Mathematical Models of Traffic Systems

Garant předmětu:

Lecturer:

Tutor:

Supervisor:

Department of Mathematics

Synopsis:

Basic macroscopic quantities and relationships among them. Fundamental relation of vehicular modelling. Microscopic description of traffic and discussion of statistical nature of micro-quantities. Headway-distributions and relationships among them. Special functions for theory of vehicular microstructure. Theorem on saddle point approximation. Discussion on empirical knowledge on macroscopic and microscopic phenomena of vehicular traffic. Methods for evaluations of traffic data. Classification of traffic models. Lighthill-Whitham model and associated solutions. Cole-Hopf transformation. Cauchy problem and its solution in generalized functions. Burgers PDE. Cellular traffic models: NaSch-model, Fukui-Ischibaschi model and models with exclusion rules. Theoretical solution of TASEP model. CF-models. Formulation of interaction dynamics in CF-models. Numerical representation of models. Thermodynamic traffic models. Interaction potentials. Analytical solutions for basic variants of models. Derivation of clearance distribution. Class of balanced distributions and its properties. Criteria for acceptance of traffic headway-distributions. Statistical rigidity a NV-statistics. Rigidity of Poissonian processes. Cluster function. Derivation of general formula for statistical rigidity. Analysis of statistical rigidity for traffic models.

Requirements:

Syllabus of lectures:

Syllabus of tutorials:

1. Macro- and micro-quantities in empirical traffic data.

2. Traffic macro-models based on description of microstructure.

3. Properties of balanced distributions.

4. Model TASEP and associated statistical description based on Matrix Product Ansatz.

5. Steady-state solution of thermodynamical traffic model.

6. Headway distribution and its properties.

7. Balancing particle system and its description.

8. Statistical rigidity.

Study Objective:

Knowledge: Theoretical formulation of traffic models, their analytical solutions and statistical prediction of their microstructure.

Skills: Statistical analysis of vehicular data or data from numerical realizations of traffic models.

Study materials:

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

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

[3] D. Chowdhury, L. Santen, and A. Schadschneider, Physics Reports 329 (2000), 199

[4] N. Rajewski, L. Santen, A. Schadschneider, M. Schreckenberg: The asymmetric exclusion process: comparison of update procedures, Journal of statistical physics 92 (1998), 151

[5] M. Krbálek, Equilibrium distributions in a thermodynamical traffic gas, J. Phys. A: Math. Theor. 40 (2007), 5813

Note:

Further information:

No time-table has been prepared for this course

The course is a part of the following study plans: