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.

翻译：我们考虑在具有未知作业大小的M/G/1队列中的调度问题。已知在该设定下，Gittins策略能够最小化平均响应时间。然而，即使在响应时间很大的极限下，Gittins策略下响应时间尾部的行为仍未得到充分理解。刻画Gittins策略的渐近尾部行为至关重要，因为如果Gittins策略具有最优尾部渐近性，那么它既能同时提供最优平均响应时间，又能保证良好的尾部性能。本文首次对Gittins策略的渐近尾部行为进行了全面阐述。对于重尾作业大小，我们发现Gittins策略始终具有渐近最优尾部。而对于轻尾作业大小的情况则不那么明确：Gittins策略的尾部可能处于最优、最差或介于两者之间的状态。为解决此问题，我们证明Gittins策略的一种改进形式能够避免最差尾部行为，同时实现接近最优的平均响应时间。