On a Rail Transportation Model with Scheduled Services

A safety precaution used in rail transportation is to ensure that only one train occupies the track connecting two stations at any time. We consider a train station where passengers arrive according to a prescheduled timetable, and the transportation authority tries to dispatch the trains on their scheduled departure times. However, if the track to the next station is occupied by a preceding train at such a departure instant, then the departure of the next train is delayed, or even cancelled, to achieve safety in transportation. Under such circumstances, the dispatcher may choose to cancel overdelayed trains intentionally so that the succeeding trains can depart on time, or to dispatch some trains with faster speed to decrease service time. We analyze this rail transportation model to identify the relationships between various dispatching strategies, the arrival pattern of the passengers, and the performance measures involving the service delay and passenger waiting time. The steady-state distribution of the service delay is characterized as the solution of a system of linear equations and an explicit computational formula for the average passenger waiting time is derived. Interesting intuitive and counterintuitive examples are also provided in the context of an optimal dispatching problem.