## 2 Modeling of Subsystem
### Model Optimization  
The reliability assessment results will differ greatly if the reliability models are different. Each subsystem will be fitted by five distribution functions, so that it is necessary to select one optimal model as the reliability model for each subsystem. The \(K\)-\(S\) hypothesis test method, correlation coefficient method, and gray correlation degree method are the normal model optimization methods [15; 16].  
The accuracy of the \(K\)-\(S\) hypothesis test is proportional to the number of reliability data. The correlation coefficient method is used to optimize the models that are linear or can be converted into a linear relationship. The standard gray correlation is mainly to judge the similarity of two models [17; 18]. Not only will the similarity be judged, but also the closeness should be calculated between sample data and the fitted model when the model optimization is executed. One modified gray correlation model optimization method is proposed based on the standard gray correlation, which is called the similar slope and close median gray correlation. The basic idea of the model optimization method is to judge the similarity degree of the slope of adjacent points between the sample data and fitted curve and the close degree of the median between the sample data and fitted curve. The algorithm of the model optimization method is shown as follows and the flow chart is shown in Figure 2.  
1. Calculate the adjacent points' slope of sample data and fitted curve, as shown in Equation (19). \[\left\{\begin{array}{c}K_{i}=\frac{\hat{F}(t_{i+1})-\hat{F}(t_{i})}{t_{i+1} -t_{i}}\\ K_{ji}=\frac{\hat{F}(t_{i+1})-\hat{F}(t_{i})}{t_{i+1}-t_{i}}\end{array}\right.\] (19) where \(\hat{F}(t_{i})\) is the function value of the \(i\)th sample data, \(K_{ji}\) is the slope of the \(i\)th and (\(i\)+1)th sample data of the \(j\)th fitting function, and \(F_{j}(t_{i})\) is the function value of the \(j\)th fitting function when the independent variable equals to the value of the \(i\)th sample data.
2. Calculate the adjacent points' median of sample data and fitted curve, as shown in Equation (20). \[\left\{\begin{array}{c}V_{i}=\frac{\hat{F}(t_{i+1})+\hat{F}(t_{i})}{2}\\ V_{ji}=\frac{\hat{F}(t_{i+1})+\hat{F}(t_{i})}{2}\end{array}\right.\] (20)
3. Calculate the absolute value of difference of slope and median, as shown in Equation (21). \[\left\{\begin{array}{c}DK_{ji}=\left|K_{i}-K_{ji}\right|\\ DV_{ji}=\left|V_{i}-V_{ji}\right|\end{array}\right.\] (21)Maximum image processing of slope and median, as shown in Equation (22).  
\[\left\{\begin{array}{l}RK_{ji}=DK_{ji}/\max(DK_{ji})\\ RV_{ji}=DV_{ji}/\max(DV_{ji})\end{array}\right. \tag{22}\]  
Calculate the gray correlation between sample data and fitted function, as shown in Equation (23).  
\[R_{kyj}=\frac{1}{n-1}\sum_{i=1}^{n-1}RK_{ji}\frac{1}{n-1}\sum_{i=1}^{n-1}RV_{ji} \tag{23}\]  
Normalization. In Equation (23), \(R_{kyj}\) is the similarity and closeness between the \(j\)th fitted curve and the sample data. The smaller the value of \(R_{kyj}\), the better the fitting. Usually, the big gray correlation degree indicates the good fitting, and its value is between 0 and 1, which needs to be normalized, as shown in Equation (24).  
\[R_{gj}=\frac{\min(R_{kyj})}{R_{kyj}+\rho\min(R_{kyj})} \tag{24}\]  
In Equation (24), \(\rho\) is the discrimination adjustment coefficient, which is between [0,1]. The smaller \(\rho\) is, the greater the degree of distinction will be. The gray correlation can be calculated using the five distribution functions. The distribution function of the maximum value of the gray correlation will be selected as the reliability model of the corresponding subsystem.