Capturing stochastic variations of train event times and process times with goodness-of-fit tests

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1 Capturing stochastic variations of train event times and process times with goodness-of-fit tests Jianxin Yuan Department of Transport and Planning, Delft University of Technology P.O. Box 5048, 2600 GA Delft, The Netherlands, Abstract In this paper, several statistical distributions selected for capturing stochastic variations of train event times and process times are assessed based on real-world data recorded at a Dutch railway station The Hague Holland Spoor. The assessment is performed using the Kolmogorov-Smirnov goodness-of-fit test. To achieve a precise assessment of those candidate distributions, a new approach to fine-tuning the distribution parameters is proposed. The distributions are compared not only for the arrival and departure times of trains at the station, but also for the arrival times of trains at the boundaries of the local railway network and the train running, dwell and track occupancy times in the network. Furthermore, the selected distributions are assessed for the conditional running and dwell times as well as track occupancy times of trains in case of (no) hinder caused by other trains. The analysis results reveal that the stochastic variations of train event times and process times can be well captured by either the lognormal distribution or the Weibull distribution. These fitted statistical distributions can be used as input models in the analysis of timetable robustness and prediction of the punctuality of train operations. Keywords Train delays, Running & dwell times, Statistical distributions, Parameter estimation, Goodness-of-fit 1 Introduction To analyze the robustness of timetables and predict the punctuality of train operations in a railway network, the distributions of initial delays at the network boundaries and the distributions of additional delays within the network are often assumed. Limited literatures [7], [13], [11], [4], [5] exist for a statistical analysis of the distributions of train delays based on real-world data. It is doubtful whether an analytical or simulation model could accurately predict the propagation of train delays and the punctuality of trains without a realistic description of stochastic variations of the initial and additional delays [6], [12]. The initial delay of a train is the difference between the actual event time and the scheduled time at a network boundary, while an additional delay within the network is the difference between the actual and scheduled time periods for completing a running or dwell process. This paper aims to capture the stochastic variations of train event times and process times using real-world traffic data, i.e. to obtain the distributions of initial and additional delays statistically. For the analysis and prediction of timetable robustness and train punctuality, it is important to make a distinction between primary delays and knock-on delays occurring in an operational process. The primary delays of trains may be due to technical failures, 1

2 running at a speed lower than scheduled, prolonged alighting and boarding times of passengers, and bad weather conditions, while the knock-on delays of trains result from tight headway, route conflicts and late transfer connections at stations. For the primary delays in train running or dwell processes, it appears that the distributions cannot be derived directly from track occupation and release records, which can only show the total delays and may include knock-on delays. Therefore, data filtering is necessary to model the distribution of primary delays. To accurately estimate the knock-on delays of trains suffered before occupying a certain track junction or station platform, the part of delays due to deceleration and acceleration in the case of tight headway and route conflicts must be taken into account. Therefore, the conditional distributions of train running and track occupancy times in case of hinder are to be studied, too. To capture the variability of train event times and process times, several statistical distributions are first selected and then assessed with goodness-of-fit tests. Following this brief introduction, the collection of data is outlined. Afterwards, a discussion on the selection of appropriate statistical distributions and an introduction to the distribution assessment approach, i.e., the goodness-of-fit test is given. Since a hypothesized distribution with grossly estimated parameters can be easily rejected by a goodness-of-fit test especially in case of a large sample size, a new approach to fine-tuning the parameters of hypothesized distributions is proposed. Furthermore, the distribution fitting results are shown. The final section of the paper covers the main conclusions. 