Asymptotic expansions for the conditional sojourn time distribution in the M/M/1-PS queue

Authors:
Qiang Zhen;Charles Knessl
Affiliations:
Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, Chicago, USA 60607-7045;Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, Chicago, USA 60607-7045
Venue:
Queueing Systems: Theory and Applications
Year:
2007

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Tail asymptotics of the waiting time and the busy period for the $${{\varvec{M/G/1/K}}}$$ queues with subexponential service times

Queueing Systems: Theory and Applications

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Abstract

We consider the M/M/1 queue with processor sharing. We study the conditional sojourn time distribution, conditioned on the customer's service requirement, in various asymptotic limits. These include large time and/or large service request, and heavy traffic, where the arrival rate is only slightly less than the service rate. The asymptotic formulas relate to, and extend, some results of Morrison (SIAM J. Appl. Math. 45:152---167, [1985]) and Flatto (Ann. Appl. Probab. 7:382---409, [1997]).