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Asymptotic Analysis of the Waiting-Time Distribution for a Large Closed Processor-Sharing System

J. A. Morrison
SIAM Journal on Applied Mathematics
Vol. 46, No. 1 (Feb., 1986), pp. 140-170
Stable URL: http://www.jstor.org/stable/2101493
Page Count: 31
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Asymptotic Analysis of the Waiting-Time Distribution for a Large Closed Processor-Sharing System
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Abstract

The service of an interactive computer system is characterized by the random waiting (or response) time perceived by users. The system consists of a bank of terminals, each of which asynchronously alternates between "thinking" and waiting for service from a central processing unit which operates under the processor-sharing discipline. It is assumed that there are (N + 1) terminals, where N is large, and that ρ = Np = O(1), where p is the ratio of the mean required service time to the mean think time. Asymptotic approximations to the equilibrium distribution of the waiting time are derived for: normal usage, $\rho < 1$; heavy usage, ρ - 1 = O(N-1/2); and very heavy usage, $\rho > 1$. Singular perturbation techniques are used, and the matching of the heavy-usage approximation with both the normal and very heavy-usage approximations is investigated. Numerical results are presented which indicate that the approximations are useful for N ≥ 100.

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