Simplification of inclusion-exclusion on intersections of unions with application to network systems reliability

TL;DR

This paper introduces a simplified mathematical expression for calculating the reliability of complex network systems, significantly reducing computational complexity and enabling more efficient analysis of safety-critical systems.

Contribution

A new simplified formula for inclusion-exclusion in reliability calculations of non series-parallel networks, reducing computational complexity from double exponential to exponential.

Findings

Reduced number of terms in reliability calculations

Complexity analysis shows exponential vs. double exponential growth

Application demonstrated on aircraft door management system

Abstract

Reliability of safety-critical systems is an important issue in system engineering and in most practical situations the reliability of a non series-parallel network system has to be calculated. Some methods for calculating reliability use the probability principle of inclusion-exclusion. When dealing with complex networks, this leads to very long mathematical expressions which are usually computationally very expensive to calculate. In this paper, we provide a new expression to simplify the probability principle of inclusion-exclusion's formula for intersections of unions, which appear when calculating reliability on non series parallel network systems. This new expression has much less terms, which reduces enormously the computational cost. We also show that the general form of the probability principle of inclusion-exclusion's formula has double exponential complexity whereas the simplified form has only exponential complexity with a linear exponent. Finally, we illustrate how to use this result when calculating the reliability of a door management system in aircraft engineering.