We consider a class of parallel server systems that are known as N-systems. In an N-system, there are two customer classes that are catered by servers in two pools. Servers in one of the pools are cross-trained and can serve customers from both classes, whereas all of the servers in the other pool can serve only one of the customer classes. A customer reneges from his queue if his waiting time in the queue exceeds his patience. Our objective is to minimize the total cost that includes a linear holding cost and a reneging cost. We prove that, when the service speed is pool dependent, but not class dependent, a cμ-type greedy policy is asymptotically optimal in many-server heavy traffic.
