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

Part of: Special processes

Published online by Cambridge University Press:  01 July 2016

Félix Belzunce ,
Eva-María Ortega  and
José M. Ruiz
Félix Belzunce*
Affiliation:
Universidad de Murcia
Eva-María Ortega*
Affiliation:
Universidad Miguel Hernández
José M. Ruiz*
Affiliation:
Universidad de Murcia
*
∗ Postal address: Departamento de Estadística e Investigación Operativa, Universidad de Murcia, Campus de Espinardo, 30100 Espinardo, Murcia, Spain.
∗∗ Postal address: Centro de Investigación Operativa, Universidad Miguel Hernández, Campus La Gal·lia, Av. Ferrocarril s/n, 03202 Elche, Alicante, Spain.
∗ Postal address: Departamento de Estadística e Investigación Operativa, Universidad de Murcia, Campus de Espinardo, 30100 Espinardo, Murcia, Spain.
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Abstract

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.

Keywords

First-passage timesincreasing Markov chainsincreasing concave orderLaplace transform orderIFR(2)NBU(2)DRL_Lt and NBU_Lt ageing classesuniformizable Markov processes

MSC classification

Primary: 60K15: Markov renewal processes, semi-Markov processes 60K10: Applications (reliability, demand theory, etc.)
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 34 , Issue 1 , March 2002 , pp. 241 - 259
Copyright
Copyright © Applied Probability Trust 2002 

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