A Correlation Inequality for Markov Processes in Partially Ordered State Spaces

We study monotone and associated Markov chains on finite partially ordered state spaces. Both discrete and continuous time, and both time-homogeneous and time-inhomogeneous chains are considered. The results are applied to binary and multistate reliability theory.

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Bounds for the Reliability of Multistate Systems with Partially Ordered State Spaces and Stochastically Monotone Markov Transitions

International Journal of Reliability Quality and Safety Engineering ◽

10.1142/s0218539303001135 ◽

2003 ◽

Vol 10 (03) ◽

pp. 235-248

Author(s):

Bo Henry Lindqvist

Keyword(s):

Markov Chain ◽

System Performance ◽

Failure Time ◽

The State ◽

Upper And Lower Bounds ◽

State Spaces ◽

Partially Ordered ◽

Perfect System ◽

Multistate System ◽

Ordered State

Consider a multistate system with partially ordered state space E, which is divided into a set C of working states and a set D of failure states. Let X(t) be the state of the system at time t and suppose {X(t)} is a stochastically monotone Markov chain on E. Let T be the failure time, i.e., the hitting time of the set D. We derive upper and lower bounds for the reliability of the system, defined as Pm(T > t) where m is the state of perfect system performance.

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Monotone and associated Markov chains, with applications to reliability theory

Journal of Applied Probability ◽

10.2307/3214099 ◽

1987 ◽

Vol 24 (3) ◽

pp. 679-695 ◽

Cited By ~ 12

Author(s):

Bo Henry Lindqvist

Keyword(s):

Markov Chains ◽

Continuous Time ◽

Reliability Theory ◽

State Spaces ◽

Partially Ordered ◽

Ordered State

We study monotone and associated Markov chains on finite partially ordered state spaces. Both discrete and continuous time, and both time-homogeneous and time-inhomogeneous chains are considered. The results are applied to binary and multistate reliability theory.

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SIEGMUND DUALITY FOR MARKOV CHAINS ON PARTIALLY ORDERED STATE SPACES

Probability in the Engineering and Informational Sciences ◽

10.1017/s0269964817000341 ◽

2017 ◽

Vol 32 (4) ◽

pp. 495-521 ◽

Cited By ~ 1

Author(s):

Paweł Lorek

Keyword(s):

Stationary Distribution ◽

The Other ◽

Stochastic Monotonicity ◽

State Spaces ◽

Partially Ordered ◽

A Chain ◽

Gambler's Ruin Problem ◽

Absorbing States ◽

Ordered State ◽

Absorption Probabilities

For a Markov chain on a finite partially ordered state space, we show that its Siegmund dual exists if and only if the chain is Möbius monotone. This is an extension of Siegmund's result for totally ordered state spaces, in which case the existence of the dual is equivalent to the usual stochastic monotonicity. Exploiting the relation between the stationary distribution of an ergodic chain and the absorption probabilities of its Siegmund dual, we present three applications: calculating the absorption probabilities of a chain with two absorbing states knowing the stationary distribution of the other chain; calculating the stationary distribution of an ergodic chain knowing the absorption probabilities of the other chain; and providing a stable simulation scheme for the stationary distribution of a chain provided we can simulate its Siegmund dual. These are accompanied by concrete examples: the gambler's ruin problem with arbitrary winning/losing probabilities; a non-symmetric game; an extension of a birth and death chain; a chain corresponding to the Fisher–Wright model; a non-standard tandem network of two servers, and the Ising model on a circle. We also show that one can construct a strong stationary dual chain by taking the appropriate Doob transform of the Siegmund dual of the time-reversed chain.

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DEPENDENCE ORDERING FOR QUEUING NETWORKS WITH BREAKDOWN AND REPAIR

Probability in the Engineering and Informational Sciences ◽

10.1017/s0269964806060359 ◽

2006 ◽

Vol 20 (4) ◽

pp. 575-594 ◽

Cited By ~ 4

Author(s):

Hans Daduna ◽

Rafał Kulik ◽

Cornelia Sauer ◽

Ryszard Szekli

Keyword(s):

Markov Processes ◽

Stochastic Ordering ◽

Queuing Networks ◽

State Spaces ◽

Jackson Networks ◽

Dependency Structure ◽

Performance Behavior ◽

Ordered State ◽

Random Breakdowns

In this article we introduce isotone differences stochastic ordering of Markov processes on lattice ordered state spaces as a device to compare the internal dependencies of two such processes. We derive a characterization in terms of intensity matrices. This enables us to compare the internal dependency structure of different degradable Jackson networks in which the nodes are subject to random breakdowns and repairs. We show that the performance behavior and the availability of such networks can be compared.

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