## 2 Basic Theories
### Dynamic Bayesian Network  
A dynamic Bayesian network (DBN) is a temporal extension of a static Bayesian network. By combining the static Bayesian network with temporal information, the DBN model's transition between two adjacent time segments can be modeled as follows [34]:  
\[P(X_{t}|X_{t-1})=\prod_{i=1}^{N}P(X_{t,i}|Pa(X_{t,i})) \tag{7}\]  
where \(X_{t}\) and \(X_{t-1}\) denote the nodes in time slice \(t\) and \(t-1\); \(X_{t,i}\) is the \(i\)th node in time slice \(t\); and \(Pa(X_{t,i})\) is the set of parent nodes of the node. In the DBN model, the joint probability distribution of multiple time segments can be calculated by the following Equation (8):  
\[P(X_{1:\text{T}})=\prod_{i=1}^{T}\prod_{i=1}^{N}P(X_{t}^{i}|Pa(X_{t}^{i})) \tag{8}\]  
where \(T\) is the number of time segments; \(P(X_{1:\text{T}})\) is the joint probability distribution of time segments 1:T.  
Multistate Modeling of Dynamic Bayesian Networks  
In order to guarantee reliable vehicle operation, the maintenance department will regularly repair and maintain the vehicle equipment and the vehicle equipment to carry out a comprehensive overhaul and repair the equipment in the existence of faults or potential problems. City metro vehicle equipment mainly takes cycle maintenance and after-the-fact maintenance. Cycle maintenance is divided into equalization repair and frame overhaul, with equalization repair cycles ranging from 1 to 12 months. The frame overhaul is a comprehensive vehicle repair with an active life of 5/10 years. After-the-fact repair is for the vehicle equipment in the main line after failure, affecting the operation of direct repair. A city subway for the vehicle equipment according to the impact of the failure of its fault state is divided, respectively, for the average working state (0), for no impact on the vehicle operation of the minor fault state (0.5), and the vehicle cannot continue to operate in the severe fault state (1). Therefore, the essential event state \(x_{i}^{a_{i}}\), representing the equipment in the T-S fault tree, is divided into three states. The corresponding intermediate and top events \(x_{i}^{a_{i}}\) are also three states (\(k_{i}=\eta_{j}=3\)). Equipment fault states in dynamic Bayesian networks are established as three states through T-S polymorphic fault trees [35]: regular operation (0), minor fault (0.5), and severe fault (1). Furthermore, as the vehicle is in operation, the equipment state may randomly shift to a worse state, i.e., fault escalation. In this paper, the following assumptions are made when DBN is used to analyze the reliability of multistate systems: (1) if the actual events in the system T-S multistate fault tree are considered as the root nodes in DBN, then there are three states of 0, 0.5 and 1 at the root nodes in DBN; (2) the actual events may randomly transition to worse states during operation; (3) the state transfer rate is constant and exponentially distributed; (4) repair of equipment in a severe fault and restoration of equipment to regular operation after repair; (5) the system can also perform planned maintenance to avoid escalation of minor faults. The system is in as-new condition after planned maintenance, i.e., after planned maintenance, the state of the equipment can be repaired from a minor fault state and severe fault state to a regulation operation state (Section 4.4 analyzes the impact of planned maintenance on system reliability). The state transfer process of the root node in DBN is shown in Figure 3. \(\lambda\) is the failure rate in the state transfer process of this node and \(u\) is the maintenance rate in the multistate process of this node. When equipment is being maintained, planned maintenance is often carried out at the same time in order to save resources. Therefore, for the purpose of subsequent calculations, it is assumed that a severe fault to a regular operation and a minor fault to a regular operation have the same maintenance rate \(u\)[14].