7.2 Framework of the algorithm
7.3.4 The Central station as bottleneck
In Table 7.6, there is a remarkable peak in the amount of propagated delays due to conflicts in the Central station. The delay scenario that is used to obtain these results specifies that half of the trains get a primary delay during their dwell action in the Central station. This delay scenario is selected based on the observation that, in practice, the planned dwell time of 1 minute is nearly always insufficient. In the simulation, all but one delay scenario use fixed dwell delays. Since a comparison of Tables 7.2 and 7.3 indicates that the difference of using fixed versus stochastic dwell delays is small, it is decided to work with fixed dwell delays.
Although adding these dwell delays gives a more realistic delay scenario, one can wonder how the results change if there are no dwell delays at the Central station or how the system behaves in case of dwell delays at other stations.
24To obtain the corresponding factors, the obtained value for the reference situation is divided by that of the final system. For example, the ratio of the value for Robstdev of 0.093
Therefore, the simulation results of three extra delay scenarios are presented in Tables 7.9-7.11. In Tables 7.9 and 7.10, no initial dwell delays are inserted in the system and half, respectively, three-quarters of the trains are delayed upon arrival. These tables are the counterpart of, respectively, Tables 7.2 and 7.4 in which the same number of trains gets an arrival delay, but also half of the trains are delayed during their stop in the Central station. The delay scenario of Table 7.11 is obtained from that of Table 7.9 by adding extended dwell times in all stations.
A comparison of the results with or without dwell delays gives that the (relative) improvement of the algorithm (column final versus reference) is rather stable and independent of the dwell delays. Without the initial disturbances in the Central station, the size of the total delays is smaller. If less trains are delayed from the start, the increase of the percentage of newly delayed trains is expected. To obtain Table 7.11, more primary delays are inserted in the system. As a consequence, the total amount of train delays is higher than in Tables 7.2 or 7.9. However, where the increase in train delays and in knock-on delays of adding dwell delays in the Central station are equal to about 18%25, adding delays
in the outer stations increases the train delays with more than 27% while the amount of knock-on delays only increased with 9%. This means that the impact of conflicts in stations Midi and North is smaller than in the Central station26.
Since all trains pass the three stations, this result confirms the logic that the Central station is the bottleneck station.
The same conclusion is obtained by looking at the delay propagation in the stations or on the grids. Where Table 7.6 contains the results for the delay scenario with dwell delays at the Central station only, no dwell delays are used to get Table 7.12, and in Table 7.13, there are primary delays at all stations. For Tables 7.6 and 7.12, all numbers, except for the ones in column Central, are more or less equal. In column Central, the impact of (leaving out) the primary dwell delays becomes clear. Although the amount of knock-on delays in the Central station decreases, it remains a lot higher than that of the other columns. Adding delays in the outer stations makes the system more vulnerable to delay propagation, but the increase is considerably smaller than that of the Central station.
25From 137 minutes of train delays for the reference system in Table 7.9 to 162 minutes in Table 7.2 is an increase of (162 − 137)/137 = 18%. Also the difference in knock-on delays equals 18%, (35.2 − 29.8)/29.8 = 18%. For the results in column final, more or less the same results are found.
26By adding delays in the two outer stations, more initial delays are inserted than by adding delays in the Central station. This explains the larger increase of train delays. Since the increase in knock-on delays is smaller, less conflicts occur or the conflicts that occur have a smaller impact. Next, where the increase of both measures was proportional for delays in the Central station, more initial delays need to be inserted in the outer stations to get the same increase in propagated delays as obtained from delays in the Central station.
