# Investigation of Railway Network Capacity by Means of Dynamic Flows
## 4 Exemplary Application to a Sample Network
### Calculation of the Input Parameters  
The necessary input parameters for the optimization model are determined with LUKS(r) [28], which is a software tool for microscopic railway operation studies, that, in addition to the analytical capacity investigation of nodes and lines, also enables the simulation of timetable creation and operational processes. The input parameters include the travel times of the different route numbers listed in Table 2, the average minimum headway times between two trains on each line section and the hourly capacity limits for each line section. The minimum headway times as well as the capacity limits of the line sections thereby depend on the train mix. Since the scenario considered in the present study assumes a fixed set of trains that is filled up with freight trains only (cf. Section 4.1), the train mix in the optimization process changes in each iteration (cf. Section 4.3 for a detailed description of the optimization process). An important note in this context is therefore that the calculation of the input parameters is based on the train mix that results from the initially specified operating program through the set \(K_{sch}\) (cf. Table 3). Furthermore, since the time resolution here is set to minute steps, all parameters must be rounded to full minutes. Travel times are calculated for every arc of the core network and each route number running on it. All dynamic vehicle properties are considered. If the resulting travel times exceed full minutes by more than six seconds (0.1 min), they are rounded up to the next full minute. Otherwise, they are rounded down. This approach ensures that the network-wide capacity is rather underestimated than overestimated in the end. Stops for passenger trains are defined in each station of the core network so that the calculated travel times correspond to realistic values for the modeling. No stops are planned for freight trains, but the optimization model allows them to stop in stations if they should be overtaken by passenger trains (for capacity reasons). To consider the time that is necessary for the start-up process and brake application in case of such stops, the travel times of freight trains on each arc of the core network are blanketly increased by one minute. The use of headway-specific minimum headway times would quickly increase the complexity of the model and thus the runtime. Therefore, average minimum headway times based on the initial train mix (cf. Table 3) are calculated for each arc of the core network. The average minimum headway times are rounded in the same way as the travel times. The minimum buffer times are set to one minute for all arcs of the core network in accordance with DB regulations [29]. \begin{table}
\begin{tabular}{l l l l} \hline \hline
**Arc (\(i\),\(j\))\(\in\)\(A_{c}\)** & \begin{tabular}{l} **Total Number of** \\ **Trains per Hour** \\ \end{tabular} & \
\begin{tabular}{l} **Number of Local** \\ **Passenger Trains** \\ \end{tabular} &
\begin{tabular}{l} **Number of Freight** \\ **Trains** \\ \end{tabular} \\ \hline (1,2), (2,1) & 4 & 3 & 1 \\ \hline (2,3), (3,2) & 3 & 3 & 0 \\ \hline (3,4), (4,3) & 4 & 3 & 1 \\ \hline (4,5), (5,4) & 2 & 1 & 1 \\ \hline (5,6), (6,5) & 2 & 2 & 0 \\ \hline (6,7), (7,6) & 2 & 2 & 0 \\ \hline (7,8), (8,7) & 3 & 2 & 1 \\ \hline (8,9), (9,8) & 3 & 2 & 1 \\ \hline (9,10), (10,9) & 7 & 5 & 2 \\ \hline (10,5), (5,10) & 5 & 5 & 0 \\ \hline (10,1), (1,10) & 6 & 4 & 2 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Initial train numbers on the arcs of the network shown in Figure 3. These numbers define the train mix used to calculate the input parameters for the optimization model. The line capacities are determined using the STRELE-formula, which is integrated in the LUKS(r) software tool, assuming an optimal operational quality according to the standards applied by DB in Germany. The calculation requires, among others, the definition of delay characteristics for the different route numbers that are assumed to take the standard values defined by DB for highly loaded lines.