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

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

2009-02-06

arxiv.org/abs/0902.1200v1

Appl. Math. Res. Express. AMRX 2009 (2010) 142-167

Mathematics

Probability

45 pages, 2 figures

Scientific paper

10.1093/amrx/abq001

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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