2 Obtaining data This statistical analysis is based on the track occupancy and release times of 10,000 trains which were recorded at a Dutch railway station The Hague Holland Spoor (The Hague HS). In the Dutch Railways, the train describers, i.e., TNV systems (Trein- NummerVolgsystemen) keep track of the progress of trains at discrete steps over their routes. The TNV-logfiles of about 25 MB ASCII-format per day per TNV-system contain chronological information about all signalling and interlocking events in a traffic control area. The TNV logfiles give an accurate description of train movements with a maximal error of 1s, but track section messages are not matched to individual train numbers. By posterior analysis of the TNV logfiles, Goverde and Hansen [3] developed the data mining tool TNV-Prepare that couples events of infrastructural elements to train numbers. The TNV-Prepare output consists of chronological TNV-tables per train line service and (sub)route. For each individual train, event times along the route are given, including train description steps, section entries and clearances, signals, and point switches. From this information other interesting process times such as running times, blocking times, and headways can be derived easily. Because the actual arrival and departure times of a train at a platform stop are not recorded in the TNV records, the tool TNV-Filter [4] was developed which estimates these event times based on data generated by TNV-Prepare. The arrival and departure delays as well as the dwell time of each individual train are then calculated with a precision in the order of a second. During the time period of data recording, 24 passenger trains arrived and departed at The Hague HS per hour corresponding to 9 different train series in both the southbound and northbound directions. The types and routes of trains may affect the distributions of train event times and process times. The assessment of those candidate distributions is to be performed per train series in each direction. A train series in one direction is hereafter referred to as a studied case. 2

3 3 Selecting statistical distributions and testing the goodness-of-fit 3.1 Select statistical distributions An appropriate distribution can be selected based on the physical properties of a random variable. However, it is often difficult to know all the properties when the variable represents a result of a great number of complicated processes influenced by human behaviours. The statistical distribution of a random variable can also be selected by inspecting the data. In this case, a histogram is a useful representation since it shows the distribution shape of the data. It is best to use both these approaches to ensure that they are in agreement [10]. Train event times and process times generally have a lower bound, i.e., the earliest or shortest time. Furthermore, the histograms are often skewed to the right especially in the case of train event times, which are subject to an accumulation of the previous delays. It appears that the lognormal, gamma, and Weibull distributions, which are very flexible in the fitting to the shape of the distribution of a data set, may capture the stochastic variations of train event times and process times with a higher accuracy than other distributions. A symmetrical distribution without a lower bound such as the normal distribution cannot be a generic candidate distribution, although this sort of distribution might approximate the variability reasonably well in some situations. The histograms of train arrival and departure times at stations are subject to the alteration of train orders in real-time operations and thus often have a heavy tail on the right. It seems that a mixture distribution may better represent the variability since this sort of distribution enables to incorporate the heavy tail. Güttler [5] fitted a normallognormal mixture distribution to excess running times of trains between two stations using the data obtained from the German Railway. Note that finite mixtures will introduce additional complexity because they have more parameters, whose estimation usually requires solving a complex system of non-linear equations. Moreover, train planners generally prefer to use a simpler distribution model if it can approximate the reality reasonably well. Thus, fitting a mixture distribution to the observed event times and process times will not be studied. Instead, we will pay much attention to conditional distributions of the running times of trains in case of freely and hindered travelling as well as the conditional distribution of the dwell times of trains in case of no hinder. This aims at a provision of input distribution models for predicting knock-on delays and the impact on train punctuality in railway stations and networks [14]. Based on the analysis above, the lognormal, gamma, and Weibull distributions are mainly selected for capturing the stochastic variations of train event times and process times. The exponential distribution, which is widely used to approximate the variability of non-negative delays, is actually incorporated since it is a special case of the gamma and Weibull distributions. The normal, uniform, and beta distributions have been applied in some literatures [1], [4] and we also consider them as the candidate distributions for modelling the variability of train event times and process times. 3.2 Test the goodness-of-fit For assessing the goodness-of-fit of statistical distributions, both graphical approaches and goodness-of-fit tests can be applied. Law and Kelton [8] presented five main graphical approaches for comparing fitted distributions with the true underlying distribution. These approaches include density/histogram overplot, frequency comparison, distribution 3

