Characterization and optimization of achievable performance in general queueing systems

This paper considers general (single facility) queueing systems with exponential service times, dealing with a finite number J of distinct customer classes. Performance of the system, as measured by the vector of steady state expected sojourn times of the customer classes (the performance vector) may be controlled by adopting an appropriate preemptive priority discipline. We show that the performance space, the set of performance vectors which are achievable under some preemptive work conserving rule, is a polyhedron described by 2^J-1 (in)equalities. The special structure of this polyhedron nevertheless allows for efficient procedures to minimize any separable convex function of the performance vector. Linear objectives are shown to be minimized by absolute priority rules, thus generalizing a well known result for M/M/1 systems. We also show that each point in the performance space may be achieved by a specific randomization of at most J + 1 absolute priority rules.