Asymptotically optimal interruptible service policies for scheduling jobs in a diffusion regime with nondegenerate slowdown

Abstract

A parallel server system is considered, with I customer classes and many servers, operating in a heavy traffic diffusion regime where the queueing delay and service time are of the same order of magnitude. Denoting by $\widehat{X}^{n}$ and $\widehat{Q}^{n}$, respectively, the diffusion scale deviation of the headcount process from the quantity corresponding to the underlying fluid model and the diffusion scale queue-length, we consider minimizing r.v.’s of the form $c^{n}_{X}=\int_{0}^{u}C(\widehat{X}^{n}(t))\,dt$ and $c^{n}_{Q}=\int_{0}^{u}C(\widehat{Q}^{n}(t))\,dt$ over policies that allow for service interruption. Here, C:ℝ^I→ℝ₊ is continuous, and u>0. Denoting by θ the so-called workload vector, it is assumed that $C^{*}(w):=\min\{C(q):q\in\mathbb{R}_{+}^{\mathbf{I}},\theta\cdot q=w\}$ is attained along a continuous curve as w varies in ℝ₊. We show that any weak limit point of $c^{n}_{X}$ stochastically dominates the r.v. $\int_{0}^{u}C^{*}(W(t))\,dt$ for a suitable reflected Brownian motion W and construct a sequence of policies that asymptotically achieve this lower bound. For $c^{n}_{Q}$, an analogous result is proved when, in addition, C ^∗ is convex. The construction of the policies takes full advantage of the fact that in this regime the number of servers is of the same order as the typical queue-length.