Mathematical Modelling of Traffic

Course guarantor:

Milan Krbálek

Lecturer:

Milan Krbálek

Tutor:

Milan Krbálek

Supervisor:

Department of Mathematics

Synopsis:

1. Basic mathematical description of vehicular traffic - macroscopic and microscopic quantities, relations between them, fundamental diagram and phase map.

2. Empirical knowledge about traffic flow - methodology of traffic data evaluation, 3s-unification procedure, two-phase theory, three-phase theory, VHM and link to capacity calculations in physics of traffic.

3. Traffic models - general overview, classification of models, examples, Greenberg’s macroscopic model and its solution, Montroll’s microscopic model and its solution.

4. Lighthill-Whitham model - formulation and theoretical solution, Cole-Hopf transformation, formulation of associate Cauchy problem and its solution in distributions, Burgers equation.

5. Cellular traffic models - Nagel-Schreckenberg model, Fukui-Ischibaschi model, model TASEP and its theoretical solution by MPA.

6. Thermodynamic traffic models - variants, classification by range and type of potential, Hamiltonian description, general solution methodology, solution of short-range model, connection between thermodynamic models and balance particle systems, solution of middle-ranged model with logarithmic potential.

7. Vehicular Headway Modeling - an insight into the issue, empirical and theoretical knowledge in a given area, criteria for admissibility of headway distributions, statistical rigidity and changes in its course, derivation of statistical rigidity for thermodynamic gas.

8. Statistical properties of traffic flow - Poisson and semi-Poisson mode of transport, supra-random traffic states, their detection.

Requirements:

Syllabus of lectures:

Syllabus of tutorials:

Study Objective:

Study materials:

Key references:

[1] Li, L., Chen, X.M.: Vehicle headway modeling and its inferences in macroscopic/microscopic traffic flow theory: A survey, Transportation Research Part C 76, 170, 2017

[2] Krbálek, M., Apeltauer, J., Apeltauer, T., Szabová, Z.: Three methods for estimating a range of vehicular interactions, Physica A 491, 112–126, 2018

[3] Treiber, M., Kesting, A.: Traffic Flow Dynamics, Springer, Berlin, 2013.

Recommended references:

[4] Krbálek, M., Krbálková, M.: 3s-Unification for Vehicular Headway Modeling, Proceedings of SPMS 2018, Dobřichovice, 2018

[5] Krbálek, M., Šleis, J.: Vehicular headways on signalized intersections: theory, models, and reality, J. Phys. A: Math. Theor. 48, 015101, 2015

[6] Helbing, D.: Traffic and related self-driven many-particle systems, Rev. Mod. Phys. 73, 1067, 2001.

Note:

Time-table for winter semester 2024/2025:

Time-table is not available yet

Time-table for summer semester 2024/2025:

Time-table is not available yet

The course is a part of the following study plans:

• Aplikované matematicko-stochastické metody (compulsory course in the program)