## The Annals of Probability

- Ann. Probab.
- Volume 11, Number 3 (1983), 760-764.

### The Reliability of $K$ Out of $N$ Systems

Philip J. Boland and Frank Proschan

#### Abstract

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].

#### Article information

**Source**

Ann. Probab., Volume 11, Number 3 (1983), 760-764.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176993520

**Digital Object Identifier**

doi:10.1214/aop/1176993520

**Mathematical Reviews number (MathSciNet)**

MR704562

**Zentralblatt MATH identifier**

0519.62086

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60C05: Combinatorial probability

Secondary: 62N05: Reliability and life testing [See also 90B25] 62E15: Exact distribution theory 26B25: Convexity, generalizations

**Keywords**

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

#### Citation

Boland, Philip J.; Proschan, Frank. The Reliability of $K$ Out of $N$ Systems. Ann. Probab. 11 (1983), no. 3, 760--764. doi:10.1214/aop/1176993520. https://projecteuclid.org/euclid.aop/1176993520