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.