Shock models with renewal failure rate properties

Abstract

For the counting process N={N(t), t≥0} and the probability that a device survives the first k shocks$\bar P_k$, the probability that the device survives beyond t that is$\bar H(t) = \sum\limits_{k = 0}^\omega {P(N(t) = k)} \bar P_k$ is considered. The survival$\bar H(t)$ is proved to have the new better (worse) than used renewal failure rate and the new better (worse) than average failure rate properties under, some conditions on N and$(\bar P_k )_{k = \rho }^\omega$. In particular we study the survival probability when N is a nonhomogeneous Poisson process or birth process. Acumulative damage model and Laplace transform characterization for properties are investigated. Further the generating functions for these renewal failure rates properties are given.