# Lifetime properties of a cumulative shock model with a cluster structure

- 355 Downloads
- 3 Citations

## Abstract

This paper studies a generalized cumulative shock model with a cluster shock structure. The system considered is subject to two types of shocks, called primary shocks and secondary shocks, where each primary shock causes a series of secondary shocks. The lifetime behavior of such a system becomes more complicated than that of a classical model with only one class of shocks. Under a non-homogeneous Poisson process of primary shocks, we analyze the lifetime behavior of the system with light-tailed and heavy-tailed distributed secondary shocks. We show some important characteristics of lifetime of this type of system. Our model, as an extension of the classical shock models, has wide applications in maintenance engineering, operations management, and insurance risk assessment.

## Keywords

Cumulative shock model Cluster point process Lifetime Light-tailed distribution Regular-tailed distribution## Notes

### Acknowledgements

This work is supported by the Natural Science Foundation of China 71171103 and 10871086 and the NSERC grant RGPIN197319 of Canada. The authors would like to thank the editor and the anonymous referees for their valuable comments and suggestions and also are grateful to Dr. Zhi Liu and Dr. Yanjun Shi for their help in creating the illustrations.

## References

- Agrafiotis, G. K., & Tsoukalas, M. Z. (1995). On excess-time correlated cumulative processes.
*Journal of the Operational Research Society*,*46*, 1269–1280. Google Scholar - Albrecherab, H., & Asmussen, S. (2006). Ruin probabilities and aggregate claims distributions for shot noise Cox processes.
*Scandinavian Actuarial Journal*,*2*, 86–110. CrossRefGoogle Scholar - Bai, J. M., Li, Z. H., & Kong, X. B. (2006). Generalized shock models based on a cluster point process.
*IEEE Transactions on Reliability*,*55*, 542–550. CrossRefGoogle Scholar - Bai, J. M., & Xiao, H. M. (2008). A class of new cumulative shock models and its application in insurance risk.
*Journal of Lanzhou University (Natural Sciences)*,*44*, 132–136. Google Scholar - Daley, D. J., & Vere-Jones, D. (2003).
*An introduction to the theory of point processes, Vol. I*(2nd ed.). New York: Springer. Google Scholar - von Elart, C. (1999). Control of production processes subject to random shocks.
*Annals of Operations Research*,*91*, 289–304. CrossRefGoogle Scholar - Finkelstein, M. S., & Zarudnij, V. I. (2001). A shock process with a non-cumulative damage.
*Reliability Engineering & Systems Safety*,*71*, 103–107. CrossRefGoogle Scholar - Goldie, C. M., & Klüppelberg, C. (1998).
*Subexponential distributions (a practical guide to heavy tails: statistical techniques for analyzing heavy tailed distributions)*(pp. 435–459). Boston: Birkhäuser. Google Scholar - Gut, A. (1990). Cumulative shock models.
*Advances in Applied Probability*,*22*, 504–507. CrossRefGoogle Scholar - Hohn, N., Veitch, D., & Abry, P. (2003). Cluster processes: a natural language for network traffic.
*IEEE Transactions on Signal Processing*,*51*, 2229–2244. CrossRefGoogle Scholar - Igaki, N., Sumita, U., & Kowada, M. (1995). Analysis of Markov renewal shock models.
*Journal of Applied Probability*,*32*, 821–831. CrossRefGoogle Scholar - Jeffrey, P. K., Steven, M. C., & Mark, E. O. (2011). Reliability of manufacturing equipment in complex environments.
*Annals of Operations Research*. doi: 10.1007/s10479-011-0839-x. Google Scholar - Lewis, P. A. W. (1964). A branching Poisson process model for the analysis of computer failure patterns.
*Journal of the Royal Statistical Society. Series B*,*26*, 398–456. Google Scholar - Li, Z. H., & Kong, X. B. (2007). A new risk model based on policy entrance process and its weak convergence properties.
*Applied Stochastic Models in Business and Industry*,*23*, 235–246. CrossRefGoogle Scholar - Mallor, F., & Omey, E. (2001). Shocks, runs and random sums.
*Journal of Applied Probability*,*38*, 438–448. CrossRefGoogle Scholar - Mallor, F., & Omey, E. (2002). Shocks, runs and random sums: asymptotic behaviour of the tail of the distribution function.
*Mathematical Sciences*,*111*, 3449–3565. Google Scholar - Mikosch, T. (1999).
*Regular variation, subexponentiality and their applications in probability theory*. Eurandom report, 1999-013. Google Scholar - Nader, A., Georgios, K. D. S., Hamid, D., Hooman, M., & Seyed, A. Y. (2012). Strategies for protecting supply chain networks against facility and transportation disruptions: an improved Benders decomposition approach.
*Annals of Operations Research*. doi: 10.1007/s10479-012-1146-x. Google Scholar - Ramirez-Perez, F., & Serfling, R. (2001). Shot noise on cluster processes with cluster marks, and studies of long range dependence.
*Advances in Applied Probability*,*33*, 631–651. CrossRefGoogle Scholar - Ramirez-Perez, F., & Serfling, R. (2003). Asymptotic normality of shot noise on Poisson cluster processes with cluster marks.
*Journal of Probability and Statistical Science*,*1*, 157–172. Google Scholar - Rolski, T., Schmidli, H., Schmidt, V., & Teugels, J. (1999).
*Stochastic processes for insurance and finance*. Chichester: Wiley. CrossRefGoogle Scholar - Sara, L., & de Jacques, M. (2011). Maintenance for reliability: a case study.
*Annals of Operations Research*. doi: 10.1007/s10479-011-0873-8. Google Scholar - Sheu, S. H., Chang, C. C., & Chien, Y. H. (2011). Optimal age-replacement time with minimal repair based on cumulative repair-cost limit for a system subject to shocks.
*Annals of Operations Research*,*186*, 317–329. CrossRefGoogle Scholar - Skoulakis, G. (2000). A general shock model for a reliability system.
*Journal of Applied Probability*,*37*, 925–935. Google Scholar - Sumita, U., & Shanthikumar, J. G. (1985). A class of correlated cumulative shock models.
*Advances in Applied Probability*,*17*, 347–366. CrossRefGoogle Scholar - Xiao, H. M., Li, Z. H., & Liu, W. W. (2008). The limit behavior of a risk model based on entrance processes.
*Computers & Mathematics With Applications*,*56*, 1434–1440. CrossRefGoogle Scholar