On the Instability of Matching Queues

TL;DR

This paper investigates the stability of matching queues modeled on graphs, revealing that even when a natural necessary condition for stability is met, these systems can still be unstable, highlighting complex dynamics in such networks.

Contribution

The paper demonstrates that matching queues can be unstable despite satisfying a natural necessary stability condition, using fluid-stability analysis.

Findings

Matching queues can be unstable even when the necessary condition for stability is satisfied.

Fluid-stability arguments reveal potential instability in matching queue systems.

The study highlights the complexity of stability analysis in graph-based matching models.

Abstract

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.