International Journal of Stochastic Analysis

International Journal of Stochastic Analysis / 1999 / Article

Open Access

Volume 12 |Article ID 673856 | https://doi.org/10.1155/S1048953399000295

Charles Knessl, "A diffusion model for two parallel queues with processor sharing: transient behavior and asymptotics", International Journal of Stochastic Analysis, vol. 12, Article ID 673856, 28 pages, 1999. https://doi.org/10.1155/S1048953399000295

A diffusion model for two parallel queues with processor sharing: transient behavior and asymptotics

Received01 Apr 1998
Revised01 Feb 1999

Abstract

We consider two identical, parallel M/M/1 queues. Both queues are fed by a Poisson arrival stream of rate λ and have service rates equal to μ. When both queues are non-empty, the two systems behave independently of each other. However, when one of the queues becomes empty, the corresponding server helps in the other queue. This is called head-of-the-line processor sharing. We study this model in the heavy traffic limit, where ρ=λ/μ1. We formulate the heavy traffic diffusion approximation and explicitly compute the time-dependent probability of the diffusion approximation to the joint queue length process. We then evaluate the solution asymptotically for large values of space and/or time. This leads to simple expressions that show how the process achieves its stead state and other transient aspects.

Copyright © 1999 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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