Distance- and Time-headway Distribution for Totally Asymmetric Simple Exclusion Process

We consider the one-dimensional totally asymmetric simple exclusion process on a lattice of N sites with both, the periodic boundary condition and the open boundary condition. Because of its simplicity, this model is often used for highway traffic simulations. Our goal is to investigate the headway distribution for this model in order to compare the microscopic structure of the model with the real highway traffic.In recent articles, the distance-headway distribution ℘(d) = ρ(1 − ρ)d−1 for this model has been presented. Using this result, the time-headway distribution is derived, which represents the probability density for the time interval of the length t between the jumps of two successive particles from the observed site. Firstly, the distribution function for the random-sequential time-discrete update is derived, from which the distribution for the time-continuous dynamics can be obtained. Furthermore, the time-headway distribution for forward- and backward-sequential update is studied by means of Monte Carlo simulations. We find out that for all variants of the considered model, the space discrete structure is reflected in the distance-headway distribution. The time-headway distribution reflects (contrary to the real traffic behavior) the “particle-hole symmetry” of the model.