Comparison results for inactivity times of k-out-of-n and general coherent systems with dependent components

Consider a coherent system with possibly dependent components having lifetime T, and assume we know that it failed before a given time t > 0. Its inactivity time t-T can be evaluated under different conditional events. In fact, one might just know that the system has failed and then consider the inactivity time (t-T | T < t), or one may also know which ones of the components have failed before time t, and then consider the corresponding system's inactivity time under this condition. For all these cases we obtain a representation of the reliability function of system inactivity time based on distortion functions, which, in turn, includes a description of the structure of depen-ence between components through the copula of the vector of components' lifetimes. Making use of these representations, new stochastic comparison results for inactivity times under the different conditional events are provided, as well as comparison results for inactivity times of systems having different structure functions. These results also apply to order statistics, being the order statistics particular cases of coherent systems (k-out-of-n systems).