Delft University of Technology
FACULTY MECHANICAL, MARITIME AND MATERIALS ENGINEERING
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This report consists of 32 pages and 1 appendix. It may only be reproduced literally and as a whole. For commercial purposes only with written authorization of Delft University of Technology. Requests for consult are only taken into consideration under the condition that the applicant denies all legal rights on liabilities concerning the contents of the advice.
Specialization: Transport Engineering and Logistics
Report number: 2015.TEL.7962
Title:
Train delay analysis
Author:
C.D. van Goeverden
Title (in Dutch) Trein vertraging analyse
Assignment: literature
Confidential: no
Initiator (university): Dr.ir. F. Corman Supervisor: Dr.ir. F. Corman
Summary
Railway network operators are always trying to improve the quality of service of their railway network and the passenger satisfaction by reducing delays and thus increasing the reliability and punctuality of then network. Without analyses from studies regarding the causations and effects of train delays, it is not possible to propose new models for delay management and delay propagation.
This report analyses and summarizes recent scientific research done regarding train delays. First it describes the different approaches to defining train reliability based on the train delays and it shows the passenger perception of these delays. Secondly it describes various methods of logging the train movements by different railway network operators. A few examples of studies using different methods to gather data are given. Thirdly a summarization of probability distribution models used to describe train delays is given and some methodologies to fitting the theoretical distributions and testing their fitness are stated. A small selection of studies is picked and a more detailed explanation is given regarding the proposed distribution models and the fitting of those models.
Contents
Summary ... 2 1. Introduction ... 4 2. Delay Definitions ... 5 Primary delay ... 5 Secondary delays ... 6 3. Performance Indicators ... 7 Punctuality ... 7 Average delay ... 8 Reliability ... 8 Passenger perception/valuation ... 9 Delay variety ... 10 Discussion ... 10 4. Data collection ... 11Train detection data ... 11
5. Delay distributions ... 14
Fitting and testing distributions... 15
Yuan ... 15 Goverde ... 19 Herrmann ... 21 Briggs ... 23 Other ... 25 Percentiles ... 26 Discussion ... 28 6. Conclusion ... 29 References ... 30 Appendix A ... 32
1. Introduction
Railway network operators are always attempting to increase the punctuality and reliability of their train services. Train delays have a large economic and social impact, for example in The Netherlands on average 1.2 million passengers travel by train each day and many railway network companies are multi-billion dollar operations. Railway managers, train operators and timetable designers are always attempting to improve the quality of service and passenger satisfaction by reducing delays and thus increasing the reliability of the railway network. Without the analyses from studies about the causations and effects of train delays, it is not possible to propose new models for delay management and delay propagation. Because of this, there is a high interest in all studies about train delays.
The goal of this report is to analyse and summarize the scientific research done on train delays. The beginning of this report states the basic definitions of train delays that are often used in studies. After these definitions several indicators that quantify the performance of railway networks are summarized and their pros and cons discussed. In the following chapter the possibilities of gathering data for studies on train delays are mentioned and at the end of the report several delay distributions models are mentioned and discussed. Because this report is a summary of available literature on train delays it will not provide any new research or results but only restate previously performed studies and discuss these studies.
2. Delay Definitions
For passengers of public transportation the delay of a train is defined by the difference in time from when a train actually departs or arrives compared to the scheduled departure or arrival time of the train. This is the most basic definition of a delay but there are many more definitions possible. This chapter will therefore be a summary of several representations and definitions of train delays.
Primary delay
“A primary delay is the deviation from a scheduled process time caused by disruption within the process” (Goverde, 2005). Because of the enormous amount of factors that can influence public transportation, the running times of trains between stations and the dwell times at stations will always slightly vary for each trip. The running time of the train can for example be delayed by a technical malfunction of the train or the infrastructure, but the delay can just as easily be caused by human-related interference of the process. Even if a train manages to arrive at the station according to schedule, there can be various reasons to have increased dwell times at the station, for example the number of passengers boarding the train or the coupling/uncoupling of trains. There are many random disruptions that are difficult to forecast and impossible to prevent. Because of this, delays will always be present in railway transportation. Table 1 gives an overview of some of the more common sources of primary delays in a railway process.
Table 1; Sources of primary delays – Goverde (2005)
A majority of the causes listed in the table will only cause minor disruptions, possibly taking only seconds and most likely not exceeding several minutes. There are also some major disruptions possible that are not listed, for example an engine breakdown, destruction of the infrastructure or even a train crash.
These incidents are however very infrequent and the delays or even cancellations of trains caused by these issues are often not considered representative of the usual stochastic behaviour.
Secondary delays
“A secondary delay is the deviation of a scheduled process time caused by conflicting train paths or waiting for delayed trains” (Goverde, 2005). Secondary delays (or knock-on delays) are delays that are a direct result of the delay of another train. A distinction between initial delays and knock-on delays is important since often primary delays are unavoidable, yet this is not the case for secondary delays. The amount and length of secondary delays are heavily influenced by the stress exerted on a railway network. Secondary delays are therefore significantly increased by using a train network closer to its capacity and the delays can be reduced or prevented by increasing the buffer times in the timetable.
