## 3 Mathematical Modeling
### Passenger Flow Dynamic Model  
To illustrate the passenger-station interaction, Figure 5 was plotted. When a train is running between stations, the number of passengers waiting on the platform increases with the number of new passengers arriving until passenger loading actions take place. However, the number of on-board passengers varies as passengers alight and board. Taking into account the time-varying passenger demand, it is critical to divide the time interval \(\Delta t\) to accurately count passenger arrivals. By using the passenger arrival rate \(P_{s,\Delta t}\) at station \(s\) during the time interval \(\Delta t\) and the OD arrival ratio matrix \(O_{sk}\), the number of passengers arriving at station \(s\) going to station \(k\) (\(Q_{i,sk}^{a}\)) between the departures of two consecutive trains can be calculated. \[Q_{i,sk}^{s}=P_{s,\Delta t}\cdot O_{sk}\cdot(t_{i,s}^{d}-t_{i-1,s}^{d}),\forall i \in I\backslash\{1\},s\in S\backslash\{n,2n\},k\geq s+1 \tag{14}\]  
The total number of passengers waiting on station \(s\) toward destination \(k\) (\(Q_{i,sk}^{all,w}\)) is the combination of the stranded passengers of the previous train (\(Q_{i-1,sk}^{l}\)) and the newly arriving passengers \(Q_{i,sk}^{a}\). \[Q_{i,sk}^{all,w}=Q_{i-1,sk}^{l}+Q_{i,sk}^{a},\forall i\in I\backslash\{1\},s \in S\backslash\{n,2n\},k\geq s+1 \tag{15}\]  
Figure 5: Diagram of dynamically varying passenger flows. \[Q_{i,s}^{all,w}=\sum_{k}Q_{i,sk}^{all,w},\forall i\in I,s\in S\backslash\{n,2n\},k \geq s+1 \tag{16}\]  
where \(Q_{i,s}^{all,w}\) is the total number of passengers waiting for train \(i\) at station \(s\). Moreover, according to Assumption 2, the number of passengers who want to board the train \(i\) at station \(s\) toward destination \(k\) is affected by the skip-stop action. The valid number of passengers waiting \(Q_{i,sk}^{uw}\), is described as follows:  
\[Q_{i,sk}^{uw}=Q_{i,sk}^{all,w}A_{i}^{k}A_{i}^{s},\forall i\in I,s\in S \backslash\{n,2n\},k\geq s+1 \tag{17}\]  
\[Q_{i,s}^{uw}=\sum_{k}Q_{i,sk}^{uw},\forall i\in I,s\in S\backslash\{n,2n\},k \geq s+1 \tag{18}\]  
where \(Q_{i,s}^{uw}\) is the total valid number of passengers waiting for train \(i\) at station \(s\). During peak hours, the actual number of passengers boarding the train \(i\) at station \(s\) (\(Q_{i,s}^{board}\)) is related to the number of valid passengers waiting at station \(s\) (\(Q_{i,s}^{uw}\)) and the remaining capacity of train \(i\) at station \(s\) (\(Q_{i,s}^{r}\)). The actual number of passengers alighting at station \(s\) (\(Q_{i,s}^{slight}\)) is the total number of passengers who boarded the train \(i\) at other stations before station \(s\) and want to alight at station \(s\). \[Q_{i,s}^{slight}=\sum_{m}Q_{i,ms}^{board},\forall i\in I,s\in S\backslash\{1,n +1\},m\leq s-1 \tag{19}\]  
\[Q_{i,s}^{r}=C-Q_{i,s-1}^{pv}+Q_{i,s}^{slight},\forall i\in I,\forall s\in S \backslash\{1,n+1\} \tag{20}\]  
\[Q_{i,s}^{pv}=Q_{i,s-1}^{pv}-Q_{i,s}^{slight}+Q_{i,s}^{board},\forall i\in I, \forall s\in S\backslash\{1,n+1\} \tag{21}\]  
\[Q_{i,s}^{board}=\min\{Q_{i,s}^{r},Q_{i,s}^{uw}\},\forall i\in I,\forall s\in S \tag{22}\]  
where \(Q_{i,ms}^{board}\) is the number of passengers successfully boarding train \(i\) from station \(m\) with destination station \(s\). \(Q_{i,s-1}^{pv}\) is the number of on-board passengers of the train \(i\) leaving station \(s\)-1. By the description of the number of passengers alighting in Equation (19), it is obvious that the number of passengers successfully boarding train \(i\) from station \(s\) with destination station \(k\) (\(Q_{i,sk}^{board}\)) must be obtained. Based on assumption 3, all passengers were mixed on the platform to wait for the train \(i\), which is consistent with the actual situation. To simplify the calculations, the number of passengers who successfully board the train \(i\) is proportional to the valid number of passengers waiting.