Scheduled waiting time from crossing on single track railway lines
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For the long term planning of railway infrastructure several analyses are necesarry. One of them is the prognosticated demand for future railway transportation. This information serves as input parameter for the capacity and benefit cost analysis. For the capacity analysis, timetable independent analytical methods that can estimate the scheduled waiting time in dependency of the number of trains running on the line is required. If the scheduled waiting time is too long, the operators risk losing customers and the infrastructure operators risk losing the operators as customers. Changes must be made in infrastructure or in train mixture to reduce the scheduled waiting time in an early planning stage. There are only a few existing models concerning scheduled waiting time. The crossing situation has hardly been investigated in studies since the main focus has been on double track lines. The scheduled waiting time as a topic so far has not been paid enough attention to since unscheduled waiting time has been the main focus. Estimating the scheduled waiting time is even important than the estimation of the unscheduled waiting time. An overbooked railway line will seldom be as successful as a balanced railway line as a transportation offer or in operation. The planner’s task is to estimate and design an infrastructure that will support the market on time in best way with regards to both scheduled and unscheduled waiting time. There is a demand to develop and to improve models for the calculation of scheduled waiting time. This thesis is an attempt to meet a part of the demand of the research within this subject. In this thesis a deterministic analysis of the crossing situation of trains on single track lines has been performed. A new model for the calculation of the number of crossings, the expected waiting time for crossing and merging has been developed on the basis of the conditions formed for incuring a crossing. This analysis states that the conditions for a train to incur a crossing is similar to the conditions given in [Schw81] for an overtaking to take place. The analytical model developed makes use of stochastics in order to estimate the expected waiting time from crossing for each train model. This ensures a timetable independent estimation. The model focuses on mixed train traffic in both directions with a strict hierarchical priority system. An exponential buffer time distribution between the requested train paths of higher priority is assumed to make it possible for trains of lower priority to merge in between the trains of higher priority. This philosophy is based on a timetable construction process where different train models are given different priority due to their trackage rights achieved. The train model with highest priority will be included into the timetable first. The expected waiting time from crossing within a train model can then be estimated. In the model deduced, one direction is chosen to take priority over the opposite direction. The prior direction can, for example, be outward traffic. This means that the trains running in the direction of lower priority risks incuring waiting time from crossing. A case study of the timetable characteristics on single track lines was carried out to investigate whether the timetables follow a cyclic or a stochatic pattern. The noncyclic timetables were further analysed by a χ2 - test of goodness of fit of the buffer time between the trains of higher priority if the buffer time distribution could be exponential, hyper-exponential or Erlang2 distributed. An asynchronous simulation tool has been used to control and evaluate the model. Random timetables were generated and served as input to the simulation tool. A dummy illustrating a railway single track line was constructed and served as the main study object. An existing train model was chosen to run the line in both directions. The number of crossings, the number of multiple crossings and the waiting times were recorded and compared with the results from the model established. The evaluation of the reliability of the model requires more research by simulation before a satisfactory statement can be fulfilled. Two different priority strategies were simulated and compared. A priority strategy with equal priority between the trains of opposite directions tend to generate more crossings than a strategy with strict priority for one direction. On the other hand a strategy with one direction priori over the opposite direction tend to generated more multiple crossings relative to the number of crossings compared to the equal priority strategy. An attempt has been made to use the same methodology for the derivation of a model for the estimation of the expected waiting time from crossing with constant buffer times between the requested train paths of higher priority. This model has some weaknesses and has therefore not been further analysed. Finally, a sensitivity analysis of the model developed illustrates that the time gap necessary for a train to reach the next station before meeting an opposing train has the most influence on the estimated result. The model is therefore probably most suited for railway lines with less variation in the occupation time between the stations.