Ordering properties of parallel and series systems with a general lifetime family of distributions for independent components under random shocks

Often, reliability systems suffer shocks from external stress factors, stressing the system at random. These random shocks may have non ignorable effects on the reliability of the system. In this article, we provide sufficient (and necessary) conditions on components’ lifetimes and their survival probabilities from random shocks for comparing the lifetimes of two parallel systems with independent components by means of usual stochastic and reversed hazard rate orders, when the matrix of parameters changes to another matrix in a mathematical sense. We also discuss stochastic comparisons of series systems with independent components under random shocks in the sense of usual stochastic and hazard rate orders, by using the concept of vector majorization and related orders. The results developed in this article generalize some known results in the literature.