## 3 Importance Measure Index and Calculation Method
### Calculation Method
Accordingly, for the power supply reliability matrix \(R_{s}\) obtained through the matrix \(\lambda_{\sim i}|\lambda_{i}\), each element in its \(j\)-th row represents a possible value of the power supply reliability of sink node \(s\) when the failure rate of component \(i\) takes the value \(\lambda_{i}^{j}\), and the mean value of the \(j\)-th row is \(\hat{R}_{s}(\lambda_{i}^{j})\) by averaging \(C\) possible values of the row. In addition, according to the law of large numbers, as the value of \(C\) increases, \(\hat{R}_{s}(\lambda_{i}^{j})\) will surely converge to the expectation \(E_{s,\lambda_{\sim i}}(R_{s}|\lambda_{i}^{j})\); moreover, when the two samples \(\lambda_{i}^{j}\) and \(\lambda_{i}^{j+1}\) of the failure rate of the \(i\)-th component are the same, the mean value \(\hat{R}_{s}(\lambda_{i}^{j})\) and \(\hat{R}_{s}(\lambda_{i}^{j+1})\) tend to be the same as \(C\) increases. That is to say, for the row mean value \(\hat{R}_{s}(\lambda_{i}^{1})\), \(\hat{R}_{s}(\lambda_{i}^{2})\),..., \(\hat{R}_{s}(\lambda_{i}^{M})\) of the matrix \(R_{s}\), the numerical difference between them is mainly caused by the different failure rate values of component \(i\). Therefore, the variance \(\hat{V}ar_{s,\lambda_{i}}[E_{s,\lambda_{\sim i}}(R_{s}|\lambda_{i})]\), calculated by Equation (7), quantifies the degree of dispersion of the power supply reliability of the sink nodes \(s\) induced by the dispersion of the failure rate of the component \(i\). Since the uncertainty of output propagated from input can be measured with the variance of the output variable [44], the variance computed by Equation (7) has the ability to quantify the uncertainty of the failure rate of component \(i\) on the contribution of the power supply reliability uncertainty of the sink node \(s\). Furthermore, because the variance calculated by Equation (8) is based on all the elements in the matrix \(R_{s}\) and the matrix includes all the possible values of the power supply reliability of the sink node \(s\), the obtained variance \(\hat{V}ar_{s}[R_{s}]\) is a measurement that quantifies the failure uncertainty contribution of all the components to the uncertainty of the power supply reliability of the sink node \(s\). In summary, first, the ratio of variance \(\hat{V}ar_{s,\lambda_{i}}\big{[}E_{s,\lambda_{i-}}(R_{s}|\lambda_{i})\big{]}\) to variance \(\hat{V}ar_{s}[R_{s}]\) calculated by Equation (9) essentially quantifies the ratio of the uncertainty that the \(i\)-th component contributes to and the whole uncertainty that all the components contribute to in theory; and according to the law of large numbers, with \(S,C\rightarrow\infty\), \(\sum\limits_{i=1}^{n}\hat{\eta}_{s,i}^{2}\to 1\) holds. This means that if the simulation results obtained by the proposed method satisfies the convergence condition: \(\sum\limits_{i=1}^{n}\hat{\eta}_{s,i}^{2}\,1\), it indicates that the sampling sizes \(M\) and \(C\) are reasonable, otherwise, the sampling size needs to be increased until the convergence condition is met. Second, from the index definition of Equation (3) and the above explanation, it can be seen that this index value varies in the range [0, 1]. Furthermore, the higher the index value of one component, the greater the influence of the uncertainty of the component's failure rate on the uncertainty of system reliability, and vice versa. Therefore, the index can help designers identify which components are more important to the system to reduce the system reliability uncertainty, and the quality and performance of these components with a higher index can be prioritized to be improved. Conversely, improving components with lower index will not achieve better results. Algorithm 1 shows the pseudocode of the proposed algorithm for computing the importance degree.