4 function differences plot, probability-probability plot and quantile-quantile plot. Graphical approaches give a good guide as to the goodness-of-fit and enable us to weed out poor candidates. Following on from a graphical analysis it is important to perform statistical tests to determine how closely a distribution fits the empirical data. The oldest goodness-of-fit test is the chi-square test, which can be thought of as a more formal comparison of a histogram with the fitted density function. Another commonly used goodness-of-fit test is the Kolmogorov-Smirnov (K-S) test, which compares an empirical distribution function with the hypothesized distribution function. For detailed description as to these goodness-of-fit tests, see Law and Kelton [8]. Both the goodness-of-fit tests have their own advantages and drawbacks. Perhaps the main advantage of the chi-square test is that when unknown parameters must be estimated from the data, a generic correction can be introduced in the statistic distribution by reducing the number of degrees of freedom. If parameters must be estimated from the data to apply the K-S test, no general adjustment is available for the critical value of the statistic distribution. However, the K-S test also has several advantages over the chisquare test. First, the K-S test does not require to group the data in any way, so no information is lost; this also eliminates the troublesome problem of interval specification, which occurs in case of the chi-square test. Secondly, the K-S test is exact for any sample size and any distribution model when the parameters are not estimated from the data, whereas the chi-square test is valid only in an asymptotic sense. For several hypothesized continuous distributions, if the parameters are specified without making use of the data, the K-S test can be applied to accurately assess and rank the goodness-of-fit of those distributions. In the following, the K-S test will be applied to compare the distributions selected for capturing the stochastic variations of train event times and process times. 4 Assessing the selected distributions based on a parameter finetuning approach Having selected the statistical distributions for train event times and process times, we need to further specify the parameters to test the goodness-of-fit. If the parameters of a selected distribution were estimated directly from the data which are used to obtain the empirical distribution in the goodness-of-fit test, the critical value of the test statistic distribution, depending on the type of the hypothesized distribution, could not be obtained accurately. To assess the selected statistical distributions, we will specify the parameters without making use of the data directly, and apply the K-S test and rank the goodness-offit. As mentioned before, if the distribution parameters are specified grossly, the hypothesized distribution can often be rejected especially in case of a large sample size of the data. Therefore, we propose a new approach to fine-tuning the parameters of the selected distributions. The assessment of the candidate distributions will be carried out based on this parameter fine-tuning approach. To specify the parameters and test a selected statistical distribution for a sort of train event times or process times, we first split the data into two subsets randomly, e.g. by assigning the chronologically recorded times alternately. Using the first data subset, an initial estimate of the distribution parameters is obtained. Furthermore, we fine-tune the parameters to improve the goodness-of-fit of the K-S test where the empirical distribution is, however, obtained from the second data subset. The sample size of these data subsets has been halved, but the all-parameters-known form of the K-S test can be applied. In this case, the critical value of the test statistic distribution can be accurately estimated and 4

5 applicable to all distribution types. In addition, randomly splitting the original data into two subsets ensures that the distribution fitted well to the second data subset is the distribution we need for capturing the variability of the original data. The parameters of a given distribution can be estimated on the basis of empirical data using the moment method, maximum likelihood method, and Bayesian estimation. The maximum likelihood method is widely used in practice because a Maximum Likelihood Estimator (MLE) is the minimum variance unbiased as the sample size increases [2]. The estimation of distribution parameters can be influenced by outliers. An outlier is a data point which deviates from the bulk of the data [4]. In the case of train event times and process times, the outliers are mostly some very large values, as has been confirmed by our data analysis. The impact of an outlier depends on the sample size and the spread of the data. However, there is no exact standard method to detect outliers. A new heuristic procedure to deal with outliers is incorporated in the following approach to fine-tuning the parameters of a candidate distribution of a data set: An initial estimate of the distribution parameters is obtained using the MLE based on the first split