Simulation results

When the Kaufman-TRP model is solved, the impact on the robustness and other performance indicators of the optimized routing can be simulated. Therefore, the simulation model of Section 3.3.4 is used. The reference system with the input timetable and routing is used for comparison. The simulation output is summarized in Tables 5.7-5.10. The difference between these tables comes from the delay scenarios that are used. According to the notation introduced in Section 3.3.4, the delay scenario of Table 5.7, E|T |/2, 0, P

(0.5) |T |/2, 0



, should be interpreted as follows. The first entry, E|T |/2, indicates that half of the

trains are delayed upon arrival at the network’s boundary and the size of these delays is drawn from the exponential distribution with each train’s average arrival delay as parameter. Next to delays upon arrival, half of the trains face a dwell delay of 30 seconds during their stop at the Central stationP_{|T |/2}(0.5). No external delays are added to the trains in the two outer stations.

One value is equal in all tables, this is the improvement in spreading cost, the objective of the algorithm. Since this value is computed within the algorithm and not in the simulation module, there is no influence of delays. For this criterion, a reduction from 100 to 76.3% is achieved by optimizing the routes of the trains through the network.

The reduced value for Rob1and the increased value for Rob2, which are computed

using (3.4) and (3.5), indicate that the robustness of the optimal routing solution is slightly better than that of the reference system. The standard deviations of the RWTT (Robstdev), the average delays per passenger (pax delays) and the

average delays of all trains (train delays) are, statistically seen, equal for both systems. For the total amount of knock-on delays, a significant reduction in the size of the standard deviation is detected in all delay scenarios. Independent of whether the standard deviations are equal or not, the reduction in delays for

COMPUTATIONAL RESULTS 85

Table 5.7: Simulation results of the optimal routing solution for delay scenario 

E|T |/2, 0, P (0.5) |T |/2, 0



. Next to the spreading cost, the two robustness scores of Section 3.3.3 and the robustness’ standard deviation (Robstdev) are given in

the first rows. Then, for the arrival delay per passenger (pax delays), the total amount of train delays, and the total amount of propagated delays (knock-on), the average values and the standard deviations (between brackets) are given. The last three rows contain the percentages of newly, respectively, extra delayed trains as defined in Section 3.3.4, and the maximal amount of train delays that were found during any iteration of the simulation run (worst case).

reference routing spreading cost (%) 100 76.3

Rob1(%) 100 99.5

Rob2(%) 100 101.1

Robstdev(min) 0.861 0.855

pax delays (min) 1.69 (0.287) 1.67 (0.285) train delays (min) 162 (23.4) 159 (23.2)

knock-on (min) 35.2 (10.1) 32.5 (9.8) newly delayed (%) 8.42 (2.85) 7.54 (2.75)

extra delayed (%) 33.5 (5.68) 30.0 (5.52) worst case (min) 298 289

Table 5.8: Simulation results of the optimal routing solution for delay scenario

E|T |/2, 0, E|T |/2, 0
.

reference routing spreading cost (%) 100 76.3

Rob1(%) 100 99.5

Rob2(%) 100 101.1

Robstdev(min) 0.878 0.872

pax delays (min) 1.70 (0.293) 1.68 (0.291) train delays (min) 164 (24.2) 161 (24.2)

knock-on (min) 37.1 (10.7) 34.5 (10.4) newly delayed (%) 10.18 (3.12) 9.19 (3.02) extra delayed (%) 34.6 (5.56) 31.2 (5.44)

Table 5.9: Simulation results of the optimal routing solution for delay scenario  E3|T |/4, 0, P (0.5) |T |/2, 0  . reference routing spreading cost (%) 100 76.3 Rob1(%) 100 99.5 Rob2(%) 100 100.9

Robstdev(min) 1.001 1.000

pax delays (min) 2.39 (0.334) 2.37 (0.333) train delays (min) 222 (26.8) 220 (26.7)

knock-on (min) 42.2 (10.4) 39.3 (10.2) newly delayed (%) 5.69 (2.28) 5.20 (2.25) extra delayed (%) 38.4 (5.43) 34.8 (5.29)

worst case (min) 369 360

Table 5.10: Simulation results of the optimal routing solution for delay scenario  P_{|T |/2}(0.5 ˆD), 0, P_{|T |/2}(0.5), 0. reference routing spreading cost (%) 100 76.3 Rob1(%) 100 99.7 Rob2(%) 100 100.8

Robstdev(min) 0.168 0.169

pax delays (min) 1.28 (0.056) 1.27 (0.056) train delays (min) 121 (4.31) 119 (4.29)

knock-on (min) 14.7 (2.64) 13.1 (2.59) newly delayed (%) 6.87 (2.36) 6.16 (2.30) extra delayed (%) 23.5 (3.24) 20.9 (3.21)

COMPUTATIONAL RESULTS 87

each of these performance indicators is found to be significant. The last rows in Tables 5.7-5.10 contain the percentages of newly delayed and extra delayed trains and also the maximal amount of train delays that have arisen during one of the iterations of the simulation module (worst case). For the percentages, the standard deviation as well as the average values are significantly smaller for the optimal routing than for the reference system.

