PhD position: Analytic traffic modeling and control on a mass transit network including passenger flows

Type de recrutement
University Gustave Eiffel, Paris, France
Fin de l'affichage


Rail-bound transport systems such as subways or metros are the backbone of public transport in urban areas and mega cities. They provide a high level of passenger capacity and transport efficiency, and typically operate on specific tracks and lines in order to ensure high operating frequency and service quality. Given the limited routing flexibility and dense operation of trains, corresponding mass transit systems are susceptible to operational perturbations, resulting from, e.g., excessive station dwell times due to train or platform crowding. To optimally control and provide a high reliability of train operations, a detailed understanding of the behavior and performance of the transport system is required. To this end, advanced mathematical modelling techniques allowing to assess the inter-dependencies of trains are vital in providing decision support to traffic planning and management. This PhD project, which is co-located at Grettia laboratory of Univ. Gustave Eiffel and the DLR – Institute of Transportation Systems in Brunswick, Germany, aims to extend and improve existing modelling techniques to improve the quality of information available to rail traffic planners.


The objective of this thesis is to extend an existing algebraic approach for traffic modeling on isolated loop metro lines, to traffic modeling and control on a network of metros and mass transit lines. Several extensions will be considered:

1. Account for stochastic effects in relation to passenger boarding / alighting behavior and corresponding train station dwell times.

2. Isolated metro lines will be connected based on the flow of passengers in transfer stations. Starting from small networks with two or three lines, models with an increasing number of metro lines will be considered.

3. Finally, passenger flow assignment will be considered. Data on the passenger travel demand can be available thanks to a collaboration with RATP (the metro line operator Paris).

Methodology and related works:

In this thesis, we propose to consider an algebraic approach for modelling the train dynamics in metro lines, which can be used both in a deterministic and stochastic setting allowing to incorporate different kinds of uncertainties, in particular concerning the passenger flow dynamics. The passenger traffic assignment is to be introduced using equilibrium and optimization models to models based on the operating status of the network. Concerning the algebraic approach for the train dynamics, the first traffic model was proposed in [1]. It is a Max-plus algebra model for the train dynamics on a loop metro line, where lower bounds for train run, dwell and safe-separation times are considered. This model permitted analytical derivation of what is called fundamental traffic diagrams (giving the train frequency as a function of the number of running trains). One of the most recent extension of this model was the one in [2], where the passenger travel demand has been considered (flows by origin-destination) and where passenger capacities of trains as well as passenger comfort inside the trains have been considered.

The current approaches for optimal control of metro systems discussed before rely on line-specific, deterministic modelling of train operations and passenger flows (cf. [1]) based on a mean-value approach. At the same time, it is well-known that variability of input parameters and the associated stochastic effects have a significant effect on the system performance (cf. [3]). The statistics of station dwell times notably depends on the level of crowding on station platforms and the passenger transfer processes between different lines, and hence, the network situation. To this end, (max,+)-models with stochastic input have been developed [4, 5, 6] and used to assess the stability and robustness of timetables in view of disruptions and input data variability. In this work, train control and traffic assessment in the deterministic case are to be extended by stochastic modelling of passenger flows and station dwell times, and transferability and limitations of modelling and solution approaches are assessed.

Challenges and Impact:

The work we propose here can have impacts on many pressing problems in railway operations on urban mass-transit networks:

- First, in terms of traffic modeling on mass transit lines, it consists in proposing realistic traffic models taking into account uncertainty and its role in the effective system performance.

- Second, it provides an integrated modeling perspective on train traffic and passenger flows including inter-dependencies between the network traffic situation and the passengers’ route choice and the assignment and reassignment of itineraries on the network.

As the work deals with intricate (possibly cyclic) dependencies within a network, it bears several major research and modelling challenges.

- The extension of existing deterministic traffic models to stochastic ones is one of the main challenges, in the sense that the main uncertainties on the traffic dynamics need to be characterized and modeled in a way which would permit a good comprehension of the traffic.

- Effective optimization models for the passenger flow assignments on the network are to be developed in order to allow the operators of the transit networks to control and regulate the passenger flows.

- Finally, a consistent coupling of the passenger flow assignment and the train dynamics modelling is required, which allows to assess passenger assignment and re-assignment based on the current network operation state.


[1] N. Farhi et al. Traffic modeling and real-time control for metro lines, IEEE ACC, 2017.

[2] N. Farhi, A discrete-event model of the train traffic on a linear metro line, Applied Math. Modelling 2021.

[3] N. Weik and N. Nießen, Quantifying the effects of running time variability on the capacity of rail corridors, Journal of Rail Transport Planning & Management, 2020.

[4] T. Büker and B. Seybold, Stochastic modelling of delay propagation in large networks, Journal of Rail Transport Planning & Management, 2012.

[5] R. Goverde, B. Heidergott and G. Merlet, Railway Timetable Stability Analysis Using Stochastic Max-Plus Linear Systems, in 3rd International Seminar on Railway Operations Modelling and Analysis, Zurich, 2009.

[6] A. de Kort, B. Heidergott and H. Ayhan, A probabilistic (max,+) approach for determining railway infrastructure capacity, European Journal of Op. Research, 2003.