Optimal threshold-based admission control in the $M/M/s$ system with heterogeneous servers and a common queue

The article discusses the $M/M/s$ system with heterogeneous servers and a common queue equipped with the mechanism to control the queue length in order to maximize the average marginal profit. The profit function includes a fee for successfully serviced customers, a fine for each rejected customer, a fine for idle period for each server, a fine for waiting (or for exceeding the allowable waiting time), and costs associated with queue maintenance. The problem is to maximize the marginal profit on a set of simple threshold-based queue length control policies. The property of convexity of the profit function is proved and conditions for existence of a finite optimal threshold of the queue length are obtained.