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Deteriorations are considered in the design phase of timetables to maintain reliability in actual operations. The response of periodic timetables to deteriorating events should be tracked and delays are desired to settle within a reasonable amount of time. We analysed an urban and intercity train service operating in separate lines in the Dutch railway network to measure their service reliability. Because trains are supposed to periodically depart from their respective terminals according to their schedules, their arrival delays and the allocated terminal slacks play a crucial role in reliability. We adopted a Markov chain model to represent the delay evolution of trains and examine their recovery patterns. The steady-state probabilities were uniquely and precisely obtained using the non-homogeneous matrices of the successive processes in combination within a cycle. It is observed that the timetables of the two train services can return to their periodicity and steady-state at most in one cycle.

A stochastic model for reliability analysis of periodic train timetables