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 the largest jobs. We consider scheduling policies that bias against large jobs, towards large jobs, and those that are fair, e.g., processor-sharing (PS). 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−ρ), where ρ denotes the load. We also find that the expected slowdown under any work conserving policy can be made arbitrarily close to that under PS, for all job sizes that are sufficiently large.