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| The '''Biham–Middleton–Levine traffic model''' is a [[self-organization|self-organizing]] [[cellular automaton]] [[microscopic traffic flow model|traffic flow 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). The two types of cars take turns to move. During each turn, all the cars for the corresponding type advance by one step if they are not blocked by another car. It may be considered the two-dimensional analogue of the simpler [[Rule 184]] model. It is possibly the simplest system exhibiting phase transitions and [[self-organization]].<ref name="dsouzaweb">{{cite web |last=D'Souza |first=Raissa |title=The Biham-Middleton-Levine traffic model |url=http://mae.ucdavis.edu/dsouza/bml.html |accessdate=14 December 2012}}</ref>
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| ==History==
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| The Biham–Middleton–Levine traffic model was first formulated by [[Ofer Biham]], A. Alan Middleton, and Dov Levine in 1992.<ref name="bml">{{cite journal |last1=Biham |first1=Ofer |authorlink1=Ofer Biham |last2=Middleton |first2=A. Alan |last3=Levine |first3=Dov |title=Self-organization and a dynamical transition in traffic-flow models |journal=Phys. Rev. A |date=November 1992 |volume=46 |issue=10 |pages=R6124–R6127 |doi=10.1103/PhysRevA.46.R6124 |url=http://pra.aps.org/abstract/PRA/v46/i10/pR6124_1 |accessdate=14 December 2012 |publisher=American Physical Society |issn=1050-2947 |pmid=9907993}}</ref> Biham ''et al'' found that as the density of traffic increased, the [[steady-state]] flow of traffic suddenly went from smooth flow to a complete jam. 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.<ref name="dsouzaPRE">{{cite journal|last=D'Souza|first=Raissa M.|title=Coexisting phases and lattice dependence of a cellular automaton model for traffic flow|journal=Phys. Rev. E|year=2005|volume=71|issue=6|doi=10.1103/PhysRevE.71.066112|pmid=16089825|url=http://pre.aps.org/abstract/PRE/v71/i6/e066112|accessdate=14 December 2012|publisher=The American Physical Society}}</ref> In the same year, Alexander Holroyd ''et al'' were the first to rigorously prove that for densities close to one, the system will always jam.<ref name="angel">{{cite journal |last1=Angel |first1=Omer |last2=Holroyd |first2=Alexander E. |last3=Martin |first3=James B. |title=The Jammed Phase of the Biham-Middleton-Levine Traffic Model |journal=Electronic Communications in Probability |date=12 August 2005 |volume=10 |pages=167–178 |doi=10.1214/ECP.v10-1148 |url=http://ecp.ejpecp.org/article/view/1148 |accessdate=14 December 2012 |issn=1083-589X}}</ref> 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.<ref name="austin">{{cite arXiv |last1=Austin |first1=Tim |last2=Benjamini |first2=Itai |eprint=math/0607759 |title=For what number of cars must self organization occur in the Biham–Middleton–Levine traffic model from any possible starting configuration? |year=2006 |accessdate=14 December 2012}}</ref>
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| ==Lattice space==
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| [[File:TorusAsSquare.svg|thumb|right|The fundamental polygon of the torus, on which the cars move]]
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| The cars are typically placed on a square lattice that is [[topology|topologically]] equivalent to a [[torus]]: that is, cars that move off the right edge would reappear on the left edge; and cars that move off the bottom edge would reappear on the top edge.
