Asymptotic convergence of scheduling policies with respect to slowdown

We explore the performance of an M/GI/1 queue under various scheduling policies from the perspective of a new metric: the slowdown experienced by largest jobs. We consider scheduling policies that bias against large jobs, towards large jobs, and those that are fair, e.g., Processor-Sharing. We prove that as job size increases to infinity, all work conserving policies converge almost surely with respect to this metric to no more than 1=(1 ), where denotes load. We also find that the expected slowdown under any work conserving policy can be made arbitrarily close to that under Processor-Sharing, for all job sizes that are sufficiently large.