A diffusion model for two parallel queues with processor sharing: transient behavior and asymptotics. (1999)

• Record Type: Journal Article
• Authors: Knessl, Charles
• Is Part Of: Journal of applied mathematics and stochastic analysis. Volume 12:Number 4(1999)
• Page Start: 311
• Page End: 338
• Publication Date: 1999
• Subjects: processor-shared queues -- diffusion model -- transient distribution; Mathematical models -- Periodicals; Computer simulation -- Periodicals; Computer science -- Mathematics -- Periodicals; Computer science -- Mathematics; Computer simulation; Mathematical models; Applied Mathematics; Periodicals; Electronic journals; 519.22
• DOI: 10.1155/S1048953399000295
• Languages: English
• ISSNs: 1048-9533

Abstract

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.