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

TL;DR

This paper proves that no policy can stabilize the critical push-pull queueing network and introduces a general condition for non-stabilizability applicable to similar networks.

Contribution

It settles the conjecture on the non-existence of stabilizing policies and provides a new sufficient condition for non-stabilizability in queueing networks.

Findings

Proves the non-existence of stabilizing policies for the critical push-pull network.

Develops a linear martingale-based criterion for non-stabilizability.

Applies the criterion to generalizations of the push-pull network.

Abstract

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.