# Reliability Modelling and Evaluation for LTD System Based on Load-Sharing Model
## 4 Numerical Simulation and Analysis
### Load-Sharing Based Level Reliability Analysis with the Increasing Number of Failure Modules  
If we only concentrate on the pulsed outputs, a certain number of normal modules in each level can guarantee the output performance to meet the expected requirement [7], which could be easily determined. However, the load-sharing mechanism results in a higher failure rate of each remaining normal modules after some modules have failed unexpectedly. In other words, we will need to bear a greater risk if we continue the LTD experiment when failures occur. Therefore, condition reliability should be evaluated to decide whether to continue the LTD experiment, where the condition reliability here refers that at least one module keeps normal in the experiment when there have been a certain number of failures happened. In this section, we will discuss the module failure threshold \(k\) by evaluating the condition reliability.  
As discussed before, a different failure sequence leads to a different load redistribution result, which could influence the sampling probability while conducting the Monte Carlo simulation. Therefore, we proposed a self-adaptive Monte Carlo simulation flow to calculate the statistical reliability according to the system structure considered, in which the failures probability for each module used to sample is updated with the failures sequence rather than remaining unchanged. The proposed simulation flow is shown in Table 3.  
Similarity, we still analyze different types of load-sharing rules. The reliability degradation curve is shown in Figure 5. The results show that there is always a marked fall as long as we consider the load-sharing mechanism, which is caused by the increasing failure rates of the remaining normal modules. Moreover, significant differences can be seen when we compare the different curves. For example, the curve of the equal rule drops sharply after 11 modules failed, while the curve of the exponential rule with the parameter \(d=0.9\) declines steadily after 21 modules failed.  
\begin{table}
\begin{tabular}{l} \hline \hline
**Simulation Flow: calculate the statistical reliability** \\ \hline
**Input:**\(numS\) (sample time) \\
**Output:**\(conRL\) (statistical reliability array) \\
**Begin** \\
01: Initalization: construct an Array variable \(sumR\) to save all the simulation results \\
02:**for**\(i=1\) to \(numS\) \\
03: construct an Array variable \(sum\) to save the single simulation result \\
04: calculate the module failure rate Array \(FM\) \\
05: update the Array variable \(sum\) \\
06:**for**\(j=1\)**to**_threshold_ \\
07: generate a random variable \(\eta\) according to \(FM\) to reflect the failure \\
08: update the module failure rate Array \(FM\) \\
09: update the Array variable \(sum\) \\
10:**end for** \\
11: \(sumR=sum+sumR\); \\
12:**end for** \\
13: \(conRL=sumR/numS\); \\
**End** \\ \hline \hline \end{tabular}
\end{table}
Table 3: Self-adaptive Monte Carlo simulation flow.  
After further analysis of these differences, we can draw the following conclusions:  
1. When just a few modules fail, there are nearly no differences in the impact of different load-sharing rules on the level of condition reliability, namely the early stage of reliability degradation curve is almost overlapped.
2. With the continuously increase of the failure modules, all the reliability degradation curve will drop significantly but the point at which the curve begins to decrease obviously and the corresponding decreasing rate is greatly affected by the load-sharing rule. The more sensitive to the distance factor the load-sharing rule is, the slower the level reliability is degraded. Accordingly, the curve of the equal rule is the first one to decrease with a sharp rate.
3. In the later stage of the reliability curve, almost all the loads are shared by the few remaining modules, thus the failure rates of each module are quite high and the level reliability is close to 0.