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

TL;DR

This paper investigates how the lifetimes of dependent series and parallel systems with heterogeneous components behave under random shocks, using Archimedean copulas to model dependencies and establishing stochastic orderings.

Contribution

It introduces new conditions for comparing system lifetimes with dependent components under random shocks using Archimedean copulas, expanding stochastic ordering theory.

Findings

Established conditions for stochastic ordering of system lifetimes

Demonstrated dependency effects on system reliability

Provided illustrative examples and graphical analysis

Abstract

In this paper, 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 on the lifetime of individual components where the dependency among components defined by Archimedean copulas, and the impact of random shocks on the overall system lifetime to get the results. We consider components whose survival functions are either increasing log-concave or decreasing log-convex functions of the parameters involved. 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.