Since the RWTT is hard to compute analytically, simulation is used to evaluate the robustness of different railway systems. In this section, a two step simulation model is introduced. First, conflicts and the propagation of delays are simulated using microscopic infrastructure data. Once all information about the exact arrival and departure times of trains, the usage of the supplements and buffers, etc., is known, the second step starts. In the second step, the output is generated. Next to the robustness scores Rob1 and Rob2, some other
performance indicators are computed. This is done by simulating, event by event, the itinerary of each passenger and evaluating the evolution of train delays.
Step 1: microscopic simulation model
This part of the simulation model is used to simulate the train traffic through a station area. As input, detailed infrastructure data and timetable, routing,
and platform information is used. All required data is provided by Infrabel or results from the optimization algorithm. Upon building this simulation model, some assumptions are made to reduce its complexity. As a consequence, the output needs to be interpreted with care. Nevertheless, the model is suitable for evaluating the performance of railway systems because all instances are evaluated using the same assumptions such that a fair comparison is guaranteed. A validation study using a commercial simulation package confirms this for a case study of Chapter 8.
In the discrete event driven simulation model, events are handled synchronously and stochastic influences are represented by input delays for the trains. Two types of delays are considered: delays upon entering the considered area and dwell delays at any of the |S| stations, with S the set of stations. The input delays are denoted with a (1 + |S|)-tuple representing, respectively, the delays upon arrival and the dwell delays at the stations. The size of the delays equals a predetermined value or, similar to Goverde et al. (2001), Jensen et al. (2013), Yuan (2006), and many others, is drawn from the exponential distribution using each train’s real average delays ( ˆD) as parameter. The latter is denoted
with E from exponential, the former with P(size) from predetermined. In both
cases, the number of delayed trains is added as index. Let T (|T |) be the set (number) of trains, then E|T |/2, P
(0.5) 3|T |/4
means that half of the trains are delayed upon arrival with the delays drawn from the exponential distribution and three-quarters of the trains gets fixed dwell delays of 0.5 minutes at the only station in the station area. To represent the daily occurring, small disturbances, only input delays smaller than 15 minutes are allowed. The upper bound is set to 15 minutes since this is the value that is used by Belgium’s main passenger railway operator NMBS/SNCB as threshold for compensations against large recurrent delays5. Moreover, for delays of this size and larger, real-time interventions become more appropriate.
Trains enter the system at their inbound line or at the platform of departure in case of a reutilization. The trains travel from signal to signal at a predetermined speed that equals the real allowed maxima within the station area. It is assumed that the time lost by slowing down or speeding up can be approximated by a constant and only affects the travel time through the first (last) block section after (before) a stop. The size of this constant is based on the difference between the expected travel time when traveling at maximum allowed speed and the scheduled travel time for that type of train through the corresponding sections. Note that the type of rolling stock, and thus the specific acceleration characteristics, is approximated by the type of train.
5Source: http://www.belgianrail.be/en/customer-service/compensation-for-delays.aspx, consulted in September 2014.
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Next to the assumptions about the speed profiles, some simplifications concerning the blocking times are made. In Pachl (2008), the intervals that are part of the total blocking time are being described. In the simulation model, however, three different blocking time subintervals are distinguished: the travel time through the section, the clearing time, and thirdly, an interval of constant size representing, among others, the signal processing. The first one is based on the ratio between the length of the block section(s) and the speed of the train together with the penalties for speeding up or slowing down (if applicable). The clearing time consists of the time needed for the tail of the train to leave the block section and is a function of the length of the train and its speed. The last subinterval captures the time needed for setting the signals, aligning the switches, and the time needed to release the section after the passage of a train. The minimum headway time between two trains on the same inbound line is set to three minutes, which is a commonly used threshold for trains on a common line.
Events are handled chronologically with time steps of 6 seconds. Conflicts are not predicted in advance but detected when a train approaches an already reserved block section. The conflicts are solved one by one on a first-come, first-serve basis meaning that, once detected, a conflict is solved immediately by postponing the next event of the approaching train until the estimated time the corresponding block section becomes available again. If multiple events become active simultaneously on a shared resource, extra priority rules, which are derived from practice, apply. This way, high speed trains get priority on local commuter trains and punctual trains may precede slightly delayed trains in the event list. If applicable, the number of trains within the bottleneck area is restricted by prioritizing trains that are about to leave the system compared to trains that want to enter the bottleneck area and are waiting on the open track. Based on the observation that a delayed train can be overtaken by another train outside the considered area, deviations for the planned arrival sequence at the border of the system are allowed. No real-time rerouting actions, platform changes, or cancelations of trains are made.
In the end, the real arrival times, the usage of the supplements and buffers, the locations of the conflicts, etc., all information with respect to the passage of the trains through the network is available. For every performance measure that will be evaluated in step 2, the average over 10 000 simulation runs is taken as result. As said above, the results from the simulation model need to be interpreted while keeping the assumptions in mind. Although some simplification is made, the interaction between trains is taken into account and rules from practice are applied when solving conflicts. Introducing more details in the simulation model and necessarily also within the entire developed algorithm, would complicate the computations a lot with a moderate gain in accuracy as a result. However,
the question remains how external effects, such as the driver’s behavior, can bias the results.
Step 2: computing the performance indicators
The output from step 1 or actual delay data can be used as input for this step. Starting from this information, together with the original (reference) timetable, the minimum necessary process times, and passenger flows from the model of Sels et al. (2011a), the passengers’ travel times and other performance indicators can be computed. This is done by simulating, event by event, the itinerary of each passenger. Doing so, the RWTT and its variants RWTTnormand RWTText
are obtained. When a missed transfer or a cancelation is detected, the number of harmed passengers is counted. Similarly, the total passengers’ arrival delays are recorded. By considering a train as unit instead of a passenger, the total delays of all trains, the evolution in knock-on delays, and the percentages of
extra or newly delayed trains can be found. A train is said to be extra delayed if
it arrives at its terminal or leaves the system with more delays than it had upon entering. The newly delayed trains are those that were not delayed initially but got delayed during their passage through the network.
In Kanai et al. (2011), it is said that the variance reflects the fairness among passengers. For example, taking the standard deviation of a performance indicator such as the passengers’ delays into account, one can distinguish between 10 times 5 minutes of arrival delays and one time an arrival delay of 50 minutes together with no delays the other 9 times. Therefore, the standard deviation of the performance indicators (between brackets) and the worst case performance with respect to the total amount of train delays are added in the result tables of the following chapters. Since the stochastic influences are equal for the different robustness measures, the standard deviation of the robustness scores is equal. The name Robstdev is used to indicate the standard deviation
of the RWTT.
To test whether an improvement is significant, statistical tests are performed and a significance level of 0.05 is used. The large number of simulation runs from step 1 support the assumption that all performance indicators are (approximately) normally distributed. First the significance of the difference in variances is tested before the means are compared. Although these tests are based on the variances and not on the standard deviations, only the standard deviations are reported.
ROBUSTNESS OF THE ENTIRE BELGIAN NETWORK 45