scholarly journals Bounding Mean First Passage Times in Population Continuous-Time Markov Chains

2020 ◽  
pp. 155-174
Author(s):  
Michael Backenköhler ◽  
Luca Bortolussi ◽  
Verena Wolf
Keyword(s):  
Markov Chains ◽  
Continuous Time ◽  
First Passage Times ◽  
First Passage ◽  
Mean First Passage Times ◽  
Continuous Time Markov Chains ◽  
Passage Times

scholarly journals Exponentiality of First Passage Times of Continuous Time Markov Chains

2013 ◽  
Vol 131 (1) ◽  
pp. 197-212 ◽  
Author(s):  
Romain Bourget ◽  
Loïc Chaumont ◽  
Natalia Sapoukhina
Keyword(s):  
Markov Chains ◽  
Continuous Time ◽  
First Passage Times ◽  
First Passage ◽  
Continuous Time Markov Chains ◽  
Passage Times

scholarly journals SIMPLE PROCEDURES FOR FINDING MEAN FIRST PASSAGE TIMES IN MARKOV CHAINS

2007 ◽  
Vol 24 (06) ◽  
pp. 813-829 ◽  
Author(s):  
JEFFREY J. HUNTER
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Stationary Distribution ◽  
Linear Equations ◽  
First Passage Times ◽  
First Passage ◽  
Mean First Passage Times ◽  
The Mean ◽  
Computational Procedures ◽  
Passage Times

The derivation of mean first passage times in Markov chains involves the solution of a family of linear equations. By exploring the solution of a related set of equations, using suitable generalized inverses of the Markovian kernel I - P, where P is the transition matrix of a finite irreducible Markov chain, we are able to derive elegant new results for finding the mean first passage times. As a by-product we derive the stationary distribution of the Markov chain without the necessity of any further computational procedures. Standard techniques in the literature, using for example Kemeny and Snell's fundamental matrix Z, require the initial derivation of the stationary distribution followed by the computation of Z, the inverse of I - P + eπT where eT = (1, 1, …, 1) and πT is the stationary probability vector. The procedures of this paper involve only the derivation of the inverse of a matrix of simple structure, based upon known characteristics of the Markov chain together with simple elementary vectors. No prior computations are required. Various possible families of matrices are explored leading to different related procedures.

scholarly journals The computation of the mean first passage times for Markov chains

2018 ◽  
Vol 549 ◽  
pp. 100-122 ◽  
Author(s):  
Jeffrey J. Hunter
Keyword(s):  
Markov Chains ◽  
First Passage Times ◽  
First Passage ◽  
Mean First Passage Times ◽  
The Mean ◽  
Passage Times

First-passage-time moments of Markov processes

1985 ◽  
Vol 22 (4) ◽  
pp. 939-945 ◽  
Author(s):  
David D. Yao
Keyword(s):  
Markov Chains ◽  
Markov Processes ◽  
Continuous Time ◽  
Fundamental Matrix ◽  
Generalized Inverse ◽  
First Passage Time ◽  
Passage Time ◽  
First Passage ◽  
Continuous Time Markov Chains ◽  
Passage Times

We consider the first-passage times of continuous-time Markov chains. Based on the approach of generalized inverse, moments of all orders are derived and expressed in simple, explicit forms in terms of the ‘fundamental matrix'. The formulas are new and are also efficient for computation.

First-passage-time moments of Markov processes

1985 ◽  
Vol 22 (04) ◽  
pp. 939-945
Author(s):  
David D. Yao
Keyword(s):  
Markov Chains ◽  
Markov Processes ◽  
Continuous Time ◽  
Fundamental Matrix ◽  
Generalized Inverse ◽  
First Passage Time ◽  
Passage Time ◽  
First Passage ◽  
Continuous Time Markov Chains ◽  
Passage Times

We consider the first-passage times of continuous-time Markov chains. Based on the approach of generalized inverse, moments of all orders are derived and expressed in simple, explicit forms in terms of the ‘fundamental matrix'. The formulas are new and are also efficient for computation.

Spectral Theory for Skip-Free Markov Chains

1989 ◽  
Vol 3 (1) ◽  
pp. 77-88 ◽  
Author(s):  
Joseph Abate ◽  
Ward Whitt
Keyword(s):  
Markov Chains ◽  
Spectral Theory ◽  
First Passage Time ◽  
Passage Time ◽  
First Passage Times ◽  
Simple Relationship ◽  
First Passage ◽  
Birth And Death Processes ◽  
The Laplace Transform ◽  
Passage Times

The distribution of upward first passage times in skip-free Markov chains can be expressed solely in terms of the eigenvalues in the spectral representation, without performing a separate calculation to determine the eigenvectors. We provide insight into this result and skip-free Markov chains more generally by showing that part of the spectral theory developed for birth-and-death processes extends to skip-free chains. We show that the eigenvalues and eigenvectors of skip-free chains can be characterized in terms of recursively defined polynomials. Moreover, the Laplace transform of the upward first passage time from 0 to n is the reciprocal of the nth polynomial. This simple relationship holds because the Laplace transforms of the first passage times satisfy the same recursion as the polynomials except for a normalization.

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