Abstract
In this paper, a generalized class of run shock models associated with a bivariate sequence {(Xi, Yi)}i≥1 of correlated random variables is defined and studied. For a system that is subject to shocks of random magnitudes X1, X2, ... over time, let the random variables Y1, Y2, ... denote times between arrivals of successive shocks. The lifetime of the system under this class is defined through a compound random variable T = ∑Nt=1 Yt , where N is a stopping time for the sequence {Xi}i≤1 and represents the number of shocks that causes failure of the system. Another random variable of interest is the maximum shock size up to N, i.e. M = max {Xi, 1≤i≤ N}. Distributions of T and M are investigated when N has a phase-type distribution.
References
[1] Anderson, K. K. (1988). A note on cumulative shock models. J. Appl. Probab. 25(1), 220-223.10.2307/3214249Search in Google Scholar
[2] Cirillo, P. and J. Hüsler (2011). Extreme shock models: An alternative perspective. Stat. Probab. Lett. 81(1), 25-30.10.1016/j.spl.2010.09.014Search in Google Scholar
[3] Eryilmaz, S. (2012). Generalized δ-shock model via runs. Stat. Probab. Lett. 82(2), 326-331.10.1016/j.spl.2011.10.022Search in Google Scholar
[4] Eryilmaz, S. (2013). On the lifetime behavior of a discrete time shock model. J. Comput. Appl. Math. 237(1), 384-388.10.1016/j.cam.2012.06.008Search in Google Scholar
[5] Eryilmaz, S. (2017a). Computing optimal replacement time and mean residual life in reliability shock models. Comput. Ind. Eng. 103, 40-45.10.1016/j.cie.2016.11.017Search in Google Scholar
[6] Eryilmaz, S. (2017b). On compound sums under dependence. Insurance Math. Econom. 72, 228-234.10.1016/j.insmatheco.2016.12.003Search in Google Scholar
[7] Gut, A. (1990). Cumulative shock models. Adv. in Appl. Probab. 22(2), 504-507.10.2307/1427554Search in Google Scholar
[8] Gut, A. (2001). Mixed shock models. Bernoulli. 7(3), 541-555.10.2307/3318501Search in Google Scholar
[9] Gut, A. and J. Hüsler (1999). Extreme shock models. Extremes 2(3), 293-305.Search in Google Scholar
[10] Li, Z., Chan L. Y. and Z. Yuan (1999). Failure time distribution under a _-shock model and its application to economic design of system. Int. J. Rel. Qual. Saf. Eng. 6(3), 237-247.10.1142/S0218539399000231Search in Google Scholar
[11] Li, Z. and X. Kong (2007). Life behavior of _-shock model. Stat. Probab. Lett. 77(6), 577-587.10.1016/j.spl.2006.08.008Search in Google Scholar
[12] Mallor, F. and E. Omey (2001). Shocks, runs and random sums. J. Appl. Probab. 38(2), 438-448.10.1017/S0021900200019951Search in Google Scholar
[13] Neuts, M. F. (1981).Matrix-Geometric Solutions in StochasticModels: An Algorithmic Approach. The Johns Hopkins University Press, Baltimore MD.Search in Google Scholar
[14] Nikoloulopoulos, A. K. and D. Karlis (2008). Fitting copulas to bivariate earthquake data: The seismic gap hypothesis revisited. Environmetrics 19(3), 251-269.10.1002/env.869Search in Google Scholar
[15] Ozkut, M. and I. Bayramloglu (Bairamov) (2014). On Marshall-Olkin type distribution with effect of shock magnitude. J. Comput. Appl. Math. 271, 150-162.10.1016/j.cam.2014.04.002Search in Google Scholar
[16] Parvardeh, A. and N. Balakrishnan (2015). On mixed _-shock models. Stat. Probab. Lett. 102, 51-60.10.1016/j.spl.2015.04.006Search in Google Scholar
[17] Sumita, U. and J. G. Shanthikumar (1985). A class of correlated cumulative shock models. Adv. in Appl. Probab. 17(2), 347-366.10.2307/1427145Search in Google Scholar
[18] Tank, F. and S. Eryilmaz (2015). The distributions of sum, minima and maxima of generalized geometric random variables. Statist. Papers. 56(4), 1191-120310.1007/s00362-014-0632-4Search in Google Scholar
© 2018 Femin Yalcin, published by De Gruyter
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