The importance of investigating the robustness of routing through the grids

Although the improvements based on only the routing module are statistically significant, we think these improvements are not significant enough for the passengers. Together with the timetabling and platforming modules of the next chapters, the size of these improvements will further increase. Moreover, the fact that the results in Tables 5.7-5.10 are not that spectacular is not unexpected since the only change that is made in comparison with the reference situation is the routing of the trains through the grids. However, if one takes a closer look at the amount of delay propagation within the grids, like is done in Table 5.11, the real impact of the routing module can be seen. The location of the grids are indicated in Figure 5.9. Since trains mainly run straight on in grid CM1, the grids between stations Central and Midi are grouped and

considered as one (grid CM ). Where in the stations, the amount of knock-on delays is more or less stable, a reduction in the amount of propagated delays of 23% to 28% (= 1 − 3.4/4.7) is measured on the grids. This endorses the quotation of the beginning of this chapter that investigating the routing of trains through station areas is highly relevant for improving the punctuality of a railway system.

Another argument to illustrate the importance of investigating the routing of trains is found by comparing the impact of a conflict within a grid (grid conflict) or at a station (station conflict). The limited capacity within the NSC, or more specifically the tunnel with the Central station, is often seen as the weakest link of the network. Nevertheless, the crisscross of routes at the grids makes that the set of hindered trains often is larger due to a conflict in one of the grids than due to an extended stop. The results of Table 5.12 confirm this. This table has the same layout as Tables 5.7-5.10. Instead of any of the delay scenarios of these tables, a single conflict of predetermined size is triggered at the Central station (column Central) or at one of the grids such as indicated in Figure 5.9 (the other columns). For the results in this table, the initial disturbance that is inserted in the system equals 5 minutes. This value is chosen because of the clear differences it gives. For other (smaller) disturbances, the trend is similar.

Table 5.11: The amount of propagated delays (in minutes) in the stations and on the grids of the optimal routing solution for delay scenarioE|T |/2, 0, P

(0.5) |T |/2, 0

 . The locations of the grids are indicated in Figure 5.9.

North grid NC Central grid CM Midi reference 7.3 4.7 16.0 4.1 3.0 routing 7.2 3.4 15.9 3.1 3.0 North Central Midi grid NC grid CM1 grid CM2 grid CM

Figure 5.9: The grids within the NSC.

Compared with the impact of one delayed train in the Central station, the worsening in RWTT and RWTText due to a grid conflict is larger and more

(knock-on) delays arise. What is remarkable, is that the total amount of passengers’ delays12_{is considerably higher when the disturbance happens in one}

of the grids. This comes from the fact that the total number of passengers that are in the train at the moment of the disturbance or will board the disturbed train, is, on average, larger for the grid conflicts than for the conflicts in the Central station. The other numbers in the table show that grid conflicts cause more harmed trains what also leads to considerably more delays in the worst case scenario. A delay in grid CM1 has a smaller effect on the system than in

grid NC or grid CM2. This is because the blocking times of the sector that

corresponds to grid CM1is smaller than that of the other grids and there are

hardly trains that make use of these switches. This also holds for the routing solution. Doing the same exercise as in Table 5.12 for the system with the optimal routing, it can be seen in Table 5.13, that the impact of the grid conflicts reduces. This is expected since the routing module improves the spreading on the grids.

12_{To get relevant numbers, the amount of passengers’ delays is summed over all passengers} instead of averaged like in the previous tables.

COMPUTATIONAL RESULTS 89

Table 5.12: Comparison of the impact of a conflict within a grid or at the Central station for the reference system. Unlike in the previous tables, the passengers’ delays are summed over all passengers to get meaningful numbers.

Central grid NC grid CM1 grid CM2

Rob1(%) 100 100.6 100.3 100.8

Rob2(%) 100 82.5 90.5 76.3

Robstdev(min) 0.093 0.130 0.128 0.151

pax delays (min) 1408 (689) 1654 (972) 1541 (955) 1742 (1123)

train delays (min) 9.7 (3.3) 10.5 (4.0) 9.6 (4.3) 10.8 (5.0)

knock-on (min) 4.7 (3.3) 5.5 (4.0) 4.6 (4.3) 5.8 (5.0)

newly delayed (%) 2.51 (1.78) 3.06 (2.13) 2.71 (2.29) 3.40 (2.64) extra delayed (%) 2.60 (1.84) 3.09 (2.13) 2.77 (2.33) 3.47 (2.68)

worst case (min) 18.7 23.8 28.2 33.5

Table 5.13: Comparison of the impact of a conflict within a grid or at the Central station for the optimal routing solution.

Central grid NC grid CM1 grid CM2

Rob1(%) 100 100.2 100.2 100.4

Rob2(%) 100 92.7 94.5 86.0

Robstdev(min) 0.091 0.125 0.116 0.125

pax delays (min) 1392 (677) 1494 (928) 1468 (861) 1586 (928)

train delays (min) 9.6 (3.3) 9.5 (3.4) 9.1 (3.4) 9.8 (3.8)

knock-on (min) 4.6 (3.3) 4.5 (3.4) 4.1 (3.4) 4.8 (3.8)

newly delayed (%) 2.39 (1.64) 2.46 (1.75) 2.33 (1.78) 2.80 (2.04) extra delayed (%) 2.43 (1.67) 2.48 (1.76) 2.36 (1.81) 2.83 (2.05)