Biham–Middleton–Levine traffic model

It consists of a number of cars represented by points on a lattice with a random starting position, where each car may be one of two types: those that only move downwards (shown as blue in this article), and those that only move towards the right (shown as red in this article).

[1] The Biham–Middleton–Levine traffic model was first formulated by Ofer Biham, A. Alan Middleton, and Dov Levine in 1992.

In 2005, Raissa D'Souza found that for some traffic densities, there is an intermediate phase characterized by periodic arrangements of jams and smooth flow.

[3] In the same year, Angel, Holroyd and Martin were the first to rigorously prove that for densities close to one, the system will always jam.

[4] Later, in 2006, Tim Austin and Itai Benjamini found that for a square lattice of side N, the model will always self-organize to reach full speed if there are fewer than N/2 cars.

For rectangles with coprime dimensions, the intermediate states are self-organized bands of jams and free-flow with detailed geometric structure, that repeat periodically in time.

[2] For low numbers of cars, the system will usually organize itself to achieve a smooth flow of traffic.

[7] Yet, on square lattices disordered intermediate phases are more frequently observed and tend to dominate densities close to the transition region.

In 2005, Alexander Holroyd et al proved that for densities sufficiently close to one, the system will have no cars moving infinitely often.

[4] In 2006, Tim Austin and Itai Benjamini proved that the model will always reach the free-flowing phase if the number of cars is less than half the edge length for a square lattice.

[5] The model is typically studied on the orientable torus, but it is possible to implement the lattice on a Klein bottle.

The behaviour of the system on the Klein bottle is much more similar to the one on the torus than the one on the real projective plane.

The mobility on the real projective plane decreases more gradually for densities from zero to the critical point.

[9] Under periodic boundaries, instead of updating all cars of the same colour at once during each step, the randomized model performs

In this case, the intermediate state observed in the usual BML traffic model does not exist, due to the non-deterministic nature of the randomized model; instead the transition from the jammed phase to the free flowing phase is sharp.

The fundamental polygon of the torus, on which the cars move
A free-flowing phase observed on a 144×89 rectangular lattice with a traffic density of 28%
A globally jammed phase observed on a 144×89 rectangular lattice with a traffic density of 60%
A 512×512 lattice with density of 27% after 64000 iterations. Traffic is at a free-flowing phase.
A 512×512 lattice with density of 29% after 64000 iterations. Traffic is at a free-flowing phase.
A 512×512 lattice with density of 38% after 64000 iterations. Traffic is at a globally jammed phase.
Mobility with respect to time for above lattice. Mobility is defined as the number of cars that can move as a fraction of the total. (The points are in the upper-left corner of the image.)
Mobility with respect to time for above lattice. Mobility is defined as the number of cars that can move as a fraction of the total. (The points are in the upper-left corner of the image.)
Mobility with respect to time for above lattice. Mobility is defined as the number of cars that can move as a fraction of the total. (The points are on the left side of the image.)
A periodic intermediate phase observed on a 144×89 rectangular lattice with a traffic density of 38%
A disordered intermediate phase observed on a 144×89 rectangular lattice with a traffic density of 39%
A 512×512 lattice with density of 31% after 64000 iterations. Traffic is at a disordered intermediate phase.
A 512×512 lattice with density of 33% after 64000 iterations. Traffic is at a disordered intermediate phase.
A 512×512 lattice with density of 37% after 64000 iterations. Traffic is at a disordered intermediate phase.
Mobility with respect to time for above lattice. Mobility is defined as the number of cars that can move as a fraction of the total.
Mobility with respect to time for above lattice. Mobility is defined as the number of cars that can move as a fraction of the total.
Mobility with respect to time for above lattice. Mobility is defined as the number of cars that can move as a fraction of the total.