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 of unknown size (a.k.a. service time) in a queue to minimize mean number-in-system, or equivalently mean response time (a.k.a. sojourn time). We know how to solve this problem in an M/G/1, provided the job size distribution is known to the scheduler. In this case, the optimal policy is the Gittins policy (a.k.a. Gittins index policy) [1].