Ordered properties on the residual life and inactivity time of (n−k+1)-out-of-n systems under double monitoring

The (n−k+1)-out-of-n system is an important structure of the reliability of technical systems. In this paper, we consider the residual lifetime and the inactivity time of the system consisting of independent and identically distributed components, under the condition that the total number of failures of the components at time t1 is r (rt1) the system is still working or it has failed. Under these conditions, some ordered properties of one system or between two systems with two sets of independent components are obtained both for the hazard rate order and the likelihood ratio order. The results obtained here are stronger than the results included in Poursaeed (in press).