There are two main causes for secondary delays that can be distinguished. The first one is a result from the fact that different vehicles use the same infrastructure. Consider a line with several trains travelling in the same direction. As described in the paragraph about primary delays, there are many possible causes for minor disruption in a scheduled train trip. Unscheduled delays and variation of train speeds can delay following trains by still occupying part of the scheduled route and preventing the passing of other trains. Besides the physical obstruction of the track there are also speed restrictions on following trains and a required minimum headways between trains. The railway network control and signalling system will alert following trains to delay in order to maintain these headways and speed restrictions. (Carey & Kwieciński, 1994)
The second cause for knock-on delays is a result from connecting trains at interchange stations that have to delay their departure when waiting for a delayed feeder train. In some cases the delay from the feeder train will be too large and the connecting train will not wait any longer, though this will limit the delay of the departing train, it can significantly increase the delay of arriving passengers from the feeder train. In some other cases the connecting train cannot depart before the arrival of the feeder train. This is the fact when for example train coaches have to be (de-)coupled and change lines or when (part of) the crew of the feeder train has to transfer trains. Table 2 summarizes some of the mentioned sources of secondary delays in in a railway process.
Table 2; Sources of secondary delays – Goverde (2005)
The distinction between primary and secondary delays is important, because since both have different causes, they also require different approaches to managing them. Primary delays are managed by including sufficient margins in running and dwell times and secondary delays are managed through the buffer times between trains that prevent or reduce the hindrance. (Goverde, 2005)
3. Performance Indicators
Punctuality
As stated in the previous chapter a delay can be defined as the deviation from a scheduled event of a train. To be able to evaluate the performance of a railway network, the use of punctuality is often used. The punctuality of a train services can be expressed as the percentage of trains arriving, departing or passing by a given location on its route no later than a certain time in minutes. (Hansen, 2001), (Hansen & Pachl, 2014)
Punctuality is often used by railway operators to represent the performance of their railway traffic. In figure 1 you can see how this can be used to compare the performance of railway networks from various countries.
Figure 1; Punctuality vs congestion of railway networks of various countries - NS Annual Report 2014
In figure 1 the punctuality is set for trains that pass with a delay of less than 5 minutes, however the definition of punctuality greatly varies per operator. While most European railway operators measure the delays within 5 minutes this is not everywhere the case. For example in Germany trains are considered late after a delay of 6 minutes, in Austria this is after 5 and a half minutes, in Norway and Switzerland the threshold lies at 4 minutes and for Japan it even has a value lower than 1 minute.
Though punctuality is often used as an indicator for the performance of railway companies, it can give an incomplete image. Because an average value is given, there often can be significant outliers, for example in The Netherlands in 2013 on average 93.6% of all the train were considered on time (within
5 minutes of scheduled time), yet the average for certain specific routes was consistently below 75%. The punctuality was not constant over time either, during rush hour the punctuality significantly decreased. Another problem with average punctuality is that it does not make a distinction between for example trains that have a delay of 6 minutes or trains that have a delay of 60 minutes. Because the threshold for delayed trains is set at more than 5 minutes late, both are regarded equally as delayed. Railway operators should be aware that even though they may only consider trains with a delay of over 5 minutes as delayed, the passengers will most likely not.
It should be noted that there is a clear difference between passenger punctuality and train punctuality. Especially taking transfer delays in account, the impact of the small train delays can be significant. When a train is 4 minutes late and therefore not considered delayed by the railway operator, it is possible that a passenger will miss his 3 minute transfer time and have to wait an hour for the next train. Railway operators should not underestimate the effect and value of smaller delays even though they may not directly improve their punctuality.
Average delay
Besides the previously described way to indicate a railway networks performance, another method many railway operators use to display punctuality and quantify their performance is the use of average delay. For example the Shinkansen (Japanese high-speed railway lines) will show in reports they had an average delay of 0.5 minutes per operational train in 2012.
However similarly as when using a percentage of trains that arrive within a certain amount of minutes to define punctuality, the use average delay can give a misleading representation of the occurring delays. Börjesson & Eliasson (2011) mention some of issues on using average delay as a performance indicator for train reliability. Imagine for example a train network where 10% of all trips are delayed by 10 minutes and another train network where 2% of all trains have a delay of 50 minutes. Calculating the average delay will indicate that both these network have a similar performance with an average delay of 1 minute per train. However, when using the punctuality defined as the percentage of trains arriving with a delay of 10 minutes or less the first network shows a punctuality of 90% as opposed to the 98% of the second network. It is important for the network operators to realize that 50 minute delays are very different compared to 10 minute delays. Both delays have different causes and a different effect on the railway network and passengers, and both require different solutions to be improved.
