Running times on railway sections with heterogeneous train traffic

In contemporary railway traffic, different train services often run on the same track of a railway section. Since these train services generally travel at different speed, this may result in delays due to fast trains being caught behind slower ones. This note investigates these delays developing a stochastic model capturing both scheduled and unscheduled train movements. Running time distributions for each train service can be obtained solving a system of linear differential equations. This system can be solved explicitly when free running times for all train services are deterministic. The mathematical model falls in the class of infinite server resequencing queues, and the results generalise existing results for such queues, allowing dependence between consecutive interarrival times, as well as between interarrival times and service times. A case study of a railway section in the Dutch railway network illustrates the practical value of the model, both for long and short term railway planning.