A CA Hybrid of the Slow-to-Start and the Optimal Velocity Models and its Flow-Density Relation
SSymmetry, Integrability and Geometry: Methods and Applications SIGMA (2015), 065, 10 pages A CA Hybrid of the Slow-to-Start and the OptimalVelocity Models and its Flow-Density Relation (cid:63)
Hideaki UJINO † and Tetsu YAJIMA ‡† National Institute of Technology, Gunma College, Maebashi, Gunma 371–8530, Japan
E-mail: [email protected] ‡ Department of Information Systems Science, Graduate School of Engineering,Utsunomiya University, Utsunomiya 321–8585, Japan
E-mail: [email protected]
Received March 31, 2015, in final form July 27, 2015; Published online July 31, 2015http://dx.doi.org/10.3842/SIGMA.2015.065
Abstract.
The s2s-OVCA is a cellular automaton (CA) hybrid of the optimal velocity(OV) model and the slow-to-start (s2s) model, which is introduced in the framework of theultradiscretization method. Inverse ultradiscretization as well as the time continuous limit,which lead the s2s-OVCA to an integral-differential equation, are presented. Several trafficphases such as a free flow as well as slow flows corresponding to multiple metastable statesare observed in the flow-density relations of the s2s-OVCA. Based on the properties of thestationary flow of the s2s-OVCA, the formulas for the flow-density relations are derived.
Key words: optimal velocity (OV) model; slow-to-start (s2s) effect; cellular automaton (CA);ultradiscretization, flow-density relation
Self-driven many-particle systems have provided a good microscopic point of view on the vehicletraffic [3, 5]. The optimal velocity model [1] gives a description of such a system with a setof ordinary differential equations (ODE). It is a car-following model describing an adaptationto the optimal velocity that depends on the distance from the vehicle ahead. Another way ofdescribing such systems is provided by cellular automata (CA). For example, the elementaryCA of Rule 184 (ECA184) [16], the Fukui–Ishibashi (FI) model [4] and the slow-to-start (s2s)model [12] are CA describing vehicle traffic as self-driven many-particle systems.Studies of the self-driven many-particle systems have been wanting a framework that com-mands a bird’s eye view of both ODE and CA models in a unified manner. Ultradiscretiza-tion [14], which gives a link between the Korteweg–de Vries (KdV) equation and integrable soli-ton CA [11], is expected to provide such a framework, for it can be applied to non-integrable sys-tems, too. As a first step toward such a framework, an ultradiscretization of the OV model [10]was presented. A specific choice of the OV function enabled the ultradiscretization of the OVmodel without any other specialization. We should note that another ultradiscretization of theOV model with essentially the same OV function as above was derived from the modified KdV(mKdV) equation, which is an effective theory around the critical point, with specializing itssolutions to traveling wave solutions [7]. The latter ultradiscretization of the OV model dependson the ultradiscretization of the mKdV equation, which is an integrable soliton equation withrich accumulation of the studies of integrable discretization and ultradiscretization. The former (cid:63) a r X i v : . [ n li n . C G ] J u l H. Ujino and T. Yajimaone, on the other hand, has nothing to do with integrability, which indicates a possibility toexpand the scope of the ultradiscretization beyond the integrable models. Thus the two ultra-discretizations make a clear contrast regarding to integrability. An early search for a CA-typeOV model