We consider scheduling to minimize mean response time of the M/G/k queue with unknown job sizes. In the single-server case, the optimal policy is the Gittins policy, but it is not known whether Gittins or any other policy is optimal in the multiserver case. Exactly analyzing the M/G/k under any scheduling policy is intractable, and Gittins is a particularly complicated policy that is hard to analyze even in the single-server case. In this work we introduce monotonic Gittins (M-Gittins), a new variation of the Gittins policy, and show that it minimizes mean response time in the heavy-traffic M/G/k for a wide class of finite-variance job size distributions. We also show that the monotonic shortest expected remaining processing time (M-SERPT) policy, which is simpler than M-Gittins, is a 2-approximation for mean response time in the heavy traffic M/G/k under similar conditions. These results constitute the most general optimality results to date for the M/G/k with unknown job sizes. Our techniques build upon work by Grosof et al., who study simple policies, such as SRPT, in the M/G/k; Bansal et al., Kamphorst and Zwart, and Lin et al., who analyze mean response time scaling of simple policies in the heavy-traffic M/G/1; and Aalto et al. and Scully et al., who characterize and analyze the Gittins policy in the M/G/1.

中文翻译：

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在繁忙的流量中具有未知作业大小的最佳多服务器调度

我们考虑调度以最小化具有未知作业大小的 M/G/k 队列的平均响应时间。在单服务器情况下，最优策略是 Gittins 策略，但不知道 Gittins 或任何其他策略在多服务器情况下是否最优。在任何调度策略下准确分析 M/G/k 都是棘手的，而 Gittins 是一个特别复杂的策略，即使在单服务器情况下也很难分析。在这项工作中，我们引入了单调 Gittins (M-Gittins)，这是 Gittins 策略的一种新变体，并表明它可以最大限度地减少大流量 M/G/k 中的平均响应时间，适用于各种有限方差作业大小分布. 我们还展示了比 M-Gittins 更简单的单调最短预期剩余处理时间 (M-SERPT) 策略，是类似条件下繁忙交通 M/G/k 中平均响应时间的 2 近似值。这些结果构成了迄今为止对于未知作业大小的 M/G/k 最一般的优化结果。我们的技术建立在 Grosof 等人的工作之上，他们研究了 M/G/k 中的简单策略，例如 SRPT；Bansal 等人、Kamphorst 和 Zwart 以及 Lin 等人，他们分析了大流量 M/G/1 中简单策略的平均响应时间缩放；和阿尔托等人。和 Scully 等人，他们在 M/G/1 中描述和分析了 Gittins 政策。谁分析了大流量 M/G/1 中简单策略的平均响应时间缩放；和阿尔托等人。和 Scully 等人，他们在 M/G/1 中描述和分析了 Gittins 政策。谁分析了大流量 M/G/1 中简单策略的平均响应时间缩放；和阿尔托等人。和 Scully 等人，他们在 M/G/1 中描述和分析了 Gittins 政策。