The Reliability of K Out of N Systems

A system with n independent components which functions if and only if at least k of the components function is a k out of n system. Parallel systems are 1 out of n systems and series systems are n out of n systems. If p = (p₁, ⋯, pₙ) is the vector of component reliabilities for the n components, then hₖ(p) is the reliability function of the system. It is shown that hₖ(p) is Schur-convex in (k - 1)/(n - 1), 1ⁿ and Schur-concave in 0, (k - 1)/(n - 1)ⁿ. More particularly if ∏ is an n × n doubly stochastic matrix, then hₖ(p) ≥ (≤)hₖ(p∏) whenever p ∈ (k - 1)/(n - 1), 1ⁿ( 0, (k - 1)/(n - 1)ⁿ). This Theorem is compared with a result on Schur-convexity and -concavity by Gleser [2] which in turn extends work of Hoeffding [4].