Optimal admission control in queues with multiple customer classes and abandonments

We consider a Markovian, finite capacity queueing system with multiple customer classes and multiple servers where customers waiting in queue may get impatient and leave without being served. There is a cost associated with abandonments and a holding cost associated with customers in the system. Admitted customers pay a class-dependent reward at the time of arrival. Under these assumptions, our objective is to characterize the optimal admission control policy that maximizes the long-run average reward. We formulate the problem as a Markov decision process problem and prove that the optimal policy is a DST (Double Set of Thresholds) policy where there is a pair of thresholds for each class, such that customers of that class are admitted only if the total number of customers in the system is between the two thresholds. We also identify sufficient conditions under which the optimal policy reduces to an SST (Single Set of Thresholds) policy where each customer class is admitted only if the total number of customers in the system is less than a certain threshold. After investigating how the optimal long-run average reward changes with respect to system parameters, we compare the performance of the optimal SST policy to the optimal policy. We conclude with a study of various extensions of our original model.