Reliability
There are many ways to define the reliability of a railway system. In general a railway system is considered reliable when most of the trains run according to schedule most of the time and passenger and goods are transported without a delay, only a minor portion of the trains experience disruptions, but the average delay and variation in delays should be low (Vromans, 2005).
Rietveld et al, (2001) has provided several measures for reliability:
2. the probability of an early departure;
3. the mean difference between the expected arrival time and the scheduled arrival time; 4. the mean delay of an arrival given that one arrives late;
5. the mean delay of an arrival given that one arrives more than x minutes late; 6. the standard deviation of arrival times;
7. an adjusted standard deviation of arrival times (ignoring the early arrivals), and various other more complex measures to represent the seriousness of unreliability.
Besides the above mentioned measures, there are many more approaches for the measurement of reliability conceivable. It can be difficult that a characteristic like reliability that is so important for transportation systems (such as a railway network) has so many quantifications. According to Vromans (2005) when the reliability objective is to minimize the positive difference between the arrival times in operations and the scheduled arrival times, this will in many cases have a beneficial effect on all other mentioned measures.
Passenger perception/valuation
When railway network operators try to increase the reliability of their trains it is mainly to satisfy the passengers and it is therefore important to realize how the passengers perceive the delays and how they value their (lost) time.
Several studies have shown that passengers do not value all delays equally. First of all the length of delays is non-linear with the dissatisfaction of passengers, passengers become relatively increasingly more agitated per minute as delays increase. This makes the use of average delay easy to underestimate the value of reducing infrequent large delays as opposed to more frequent shorter delays since passengers experience a significant greater discomfort from longer delays. Passengers also value a delay caused by waiting at the platform (before departure of the train) significantly worse than an equally long delay experienced in vehicle en route. And passengers are generally risk-averse towards travel time, a 50% probability of a 2 minute delay is considered more than twice as bad as a guaranteed delay of 1 minute. (Börjesson & Eliasson, 2011), (Vincent, 2008). (Rietveld et al, 2001).
Besides the actual (length of) delays there are more factors in the (dis)satisfaction of travellers. The comfort of passengers will heavily influence their valuation of train travelling. Having to stand in very crowded trains can have a similar dissatisfaction as travelling seated with almost half an hour of delay. While on the other hand providing clear information about the duration and cause of delays is valued similar to a 10 minute reduction in travel time. (Kroes et al, 2005)
Finally it is important to emphasize that generally passengers do not care about the difference between primary or secondary delays, for them a delay is a delay. As mentioned before a passenger will experience great discomfort when they miss a transfer that can turn a 5 minute delay in an hour long delay. It is also important to realize that some measure to improve punctuality can even have passenger unfriendly effects. For example spreading arrival and departure times of all trains at a station more
equally will decrease the interdependence of the trains, and most likely increase the punctuality, yet at the same time will also increase the transfer times for passengers (Vromans, 2005).
Delay variety
There is a great variety of uses for rail transportation and each have their own delay characteristics. Passenger transport delays will differ from freight transport delays and large railway networks will show different delay characteristics than urban or suburban networks. Freight trains do not have to stop as often from origin to final destination as passenger trains, therefore only the final arrival time is often important. Even though freight trains stop less often they have longer delays when the freight routes consist of a single track, in these cases the delay of freight services is heavily dependent of the amount of sidings. Passenger trains, particular in urban area networks, are nowadays almost always running on double track and will not be delayed by trains from the opposite direction. (Carey & Kwieciński, 1994), (Higgins, 1996), (Higgins & Kozan, 1998)
It is also a possibility that there is no fitting measure for delays to indicate the performance of a network. It is a possibility for freight transport that there is no set timetable and there is only transport when there is a demand for the goods. Because there is standard time schedule and no set arrival time it could be argued that the train cannot be delayed. In this case the delay could be defined by the deviation of the minimum time it takes the train to travel the complete route taking into account speed limits etc. Another case where delays are not clearly defined is during complete cancellations of a train. Because delays through cancellation are often not structural and too uncommon to represent the usual behavior of a network, not all studies include them in statistical analysis and rather view them as outliers (Goverde 2005).
Discussion
There has been a lot of research done to the causes of train delays and its results. Railway network operators and researchers use various ways to describe the performance of a network. The most common ways to describe the reliability of a network is by defining the punctuality. This can be done by calculating the average delay of a network or by analysing the amount of trains that arrive or depart within x minutes of the scheduled time. Both these indicators are easily understandable and can represent the performance with a single number. This is also why neither are perfect, they both provide an incomplete or even misleading representation of the total performance.
In general the reliability will be improved when the objective for operators is to minimize the positive difference between the arrival times in operations and the scheduled arrival times. However they should realize not all delays are equal and that indicators like average delay can easily allow to underestimate the effect of larger delays on passengers. It can even be possible that attempts to improve train punctuality will have a negative effect on the travel time of passengers. Before operators attempt to improve their network they should take into account all the performance indicators and all the results of their improvements on the passengers’ satisfaction.
