Discrete time shock model with varying success probability

Abstract

    Let us consider a system fails when the time between two consecutive shocks falls below a fixed threshold $\delta \in N$ and the system's  lifetime is measured as the time up to the occurrence of  this event. In this paper, we consider the interarrival times between $(i-1)$ i-th and $i$- th successive shocks follow a geometric distribution with mean 1/p_i ,where p_i =theta*p^i-1, i=1,2,..., ,0<theta<1, 0<p<1. Under the above considerations, the distribution of system lifetime is obtained. Probability generating function and than also moments of system are derived. The proportion estimates of distribution parameters are studied. A numerical example is also presented buy using real data.

Keywords

• q-distributions,delta-Shock model,probabilty generating function

References

1. Sumita, U. and Shanthikumar J.G. (1985). A class of correlated cumulative shock models. Advances in Applied Probability, 17, 347-366.
2. Gut, A. (1990). Cumulative shock models. Advances in Applied Probability, 22, 504-507.
3. Mallor, F. and Omey E. (2001). Shocks, runs and random sums. Journal of Applied Probability, 38, 438-448.
4. Wang, G.J. and Zhang Y.L. (2005). A shock model with two-type failures and optimal replacement policy. International Journal of Systems Science, 36, 209-214.
5. Bai, J-M., Li, Z-H. and Kong, X-B. (2006). Generalized shock models based on a cluster point process. IEEE Transactions on Reliability, 55, 542-550.
6. Li, Z.H. and Kong, X.B. (2007). Life behavior of δ-shock model. Statistics and Probability Letters, 77, 577-587.
7. Li, Z.H. and Zhao, P. (2007). Reliability analysis on the δ-shock model of complex systems. IEEE Transactions on Reliabilit, 56, 340-348.
8. Eryilmaz, S. (2012). Generalized δ-shock model via runs. Statistics and Probability Letters, 82, 326-331.

9. Eryilmaz, S. (2013). On the lifetime behavior of a discrete time shock model. Journal of Computational and Applied Mathematics, 237, 384-388.
10. Charalambides, C.A. (2010). The q-Bernstein basis as a q-binomial distributions. Journal of Statistical Planning and Inference, 140, 2184-2190.
11. Yalcin, F. and Eryilmaz, S. (2014). q-Geometric and q-Binomial distributions of order k. Journal of Computational and Applied Mathematics, 271, 31-38.
12. Khan, M.S.A., Khalique. A. and Abouammoh, A.M. (1989). On estimating parameters in a discrete Weibull distribution. IEEE Transactions on Reliability, 38 (3), 348-350.
13. Phyo, I. (1973). Use of a chain binomial in epidemiology of caries. Journal of Dental Research 52, 750-752.
14. Krishna, H. and Pundir, P.S. (2009). Discrete Burr and discrete Pareto distributions. Statistical Methodology, 6, 177-188.
15. Nakagawa, T. and Osaki, S. (1975). The discrete Weibull distribution. IEEE Transactions on Reliability, 24, 300-301.