# System Reliability Assessment with Imprecise Probabilities
## 5 Conclusions  
Modern systems are becoming more and more complicated. At the same time, the exact statistical characteristics of some components may be unavailable because of the limited sample information in practical engineering. The lack of information about a true form of life distribution leads to the presence of epistemic uncertainty over the distributional parameters (e.g., mean and variance). Previous analyses have shown that, if epistemic uncertainty is not taken into account during reliability prediction, the results may be seriously biased and fail to give a useful and reliable estimate. Therefore, it is increasingly important to propose a methodology for reliability assessment of complex systems under aleatory and epistemic uncertainties.  
In this paper, an efficient and effective method for reliability analysis of complex systems is proposed based on the theories of p-box and survival signature. Bayesian inference is used to construct the p-boxes and the life distributions of the components with a limited number of samples. The theory of survival signature is applied to calculate the reliability of the system with a mixture of precise and imprecise knowledge of the life distributions of the components. The epistemic uncertainty contained in the system is effectively quantified by using the theory of p-box. Furthermore, reliability analysis of a system using survival signature could separate the system structure from the component probabilistic  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Component Type** & **Confidence Level** & \(\lambda\) & \(\alpha\) & \(\beta\) \\ \hline T2 & 95\% & [0.1384, 0.1940] & - & - \\ T4 & 95\% & - & [14.8610, 15.5449] & 8 \\ T6 & 95\% & [0.2441, 0.3422] & - & - \\ T8 & 95\% & [0.0636, 0.0892] & - & \\ \hline \hline \end{tabular}
\end{table}
Table 8: Bayesian parameter confidence interval.  
Figure 7: Reliability of the system.  
failure distribution. Therefore, the proposed approach is easy to be implemented in practice and has high computational efficiency.  
It should be noted that, because we calculated the p-box based on Bayesian interval estimation in this paper, we need to know the lifetime distribution types of the components. In reality, we may obtain prior information of lifetime distributions of the components from different sources, such as expert experience, failure data, engineering knowledge, etc. How to give a reasonable prior for Bayesian interval estimation using these information sources is a topic of future research for the authors. In general, however, based on p-box and survival signature, this paper presents a new and practical method for system reliability analysis with limited samples, which quantifies the uncertainty in system reliability analysis effectively.  
Methodology, P.D.; Software, Y.L.; Writing--original draft, G.Y.; Writing--review & editing, X.H.  
This research was funded by National Natural Science Foundation of China (grant number 51975110, 51575094, 51875094)  
The authors thank the anonymous reviewers for their valuable comments.  
The authors declare that there is no conflict of interests regarding the publication of this paper.  
The data used to support the findings of this study are included within the article.