On first-passage times in increasing Markov processes

1996 ◽  
Vol 26 (3) ◽  
pp. 199-203 ◽  
Author(s):  
Rafael Pérez-Ocón ◽  
M.Luz Gámiz-Pérez
Keyword(s):  
Markov Processes ◽  
First Passage Times ◽  
First Passage ◽  
Passage Times

Ageing first-passage times of Markov processes: a matrix approach

1997 ◽  
Vol 34 (1) ◽  
pp. 1-13 ◽  
Author(s):  
Haijun Li ◽  
Moshe Shaked
Keyword(s):  
Markov Process ◽  
Markov Processes ◽  
First Passage Time ◽  
Passage Time ◽  
Critical Value ◽  
First Passage Times ◽  
Matrix Approach ◽  
First Passage ◽  
Passage Times

Using a matrix approach we discuss the first-passage time of a Markov process to exceed a given threshold or for the maximal increment of this process to pass a certain critical value. Conditions under which this first-passage time possesses various ageing properties are studied. Some results previously obtained by Li and Shaked (1995) are extended.

On ageing properties of first-passage times of increasing Markov processes

2002 ◽  
Vol 34 (01) ◽  
pp. 241-259
Author(s):  
Félix Belzunce ◽  
Eva-María Ortega ◽  
José M. Ruiz
Keyword(s):  
Markov Chain ◽  
Markov Processes ◽  
Continuous Time ◽  
Transition Matrix ◽  
Sufficient Conditions ◽  
First Passage Times ◽  
Generator Matrix ◽  
First Passage ◽  
Finite Case ◽  
Passage Times

The purpose of this paper is to study ageing properties of first-passage times of increasing Markov chains. We extend the literature to some new ageing classes, such as the IFR(2), NBU(2), DRLLt and NBULt classes. We also give sufficient conditions in the finite case, that are more efficient computationally, just in terms of the transition matrix K, in the discrete case, or the generator matrix Q, in the continuous case. For the uniformizable, continuous-time Markov processes, we derive conditions in terms of the discrete uniformized Markov chain for the NBU(2) and the NBULt classes. In the last section, a review of the main results in this direction in the literature is given, and we compare some of the conditions stated in this paper with others given in the literature about some other ageing classes. Some examples where these results are applied are given.

On ageing properties of first-passage times of increasing Markov processes

2002 ◽  
Vol 34 (1) ◽  
pp. 241-259 ◽  
Author(s):  
Félix Belzunce ◽  
Eva-María Ortega ◽  
José M. Ruiz
Keyword(s):  
Markov Chain ◽  
Markov Processes ◽  
Continuous Time ◽  
Transition Matrix ◽  
Sufficient Conditions ◽  
First Passage Times ◽  
Generator Matrix ◽  
First Passage ◽  
Finite Case ◽  
Passage Times

The purpose of this paper is to study ageing properties of first-passage times of increasing Markov chains. We extend the literature to some new ageing classes, such as the IFR(2), NBU(2), DRLLt and NBULt classes. We also give sufficient conditions in the finite case, that are more efficient computationally, just in terms of the transition matrix K, in the discrete case, or the generator matrix Q, in the continuous case. For the uniformizable, continuous-time Markov processes, we derive conditions in terms of the discrete uniformized Markov chain for the NBU(2) and the NBULt classes. In the last section, a review of the main results in this direction in the literature is given, and we compare some of the conditions stated in this paper with others given in the literature about some other ageing classes. Some examples where these results are applied are given.

First passage times and lumpability of semi-Markov processes

1988 ◽  
Vol 25 (04) ◽  
pp. 675-687 ◽  
Author(s):  
Ushio Sumita ◽  
Maria Rieders
Keyword(s):  
Markov Processes ◽  
First Passage Times ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
First Passage ◽  
The Laplace Transform ◽  
First Exit Times ◽  
Semi Markov Processes ◽  
Necessary And Sufficient ◽  
Passage Times

A necessary and sufficient condition of Serfozo (1971) for lumpability of semi-Markov processes is reinterpreted in terms of first-exit times. Furthermore, a new necessary and sufficient condition is developed by establishing relationships between first-passage times and lumpability of semi-Markov processes. The approach taken in this paper is entirely based on the Laplace-transform domain.

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