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

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 Xⁿ and Qⁿ, 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, 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, is attained along a continuous curve as w varies in ℝ₊. We show that any weak limit point of cⁿ_X stochastically dominates the r.v. ∫^u₀ C*(W(t)) for a suitable reflected Brownian motion W and construct a sequence of policies that asymptotically achieve this lower bound. For cⁿ_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.