Conditional sojourn time of optimal scheduling policy in a multi - class single - server queue ⋆

Based on the groundbreaking result of Gittins on the multiarmed bandit problem we provide a characterization of the optimal nonanticipating scheduling policy in a multi-class single-server queue. We apply Gittins’ framework to characterize the optimal policy when the service time distribution of the various classes belong to the set of Decreasing Hazard Rate (DHR) distributions, like Pareto or hyper-exponential. When there is only one class it is known that the Least Attained Service (LAS) policy is optimal. We show that in the multi-class case the optimal policy is a priority discipline, where jobs of the various classes depending on their attained service are classified into several priority levels. Using a tagged-job approach and the collective mark method we obtain, for every class, the mean sojourn time conditioned on the service requirement. Numerical computations show that the performance gain of Gittins’ policy can be significant.