Counter-Examples to the Optimality of MWS-α Policies for Scheduling in Generalized Switches

It has been conjectured that MWS-α scheduling policies with α going to zero are heavy-traffic optimal for scheduling in a generalized switch when the objective is to minimize the number of backlogged packets in the system. We examine this conjecture by first deriving optimal or heavy-traffic optimal policies for small switches and then comparing them to MWS-α policies. Our conclusion is that the conjecture is not true in general, i.e., there are simple topologies for which there exist policies which outperform MWS-α in heavy traffic.