## 4 Simulation Comparative Study  
This section employs the Monte Carlo method to compare the proposed method with the conventional LM method. The LM method is widely accepted in the engineering community due to its simplicity and applicability to cases involving zero-failure components. Numerous simulation comparisons verify that the proposed method outperforms the LM method. For illustration, a series system consisting of three independent Weibull-distributed components is presented as a representative example. Two combinations of distribution parameters are designed to represent two typical scenarios, as described in the second column of Table 1. In Scenario 1, the series system  
Figure 3: Framework of the system reliability assessment and update algorithm. consists of a mixture of exponential and Weibull distributions. In Scenario 2, the series system comprises Weibull distributions with different shape parameters, all of which are greater than one. Given the component distribution parameters, the real value of system reliability is calculated using Equation (2), where \(m=3\). In each scenario, Weibull zero-failure data are generated for all components. For a given sample size \(n_{i}\), the simulation steps for the GTT \(T_{i}\) are as follows:  
1. Generate \(N_{i}\) (where \(N_{i}\) should be much larger than \(n_{i}\)) random numbers from the distribution model \(W(\alpha_{i},\beta_{i})\) of component \(i\) and arrange them in ascending order;
2. Select the first \(n_{i}\) data as the failure data \(\tau_{i1},\tau_{i2},\cdots,\tau_{in_{i}}\) of component \(i\), where \(i=1,2,3\);
3. Take the integer slightly less than the failure data \(\tau_{ij}\) as the zero-failure data \(t_{ij}\) of component \(i\), where \(i=1,2,3,j=1,2,\cdots,n_{i}\);
4. Calculate the \(T_{i}\) for component \(i\) from Equation (5). Following the above steps; the GTTs of all components are simulated and listed in the last three columns of Table 1. Then, at a confidence level of \(\gamma=0.9\), both the LM and the proposed methods are used to calculate the LCLs of the series system reliability at different time moments, ranging from 0 to 10. The simulation comparison results under the two scenarios are shown in Figure 4.  
A comparison of the two scenarios in Figure 4 reveals that the shapes of the system reliability curves differ: Scenario 1 is approximately linear, while Scenario 2 shows a slow and then fast curve. This difference arises because the life of the series system is limited by the weakest component, and its lifetime distribution directly determines the overall reliability characteristics of the system. Numerous simulations show that the system reliability curves follow one of these two patterns, regardless of the combination of component lifetime distribution parameters. Therefore, the two selected sets of distribution parameters are typically representative. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Scenario**} & \multirow{2}{*}{**Distribution Parameters**} & \multicolumn{3}{c}{**GTT of Components**} \\ \cline{3-5}  & & \(\mathbf{T_{1}}\) & \(\mathbf{T_{2}}\) & \(\mathbf{T_{3}}\) \\ \hline
1 & \(\begin{array}{c}(\alpha_{1},\alpha_{2},\alpha_{3})=(1.5,1,1)\\ (\beta_{1},\beta_{2},\beta_{3})=(400,600,1000)\\ \end{array}\) & 3907 & 642 & 574 \\
2 & \(\begin{array}{c}(\alpha_{1},\alpha_{2},\alpha_{3})=(3,2.5,2)\\ (\beta_{1},\beta_{2},\beta_{3})=(60,60,60)\\ \end{array}\) & 80,266 & 14,045 & 3736 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Distribution parameters and GTT simulation results in the two scenarios. Figure 4: Simulation comparison results in the two scenarios. In both scenarios, the LCL curves for system reliability calculated by both the LM and the proposed approaches lie below the real reliability curve, indicating that the evaluation results of both methods are safe and conservative in engineering applications. Furthermore, the LCLs obtained by the proposed method are consistently higher than those from the LM method and closer to the real value at any given time.