Optimal Trunk Reservation for an Overloaded Link

We consider a single link with a large number of circuits which is offered two types of calls. The calls arrive as Poisson processes, and they have exponentically distributed holding times with the same mean. The traffic intensity of the primary calls is assumed to be strictly less than 1, but the total traffic intensity is assumed to be strictly greater than 1, corresponding to an overloaded link. R of the circuits are reserved for the primary calls, so that if fewer than R + 1 circuits are idle when a secondary call arrives it is not accepted. If all circuits are busy when a primary call arrives it is not accepted. Rewards are generated when calls are accepted, and it is assumed that the reward for primary calls is greater than that for secondary calls. We determine the trunk reservation parameter R which asymptotically maximizes the long run average reward.