## 5 Case Studies
### Time-Phased Dynamic Demand Simulation  
Before simulating the non-homogeneous Poisson ticket-purchasing process in each OD section, the ticket-purchasing rates' curve needs to be fitted first. We use the regression analysis method to fit the ticket-purchasing rate of each OD section via MATLAB R2016B software for the primary function, quadratic function, cubic function, linear function, exponential function, and power function. Then, many varieties of curve-fitting functions are obtained. The fitting parameter statistics of the Beijing South-Nanjing South section g\({}_{13}\)(t) are shown in Table 5.  
The SSE is the sum of the squared errors of the corresponding points of the fitting data and the original data. The closer the value is to 0, the better the fitting effect will be. The R-square is the coefficient of determination, which is used to determine the degree of curve regression fitting; the closer the value is to 1, the better the fitting effect will be. The adjusted-square is the adjusted fitting coefficient; the closer the value is to 1, the better the fitting effect will be. The RMSE is based on the average of the predicted value and the original data; the closer the value is to 0, the better the fitting effect will be. Comparing the statistical parameters of each fitting function, the exponential function with a better statistical effect is selected as the fitting function, as shown in Figure 5.  
By fitting the ticket data of the Beijing South-Nanjing South route, we can obtain the ticket-purchasing probability density function during the pre-sale period of this OD section, which is expressed as Equation (21). \[\mathrm{g}_{13}(\mathrm{t})=\left\{ \begin{aligned} & 0.0003803*\mathrm{e}^{(0.4778 *\mathrm{t})}&\mathrm{t}\leq 14\\ & 1-0.0003803*\mathrm{e}^{(0.4778*\mathrm{t})}& \mathrm{t}=15\end{aligned} \right. \tag{21}\] \begin{table}
\begin{tabular}{c c c c} \hline
**OD Section** & **Upper Limit/CNY** & **Actual Price/CNY** & **Lower Limit/CNY** \\ \hline (1,2) & 223 & 211 & 202 \\ (1,3) & 533 & 504 & 481 \\ (1,4) & 662 & 626 & 598 \\ (2,3) & 333 & 315 & 301 \\ (2,4) & 479 & 453 & 433 \\ (3,4) & 162 & 153 & 146 \\ \hline \end{tabular}
\end{table} 
Table 4: Table of upper and lower prices for each OD section. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Function Types**} & \multicolumn{4}{c}{**Statistical Parameters**} \\ \cline{2-5}  & **SSE** & **R-Square** & **Adjusted-Square** & **RMSE** \\ \hline Primary Function & 0.039480 & 0.0169 & 0.5850 & 0.057530 \\ Quadratic Function & 0.000332 & 0.9120 & 0.8959 & 0.028720 \\ Cubic Function & 0.00759 & 0.9926 & 0.9904 & 0.008710 \\ Linear Function & 0.083200 & 0.1928 & 0.4606 & 0.086970 \\ Exponential Function & 0.000380 & 0.9963 & 0.9960 & 0.005626 \\ Power Function & 0.000655 & 0.9936 & 0.9931 & 0.007385 \\ Primary Function & 0.039480 & 0.0169 & 0.5850 & 0.057530 \\ \hline \end{tabular}
\end{table} 
Table 5: Results of ticket-purchasing rates’ curve fitting. The corresponding ticket-purchasing probability distribution function of this OD section is expressed as Equation (22). \[\text{G}_{13}(\text{t})=\left\{ \begin{matrix}\int_{0}^{\text{t}}0.0003803*\text{ e}^{(0.4778*\text{t})}\text{dt}&\text{t}\leq 14\\ 1&\text{t}=15\end{matrix} \right. \tag{22}\]  
Similarly, the g\({}_{12}\)(t) for the Beijing South-Jinan West section, g\({}_{14}\)(t) for the Beijing South-Shanghai Hongqiao section, g\({}_{23}\)(t) for the Jinan West-Nanjing South section, g\({}_{24}\)(t) for the Jinan West-Shanghai Hongqiao section, and g\({}_{34}\)(t) for the Nanjing South-Shanghai Hongqiao section are fitted; then, the ticket-purchasing probability distribution functions for the corresponding OD sections are obtained in turn. According to the above fitted ticket-purchasing rate function g\({}_{\overline{\text{ij}}}(\text{t})\), the intensity function \(\lambda_{\overline{\text{ij}}}(\text{t})\) can be obtained through \(\lambda_{\overline{\text{ij}}}(\text{t})=\text{Q}_{\overline{\text{ij}}} \cdot\text{g}_{\overline{\text{ij}}}(\text{t})\).