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We consider the problem of scheduling to minimize asymptotic tail latency in an M/G/1 queue with unknown job sizes. When the job size distribution is heavy-tailed, numerous policies that do not require job size information (e.g. Processor Sharing, Least Attained Service) are known to be strongly tail optimal, meaning that their response time tail has the fastest possible asymptotic decay. In contrast, for light-tailed size distributions, only in the last few years have policies been developed that outperform simple First-Come First-Served (FCFS). The most recent of these is γ-Boost, which achieves strong tail optimality in the light-tailed setting. But thus far, all policies that outperform FCFS in the light-tailed setting, including γ-Boost, require known job sizes. In this paper, we design a new scheduling policy that achieves strong tail optimality in the light-tailed M/G/1 with unknown job sizes. Surprisingly, the optimal policy turns out to be a variant of the Gittins policy, but with a novel and unusual feature: it uses a negative discount rate. Our work also applies to systems with partial information about job sizes, covering γ-Boost as an extreme case when job sizes are in fact fully known. 

Service level objectives (SLOs) for queueing systems typically relate to the tail of the system's response time distribution T. The tail is the function mapping a time t to the probability P[T > t]. SLOs typically ask that high percentiles of T are not too large, i.e. that P[T > t] is small for large t.