Performance of the Gittins Policy in the G/G/1 and G/G/k, With and Without Setup Times

We consider the classic problem of preemptively scheduling jobs in a queue to minimize mean number-in-system, or equivalently mean response time. Even in single-server queueing models, this can be a nontrivial problem whose answer depends on the information available to the scheduler. The simplest case is when the scheduler knows each job's size, for which the optimal policy is Shortest Remaining Processing Time (SRPT) [9]. In the more realistic case of scheduling with unknown or partially known job sizes, people consider the Gittins policy [1-3, 12]. Roughly speaking, Gittins assigns each job a scalar rank based on an estimate of its remaining work, then serves the job of least rank.