Threshold control policies for heterogeneous server systems

We study the problem of optimally controlling a multiserver queueing system. Customers arrive in a Poisson fashion and join a single queue, served by N servers, S-1, S-2,..., S-N. The servers have different rates. The service times at each server are independent and exponentially distributed. The objective is to determine the policy which minimizes the average number of customers in the system. We show that any optimal, nonpreemptive policy is of threshold type, i.e., it assigns a customer to server Si, if this server is the fastest server available and the number of customers in the queue is m(i) or more. The threshold m(i) may depend on the condition of other (slower) servers at the decision instant. In order to establish the results, we reformulate the optimal control problem as a linear program and use a novel argument based on the structure of the constraint matrix.