Two tandem queues with general renewal input - I: Diffusion approximation and integral representations

We consider two tandem queues with exponential servers. Arrivals to the fir st queue are governed by a general renewal process. If the arrivals were al so exponentially distributed, this would be a simple example of a Jackson n etwork. However, the structure of the model is much more complicated for ge neral arrivals. We analyze the joint steady-state queue length distribution for this network, in the heavy traffic limit, where the arrival rate is on ly slightly less than the service rates. We formulate and solve the boundar y value problem for the diffusion approximation to this model. We obtain si mple integral representations for the (asymptotic) steady-state queue lengt h distribution.