A discrete event traffic model explaining the traffic phases of the train dynamics on a linear metro line with demand-dependent control

In this paper we present a mathematical model of the train dynamics in a linear metro line system with demand-dependent run and dwell times. On every segment of the line, we consider two main constraints. The first constraint is on the travel time, which is the sum of run and dwell time. The second one is on the safe separation time, modeling the signaling system, so that only one train can occupy a segment at a time. The dwell and the run times are modeled dynamically, with two control laws. The one on the dwell time makes sure that all the passengers can debark from and embark into the train. The one on the run time ensures train time-headway regularity in the case where perturbations do not exceed a run time margin. We use a Max-plus algebra approach which allows to derive analytic formulas for the train time-headway and frequency depending on the number of trains and on the passenger demand. The analytic formulas, illustrated by 3D figures, permit to understand the phases of the train dynamics of a linear metro line being operated as a transport on demand system.