Delay in a Railway Network 129

7.3. Delay in a Railway Network 129

Table 7.4: Information on Lines and Segments in the Railway Network

Line Seg. Orig. Dest. Dep. Run. Stop. Buff. t r

who need to transfer to a train departing from the origin city of a connecting segment (column 3–4). The last column of the table lists the time (in minutes) needed for the

transfer. The listed time must be made available between the arrival of the feeder train to the departure of the connecting train. This time duration can overlap with the stop time if the feeder train arrives beforehand. As an example, from the fifth row of the table we conclude that a train arriving in Deventer from Enschede at segment 51 of line 3 feeds a transfer to the connecting train departing from Deventer to Zwolle at segment 54 of line 2 with a transfer time of 2 minutes.

Several synchronizations are two ways, i.e., both trains are simultaneously a feeder and a connecting train. This is the case for trains travelling on lines 1 and 3 synchro- nizing in Amersfoort (row 1 and 4) and trains travelling on lines 3 and 1 synchronizing in Deventer (row 2 and 7).

The data we described so far specify the schedule that the trains travelling on the networks are expected to meet. In this sense, they are deterministic. In real life, how- ever, the actual departure, running, and stopping times deviate from their scheduled times. If the actual running time of a train in some segment exceeds its scheduled running time, a primary delay occurs. Because of the interdependence of segments

130 Chapter 7. Case Studies

Table 7.5: Synchronization in the Railway Network

Line Segment Line Segment

on the same or different lines, this delay may be passed on and may cause delay in other segments, which is referred to as a secondary delay [MM07]. The purpose of the model is to analyze the propagation of delays by modelling the deviations from the scheduled times as random variables. In the following analysis, we follow closely the method of [MM07].

In the penultimate column of Table 7.4, the minimal running times (in minutes) of all segments in the railway network are listed. To each segment we assign a running- time distribution, which is described by t+T , where t is the segment’s minimal running time and T is a continuous nonnegative random variable describing the actual time needed in addition to the minimal running time. The distributions of the random variables T are parameterized by the values r listed in the last column of the table. For

a segment, the value of r is chosen such that it is constituted by the segment’s buffer time and an additional (around) 10% of the minimal running time. Overall, t+r should be equal to the sum of the expected running time and the expected buffer time for each segment, except for line 1 segment 03, line 2 segment 04, and line 3 segment

03, where the buffer times are shortened to 5 minutes. We can see in the table that there are 10 different values of r. In our models, then, we have 10 random variables T and each describes a basic additional running-time distribution.