scholarly journals Linear Least Square Method for the Computation of the Mean First Passage Times of Ergodic Markov Chains

2018 ◽  
Vol 28 (5) ◽  
pp. 1-9
Author(s):  
Yaming Chen
Keyword(s):  
Markov Chains ◽  
Least Square Method ◽  
Least Square ◽  
First Passage Times ◽  
First Passage ◽  
Mean First Passage Times ◽  
The Mean ◽  
Linear Least Square ◽  
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.

Weighted trapping time of weighted directed treelike network

2020 ◽  
Vol 31 (08) ◽  
pp. 2050108
Author(s):  
Meifeng Dai ◽  
Yongbo Hou ◽  
Tingting Ju ◽  
Changxi Dai ◽  
Yu Sun ◽  
...  
Keyword(s):  
Complex Networks ◽  
First Passage Times ◽  
First Passage ◽  
Trade Network ◽  
Trapping Time ◽  
Undirected Network ◽  
Mean First Passage Times ◽  
Central Node ◽  
The Mean ◽  
Passage Times

With the deepening of research on complex networks, many properties of complex networks are gradually studied, for example, the mean first-passage times, the average receive times and the trapping times. In this paper, we further study the average trapping time of the weighted directed treelike network constructed by an iterative way. Firstly, we introduce our model inspired by trade network, each edge [Formula: see text] in undirected network is replaced by two directed edges with weights [Formula: see text] and [Formula: see text]. Then, the trap located at central node, we calculate the weighted directed trapping time (WDTT) and the average weighted directed trapping time (AWDTT). Remarkably, the WDTT has different formulas for even generations and odd generations. Finally, we analyze different cases for weight factors of weighted directed treelike network.

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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