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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 $(n-1)$-out-of-$n$ systems in two cases: (i) when components are independent, and then (ii) when components are dependent.  

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Adding parameters to a known distribution is a valuable way of constructing flexible families of distributions. In this paper, we introduce a new model, the modified additive hazard rate model, by replacing the additive hazard rate distribution in the general proportional add ratio model. Next, when two sets of random variables follow the modified additive hazard model, we establish stochastic comparisons between the series and parallel systems comprising these components.

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In this article, we investigate stochastic comparisons of the lifetimes of series and parallel systems comprising components following the scale-additive hazard rate model, subject to random shocks, where dependence among component lifetimes is modeled via Archimedean copulas. By imposing suitable conditions on the baseline distribution, model parameters, generator functions, and shock occurrence probabilities, we establish sufficient criteria for comparing the lifetimes of two systems under both the usual stochastic order and the hazard rate order. The results demonstrate how parameter heterogeneity and dependence structure simultaneously influence system reliability. Several numerical examples are also provided to substantiate the theoretical findings.