The paper investigates multi-class queueing systems with dynamic flow-control and dynamic scheduling. A simple M/M/1 queue with two classes of customers is considered. Delay and throughput are measured, respectively, by holding cost rates for each job in queue and admission rewards for each job accepted into the system. The flow-control and the service scheduler jointly optimizing the performance of the queue are obtained numerically (by policy iteration on the dynamic programming optimality equation). When holding cost rates and rewards are class-independent, this computation indicates that the long-run average operating costs are minimized by combining a preemptive scheduler giving priority to the class with shortest mean service time (Shortest-Expected-Job-First) with monotonic admission policies (one for each class). Moreover, it turns out that this monotonic flow-controller admits short jobs according to a threshold on the total number of jobs in queue; on the other hand, the optimal admission of long jobs is closely approximated by a control-limit policy on the total expected work in the queue. Based on this, a steady-state approximation is considered which greatly simplifies the optimization problem and is capable of remarkable accuracy.
