A Formula for the Reliability of a $d$-dimensional Consecutive-$k$-out-of-$n$:F System

Abstract

We derive a formula for the reliability of a $d$-dimensional consecutive-$k$-out-of-$n$:F system. That is, a formula for the probability that an $n_1 \times \ldots \times n_d$ array whose entries are (independently of each other) $0$ with probability $p$ and $1$ with probability $q = 1 - p$ does not include a contiguous $s_1 \times \ldots \times s_d$ subarray whose every entry is $1$.