A hybrid of the optimal velocity and the slow-to-start models and its ultradiscretization
aa r X i v : . [ n li n . C G ] O c t A hybrid of the optimal velocity and the slow-to-start models andits ultradiscretization ∗ Kazuhito Oguma † Department of Mathematical Engineering and Information Physics,Faculty of Engineering, The University of Tokyo,7–3–1 Hongo, Bunkyo-ku, Tokyo 113–8656, Japan
Hideaki Ujino ‡ Gunma National College of Technology,580 Toriba, Maebashi, Gunma 371–8530, Japan
Abstract
Through an extension of the ultradiscretization for the optimal velocity (OV) model, we introducean ultradiscretizable traffic flow model, which is a hybrid of the OV and the slow-to-start (s2s)models. Its ultradiscrete limit gives a generalization of a special case of the ultradiscrete OV (uOV)model recently proposed by Takahashi and Matsukidaira. A phase transition from free to jamphases as well as the existence of multiple metastable states are observed in numerically obtainedfundamental diagrams for cellular automata (CA), which are special cases of the ultradiscrete limitof the hybrid model.
Keywords: optimal velocity (OV) model, slow-to-start (s2s) effect, ultradiscretization ∗ Accepted for publication in JSIAM Letters. † Also at Gunma National College of Technology, 580 Toriba, Maebashi, Gunma 371–8530, Japan ‡ Electronic address: [email protected] . INTRODUCTION Studies on microscopic models for vehicle traffic provided a good point of view on thephase transition from free to congested traffic flow. Related self-driven many-particle systemshave attracted considerable interests not only from engineers but also from physicists [1, 2].Among such models, the optimal velocity model [3], which successfully shows a formationof “phantom traffic jams” in the high-density regime, is a car-following model describing anadaptation to the optimal velocity that depends on the distance from the vehicle ahead.Whereas the OV model consists of ordinary differential equations (ODE), cellular au-tomata (CA) such as the Nagel–Schreckenberg model [4], the elementary CA of Rule 184(ECA184) [5], the Fukui–Ishibashi (FI) model [6] and the slow-to-start (s2s) model [7] areextensively used in analyses of traffic flow. Recently, Takahashi and Matsukidaira proposeda discrete OV (dOV) model, which enables an ultradiscretization of the OV model [8]. Theresultant ultradiscrete OV (uOV) model includes both the ECA184 and the FI model as itsspecial cases. However, the s2s effect remains to be included in their ultradiscretization. Theaim of this letter is to present an ultradiscretizable hybrid of the OV and the s2s models.
II. THE OV MODEL AND THE S2S EFFECT
Imagine many cars running in one direction on a single-lane highway. Let x k ( t ) denotethe position of the k -th car at time t . No overtaking is assumed so that x k ( t ) ≤ x k +1 ( t )holds for arbitrary time t . The time-evolution of the OV model [3] is given byd v k ( t )d t = 1 t (cid:16) v opt (cid:0) ∆ x k ( t ) (cid:1) − v k ( t ) (cid:17) , (1)where v k := d x k d t and ∆ x k := x k +1 − x k are the velocity of the k -th car and the intervalbetween the cars k and k + 1, respectively. A function v opt and a constant t represent anoptimal velocity and sensitivity of drivers, or the delay of drivers’ response, in other words.Since the current velocity and the current interval between the car ahead determinethe acceleration through the time-evolution and the optimal velocity, we classify the OVmodel (1) as the acceleration-control type (aOV). On the other hand, the OV model of thevelocity-control type (vOV) was proposed in earlier studies of the car-following models [9], v k ( t ) = v opt (cid:0) ∆ x k ( t − t ) (cid:1) . (2)2eplacement