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

We consider a general single-server multiclass queueing system that incurs a delay cost Cₖ(τₖ) for each class k job that resides τₖ 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/μₖ, 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 μₖ(t)cₖ(aₖ(t)), the generalized cμ rule. As a dynamic priority rule that depends on very little data, the generalized cμ rule is attractive to implement. We show that, with nondecreasing convex delay costs, the generalized cμ 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.