Optimal multiserver scheduling with unknown job sizes in heavy traffic

We consider scheduling to minimize mean response time of the M/G/k queue with unknown job sizes. In single-server k=1 case, optimal policy is Gittins policy, but it not known whether or any other in multiserver case. Exactly analyzing under intractable, and a particularly complicated that hard analyze even this work we introduce monotonic (M-Gittins), new variation show minimizes heavy-traffic for wide class finite-variance size distributions. also shortest expected remaining processing (M-SERPT) which simpler than M-Gittins, 2-approximation heavy traffic similar conditions. These results constitute most general optimality date Our techniques build upon by Grosof et al. (2018), who study simple policies, such as SRPT, M/G/k; Bansal Kamphorst Zwart (2020), Lin (2010), scaling policies M/G/1; Aalto (2009,2011) Scully (2018,2020), characterize M/G/1.