Dynamic Scheduling with Convex Delay Costs: The Generalized $c|mu$ Rule

Abstract

We consider a general single-server multiclass queueing system that incurs a delay cost Ck($τ$k) for each class k job that resides $τ$k units of time in the system. This paper derives a scheduling policy that minimizes the total cumulative delay cost when the system operates during a finite time horizon. Denote the marginal delay cost function and the (possibly nonstationary) average processing time of class k by ck = C'k and 1/$μ$k, respectively, and let ak(t) be the "age" or time that the oldest class k job has been waiting at time t. We call the scheduling policy that at time t serves the oldest waiting job of that class k with the highest index $μ$k(t)ck(ak(t)), the generalized c$μ$ rule. As a dynamic priority rule that depends on very little data, the generalized c$μ$ rule is attractive to implement. We show that, with nondecreasing convex delay costs, the generalized c$μ$ rule is asymptotically optimal if the system operates in heavy traffic and give explicit expressions for the associated performance characteristics: the delay (throughput time) process and the minimum cumulative delay cost. The optimality result is robust in that it holds for a countable number of classes and several homogeneous servers in a nonstationary, deterministic or stochastic environment where arrival and service processes can be general and interdependent.