Stochastic ordering of series and parallel systems lifetime in Archimedean copula under random shock

In this manuscript, we studied the stochastic ordering behavior of series as well as parallel systems lifetimes comprising dependent and heterogeneous components, experiencing random shocks, and exhibiting distinct dependency structures. We establish certain conditions for the lifetime of individual components, the dependency among components defined by Archimedean copulas, and the impact of random shocks on the overall system lifetime to reach the conclusion. We consider components whose survival functions are either increasing log-concave or decreasing log-convex functions of the parameters involved and systems exhibit different dependency structures. These conditions make it possible to compare the lifetimes of two systems using the usual stochastic order framework. Additionally, we provide examples and graphical representations to elucidate our theoretical findings.