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Mean delay optimization for the M/G/1 queue with pareto type service times

We consider the optimal scheduling problem in the M/G/1 queue with the objective to minimize the mean delay (i.e., sojourn time). We assume that jobs are served according to a work conserving and nonanticipating (scheduling) discipline π. A discipline is work conserving if it does not idle when there are jobs waiting, and nonanticipating if the remaining service times of jobs are unknown for the scheduler. Let Π denote the family of such disciplines. In particular we note that the Shortest-Remaining-Processing-Time (SRPT) discipline does not belong to Π. It is known that for Decreasing Hazard Rate (DHR) service times, the Foreground-Background (FB) discipline is optimal in Π, whereas the First-Come-First-Served (FCFS) discipline minimizes the mean delay for the service time distributions that belong to the New Better than Used in Expectation (NBUE) class [6, 4, 5, 1]. FB is an age-based discipline giving full priority to the job with least amount of attained service, see [3, Section 4.6]. Our goal is to shed light on the impact of designing an optimal scheduling discipline when the hazard rate of the service time distribution is nonmonotone. In particular, we concentrate on distributions for which the hazard rate is constant (zero or positive) for small values while decreasing for larger values. An example is the Pareto distribution,