RESULTS 135
Table 7.9: Simulation results for the full algorithm on the NSC case study for delay scenario E|T |/2, 0, 0, 0. The results in this table can be compared with
those in Tables 7.2 and 7.3 to find the impact of external dwell delays in the Central station.
reference routing timetabling final spreading cost (%) 100 76.3 34.7 28.1
Rob1(%) 100 99.5 97.6 96.9
Rob2(%) 100 101.2 105.4 106.9
Robstdev(min) 0.843 0.843 0.832 0.809
pax delays (min) 1.54 (0.281) 1.52 (0.281) 1.45 (0.277) 1.43 (0.270)
train delays (min) 137 (22.7) 135 (22.6) 125 (21.3) 123 (20.7)
knock-on (min) 29.8 (9.1) 27.4 (8.9) 18.0 (6.5) 15.3 (5.7)
newly delayed (%) 14.18 (3.67) 12.53 (3.55) 8.42 (2.93) 7.15 (2.68)
extra delayed (%) 30.5 (5.53) 27.0 (5.33) 19.1 (4.65) 15.9 (4.17)
worst case (min) 267 261 244 241
Table 7.10: Simulation results for the full algorithm on the NSC case study for delay scenario E3|T |/4, 0, 0, 0. This delay scenario is the same as in Table 7.4,
except for the dwell delays in the Central station.
reference routing timetabling final spreading cost (%) 100 76.3 34.7 28.1
Rob1(%) 100 99.5 97.7 97.0
Rob2(%) 100 100.9 104.3 105.6
Robstdev(min) 0.976 0.969 0.965 0.941
pax delays (min) 2.23 (0.325) 2.21 (0.323) 2.14 (0.322) 2.11 (0.314)
train delays (min) 197 (26.2) 194 (25.9) 183 (25.0) 180 (24.3)
knock-on (min) 36.3 (9.6) 33.4 (9.3) 22.4 (7.1) 19.2 (6.3)
newly delayed (%) 9.85 (2.80) 8.80 (2.74) 6.45 (2.43) 5.54 (2.31)
extra delayed (%) 35.6 (5.36) 31.8 (5.25) 23.5 (4.78) 19.6 (4.40)
Table 7.11: Simulation results for the full algorithm on the NSC case study for delay scenarioE|T |/2, P
(0.5) |T |/2, P (0.5) |T |/2, P (0.5) |T |/2
. To obtain these results, the same number of trains are delayed upon arrival as in the delay scenarios of Tables 7.2, 7.3, and 7.9. The differences come from the inserted dwell delays.
reference routing timetabling final spreading cost (%) 100 76.3 34.7 28.1
Rob1(%) 100 99.4 97.2 96.4
Rob2(%) 100 101.1 105.6 107.1
Robstdev(min) 0.873 0.865 0.845 0.829
pax delays (min) 1.94 (0.291) 1.92 (0.288) 1.84 (0.282) 1.81 (0.276)
train delays (min) 207 (24.2) 204 (23.9) 192 (22.4) 189 (21.7)
knock-on (min) 38.4 (11.0) 35.6 (10.7) 23.2 (7.9) 20.1 (7.1)
newly delayed (%) 2.36 (1.63) 2.11 (1.55) 1.43 (1.28) 1.31 (1.24)
extra delayed (%) 35.4 (5.62) 31.8 (5.56) 22.5 (5.07) 19.1 (4.60)
worst case (min) 351 351 322 309
Table 7.12: The amount of propagated delays (in minutes) in the stations and on the grids for the final solution for delay scenario E|T |/2, 0, 0, 0.
North grid NC Central grid CM Midi reference 7.6 4.8 10.3 4.0 3.0
routing 7.5 3.5 10.3 3.1 2.9
timetabling 4.7 2.2 7.2 2.0 1.8
final 4.9 1.7 6.3 1.6 0.9
Table 7.13: The amount of propagated delays (in minutes) in the stations and on the grids for the final solution for delay scenarioE|T |/2, P
(0.5) |T |/2, P (0.5) |T |/2, P (0.5) |T |/2 . North grid NC Central grid CM Midi
reference 9.4 5.2 15.8 4.3 3.7
routing 9.2 3.8 15.8 3.3 3.6
timetabling 5.8 2.3 11.0 2.0 2.1
RESULTS 137