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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.
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When Does the Gittins Policy Have Asymptotically Optimal Response Time Tail in the M/G/1?
We consider scheduling in the M/G/1 queue with unknown job sizes. It is known that the Gittins policy minimizes mean response time in this setting. However, the behavior of the tail of response time under Gittins is poorly understood, even in the large-response-time limit. Characterizing Gittins’s asymptotic tail behavior is important because if Gittins has optimal tail asymptotics, then it simultaneously provides optimal mean response time and good tail performance. In this work, we give the first comprehensive account of Gittins’s asymptotic tail behavior. For heavy-tailed job sizes, we find that Gittins always has asymptotically optimal tail. The story for light-tailed job sizes is less clear-cut: Gittins’s tail can be optimal, pessimal, or in between. To remedy this, we show that a modification of Gittins avoids pessimal tail behavior, while achieving near-optimal mean response time.
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- Award ID(s):
- 2307008
- PAR ID:
- 10532338
- Publisher / Repository:
- INFORMS
- Date Published:
- Journal Name:
- Operations Research
- ISSN:
- 0030-364X
- Subject(s) / Keyword(s):
- queueing theory scheduling M/G/1 queue Gittins index response time tail heavy-tailed distributions light-tailed distributions
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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We study the problem of scheduling jobs in a queueing system, specifically an M/G/1 with light-tailed job sizes, to asymptotically optimize the response time tail. This means scheduling to make P[T > t], the chance a job's response time exceeds t, decay as quickly as possible in the t \to \infty limit. For some time, the best known policy was First-Come First-Served (FCFS), which has an asymptotically exponential tail: P[T > t] ~ C e^-γ t . FCFS achieves the optimal decay rate γ, but its tail constant C is suboptimal. Only recently have policies that improve upon FCFS's tail constant been discovered. But it is unknown what the optimal tail constant is, let alone what policy might achieve it. In this paper, we derive a closed-form expression for the optimal tail constant C, and we introduce γ-Boost, a new policy that achieves this optimal tail constant. Roughly speaking, γ-Boost operates similarly to FCFS, but it pretends that small jobs arrive earlier than their true arrival times. This significantly reduces the response time of small jobs without unduly delaying large jobs, improving upon FCFS's tail constant by up to 50% with only moderate job size variability, with even larger improvements for higher variability. While these results are for systems with full job size information, we also introduce and analyze a version of γ-Boost that works in settings with partial job size information, showing it too achieves significant gains over FCFS. Finally, we show via simulation that γ-Boost has excellent practical performance.more » « less
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1287/opre.2022.0038
