Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-23T05:39:46.787Z Has data issue: false hasContentIssue false
  • English
  • Français

Stochastic properties of a cumulative damage threshold crossing model

Part of: Distribution theory - Probability Special processes Operations research and management science

Published online by Cambridge University Press:  14 July 2016

Nader Ebrahimi
Nader Ebrahimi*
Affiliation:
Northern Illinois University
*
∗Postal address: Division of Statistics, Northern Illinois University, DeKalb, IL 60115, USA. Email address: [email protected].
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we describe a model for survival functions. Under this model a system is subject to shocks governed by a Poisson process. Each shock to the system causes a random damage that grows in time. Damages accumulate additively and the system fails if the total damage exceeds a certain capacity or threshold. Various properties of this model are obtained. Sufficient conditions are derived for the failure rate (FR) order and the stochastic order to hold between the random lifetimes of two systems whose failures can be described by our proposed model.

Keywords

Poisson processstochastic orderingfailure rate orderingIFRNBUPF2 propertyTP2 property

MSC classification

Primary: 60K10: Applications (reliability, demand theory, etc.)
Secondary: 60E15: Inequalities; stochastic orderings 90B25: Reliability, availability, maintenance, inspection
Type
Research Papers
Information
Journal of Applied Probability , Volume 36 , Issue 3 , September 1999 , pp. 720 - 732
Copyright
Copyright © Applied Probability Trust 1999 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

A-Hameed, M. S., and Proschan, F. (1973). Nonstationary shock models. Stoch. Proc. Appl. 1, 383–404.CrossRefGoogle Scholar
A-Hameed, M. S., and Proschan, F. (1975). Shock models with underlying birth process. J. Appl. Prob. 12, 18–28.CrossRefGoogle Scholar
Esary, J. D., Marshall, A. W., and Proschan, F. (1973). Shock models and wear processes. Ann. Prob. 1, 627–649.Google Scholar
Fagiuoli, E., and Pellerey, F. (1994). Preservation of certain classes of life distributions under Poisson shock models. J. Appl. Prob. 31, 458–465.Google Scholar
Karlin, S. (1968). Total Positivity. Stanford University Press, Palo Alto, CA.Google Scholar
Keilson, J., and Sumita, U. (1982). Uniform stochastic ordering and related inequalities. Canad. J. Statist. 10, 181–198.Google Scholar
Klüppelberg, C., and Mikosch, T. (1995). Explosive shot noise processes with applications to risk retention. Bernoulli 23, 125–147.Google Scholar
Pellerey, F. (1993). Partial orderings under cumulative damage shock models. Adv. Appl. Prob. 25, 939–946.CrossRefGoogle Scholar
Pellerey, F. (1994). Shock models with underlying counting processes. J. Appl. Prob. 31, 156–166.Google Scholar