Biham-Middleton-Levine Traffic Model Research Articles

Loading conditions for self‐organization in the BML model with stochastic direction choice

A dynamical system is considered such that, in this system, particles move on a toroidal lattice of the dimension according to a version of the rule of particle movement in Biham–Middleton–Levine traffic model. We introduce a stochastic case with direction choice for particles. Particles of the first type move along rows, and the particles of the second type move along columns. The goal is to find conditions of self‐organization system for any lattice dimension. We have proved that the BML model as a dynamical system is a special case of Buslaev nets. This equivalence allows us to use of Buslaev net analysis techniques to investigate the BML model. In Buslaev nets conception, the self‐organization property of the system corresponds to the existence of velocity single point spectrum equal to 1. In the paper, we consider the model version when one notable aspect is that a particle may change its type. Exactly, we assume a constant probability that a particle changes type at each step. In the case where , the system corresponds to the classical version of the BML model. We define a state of the system where all particles continue to move indefinitely, in both the present and the future, as a state of free movement. A sufficient condition for the system to result in a state of free movement from any initial state (condition for self‐organization) has been found. This condition is that the number of particles be not greater than half the greatest common divisor of the numbers . It has been proved that, if , and whether or and there are both at least one particle of the first type and at least particle of the second type, then a necessary condition for a state of free movement to exist is the greatest common divisor of and be not less than 3. The theorems are formulated in terms of the algebraic structures and terms of the dynamical system. The spectrum of particle velocities has been found for the net . This approach allows us to hope that the spectrum of dynamical system can be studied for an arbitrary dimension of the net.

BML model on non-orientable surfaces

Two-dimensional Biham–Middleton–Levine traffic model on a N × N square lattice embedded on a Klein bottle and a projective plane is investigated by computer simulations. The behavior of the model with these boundary conditions is compared with the model under toroidal boundary conditions. Our numerical results show that the phase diagram depends strongly on the underlying topology. We also investigate the influence of the ratio between the number of red and blue particles has over the speed of the system and conclude that the projective plane case differs considerably from the other cases reviewed here.

EVACUATION PROCESS IN TWO-DIMENSIONAL TRAFFIC FLOW MODELS

Within the framework of Biham–Middleton–Levine traffic model with origin–destination trips, we study the evacuation processes of cars in cities. Cars move from the origin to the destination points. A driver which reaches its destination disappears with rate β. It is found that the evacuation processes are greatly influenced by the origin–destination distance probability distribution. We also find that the evacuation time of drivers diverges in the form of a power law τ ∝ β-ν, with ν = 1.

An improved upper bound for the critical car density of the two-dimensional Biham–Middleton–Levine traffic model

By means of non-linear dynamics, we show that the critical car densities of both the two-dimensional Biham-Middleton-Levine and the green wave traffic models are strictly less than 1/2.