System reliability and weighted lattice polynomials

TL;DR

This paper investigates the distribution of system lifetime modeled as a weighted lattice polynomial of component lifetimes, providing theoretical results for both independent and dependent cases, with applications to reliability analysis.

Contribution

It introduces a comprehensive framework for analyzing the distribution of weighted lattice polynomial systems, extending existing models to dependent component lifetimes.

Findings

Derived formulas for the distribution of weighted lattice polynomial systems.

Established connections between system lifetime and order statistics.

Provided techniques for handling dependent component lifetimes.

Abstract

The lifetime of a system of connected units under some natural assumptions can be represented as a random variable Y defined as a weighted lattice polynomial of random lifetimes of its components. As such, the concept of a random variable Y defined by a weighted lattice polynomial of (lattice-valued) random variables is considered in general and in some special cases. The central object of interest is the cumulative distribution function of Y. In particular, numerous results are obtained for lattice polynomials and weighted lattice polynomials in case of independent arguments and in general. For the general case, the technique consists in considering the joint probability generating function of "indicator" variables. A connection is studied between Y and order statistics of the set of arguments.