# Reliability Analysis of Vehicle Braking System Based on Hyperellipsoidal Dynamic Bayesian Network
## 3 Reliability Analysis Model of Braking System
### Hyper-Ellipsoidal Dynamic Bayesian Network Modeling of the Braking System  
There are two different repair strategies for equipment that has failed. First, if the equipment is in a severe failure state (S), the equipment is repaired through replacement, and usually the repaired equipment will return directly to its normal operating condition (N). This maintenance model has been widely adopted and is hereafter referred to as \"with maintenance\". In addition, this paper presents a contrasting maintenance scenario called \"no maintenance\". In the no-maintenance condition, once the equipment enters a fault state, no repair operation is performed on it, whether it is a minor fault (M) or a Severe fault (S). This means that the faulty condition of the equipment may persist and even worsen and escalate in the subsequent time period. By setting up this scenario, the goal is to deeply analyze and compare the reliability performance of equipment components with and without repair strategies [7]. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Rule 1** & \(\mathbf{y_{6}}\) & \(\mathbf{y_{7}}\) & \multicolumn{3}{c}{\(\mathbf{y_{2}}\)} \\  & & \(\mathbf{y_{2}^{b_{j}=1}}\)=\(N\) & \(\mathbf{y_{2}^{b_{j}=2}}\)=\(M\) & \(\mathbf{y_{2}^{b_{j}=3}}\)=\(S\) \\ \hline
1 & N & N & 1 & 0 & 0 \\
2 & N & M & 0.603 & 0.313 & 0.084 \\
3 & N & S & 0.264 & 0.237 & 0.499 \\
4 & M & N & 0.616 & 0.272 & 0.112 \\
5 & M & M & 0.252 & 0.592 & 0.156 \\
6 & M & S & 0.183 & 0.24 & 0.577 \\
7 & S & N & 0.225 & 0.334 & 0.441 \\
8 & S & M & 0.211 & 0.277 & 0.512 \\
9 & S & S & 0.086 & 0.128 & 0.786 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Forms of T-S gate rules for two component states (Result). \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Rule 1** & \(\mathbf{y_{6}}\) & \(\mathbf{y_{7}}\) & \multicolumn{3}{c}{\(\mathbf{y_{2}}\)} \\  & & & \(\mathbf{y_{2}^{b_{j}=1}}\)=\(N\) & \(\mathbf{y_{2}^{b_{j}=2}}\)=\(M\) & \(\mathbf{y_{2}^{b_{j}=3}}\)=\(S\) \\ \hline
1 & N & N & \(P^{l=1 In DBN, node state evolution follows Markov and chi-squaredness assumptions, which provide the basis for time dimension extension. These assumptions imply that state transitions depend only on the current state with constant transfer probability, enabling the construction of directed connections between time segments and the extension from t to (t + \(\Delta\)t). Figure 6 illustrates the multi-state transition graph for the root node, labeled with the key parameters. \(\alpha_{1},~{}\alpha_{2},~{}\alpha_{3}\) are the failure rates, which quantify the probability of the system to deviate from the current state, and \(\mu_{1},~{}\mu_{2}\) are the maintenance rates, which reflect the effectiveness of the system maintenance, i.e., the probability of recovering a state. Based on the assumption that the node failure states follow the exponential distribution of the failure rate, the transfer relationship between the non-maintenance state and the maintenance state in the vehicle operation process can be deduced, and the transfer relationship table shown in Tables 5 and 6 is constructed accordingly. This table elaborates the probability formula for the transfer of each state of the node at time t to each state at time (t + \(\Delta\)t), and its intrinsic logical relationship is clear and explicit [16]. Dynamic Bayesian network can be assigned to the root node in the adjacent time slice of the state transfer probability matrix, based on the failure data of a subway company with nearly 4 years of operation, using the confidence index modified expert survey method combined with fuzzy numbers for processing and analysis, which can be obtained at the bottom of the event failure efficiency possibility interval as Table 7.