Monotone and associated Markov chains, with applications to reliability theory

1987 ◽  
Vol 24 (03) ◽  
pp. 679-695 ◽  
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.

Monotone and associated Markov chains, with applications to reliability theory

1987 ◽  
Vol 24 (3) ◽  
pp. 679-695 ◽  
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.

scholarly journals Strong Stationary Duality and Algebraic Duality for Continuous Time Möbius Monotone Markov Chains

2021 ◽  
Vol 15 (2) ◽  
pp. 69-86
Author(s):  
Pan Zhao ◽  
Keyword(s):  
Markov Chains ◽  
State Space ◽  
Continuous Time ◽  
Two Dimensional ◽  
Partially Ordered ◽  
Continuous Time Markov Chains ◽  
Strong Stationary Duality ◽  
Birth And Death Chain ◽  
Algebraic Duality ◽  
Ordered State

Under the assumption of Möbius monotonicity, we develop the theory of strong stationary duality for continuous time Markov chains on the finite partially ordered state space, we also construct a nonexplosive algebraic duality for continuous time Markov chains on Finally, we present an application to the two-dimensional birth and death chain.

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

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.

Stochastic domination and Markovian couplings

2000 ◽  
Vol 32 (4) ◽  
pp. 1064-1076 ◽  
Author(s):  
F. Javier López ◽  
Servet Martínez ◽  
Gerardo Sanz
Keyword(s):  
Markov Chains ◽  
Continuous Time ◽  
Partially Ordered Set ◽  
Ordered Set ◽  
Sufficient Conditions ◽  
Stochastic Domination ◽  
Jackson Networks ◽  
Partially Ordered ◽  
Continuous Time Markov Chains ◽  
Necessary And Sufficient

For continuous-time Markov chains with semigroups P, P' taking values in a partially ordered set, such that P ≤ stP', we show the existence of an order-preserving Markovian coupling and give a way to construct it. From our proof, we also obtain the conditions of Brandt and Last for stochastic domination in terms of the associated intensity matrices. Our result is applied to get necessary and sufficient conditions for the existence of Markovian couplings between two Jackson networks.

Markovian couplings staying in arbitrary subsets of the state space

2002 ◽  
Vol 39 (01) ◽  
pp. 197-212 ◽  
Author(s):  
F. Javier López ◽  
Gerardo Sanz
Keyword(s):  
Markov Chains ◽  
State Space ◽  
Continuous Time ◽  
Sufficient Conditions ◽  
Transition Rates ◽  
Arbitrary Subset ◽  
State Spaces ◽  
Continuous Time Markov Chains ◽  
Countable State ◽  
Necessary And Sufficient

Let (X t ) and (Y t ) be continuous-time Markov chains with countable state spaces E and F and let K be an arbitrary subset of E x F. We give necessary and sufficient conditions on the transition rates of (X t ) and (Y t ) for the existence of a coupling which stays in K. We also show that when such a coupling exists, it can be chosen to be Markovian and give a way to construct it. In the case E=F and K ⊆ E x E, we see how the problem of construction of the coupling can be simplified. We give some examples of use and application of our results, including a new concept of lumpability in Markov chains.

SIEGMUND DUALITY FOR MARKOV CHAINS ON PARTIALLY ORDERED STATE SPACES

2017 ◽  
Vol 32 (4) ◽  
pp. 495-521 ◽  
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.

Markovian couplings staying in arbitrary subsets of the state space

2002 ◽  
Vol 39 (1) ◽  
pp. 197-212 ◽  
Author(s):  
F. Javier López ◽  
Gerardo Sanz
Keyword(s):  
Markov Chains ◽  
State Space ◽  
Continuous Time ◽  
Sufficient Conditions ◽  
Transition Rates ◽  
Arbitrary Subset ◽  
State Spaces ◽  
Continuous Time Markov Chains ◽  
Countable State ◽  
Necessary And Sufficient

Let (Xt) and (Yt) be continuous-time Markov chains with countable state spaces E and F and let K be an arbitrary subset of E x F. We give necessary and sufficient conditions on the transition rates of (Xt) and (Yt) for the existence of a coupling which stays in K. We also show that when such a coupling exists, it can be chosen to be Markovian and give a way to construct it. In the case E=F and K ⊆ E x E, we see how the problem of construction of the coupling can be simplified. We give some examples of use and application of our results, including a new concept of lumpability in Markov chains.

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