## 4 Case Studies Based on Real Data
### Numerical Results
\[g_{p}^{T}(t_{k})=\frac{1}{|N|}\sum_{n\in N}g_{p}^{T}(t_{k},n) \tag{35}\]  
All linear functions are connected to approximate the curve of the change in average occupied area during the maximum allowable travel time \(\theta_{p}^{T}\); the range of exit widths for each link is obtained and shown in Table 7.  
Finally, the reasonable exit width is between 6.43 m and 7.01 m (Table 8). Table 8 shows that the reasonable exit width range is from 6.43 m to 7.01 m, which is 0.43 m larger than the actual exit width of 6 m. The width of 6 m meets the maximum allowable time constraint on relevant paths for passengers traveling from nodes A and C to node B through the bottleneck (6 < 7.01 < 7.67). However, considering the situation where passengers travel from node B to exits D and E (\(p=11,12\)), 6 m still cannot ensure that passengers complete their exit within the maximum allowable travel time during period 1 (\(T=1\)). This demonstrates that the current exit width of 6 m on Chinese high-speed railway \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multicolumn{1}{c}{\multirow{2}{*}{**Maximum Allowable**}} & \multicolumn{5}{c}{**Link (\(p\))**} \\ \cline{2-7}  & **1** & **2** & **3** & **4** & **5** & **6** \\ \hline Period (\(T\)) & 1 & 82 & 75 & 92 & 92 & 88 & 80 \\
2 & / & / & / & / & / & / \\ \hline \multicolumn{1}{c}{\multirow{2}{*}{**Maximum Allowable**}} & \multicolumn{5}{c}{**Link (\(p\))**} \\ \cline{2-7}  & **Travel Time (\(\theta_{p}^{T}\))** & 7 & 8 & 9 & **10** & **11** & **12** \\ \hline Period (\(T\)) & 1 & 99 & 99 & 67 & 62 & 75 & 75 \\
2 & / & / & / & / & 113 & 113 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Maximum allowable travel time for links with different periods (intermediate results). \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{3}{c}{**Link (\(p\))/Period (\(T\))**} \\ \cline{2-5}  & **3/1** & **4/1** & **7/1** & **8/1** \\ \hline
**Range of widths of exit (\(y_{p}^{T}\))** & (0, 7.01) & (0, 7.01) & (0, 7.67) & (0, 7.67) \\ \hline \hline \multicolumn{5}{c}{**Link (\(p\))/Period (\(T\))**} \\ \cline{2-5}  & **11/1** & **12/1** & **11/2** & **12/2** \\ \hline
**Range of widths of exit (\(y_{p}^{T}\))** & (6.43, 10) & (6.43, 10) & (3.96, 10) & (3.96, 10) \\
**(Unit: m)** & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 7: The range of widths of the high-speed railway platform exit (final results). \begin{table}
\begin{tabular}{c c c c} \hline \hline \multicolumn{1}{c}{\multirow{2}{*}{**The exit width range (\(y\))**}} & \multicolumn{2}{c}{**(\(6.43,7.01\))**} \\ \cline{2-5}  & **1** & **2** & **3** & **4** & **5** & **6** \\ \hline Period (\(T\)) & 1 & 82 & 75 & 92 & 92 & 88 & 80 \\
2 & / & / & / & / & / & / \\ \hline \multicolumn{1}{c}{\multirow{2}{*}{**Maximum Allowable**}} & \multicolumn{5}{c}{**Link (\(p\))**} \\ \cline{2-5}  & **Travel Time (\(\theta_{p}^{T}\))** & 7 & **8** & **9** & **10** & **11** & **12** \\ \hline Period (\(T\)) & 1 & 99 & 99 & 67 & 62 & 75 & 75 \\
2 & / & / & / & / & 113 & 113 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: The reasonable width of the high-speed railway platform exit (final results). platforms is insufficient and will inevitably lead to passenger congestion around the area near node B in extreme cases. As the number of congested passengers increases, the risk of falling from the platform at the location is exacerbated. According to our calculation results, the platform exit should be increased from 6 m to 6.43 m at least. This ensures that passengers can successfully leave the platform within the maximum allowable travel time in 90% of cases, and improves the safety of passengers on the platform.