Orderings of fail-safe systems with distribution-free 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 paper‎, ‎we provide sufficient and necessary conditions on components' lifetimes and their survival probabilities from random shocks for comparing the lifetimes of two fail-safe systems in two cases‎: ‎(i) when components are independent‎, ‎and then (ii) when components are dependent‎. ‎We then apply the results for some distribution-free random variables with possibly different parameters to illustrate the established results.