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

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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.