A Processor-Shared Queue that Models Switching Times: Heavy Usage Asymptotics

A closed processor-sharing queueing model that corresponds to M terminals in series with a central processing unit (CPU) is considered under heavy usage conditions. The effects of switching times between the CPU and the individual jobs are included by assuming that the effective service rate that the CPU provides is a decreasing function of the instantaneous number of jobs being processed. Using perturbation methods, asymptotic expansions are constructed for the first two moments of a customer's response (sojourn) time, conditioned on that customer's total required service from the CPU. The asymptotic limit assumes that the number of terminals M is large $( M \gg 1 )$, that the rate at which the CPU provides service is fast $( O( M ) )$, and that the duration of a typical switching time is small compared to the time the CPU actually spends servicing the individual jobs. Extensive numerical comparisons show the quality of our asymptotic approximations.

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