dates back to 1999 [5, 6], which was done from a phenomenological point of view tohighway traffic.The former ultradiscretization of the OV model [10] provided a foundation toward a hybri-dization of the OV model and the s2s model, and it lead to the s2s-OVCA [9] indeed, withoutthe aid of the integrable models. The s2s-OVCA is a CA-type hybrid of the OV model andthe s2s model. As we shall see, the equation of the s2s-OVCA generally involves three or moretimes, or higher order time-differences, in other words. As far as the authors know, differenceequations involving higher order time-differences yet want a thorough study from a point ofview of the ultradiscretization. Besides an interest from the traffic theoretical point of view, aninterest toward a new horizon of the scope of the ultradiscretization motivates us to introduceand study the s2s-OVCA. As we shall show in Section 2, the s2s-OVCA reduces to an ODE thatis an extension of the OV model in the inverse-ultradiscrete and the time-continuous limits.It was observed by numerical experiments that motion of the vehicles described by the s2s-OVCA went to stationary flow in the long run, irrespectively of the initial configuration [9, 15].It was also observed by numerical experiments that the flow-density relation for the stationaryflow of the s2s-OVCA was piecewise linear and flipped- λ shaped with several metastable slowbranches [9]. Exact expression for the flow-density relation was given by a set of exact solutionsgiving stationary flows of the s2s-OVCA [15]. The flipped- λ shaped diagram captures thecharacteristic of observed flow-density relations [3, 5]. Some other CA type models that gavea flipped- λ shaped flow-density relation with a metastable branch was also reported [2]. Weshall explain in Section 3 the flow-density relation of the s2s-OVCA based on the properties ofthe stationary flow which was numerically observed [9]. The s2s-OVCA is given by a set of difference equations below x n +1 k = x nk + min (cid:16) n min n (cid:48) =0 (cid:0) x n − n (cid:48) k +1 − x n − n (cid:48) k − (cid:1) , v (cid:17) , (1)where the integers n ≥ v ≥ x nk , k = 1 , , . . . , K , are the monitoring period, the topspeed and the position of the car k at the n -th discrete time. Note that the definition of thesymbol N min k =0 is N min k =0 ( a k ) := min( a , a , a , . . . , a N ) . The equation (1) is called an ultra-discrete equation in the sense that it is a difference equationwhich is piecewise linear with respect to the dependent variables x nk . The s2s-OVCA includesthe ECA184 ( n = 0, v = 1) [16], the FI model ( n = 0) [4] and the s2s model ( n = 1, v = 1) [12] as its special cases.Since the second term in the right hand side of equation (1) gives the speed of the car k at the time n , the s2s-OVCA describes many cars running on a single lane highway in onedirection, which is driven by cautious drivers requiring enough headway to go on at least for n time steps before they accelerate their cars. The equation (1) also means that the car slowsdown immediately when its headway becomes less than its velocity. Thus the monitoring pe-riod n describes asymmetry between acceleration and deceleration of the cars. It is said thatthe acceleration times are about five to ten times larger than the braking times [5].2s-OVCA and Flow-Density Relation 3Without loss of generality, we can assume that the cars are arrayed in numerical order, x < x < · · · < x K , which is also assumed throughout below. Then the number of empty cellsbetween the cars k and k + 1 for any k is always non-negative, i.e., x