Open Access
August, 1983 The Reliability of $K$ Out of $N$ Systems
Philip J. Boland, Frank Proschan
Ann. Probab. 11(3): 760-764 (August, 1983). DOI: 10.1214/aop/1176993520


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 $\mathbf{p} = (p_1, \cdots, p_n)$ is the vector of component reliabilities for the $n$ components, then $h_k(\mathbf{p})$ is the reliability function of the system. It is shown that $h_k(\mathbf{p})$ is Schur-convex in $\lbrack(k - 1)/(n - 1), 1\rbrack^n$ and Schur-concave in $\lbrack 0, (k - 1)/(n - 1)\rbrack^n$. More particularly if $\prod$ is an $n \times n$ doubly stochastic matrix, then $h_k(\mathbf{p}) \geq (\leq)h_k(\mathbf{p}\prod)$ whenever $\mathbf{p} \in \lbrack(k - 1)/(n - 1), 1\rbrack^n(\lbrack 0, (k - 1)/(n - 1)\rbrack^n)$. This Theorem is compared with a result on Schur-convexity and -concavity by Gleser [2] which in turn extends work of Hoeffding [4].


Download Citation

Philip J. Boland. Frank Proschan. "The Reliability of $K$ Out of $N$ Systems." Ann. Probab. 11 (3) 760 - 764, August, 1983.


Published: August, 1983
First available in Project Euclid: 19 April 2007

zbMATH: 0519.62086
MathSciNet: MR704562
Digital Object Identifier: 10.1214/aop/1176993520

Primary: 60C05
Secondary: 26B25 , 62E15 , 62N05

Keywords: $k$ out of $n$ systems , independent Bernoulli trials , majorization , Schur-convexity and Schur-concavity

Rights: Copyright © 1983 Institute of Mathematical Statistics

Vol.11 • No. 3 • August, 1983
Back to Top