data subset. The large delays in the data subset are omitted iteratively one by one estimating the distribution parameters correspondingly using the MLE. In each iteration, we compute the p-value of the K-S test where the empirical distribution is obtained from the second split data subset. The iterative procedure terminates if the p-value cannot be increased any more. After the iterative procedure, we fine-tune the parameter estimate again to further improve the estimation by maximizing the p-value of the K-S test, where of course the empirical distribution remains unchanged. In detail, a neighbourhood ±1.0 and a fine-tuning step 0.1 are considered for the parameters of the lognormal distribution as well as the shape parameters of the gamma, Weibull and beta distributions. In case of the other distribution parameters, a neighbourhood ±10 and a fine-tuning step 1.0 are considered. Herein, the time unit is in seconds. The parameters of a statistical distribution can be classified, on the basis of their physical or geometric interpretation, as one of three basic types: location, scale and shape parameters. If the location parameter of a statistical distribution is the lower endpoint of the distribution s range and is nonzero, this location parameter is also called shift parameter and the distribution is called a location-shifted distribution. Train event times and process times generally have a lower bound. Therefore, a location-shifted distribution will be mostly applied. It should be mentioned that we take the shift parameter of a statistical distribution as the minimal value of the data. To assess those candidate distributions, the K-S tests are performed at a commonly adopted significance level of Each type of the distributions is ranked according to the p-value of the K-S test where the hypothesized distribution is specified with the finetuned parameters. In detail, we give rank 1 to the candidate distribution with the biggest p- value, rank 2 to the distribution with the second biggest p-value, and so on. The candidate distribution ranked 1 is considered to be the best one among the selected distributions. The quality of the distribution fitting for train event times and process times is also visualized by comparing the fitted distribution density curve with the kernel density estimate [2] and the histogram and by applying the distribution differences plot for the fitted distribution and the empirical one. 5

6 5 Distribution assessment results 5.1 Train event times The statistical fitting of train arrival time distribution is a prerequisite for predicting the propagation of train delays in stations. To incorporate the impact of knock-on delays suffered before occupying the platform track in a delay propagation model, we need to distinguish the arrival times of trains at the platform track from those at the station approach signal. We have compared the K-S goodness-of-fit among the candidate distributions for the arrival times at those locations. It has been found that the lognormal distribution gives the best fit in 9 and 11 out of 14 studied cases (a case represents a train series in one direction), respectively. Additionally, the lognormal fits have been accepted by the K-S test in most cases. It should be mentioned that the lognormal fits have been specified with a shift parameter corresponding to the earliest arrival time. Figure 1 shows the lognormal density fit, the second best, i.e. gamma density fit, kernel density estimate and empirical histogram for the arrival delays of an intercity train series in the northbound direction at the platform track. The distribution differences plots are also given in Figure 2. Both the density fits match with the kernel density estimate and the histogram rather well. It appears that the lognormal fit is more attractive than the gamma fit since the former has a greater probability around the distribution mode than the latter. The differences between the lognormal distribution and the empirical one are overall smaller than the differences between the gamma distribution and the empirical one, too. Figure 1: Lognormal and gamma density fits, kernel density estimate and histogram for the arrival delays of an intercity train series IC2100N at the platform track Figure 2: Distribution difference plot for the lognormal fit and the arrival delays of IC2100N at the platform track and the plot for the gamma fit and the delays Early arriving trains generally have a longer dwell time than late arriving trains. To estimate the distribution of departure times more realistically by distinguishing the dwell times of late arriving trains from those of early arriving trains, the distribution of nonnegative arrival delays is needed [12]. We have compared the K-S goodness-of-fit among the candidate distributions for non-negative arrival delays at the platform track. The analysis results reveal that the Weibull distribution gives the best fit to the data in 12 out 6

