# Reliability Analysis of Vehicle Braking System Based on Hyperellipsoidal Dynamic Bayesian Network
## 2 Basic Theory of Reliability Analysis  
### Dynamic Bayesian Networks  
In evaluating the dependability of dynamic or intricate systems, Bayesian Networks (BN), as static models, may exhibit certain constraints [4]. In contrast, a Dynamic Bayesian Network (DBN) is better able to capture the dynamic behavior of a system with greater accuracy. When forecasting future reliability, DBN nodes must be configured due to the temporal characteristics involved. A DBN is an extension of BN that is capable of representing dynamic systems by interrelating variables between neighboring time steps [5]. In particular, a DBN comprises multiple instances of the same variable, each denoting a distinct temporal state of the variable. As a robust tool for representing dynamic systems, the DBN has been extensively employed in the domain of reliability prediction. In the construction of a first-order dynamic Bayesian network, the Markov assumption and the chi-squaredness assumption are typically adhered to. This entails that the state at the current moment is solely related to the state at the previous moment, and that the probability of state transfer at any given moment remains constant. This form of DBN is frequently designated as 2 TBN (2 Time-slice Temporal Bayesian Network) [6]. Consequently, a DBNmodel is defined as a pair (B1, B-), where B1 represents a time-slice of the BN model with a priori probabilities defined, and B- represents a 2 TBN, defined through a table of state transfer probabilities, as illustrated in Equation (1) [7]. The three-parameter Weibull function is a continuous probability distribution with a probability density function:  
\[P(X_{t}|X_{t-1})=\prod_{i=1}^{N}P(X_{t}^{i}\Big{|}Pa(X_{t}^{i})) \tag{1}\]  
where \(X_{t}^{i}\) refers to the \(i\)th node at the moment \(t\)(\(i\) = 1, 2, \(\ldots\), \(N\)), the \(i\)th node and \(Pa(X_{t}^{i})\) refer to the parent node of \(X_{t}^{i}\). By expanding 2 TBN to \(T\) time slices, the joint distribution probability is obtained as the following:  
\[P(X_{1-T})=\prod_{t=1}^{T}\prod_{i=1}^{N}P(X_{t}^{i}\Big{|}Pa(X_{t}^{i})) \tag{2}\]  
where \(T\) is the number of time slices and \(P(X_{1-T}))\) is the joint distribution probability from time slice 1 to \(T\). This paper examines the application of dynamic Bayesian networks to the reliability analysis of multi-state systems, with a particular emphasis on the definition of root node fault states and their evolutionary properties over operational time [8]. Specifically, the theoretical framework of the T-S multistate fault tree is integrated to categorize the fault states of the root node into three levels: normal operating state (N), minor fault state (M), and severe fault state (S). This categorization provides a clear description of the diversity and complexity of fault modes and establishes a robust foundation for subsequent fault prediction and management [9]. In developing the DBN model to simulate the dynamic alterations in equipment status throughout the course of vehicle operation, this paper makes the following pivotal assumptions, which are based on an analysis of the existing literature [7; 8; 9]:  
1. Random transitions of basic event states: It is assumed that the basic events in the system (i.e., the root nodes of the DBN) have the potential to transition randomly to a severe failure state during the operation cycle. This assumption reflects the objective law of natural degradation of equipment performance over time in the actual operating environment, emphasizing the randomness and unpredictability of fault escalation. 2. Instant observability: The current degradation state of the equipment can be observed with precision and expediency through a series of system parameters, with the associated time cost deemed to be insignificant. This assumption guarantees the punctuality and efficacy of condition monitoring and offers the opportunity to implement timely maintenance procedures. 3.