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Mixture Representations of Reliability in Coherent Systems and Preservation Results Under Double Monitoring

The mixture representations of the reliability functions of the residual life and inactivity time of a coherent system with n independent and identically distributed components are obtained, given that before time t 1 (t 1 ≥ 0), exactly r (r < n) components have failed and at time t 2 (t 2 ≥ t 1), the system is either still working or has failed. Based on the stochastically ordered coefficient vectors between systems, some preservation results of the residual live and the inactivity time of the system are obtained. The results in this article extend previous results in the literature and are useful for comparing similar systems that have different structure functions.