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Shocks, runs and random sums

Part of: Operations research and management science Stochastic processes Special processes Limit theorems Real functions

Published online by Cambridge University Press:  14 July 2016

F. Mallor  and
E. Omey
F. Mallor*
Affiliation:
Universidad Pública de Navarra
E. Omey*
Affiliation:
Economische Hogeschool Sint-Aloysius
*
∗ Postal address: Department of Statistics and Operations Research, Universidad Pública de Navarra, Campus Arrosadia, 31006 Pamplona, Spain. Email address: mallor@unavarra.es
∗∗ Postal address: Department of Mathematics and Statistics, Economische Hogeschool Sint-Aloysius, Stormstraat 2, 1000-Brussels, Belgium.
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Abstract

In this paper we study random variables related to a shock reliability model. Our models can be used to study systems that fail when k consecutive shocks with critical magnitude (e.g. above or below a certain critical level) occur. We obtain properties of the distribution function of the random variables involved and we obtain their limit behaviour when k tends to infinity or when the probability of entering a critical set tends to zero. This model generalises the Poisson shock model.

Keywords

shock modelrandom sumsreliabilityregular variation

MSC classification

Primary: 60F05: Central limit and other weak theorems 60K10: Applications (reliability, demand theory, etc.) 90B25: Reliability, availability, maintenance, inspection
Secondary: 26A12: Rate of growth of functions, orders of infinity, slowly varying functions 60G50: Sums of independent random variables; random walks
Type
Research Papers
Information
Journal of Applied Probability , Volume 38 , Issue 2 , June 2001 , pp. 438 - 448
Copyright
Copyright © Applied Probability Trust 2001 

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References

Anderson, K. K. (1987). Limit theorems for general shock models with infinite mean intershock times. J. Appl. Prob. 24, 449–456.CrossRefGoogle Scholar
Bingham, N. H., Goldie, C. M., and Teugels, J. L. (1987). Regular Variation. Cambridge University Press.CrossRefGoogle Scholar
Esary, J. D., Marshall, A. W. and Proschan, F. (1973). Shock models and wear processes. Ann. Prob. 1, 627–650.CrossRefGoogle Scholar
Feller, W. (1968). An Introduction to Probability Theory and its Applications, Vol. 1. John Wiley, New York.Google Scholar
Gaver, D. P. (1963). Random hazard in reliability problems. Technometrics 5, 211–226.CrossRefGoogle Scholar
Keilson, J. (1966). A limit theorem for passage times in ergodic regenerative processes. Ann. Math. Statist. 37, 866–870.CrossRefGoogle Scholar
Keilson, J. (1979). Markov Chain Models—Rarity and Exponentiality. Springer, New York.CrossRefGoogle Scholar
Mallor, F., and Omey, E. (2001). Shocks, runs and random sums II. Asymptotic behaviour of the distribution function. Submitted.Google Scholar
Ross, S.M. (1981). Generalized Poisson shock models. Ann. Prob. 9, 896–898.CrossRefGoogle Scholar
Seneta, E. (1976). Regularly Varying Functions (Lecture Notes Math. 508). Springer, Berlin.Google Scholar
Serfozo, R., and Stidham, S. (1978). Semi-stationary clearing processes. Stoch. Proc. Appl. 6, 165–178.CrossRefGoogle Scholar
Shanthikumar, J. G., and Sumita, U. (1983). General shock models associated with correlated renewal sequences. J. Appl. Prob. 20, 600–614.CrossRefGoogle Scholar
Stidham, S. (1974). Stochastic clearing systems. Stoch. Proc. Appl. 2, 85–113.CrossRefGoogle Scholar