A Queue-Length Cutoff Model for a Preemptive Two-Priority M/M/1 System

Abstract. We consider a two-priority, preemptive, single-server queueing model. Each customer is classified into either a high priority class or a low priority class. The arrivals of the two priority classes follow independent Poisson processes and service time is assumed to be exponentially distributed. A queue-length-cutoff method is considered. Under this discipline the server responds only to high priority customers until the queue length of the other class exceeds a threshold L. After that the server switches to handle only the low priority queue. Steady-state balance equations are established for this system. Then we introduce two-dimensional generating functions to obtain the average number of customers for each priority class. We then focus on the preemptive resume case while allowing for weights associated with both priority class queues. We develop methodologies to obtain the optimal cutoffs for the situation when the weights of both queues are constant (i.e., not a function of queue length) and the situation when the weights change linearly with the queue lengths. It is important to point out that our method does not lead to a closed-form exact solution, but rather a numerical approximation, from which cutoff policies are analyzed.