This article mediates an mathematical insight to the theory of vehicular headways measured on signalized crossroads. Considering both, mathematical and empirical substances of the socio-physical system studied, we firstly formulate several theoretica
l and empirically-inspired criteria for 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 experimental time-clearances gauged among neighboring vehicles leaving signal-controlled crossroads after a green signal appears. Another goal of this paper is, however, to decide (by means of three completely different numerical schemes) on the origin of statistical distributions recorded by stop-line-detectors. Specifically, we intend to examine whether an arrangement of vehicles is a consequence of traffic rules, drivers estimation-processes, and decision-making procedures or, on contrary, if it is a consequence of general stochastic nature of queueing systems.
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.
We consider the one-dimensional totally asymmetric simple exclusion model (TASEP model) with open boundary conditions and present the analytical computations leading to the exact formula for distance clearance distribution, i.e. probability density f
or a clear distance between subsequent particles of the model. The general relation is rapidly simplified for middle part of the one-dimensional lattice using the large $N$ approximation. Both the analytical formulas and their approximations are successfully compared with the numerical representation of the TASEP model. Furthermore, we introduce the pertinent estimation for so-called spectral rigidity of the model. The results obtained are sequentially discussed within the scope of vehicular traffic theory.
Using the methods originally developed for Random Matrix Theory we derive an exact mathematical formula for number variance (introduced in [4]) describing a rigidity of particle ensembles with power-law repulsion. The resulting relation is consequent
ly compared with the relevant statistics of the single-vehicle data measured on the Dutch freeway A9. The detected value of an inverse temperature, which can be identified as a coefficient of a mental strain of car drivers, is then discussed in detail with the respect to the traffic density and flow.
We investigate a microscopical structure in a chain of cars waiting at a red signal on signal-controlled crossroads. Presented is an one-dimensional space-continuous thermodynamical model leading to an excellent agreement with the data measured.Moreo
ver, 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.
