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

Charles Knessl
1999 Journal of Applied Mathematics and Stochastic Analysis  
We consider two identical, parallel M/M/1 queues. Both queues are fed by a Poisson arrival stream of rate I and have service rates equal to #. 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-@the-line processor sharing. We study this model in the heavy traffic limit, where p 1/#1. We formulate the heavy traffic diffusion approximation and
more » ... citly 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.
doi:10.1155/s1048953399000295 fatcat:fmixb2737bhvfgfba2s4j6hjiu