Improved Ordering Results for Fail-Safe Systems with Exponential Components

Let X2: n and Y2: m be the second order statistics from n independent exponential variables with hazards λ1, …, λn, and an independent exponential sample of size m with hazard change to λ, respectively. When m ⩾ n, we obtain necessary and sufficient conditions for comparing X2: n and Y2: m in mean residual life, dispersive, hazard rate, and likelihood ratio orderings based on some inequalities between λi’s and λ. The established results show how one can compare an (n − 1)-out-of-n system consisting of heterogeneous components with exponential lifetimes with any (m − 1)-out-of-m system consisting of homogeneous components with exponential lifetimes.