## Headway and traffic modelling

**Headway and traffic modelling**

Physics of Traffic is a complex scientific discipline having many interesting components. One of them is Vehicular Headway Modeling (VHM) analyzing and predicting changes in a vehicular microstructure forced by external conditions (traffic density, intensity, or global velocity). Typical representatives of vehicular micro-quantities are individual velocities, time/space headways, or time/space clearances. Mathematically, all these quantities represent random variables and associated probability densities (for headways and clearances) belong to a specific family of distributions. As is well known, parameters of these distributions evolve rapidly over time and are therefore markedly dependent on actual values of density, intensity, and global velocity. It excludes the possibility of applying standard statistical approaches to vehicle-by-vehicle data structures and, on contrary, it enforces the use of more complex statistical procedures applied to partial data samples. The development and innovations of these methods is the main content of VHM.

Our research team deals with mathematical theories standing behind the vehicular headway modelling, like level processes, balancing particle systems, theory of statistical corigidity and so on. The main goal is to predict empirical features of vehicular systems by means of sophisticated mathematical tools. Moreover, members of our team investigate also many related topics including Totally Asymmetric Simple Exclusion Process, Dyson’s thermodynamic gases, random matrix ensembles, dynamics of vehicular flows, or mathematical modelling of partial traffic problems (merging processes, capacity of intersections, e.g.).

** ****Members**

**Milan Krbálek**

**associate professor at FNSPE CTU in Prague**

- mathematical theories for vehicular headway modelling

**Ondřej Kollert**

**Ph.D. student at FNSPE CTU in Prague**

- theory of level processes and applications in transportation science

**Zuzana Szabová**

**Ph.D. student at FNSPE CTU in Prague**

- detection of strategies in socio-dynamical systems

**František Šeba**

**Ph.D. student at University of Hradec Králové**

- microstructure in thermal particle gases with composite potentials

**Michaela Krbálková**

**assistant professor at University of Pardubice, **

**Ph.D. student at University of Hradec Králové**

- non-equilibrium states of traffic gases with middle-ranged interactions

**Nikola Groverová**

**bachelor student at FNSPE CTU in Prague**

- non-equilibrium states in classical Dyson’s Coulomb gases

**Graduates**

**Ondřej Faltys**

(study of microstructure in particle ensembles with middle-ranged potentials)

**Pavel Hrabák**

(asymmetric simple exclusion process, solution and modifications)

**Václav Kautský**

(cluster function of DUE random matrices)

**Michael Matějů**

(alternative formulation of thermodynamic traffic model)

**Ondřej Sláma**

(correlation analysis of vehicular flows)

**Jiří Šleis**

(theory of balanced distributions and associated socio-physical applications)

**Jana Vacková**

(perturbation theory of statistical rigidity in particle systems)

**Significant papers**

- Krbálek, J. Apeltauer, T. Apeltauer, Z. Szabová, Three methods for estimating a range of vehicular interactions. Physica A 491(2018) 112–126.
- Krbálek and T. Hobza, Inner structure of vehicular ensembles and random matrix theory. Physics Letters A 380/21(2016) 1839–1847.
- Krbálek and J. Šleis, Vehicular headways on signalized intersections: theory, models, and reality. J. Phys. A: Math. Theor. 48(2015) 015101.
- Krbálek, Theoretical predictions for vehicular headways and their clusters. J. Phys. A: Math. Theor. 46(2013) 4451011.
- Krbálek, Equilibrium distributions in a thermodynamical traffic gas. J. Phys. A: Math. Theor. 40(2007) 5813-5821.
- Krbálek and D. Helbing, Determination of interaction potentials in freeway traffic from steady-state statistics. Physica A 333 (2004) 370 – 378.
- Krbálek, P. Šeba, and P. Wagner, Headways in the traffic flow – remarks from a physical perspective. Phys. Rev. E 64(2001), 066119 – 066125.
- Krbálek and P. Šeba, Statistical properties of the city transport in Cuernavaca (Mexico) and random matrix ensembles. J. Phys. A: Math. Gen. 33(2000), L229 – L233.