On the asymptotic optimality of the cμ/θ rule under ergodic cost

We consider an overloaded multi-server multi-class queueing model where customers may abandon while waiting to be served. For class i, service is provided at rate μi, and abandonment occurs at rate θi. In a many-server fluid regime, we show that prioritizing the classes in decreasing order of ciμi/θi asymptotically minimizes an ergodic holding cost, where ci denotes the equivalent holding cost per unit time for class i.