We consider a single stage queueing system with c heterogeneous servers. Customers arrive at this system according to a renewal process with mean 1/λ and squared coefficient of variation (scv) C2a. An incoming customer is routed to server i with probability θi, Σci=1 θi = 1. The service times at server i are independent, identically distributed random variables with mean 1/μi and scv C2Si. The holding cost rate of queue i is hi per customer, i = 1, 2,…, c. The problems of interest are twofold: (a) for a fixed service rate allocation μi, Σci=1 μi = μ, find the routing probabilities, θ*i, Σci=1 θ*i = 1, that minimize the average total holding cost; and (b) for fixed routing probabilities θi, Σci=1 θi, and total service rate μ, find the service rate allocation μ*i = μδ*i, Σci=1 δ*i = 1, that minimizes the average total holding cost of the system. For each problem, we characterize the optimal policy under heavy traffic conditions. We also derive the routing probabilities, &thetacrc;i (proportions &deltacrc;i), i = 1,…, c, that are strongly asymptotically optimal. That is, the difference between the average total holding costs under &thetacrc;i, i = 1,…, c, and θ*i, i = 1,…, c (&deltacrc;i, i = 1,…, c, and δ*i, i = 1,…, c) is bounded by a fixed constant independent of the routing probabilities (proportions) and the arrival rate. In addition, we discuss the necessity and sufficiency of the accurate knowledge of the means and scvs of the interarrival and service times in obtaining asymptotically optimal policies.
