Heavy Traffic Analysis of Two Coupled Processors

We consider two parallel queues. When both are non-empty, they behave as two independent M/M/1 queues. If one queue is empty the server in the other works at a different rate. We consider the heavy traffic limit, where the system is close to instability. We derive and analyze the heavy traffic diffusion approximation for this model. In particular, we obtain simple integral representations for the joint steady state density of the (scaled) queue lengths. Asymptotic and numerical properties of the solution are studied.