Stochastic Comparison Of Coherent System Based On The Inactivity Time

Coherent systems play an important role in reliability theory and survival analysis.Some simple coherent systems:such as k-out-of-n systems,parallel system has widely applications in electrion engineering,aerospace and other related fields.Based on signature representations of the reliability function of coherent systems with independent and identical components have been poved to be very useful in comparing the performance of different coherent systems.In this paper,we obtained random properties of conditional inactivity time of different coherent systems.This paper mainly includes two parts:The first part mainly studies the inactivity time of the failured components of the coherent system with n independent and identical components at time t.We build a mixture representation of the reliability function of the inactivity time of system in terms of the reliability function of the conditional inactivity time of k-out-of-n system,and then we carry out stochastic comparisons between the inactivity times of two coherent systems having either different structures or different component lifetimes based on the above mixture representation.Similarly,the reliability function and the stochastic comparisons of the inactivity times of the coherent system with n independent and identical components are studied,under the condition of the system fails at time t and at least k components are operating normally.The second part mainly studies the inactivity time of failed components of the coherent system with n independent and identical components under double monitoring has failed at time t1,and at least n-j+1 components are working at time t2.Using the corresponding reliability functions of the conditional inactivity time of k-out-of-n system,we carry out stochastic comparisons between the inactivity times of two coherent systems having either different structures or different component lifetimes.