4. Data collection
Data collection is essential for the analysis of train delays. With some of the earliest performed studies the data was collected manually, however manual records most often only include passenger and train movements at stations and are not always very accurate. With computers and software playing an increasingly larger role in the last few decades there are faster and more precise ways of tracking train delays than just from manual data collection. Track occupancy and release records from the signalling and safety systems provide a very precise collection of all data necessary for train delay analyses. With most of the newer train systems also including precise positioning via GPS and logging all movements in train event recorder data, it is even easier to very accurately track all the train movements including arrival and departure times.
Train detection data
Train describer systems are essential for traffic management. A train describer system keeps track of the positions of all trains at discrete steps along their route. Whenever a new position signal is observed all information is updated in a database that describes all the current train movements. The log files will contain all the chronological signalling information and are able to accurately provide all the train movements of a certain period.
Even though these systems are able to describe train movements within the accuracy of seconds, smaller delays and in particular the area of station stops, have not been well-researched. This is for a number of reasons. Firstly, some systems may only store the arrival and departure times at stations or the occupancy and release times of all tracks. If the latter is true then the arrival and departure times are estimated using the occupancy and release times of the platform track sections rather than individually measured. Secondly, many describer systems only signal when a train passes a specific measurement point and do not measure the time during which trains remain stationary. Thirdly, even though the system is capable to describe movements with a maximal error of seconds, infrastructure managers do not always record delays of less than a minute or choose to round arrival times to a minute (Harris et al. 2013).
Hansen & Pachl (2014) gives some examples of programs developed and studies done specifically to allow the use of available log-files from train describer systems of various countries for accurate statistical analysis. For example Goverde & Hansen (2000) developed the data mining tool TNV-Prepare. TNV (TreinNummerVolgsysteem) is the train describer system that was used in The Netherlands for over 20 years until it was replaced by a newer similar system (TROTS) in 2011. The TNV log-files contain the chronological information of all signalling controls and monitoring information (track, signal, switch and route relays), but the system does not automatically match these occupancy messages with the individual train numbers. TNV-Prepare matches all the individual events with the corresponding trains and is therefore able to precisely derive all the running times, blocking times and headways of the trains. Hermann (2006) found a difference between the timetable points and the measurement points in the Area of Frankfurt in Germany. He used data from the train describer system RZÜ (Rechnerunterstützte Zugüberwachung). The timetable points are at the station platforms, yet the actual location of the
measurements are at the main signals and insulation joints at a certain distance from the platforms. He approximated the off-set of these measurements and the actual platform arrival and departure times. The Swiss Federal Railways (SBB) in cooperation with ETH Zürich have developed their own timetable
analysis tool OpenTimeTable. This tool provides many possibilities to visualize and compare arrival and departure times. The Swiss train describer system SURF (Système Unifié de Régulation Ferroviaire) also takes measurements at main signals instead of at the platforms where the timetable points are, however the times have a higher accuracy because the SBB uses correction terms differentiated by route, activity, and train type, and moreover locations of signals are optimized (Goverde 2005).
Higher accuracy of time measurements are helpful in research, as they allow more exact analysis of delays and a better fitting of possible distribution models. For example figure 2 shows Ullius (2004) using OpenTimeTable to analyse arrival delays in Switzerland presents train delays rounded to 1/10 minute.
Figure 2; Distribution of number of arrival delays in Bern (BN), Konolfingen (KF), Langnau (LN), Wiggen (WIG), Escholzmatt (ESCH), Schupfheim (SCHH), Entlebuch (ENT), Wolhusen (WH), Schachen (SCHA), Gutsch (GTS) and Luzern (LZ) – Ullius (2004)
Studies done with the Dutch TNV train describer data (e.g. Yuan, 2006) often display the data rounded to a full minute, see figure 3. However Goverde (2005) developed the tool TNV-Filter to more accurately estimates arrival and departure times in (large) stations based on tables generated by TNV-Prepare. This tool takes into account speed limits, standard maximum acceleration and deceleration characteristics of each train, the trains’ length, speed profiles etc. to make estimations with an accuracy in order of seconds.
Figure 3; Log-normal and gamma density fits, kernel density estimate and histogram for the arrival delays of IC2100N at the platform track at the platform track in The Hague HS – Yuan (2006)
Recently railway operators have started to install GPS devices and digital train event recorders in all new trains and replacing the analogue systems in older trains. The train event recorder is similar to the flight recorder in airplanes and logs among other things the speed of the train, throttle and brake percentage use. This data can for example be used to monitor the performance of the train or in the investigation of an accident (Hansen & Pachl, 2014).
Medeossi et al. (2011) used data obtained from GPS tracking data collected on-board the trains. This data was used to calibrate a motion equation used to investigate the behaviour of individual trains. To be able to use the GPS tracking data they first had to prepare the data by processing it through a Kalman filter after which they were able to compare the recorded speed profile to a planned speed profile. GPS and train event recorder data are therefore better for accurately creating the speed profile of a train, and allows you to clearly show at what point the train is accelerating, cruising, coasting or braking.
