Optimal multiserver scheduling with unknown job sizes in heavy traffic

Abstract

We consider scheduling to minimize mean response time of the M/G/k queue with unknown job sizes. In the single-server 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. (2018), who study simple policies, such as SRPT, in the M/G/k; Bansal et al. (2018), Kamphorst and Zwart (2020), and Lin et al. (2010), who analyze mean response time scaling of simple policies in the heavy-traffic M/G/1; and Aalto et al. (2009,2011) and Scully et al. (2018,2020), who characterize and analyze the Gittins policy in the M/G/1.