Structure of a k-out-of-n system

Topic: Reliability · more questions on this topic
‹ Three components, one of them essential Pivoting the bridge ›

Write the structure function of a system that works if and only if at least k of its n components work.

Worked answer (click to show)

\varphi(\mathbf x)=\begin{cases}1,&\sum_{i=1}^nx_i\ge k,\\0,&\text{otherwise}.\end{cases}

As a polynomial, the minimal path sets are the \binom nk subsets of size k, so \varphi(\mathbf x)=1-\prod_{|S|=k}\bigl(1-\prod_{i\in S}x_i\bigr). The cases k=n and k=1 give the series and parallel systems.