5. Delay distributions
In recent literature a variety of theoretical distributions have been used to describe the distributions of train delays. These distributions have been adopted to model both the stochasticity of train events and process times as well as the statistical modelling based on empirical data. The studies often include specific distributions for specific types of train delays including arrival, departure, dwell times and running time delays. With empirical studies it is important to realize it can be difficult to accurately derive the original delays from data of track occupancy and measured delays since they will often contain knock-on delays.
One of the first models used to describe delays in a railway system is the negative exponential distribution introduced by Schwanhäuβer (1974). He was likely the first who tried to approximate the variability of positive arrival delays by applying this exponential distribution. It is widely used in literature to describe delays and has been validated in numerous other analyses and studies. (Ferreira & Higgins 1996), (Goverde & Lopuhaa, 2001), (Yuan, Goverde, & Hansen, 2002), (Haris, 2006), (Goverde, 2005). The negative exponential is most often used when describing the delays of a train process or event that has a lower bound. Besides the positive arrival delays it is also fit to use when analysing the departure delay of trains. The lower bound is in this case the minimal delay, meaning the train departs the station on time. The distribution is skewed to the right where short delays are more likely to occur than longer delays.
The negative exponential distribution is also used as basis for other models and adjusted to be better fit to describe certain distributions. Examples are the weighted exponential distribution (Schwanhäuβer, 1974), (Ullius, 2004), (Wendler & Naehri, 2004), (Hermann, 2006) or the q-exponential function (Briggs and Beck, 2007).
Other distributions that are often used to display stochasticity of arrival delays of trains are for example the normal, Erlang, Weibull, gamma and log-normal distributions (Steckel 1991), (Hermann 1996), (Higgins 1998), (Yuan 2006), (Bruinsma, Rietveld, & Vuuren, 1999). These distributions also take into the model the negative arrival delays (meaning the trains arrive ahead of their scheduled time at the station or any other measured point on the track). Just like the exponential distribution, these distributions are used as basis for other models, for example Güttler (2006) uses a normal-lognormal mixed distribution to evaluate running times of trains between two stations in Germany.
(Büker & Wendler, 2009) present an approach to modelling distribution functions that adapt to the complex real world delays of major railway networks whilst still managing to be accurately and efficiently used to compute delay propagation. (Yuan, 2006) compared the Kolmogorov-Smirnov goodness-of-fit among several distribution models.
This chapter will discuss a small selection of some of the aforementioned studies and will exemplify how they applied specific distributions in their research.
Fitting and testing distributions
When the stochasticity of train delays is measured, any of the fitting distribution models has to be selected and the parameters specified. Hansen & Pachl (2014) describes approaches to the parameter estimations and testing the goodness-of-fit of a distribution. Two common methods for estimation of parameters are the method of moments and the maximum likelihood method. The method of moments derives equations that equate sample moments with theoretical moments. The equations are then solved for the parameters of interest. The maximum likelihood estimation (MLE) uses a likelihood function to estimate the values of the unknown parameters that maximizes the likelihood of getting the data that is observed. The advantage of the method of moments is that it is fairly simple and estimators could be calculated by hand whereas using the likelihood equations can be cumbersome without the use of computers. Even though the method of moments is the simpler method, the MLE is used more widely since the method of moments’ estimators are often biased and the MLE is a better unbiased estimator as the sample size increases.
To investigate how well the distributions represent the measured data, the validity of the selected distributions can be tested through graphical techniques and quantitative techniques (Hansen & Pachl, 2014), (Yuan, 2006). Law & Kelton (2000) presents five graphical approaches used to evaluate the goodness-of-fit: the density/histogram overplot, the frequency comparison, the distribution difference plot, probability-probability plot and quantile-quantile plot. With the histogram plot a kernel density estimation is often used to compare the data distribution more clearly with other distributions. After the graphical representation it is possible to apply goodness-of-fit hypothesis tests as quantitative tests. A goodness-of-fit test is meant to assess whether the data fits any of the assumed distributions. Examples of goodness-of-fit tests are the chi-squared test and the Kolmogorov-Smirnov (K-S) test.
Yuan
Yuan (2006) presented a new approach for fine-tuning the parameters of several distributions models based on real-world data. He evaluated candidate distributions for train events and process times based on data recorded in The Netherlands Den Haag HS station. The normal, uniform, exponential, gamma, beta, Weibull and log-normal distributions have been evaluated. The evaluation of the distribution models was performed on a large set of different train events and process times and included:
The arrival delay at the station
The arrival delay of trains at the local railway network
The departure delay at the station
The train running times
The train dwell times
The track occupancy times
The running and dwell times and track occupancy times in case of (no) hinder caused by other trains
After using the newly proposed method of fine-tuning the parameters, each candidate distribution is ranked according to the p-value of the Kolmogorov-Smirnov test where the hypothesized distribution is specified with the fine-tuned parameters. The quality of the distribution fitting is also visualized by first
comparing the kernel density estimate, the fitted proposed density distributions and the empirical histogram and secondly by applying the distribution differences plot for the fitted distributions and the empirical one.