nk +1 − x nk − x nk − ≥ . (2)It is obvious that the inequality holds for n = 0. We assume that the inequality holds up tosome n , as the induction hypothesis. The induction hypothesis as well as the definition of minassure the inequality0 ≤ min (cid:16) n min n (cid:48) =0 (cid:0) ∆ x n − n (cid:48) k − (cid:1) , v (cid:17) ≤ ∆ x nk − k . Using equation (1), we get an expression of ∆ x nk as∆ x n +1 k − x nk − (cid:16) n min n (cid:48) =0 (cid:0) ∆ x n − n (cid:48) k +1 − (cid:1) , v (cid:17) − min (cid:16) n min n (cid:48) =0 (cid:0) ∆ x n − n (cid:48) k − (cid:1) , v (cid:17) = min (cid:16) n min n (cid:48) =0 (cid:0) ∆ x n − n (cid:48) k +1 − (cid:1) , v (cid:17) + (cid:104) ∆ x nk − − min (cid:16) n min n (cid:48) =0 (cid:0) ∆ x n − n (cid:48) k − (cid:1) , v (cid:17)(cid:105) . (4)The inequality (3) and the equation (4) show that the inequality (2) holds for n + 1. Theinequality (2) means that both overtake and clash are prohibited by the s2s-OVCA.We should note that the s2s-OVCA is obtained from a difference equation by a limiting proce-dure named ultradiscretization [14], which generates a piecewise-linear equation from a differenceequation via the limit formulalim δx → +0 δx log (cid:18) N (cid:88) k =0 b k e a k /δx (cid:19) = max( a , a , a , . . . , a N ) =: N max k =0 ( a k ) , (5)where arbitrary numbers b k must be positive. The equation (5) is rewritten aslim δx → +0 δx log (cid:18) N (cid:88) k =0 b k e − a k /δx (cid:19) − = N min k =0 ( a k ) , for min( a , a , a , . . . , a N ) = − max( − a , − a , − a , . . . , − a N ).For the sake of convenience in the calculation below, we introduce two parameters, x and δt ,in the s2s-OVCA x n +1 k = x nk + min (cid:16) n min n (cid:48) =0 (cid:0) ∆ x n − n (cid:48) k (cid:1) − x , v δt (cid:17) =: x nk + v uopt (∆ eff x nk ) δt, (6)where ∆ eff x nk := min n n (cid:48) =0 (cid:0) ∆ x n − n (cid:48) k (cid:1) . The parameters x and δt are the length of a cell, whichcorresponds to the space occupied by a single car or the length of a car itself in the shortestcase imaginable, and the discrete time-step, respectively. For simulation of highway traffics,the length of a cell x and the discrete time-step δt are usually chosen as x = 7 . v = 5 × x δt [5]. But we do not consider specific values of the parameters in the s2s-OVCA andregard them as generic. The two parameters x and δt were set to be unity in equation (1).Introduction of x into the inequality (3) gives∆ x nk − x ≥ n and k , which is shown by induction with the aid of an inequality0 ≤ min (cid:16) n min n (cid:48) =0 (cid:0) ∆ x n − n (cid:48) k − x (cid:1) , v δt (cid:17) ≤ ∆ x nk − x H. Ujino and T. Yajimaand an expression of ∆ x nk derived from equation (6)∆ x n +1 k − x = min (cid:16) n min n (cid:48) =0 (cid:0) ∆ x n − n (cid:48) k +1 − x (cid:1) , v (cid:17) + (cid:104) ∆ x nk − x − min (cid:16) n min n (cid:48) =0 (cid:0) ∆ x n − n (cid:48) k − x (cid:1) , v (cid:17)(cid:105) that correspond equations (3) and (4), respectively.Since we have the inequality (7) for the headway ∆ x nk − x , the effective headway ∆ eff x nk − x is also always non-negative, ∆ eff x nk − x ≥
0, for any k . With the aid of the identitymin( A, B ) = A − max(0 , A − B ) = max(0 , A ) − max(0 , A − B )for any A ≥
0, the optimal velocity function v uopt ( x ) δt := min( x − x , v δt ) in the s2s-OVCA isexpressed as v uopt ( x ) δt = max(0 , x − x ) − max(0 , x − x − v δt ) , for any x >