7 of 18 studied cases and it ranks second or third in the other 6 cases. In addition, the Weibull fits have been accepted by the K-S test in all the cases. Figure 3 shows the Weibull density fit, the second best, i.e. gamma density fit as well as the empirical histogram for the non-negative arrival delays of an interregional train series in the northbound direction. It appears that the Weibull density fit matches with the histogram better than the gamma fit. Figure 4 displays the distribution differences plots. The differences between the Weibull distribution and the empirical distribution are overall smaller than the differences between the gamma distribution and the empirical one. Probability density Weibull fit gamma fit Fitted distribution Empirical distribution Weibull fit gamma fit Non negative arrival delay IR2200N [s] Figure 3: Fitted Weibull and gamma density curves and histogram of the nonnegative arrival delays of an interregional train series IR2200N Non negative arrival delay IR2200N [s] Figure 4: Distribution differences plot for the Weibull fit and the non-negative arrival delays of IR2200N and the plot for the gamma fit and the delays The distribution of departure times at stations can be used to predict the distribution of outbound track release times and the distribution of arrival times at the following stations. For the departure delays of trains, the Weibull distribution fits best to the data in 11 out of 18 studied cases. Furthermore, the Weibull fits have been accepted by the K-S test in most cases. Figure 5 and Figure 6 visualize the goodness-of-fit of the Weibull and gamma distributions for the departure delays of an intercity train series in the southbound direction. It appears that the Weibull density fit matches with the data better than the gamma fit and the maximum difference between the Weibull distribution and the empirical distribution is smaller than the maximum difference between the gamma distribution and the empirical one. For both non-negative arrival delays and departure delays, the Weibull distribution fits the data generally better than the gamma distribution. This may result from a more flexible property of the Weibull model than the gamma model for capturing the distribution shape of a data set. The kernel density curves estimated for non-negative arrival delays and departure delays are decreasing except for few cases. As a result, the shape parameter of a Weibull distribution fit is generally smaller than 1.0. If the finetuned shape parameter is around 1.0, the exponential distribution, as a special type of the Weibull distribution, can be used to capture the stochastic variations of both non-negative arrival delays and departure delays. 7

8 Probability density Weibull fit gamma fit Fitted distribution Empirical distribution Weibull fit gamma fit Departure delay IC2400S [s] Figure 5: Fitted Weibull and gamma density curves and the empirical histogram of the departure delays of an intercity train series IC2400S Departure delay IC2400S [s] Figure 6: Distribution differences plot for the Weibull fit and the departure delays of IC2400S and the plot for the gamma fit and the delays 5.2 Train process times Train process times include the dwell times, running times, and track occupancy times of trains. The dwell times of trains at a station are the difference between the arrival and departure times. To estimate knock-on delays and departure delays of trains at a station, we need to know the necessary dwell times of trains for passenger alighting and boarding in the absence of hindrance from other trains. These necessary dwell times of trains are defined as the free dwell times in Yuan [12]. To accurately incorporate the impact of the knock-on delays of trains suffered before occupying a platform track or a junction in the modelling of delay propagation in a railway network, the conditional distributions of train running times and track occupancy times in case of (no) hinder are needed. To obtain the distribution of the free dwell times of trains, we need the corresponding data. However, it is hardly possible to measure the free dwell time of an individual train which is hindered at the platform track due to occupancy of the next signal block only based on the track occupancy and release records, i.e. TNV data. Therefore, the free dwell time distributions will be only fitted for early and late arriving trains respectively by using the observed dwell times of a subset of those trains which are not hindered at the station. In case of early arriving trains, the subset contains the trains whose outbound route has been set to Free and the departure signal to Go before it is ready for departure assumed at the scheduled departure time. In case of late trains, the subset includes the trains whose outbound route has been set to Free and the departure signal to Go before it is ready for departure after an assumed minimal dwell time of 30s. In fact, some trains which do not belong to the subsets are not hindered by other trains, too, when they have a longer necessary dwell time. Note that the free dwell time of an early arriving train is influenced by the arrival earliness significantly, and the knock-on delay and the departure delay may be estimated based on the scheduled arrival time and the free dwell time excluding the arrival earliness [12]. Therefore, we take only this part of the free dwell time as the free dwell time for an early train in this analysis. 8

