Optimal routing to parallel heterogeneous servers-small arrival rates

Consider a set of k(⩾2) heterogeneous and exponential servers that operate in parallel. Customers arrive into a single infinite capacity buffer according to a Poisson process, and are routed to available servers in accordance with some routing policy. It is shown that for arrival rates in some possible interval (0,λ₀), ever routing policy which minimizes the long-run expected holding cost is contained in the set of routing policies that minimize the expected flow time for a system with fixed initial population and no new arrivals