Is Tail-Optimal Scheduling Possible?

This paper focuses on the competitive analysis of scheduling disciplines in a large deviations setting. Though there are policies that are known to optimize the sojourn time tail under a large class of heavy-tailed job sizes (e.g. Processor Sharing and Shortest Remaining Processing Time) and there are policies known to optimize the sojourn time tail in the case of light-tailed job sizes (e.g. First Come First Served), no policies are known that can optimize the sojourn time tail across both light-tailed and heavy-tailed job size distributions. We prove that no such work-conserving, non-anticipatory, non-learning policy exists and thus that a policy must learn (or know) the job size distribution in order to optimize the sojourn time tail.