  The push-pull queueing network is a simple example in which servers either
serve jobs or generate new arrivals. It was previously conjectured that there
is no policy that makes the network positive recurrent (stable) in the critical
case. We settle this conjecture and devise a general sufficient condition for
non-stabilizability of queueing networks which is based on a linear martingale
and further applies to generalizations of the push-pull network.
