Journal of Applied Mathematics and Stochastic Analysis
Volume 12 (1999), Issue 4, Pages 311-338

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

Charles Knessl

University of Illinois at Chicago, Dept. of Mathematics, Statistics, and Computer Science (M/C 249), Chicago 60607, IL, USA

Received 1 April 1998; Revised 1 February 1999

Copyright © 1999 Charles Knessl. 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.


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.