On Sojourn Times in the $M/M/1$-PS Model, Conditioned on the Number of Other Users

Qiang Zhen; Charles Knessl

doi:10.1093/amrx/abq001

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arXiv · 0902.1200

On Sojourn Times in the $M/M/1$-PS Model, Conditioned on the Number of Other Users

Qiang Zhen·Charles Knessl

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Abstract

We consider the $M/M/1$-PS queue with processor sharing. We study the conditional sojourn time distribution of an arriving customer, conditioned on the number of other customers present. A new formula is obtained for the conditional sojourn time distribution, using a discrete Green's function. This is shown to be equivalent to some classic results of Pollaczeck and Vaulot from 1946. Then various asymptotic limits are studied, including large time and/or large number of customers present, and heavy traffic, where the arrival rate is only slightly less than the service rate.

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Qiang Zhen, Charles Knessl. 2009-02-06. On Sojourn Times in the $M/M/1$-PS Model, Conditioned on the Number of Other Users. https://doi.org/10.1093/amrx/abq001

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