Strong stationary duality for Möbius monotone Markov chains

Paweł Lorek,Ryszard Szekli

doi:10.1007/s11134-012-9284-z

Abstract

For Markov chains with a finite, partially ordered state space, we show strong stationary duality under the condition of Möbius monotonicity of the chain. We give examples of dual chains in this context which have no downwards transitions. We illustrate general theory by an analysis of nonsymmetric random walks on the cube with an interpretation for unreliable networks of queues.

Highlights

The motivation of this paper stems from a study on the speed of convergence to stationarity for unreliable queueing networks, as in Lorek and Szekli [13]
In order to give bounds on the speed of convergence for some unreliable queueing networks, it is necessary to study the availability vector of unreliable network processes. This vector is a Markov chain with the state space representing sets of stations with down or up status via the power set of the set of nodes
We introduce two versions of Möbius monotonicity, and we define a new notion of Möbius monotone functions which appear in a natural way in our main result on Strong Stationary Dual (SSD)

Summary

Introduction

The motivation of this paper stems from a study on the speed of convergence to stationarity for unreliable queueing networks, as in Lorek and Szekli [13]. One of the most basic and interesting ones is given by Diaconis and Fill [7] (Theorem 4.6) when the state space is linearly ordered In this case, under the assumption of stochastic monotonicity for the time reversed chain, and under the condition that for the initial distribution ν, ν ≤mlr π (that is, for any k1 > k2, ν(k1) π (k1 ). Fill [11] uses the theory of strong stationary duality to give a stochastic proof of an analogous result for discrete-time birth-and-death chains and geometric random variables He shows a link for the parameters of the distributions to eigenvalue information about the chain. Instead of a linearly ordered one and utilize Möbius monotonicity instead of the usual stochastic monotonicity This construction opens new ways to study particular Markov chains by a dual approach and is of independent interest. It is worth mentioning that Möbius monotonicity of nonsymmetric nearest neighbor walks is a stronger property than the usual stochastic monotonicity for this chain

Time to stationarity and strong stationary duality

Möbius monotonicities

Main result

Nearest neighbor Möbius monotone walks on a cube

Further research