# Real-Time Adjustment Method for Metro Systems with Train Delays Based on Improved Q-Learning
## 3 Model Construction  
### Model Notations and Assumptions  
In this section, we will provide the mathematical description of the TTR problem. Table 2 presents the list of symbols to be used. The metro line studied in this paper involves regular bi-directional operations, including both upstream and downstream directions with turna \begin{table}
\begin{tabular}{c l} \hline \hline
**Notations** & **Definition** \\ \hline Set & \\ \(K\) & Set of train services. \(K=\{1,2,\ldots,|K|\}\). \(|K|\) is the maximum index of train services \\ \(I\) & Set of stations, including physical and virtual stations. \(I=\{1,2,\ldots,2|I|\}\). \(2|I|\) is the end index of stations. \\ \(I^{\prime}\) & Set of physical stations. \\ \(M\) & Set of block running time levels. \(M=\{1,2,\ldots,|M|\}\). \(|M|\) is the maximum index of block running time levels. \\ \(k\) & Index of train service, \(\forall k\in K\) \\ \(i\) & Index of station, \(\forall i\in I\) \\ \(m\) & Index of block running time level, \(\forall m\in M\) \\ Input Parameter & \\ \(G_{k,l}^{arr}\) & Scheduled arrival time of train service \(k\) at physical station \(i\), \(i\in I^{\prime}\) \\ \(G_{k,i}^{dep}\) & Scheduled departure time of train service \(k\) at physical station \(i\), \(i\in I^{\prime}\) \\ \(\lambda_{k,i}^{in}\) & The running time of train service \(k\) using block running time level \(m\) from station \(i\) to \(i+1\) \\ \(\mu_{k,i}^{in}\) & The Boolean variable to decide whether block running time level \(m\) from station \(i\) to \(i+1\) for train service \(k\) \\ \(U_{\min}\) & Minimum train headway on the truck line \\ \(B_{k}\) & The block where train service \(k\) locates \\ \(S_{\min}\) & Minimum station dwelling time on the truck line \\ \(t_{\text{open}}\) & Time for door opening after train’s arrival \\ \(t_{\text{close}}\) & Time for door closing before train’s departure \\ \(t_{\text{min}}\) & Minimum boarding and alighting time for passengers \\ \(n_{\text{max}}\) & Maximum station passenger capacity \\ \(n_{\text{aboard}}\) & Number of passengers entering the station \\ \(T_{\text{turn}}^{\min}\) & Minimum turnaround time at the turnaround station \\ Intermediate variables & \\ \(\overline{\sigma}_{k,i}^{dir}\) & Rescheduled arrival time of train service \(k\) at station \(i\) \\ \(\overline{\sigma}_{k,i}^{dep}\) & Rescheduled departure time of train service \(k\) at station \(i\) \\ \(t_{\text{on}}\) & Boarding time for passengers \\ \(t_{\text{off}}\) & Alighting time for passengers \\ \(t_{\text{add}}\) & Additional boarding and alighting time for passengers \\ \(D_{k,i}^{delay}\) & Delay time of train service \(k\)at physical station \(i\), \(i\in I^{\prime}\) \\ \(G_{\text{time}}\) & Current checking time \\ \(G_{k,i}^{all}\) & Additional dwelling time of train service \(k\) at physical station \(i\), \(i\in I^{\prime}\) \\ \(a_{i}^{random}(k)\) & The random action for train service \(k\) at physical station \(i\), \(i\in I^{\prime}\) \\ \(\dot{a}_{i}^{best}(k)\) & The optimal action for train service \(k\)at physical station \(i\), \(i\in I^{\prime}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Notations and parameters used in the formulation. in Figure 3. In the figure, the set of stations is represented by \(I\), while the set of physical stations is denoted by \(I^{\prime}\). Solid circles represent physical stations, while hollow circles represent virtual stations. The stations between station \(|I|\) and \(|I|+1\), as well as between station \(2|I|\) and \(1\), serve as turnaround stations. Under normal circumstances, metro trains operate in CBTC (communication-based train control) mode. However, the metro signal system utilizes a fixed-block system under emergency situations, as illustrated in Figure 4, to maintain the safe spacing of trains and ensure route protection. Different colors in the metro blocks represent the positions of trains, with red indicating occupied blocks that require additional stops, and yellow indicating slow blocks to ensure the safety of the preceding train.