## 3 Importance Measure Index and Calculation Method
### Calculation Method
It is clear that \(R_{s}^{j,k}\) (\(j=1,\,2,\,\ldots,\,M,k=1,\,2,\,\ldots,\,C\)) can form an output matrix \(\mathbf{R}_{s}\) with size \(M\times C\);```
1:Input: array Lambda with size S\(\times\)N, Distribution_type with size N, Path_sets with size Num \(\times\) N, variable s and T. \(N\) is the component number in the MPSs of sink node s. S is the max number of parameters that used to describe the probability distribution of each component's failure rate, e.g., there are two components in the MPSs of sink node s, and the failure rates of the first and the second component follow lognormal and triangular distributions, respectively, which are described with two and three parameters in separate, so S = 3. The \(i\)th element represents the distribution type of component \(i\)'s failure rate, and the \(i\)th column of Lanbuda stores the probability distribution parameters of node \(i\)'s failure rate. Num represents the number of MPSs of sink node s and the \(i\)th row of Path_sets stores the components in the \(i\)th MPS. T represents system time.//Output: array RELI_var with size N. Its \(i\)th element represents component \(i\)'s importance degree that measures the contribution to the uncertainty of node \(s\)'s power supply reliability. 2:
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Step 4: Estimate the value of \(E_{s,\lambda_{\sim i}}(R_{s}|\lambda_{i}^{j})\). For \(j\)-th row of the output matrix \(R_{s},j=1,2,\ldots,M\), use Equation (5) to calculate the mean reliability. \[\hat{R}_{s}(\lambda_{i}^{j})=\frac{1}{C}\sum_{k=1}^{C}R_{s}^{ijk}\approx E_{s, \lambda_{\sim i}}[R_{s}|\lambda_{i}^{j}] \tag{5}\]  
The term \(E_{s,\lambda_{\sim i}}(R_{s}|\lambda_{i}^{j})\) indicates the expected value of the power supply reliability of the sink nodes \(s\) when the failure rate of the component \(i\) takes the value \(\lambda_{i}^{j}\). Step 5: Estimate the unconditional variance of power supply reliability and the variance of \(E_{s,\lambda_{\sim i}}(R_{s}|\lambda_{i}^{j})\). For the power supply reliability of sink nodes \(s\), estimate its value of expectation using Equation (6). Then, with the help of Equation (7), calculate the estimation value \(\hat{V}ar_{s,\lambda_{i}}[E_{s,\lambda_{\sim i}}(R_{s}|\lambda_{i})]\) of variance \(Var_{s,\lambda_{i}}[E_{s,\lambda_{\sim i}}(R_{s}|\lambda_{i})]\). Finally, use Equation (8) to calculate the estimation value \(\hat{V}ar_{s}[R_{s}]\) of variance \(Var_{s}[R_{s}]\):  
\[\overline{R}_{s}=\frac{1}{M}\sum_{j=1}^{M}\hat{R}_{s}(\lambda_{i}^{j})\approx E _{s}[R_{s}] \tag{6}\]  
\[\hat{V}ar_{s,\lambda_{i}}[E_{s,\lambda_{\sim i}}(R_{s}|\lambda_{i})]=\frac{1} {M-1}\sum_{j=1}^{M}\left[\hat{R}_{s}(\lambda_{i}^{j})-\overline{R}_{s}\right]^ {2} \tag{7}\]  
\[\hat{V}ar_{s}[R_{s}]=\frac{1}{MC-1}\sum_{j=1}^{M}\sum_{k=1}^{C}\left(R_{s}^{jk }-\overline{R}_{s}\right)^{2} \tag{8}\]  
Step 6: Compute the estimation value of the proposed importance measure index \(\hat{\eta}_{s,j}^{2}\). Substitute the results of Equations (7) and (8) into Equation (9), the importance measure index of component \(l\), which can quantify its failure uncertainty contribution to the power supply reliability uncertainty of the sink node \(s\), is estimated. \[\hat{\eta}_{s,i}^{2}=\frac{\hat{V}ar_{s,\lambda_{i}}[E_{s,\lambda_{\sim i}}(R _{s}|\lambda_{i})]}{\hat{V}ar_{s}[R_{s}]} \tag{9}\]  
The physical meaning of the six steps is explained as follows. First, each element in \(j\)-th row of the matrix \(\lambda_{\sim i}|\lambda_{i}\) stands for the failure rate sample of the remaining components when the \(i\)-th component takes the value \(\lambda_{i}^{j}\).