scholarly journals On the First Passage Time Distribution for a Class of Markov Chains

1983 ◽  
Vol 11 (4) ◽  
pp. 1000-1008 ◽  
Author(s):  
Mark Brown ◽  
Narasinga R. Chaganty
Keyword(s):  
Markov Chains ◽  
Time Distribution ◽  
First Passage Time ◽  
Passage Time ◽  
First Passage

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.

Identifying Coefficients in the Spectral Representation for First Passage Time Distributions

1987 ◽  
Vol 1 (1) ◽  
pp. 69-74 ◽  
Author(s):  
Mark Brown ◽  
Yi-Shi Shao
Keyword(s):  
Markov Processes ◽  
Time Distribution ◽  
Infinitesimal Generator ◽  
Spectral Representation ◽  
First Passage Time ◽  
Passage Time ◽  
Spectral Approach ◽  
Generator Matrix ◽  
Eigenvalues And Eigenvectors ◽  
First Passage

The spectral approach to first passage time distributions for Markov processes requires knowledge of the eigenvalues and eigenvectors of the infinitesimal generator matrix. We demonstrate that in many cases knowledge of the eigenvalues alone is sufficient to compute the first passage time distribution.

scholarly journals Sojourn time distributions in a Markovian G-queue with batch arrival and batch removal

1999 ◽  
Vol 12 (4) ◽  
pp. 339-356 ◽  
Author(s):  
Yang Woo Shin
Keyword(s):  
Markov Chains ◽  
Sojourn Time ◽  
First Passage Time ◽  
Passage Time ◽  
Batch Arrival ◽  
Single Server ◽  
Negative Customers ◽  
First Passage ◽  
Markovian Queue ◽  
Sojourn Time Distributions

We consider a single server Markovian queue with two types of customers; positive and negative, where positive customers arrive in batches and arrivals of negative customers remove positive customers in batches. Only positive customers form a queue and negative customers just reduce the system congestion by removing positive ones upon their arrivals. We derive the LSTs of sojourn time distributions for a single server Markovian queue with positive customers and negative customers by using the first passage time arguments for Markov chains.

Evaluation of the first-passage time probability to a square root boundary for the Wiener process

1977 ◽  
Vol 14 (4) ◽  
pp. 850-856 ◽  
Author(s):  
Shunsuke Sato
Keyword(s):  
Wiener Process ◽  
Time Distribution ◽  
First Passage Time ◽  
Passage Time ◽  
Square Root ◽  
First Passage ◽  
Uhlenbeck Process ◽  
Ornstein Uhlenbeck Process ◽  
Asymptotic Evaluation

This paper gives an asymptotic evaluation of the probability that the Wiener path first crosses a square root boundary. The result is applied to estimate the moments of the first-passage time distribution of the Ornstein–Uhlenbeck process to a constant boundary.

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