Stochastic Comparison Of Residual Lifetime Of (n-m+1)-out-of-n Systems With Nonidentical Components

Stochastic comparison theory has been widely applied in applied statistics, reliability theory, industrial manufacturing and other fields. In this article, by considering parallel system and ?n-m+l?-out-of-n system which are composed of independent and nonidentical components, we discuss their residual life, mean residual life and the stochastic properties of the residual life.The first part mainly studies stochastic properties of residual lifetimes of parallel system with independent and nonidentical components. First, we study the residual life function and the mean residual life function of a parallel system given that at least r components and at most ?m-1? components has failed at time t, then we present some stochastic monotone properties of residual life of above considered system with respect to parameters r and m. And then we discuss stochastic comparisons of the residual life from two similar parallel systems.In the end , we draw a system residual life boundary value through the relationship between failure rate.The second part mainly studies stochastic properties of residual lifetimes of?n-m+l?-out-of-n system with independent and nonidentical components under condition that:1. r components have failed at time t₁ and at most ?n-k+l? components have failed at time t₂; 2.r components have failed at time t₁ and at least k components have failed at time t₂. First, we study the residual life survival functions'mixture representations of the ?n-m+l?-out-of-n system under the above two cases; then we present some monotone stochastic properties of residual life of above considered system. And then we discuss stochastic comparisons of the residual life from two similar ?n-m+l?-out-of-n systems.In the end ,we draw a system residual life survival function boundary value through the relationship between failure rate.