Reliability Model and Sensitivity Analysis for General Electronic Systems with Failure Types based on Non-identical Correlated Components

Abstract

Reliability is the most important qualitative properties of products and industrial electronic systems. Today, industrial electronic systems need to produce products with maximum reliability. In every industry, when a system fails, it becomes harmful in various aspects such as economic, human and political; therefore, accurate estimation of system reliability is very important. Previous methods to calculate system reliability assumed that a large number of components failures are statistically independent. Considering such a hypothesis makes it possible to calculate probability and mathematical computation, but it does not provide perfect system reliability. This paper presents a simple and new technique for reliability analysis by considering unequal reliability and non-identical distribution of correlated components for k-out-of-n and coherent systems. The efficiency of our proposed method is demonstrated by computing the reliability of Bridge system. The results show that the function of existing components correlation has a major impact, and that ignoring it has a significant effect on system safety.

Appendix 1: Proof of Eqs. (5), (6) and (7)

Consider steps 1 through 4 to prove Eqs. (5), (6) and (7)

$$ E\left({X}_i\right)=E\Big\{{I}_{\left\{0\right\}}\left\{{Y}_i\left({\lambda}_i-{\lambda}_{i,j}\right)+{Y}_{i,j}\left({\lambda}_{i,j}\right)\right\}=\mathit{\Pr}\left\{Y{\left({\lambda}_i-{\lambda}_{i,j}\right)}_i=0\right\}\mathit{\Pr}\left\{{Y}_{i,j}\left({\lambda}_{i,j}\right)=0\right\}={e}^{-\left({\lambda}_i-{\lambda}_{i,j}\right)}{e}^{-{\lambda}_{i,j}}={e}^{-{\lambda}_i}={R}_i. $$

$$ E\left({X}_i{X}_j\right)=\mathit{\Pr}\left({Y}_i\left({\lambda}_i-{\lambda}_{i,j}\right)=0\&{Y}_j\left({\lambda}_j-{\lambda}_{i,j}\right)\&{Y}_{i,j}\left({\lambda}_{i,j}\right)=0\right)={e}^{-\left({\lambda}_i-{\lambda}_{i,j}\right)}.{e}^{-\left({\lambda}_j-{\lambda}_{i,j}\right)}{e}^{-{\lambda}_{i,j}}={e}^{-\left({\lambda}_i-{\lambda}_{i,j}\right)}.{e}^{-\left({\lambda}_j-{\lambda}_{i,j}\right)}{e}^{-{\lambda}_{i,j}}={R}_i.{R}_j.{e}^{\lambda_{i,j}} $$

$$ \mathit{\operatorname{var}}\left({X}_i\right)=E\left({X_i}^2\right)-{E}^2\left({X}_i\right)=\mathit{\Pr}\left({X}_i=1\right)-{R^2}_i={R}_i\left(1-{R}_i\right) $$

Therefore:

$$ \mathit{\Pr}\left({X}_i=1\&{X}_j=1\left)=\mathit{\Pr}\right({Y}_i\left({\lambda}_i-{\lambda}_{i,j}\right)+{Y}_{i,j}\left({\lambda}_{i,j}\right)=0\&{Y}_j\left({\lambda}_j-{\lambda}_{i,j}\right)+{Y}_{i,j}\left({\lambda}_{i,j}\right)=0=\mathit{\Pr}\left({Y}_i\left({\lambda}_i-{\lambda}_{i,j}\right)=0\&{Y}_j\left({\lambda}_j-{\lambda}_{i,j}\right)\&{Y}_{i,j}\left({\lambda}_{i,j}\right)=0\right)={e}^{-\left({\lambda}_i-{\lambda}_{i,j}\right)}.{e}^{-\left({\lambda}_j-{\lambda}_{i,j}\right)}{e}^{-{\lambda}_{i,j}}={R}_i{R}_j\left(1+{\rho}_{i,j}\sqrt{\frac{\left(1-{R}_i\right)\left(1-{R}_j\right)}{R_i.{R}_j}}\right)\right) $$

1.1 Appendix 2: Proof of Eq. (21)

The following relation is used to prove this equation:

$$ \Pr \left({E}_1\cup {E}_2\cup \dots \cup {E}_n\right)=\sum \limits_{i=1}^nP\left({E}_i\right)-\sum \limits_{i_1<{i}_2}P\left({E}_{i_1}\cap {E}_{i_2}\right)+\dots +{\left(-1\right)}^{r+1}\sum \limits_{i_1<{i}_2<\dots <{i}_r}P\left({E}_{i_1}\cap {E}_{i_2}\cap \dots \cap {E}_{i_r}\right)+\dots +{\left(-1\right)}^{n+1}P\left({E}_{i_1}\cap {E}_{i_2}\cap \dots \cap {E}_{i_n}\right) $$