Note--Optimal Assignment of Customers in a Two Server Congestion System with No Waiting Room

We consider a two server congestion system which is heterogeneous in the sense that the reward received depends on the match between server and customer. In particular, if a type s customer (s = 1, 2, 3, ...) is assigned to server i (i = 1, 2) a reward R_st is earned. Service time is assumed to depend only on customer type. Upon arrival a customer must be assigned to a server, (if one is available) with all customers who find both servers occupied being turned away. The policy that maximizes the long-run expected reward earned over an infinite horizon is shown to depend on a single critical number. Applications to the deployment of fire engines and assignment of patients to coronary care units are briefly discussed.