A class of correlated cumulative shock models

Abstract

In this paper we define and analyze a class of cumulative shock models associated with a bivariate sequence {Xn , Yn }∞ n =0 of correlated random variables. The {Xn } denote the sizes of the shocks and the {Yn } denote the times between successive shocks. The system fails when the cumulative magnitude of the shocks exceeds a prespecified level z. Two models, depending on whether the size of the nth shock is correlated with the length of the interval since the last shock or with the length of the succeeding interval until the next shock, are considered. Various transform results and asymptotic properties of the system failure time are obtained. Further, sufficient conditions are established under which system failure time is new better than used, new better than used in expectation, and harmonic new better than used in expectation.