PHYSICAL REVIEW E, VOLUME 63, 012101

Low-density limit of the Nagel-Schreckenberg model

Ding-wei Huang and Chung-wei TsaiDepartment of Physics, Chung Yuan Christian University, Chung-li, Taiwan

~Received 21 August 2000; published 18 December 2000!

We study the low-density limit of the Nagel-Schreckenberg model with a large speed limit. Three distinctregions in the fundamental diagram are identified. Analytical approximations are obtained. The dependence onthree parameters~speed limit vmax, random brakingp, and lattice sizeL! is discussed. The relation tofinite-size effect is also discussed.

DOI: 10.1103/PhysRevE.63.012101 PACS number~s!: 05.40.2a, 45.70.Vn, 89.40.1k

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I. INTRODUCTION

Recently, traffic related problems have attracted considable attention from a rapidly growing community of physcists @1#. This is not only due to their practical implicationfor optimizing freeway traffic, but even more because ofobserved nonlinear dynamical phenomena and nonequrium phase transitions. Traffic dynamics has been studiedsome classical models: the hydrodynamic models@2#, the carfollowing models@3#, and the gas kinetic models@4#. Beyondthese classical approaches, a new class of discrete ceautomaton models has attracted much interest becausesimplicity and computational efficiency@5#. On the onehand, this model is quite successful in reproducing mafeatures observed in real traffic@6#. On the other hand, thebehavior of this simple model is very complex and up to nis still not well understood. Many efforts have been madereproduce analytically at least the fundamental diagra~flow versus density! @7#. However, up to now only a fewexact results are known, even for the simplest cases oflow-density limit. Recently, a peculiar enhancement of traflow in the zero-density limit has been noticed@8,9#. In thispaper, we will focus on the features of the fundamental dgrams in the low-density limit.

The basic model consists ofN cars moving on a onedimensional lattice ofL sites with periodic boundary condtions. Thus the densityr5N/L is a conserved quantity. Eacsite can be either empty, or occupied by exactly one car.cars can only move in one direction. The velocity of eachis an integer between 0 andvmax, wherevmax is the speedlimit. The position and velocity of thei th car at timet aredenoted byx( i ,t) and v( i ,t), respectively. The position othe next car ahead at timet is thenx( i 11,t). In the deter-ministic version of the model, the system evolves accordto the following synchronous rules:

v~ i ,t11!5min$x~ i 11,t !2x~ i ,t !21,v~ i ,t !1a,vmax%,~1!

x~ i ,t11!5x~ i ,t !1v~ i ,t11!. ~2!

The three termsx( i 11,t)2x( i ,t)21, v( i ,t)1a, and vmaxrepresent the driving schemes respecting the safety distathe accelerationa, and the speed limitvmax, respectively. Tocompare with real traffic, often the following parameters a

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taken:vmax55 anda51. The lattice constant is about 7.5and the time step 1 sec. Thus the speed limitvmax55 corre-sponds to 135 km/hr.

To account for the fluctuations in real traffic, a stochasnoise is further introduced. A moving car is subjected torandomized braking that will decrease the velocity by 1 wa certain probabilityp, i.e., Eq.~2! is replaced by

x~ i ,t11!5x~ i ,t !1max$v~ i ,t11!21,0%. ~3!

A convenient way to investigate the model is to drawdiagram of flow versus density, the so-called fundamendiagram. It is a smooth curve with a well-defined maximuat a certain densityrc . At low densities the flow is ‘‘free’’with very few density waves due to fluctuations, which dout quickly; at high densities start-stop waves dominatesystem, which is in the ‘‘jammed’’ state. Due to a ‘‘particlehole’’ symmetry, the value ofrc is 0.5 for the case ofvmax51. The value ofrc decreases with the increase ofvmax. Inthe special case ofp50, an analytical expression is known

rc51

vmax11. ~4!