0. It is given by the ultradiscrete limit δx → +0 of a function v dopt ( x ) δt = δx log ( x − x /δx − x /δx ( x − x − v δt ) /δx − ( x v δt ) /δx , which is an inverse-ultradiscretization of the optimal velocity function v uopt . Note that we haveintroduced arbitrary coefficients so as to make v dopt (0) = 0. In a similar way to the abovecalculation, an inverse-ultradiscretization of the effective interval ∆ ueff x nk is also obtained as∆ deff x nk := δx log (cid:32) n (cid:88) n (cid:48) =0 e − ∆ x n − n (cid:48) k /δx n + 1 (cid:33) − . Therefore an inverse-ultradiscretization of the us2s-OVCA is given by x n +1 k = x nk + v dopt (∆ deff x nk ) δt ,which is explicitly written as x n +1 k = x nk + δx (cid:40) log (cid:20) (cid:18) n (cid:88) n (cid:48) =0 e − (∆ x n − n (cid:48) k − x ) /δx n + 1 (cid:19) − (cid:21) − log (cid:0) − x /δx (cid:1) − log (cid:20) (cid:18) n (cid:88) n (cid:48) =0 e − (∆ x n − n (cid:48) k − x − v δt ) /δx n + 1 (cid:19) − (cid:21) + log (cid:0) − ( x + v δt ) /δx (cid:1)(cid:27) . (8)In other words, the s2s-OVCA is given by the ultradiscrete limit δx → +0 of the above differenceequation (8).Since equation (8) is rewritten as x n +1 k − x nk δt = δx (cid:40) − δt (cid:18) log (cid:20) (cid:18) n (cid:88) n (cid:48) =0 e − (∆ x n − n (cid:48) k − x − v δt ) /δx n + 1 (cid:19) − (cid:21) − log (cid:20) (cid:18) n (cid:88) n (cid:48) =0 e − (∆ x n − n (cid:48) k − x ) /δx n + 1 (cid:19) − (cid:21)(cid:19) + log (cid:0) − ( x + v δt ) /δx (cid:1) + log (cid:0) − x /δx (cid:1) δt (cid:41) = v (cid:18) n (cid:88) n (cid:48) =0 e − (∆ x n − n (cid:48) k − x ) /δx n + 1 (cid:19) − − v (cid:0) x /δx (cid:1) − + O ( δt ) , δt → x k d t = v (cid:18) t (cid:90) t e − (∆ x k ( t − t (cid:48) ) − x ) /δx d t (cid:48) (cid:19) − − v (cid:0) x /δx (cid:1) − , (9)where t := n δt , d x k d t = lim δt → x n +1 k − x nk δt andlim δt → n (cid:88) n (cid:48) =0 e − (∆ x n − n (cid:48) k − x ) /δx n + 1 = 1 t (cid:90) t e − (∆ x k ( t − t (cid:48) ) − x ) /δx d t (cid:48) . In terms of an optimal velocity function and an effective distance v opt ( x ) := v (cid:18)
11 + e − ( x − x ) /δx −
11 + e x /δx (cid:19) , ∆ eff x k ( t ) := δx log (cid:18) t (cid:90) t e − ∆ x k ( t − t (cid:48) ) /δx d t (cid:48) (cid:19) − , the above integral-differential equation is expressed asd x k d t = v opt (cid:0) ∆ eff x k ( t ) (cid:1) . Since the effective distance ∆ eff x k ( t ) goes to ∆ x k ( t ) in the limit below∆ x k ( t − t ) = lim h → t δx log (cid:18) t − h (cid:90) t h e − ∆ x k ( t − t (cid:48) ) /δx d t (cid:48) (cid:19) − , this integral-differential equation is an extension of the Newell model [8]d x k d t = v opt (cid:0) ∆ x k ( t − t ) (cid:1) , (10)which is a car-following model dealing with retarded adaptation to the optimal velocity deter-mined by the headway in the past.Replacement of t with t + t in equation (10) and the Taylor expansion of ˙ x k ( t + t ) = v k ( t + t )yield v opt (∆ x k ( t )) = v k ( t + t ) = v k ( t ) + d v k d t · t + 12 d v k d t · t + · · · , which is equivalent tod v k d t + 12 d v k d t · t + · · · = 1 t (cid:0) v opt (∆ x k ( t )) − v k ( t ) (cid:1) . (11)The equation of motion of the OV modeld v k d t = 1 t (cid:0) v opt (∆ x k ( t )) − v k ( t ) (cid:1) is given by neglecting the higher order terms in the left hand side of the equation (11).The discussion shown above in this section shows how the inverse ultradiscretization and thetime continuous limit connect the s2s model and the Newell model, which approximates the OVmodel, through the s2s-OVCA. H. Ujino and T. Yajima P o s iti on o f V e h i c l e s Time F l o w Density
Figure 1.
The spatio-temporal pattern (left) and the flow-density relation (right) of the s2s-OVCA [9].
