Reliability of interconnected scheduled services

In this paper we consider the performance of interconnected scheduled transport services for which the service times have a significant random element. That is, the arrival and departure times vary randomly from trip to trip. Any delay to a service can affect its punctuality at later stops and can affect all following services. For specificity we refer mainly to scheduled train services throughout. The reliability and performance of schedules is often investigated by stochastic simulation. However, by adopting a more analytic approach we are able to: (a) derive some properties of various measures of performance and, (b) more easily find improved or optimized schedules. Both (a) and (b) can be difficult or intractable using a simulation approach. We define various measures of reliability and performance for scheduled services, and show how to compute these. We also show how to compute or adjust schedules so as to satisfy reliability goals or targets, or optimize costs or benefits, including reliability costs. We derive various properties of these measures of reliability. In particular, we consider how reliability and costs vary in response to varying the scheduled trip times and station dwell times, or arrival and departure times. To ensure generality, we do not specify any particular forms for any of the distributions of trip times, etc. – they can be theoretical or empirical.