic results and examples of their application. The only mathematical background that is assumed is basic calculus. The necessary fundamentals of probability theory are included, as well as an introduction to renewal, Poisson and Markov processes. Models arising in the fields of manufacturing, computing and communications, involving single or multiple service stations and one or more customer classes, are examined in some detail. Both exact and approximate solution methods are discussed, including recent techniques such as spectral expansion. Special attention is devoted to models of systems subject to breakdowns and repairs. Throughout the book, strong emphasis is placed on explaining the ideas behind the results and helping the reader to use them, making the book ideal for students in computer science, engineering or operations research taking courses in modern system design.

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