of t in the above equation (2) with t + t and the Taylor series of v k ( t + t )yield v opt (cid:0) ∆ x k ( t ) (cid:1) = v k ( t + t ) = v k ( t ) + d v k ( t )d t t + 12 d v k ( t )d t t + · · · , which is rewritten asd v k ( t )d t + 12 d v k ( t )d t t + · · · = 1 t (cid:16) v opt (cid:0) ∆ x k ( t ) (cid:1) − v k ( t ) (cid:17) . Thus we note that the aOV model (1) is given by neglection of the higher derivatives in theTaylor series of the vOV model (2). Though the aOV model is more common in the studieson vehicle traffic, we shall concentrate on an ultradiscretizable hybrid of the vOV and thes2s models. Thus we call the vOV model (2) simply as the OV model, hereafter.Note that the input to the OV function v opt ( x ) in the OV model (2) is the headway ata single point of time t − t that is prior to the present time t . Thus we may say that theOV model describes, in a sense, “reckless” drivers since the model pays no attention to theheadway between the time t − t and the present time t . On the other hand, “cautious”drivers governed by the s2s model [7] keep watching and require enough length of headwayto go on for a certain period of time before they restart their cars. The contrast betweenthe two models suggests the idea that the s2s effect and the OV model can be broughttogether by appropriately choosing an effective distance ∆ eff x k ( t ) containing information onthe headway for a certain period of time going back from the present as an input to the OVfunction v opt ( x ). We shall see this idea works in what follows.What is crucial in the ultradiscretization of the aOV model [8] is the choice of the OVfunction, v opt ( x ) := v (cid:16)
11 + e − ( x − x ) /δx −
11 + e x /δx (cid:17) , (3)where v , x and δx are positive constants. In terms of the auxiliary functions, e v opt ( x ) := v d˜ x opt ( x )d x (4) e x opt ( x ) := δx log (cid:0) ( x − x ) /δx (cid:1) (5)the OV function (3) is expressed as v opt ( x ) = e v opt ( x ) − e v opt ( x = 0) . e v dopt ( x ) := ˜ x opt ( x ) − ˜ x opt ( x − v δt ) δt , introduces the OV function for the discrete OV (dOV) model, v dopt ( x ) = e v dopt ( x ) − e v dopt ( x = 0) = δxδt log (cid:20) ( x − x ) /δx − x /δx (cid:30) ( x − x − v δt ) /δx − ( x + v δt ) /δx (cid:21) , (6)which is found to be ultradiscretizable. [8]Let x nk := x k ( t = nδt ) and v nk := ( x n +1 k − x nk ) /δt where n (= 0 , , · · · ) and δt ( >
0) are theintegral time and the discrete time-step, respectively. Employing the effective distance as∆ deff x nk := δx log (cid:16) n X n ′ =0 e − ∆ x n − n ′ k /δx n + 1 (cid:17) − , (7)where n := t /δt , we extend the OV model (2) in a time-discretized form as v nk = v dopt (cid:0) ∆ deff x nk (cid:1) , (8)which is equivalent to x n +1 k = x nk + δx ( log (cid:20) (cid:16) n X n ′ =0 e − (∆ x n − n ′ k − x ) /δx n + 1 (cid:17) − (cid:21) − log (cid:0) − x /δx (cid:1) − log (cid:20) (cid:16) n X n ′ =0 e − (∆ x n − n ′ k − x − v δt ) /δx n + 1 (cid:17) − (cid:21) + log (cid:0) − ( x + v δt ) /δx (cid:1)(cid:27) . It is straightforward to confirm that the continuum limit δt → x k ( t )d t = v opt (cid:0) ∆ eff x k ( t ) (cid:1) = v (cid:16) t Z t e − (∆ x k ( t − t ′ ) − x ) /δx d t ′ (cid:17) − − v (cid:0) x /δx (cid:1) − , (9)where the corresponding effective distance is given by∆ eff x k := δx log (cid:16) t Z t e − ∆ x k ( t − t ′ ) /δx d t ′ (cid:17) − . We shall see that the s2s effect is indeed built into the OV model in the ultradiscrete limitof the ds2s–OV model. 4
II. ULTRADISCRETIZATION
Ultradiscretization [10] is a scheme for getting a piecewise-linear equation from a differ-ence equation via the limit formulalim δx → +0 δx (e A/δx + e