Fig. 1 gives typical examples of the spatio-temporal pattern showing jams and the flow-densityrelation of the s2s-OVCA [9]. In the numerical calculation, the periodic boundary condition isimposed and the length of the circuit L , which is the same as the number of all the cells, is fixedat L = 100. The maximum velocity v and the monitoring period n are v = 3 and n = 2.The number of the cars K in the spatio-temporal pattern is set at K = 30.The spatio temporal pattern shows the trajectories of the cars. As we can see, irregularmotion of cars is observed in the early stage of the time evolution, 0 ≤ n ≤
30, where n is thetime. But after that, the flow of the cars become stationary in the sense that length of the jamis almost constant and that cars with intermediate speeds appear only temporarily.The flows Q in the flow-density relation are computed by averaging over the time period800 = n i ≤ n ≤ n f = 1000, Q := 1( n f − n i + 1) L K (cid:88) k =1 n f (cid:88) n = n i v nk , v nk := x n +1 k − x nk , (12)in which the traffic is expected to be stationary in the above mentioned sense. The car density ρ is given by ρ := KL . As we have mentioned before, the flow-density relation of the s2s-OVCA ispiecewise linear and flipped- λ shaped with several metastable slow branches.The flow-density relation shown above is derived by admitting the features of the flow ofthe s2s-OVCA. Namely, the flow of the s2s-OVCA goes to one of the stationary flows in thelong run. The stationary flows consist of the free flow in which all the cars run at the topspeed v and the slow flows that always contain slow cars running at the minimum speed v ∞ min ,0 ≤ v ∞ min < v , which remains constant. Formation of the line of slow cars corresponds to that oftraffic jam. In the slow flows, lengths of the jams are almost constant and fluctuate periodically.Our previous paper [15] gives a set of such stationary flows.First we shall deal with the free flow and its flow-density relation. Since all the cars run at thetop speed, v nk = v , ∀ k , all the headways respectively remain constant, ∆ x n +1 k = ∆ x nk ≥ v + 1, ∀ k . Hence the flow Q is also constant in the future. Using the definition of the flow (12) with n i = n f = n , we have Q = 1 L K (cid:88) k =1 v = ρv , which gives the straight line in the flow-density relation with a positive inclination that is equal2s-OVCA and Flow-Density Relation 7 F l o w Density
Figure 2.
The formula for the flow-density relation of the s2s-OVCA [15]. to the top speed v . For example in Fig. 2, the flow-density relation of the free flow which islabeled with v = 3 is Q = 3 ρ , since v = 3 in this case.Next we shall consider the slow flows and their flow-density relations. Let us see a specificsolution of the s2s-OVCA starting from the following initial configuration0 : . (13)Note that the number 0 at the leftmost shows the time. The digits and the blank symbols (cid:32) in the above configuration mean the indices of the cars and the empty cells, respectively. Thusthe number of the cars K is 10 and the length of the circuit L is 38 in this case. We set themonitoring period n at 2. The speed of the cars 4, 9 and 0 is 3, which is the top speed v ofthis case. The speed of the car 5 is 2, whose headway is also 2. All the other cars’ speeds are 1,whose headways are also 1 except for the car 3. Thus the headways of the cars in tha past havenothing to do with the motion of the cars in the future except for the car 3. The headway ofthe car 3 at the time − , (cid:32)1(cid:32)2(cid:32)3(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)4(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)5(cid:32)6(cid:32)7(cid:32)8(cid:32)(cid:32)(cid:32)9(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)0(cid:32) , , (cid:32)0(cid:32)1(cid:32)2(cid:32)(cid:32)(cid:32)3(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)4(cid:32)(cid:32)5(cid:32)6(cid:32)7(cid:32)8(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)9(cid:32)(cid:32)(cid:32) , (cid:32)(cid:32)0(cid:32)1(cid:32)2(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)3(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)4(cid:32)5(cid:32)6(cid:32)7(cid:32)(cid:32)(cid:32)8(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)9 , (cid:32)9(cid:32)0(cid:32)1(cid:32)2(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)3(cid:32)(cid:32)(cid:32)(cid:32)4(cid:32)5(cid:32)6(cid:32)7(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)8(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) , (cid:32)(cid:32)9(cid:32)0(cid:32)1(cid:32)(cid:32)(cid:32)2(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)3(cid:32)(cid:32)4(cid:32)5(cid:32)6(cid:32)7(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)8(cid:32)(cid:32) . Note that the minimum speed of the cars v ∞ min is 1 in the flow above. We notice that theconfiguration at the time 3 is obtained by moving all the cells of the initial configuration onecell rightward as well as changing the car indices k to k − ρ and the average flow Q H. Ujino and T. Yajima
Two cells are the same under PBC.
Figure 3.
