# Competing Failure Modeling for Systems under Classified Random Shocks and Degradation
## 5 Numerical Case: An MEMS Application
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameters** & **Values** & **Sources** \\ \hline \multirow{3}{*}{\(\beta\)} & \(\sim\)\(N\) (\(\mu_{\beta}\), \(\sigma_{B}\)\({}^{2}\)) & \\  & \(\mu_{\beta}\) = 8.4823 \(\times\) 10\({}^{-9}\)\(\mu\)m\({}^{3}\) & Tanner [2] \\  & \(\sigma_{B}\) = 6.0016 \(\times\) 10\({}^{-10}\)\(\mu\)m\({}^{3}\) & \\ \hline \multirow{3}{*}{\(W_{i}\)} & \(\sim\)\(N\) (\(\mu_{W}\), \(\sigma_{W}\)\({}^{2}\)) & \\  & \(\mu_{W}\) = 1.2 Gpa & Jiang L. [4] \\  & \(\sigma_{W}\) = 0.2 Cpa & \\ \hline \multirow{3}{*}{\(B_{i}\)} & \(\sim\)\(N\) (\(\mu_{\beta}\), \(\sigma_{B}\)\({}^{2}\)) & \\  & \(\mu_{B}\) = 1.2 \(\times\) 10\({}^{-4}\)\(\mu\)m\({}^{3}\) & Assumption \\  & \(\sigma_{B}\) = 2 \(\times\) 10\({}^{-10}\)\(\mu\)m\({}^{3}\) & \\ \hline \multirow{3}{*}{\(C_{i}\)} & \(\sim\)\(N\) (\(\mu_{C}\), \(\sigma_{C}\)\({}^{2}\)) & \\  & \(\mu_{C}\) = 1.8\(\times\)10\({}^{-4}\)\(\mu\)m\({}^{3}\) & Assumption \\ \cline{1-1}  & \(\sigma_{C}\) = 2\(\times\)10\({}^{-10}\)\(\mu\)m\({}^{3}\) & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: _Cont._  
Figure 3: Comparing probability and modified probability. (**A**): System Reliability Probability; (**B**): Competing failure probability 1 (probability of hard failure while there is no soft failure); (**C**): Competing failure probability 2 (probability of soft failure while there is no hard failure). In Figure 5, the probabilities with different rates \(\lambda\) are calculated. It can be observed that larger \(\lambda\) will lead to lower system reliability and higher competing failure probability 2. The reason is that larger \(\lambda\) means more frequent shocks, including those that can destroy the system. However, the MCFP 1 increases with \(\lambda\) at the initial working time, while it decreases with \(\lambda\) at the later working period after a critical point. Figures 6 and 7 show the impacts on MSRP and MCFP of different \(D_{1}\) and \(D_{2}\). The impacts on MSRP and MCFP 2 are the same for \(D_{1}\) and \(D_{2}\). The increase of \(D_{1}\) or \(D_{2}\) will increase MSRP and decrease MCFP 2. By comparison, MCFP 1 decreases with \(D_{1}\) increasing, while it increases with \(D_{2}\) increasing. Particularly, the MCFP 1 in Figure 6B is so sensitive to \(D_{1}\) that the maximum value when \(D_{1}\) equals 1.0 is almost four times larger than it is at 1.4. Figure 4: Probability with different H. (**A**): MSRP; (**B**): MCFP 1 (Probability of hard failure while there is no soft failure); (**C**): MCFP 2 (Probability of soft failure while there is no hard failure). Figure 5: Probability with different \(\lambda\). (**A**): MSRP; (**B**): MCFP 1 (Probability of hard failure while there is no soft failure); (**C**): MCFP 2 (Probability of soft failure while there is no hard failure). In conclusion, the reliability performance will be different with different parameter values. The threshold \(H\), \(D_{1}\), and \(D_{2}\) are properties of the product itself. Higher and more credible threshold values will lead to better system reliability performance, according to our results. The rate \(\lambda\) of the Poisson process indicated the frequency of external shocks. Larger \(\lambda\) means more frequent shocks, which will decrease the system reliability performance. An adaptive working environment with a small enough \(\lambda\) will benefit the reliability.