scholarly journals Gaussian Elimination, Perturbation Theory, and Markov Chains

1993 ◽  
pp. 59-69 ◽  
Author(s):  
G. W. Stewart
Keyword(s):  
Perturbation Theory ◽  
Markov Chains ◽  
Gaussian Elimination

Perturbation theory and finite Markov chains

1968 ◽  
Vol 5 (2) ◽  
pp. 401-413 ◽  
Author(s):  
Paul J. Schweitzer
Keyword(s):  
Perturbation Theory ◽  
Markov Chain ◽  
Markov Chains ◽  
Stationary Distribution ◽  
Fundamental Matrix ◽  
Transition Probabilities ◽  
Semi Group ◽  
Partial Derivatives ◽  
Finite Markov Chains ◽  
Derivatives Of

A perturbation formalism is presented which shows how the stationary distribution and fundamental matrix of a Markov chain containing a single irreducible set of states change as the transition probabilities vary. Expressions are given for the partial derivatives of the stationary distribution and fundamental matrix with respect to the transition probabilities. Semi-group properties of the generators of transformations from one Markov chain to another are investigated. It is shown that a perturbation formalism exists in the multiple subchain case if and only if the change in the transition probabilities does not alter the number of, or intermix the various subchains. The formalism is presented when this condition is satisfied.

Perturbation theory and finite Markov chains

1968 ◽  
Vol 5 (02) ◽  
pp. 401-413 ◽  
Author(s):  
Paul J. Schweitzer
Keyword(s):  
Perturbation Theory ◽  
Markov Chain ◽  
Markov Chains ◽  
Stationary Distribution ◽  
Fundamental Matrix ◽  
Transition Probabilities ◽  
Semi Group ◽  
Partial Derivatives ◽  
Finite Markov Chains ◽  
Derivatives Of

A perturbation formalism is presented which shows how the stationary distribution and fundamental matrix of a Markov chain containing a single irreducible set of states change as the transition probabilities vary. Expressions are given for the partial derivatives of the stationary distribution and fundamental matrix with respect to the transition probabilities. Semi-group properties of the generators of transformations from one Markov chain to another are investigated. It is shown that a perturbation formalism exists in the multiple subchain case if and only if the change in the transition probabilities does not alter the number of, or intermix the various subchains. The formalism is presented when this condition is satisfied.

scholarly journals Parametric Markov Chains: PCTL Complexity and Fraction-free Gaussian Elimination

2017 ◽  
Vol 256 ◽  
pp. 16-30 ◽  
Author(s):  
Lisa Hutschenreiter ◽  
Christel Baier ◽  
Joachim Klein
Keyword(s):  
Markov Chains ◽  
Gaussian Elimination

Analysis of Queueing Networks in Equilibrium

2020 ◽  
Vol 12 (4) ◽  
pp. 1-17
Author(s):  
Izabella V. Lokshina ◽  
Cees J. M. Lanting
Keyword(s):  
Markov Chains ◽  
Communication Networks ◽  
Queueing Networks ◽  
Numerical Solutions ◽  
Gaussian Elimination ◽  
Steady State Analysis ◽  
Numerical Examples ◽  
Queuing Models ◽  
Stochastic Techniques ◽  
Computational Procedures

Equilibria of queueing networks are a means for performance analysis of real communication networks introduced as Markov chains. In this paper, the authors developed, evaluated, and compared computational procedures to obtain numerical solutions for queueing networks in equilibrium with the use of direct, iterative, and aggregative techniques in steady-state analysis of Markov chains. Advanced computational procedures are developed with the use of Gaussian elimination, power iteration, Courtois' decomposition, and Takahashi's iteration techniques. Numerical examples are provided together with comparative analysis of obtained results. The authors consider these procedures are also applicable to other domains where systems are described with comparable queuing models and stochastic techniques are sufficiently relevant. Several suitable domains of applicability are proposed.

scholarly journals Regular Perturbation of V-Geometrically Ergodic Markov Chains

2013 ◽  
Vol 50 (01) ◽  
pp. 184-194 ◽  
Author(s):  
Déborah Ferré ◽  
Loïc Hervé ◽  
James Ledoux
Keyword(s):  
Perturbation Theory ◽  
Asymptotic Expansion ◽  
Probability Density Function ◽  
Markov Chains ◽  
Invariant Probability Measure ◽  
Regular Perturbation ◽  
Regularity Properties ◽  
The Stability ◽  
Higher Regularity ◽  
Standard Perturbation Theory

In this paper, new conditions for the stability ofV-geometrically ergodic Markov chains are introduced. The results are based on an extension of the standard perturbation theory formulated by Keller and Liverani. The continuity and higher regularity properties are investigated. As an illustration, an asymptotic expansion of the invariant probability measure for an autoregressive model with independent and identically distributed noises (with a nonstandard probability density function) is obtained.

scholarly journals Parametric Markov chains: PCTL complexity and fraction-free Gaussian elimination

2020 ◽  
Vol 272 ◽  
pp. 104504
Author(s):  
Christel Baier ◽  
Christian Hensel ◽  
Lisa Hutschenreiter ◽  
Sebastian Junges ◽  
Joost-Pieter Katoen ◽  
...  
Keyword(s):  
Markov Chains ◽  
Gaussian Elimination
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