Stability and instability of the maxweight policy

Consider a switched queueing network with general routing among its queues. The MaxWeight policy assigns available service by maximizing the objective function ^∑_jQ_jσ_j among the different feasible service options, where Q_j denotes queue size and σ_j denotes the amount of service to be executed at queue j. MaxWeight is a greedy policy that does not depend on knowledge of arrival rates and is straightforward to implement. These properties and its simple formulation suggest MaxWeight as a serious candidate for implementation in the setting of switched queueing networks; MaxWeight has been extensively studied in the context of communication networks. However, a fluid model variant of MaxWeight was previously shown not to be maximally stable. Here, we prove that MaxWeight itself is not in general maximally stable. We also prove MaxWeight is maximally stable in a much more restrictive setting, and that a weighted version of MaxWeight, where the weighting depends on the traffic intensity, is always stable.