  Consider a switched queueing network with general routing among its queues.
The MaxWeight policy assigns available service by maximizing the objective
function $\sum_j Q_j \sigma_j$ among the different feasible service options,
where $Q_j$ denotes queue size and $\sigma_j$ 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, as well as 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 shown by
Andrews--Zhang (2003) 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.
