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STOCHASTIC SURVIVAL MODELS WITH EVENTS TRIGGERED BY EXTERNAL SHOCKS

Published online by Cambridge University Press:  27 April 2012

Ji Hwan Cha  and
Maxim Finkelstein
Ji Hwan Cha
Affiliation:
Department of Statistics, Ewha Womans University, Seoul 120-750, Korea E-mail: [email protected]
Maxim Finkelstein
Affiliation:
Department of Mathematical Statistics, University of the Free State, 339 Bloemfontein 9300, South Africa and Max Planck Institute for Demographic Research, Rostock, Germany E-mail: [email protected]
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Abstract

In most conventional settings, the events caused by an external shock are initiated at the moments of its occurrence. In this paper, we study the new classes of shock models: (i) When each shock from a nonhomogeneous Poisson processes can trigger a failure of a system not immediately, as in classical extreme shock models, but with delay of some random time. (ii) When each shock from a nonhomogeneous Poisson processes results not in an ‘immediate’ increment of wear, as in classical accumulated wear models, but triggers its own increasing wear process. The wear from different shocks is accumulated and the failure of a system occurs when it reaches a given boundary. We derive the corresponding survival and failure rate functions. Furthermore, we study the limiting behavior of the failure rate function where it is applicable.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 26 , Issue 2 , April 2012 , pp. 183 - 195
Copyright
Copyright © Cambridge University Press 2012

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References

REFERENCES

1.Cha, J.H. & Finkelstein, M. (2009). On a terminating shock process with independent wear increments. Journal of Applied Probability 46: 353–362.CrossRefGoogle Scholar
2.Cha, J.H. & Finkelstein, M. (2011). On new classes of extreme shock models and some generalizations. Journal of Applied Probability 48: 258–270.CrossRefGoogle Scholar
3.Esary, J.D., Marshal, A.W. & Proschan, F. (1973). Shock models and wear processes. Annals of Probability 1: 627–649.CrossRefGoogle Scholar
4.Finkelstein, M. (2008). Failure rate modelling for reliability and risk. London: Springer.Google Scholar
5.Finkelstein, M. & Marais, F. (2010). On terminating Poisson processes in some shock models. Reliability Engineering and System Safety 95: 874–879.CrossRefGoogle Scholar
6.Gut, A. (1990). Cumulated shock models. Advances in Applied Probability 22: 504–507.CrossRefGoogle Scholar
7.Gut, A. & Hüsler, J. (2005) Realistic variation of shock models. Statistics and Probability Letters 74: 187–204.CrossRefGoogle Scholar
8.Lemoine, A.J. & Wenocur, M.L. (1986). A note on shot-noise and reliability modeling. Operations Research 34: 320–323.CrossRefGoogle Scholar
9.Nakagawa, T. (2007). Shock and damage models in reliability theory. London: Springer.Google Scholar
10.Rice, J. (1977). On generalized shot noise. Advances in Applied Probability 9: 553–565.CrossRefGoogle Scholar
11.Ross, S.M. (1996). Stochastic processes. New York: John Wiley.Google Scholar
12.Shaked, M. & Shanthikumar, J. (2007). Stochastic orders. New York: Springer.CrossRefGoogle Scholar