# Delay Management in Regional Railway Transport
## 2 Materials and Methods
### Waiting Time Issue
#### 2.2.2 Bindings between Waiting Times, Infrastructure Capacity, and Timetable Stability  
In determining transfer waiting times, it is necessary to consider that the waiting causes partial delay transmission for connecting trains. Such delays are transmitted from one train to another due to trains' mutual interactions [18]. The number of mutual interactions increases with the number of trains on the corresponding railway line and with the growing heterogeneity of the timetable [19]. The more interactions there are, and the more heterogeneous a timetable is, the more potential conflicts there will be between trains, with a rise in and potential sources of delay. Considering the number of interactions between trains, partially segregated railways in locations where regional transport is managed separately from long-distance transport (e.g., the German S-Bahn system) can provide an intermediate resolution. However, in the Czech Republic, long-distance transport and regional transport are operated on the same railway lines, resulting in delay transmissions among a large number of mutually interacting trains. Generally, the dependencies pertaining to the number of trains, timetable heterogeneity, the average speed of trains, and timetable stability are described by the capacity balance [20; 21] shown in Figure 1. The perimeter of the quadrangle, which limits the capacity balance, is constant for the railway with unchanging technical parameters. Therefore, an increase in the value of one of the four represented parameters leads to a decrease in the value at least one of the remaining three parameters. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Occupancy Conflict** & **Vehicle Circulation Conflict** & **Transfer Connection Conflict** \\ \hline \multirow{2}{*}{**Basic solution 1**} & One of two trains uses other parts of the infrastructure. & Cancellation of a circulation binding (operational standby necessity) & Cancellation of a transfer binding (connection is lost) \\  & One of the trains will be managed & One of the trains will be managed & The second train will be managed via conflict place later (sequence of train paths cannot be changed) \\ \cline{1-1}
**Basic solution 2** & via conflict place later (sequence of train paths may be changed) & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The basic solution for occupancy conflict, vehicle circulation conflict, and transfer connection conflict [15]. The number and size of reserves and spaces in a timetable determine timetable stability. The reserve means time supplements (i.e., margin times) in technological times, which influence transportation times [18; 19; 21; 22; 23; 24; 25; 26]. Reserves can be in run times, dwell times, transfer times, etc. Reserves in run times arise by using additional time supplemental to the technical run time. These additional time supplements can have either point character or linear character (depending on the run time or the distance). The sum of the technical run time and the time supplement is the regular (scheduled) run time, which is then used for timetable construction. In contrast, spaces (or headway time margins) indicate the empty time windows between individual trains' paths, which limit primary delay transmission from one train to another. The margin time between two train paths can be deleted, and trains can be moved to each other's timetable without influencing either of them [18; 19; 21; 22; 23; 24; 25; 26]. The spaces are not parts of transportation times; rather, they are important margin times, which influence transportation times and quality. They are buffer times in minimum headways, train set reversion times, etc. Although the reserves and spaces reduce infrastructure capacity, they are important in achieving a stable timetable [19; 27; 28; 29]. The stability of a timetable can be assessed through time for the total elimination of an input delay of a certain amount [23; 25; 26]. In the case of periodic (or interval) timetables, this time may be related to the interval/period [25].