Optimal dispatching to parallel heterogeneous servers in light traffic

Customers arrive at a single queue according to a Poisson process and are dispatched to several parallel heterogeneous and exponential servers. It is shown that the set of optimal dispatching policies which minimize the total expected holding cost for a system with a fixed initial population and no new arrivals contains every optimal policy which minimizes the long-run expected holding cost in a system with an arrival rate in some small positive interval.<>