Very recently, a peculiar behavior in the low-density limitthe fundamental diagram was noticed@8#. In the limit ofvmax→`, as the density decreases, the flow does notcrease as expected, but has an abrupt increase to a valuesee Fig. 1. This peculiar behavior was totally missed bymean-field theory in Ref.@5# and was linked to a phase transition at r50. This paper is devoted to a detailed studythis peculiar behavior of the fundamental diagram in tlimit of zero density and infinite velocity (r→0, vmax→`).

II. THE CASE r™1 AND vmaxš1

With a large speed limit, the typical behavior of the fudamental diagram in the low-density region is shown in F2. Three distinct regions are observed. In region~I!, the flowincreases linearly with the increase of density. In region~II !,the flow decreases abruptly with the increase of densityregion~III !, the flow reaches a plateau with a further increaof density. Compared to this peculiar enhancement, thehavior of the fundamental diagram is much smoother inmedium- and high-density regions. As we further increa

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BRIEF REPORTS PHYSICAL REVIEW E 63 012101

the density to its maximum value of 1, the flow decreagradually to its corresponding value of 0.

The linear behavior in region~I! can be understood asconsequence of the speed limit. As the interactions betwvehicles can be neglected, the velocity is constrained onlythe speed limit. Taking into account the random braking,

FIG. 1. Fundamental diagram~flow f vs densityr). Typicalbehaviors forvmax55 (p50.5) are shown by open circles. Thpeculiar behaviors forvmax→` (p50.5) are shown by filledcircles.

FIG. 2. Fundamental diagram in semilog scale to emphasizelow-density region. Parameters arevmax5100, p50.5, and L51000. The solid line is from the analytical expressions descriin the text.

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average velocity is given by (vmax2p). Thus the flow be-comes

f 5r~vmax2p!. ~5!

As we are consideringvmax@1 andp,1, the linear relationis defined by the speed limitvmax; the random brakingponly plays an insignificant role.

As the density increases, in region~II !, the interactionsbetween vehicles can no longer be neglected. The abdecrease of flow implies a strong correlation among vehicThe average velocity drops significantly with one more vhicle added on the road. The fundamental diagram canwell described by a quadratic form

f 512r2Xr2, ~6!

where the coefficientX is independent of the speed limvmax. With fixed lattice sizeL and random brakingp, theflow-density relations for various speed limitsvmax fallnicely on the same curve. When the stochastic noise isglected, we haveX50. Instead of an abrupt decrease, a lear decrease of flow is observed. In this model, the veloof a vehicle is constrained by its headway to the next vehahead. The average headway decreases with the increadensity and is given by (1/r21). Without the random brak-ing, each vehicle should be able to keep up with a spequal to its headway. The average velocity is then equathe average headway, i.e.,f 512r andX50. With stochas-tic noise, this is no longer possible. If vehicles brake radomly, the average velocity will be smaller than the averaheadway. When there arem cars braking randomly, theavailable headway for the following vehicle will be reduceby m and the average velocity can only assumes a value(1/r212m). On average, we havem5Np. As the averagevelocity reduces byNp, the flow will be reduced byrNp,which corresponds a coefficientX5Lp. One more modifica-tion is required. With periodic boundary conditions, the vhicles are moving on a ring and the value of random brakshould be renormalized as the following:

p→p1p21p31•••5p

12p. ~7!

Thus the flow becomes

f 512r2Lp

12pr2, ~8!

which is independent ofvmax. The flow decreases morabruptly with a larger value ofp and also with a larger valueof L.

In region ~III !, the flow reaches a plateau. The averavelocity is inversely proportional to the density. This cantaken as a signature of traffic jams, where correlations amvehicles are no longer infinitely ranged. The appearancejams would block out the long-range interactions. Thus

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BRIEF REPORTS PHYSICAL REVIEW E 63 012101

average velocity will not change rapidly with the variationdensity. With the appearance of jams, the flow can be demined by the resolvability of the jams. The first vehicwithin a jam has a probability of (12p) to move, and theflow is naively expected as (12p). However, it is not pos-sible for the second vehicle to move along if the first vehiremains stopped, which has a probabilityp. Thus the ex-pected flow should be

f 512p

11p. ~9!

