A diffusion model for two tandem queues with general renewal input

We consider two tandem queues with exponential servers. Arrivals to the first queue are governed by a general renewal process. If the arrivals were also exponentially distributed, this would be a simple example of a Jackson network. However, the structure of the model is much more complicated for general arrivals. We analyze the joint steady-state queue length distribution for this network, in the heavy traffic limit, where the arrival rate is only slightly less than the service rates. We formulate and solve the boundary value problem for the diffusion approximation to this model. We obtain simple integral representations for the (asymptotic) steady-state queue length distribution. We also do a detailed study of the tail behavior of the diffusion approximation.