1. What is the importance of numerical stability analysis in railway planning?
Numerical stability analysis is crucial in railway planning as it helps identify parts of a timetable that are susceptible to delays or chain reactions of delays. By applying max-plus algebra to timed discrete event systems, railway companies can optimize system stability. This approach allows for the evaluation of the traffic system under consideration, considering operational trip times, headways, stopping times, transfer times, crossing times, transit times, and turning times. The system model created can be examined for its stability in case of disruptions. However, the typical algebraic calculation time of an evaluation process for a moderately sized part network of the RhB is still too high for practical integration into optimization procedures. Therefore, a method for contracting the network graph before applying max-plus stability analysis has been proposed to speed up the evaluation process without compromising the quality of the results. This method has been applied to a timetable scenario of a representative part of the RhB network, and the results of the accelerated procedure have been compared with those of the original procedure based on the uncontracted EAN. The reduced EAN size, calculation times, and calculated indicators are analyzed, evaluated, and discussed.
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2. What is the critical cycle in a system?
The critical cycle in a system is the one with the largest cycle mean or least buffer time to the timetable periodicity. It represents the length of time in which all cycles of the system can be carried out at least once. This cycle is significant for evaluating timetable stability indicators such as CDS and CDI. The formal descriptions of the critical cycle are detailed in Wust et al. [7] and Steiner [8].
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3. How does graph contraction impact system stability?
Graph contraction, based on first-or second-degree events, does not influence system stability. It allows removal and re-linking of upstream or downstream activities, assuming the recovery matrix provides the same result. This approach, proposed by Goerigk and Liebchen, optimizes PESP problems and is applicable to periodic EANs with exclusively accessible events. Second-degree events are contracted by removing them and linking their activities, adding process times at the new link. This step is illustrated in an example EAN.
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4. How is the quality of results measured in the contracted network?
The quality of results is measured by comparing the absolute and relative deviation of stability measures between the contracted and original network. Critical cycles length represents an absolute quality measure, while cumulative delay impact (CDIs) weighted by the number of events provides a relative quality measure. The plots in Figure 4 illustrate the good relative agreement between the two methods for train run 1125 Chur - St.Moritz.
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