B/δx + · · · ) = max( A, B, · · · ) . In order to go forward to the ultradiscretization of the ds2s–OV model (8), it will be a goodchoice for us to begin with the ultradiscrete limit δx → +0 of the auxiliary function (5): e x uopt ( x ) := lim δx → +0 e x opt ( x ) = max(0 , x − x ) . (10)In the same way to make the OV function for the dOV model (6) from the auxiliary func-tion (5), we obtain the OV function for the uOV model [8] as v uopt ( x ) = e v uopt ( x ) − e v uopt ( x = 0) = max (cid:16) , x − x δt (cid:17) − max (cid:16) , x − x δt − v (cid:17) , (11)where e v uopt ( x ) := (cid:0)e x uopt ( x ) − e x uopt ( x − v δt ) (cid:1) /δt . The effective distance (7), on the other hand,is ultradiscretized in the same manner:∆ ueff x nk := lim δx → +0 ∆ deff x nk = − n max n ′ =0 (cid:0) − ∆ x n − n ′ k (cid:1) = n min n ′ =0 (cid:0) ∆ x n − n ′ k (cid:1) . (12)Thus we obtain an ultradiscrete equation v nk = v uopt (cid:0) ∆ ueff x nk (cid:1) , (13)which is equivalent to x n +1 k = x nk + max (cid:16) , n min n ′ =0 (cid:0) ∆ x n − n ′ k (cid:1) − x (cid:17) − max (cid:16) , n min n ′ =0 (cid:0) ∆ x n − n ′ k (cid:1) − x − v δt (cid:17) , as the ultradiscrete limit of the ds2s–OV model (8). We name it the ultradiscrete s2s–OV(us2s–OV) model. When the monitoring period n is fixed at zero, the us2s–OV modelreduces to a special case of the uOV model [8]. As we can see from eqs. (11), (12) and(13), the velocity v nk is determined by the optimal velocity for the minimum headway inthe period between n − n and n . Thus cars will not restart nor accelerate, unless enoughclearance goes on for a certain period of time. On the other hand, cars immediately stopor slow down when their headways become too small to keep their velocities. The s2s effectand a “cautious” manner of driving are built into the uOV model in this way.5ow let us see how a CA comes out from the us2s–OV model. Let x be the discretizationstep of the headway ∆ x nk , or equivalently, the size of the unit cell of the CA. Then withno loss of generality, we may set x = 1. Assume that the number of vacant cells betweenthe cars k and k + 1, e ∆ x nk := ∆ x nk − x , must be non-negative, e ∆ x nk ≥
0, which prohibitscar-crash. Then the us2s–OV model (13) reduces to x n +1 k = x nk + min (cid:16) n min n ′ =0 (cid:0) e ∆ x n − n ′ k (cid:1) , v δt (cid:17) . (14)Fixing v δt at an integer, we call this model the s2s–OV cellular automaton (CA). Thes2s–OV CA reduces to the FI model [6] when n = 0 and to the ECA184 [5] when n = 0and v δt = 1(= x ). The s2s model [7] also comes out from the s2s–OV CA by choosing n = 1 and v δt = 1(= x ). Thus the s2s–OV CA is regarded as a hybrid of the FI modeland an extended s2s model. IV. NUMERICAL EXPERIMENTS
We shall numerically investigate the s2s–OV CA (14). Throughout this section, the lengthof the circuit L is fixed at L = 100 and the periodic boundary condition is assumed as wellso that x nk + L is identified with x nk .Spatio-temporal patterns showing trajectories of each vehicle are given in fig. 1. Wechoose the parameters and initial conditions so that jams appear in the trajectories. Thetwo figures in the top share the same monitoring period n = 2 but their maximum velocitiesare different. The top left trajectories show that the velocities of the vehicles are zero or one,which is less than or equal to its maximum velocity v δt = 1. In the top right trajectorieswhose maximum velocity v δt = 3, on the other hand, the velocities of the vehicles readzero, one, two and three. Thus we notice that the vehicles driven by the s2s–OV CA can runat any allowed integral velocity which is less than or equal to its maximum velocity v δt .The other two figures in the bottom in fig. 1 share the same maximum velocity v δt = 2,but their monitoring periods are different. As is observed in the bottom two figures, thelonger the monitoring period is, the longer it takes for the cars to get out of the trafficjam. The jam front is observed to propagate against the stream of vehicles at constantvelocity x ( n +1) δt , since cars have to wait n + 1 time-steps to restart after their precedingcars restarted, as is depicted in fig. 2. 6 PSfrag replacements
