# System Reliability Assessment with Imprecise Probabilities
## 3 System Reliability Analysis Based on p-box and Survival Signature  
The research on the reliability of complex systems is of great significance in many application fields and has attracted extensive attention from scholars. In recent years, various effective system reliability analysis methods have been proposed, such as the survival signature method [31; 32; 33; 34], which can be used to easily estimate the reliability function of systems with multiple component types. It completely separates the information about system structure from that of the components' failure times, which greatly simplifies the calculation of complex systems' reliability. In this section, the theory of survival signature and p-box are applied  
The survival signature of this system can be obtained using Equation (8). The relevant results are shown in Table 5. Since the results of \(\Phi\)(\(l_{1}\),\(l_{2}\),\(l_{3}\),\(l_{4}\)) = 0 have no effect on the reliability of the system, the corresponding rows are omitted. Finally, the lower and upper reliability limit functions of the system can be obtained using Equations (12) and (13). Figure 5 shows the corresponding results. For the purposes of comparison, the precise reliability function of the system is also given in Figure 5.  
It can be seen from Figure 5 that the real system reliability is bounded by the lower and upper reliability limits, and the change trend of the limit functions are basically consistent with the true reliability, while the region between them represents the size of the uncertainty. The quantification of system uncertainty helps in effective uncertainty management and increases confidence in the results. Example 2 (Auxiliary power supply system): In this example, reliability analysis of the auxiliary power supply system for a train is studied based on p-box and survival signature. The reliability block diagram is a complex structure, as shown in Figure 6; the information of each type of component is shown in Table 6. For the exponential distribution, \(\lambda\) represents the failure rate; for the Weibull distribution, \(\alpha\) and \(\beta\) represent the scale and shape parameters, respectively. The failure time data the components of types T2, T4, T6, and T8 are shown in Table 7.  
Figure 5: Reliability of the system. \begin{table}
\begin{tabular}{c c c c c c} \hline
**Type** & **Unit** & **Distribution** & \(\Lambda\) & \(\alpha\) & \(\beta\) \\ \hline T1 & Pantograph subsystem & Exponential & 0.42188 & - & - \\ T2 & Insulation subsystem & Exponential & 0.16262 & - & - \\ T3 & Transformer subsystem & Exponential & 0.30944 & - & - \\ T4 & Taction converter subsystem 1 & Weibull & - & 15.0468 & 7.5118 \\ T5 & Taction converter subsystem 2 & Exponential & 0.14949 & - & - \\ T6 & Cable subsystem & Exponential & 0.30610 & - & - \\ T7 & Single auxiliary converter & Weibull & - & 3.8501 & 2.4975 \\ T8 & Power bus & Exponential & 0.06572 & - & - \\ \hline \end{tabular}
\end{table} 
Table 6: Information about subsystems or components. \begin{table}
\begin{tabular}{c c} \hline
**Component Type** & **Failure Data** \\ \hline T2 & 0.0349, 0.5847, 1.4434, 2.7405, 3.1757, 3.6475, 3.6797, 5.3652, 6.1297, \\  & 7.1585, 7.4751, 8.8771, 11.2241, 12.3072, 15.5112 \\ T4 & 9.9450, 12.0991, 12.2727, 12.3634, 12.6926, 13.1705, 13.2748, 13.5465, \\  & 13.9767, 14.0897, 14.7375, 14.8264, 15.8430, 16.0566, 17.4414 \\ T6 & 0.1273, 0.3411, 0.4317, 0.4856, 0.7124, 1.0251, 2.2016, 2.2057, 2.3404, \\  & 2.4718, 2.5091, 2.9371, 3.9053, 6.1177, 9.3392 \\ T8 & 0.3468, 0.6107, 1.3017, 1.8569, 3.2337, 3.2970, 7.0740, 7.8945, 8.7717, \\  & 14.3528, 15.3607, 17.4318, 19.0361, 21.7302, 26.6873 \\ \hline \end{tabular}
\end{table} 
Table 7: Component failure data. Figure 6: Auxiliary power supply system of a train. Following the steps for calculating the p-box, the 95% Bayesian confidence interval of distribution parameters of type T2, T4, T6, and T8 components are listed in Table 8.  
The survival signature of this system can be obtained from Equation (8).