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We consider scheduling to minimize mean response time of the M/G/k queue with unknown job sizes. In the singleserver k = 1 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. [6], who study simple policies, such as SRPT, in the M/G/k; Bansal et al. [2], Kamphorst and Zwart [7], and Lin et al. [9], who analyze mean response time scaling of simple policies in the heavy-traffic M/G/1; and Aalto et al. [1] and Scully et al. [11, 13], who characterize and analyze the Gittins policy in the M/G/1.

Optimal Multiserver Scheduling with Unknown Job Sizes in Heavy Traffic