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

Non-Existence of Stabilizing Policies for the Critical Push-Pull Network and Generalizations