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

Charles Knessl

International Journal of Stochastic Analysis, 1999, vol. 12, 1-28

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 / μ → 1 . 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.

Date: 1999
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jnijsa:673856

DOI: 10.1155/S1048953399000295

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