A discrete event traffic model explaining the traffic phases of the train dynamics in a metro line system with a junction

Abstract

This paper presents a mathematical model for the train dynamics in a mass-transit metro line system with one symmetrically operated junction. We distinguish three parts: a central part and two branches. The tracks are spatially discretized into segments (or blocks) and the train dynamics are described by a discrete event system where the variables are the k th departure times from each segment. The train dynamics are based on two main constraints: a travel time constraint modeling theoretic run and dwell times, and a safe separation constraint modeling the signaling system in case where the traffic gets very dense. The Max-plus algebra model allows to analytically derive the asymptotic average train frequency as a function of many parameters, including train travel times, minimum safety intervals, the total number of trains on the line and the number of trains on each branch. This derivation permits to understand the physics of traffic. In a further step, the results will be used for traffic control.

Physical Models and Control of the Train Dynamics in a Metro Line Without Junction

“…A dynamic model of the stock of passengers on the platforms would improve the traffic dynamics, particularly by taking into account the capacity limits of the platforms. Another direction of research is to extend the approach to metro lines with junctions (first results have already been obtained in [22]). Finally, other control parameters such as the train running times (speed profiles) can be considered in addition to the train dwell times at platforms (first results have already been obtained in [23]).…”

“…and Condition (21) tells us that if for any j , the passenger arrival rate λ j is higher than κ f max , then, the max-plus train dynamics cannot be retrieved, whatever the number of trains we made in the metro line. Under the condition (21), condition (22) gives the number of trains to run on the metro line in order that the train dynamics behaves as the max-plus one. Here, we notice that the first inequality [ρ ≥ λ j /(vκ), ∀ j ] in (22) is more important than the second one (ρ ≤ρ − λ j /(w κ), ∀ j ), because the latter concerns high number of running trains, and corresponds to the congestion traffic phase of the train dynamics, which is better to avoid (see [13]).…”

Comprehensive passenger demand-dependent traffic control on a metro line with a junction and a derivation of the traffic phases

“…The model we present in this article extends both models of [13] and [14] in the following way. We show that the results on the stability of the train dynamics for a metro line with a junction still hold under the following changes.…”

“…We distinguish three parts of the metro line: a central part and two branches crossing at the junction. The metro line is discretized in space into a number of segments as in [7,8,9,10,13,14,15], all following the same discrete event modeling approach. A first model for the train dynamics on a metro line with a junction has been proposed in [13], with train dwell and run times respecting given lower bounds, i.e.…”

“…The metro line is discretized in space into a number of segments as in [7,8,9,10,13,14,15], all following the same discrete event modeling approach. A first model for the train dynamics on a metro line with a junction has been proposed in [13], with train dwell and run times respecting given lower bounds, i.e. they are constant and independent of the passenger travel demand.…”