One example is the evaluated candidate distributions for both the arrival delays of trains at the platform track and the arrival delays at the approach signal of the station. 14 train series were taken into account (all in both southbound and northbound directions). For the arrival times of trains at the approach signal the log-normal was the best model in 9 out of the 14 studied cases. For the arrival times of trains at the platform tracks the log-normal distribution gave the best fit in 11 out of 14 studied cases. In the other cases the gamma, Weibull or normal distribution were best fit and the log-normal was often ranked second or third.
Figure 4 shows the kernel density estimate, histogram and fitted log-normal and gamma distribution of the arrival delay one of the train series, the northbound intercity train IC2100N at the platform track. The log-normal seems to be the better fit since the difference between the log-normal fit and the empirical distribution appears to be smaller than the difference between the gamma fit and the empirical distribution. This is confirmed by the distribution differences plot given in figure 5.
Figure 4; Log-normal and gamma density fits, kernel density estimate and histogram for the arrival delays of IC2100N at the platform track – Yuan (2006)
Figure 5; Distribution function difference plot for the log-normal fit and the arrival delays of IC2100N at the platform track and the plot for the gamma fit and the empirical delays - Yuan (2006)
Another example is the evaluated candidate distributions for the non-negative arrival time delays at the station. In this case it was revealed that the Weibull model was fit in 12 out of the 18 studied cases and ranked second in the other 6 cases. In figure 6 the empirical histogram, gamma and Weibull density fit are plotted for the non-negative arrival delays of the northbound interregional trains IR2200N at the platform track. In figure 7 the distribution differences plot is given. These visuals confirm that the Weibull distribution is the best fit to the empirical distribution. In addition it has also been found that the shape parameter of the Weibull fit was in general not larger than 1, but if the shape parameter can be fine-tuned to nearly 1 the exponential distribution would be best fit since the exponential distribution is a special type of Weibull distribution where the shape factor is 1.
In conclusion Yuan (2006) found that the log-normal distribution can be considered the best model for the arrival times at the approach signal and platform track. The Weibull distribution generally can be considered the best model for non-negative arrival delays, departure delays and the free dwell times of trains. The shape factor of the Weibull distribution for non-negative arrival delays and departure delays was in general lower than 1 and for dwell times it generally was higher than 1. For the conditional train running and track occupancy times in case of (no) hinder no good generic distribution was found.
Figure 6; Fitted Weibull and gamma density curves and histogram of the non-negative arrival delays for IR2200N - Yuan (2006)
Figure 7; Distribution differences plot for the Weibull fit and the non-negative arrival delays for IR2200N and the plot for the gamma fit and the empirical delays - Yuan (2006)
Goverde
Goverde (2005) evaluated candidate distributions for train events and process times based on data recorded in The Netherlands Eindhoven station. Part of the analysis on these train delays had been published in earlier studies (e.g. Goverde et al, 2001). The study includes arrival delays, departure delays, dwell times and transfer times.
As an example the arrival delays will be discussed again. All the train arrival times have been obtained using TNV-Prepare and TNV-Filter as described in the previous chapter. The arrival delays were defined as the difference between the measured arrival times and the scheduled arrival times given in the train timetable, which again resulted in both negative and non-negative arrival delays.
It was observed that the measured arrival delay probability density functions were mostly bell-shaped with some slightly skewed to the right due to a higher amount of larger delays, this can be seen in figure 8 and figure 9. Figure 8 displays the empirical histogram, kernel estimate and proposed normal density fit of arrival delays of the Intercity IC1500 train and figure 9 displays the histogram and kernel estimate of the arrival delay of the intercity IC800 train, where the latter is more skewed towards the right. When the skewness is small, the empirical distributions can be accurately approximated by a normal distribution. 13 train series were investigated and using a Kolmogorov-Smirnov goodness of fit test the proposed normal distribution with estimated parameters were tested. With a 5% significance level the distribution was accepted for the arrival delays of 8 out of the 13 studied cases. The 5 train lines that were rejected all had a coefficient of skewness of +1 which means they were all skewed to the right. It was suggested a Weibull, Erlang or lognormal distribution would be more appropriate in these cases however this was not further investigated.
Figure 8; Histogram, kernel estimate, and normal density fit of Arrival delays of IC1500 Hrl-Gvc - Goverde (2005)
Figure 9; Histogram, kernel estimate, and normal density fit of Arrival delays of IC800 Mt-Hlm - Goverde (2005)
Since often the non-negative arrival delays are the primary interest and the distribution is significantly different from all arrival delays, a new distribution model was considered exclusively for the non-negative arrival delays. It was observed that the empirical histogram and probability density functions were negative-exponential shaped, which can be seen in Figure 10. A negative-exponential distribution was proposed and tested using the Kolmogorov-Smirnov goodness-of-fit test. This proposed distribution model was accepted with a significance level of 5% in 11 out of the 13 studied cases. In the two rejected cases the fraction of late arrivals was very high; one of these cases was IC800 Mt-HLm which can be seen in figure 9.
