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 at a heavy traffic diffusion regime where the queueing delay and service time are of the same order of magnitude. Denoting by X̂ and Q̂ the diffusion scale deviation of the headcount process from the quantity corresponding to the underlying fluid model and, respectively, the diffusion scale queue-length, we consider minimizing r.v.’s of the form cX = ∫ u 0 C(X̂(t))dt and cQ = ∫ u 0 C(Q̂(t))dt over policies that allow for service interruption. Here, C : R → R+ is continuous and u > 0. Denoting by θ the so called workload vector, it is assumed that C∗(w) := min{C(q) : q ∈ R+, θ · q = w} is attained along a continuous curve as w varies in R+. We show that any weak limit point of cX stochastically dominates the r.v. ∫ u 0 C∗(W (t))dt for a suitable reflected Brownian motion W , and construct a sequence of policies that asymptotically achieve this lower bound. For cQ, 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.