A FORK-JOIN QUEUING MODEL - DIFFUSION-APPROXIMATION, INTEGRAL-REPRESENTATIONS AND ASYMPTOTICS

We consider two parallel M/M/1 queues which are fed by a single Poisso n arrival stream. An arrival splits into two parts, with each part joi ning a different queue. This is the simplest example of a fork-join mo del. After the individual parts receive service, they may be joined ba ck together, though we do not consider the join part here. We study th is 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 dis tribution has a very simple form. In the non-symmetric case we derive several integral representations for the distribution. We then evaluat e these integrals asymptotically, which leads to simple formulas which show the basic qualitative structure of the joint distribution functi on.