## 4 Case Studies Based on Real Data  
### Numerical Results  
We have collected the data of Nanjing South railway station in China's high-speed railway, from 1 March 2024 to 31 May 2024, as an example. These data are used to estimate parameters in the following texts. According to the actual situation, a typical platform has a length of 450 m and two exits. The length parameters of each segment are as follows: \(l_{AB}=120\) m, \(l_{CB}=120\) m, and \(l_{B}=210\) m. Due to the complexity of the investment plan and approval involved in the renovation of the platform, the width of the exit will remain unchanged at 10 m in the short term. Thus, the entire platform width is \(\lambda=10\) m and the length of the exit is \(\phi=20\) m. The current exit width is 6 m.  
Passenger flow sample data from 1 March 2024 to 31 May 2024 show that the number of disembarking passengers for trains stopping at track 1 follows a normal distribution \(N(1283,97)\), and for track 2, it also follows a normal distribution \(N(1103,81)\), as shown in Table 2. The risk probability is set as 10%. All these are shown in Tables 2 and 3. According to Figure 1, since the width, length, and positions of the two exits on the platform are symmetrically distributed, the path utility obtained by passengers facing multiple paths between the same adjacent node pair is equal. Therefore, the assigned passenger flow for all paths and links is shown in Tables 4 and 5. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Passenger Flow \(q_{rs}^{T}\)**} & \multicolumn{6}{c}{**Node Pair (\(r\),s)**} \\ \cline{2-7}  & \(A_{1}A\) & \(A_{2}A\) & \(B_{1}B\) & \(B_{2}B\) & \(C_{1}C\) \\ \hline \multirow{2}{*}{Period (\(T\))} & 1 & 384 & 329 & 673 & 576 & 384 \\  & 2 & / & / & / & / & / \\ \hline \multirow{2}{*}{**Passenger Flow \(q_{rs}^{T}\)**} & \multicolumn{6}{c}{**Node Pair (\(r\),s)**} \\ \cline{2-7}  & \(AB\) & \(CB\) & \(BD\) & \(BE\) & \(C_{2}C\) \\ \hline \multirow{2}{*}{Period (\(T\))} & 1 & 714 & 714 & 625 & 625 & 329 \\  & 2 & / & / & 714 & 714 & / \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Passenger flow of node pairs (intermediate results under passenger flow assignment). \begin{table}
\begin{tabular}{c c c} \hline \hline \multicolumn{1}{c}{**Track (\(i\))**} & **1** & **2** \\ \hline Mean (\(\mu\))/Standard & 1283/97 & 1103/81 \\ deviation (\(\delta\)) & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Mean and standard deviation of the number of passengers getting off the trains (calculated from sample data). \begin{table}
\begin{tabular}{c c c c} \hline \hline \multicolumn{1}{c}{**Passenger Flow \(f\)(\(p\),\(T\))**} & \multicolumn{6}{c}{**Link (\(p\))**} \\ \cline{2-5}  & **1** & **2** & **3** & **4** & **5** & **6** \\ \hline \multirow{2}{*}{Period (\(T\))} & 1 & 384 & 329 & 357 & 357 & 384 & 329 \\  & 2 & / & / & / & / & / \\ \hline \multirow{2}{*}{**Passenger Flow \(f\)(\(p\),\(T\))**} & \multicolumn{6}{c}{**Link (\(p\))**} \\ \cline{2-5}  & **7** & **8** & **9** & **10** & **11** & **12** \\ \hline \multirow{2}{*}{Period (\(T\))} & 1 & 357 & 357 & 673 & 576 & 625 & 625 \\  & 2 & / & / & / & / & 714 & 714 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Passenger flow of links with different periods (intermediate results under passenger flow assignment). Based on actual operational experience, a passenger's speed is limited when carrying luggage, with a maximum flow speed of \(v_{f}=1\) m/s. The minimum area occupied by a moving passenger is 0.5 square meters, i.e., \(\overset{\sim}{g}_{p}^{T}=0.5,\forall p,T\). All instances of congestion resulting from passengers failing to pass though the links during the maximum allowable travel time are taken as the scenario objects to sample the average occupied area data. Then, the average of all samples is calculated to fit linear functions at sample point \(t_{k}\).