A class of equivalent idle-time-order-based routing policies for heterogeneous multi-server systems

We consider an M/M/N/K/FCFS system (N>0, K>=N), where the servers operate at (possibly) heterogeneous service rates. In this situation, the steady state behavior depends on the routing policy that is used to select which idle server serves the next job in queue. We define a class of idle-time-order-based policies (including, for example, Longest Idle Server First (LISF)) and show that all policies in this class result in the same steady state behavior. In particular, they are all equivalent to the naive Random routing policy.