  A matching queue is described via a graph $G$ together with a matching
policy. Specifically, to each node in the graph there is a corresponding
arrival process of items which can either be queued, or matched with queued
items in neighboring nodes. The matching policy specifies how items are matched
whenever more than one matching is possible. Motivated by the increasing
theoretical interest in such matching models, we investigate the question of
(in)stability of matching queues which satisfy a natural necessary condition
for stability, which can be thought of as an analogue of the usual traffic
condition for traditional queueing networks (namely, $\rho_i < 1$ in each
service station $i$). We employ fluid-stability arguments to show that matching
queues can in general be unstable, even though the necessary stability
condition is satisfied.
