Mathematical Modelling of Traffic
Code  Completion  Credits  Range  Language 

01MMD  Z,ZK  5  2P+2C  Czech 
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 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, 3sunification procedure, twophase theory, threephase 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. LighthillWhitham model  formulation and theoretical solution, ColeHopf transformation, formulation of associate Cauchy problem and its solution in distributions, Burgers equation.
5. Cellular traffic models  NagelSchreckenberg model, FukuiIschibaschi 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 shortrange model, connection between thermodynamic models and balance particle systems, solution of middleranged 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 semiPoisson mode of transport, suprarandom traffic states, their detection.
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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.: 3sUnification 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 selfdriven manyparticle systems, Rev. Mod. Phys. 73, 1067, 2001.
 Note:
 Further information:
 No timetable has been prepared for this course
 The course is a part of the following study plans:

 Aplikované matematickostochastické metody (compulsory course in the program)