In conclusion, Goverde (2005) found that in general an exponential distribution is a good model to describe non-negative train arrival delays and departure delays. An exponential distribution was accepted in all cases for the excess dwell time delays. The normal distribution can be considered a good model in most arrival delays and was accepted as a good fit in all dwell time distributions and transfer times.
Herrmann
Herrmann (2006) described arrival and departure delays based on data provided by Ullius (2004). The data is recorded from the Swiss Federal Railways (SBB). Since railway schedules are usually not so tight that every second counts, it is observed that it could be possible to use both continuous and discrete models to describe the measured probability density function. The advantage of discrete distributions is that fewer assumptions have to be made concerning the empirical data and no parameters have to be estimated. A drawback of a discrete distribution is that a significant higher amount of data must be available while for a continuous distribution the parameters can be estimated with relatively fewer available measurements.
The delay distribution is described by the cumulative probability density function, for the continuous probability distribution a weighted-exponential distribution is proposed and as discrete delay distribution the pulse delay distribution is considered. As an example the arrival delays will be discussed again. Figure 11 displays the cumulative distribution function of the empirical arrival delays.
The weighted exponential distribution is similar compared to the standard exponential distribution but includes an extra parameter 0 ≤ µ ≤ 1, where the wexp(1,1)-distribution is equal to the exp (1)-distribution. The parameters are estimated from the empirical data and compared with theoretical values calculated from the quality target set by the SBB. Figure 12 shows the empirical distribution the estimated and the target weighted exponential distributions of the non-negative arrival delays. The target distribution is based on the target set by the SBB. This target states that at least 75% of all trains arrive with a delay of at most one minute and 95% of all trains arrive with at most 4 minutes delay. It is observed that the weighted exponential distribution fits quite well however is slightly off for all delays lower than one minute.
Figure 12; Sample, estimated and target distribution using the wexp-distribution with calculated and estimated parameters – Herrmann (2006)
For the discrete delay distributions the empirical data is grouped in delay intervals of 15 seconds. By describing the delays in a discrete manner, the idea is that through adjusting the velocities of train en route it is possible to create “exact” delays, and give a train a new time window where-in it can be handled. Since the time interval between two trains using the same resource in Switzerland is 90 seconds the use of multiples of 90 seconds is used to describe the pulse delay distribution. The result of fitting this pulse delay distribution is shown in figure 13.
Figure 13; Probability distribution for the pulse delay distribution of arrival delays – Herrmann (2006)
Briggs
Briggs & Beck (2006) analysed the probability distributions of delays occurring in the British railway network. The so called q-exponential distribution is proposed as an accurate description for the distribution of train delays. The q-exponential is defined as 𝑒𝑞,𝑏,𝑐(𝑡) = 𝑐(1 + 𝑏(1 − 𝑞)𝑡)
1
1−𝑞 where q is a real parameter. This distribution is similar to the negative exponential distribution since lim
q→1𝑒𝑞,𝑏,𝑐(𝑡) = 𝑐 exp (−𝑏𝑡). The data used for this study consisted of the departure delays for 23 major cities collected through the British railways website that provides and updates real-time all current delay information. As each train departs the data is added to a database, which in the end consists of over 2 million train departures.
The fitted models only included positive delays, which means in this case that the model represents the probability distribution of a departure delay, given the fact that the train is delayed at least one minute. Figure 14 shows the q-exponential distribution fitted to all the departure delay data and assumes all stations have the same delay distribution.
The q and b values are also separately calculated for every station and are displayed in Figure 15. This allows a quality comparison of the stations with the same q values: the larger the value of b the better the performance of this station under the given environmental conditions.
Figure 14; All train data and best-fit q-exponential: q = 1.355 ± 8.8 × 10-5, b = 0.524 ± 2.5 × 10-5 -Briggs & Beck (2006)
Other
Ferreira & Higgins (1996) proposed yet another fit to describe a delay distribution namely the Erlang distribution, which is similar to the exponential distribution and a special case of the gamma distribution. For the data there were 265 measurements for train/track delay and 261 measurements for terminal/station delay which were compared to the actual schedules over a five week period. In figure 16 the best fit distribution for the delay distributions of freight trains is shown.
Figure 16; Delay distributions of freight trains with a best fit erlang distribution - Ferreira & Higgins (1996)
Bruinsma et al. (1999) investigated the reliability of public transport modes and included train delays as well as delays in urban busses, inter-urban busses and underground transport in the Netherlands. In the data concerning train delays a distinction is made between two types of trains: the intercity and the stopping train. A distinction has also been made during peak hours and off-peak hours and delays on weekdays or delays on Sundays. The Weibull, gamma and log-normal distributions have been proposed as models to describe the empirical delay distribution. The maximum likelihood estimation has been used to select the distributions that give the best fit to the empirical data. Figure 17 shows the estimated delay distribution of an intercity train at peak hours and figure 18 shows the estimated delay distributions of the underground RET in Rotterdam. It shows the underground is more reliable with the mean closer to zero and a smaller variance.
