## 3 Mathematical Model
### Passenger Dynamic Demand Analysis
#### 3.2.1 Passenger Ticket-purchasing process
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Notations** & **Definition** & **Unit** \\ \hline m & The number of stations on a certain high-speed railway line & stations \\ H & The number of segments on this HSR line; \(\text{H}=\text{m}-1\) & segments \\ h & The segment number is \(1,2,3\dots\text{H}\) & \\ \(\varphi_{\text{h}}\) & A 0-1 variable indicates the occupancy of the current segment—if the & \\  & current segment is occupied, the value is 1; otherwise, the value is 0 & \\  & The OD section from station i to station j on this HSR line, where i & \\ (i,j) & represents the passenger origin station and j represents the passenger & \\  & destination station: & \\  & \(\text{i}=1,2,3\dots\text{m}-1,\text{j}=\text{i}+1\dots\text{m}\) & \\ T\({}_{\text{ij}}\) & A collection of divided pre-sale periods in section (i,j) & \\ t & The tth period of the pre-sale period, \(\text{t}=1,2,3\dots\text{T}_{\text{ij}}\) & \\ t\({}_{\text{ij}}^{\text{k}}\) & The tth period of the pre-sale period in section (i,j) contains k days & \\ \(\overline{\text{P}_{\text{ij}}}\) & The published ticket price of the section (i,j) & CNY \\ P\({}_{\text{ij}}^{\text{n}}\) & The actual ticket price of the section (i,j) & CNY \\ C & The rated passenger capacity of this type of train & passengers \\  & The seat capacity of the train in segment h & \\ C\({}_{\text{h}}\) & (The seat capacity of the train in each segment is equal to the rated & \\  & passenger capacity C of the train, without considering the condition of & \\  & overcrowding and overselling.) & \\ l\({}_{\text{ij}}\) & The mileage of the section (i,j) & km \\ L & The full mileage of the train & km \\ R & The revenue of the train & CNY \\ E\({}_{\text{P}}^{\text{t}}\) & The flexibility coefficient of the price at the tth period of the pre-sale period & \\  & & period \\ X\({}_{\text{ij}}\left(\text{t}_{\text{ij}}^{\text{k}}\right)\) & The stochastic ticket-purchasing demand in section (i,j) at the tth & tickets \\  & period of the pre-sale period at the actual price \(\text{p}_{\text{ij}}^{\text{a}}\) & \\ E\(\left[\text{X}_{\text{ij}}\left(\text{t}_{\text{ij}}^{\text{k}}\right)\right]\) & The expected value of stochastic ticket-purchasing demand in section & \\  & (i,j) at the tth period of the pre-sale period at the actual price \(\text{p}_{\text{ij}}^{\text{a}}\) & \\ d\({}_{\text{ij}}^{\text{t}}\) & The dynamic passenger demand of section (i,j) at the tth period of the & tickets \\  & pre-sale period & \\ N\({}_{\text{ij}}\) & The number of tickets pre-allocated in section (i,j) & tickets \\ M\({}_{\text{ij}}^{\text{T}-1}\) & The passenger flow volume exceeding the number of tickets allocated & passengers \\ \(\delta_{\text{ij}}\) & A 0-1 variable, if M\({}_{\text{ij}}^{\text{T}-1}\) is non-negative, \(\delta_{\text{ij}}=1\); otherwise, \(\delta_{\text{ij}}=0\) & \\  & If the ticket-purchasing demands of passengers are not satisfied in the & \\ \(\theta_{\text{ij}}\) & former T-1 periods of the pre-sale period in section (i,j), the passengers & \\  & will choose to purchase standby tickets with a probability of \(\theta_{\text{ij}}\) & \\ \hline
**Decision** & **Definition** & **Unit** \\
**Variables** & The price at the tth period of the pre-sale period in section (i,j) & CNY \\ \hline P\({}_{\text{ij}}^{\text{t}}\) & The number of tickets allocated at the tth period of the pre-sale period & tickets \\ x\({}_{\text{ij}}^{\text{t}}\) & in section (i,j) & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Notations in this model. Then, \(\{X_{\rm t},t\geq 0\}\) is called a compound non-homogeneous Poisson process.