An approximation for the sojourn-time distribution for the gi/g/1 processor-sharing queue

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Advanced Queuing Theory Analysis Probability and Risk Models Age of Information Optimization

Abstract

In this paper, we propose an approximation for the sojourn-time distribution for the GI/G/1 processor-sharing queue and derive the heavy traffic theory for the special case of the M/G/1 queue. The main motivation for proposing this approximation is that existing known exact methods for the M/G/1 and GI/M/1 cases are far too complicated for use by practitioners who may want numerical information about the distribution. The approximation consists of expressing the sojourn-time as the product of two independent random variables. One of them is the service time and the other has the Gamma distribution. The approximation relies on the heavy traffic theory for the M/G/1 and GI/M/1 cases. We show that this distribution is exact in many limiting cases (such as heavy and light traffic) and has some other desirable properties. We compare the approximation against simulations and show that it performs very well.

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References

1983A Guide to Simulation 1983The Single Server Queue.1970Waiting Time Distributions for Processor-Sharing Systems 1987Processor-sharing queues: Some progress in analysis 1989Markov processes whose steady state distribution is matrix-exponential with an application to theGI/PH/1 queue 1985Response-Time Distribution for a Processor-Sharing System

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2005AnRG-Factorization Approach for a BMAP/M/1 Generalized Processor-Sharing Queue 2003Sojourn time distribution in a MAP/M/1 processor-sharing queue 2004Sojourn time distribution in the M/M/1 queue with discriminatory processor-sharing 1994GI/G/1 processor sharing queue in heavy traffic 1991The gi/m/1 processor-sharing queue and its heavy traffic analysis 2003Resource sharing with subexponential distributions 2006The M/G/1 processor-sharing queue with disasters 2006A sample path relation for the sojourn times in G/G/1−PS systems and its applications

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