# Delay Management in Regional Railway Transport
## 2 Materials and Methods
### The Maximum Transfer Waiting Times Principle Based on Modified CPM
All edges of the graph have evaluations \(mtt\), which mean the minimum value of corresponding technological times:  
* \(mtt\big{(}r_{t,i-j}\big{)}\) means the minimum run time of train \(t\) from traffic point \(i\) to traffic point \(j\), increased by the minimum dwell times of this train at intermediate stops between traffic points \(i\) and \(j\);
* \(mtt(dw_{t,i})\) means the minimum dwell time of train \(t\) at traffic point \(i\) (whether due to transportation or traffic reasons);* \(mtt(t_{t-u,i})\) means the minimum transfer time between trains \(t\) and \(u\) at traffic point \(i\);
* \(mtt\big{(}h_{t,i-u,j}\big{)}\) means the minimum headway between train \(t\) at traffic point \(i\) and train \(u\) at traffic point \(j\) (when \(i=j\), it is a minimum station headway, a minimum crossing headway, a minimum departure headway, or a minimum arrival headway; when \(i\
eq j\), it is a minimum track headway);
* \(mtt(c_{t-u,i})\) means the minimum reversion time of train sets between train \(t\) a train \(u\) at traffic point \(i\), or the minimum technological time of another circulation binding (e.g., the time of train sets coupling or decoupling where a part of train set switches to another train). We designed the operational/technological processes graph so that it would reflect transportation and operational interactions between the trains, as established by the timetable, or the interaction between the trains while considering used disposal measures. Assuming that delays caused by waiting for delayed connecting trains from the end of an operational day do not transmit to trains at the beginning of the following operational day, we formed the operational/technological processes graph for one operational day. If this assumption was not applicable, we formed the graph for two operational days in a row, eventually for the number of operational days until the transmitted delays from the first operational day are expected to be eliminated. In the case of the integrated periodic timetable (IPT) [32; 33], it was sufficient to form the operational/technological processes graph only for the number of periods in a row, so that all operations and transportation periodically repeated during interactions between trains in a solved subnet were taken into consideration. The resulting transfer waiting times for the certain combination of line, direction, and traffic point were assigned as the minimum of the calculated transfer waiting times through all vertices of the corresponding combination in the graph. When there were different timetables for different operational days, it was necessary to form a graph for each type of operational day, as each graph is intended for calculating the transfer waiting times for corresponding operational days. In forming the operational/technological processes graph, it was necessary to consider that the interaction of a train passing through a traffic point with another train that is not in time (i.e., it was delayed or ran earlier) can lead to a forced stop of the passing train for traffic reasons and/or to extend some of the minimum technological times. The algorithm for assigning the evaluation of the operational/technological processes graph vertices, and for the following calculation of the train transfer waiting times, is comprised of the following steps:  
1. For all combinations of train \(t\) and traffic point \(i\) for which there are known the maximum transfer waiting times \(w_{IN,t,i}\) (which are the inputs for assigning other trains' transfer waiting times), we pasted the corresponding vertex \(d_{t,i}\) onto set \(I_{w}\). For all elements of set \(I_{w}\), the latest necessary departure time \(l_{IN}(d_{t,i})\) was calculated by Equation (3). \[l_{IN}(d_{t,i})=s(d_{t,i})+w_{IN,t,i}\] (3)
2.