Stochastic Comparisons on the Residual Lifetimes of Series Systems with Arbitrary Components using Copulas

In this paper, we consider series systems consisting of arbitrary dependent components. We study the residual lifetimes of such systems based on copulas family from a new point of view. First, we extract a new explicit expression for the reliability functions of residual lifetimes of the systems. Moreover, we give some stochastic ordering properties for the residual lifetimes of series systems based on the dependence structure of the components and the corresponding mean functions. The results are expanded for series systems having used components of age $t>0$. Subsequently, the problem of the stochastic comparison of a series system having used components and a used series system has been considered. To show the application of results, we provide some numerical examples. Finally, we present some dependence properties of the residual lifetimes of series system based on the properties of the lifetimes of components.