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TADAKI, Shin-ichi

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TitleSELF-ORGANIZATION IN A TWO-DIMENSIONALCELLULAR AUTOMATON MODEL OF TRAFFICFLOW(Session IV : Structures & Patterns, The 1st TohwaUniversity International Meeting on Statistical PhysicsTheories, Experiments and Computer Simulations)Author(s)TADAKI, Shin-ichiCitation物性研究 (1996), 66(3): 610-611Issue Date1996-06-20URL http://hdl.handle.net/2433/95754RightType Departmental Bulletin PaperTextversionpublisherKyoto UniversitySELF-ORGANIZATIO.N IN A Two-DIMENSIONAL CELLULARAUTOMATON MODEL OF TRAFFIC FLOWShin-ichi TADAKI •Department of Information Science, Saga University, Saga 840, JAPAN1 IntroductionThe traffic flow problems have been studied mainly through three modelings. The first is thefluid dynamical model. This method has been studied, for example, with the Burgers equationin one-diI:nensional cases. The second is the so-called car-following model. Each car controlsits velocity by the velocity or distance relative to the car preceding it. The third is the cellularautomaton (CA) model. The advantages of the CA models are their computational simplicity.The simplicity of the models enables us to change the models to take many realistic featuresof traffic problems into account. In spite of the simplicity of the models, they show interestingphenomena as phase transitions and self-organizations.One ofthe simplest CA models of traffic flow in one-way expressways is the rule-184 ele-mentary CA, which is a simple asymmetric exclusion rule. There have been great developmentsin one-dimensional CA models (for example, see [1]). Two-dimensional models, howeve,r, havefewer direct connections to real traffic flow problems. They seem to be abstract models withregard to a traffic systems in a whole city or an expressway network.2 The ModelI study here one of the simplest two-dimensional traffic flow model introduced by Biham,Middleton, Levine (BML)[2]. It is a simple extension of the rule-184 CA model and modelssimple two-dimensional exclusion processes. Cars are distributed on a square lattice of N x Nsites with periodic boundary conditions in both horizontal and vertical directions. Each car isrepresented as an arrow directed up or right. A traffic light controls the dynamics, such thatthe right arrow move only at odd time steps and the up arrows move at even time steps. Atodd time steps, each right arrow moves one site to its right neighborhood if and only if thedestination is empty. The corresponding rule is applied for up arrows. The density of right(up) cars is given byn..... nrp..... = N2' PT = N2where n..... (nr) denotes the. number of right (up) arrows. I examine the isotropic case wherep ..... = Pr = p/2 following Biham et al. They reported the existence of the transition pointPc I'V 0.35. Below Pc all cars move freely and the average velocity is v = 1. All cars are blockedand the average velocity vanishes above Pc.3 Two Types of Traffic JamTwo types of traffic jam configurations were found in the BML model (Fig. 1) [3]. In the low-density region (Pc < P < Pt), there is a single global cluster of jam. The backbone of thejam lies·Electronic Address: tadaki@aLis.saga-u.ac.jp- 610-diagonally and branches spread 'horizontally and vertically, The diagonal correlation functionobeys a power law. So the jam state at the low density is far from random configuration and.well self-organized. In the high-density region (Pt < p), on the other hand, patches of small localclusters of jam cover the whole system. There is no apparent global structure. The diagonalcorrelation decays' exponentially. So the jam state at the high density is random.The fractal dimension of jam states is studied by the box counting method (Fig. 2)[4]. In thehigh density region, jam configurations do not show fractal structure reflecting· the randomnessinstalled at the initial configuration. Fractal structures appear in the low density jam. Fractaldimension D seems to obey2 - D '" Ip - ptl-a .The transition point Pt is close to that of point percolation.The BML m9del has three phases. The first is the freely moving phase. The. second isthe self-organized jam phase. In this phase, the spatially diagonal correlation function obeys apower law. The spatial structure of jams is fractal. The third is the random jam phase. Thespatially diagonal correlation function decays exponentially.1 o System Size =8oox800 (for ~ and t directions)• System Size = 4oox400 (for ~ and t directions)• System Size =200x2oo (for ~ and t directions)2.00 I': . ••..P Random Jam .0"1.90 0"Pt I1.80 :i\:l.Self-Organized Jam 1.70Pc1.60Freely Moving 1.500.00 0.20 0.40 0.60 0.80 1.000 pFigure 1: Schematic phase diagram Figure 2: Fractal dimension by box countingReferences[1] S. Yukawa, M. Kikuchi and S. Tadaki, J. Phys. Soc. Japan 63, 3609 (1994).[2] O. Biham, A. A. Middleton, and D. Levine, Phys. Rev. A46, 6124 (1992).[3] S. Tadaki and M. Kikuchi, Phys. Rev. ESO, 4564 (1994).[4] S. Tadaki and M. Kikuchi, J. Phys. Soc. Japan 64 (1995) in press.- 611-

SELF-ORGANIZATION IN A TWO-DIMENSIONAL CELLULAR AUTOMATON MODEL OF TRAFFIC FLOW(Session IV : Structures & Patterns, The 1st Tohwa University International Meeting on Statistical Physics Theories, Experiments and Computer Simulations)

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SELF-ORGANIZATION IN A TWO-DIMENSIONAL CELLULAR AUTOMATON MODEL OF TRAFFIC FLOW(Session IV : Structures & Patterns, The 1st Tohwa University International Meeting on Statistical Physics Theories, Experiments and Computer Simulations)

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