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

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.