Stochastic approximation to the effects of headways on knock-on delays of trains

In train planning and timetabling, the trip time on each link is assumed to depend on the type of train and characteristics of the link, but is usually treated as independent of the time interval (headway) between trains. However, in practice trains are subject to delays from a variety of causes, and since normally they are not allowed to pass each other on a link, any delay to one train may cause "knock-on" delays to following trains. This is especially true of the high density double and multiple track railways in Britain and Europe. The shorter the scheduled headway between trains, the greater is the expected knock-on delay and hence the greater the expected trip times of following trains. We develop simple stochastic approximations to these knock-on train delays. To test and calibrate the approximations, we conduct detailed stochastic simulation of the interaction between trains as they traverse sections of the link. The approximate relationships that we derive between scheduled headways and knock-on delays can be used, for example, (a) to provide correction factors for other stochastic or deterministic models of train planning, dispatching, or control; (b) to adjust train timetables, which are currently produced without explicitly considering the expected knock-on effects, and (c) to make it feasible to conduct larger scale simulations of train networks, by reducing or removing the need to simulate behaviour within each link.