Strong Stationary Duality for M\"obius Monotone Markov Chains: Unreliable Networks

TL;DR

This paper establishes strong stationary duality for M"obius monotone Markov chains on finite partially ordered spaces, with applications to queue networks and random walks, highlighting the chain's upward-only transitions.

Contribution

It introduces strong stationary duality for M"obius monotone chains and explores their properties and applications in network models and random walks.

Findings

Dual chains have transitions only upwards.

M"obius monotonicity relates to other monotonicity definitions.

Application to queue network analysis.

Abstract

For Markov chains with a partially ordered finite state space we show strong stationary duality under the condition of M\"obius monotonicity of the chain. We show relations of M\"obius monotonicity to other definitions of monotone chains. We give examples of dual chains in this context which have transitions only upwards. We illustrate general theory by an analysis of nonsymmetric random walks on the cube with an application to networks of queues.