The independence ofvmax is expected. As one enters intotraffic jam, the constraint from the speed limit shouldirrelevant. As the range of correlation is shortened, thependence on lattice size will also be lifted. Thus the valuethe plateau is solely determined by the random brakingp.

In summary, Eqs.~5!, ~8!, and ~9! provide a simple yetsatisfactory description of the fundamental diagram; see2.

III. DISCUSSION

In this paper, we study the low-density limit of the NageSchreckenberg model with a high speed limit. Three distiregions in the fundamental diagram are identified. Analytiexpressions are obtained with phenomenological consiations. Numerical results can be well described. In region~I!,the linear relation between flow and density is controlledthe speed limitvmax. Vehicles are independent of eacother. In region~II !, a quadratic relation between flow andensity is observed. The relevant parameters are the ranbrakingp and the lattice sizeL. Vehicles are strongly correlated. In region~III !, the flow reaches a plateau whose valis solely determined by the random brakingp. The trafficjams start to emerge and the correlations among vehiclesweakened.

When the stochastic noise is neglected,p50, only tworegions can be observed. As the density increases, theincreases linearly in region~I! and then decreases linearlyregion~II !, which becomes independent of the lattice sizeL.The domain of region~II ! expands as the speed limitvmaxincreases. In the limit of largevmax, region ~II ! constitutesthe fundamental diagram. No peculiar behavior is observexcept the discontinuity of flow at zero density. Asvmaxincreases, the flow at zero density assumes a value of 0has a limiting value of 1.

The origin of singularity in the zero-density limit can beasily understood in the case of a single vehicle. In thetionary state, the velocity is equal to its maximum value, ivmax. Thus the flow is given byf 5vmax/L, where the den-sity is given by 1/L. With a periodic boundary conditionimposed, the limitvmax→` is realized byvmax→(L21).Then the flow has a limiting value of unity for a large sytem. This picture is also true with the stochastic noise intduced. Thus the discontinuity of flow at zero density cantaken as the competition between two scales:L andvmax.

With stochastic noise, the flow decreases much faster wthe increase of density in region~II !. An approximate qua-

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dratic relation is observed, which can be related to the inite ranged correlations involved. With a further increasedensity, a plateau is observed in region~III !. The approxi-mate independence of flow from density can be related tofinite ranged correlations involved. In the limitvmax→`,region~I! disappears as expected. Regions~II ! and~III ! con-stitute the fundamental diagram. The existence of a platin region ~III ! and the peculiar enhancement of flow~as thedensity decreases! in region ~II ! are direct manifestations ostochastic noise.

With the analytical expressions, the critical densities cbe easily obtained as

r I ,II ;1

vmax, ~10!

r II ,III ;A2S 12p

11pD 1/2 1

AL, ~11!

where the approximationvmax;L@1 is assumed. We notethat asvmax@1, the critical densityr I ,II can be easily missedin numerical works. In Ref.@8#, only the critical densityr II ,III was reported.

It is interesting to note that there is an explicit dependeof lattice-sizeL in region~II !. Thus the peculiar enhancemeof flow can also be taken as a finite-size effect. In previoworks @10,11#, finite-size effects have only been noted in tcase of smallvmax. With a fixedvmax, the domain of region~II ! shrinks with the increase of lattice sizeL. In the limitL→`, region~I! transits into region~III ! smoothly. The pla-teau becomes the maximum in the fundamental diagram.peculiar enhancement of flow is observed. Similarly, withfixed L, the domain of region~II ! shrinks with the decreasof vmax. A similar transition can also be observed. Resu

FIG. 3. Fundamental diagram for various values ofvmax. Fixedparameters arep50.5 andL51000.

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BRIEF REPORTS PHYSICAL REVIEW E 63 012101

are shown in Fig. 3. Though we obtain the analytic exprsions assumingvmax@1, they still provide a good approximation whenvmax is small. Only then would the renormaization in Eq.~7! be unnecessary. The coefficientX will then

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-decrease with the decrease ofvmax and the curve will shifttoward the high density. The abrupt decrease of flow willlonger follow the scaling curve. The peculiar enhancemwill shift to a higher density, as shown in Fig. 3.

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M.

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