# Subway Station Capacity Maintained by Optimizing a Maintenance Schedule of Key Equipment
## 3 DSSC Model with Equipment Failure
### Expected DSSC Model  
When the key equipment fails, the number passengers served per unit time decreases, and DSSC also declines. In terms of the different functions of equipment at a subway station, there are four types of subsystems: the inbound gate subsystem (IGS), the outbound gate subsystem (OGS), the upward escalator subsystem (UGS) and the downward escalator subsystem (DGS), respectively, regardless of the equipment layout.  
Assume that the reliability of the gate and the escalator is \(R_{1}(t)\) and \(R_{2}(t)\), respectively, and the number of the inbound gates, the outbound gates, the upward escalators, the downward escalators is \(m_{1}\), \(m_{2}\), \(m_{3}\) and \(m_{4}\), respectively. The number of normal gates of the inbound gate subsystem is \(g\), \(g=0,1,2,\ldots,m_{1}\). Therefore, there are \(m_{1}+1\) operating states for IGS, and the probability of each operating state is \(P_{1g}\). Similarly, the numbers of operating states of OGS, UGS and DGS are \(m_{2}+1\), \(m_{3}+1\) and \(m_{4}+1\), respectively, and the corresponding probability are \(P_{2h}\), \(P_{3d}\) and \(P_{4q}\), respectively. The values of \(g\), \(h\), \(d\) and \(q\) are independent of each other, and the number of key equipment working normally at the station is shown in Figure 3.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**The Number of Normal Equipment/\(k\)** & **0** & **1** & \(\ldots\) & **\(n\)** \\ \hline Probability/\(P_{k}\) & \((1-R_{e}(t))^{n}\) & \(C_{n}^{1}*(1-R_{e}(t))^{n-1}*R_{e}(t)\) & \(\ldots\) & \((R_{e}(t))^{n}\) \\ The max capacity of the subsystem/\(C_{k}\) & 0 & \(C_{e}\) & \(\ldots\) & \(n*C_{e}\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Capacity parameters of the subsystem.  
There are \(m\) operation states for subway stations and the corresponding probability and DSSC for each operation state is \(P_{i}\) and \(C_{i}\), respectively. To estimate the DSSC of the station, the corresponding capacity indicator such as ATT is calculated by simulation experiments with AnyLogic and Equation (1) [13]. Assuming that the passengers served by the equipment is shared equally by other facilities which play the same role at the station after it fails. For example, when an upward escalator fails, the passengers served by this escalator previously are shared equally by the other upward escalators and stairs.  
The expected DSSC \(C_{S}\) is defined as the expected value of DSSC considering equipment failure, which is the sum of the product of the probability of each station operation state and the corresponding DSSC. The expected travel time \(T_{S}\) is an evaluation indicator of the expected DSSC, which is the expected value of AAT corresponding to each operation state of station. \(C_{S}\) and \(T_{S}\) are formulated as follows:  
\[\text{m}=\prod_{j=1}^{4}(m_{j}+1) \tag{2}\]  
\[P_{i}=P_{1g}\cdot P_{2h}\cdot P_{3d}\cdot P_{4q} \tag{3}\]  
\[C_{S}=\sum_{i=1}^{n}(C_{i}\cdot P_{i}) \tag{4}\]  
\[T_{S}=\sum_{i=1}^{n}(T_{i}\cdot P_{i}) \tag{5}\]  
where \(C_{S}\) is the expected dynamical capacity of subway station, \(T_{s}\) is the expected travel time, \(n\) is the total number of operation states of subway stations, \(C_{i}\) is the DSSC corresponding to each operation state, \(T_{i}\) is the ATT corresponding to each operation state of station which is an indicator of \(C_{i}\), \(P_{i}\) is the probability corresponding to each operation state of the station.