A semidefinite programming approach to the optimal control of a single server queueing system with imposed second moment constraints

Classical analyses of the dynamic control of multi-class queueing systems frequently yield simple priority policies as optimal. However, such policies can often result in excessive queue lengths for the low priority jobs/customers. We propose a stochastic optimisation problem in the context of a two class M/M/1 system which seeks to mitigate this through the imposition of constraints on the second moments of queue lengths. We analyse the performance of two families of parametrised heuristic policies for this problem. To evaluate these policies we develop lower bounds on the optimum cost through the achievable region approach. A numerical study points to the strength of performance of threshold policies and to directions for future research.