9 The statistical analysis results reveal that for the free dwell times of early arriving trains, the Weibull distribution gives the best fit in 8 out of 15 studied cases. For the free dwell times of late arriving trains, the Weibull distribution fits best to the data in 7 out of 18 studied cases. Additionally, the Weibull distributions fitted to the free dwell times of both early and late arriving trains have been accepted by the K-S test except for one studied case. Figure 7 shows the Weibull density fit, the second best, i.e. normal density fit, kernel density estimate and empirical histogram for the free dwell times of the late arriving trains of an interregional train series in the northbound direction. The fitted Weibull distribution with a shape parameter of 1.8 matches with the kernel estimate for the empirical data much better than the normal fit with respect to overall shape and the tails of the distribution. The better fitting of the Weibull model is also illustrated by the distribution differences plots given in Figure 8. Figure 7: Weibull and normal fits, kernel estimate and histogram for the free dwell times of late arriving trains of an interregional train series IR2200N Figure 8: Distribution differences plots for the Weibull and normal fits and the free dwell times of late arriving trains of IR2200N The density estimate and empirical histogram of the free dwell times of trains are generally skewed to the right. The free dwell times of early arriving trains cannot be shorter than the scheduled dwell time. The free dwell times of late arriving trains may be shorter than the scheduled time, but they are longer than the minimum time for opening the doors of a train, announcing the departure, and closing the doors. Taking into account the skewness to the right and the lower bound, a location-shifted Weibull distribution better captures the variability of the free dwell times of trains than a normal distribution. In case of an early arrival, the train driver may not feel in hurry and hence wait for some late passengers boarding the train. In case of a late arrival, the train may stop at the platform longer than the minimum dwell time due to an increased number of passengers boarding the train. Obviously, the probability of large free dwell times is very small. Therefore, the probability density curve of the free dwell times of trains is generally nondecreasing and unimodal, and the shape parameter of the Weibull distribution fit is mostly bigger than 1.0. Only if the scheduled dwell time is much longer than the necessary times for passenger alighting and boarding, the shape parameter of the Weibull distribution fitted to the free dwell times of early arriving trains (excluding the arrival earliness) can be smaller than

10 In the following, we deal with the conditional distributions of the running times of trains on the preceding block of The Hague HS station and those of the occupancy times of the adjacent junctions around this station. Considering an approaching train, if the inbound route is released earlier than the train arrives at sight distance of the approach signal of the station home signal, the train proceeds freely to the platform. Otherwise, the train is hindered and has to decelerate and even to stop before the home signal. The conditional distributions of train running and track occupancy times in case of (no) hinder differ from each other significantly. To obtain those conditional distributions, the first step is to classify the data. By comparing the arrival time of each train at the station approach signal to the clearance time of the inbound route, we have extracted a data set suited for fitting the distribution of freely running times in each studied case. A hindered approaching train will pass the station home signal at a reduced speed if it may proceed to this signal. Otherwise, the hindered train will pass the home signal accelerating after a stop before this signal. Since the standstill of a train on a track is not recorded, we cannot directly identify whether or not a hindered train stops before the home signal based on track occupancy and release records. However, we have realized that the difference between the arrival time of a hindered train at the station approach signal and the clearance time of the inbound route is generally longer in case the train stops before the home signal. On the other hand, the difference between the route clearance time and the passing time of the train at the station home signal is generally shorter in case the train stops before the home signal than if it may proceed. Adopting the k-means data clustering routine included in the statistical analysis tool S-Plus [9], we split the data sample of hindered trains for each studied train series into two separate parts which correspond approximately to the aforementioned two cases. Applying the k-means data clustering method, the data sets suited for fitting the conditional distributions of inbound junction occupancy times by each relevant passing train series were also extracted. In case of a departing train, if it is hindered due to outbound route conflicts, it dwells at the station for a longer time. However, it will not be hindered on the next track sections. Thus, the conditional distributions are not applicable to the running times of trains on the outbound track sections and outbound track occupancy times. For the distributions and conditional distributions of train running and track occupancy times, either the Weibull or normal model may be the best fit to the data. However, a generic distribution model suitable for capturing the variability has not been found. This might be caused by the big variation of train speeds on the short track sections in the complicated station and interlocking area. The data classification and the further data split, which reduce the size of the data sample, also affect the determination of a generic distribution model for the conditional train running and track occupancy times. 