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

TL;DR

This paper derives a mathematical formula to calculate the reliability of a multi-dimensional system where failure occurs if a contiguous subarray of a certain size is all failed, extending reliability analysis to higher dimensions.

Contribution

The paper introduces a new formula for the reliability of $d$-dimensional consecutive-$k$-out-of-$n$:F systems, generalizing previous one-dimensional models.

Findings

Provides an explicit reliability formula for multi-dimensional systems.

Enables analysis of complex systems with higher-dimensional failure modes.

Facilitates design and assessment of multi-dimensional reliability systems.

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 $1$.