A New Device for the Synthesis Problem of Optimal Control of Admission to an M/M/c Queue

The problem of finding an optimal admission policy to an M/M/c queue with customers with deadlines is addressed in this paper. There are two streams of customers (customers of class 1 and 2) that are generated according to independent Poisson processes with constant arrival rates. All service times are exponentially distributed with a class independent service rate. Upon arrival a class 1 customer may be admitted or rejected, while incoming class 2 customers are always admitted. Associated with the n-th class 1 customer is a deadline, Dn, by which it must complete service where {Dn}( is a sequence of i.i.d. random variables. We are interested in the throughput of class 1 customers 1that complete their service before their deadline (usually referred to as goodput), and we wish to determine an admission control that maximizes this throughput. At each decision epoch we assume that the controller has available to it the complete history of the total queue length process as well as all of the past decisions that have been made up to that epoch. We show that, for a large class of deadline distributions, there exists a stationary admission policy of a threshold type that maximizes a discounted cost function (for small discount factors) that corresponds to the discounted goodput of customers that make their deadline. The proof relies on a new device that consists in a partial construction of the solution of the dynamic programming equation. In addition, we show that there also exists a threshold admission control that maximizes the long-run average problem (i.e., maximizes the goodput). The proposed method is of independent interest and should apply to many queueing control problems.