Figure 17; Estimated departure and arrival delays of an intercity train during peak hours - Bruinsma et al. (1999)
Figure 18; Estimated departure and arrival delays of the underground - Bruinsma et al. (1999)
Percentiles
Whenever it is not possible to fit a specific distribution to the measured delays that does not get rejected by the goodness-of-fit tests, it can be decided to only focus on a part of the delay. For example by only attempting to estimate the distribution of the measured data that falls above or below a certain value or percentile.
Fakhraei (2011) did not manage to fit a distribution to the entire database of measurements that would get accepted by any of the goodness-of-fit tests (Kolmogorov-Smirnov, Anderson-Darling and Chi-squared tests). Though the Generalized Pareto distribution seemed to be considered the best distribution among all the tested ones, it also got rejected by the quantitative goodness-of-fit tests and a graphical representation, the quantile-quantile plot in figure 19 showed none of the distributions could be considered a good fit.
Figure 19; The quantile-quantile (Q-Q) plot is a graph of the input (observed) data values plotted against the theoretical (fitted) distribution quantiles and the best fit should follow 45 degree line. Three candidate’s distribution Q-Q plots are compared and none follow the linear line. - Fakhraei (2011)
It was therefore chosen to focus only on specific percentiles of the measured delays. It can be seen in table 3 that for example for the 80th percentile it is shown that 74% of the total delay time is caused by only 20% of the delays and looking at the 99th percentile shows that only 1% of the total delays is responsible for 13% of the total delay time. This shows how important the extreme delays are and how they affect the total delay time, and that it is justified to just focus on large delays.
Table 3; Percentile share in percentage of total delay minutes - Fakhraei (2011)
When attempting to fit a distribution to these more extreme delay values, both a power law and exponential distribution were reviewed. The standard way to define power law behaviour is by being able to plot a straight linear line over the quantity histogram on a double logarithmic scale. Figure 20 shows that power law can be seen on the distribution trail but does not apply to the complete range of data. It seems the data follows power law behaviour up to some limit values after which it decays exponentially. After applying goodness of fit tests it was confirmed that an exponential distribution could be selected to describe the arrival delays.
Figure 20; Power law and exponential distribution test on arrival delays at final destination. - Fakhraei (2011)
Discussion
Only a selection of all the studies that performed research on train delays and proposed models to describe distributions has been discussed in this chapter. Many different probability distributions have been proposed in recent literature, examples are the Weibull, normal, log-normal, beta, gamma, (weighted) exponential, Erlang, q-exponential, uniform and even discrete distribution models. Many distributions however are closely connected to each other and only have minor differences in their describing models as can be seen in Appendix A (Leemis & McQueston, 2008).
Almost all studies use different data sets. The distribution models are identified for the delays of trains at a specific station or country, and the delays are from different times (year or time of day) and a different local infrastructure. Even though there is this large variety of data sets used in all studies, the theoretical expectation is based on the flexibility of those distribution models, where the models are widely applicable to many cases and the parameters are to be specifically decided for each individual case. It can therefore not be said that one model is the best fit, but for every case it will require an individual assessment of what model seems to be the best fit.
All research done on distribution models is important since modelling of delay distributions is a prerequisite for predicting the propagation of train delays at stations, and accurate delay propagation is vital for efficient delay management and improving the quality of railway networks.
6. Conclusion
This report has analysed and summarized recent scientific research done regarding train delays. There are two main type of delays differentiated. Primary delays are caused directly by a disruption from within the trains’ scheduled process. The causes of these delays can vary greatly between technical malfunctions, human-related interference or weather conditions. Secondary delays are caused by the disruptions from other trains that share the same infrastructure. The distinction between these types of delays is important since both require separate measures to manage the delays.
Railway network operators and passengers are interested in train delay data because it allows them to define the performance of their networks. There are various ways to indicate the performance and reliability of a railway network, no indicator however is perfect or able to fully describe all aspects of train delays and the passengers’ valuation of the delays in a network.
To be able to perform a scientific study (regarding train delays) it is vital to gather data. There are many different railway operators worldwide and they all use different technologies to log the train movements. The use of certain (older) technologies can create limitations to the accuracy of the data. If there is no public accessible database with train movements or delay measurements available, it may be required to gather and filter the desired data personally. Some methods of data collection from studies about delays in various railway networks are mentioned.
A large variety of theoretical distributions have been used to describe the distributions of train delays. It is impossible to designate one distribution as the “best” distribution model for all train delays since all delays show different characteristics. Delay characteristics are different for arrival delays and departure delays, different per country, different per year, different on routes with single or double track etc. Despite this large variety in all delay characteristics, many studies have succeeded in fitting a theoretical distribution to (part of) the measured delays. These studies are vital for further research in to the propagation and management of train delays.
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