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 μ i , and abandonment occurs at rate θ i . In a many-server fluid regime, we show that prioritizing the classes in decreasing order of c i μ i / θ i asymptotically minimizes an ergodic holding cost, where c i denotes the equivalent holding cost per unit time for class i .