Minimizing the mean slowdown in the M/G/1 queue

We consider the optimal scheduling problem in the M/G/1 queue. While this is a thoroughly studied problem when the target is to minimize the mean delay, there are still open questions related to some other objective functions. In this paper, we focus on minimizing mean slowdown among non-anticipating polices, which may utilize the attained service of the jobs but not their remaining service time when making scheduling decisions. By applying the Gittins index approach, we give necessary and sufficient conditions for the jobs’ service time distribution under which the well-known scheduling policies first come first served and foreground background are optimal with respect to the mean slowdown. Furthermore, we characterize the optimal non-anticipating policy in the multi-class case for certain types of service time distributions. In fact, our results cover a more general objective function than just the mean slowdown, since we allow the holding costs for a job to depend on its own service time S via a generic function c(S). When minimizing the mean slowdown, this function reads as c(x) = 1 / x .