Optimal Control of a Heterogeneous Two Server Queue in Light Traffic.

We consider a queueing system with two servers and an infinite buffer. Service times are exponentially distributed and customers arrive in a Poisson process. The two servers have different service rates. We are interested in assigning idle servers to waiting customers in order to minimize the long run average sojourn time of customers. Customers in service cannot be preempted. It is known that the optimal control has threshold form: always use the faster server, and place a customer into the slower server if the queue length is larger than some threshold. The transient problem, having a finite initial population and no arrivals, is also known to have an optimal policy of threshold form. We show that, for a small enough arrival rate, the optimal threshold for the problem with arrivals becomes that of the no-arrivals case.