## 4 Case Studies
### Uncertainty Analysis of Series-Parallel Systems
#### 4.1.2 Calculation and Analysis  
As a seven-hour flight is general in the real world, system time \(T=7\) h was set here to verify the proposed method, and then the stated Monte Carlo simulation method was applied to the four cases, with the results shown in Figure 5 and Table 1. The Monte Carlo simulation method includes three steps: first, given the system time \(t_{0}\), according to the probability density function of the component failure rate \(\lambda\), generate \(N\) group samples \(\lambda_{k}=(\lambda_{k1},\,\lambda_{k2},\,\ldots,\,\lambda_{kn})\), \((k=1,\,\ldots,\,N)\), and substitute \(\lambda_{k}\) and \(t_{0}\) into Equation (1) to obtain the vector \(\mathbf{p}_{k}=(p_{k1},\,p_{k2},\,\ldots,\,p_{kn})\); second, substitute \(\mathbf{p}_{k}\) into Equation (2) to compute the sink node's power supply reliability \(R_{k}\), forming a reliability vector \(R(t_{0})=(R_{1},\,R_{2},\,\ldots,\,R_{kn})\); finally, sort the elements in vector \(R(t_{0})\) from small to large, and calculate its confidence interval under the given confidence interval as well as plot the kernel density function curve. Furthermore, the proposed algorithm for computing the importance degree index of each component based on the variance decomposition was performed on the four cases separately, and the results are shown in Table 2. The results of the four cases are discussed below. Figure 5: \(Cont.\)  
(1) From Table 2, it can be seen that all of the components' importance measure indicators equal to zero in case 1. That is because the failure rate of each component in the system was constant and had no effect on the uncertainty of system reliability. It is clear that this result is consistent with engineering practice. (2) In case 2, the importance measure indicators of components C3 and C4 were equal, and the importance measure indicators of components C1 and C2 were equal. The result correctly reflects the fact that components C1 and C2 took the same structural position and had the same failure parameters in the system, and similarly, components C3 and C4 took the same structural position and had the same failure parameters in the system. In addition, since components C1 and C2 formed a parallel structure, the system will only fail when both of them fail, while components C3 and C4 were in a series structure, and the system will fail when either of them fails. That is to say, C3 and C4 are more important than C1 and C2, so under the condition that the failure rate parameters of all the four components are identical, the components C3 and C4 definitely will cause a greater impact on the system reliability uncertainty than components C1 and C2, which is consistent with the results of Table 2.  
(3) In case 3, component C4 had the highest degree of uncertainty in failure rate and its structural position was more important than components C1 and C2, so the value \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Case 1** & **Case 2** & **Case 3** & **Case 4** \\ \hline Mean & 0.6335 & 0.6332 & 0.6327 & 0.6332 \\ Variance & 8.7879 \(\times\) 10\({}^{-28}\) & 9.0439 \(\times\) 10\({}^{-5}\) & 2.3694 \(\times\) 10\({}^{-4}\) & 5.7243 \(\times\) 10\({}^{-5}\) \\ Confidence intervals & [0.6335, 0.6335] & [0.6144, 0.6519] & [0.6016, 0.6612] & [0.6182, 0.6477] \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Means and variances of system reliability in four cases (\(T\) = 7 h, \(M\) = 1000). \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Case 1** & **Case 2** & **Case 3** & **Case 4** \\ \hline \(\eta_{1}^{2}\) & 0.0000 & 0.0124 & 0.0059 & 0.0199 \\ \(\eta_{2}^{2}\) & 0.0000 & 0.0121 & 0.0207 & 0.0055 \\ \(\eta_{3}^{2}\) & 0.0000 & 0.4865 & 0.1905 & 0.7713 \\ \(\eta_{4}^{2}\) & 0.0000 & 0.4894 & 0.7961 & 0.1915 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Importance measure indicators of components in four cases (\(T\) = 7 h, \(S\) = \(C\) = 1000). Figure 5: Probability density functions of system reliability for case 1 to case 4.