Stabilization of Branching Queueing Networks

TL;DR

This paper introduces extended Jackson queueing networks with branching and control features, providing polynomial-time algorithms to determine stability and compute controllers that ensure finite queue moments.

Contribution

It extends Jackson networks with new branching and control capabilities and offers efficient methods to verify stability and design stabilizing controllers.

Findings

Decidability of network stability in polynomial time.

Existence of a static randomized controller when stability is achievable.

All moments of queue sizes are finite under the proposed control.

Abstract

Queueing networks are gaining attraction for the performance analysis of parallel computer systems. A Jackson network is a set of interconnected servers, where the completion of a job at server i may result in the creation of a new job for server j. We propose to extend Jackson networks by "branching" and by "control" features. Both extensions are new and substantially expand the modelling power of Jackson networks. On the other hand, the extensions raise computational questions, particularly concerning the stability of the networks, i.e, the ergodicity of the underlying Markov chain. We show for our extended model that it is decidable in polynomial time if there exists a controller that achieves stability. Moreover, if such a controller exists, one can efficiently compute a static randomized controller which stabilizes the network in a very strong sense; in particular, all moments of the queue sizes are finite.