Time P o s i t i o n s o f V e h i c l e s PSfrag replacements
Time P o s i t i o n s o f V e h i c l e s PSfrag replacements
Time P o s i t i o n s o f V e h i c l e s PSfrag replacements
Time P o s i t i o n s o f V e h i c l e s FIG. 1: The spatio-temporal patterns of the s2s–OV CA. For all four patterns, the number ofcars K is fixed at K = 30 The maximum velocities v δt and the monitoring periods n for thesepatterns are (top left) v δt = 1, n = 2, (top right) v δt = 3, n = 2, (bottom left) v δt = 2, n = 1and (bottom right) v δt = 2, n = 3, respectively. Fig. 3 shows fundamental diagrams giving the relation between the vehicle flow Q := 1( n − n + 1) L K X k =1 n X n = n x n +1 k − x nk δt , which is equivalent to the total momentum of vehicles per unit length, and the vehicle density ρ := KL , where K is the number of vehicles. The fundamental diagrams clearly show phasetransitions from free to jam phases as well as metastable states, which are also observed inempirical flow-density relations [1, 2]. It is remarkable that the fundamental diagrams havemultiple metastable branches. This feature is similar to that reported by Nishinari et al. [11]By observation, we note that each fundamental diagram has v δt metastable branches and ajamming line. The branches and the jamming line correspond to integral velocities that are7 Sfrag replacements direction of the streamjam fronttime x (= 1) (cid:0) n (= 3) + 1 (cid:1) δt (= 4)012345FIG. 2: Backward propagation of the jam front at constant velocity x ( n +1) δt = for the case v δt = 2, n = 3 and x = 1. less than or equal to the maximum velocity v δt . Let us confirm it with fig. 3. The top twofigures share the same monitoring period n = 3, but their maximum velocities are different.The top left diagram corresponding to v δt = 2 has three branches. This number equalsto that of all the integral velocities, two, one and zero, as is depicted in the diagram. Thenumber of the metastable branches in the top right diagram as well as those of the bottomtwo are explained in the same manner. This observation also suggests that the monitoringperiod is irrelevant to the number of metastable branches.All the end points of the branches as well as the jamming line are on the line ρ + Q (= ρ + Q δtx ) = 1. This is because the density at the end point is the maximum density ρ max ( v )that allows the velocity of the slowest car to be vδt . The maximum density ρ max ( v ) isdetermined by ρ max ( v ) = x vδt + x . Since all the cars flow at the velocity vδt when ρ = ρ max ( v ), the corresponding flow is givenby Q ( ρ max ) = vρ max . Thus the relation ρ max + Q ( ρ max ) δtx = 1 holds.The free line is a branch whose inclination equals to the maximal velocity v δt . Any othermetastable branch and the jamming line branch out from the free line. By observation, thedensity of the branch point of the branch corresponding to the velocity vδt reads ρ b = x ( v δt − vδt ) n + v δt + x . This observation is explained as follows. Suppose one car, say the car k , runs at the velocity v and the other K − v . At the moment the k -th car8 Sfrag replacements F l o w Density ......... .........9 98 87 76 65 54 43 32 2 211 11 1000000000 0000000000 0
PSfrag replacements F l o w Density ......... .........9 98 87 76 65 54 443 332 2 211 11 1000000000 0000000000 0
