Milan Krbálek

Determination of interaction potentials in freeway traffic from steady-state statistics

Many-particle simulations of vehicle interactions have been quite successful in the qualitative reproduction of observed traffic patterns. However, the assumed interactions could not be measured, as human interactions are hard to quantify compared to interactions in physical and chemical systems. We show that progress can be made by generalizing a method from equilibrium statistical physics we learned from random matrix theory. It allows one to determine the interaction potential via distributions of the netto distances s of vehicles. Assuming power-law interactions, we find that driver behavior can be approximated by a forwardly directed 1/s potential in congested traffic, while interactions in free traffic are characterized by an exponent of α≈4. This is relevant for traffic simulations and the assessment of telematic systems.

Headway statistics of public transport in Mexican cities

We present a cellular automaton simulating the behaviour of public bus transport in several Mexican cities. The headway statistics obtained from the model is compared to the measured time intervals between subsequent bus arrivals to a given bus stop and to a spacing distribution resulting from a random matrix theory.

Inter-vehicle gap statistics on signal-controlled crossroads

We investigate a microscopical structure in a chain of cars waiting at a red signal on signal-controlled crossroads. A one-dimensional space-continuous thermodynamical model leading to an excellent agreement with the data measured is presented. Moreover, we demonstrate that an inter-vehicle spacing distribution disclosed in relevant traffic data agrees with the thermal-balance distribution of particles in the thermodynamical traffic gas (discussed in [1]) with a high inverse temperature (corresponding to a strong traffic congestion). Therefore, as we affirm, such a system of stationary cars can be understood as a specific state of the traffic sample operating inside a congested traffic stream.

Equilibrium distributions in a thermodynamical traffic gas

We derive the exact formula for thermal-equilibrium spacing distribution of a one-dimensional particle gas with a repulsive potential V(r) = r−α(α > 0) depending on the distance r between the neighbouring particles. The calculated distribution (for α = 1) is successfully compared with the highway-traffic clearance distributions, which provides a detailed view of changes in the microscopical structure of a traffic sample depending on the traffic density. In addition to that, the observed correspondence is a strong support of studies applying the equilibrium statistical physics to traffic modelling.

Spectral rigidity of vehicular streams (random matrix theory approach)

Using a method originally developed for the random matrix theory, we derive an approximate mathematical formula for the number variance ΔN(L) describing the rigidity of particle ensembles with a power-law repulsion. The resulting relation is compared with the relevant statistics of the single-vehicle data measured on the Dutch freeway A9. The detected value of the inverse temperature β, which can be identified as a coefficient of the mental strain of the car drivers, is then discussed in detail with the relation to the traffic density ρ and flow J.

Theoretical predictions for vehicular headways and their clusters

This article presents a derivation of analytical predictions for steady-state distributions of netto time gaps among clusters of vehicles moving inside a traffic stream. Using the thermodynamic socio-physical traffic model with short-ranged repulsion between particles (originally introduced in [Physica A \textbf{333} (2004) 370]) we firstly derive the time-clearance distribution in the model. Consecutively, the statistical distributions for the so-called time multi-clearances are calculated by means of theory of functional convolutions. Moreover, all the theoretical surmises used during the above-mentioned calculations are proven by the statistical analysis of traffic data. The mathematical predictions acquired in this paper are thoroughly compared with relevant empirical quantities and discussed in the context of three-phase traffic theory.

Experimental Study of Phase Transition in Pedestrian Flow

The transition between low and high density phases is a typical feature of systems with social interactions. This contribution focuses on simple evacuation design of one room with one entrance and one exit; four passing-through experiments were organized and evaluated by means of automatic image processing. The phase of the system, determined by travel time and occupancy, is evaluated with respect to the inflow, a controlled boundary condition. Critical values of inflow and outflow were described with respect to the transition from low density to congested state. Moreover, microscopic analysis of travel time is provided.

Experimental Analysis of Two-Dimensional Pedestrian Flow in Front of the Bottleneck

This contribution presents an experimental study of two-dimensional pedestrian flow with the aim to capture the pedestrian behaviour within the cluster formed in front of the bottleneck. Two experiments of passing through a room with one entrance and one exit were arranged according to phase transition study in Ezaki and Yanagisawa (Metastability in pedestrian evacuation. In: Cellular automata, ed. by G. Sirakoulis, S. Bandini. LNCS, vol 7495. Springer, Berlin/Heidelberg, pp 776–784, 2012), the inflow rate was regulated to obtain different walking modes. By means of automatic image processing, pedestrians’ paths are extracted from camera recordings to get actual velocity and local density. Macroscopic information is extracted by means of a virtual detector and leaving times of pedestrians. The pedestrian’s behaviour is evaluated by means of density and velocity. Different approaches of measurement are compared using several fundamental diagrams. Two phases of crowd behaviour have been recognized and the phase transition is described.

Vehicular headways on signalized intersections: theory, models, and reality

We discuss statistical properties of vehicular headways measured on signalized crossroads. On the basis of mathematical approaches, we formulate theoretical and empirically inspired criteria for the acceptability of theoretical headway distributions. Sequentially, the multifarious families of statistical distributions (commonly used to fit real-road headway statistics) are confronted with these criteria, and with original empirical time clearances gauged among neighboring vehicles leaving signal-controlled crossroads after a green signal appears. Using three different numerical schemes, we demonstrate that an arrangement of vehicles on an intersection is a consequence of the general stochastic nature of queueing systems, rather than a consequence of traffic rules, driver estimation processes, or decision-making procedures.

Cellular Model of Pedestrian Dynamics with Adaptive Time Span

A cellular model of pedestrian dynamics based on the Floor Field model is presented. Contrary to the parallel update in Floor Field, the concept of adaptive time span is introduced. This concept, together with the concept of bounds, supports the spontaneous line formation and chaotic queue in front of the bottleneck. Model simulations are compared to the experiment “passing through”, from which a phase transition from low to high density is observed.