## 4 Evaluation of Safety Indicators  
The braking distance of the train braking system can be regarded as the safety indicator of the train. Our goal is to couple the braking distance with the failing state and the information on preventive maintenance of the braking system and to give the corresponding mathematical expressions. The train braking distance \(d_{b}\) consists of the train braking lag distance \(d_{k}\) and the effective braking distance \(d_{e}\).  
\[d_{b}=d_{k}+d_{e} \tag{13}\]  
The braking lag distance \(d_{k}\) depends on the train braking lag time \(t_{k}\) and the initial braking speed \(v_{0}\), and the train travels at a uniform speed during the lag braking process.  
\[d_{k}=\frac{t_{k}\mathbf{\bullet}v_{0}}{3.6} \tag{14}\]  
\(v_{1}\) is the initial velocity before braking and \(v_{2}\) is the final velocity after braking. The effective braking distance \(d_{e}\) of the train is the deceleration distance after the brake is activated, which is given by the following formula [38]:  
\[d_{e}=\sum\frac{4.17(v_{1}^{2}-v_{2}^{2})}{\frac{1000\beta\cdot B}{G\cdot g}+w _{0}+i_{j}} \tag{15}\]  
In the model of this paper, it can be assumed that the subway travels at a horizontal height with a constant maximum operating speed \(v_{0}\), which means that the slope is 0. Therefore, the equation can be abbreviated as:  
\[d_{e}=\frac{4.17v_{0}^{2}}{\frac{1000\beta\cdot B}{G\cdot g}+w_{0}} \tag{16}\]  
The operational risk of the train braking system is related to the braking distance. When the braking distance of the train exceeds the given safe braking distance, the braking system can be considered as having a safety risk at this point. The braking force of electric braking equipment is provided by multiple motors, and the multiple motors can be viewed as a K-out-of-N system. Each motor provides the same braking output \(B_{m}\). If the system has \(n(t)\) motors working properly in the current state, the total braking force is related to the output of a single motor and the number of brake motors in some function.  
\[B=f(B_{m},n(t)) \tag{17}\]  
In this paper, it is assumed that they satisfy a linear relationship, then the braking distance of the system at time \(t\) is:  
\[d_{e}(t)=\frac{4.17v_{0}^{2}}{\frac{1000\beta\cdot B_{m}\cdot n(t)}{G\cdot g}+ w_{0}} \tag{18}\]Then, if the safe braking distance \(d_{s}\) is to be met, the minimum number of motors that are able to work properly is:  
\[k_{b}\geq\frac{\frac{4.17v_{0}^{2}}{d_{s}-d_{b}}-w_{0}}{1000\beta\cdot B_{m}}.G \cdot g \tag{19}\]  
Then, the safety index of the K-out-of-N system at time t can be expressed as:  
\[P_{s}(t)=P(n(t)-k_{b}\geq 0) \tag{20}\]  
In the previous section, we have obtained the state probability \(\overset{\rightarrow}{P_{t}}\) coupled with fault information and two kinds of preventive maintenance information, where \(P_{t}(i)\) is obtained from Equations (9) and (12). Then, the mean train braking distance will be able to be expressed as follows:  
\[d_{e}(t)=\sum_{i=1}^{n}P_{t}(i)*d_{ei} \tag{21}\]  
The braking distance in each state \(i\) can be expressed:  
\[d_{ei}=\frac{\frac{4.17v_{0}^{2}}{1000\beta\cdot B_{m}(n-i+1)}+w_{0}}{C\cdot g }+w_{0} \tag{22}\]  
Therefore, the braking distance under a certain moment is related to the braking system state probability. Additionally, the state probability of the braking system is related to the failure rate and repair rate of the motor. The average value of the braking distance during the overhaul interval is expressed as follows:  
\[E(d_{e}(t))=\frac{1}{T}\cdot\sum_{t=1}^{T}\sum_{i=1}^{n}pt(i)*d_{ei} \tag{23}\]