Scheduling Flexible Servers with Convex Delay Costs: Heavy-Traffic Optimality of the Generalized cμ-Rule

We consider a queueing system with multitype customers and flexible (multiskilled) servers that work in parallel. If Q i is the queue length of type i customers, this queue incurs cost at the rate of C i (Q i ), where C i (·) is increasing and convex. We analyze the system in heavy traffic (Harrison and Lopez 1999) and show that a very simple generalized cμ-rule (Van Mieghem 1995) minimizes both instantaneous and cumulative queueing costs, asymptotically, over essentially all scheduling disciplines, preemptive or non-preemptive. This rule aims at myopically maximizing the rate of decrease of the instantaneous cost at all times, which translates into the following: when becoming free, server j chooses for service a type i customer such that i ε arg max i Cμ i (Q i )μ ij , where μ ij is the average service rate of type i customers by server j. An analogous version of the generalized cμ-rule asymptotically minimizes delay costs. To this end, let the cost incurred by a type i customer be an increasing convex function C i (D) of its sojourn time D. Then, server j always chooses for service a customer for which the value of C′ i (D) μ ij is maximal, where D and i are the customer's sojourn time and type, respectively.