Multicoloured Hardcore Model: Fast Mixing and Queueing

TL;DR

This paper introduces a multicoloured hardcore model to represent multi-channel resource sharing, analyzes Glauber dynamics for it, and derives conditions for rapid mixing to control queue lengths in such systems.

Contribution

It extends the hardcore model to multiple colours, providing new analysis of Glauber dynamics and conditions for fast mixing in multi-channel resource sharing contexts.

Findings

Identifies conditions for fast mixing of Glauber dynamics.

Provides bounds on queue lengths in equilibrium.

Models multi-channel resource sharing with a multicoloured hardcore framework.

Abstract

We extend the hardcore model to a multicoloured version: a subset of vertices of a graph are coloured such that no vertex is adjacent to one of the same colour; uncoloured vertices do not constrain neighbours. This mathematically models multi-channel resource sharing, such as fibreoptic routing. We analyse certain simple Glauber-type dynamics on such configurations, and find conditions which ensure fast mixing. These dynamics model a queueing system: customers queue for service at vertices, who only serve customers whilst they are coloured in the underlying configuration; uncoloured vertices sit idle. The mixing estimates are applied to control queue lengths in equilibrium.