PSfrag replacements F l o w Density ......... .........9 98 87 76 65 54 43 32 211 11000000000 0000000000
PSfrag replacements F l o w Density ......... .........9 98 87 76 65 54 43 32 211 11000000000 0000000000
FIG. 3: The fundamental diagrams of the s2s–OV CA. The flows Q are computed by averagingover the time period 800 ≤ n ≤ v δt and the monitoring periods n for these patterns are (top left) v δt = 2, n = 3, (top right) v δt = 4, n = 3, (bottom left) v δt = 3, n = 2 and (bottom right) v δt = 3, n = 4, respectively. The inclination of the free lineequals to the maximum velocity v δt . The jamming line has a negative inclination. slows down to v , the headway between the cars k and k + 1 is vδt + x . Since it takes at least n + 1 time-steps for the car k to speed up to v , the headway between the cars k and k + 1expands up to H = ( v δt − vδt )( n + 1) + vδt + x = x /ρ b ≥ v δt by the time the k -th carspeeds up to v . If all the cars can obtain the headway H , slow cars running at the velocity v disappear in the end. Thus the density at the branch point of the branch corresponding tothe velocity vδt is given by ρ b = x /H . Note that the density at the branch point becomessmaller as the monitoring period becomes larger.9 . CONCLUDING REMARKS Through an extension of the ultradiscretization for the OV model [8], we introduced theds2s–OV (8) and s2s–OV (9) models as ultradiscretizable traffic flow models. The model isa hybrid of the OV [3] and the s2s [7] models whose ultradiscrete limit gives a generalizationof a special case of the uOV model by Takahashi and Matsukidaira [8]. The phase transitionfrom free to jam phases as well as the existence of multiple metastable states were observedin numerically obtained fundamental diagrams for the s2s–OV CA (14), which are specialcases of the us2s–OV model (13).Detailed studies on the properties of the hybrid models (8), (9), (13) and (14) such asexact solutions, comparison with other traffic flow models as well as empirical data remainto be investigated.
Acknowledgments
The authors are grateful to D. Takahashi, J. Matsukidaira, A. Tomoeda, D. Yanagisawaand R. Nishi for their valuable comments at the spring meeting of JSIAM in March, 2009. [1] D. Chowdhury, L. Santen and A. Schadschneider, Statistical physics of vehicular traffic andsome related systems, Phys. Rep. (2000), 199–329.[2] D. Helbing, Traffic and related self-driven many-particle systems, Rev. Mod. Phys., (2001),1067–1141.[3] M. Bando, K. Hasebe, A. Nakayama, A. Shibata and Y. Sugiyama, Dynamical model of trafficcongestion and numerical simulation, Phys. Rev. E, (1995), 1035–1042.[4] K. Nagel and M. Schreckenberg, A cellular automaton model for freeway traffic, J. PhysiqueI, (1992), 2221–2229.[5] S. Wolfram, Theory and applications of cellular automata, World Scientific, Singapore, 1986.[6] M. Fukui and Y. Ishibashi, Traffic flow in 1D cellular automaton model including cars movingwith high speed, J. Phys. Soc. Jpn. (1996), 1868–1870.[7] M. Takayasu and H. Takayasu, 1 /f noise in a traffic model, Fractals (1993), 860–866.
8] D. Takahashi and J. Matsukidaira, On a discrete optimal velocity model and its continuousand ultradiscrete relatives, JSIAM Letters (2009), 1–4.[9] G. F. Newell, Nonlinear effects in the dynamics of car following, Oper. Res. (1961), 209–229.[10] T. Tokihiro, D. Takahashi, J. Matsukidaira and J. Satsuma, From soliton equations to inte-grable cellular automata through a limiting procedure, Phys. Rev. Lett. (1996), 3247–3250.[11] K. Nishinari, M. Fukui and A. Schadschneider, A stochastic cellular automaton model fortraffic flow with multiple metastable states, J. Phys. A. Math. Gen. (2004), 3101–3110.(2004), 3101–3110.