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| There has also been research in rectangular lattices instead of square ones. 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.<ref name=dsouzaPRE/> In non-coprime rectangles, the intermediate states are typically disordered rather than periodic.<ref name=dsouzaPRE/>
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| ==Phase transitions==
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| Despite the simplicity of the model, it has two highly distinguishable phases – the '''jammed phase''', and the '''free-flowing phase'''.<ref name=bml/> For low numbers of cars, the system will usually [[self-organization|organize itself]] to achieve a smooth flow of traffic. In contrast, if there is a high number of cars, the system will become jammed to the extent that no single car will move. Typically, in a square lattice, the transition density is when there are around 32% as many cars as there are possible spaces in the lattice.<ref name="Holroydweb">{{cite web |last=Holroyd |first=Alexander E. |title=The Biham-Middleton-Levine Traffic Model |url=http://www.math.ubc.ca/~holroyd/bml/ |accessdate=14 December 2012}}</ref>
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| {|
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| |[[File:Biham-Middleton-Levine traffic model self-organized to a free flowing phase.ogv|thumb|center|upright=1.52|A '''free-flowing phase''' observed on a 144×89 rectangular lattice with a traffic density of 28%]]
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| |[[File:Biham-Middleton-Levine traffic model self-organized to a globally jammed phase.ogv|thumb|center|upright=1.52|A '''globally jammed phase''' observed on a 144×89 rectangular lattice with a traffic density of 60%]]
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| |[[File:Bml x 512 y 512 p 27 iterated 32000.png|thumb|center|upright=1|A 512×512 lattice with density of 27% after 64000 iterations. Traffic is at a free-flowing phase.]]
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| |[[File:Bml x 512 y 512 p 29 iterated 32000.png|thumb|center|upright=1|A 512×512 lattice with density of 29% after 64000 iterations. Traffic is at a free-flowing phase.]]
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| |[[File:Bml x 512 y 512 p 38 iterated 32000.png|thumb|center|upright=1|A 512×512 lattice with density of 38% after 64000 iterations. Traffic is at a globally jammed phase.]]
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| {|
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| |[[File:Bml x 512 y 512 p 27 MOBILITY.png|thumb|center|upright=1|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.]]
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| |[[File:Bml x 512 y 512 p 29 MOBILITY.png|thumb|center|upright=1|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.]]
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| |[[File:Bml x 512 y 512 p 38 MOBILITY.png|thumb|center|upright=1|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.]]
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| ===Intermediate phase===
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| The intermediate phase occurs close to the transition density, combining features from both the jammed and free-flowing phases. There are principally two intermediate phases – '''disordered''' (which could be [[meta-stable]]) and '''periodic''' (which are provably stable).<ref name=dsouzaPRE/> On rectangular lattices with [[coprime]] dimensions, only periodic orbits exist.<ref name=dsouzaPRE/> In 2008 periodic intermediate phases were also observed in square lattices.<ref name="linesch">{{cite journal |last1=Linesch |first1=Nicholas J. |last2=D'Souza |first2=Raissa M. |title=Periodic states, local effects and coexistence in the BML traffic jam model |journal=Physica A |date=15 October 2008 |volume=387 |issue=24 |pages=6170–6176 |doi=10.1016/j.physa.2008.06.052 |url=http://www.sciencedirect.com/science/article/pii/S0378437108005839 |accessdate=14 December 2012 |issn=0378-4371}}</ref> Yet, on square lattices disordered intermediate phases are more frequently observed and tend to ''dominate'' densities close to the transition region.
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| {|
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| |[[File:Biham-Middleton-Levine traffic model self-organized to a periodic intermediate phase.ogv|thumb|center|upright=1.52|A '''periodic''' intermediate phase observed on a 144×89 rectangular lattice with a traffic density of 38%]]
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| |[[File:Biham-Middleton-Levine traffic model self-organized to a disordered intermediate phase.ogv|thumb|center|upright=1.52|A '''disordered''' intermediate phase observed on a 144×89 rectangular lattice with a traffic density of 39%]]
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| |[[File:Bml x 512 y 512 p 31 iterated 32000.png|thumb|center|upright=1|A 512×512 lattice with density of 31% after 64000 iterations. Traffic is at a disordered intermediate phase.]]
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| |[[File:Bml x 512 y 512 p 33 iterated 32000.png|thumb|center|upright=1|A 512×512 lattice with density of 33% after 64000 iterations. Traffic is at a disordered intermediate phase.]]
