The Gittins Policy is Nearly Optimal in the M/G/k under Extremely General Conditions

Abstract

The Gittins scheduling policy minimizes the mean response in the single-server M/G/1 queue in a wide variety of settings. Most famously, Gittins is optimal when preemption is allowed and service requirements are unknown but drawn from a known distribution. Gittins is also optimal much more generally, adapting to any amount of available information and any preemption restrictions. However, scheduling to minimize mean response time in a multiserver setting, specifically the central-queue M/G/k, is a much more difficult problem. In this work we give the first general analysis of Gittins in the M/G/k. Specifically, we show that under extremely general conditions, Gittins's mean response time in the M/G/k is at most its mean response time in the M/G/1 plus an $O(łog(1/(1 - ρ)))$ additive term, where ρ is the system load. A consequence of this result is that Gittins is heavy-traffic optimal in the M/G/k if the service requirement distribution S satisfies $\mathbfE [S^2(łog S)^+] < \infty$. This is the most general result on minimizing mean response time in the M/G/k to date. To prove our results, we combine properties of the Gittins policy and Palm calculus in a novel way. Notably, our technique overcomes the limitations of tagged job methods that were used in prior scheduling analyses.