Comparisons of coherent systems under the time-transformed exponential model

TL;DR

This paper analyzes coherent systems' reliability using the time-transformed exponential model, accounting for dependent component lifetimes through a frailty approach, and derives explicit reliability expressions and comparison results.

Contribution

It introduces a novel application of the TTE frailty model to coherent systems, providing explicit reliability formulas and stochastic order comparisons under dependence.

Findings

Explicit reliability functions for systems under TTE model

Comparison results for stochastic orders of system lifetimes

Analysis of residual lifetime under dependence

Abstract

The coherent systems are basic concepts in reliability theory and survival analysis. They contain as particular cases the popular series, parallel and $k$-ou-of-$n$ systems (order statistics). Many results have been obtained for them by assuming that the component lifetimes are independent. In many practical cases, this assumption is unrealistic. In this paper we study them by assuming a Time Transformed Exponential (TTE) model for the joint distribution of the component lifetimes. This model is equivalent to the frailty model which assumes that they are conditionally independent given a common risk parameter (which represents the common environment risk). Under this model, we obtain explicit expressions for the system reliability functions and comparison results for the main stochastic orders. The system residual lifetime (under different assumptions) is studied as well.