The cµ/θ rule for many-server queues with abandonment

We consider a multi-class queueing system with multiple homogeneous servers and customer abandonment. For each customer class i, the holding cost per unit time, the service rate and the abandonment rate are denoted by ci, µi and θi, respectively. We prove that under a many-server fluid scaling and overload conditions, a routing policy that assigns priority to classes according to their index ciµi/θi, is asymptotically optimal for minimizing the overall long run average holding cost. An additional penalty on customer abandonment is easily incorporated into this model and leads to a similar index rule.