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

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

Strong stationary duality for Möbius monotone Markov chains: Examples

We construct strong stationary dual chains for nonsymmetric random walks on square lattice, for random walks on hypercube and for some Ising models on the circle. The strong stationary dual chains are all sharp and have the same state space as original chains. We use Möbius monotonicity of these chains with respect to some natural orderings of the corresponding state spaces. This method provides an alternative way to study mixing times for studied models.We construct strong stationary dual chains for nonsymmetric random walks on square lattice, for random walks on hypercube and for some Ising models on the circle. The strong stationary dual chains are all sharp and have the same state space as original chains. We use Möbius monotonicity of these chains with respect to some natural orderings of the corresponding state spaces. This method provides an alternative way to study mixing times for studied models

Stochastic Comparison of Repairable Systems by Coupling

Stochastic comparison results for replacement policies are shown in this paper using formalism of point processes theory. At each failure moment a repair is allowed performed with a random degree of repair including as special cases perfect, minimal, imperfect repair models. Results for such repairable systems with schemes of planned replacements are also shown. The results are obtained by coupling methods for point processes

Dependence orderings for some functionals of multivariate point processes

We study dependence orderings for functionals of k-variate point processes [Phi] and [Psi]. We view the first process as a collection of counting measures, whereas the second as the sequences of interpoint distances. Subsequently, we establish regularity properties of stationary sequences which generalize known results for iid case. The theoretical results are illustrated by many special cases including comparison of multivariate sums and products, comparison of multivariate shock models and queueing systems.Ordering of point process Supermodular order Directionally convex order Dependence ordering Comparison of workload Multivariate shock models