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| |[[File:Bml x 512 y 512 p 37 iterated 32000.png|thumb|center|upright=1|A 512×512 lattice with density of 37% after 64000 iterations. Traffic is at a disordered intermediate phase.]]
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| |}
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| {|
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| |[[File:Bml x 512 y 512 p 31 MOBILITY.png|thumb|center|upright=1|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.]]
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| |[[File:Bml x 512 y 512 p 33 MOBILITY.png|thumb|center|upright=1|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.]]
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| |[[File:Bml x 512 y 512 p 37 MOBILITY.png|thumb|center|upright=1|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.]]
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| |}
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| ==Rigorous analysis==
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| Despite the simplicity of the model, rigorous analysis is very nontrivial.<ref name=Holroydweb/> Nonetheless, there have been [[mathematical proof]]s regarding the Biham–Middleton–Levine traffic model. Proofs so far have been restricted to the extremes of traffic density. In 2005, Alexander Holroyd ''et al'' proved that for densities close to one, the system will always jam.<ref name=angel/> 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.<ref name=austin/>
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| It would be ideal to formulate a rigorous method to predict the end result of any starting position, especially in the intermediate phases.<ref name=linesch/> To that end, this model has been the subject of research for several scientists.{{cn|date=December 2012}}
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| ==Non-orientable surfaces==
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| The model is typically studied on the orientable [[torus]], but it is possible to implement the lattice on a [[Klein bottle]].<ref name=campora>{{cite journal |last1=Cámpora |first1=Daniel |last2=de La Torre |first2=Jaime |last3=García Vázquez |first3=Juan Carlos |last4=Caparrini |first4=Fernando Sancho |title=BML model on non-orientable surfaces. |journal=Physica A |date=August 2010 |volume=389 |issue=16 |pages=3290–3298 |doi=10.1016/j.physa.2010.03.037 |url=https://www.sciencedirect.com/science/article/pii/S0378437110002827 |accessdate=12 June 2013}}</ref> When the red cars reach the right edge, they reappear on the left edge except flipped vertically; the ones at the bottom are now at the top, and vice versa. More formally, for every <math>y\in \lbrace 0, ..., N-1\rbrace</math>, a red car exiting the site <math>(N-1, y)</math> would enter the site <math>(0, N-y-1)</math>. It is also possible to implement it on the [[real projective plane]].<ref name=campora/> In addition to flipping the red cars, the same is done for the blue cars: for every <math>x\in \lbrace 0, ..., N-1 \rbrace</math>, a blue car exiting the site <math>(x, N-1)</math> would enter the site <math>(N-x-1, 0)</math>.
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| 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.<ref name=campora/> For the Klein bottle setup, the mobility as a function of density starts to decrease slightly sooner than in the torus case, although the behaviour is similar for densities greater than the critical point. The mobility on the real projective plane, decreases more gradually for densities from zero to the critical point. In the real projective plane, local jams may form at the corners of the lattice even though the rest of the lattice is free-flowing.<ref name=campora/>
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| ==References==
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| {{reflist}}
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| ==External links==
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| *[https://github.com/dllu/bml-cuda CUDA implementation] by Daniel Lu
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| *[http://www.jasondavies.com/bml/#0.31/512/512 WebGL implementation] by Jason Davies
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| *[https://github.com/MaciekBaron/BMLTrafficJS JavaScript implementation] by Maciej Baron
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| {{DEFAULTSORT:Biham-Middleton-Levine Traffic Model}}
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| [[Category:Cellular automaton rules]]
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| [[Category:Lattice models]]
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Geographically Europe was the largest industrial and commercial LED lighting marketplace and accounted for 33.1% in 2012 owing to improve in investment by government. In addition, nations such as Russia, Germany, Denmark and the U.K. are focusing on deployment of efficient lighting to boost lighting excellent and to reduced all round operational cost, which will further improve the demand for LED lighting goods and solutions.
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