6 Conclusions We have compared the K-S goodness-of-fit among several distribution models selected for train event times and process times by fine-tuning the distribution parameters. The track occupancy and release times recorded at the Dutch railway station The Hague HS were used. It has been found that a location-shifted lognormal distribution can be considered as the best model among the candidate distributions for both the arrival times of trains at the platform track and at the station approach signal. The Weibull distribution can generally be considered as the best distribution model for non-negative arrival delays, 10

11 departure delays and the free dwell times of trains. The shape parameter of a Weibull distribution fitted to either non-negative arrival delays or departure delays is mostly smaller than 1.0. However, the shape parameter of a Weibull distribution fitted to the free dwell times of trains is generally bigger than 1.0. These distribution fitting results can be used as input models of any analytical or simulation models for predicting the propagation of train delays and the punctuality of trains at certain stations. Based on this prediction, we are able to maximize railway capacity utilization while assuring a desired reliability and punctuality level of train operations. Although the distribution assessment has been done using the data recorded at a case station, the fitted lognormal and Weibull distributions can be applicable to other existing and new stations thanks to the flexibility of these two distribution models. However, it should be kept in mind that the distribution parameters of train event times and process times may vary in time and space. When timetables for a new infrastructure project are designed and evaluated and real data of train event times and process times are not available, the distribution parameters must be estimated as accurately as possible by adopting advises from experts. Moreover, a sensitivity study of the modelling results to the estimated parameters of the input distributions may be required. The distributions and conditional distributions of train running and track occupancy times have also been studied. However, a generic distribution model suitable for capturing the variability has not been found. Further research work will be carried out to capture the variability by collecting a bigger size of data sample. The variability of train running times between stations will also be studied. References [1] Carey, M., Carille, S, Testing schedule performance and reliability for train stations, Journal of the Operational Research Society 51, pp , [2] Dekking, F.M., Kraaikamp, C., Lopuhaä, H.P., Meester, L.E., A Modern Introduction to Probability and Statistics, Springer, London, [3] Goverde, R.M.P., Hansen, I.A., TNV-Prepare: Analysis of Dutch Railway Operations Based on Train detection Data', In: Allan, J., Hill, R.J., Brebbia, C.A., Sciutto, G., Sone, S. (eds.), Computers in Railways VII, pp , WIT Press, Southampton, [4] Goverde, R.M.P., Punctuality of Railway Operations and Timetable Stability Analysis, Ph.D. thesis, Delft University of Technology, [5] Güttler, S., Statistical modelling of Railway Data, M.Sc. thesis, Georg-August- Universität zu Göttingen, [6] Hansen, I.A., Improving Railway Punctuality by Automatic Piloting, In: 2001 IEEE Intelligent Transportation Systems Conference Proceedings, Oakland (CA), USA, pp , [7] Hermann, U., Untersuchung zur Verspätungsentwicklung von Fernreisezügen auf der Datengrundlage der Rechnerunterstützten Zugüberwachung Frankfurt am Main (Investigation of the Development of Delays of Long-distance Passenger Trains Based on Data from the Computer-aided Train Monitoring Frankfurt am Main), Ph.D. thesis, Technischen Hochschule Darmstadt, [8] Law, A. M., Kelton, W.D., Simulation Modeling and Analysis, McGraw-Hill Higher Education, [9] MathSoft, S-Plus 2000, User s Guide, Seattle, USA, [10] Robinson, S., Simulation: The Practice of Model Development and Use, John Wiley 11

12 & Sons, Ltd, Chichester, [11] Wendler, E., Naehrig, M., Statistische Auswertung von Verspätungsdaten (Statistical Analysis of Delay Data), Eisenbahningenieurkalender EIK 2004, pp , [12] Yuan, J., Stochastic Modelling of Train Delays and Delay Propagation in Stations, Ph.D. thesis, Delft University of Technology, [13] Yuan, J., Goverde, R.M.P. & Hansen, I.A., Propagation of Train Delays in Stations, In: Allan, J., Hill, R.J., Brebbia, C.A., Sciutto, G., Sone, S. (eds.), Computers in Railways VIII, pp , WIT Press, Southampton, [14] Yuan, J., Hansen, I.A., Optimizing Capacity Utilization of Stations by Estimating Knock-on Delays, Transportation Research Part B 41(2), pp ,