  In this paper we investigate real-time, dynamic traffic optimization in
railway systems. In order to enable practical solution times, we operate the
optimizer in a receding horizon fashion and with optimization horizons that are
shorter than the full path to destinations, using a model predictive control
(MPC) approach. We present new procedures to establish safe prediction
horizons, providing formal guarantees that the system is operated in a way that
satisfies hard safety constraints despite the fact that not all future train
interactions are taken into account, by characterizing the minimal required
optimization horizons. We also show that any feasible solution to our proposed
models is sufficient to maintain a safe, automated operation of the railway
system, providing an upper bound on the computations strictly required.
Additionally, we show that these minimal optimization horizons also
characterize an upper bound on computations required to construct a feasible
solution for any arbitrary optimization horizon, paving the way for anytime
algorithms. Finally, our results enable systematic solution reuse, when
previous schedules are available. We test our approach on a detailed simulation
environment of a real-world railway system used for freight transport.
