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Response-Time Distribution for a Processor-Sharing System

J. A. Morrison
SIAM Journal on Applied Mathematics
Vol. 45, No. 1 (Feb., 1985), pp. 152-167
Stable URL: http://www.jstor.org/stable/2101088
Page Count: 16
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Response-Time Distribution for a Processor-Sharing System
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Abstract

We consider the response time for jobs in a processor-sharing system with a Poisson arrival process and exponentially distributed required service time, i.e. an M/M/1 - PS queue. The response time W̃ is the sum of the delay and the required service time. We derive an integral representation for the equilibrium response-time distribution $Pr\{\tilde W > t\}$, and evaluate this integral numerically for several values of the traffic intensity $\rho < 1$. We also investigate the behavior of $Pr\{\tilde W > t\}$ in the heavy-traffic case when ρ is close to 1. For $\rho = 1 - \alpha, 0 < \alpha \ll 1$, it is shown that a singular perturbation occurs at t = 0, and we derive a composite approximation to $Pr\{\tilde W > t\}$. For 0.75 ≤ ρ ≤ 0.99, the relative error of this approximation is less than 0.62α2 in the range for which $1 \geqq Pr\{\tilde W > t\} \geqq 0.01$.

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