A note on the mean past and the mean residual life of a (n − k + 1)-out-of-n system under multi monitoring

In the study of the reliability of technical systems, k -out-of- n systems play an important role. In the present paper, we consider a ( n − k + 1)- out -of- n system consisting of n identical components such that the lifetimes of components are independent and have a common distribution function F . It is assumed that the number of monitoring is l and the total number of failures of the components at time t i is m i , i = 1, . . . , l − 1. Also at time t l ( t 1 < . . . < t l ) the system have failed or the system is still working. Under these conditions, the mean past lifetime, the mean residual lifetime of system and their properties are investigated.