A fork-join queueing model: Diffusion approximation, integral representations and asymptotics

We consider two parallel M/M/1 queues which are fed by a single Poisson arrival stream. An arrival splits into two parts, with each part joining a different queue. This is the simplest example of a fork-join model. After the individual parts receive service, they may be joined back together, though we do not consider the join part here. We study this model in the heavy traffic limit, where the service rate in either queue is only slightly larger than the arrival rate. In this limit we obtain asymptotically the joint steady-state queue length distribution. In the symmetric case, where the two servers are identical, this distribution has a very simple form. In the non-symmetric case we derive several integral representations for the distribution. We then evaluate these integrals asymptotically, which leads to simple formulas which show the basic qualitative structure of the joint distribution function.