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Maintaining high on-time performance and at the same time having high capacity utilization is a challenge for several railway traffic systems. The system becomes sensitive to disturbances and delays are easily propagating in the network. One way to handle this problem is to create more robust timetables; timetables that can absorb delays and prevent them from propagating. This paper presents an optimization approach to reduce the propagating of delays with a more efficient margin allocation in the timetable. A Mixed Integer Linear Programming (MILP) model is proposed in which the existing margin time is re-allocated to increase the robustness for an existing timetable. The model re-allocates both runtime margin time and headway margin time to increase the robustness at specific delay sensitive points in a timetable. We illustrate the model’s applicability for a real-world case where an initial, feasible timetable is modified to create new timetables with increased robustness. These new timetables are then evaluated and compared to the initial timetable. We evaluate how the MILP approach affects the initial timetable structure and its capability to handle disturbances by exposing the initial and the modified timetables to some minor initial disturbances of the range 1 up to 7 minutes. The results show that it is possible to reduce the delays by re-allocating margin time, for example, the total delay at end station decreases with 28 % in our real-world example. Robusta tidtabeller för järnvägstrafik

In this paper we present a method for modelling uncertainty of train delays based on a Markov stochastic process. The dynamics of a train delay over time and space is presented as a stochastic process that describes the evolution of the time-dependent random variable. Probability distribution of an arrival delay in a station changes over time in discrete steps as more information becomes available. We consider and compare the results and computational requirements of two discrete state space formulations. Moreover, we test the applicability of modelling train delays as a non-stationary Markov chain, meaning that the probability of a state change depends on the moment of transition. The model is applied on a set of historical traffic realisation data from the part of the high-speed corridor between Beijing and Shanghai. We analyse the accuracy of predictions as well as the evolution of probability distributions of all events over time. The presented method is important for making better predictions for train traffic, that are not only based on static, offline collected data, but are able to positively include the dynamic characteristics of the ever-changing delays, thus increasing the reliability of prediction by 71%.

The growing demand for railway capacity has led to high capacity consumption at times and a delay-sensitive network with insufficient robustness. The fundamental challenge is therefore to decide how to increase the robustness. To do so there is a need for accurate measures that return whether the timetable is robust or not and indicate where improvements should be made. Previously presented measures are useful when comparing different timetable candidates with respect to robustness, but less useful to decide where and how robustness should be inserted. In this paper, we focus on points where trains enter a line, or where trains are being overtaken, since we have observed that these points are critical for the robustness. The concept of critical points can be used in the practical timetabling process to identify weaknesses in a timetable and to provide suggestions for improvements. In order to quantitatively assess how crucial a critical point may be, we have defined the measure RCP (Robustness in Critical Points). A high RCP value is preferred, and it reflects a situation at which train dispatchers will have higher prospects of handling a conflict effectively. The number of critical points, the location pattern and the RCP values constitute an absolute value for the robustness of a certain train slot, as well as of a complete timetable. The concept of critical points and RCP can be seen as a contribution to the already defined robustness measures which combined can be used as guidelines for timetable constructors.

Several European railway traffic networks experience high capacity consumption during large parts of the day resulting in delay-sensitive traffic system with insufficient robustness. One fundamental challenge is therefore to assess the robustness and find strategies to decrease the sensitivity to disruptions. Accurate robustness measures are needed to determine if a timetable is sufficiently robust and suggest where improvements should be made. Existing robustness measures are useful when comparing different timetables with respect to robustness. They are, however, not as useful for suggesting precisely where and how robustness should be increased. In this paper, we propose a new robustness measure that incorporates the concept of critical points. This concept can be used in the practical timetabling process to find weaknesses in a timetable and to provide suggestions for improvements. In order to quantitatively assess how crucial a critical point may be, we have defined the measure Robustness in Critical Points (RCP). In this paper, we present results from an experimental study where a benchmark of several measures as well as RCP has been done. The results demonstrate the relevance of the concept of critical points and RCP, and how it contributes to the set of already defined robustness measures.

A tendency seen for quite some time in the Swedish railway network is a growing demand for capacity which no longer can be accommodated. This causes congestion and delays, and the relationships between the trains and how they affect eachother are significantly harder to overview and analyse. Railway traffic timetables normally contain margins to make them robust, and enable trains to recover from certain delays. How effective these margins are, depends on their size and location as well as the frequency and magnitude of the disturbances that occur. Hence, it is important to include marigns so, that they can be used operationally to recover from a variety of disturbances and not restricted to a specific part of the line and/or the timetable. In a case study we compare the performance of a selection of passenger train services to the different prerequisites given by the timetable (e.g. available margins and their location, critical train dependencies). The study focuses on the Swedish Southern mainline between Stockholm and Malmö on which a wide variety of train services operate, e.g. freight trains, local and regional commuter train services as well as long-distance trains with different speed profiles. The analysis shows a clear mismatch between where margins are placed and where delays occur. We also believe that the most widely used performance measure, which is related to the delay when arriving at the final destination, might give rise to an unnecessarily high delay rate at intermediate stations.