A Two Dimensional Diffusion Approximation for a Loss Model with Trunk Reservation

We consider a loss model with C servers, and arriving customers are split into two classes. Of the C servers, R may be used only by the high priority class. Thus if a high priority customer sees all C servers occupied that customer is lost, while a low priority customer is lost if they see >_ C-R servers occupied. Assuming Poisson arrivals of both customer types and exponential service, we study the problem asymptotically, with C -> infinity and the arrival rates comparably large. 

We assume that the total load is roughly equal to the number of servers, and obtain a two dimensional diffusion equation satisfied by the joint steady state probability distribution of the numbers of servers occupied by the two customer classes. We analyze this equation by a combination of analytic and numerical methods.