The slow flow of the s2s-OVCA. Cars in the cells drawn by dashed lines are omitted exceptfor the car 1 at the time n + n + 1. over the period, or the n + 1 = 3 steps, are calculated as ρ = KL = 519 ,Q = 1( n + 1) L K (cid:88) k =0 n (cid:88) n (cid:48) =0 v n (cid:48) k = 6 + 3 + 3 + 5 + 9 + 4 + 3 + 3 + 3 + 93 ×
38 = 819 , which will be verified with the formula we shall derive shortly.Let us consider such slow flows as we have seen above as the specific solutions in a moregeneral manner. Fig. 3 shows configurations of a slow flow at times n and n + n + 1. Sincewe employ the periodic boundary condition, two cells containing the car K are identified. Asa property of the slow flow, we assume that the slow flow is periodic in the sense that theconfiguration at the time n + n + 1, which is shown in the box in Fig. 3, is given by therightward displacement of the entire configuration at the time n in the box by n v ∞ min − n + 1 time steps is n v ∞ min − n +1 ρ . For example, the rightward displacement mentioned above for the slow flow in Fig. 3is 2 × − K by L cells, namely wholethe circuit length, which is fictitiously introduced to make the shifted initial configuration fromthe real configuration at the time n + 1 in the sense that the numerical order of the car arraysis maintained. In order to compensate the underestimation of the flow brought about by thisleftward displacement, we have to add the flow corresponding to the rightward displacement ofthe car K by L cells in n + 1 time steps, n +1) L · L = n +1 . Thus the flow of the slow flowwith the minimum speed v ∞ min is given by Q = n v ∞ min − n + 1 ρ + 1 n + 1 , ≤ v ∞ min < v . (14)For example, substitution of ρ = , n = 2 and v ∞ min = 1 into equation (14) yields Q = 2 × −
12 + 1 ×
519 + 12 + 1 = 819 , which agrees with the flow Q = for the slow flow given above as an specific solution. Theformula (14) agrees with the flow-density relation given by numerical experiments, as we cansee in Fig. 2. Three branches labeled with v = 2 , v ∞ min = v in Fig. 2. Roughly speaking, the traffic that forms the branchcorresponding to v ∞ min consists of groups of cars running at the top speed v and other groupsof cars running at the minimum speed v ∞ min , as the specific solution evolving from the initial2s-OVCA and Flow-Density Relation 9configuration (13) has shown. A set of specific solutions that correspond to the branches in thefundamental diagram is given in [15].The maximum density ρ max ( v ∞ min ) that allows the minimum speed to be v ∞ min is ρ max ( v ∞ min ) = 1 v ∞ min + 1 . (15)The flow Q ( ρ max ( v ∞ min )) corresponding to the maximum density ρ max ( v ∞ min ) is then given by Q ( ρ max ( v ∞ min )) = ρ max ( v ∞ min ) v ∞ min . (16)Since the two equations (15) and (16) holds at the same time, they leads to Q ( ρ max ( v ∞ min )) + ρ max ( v ∞ min ) = 1. Thus all the end points of the branches must be on the line Q + ρ = 1 . (17)The branching point, or the minimum density, of the flow-density relation of the slow flowcorresponding to the minimum speed v ∞ min is determined by the intersection of the flow densityrelations of the free flow and the slow flow ρ min ( v ∞ min ) = 1 n ( v − v ∞ min ) + v + 1 . (18)In Fig. 2, the branching points corresponding to v ∞ min = 2 , ρ needs to be sufficiently largeso as to form the slow flow with the minimum speed v ∞ min . The branching point gives the lowerbound of such density.Since the s2s-OVCA (1) is a deterministic CA, the initial configuration determines the finalstate. Its numerical simulation is very robust against, or more precisely speaking, free fromnumerical errors. Thus all the stationary flows including the free flow and the slow flowsbeyond the minimum density ρ min ( v ∞ min ) are stable in the numerical simulation. That is whywe were able to obtain the flow-density relation with several “metastable” states consisted bythe free flow and the slow flows beyond the minimum density, as was shown in Figs. 1 and 2.Though these metastable states are robust against numerical errors in simulation, but they aregenerally unstable against perturbation, which gives a reason of their name. For example, whenone gives a perturbation to the headways that is equal to the minimum speed v ∞ min in the groupsof cars running at the minimum speed v ∞ min , a car with a velocity that is less than v ∞ min appears.And such a perturbed car generally becomes a seed of a group of cars running at a velocityslower than v ∞ min , which eventually slows down whole the traffic. That is why we call thesestates metastable, except for the slow line with v ∞ min = 0, which we cannot make slower.When the monitoring period n is zero, all of the slow flows (14) goes to the line of the endpoints of the metastable branches (17). Thus the monitoring period plays an essential role inthe formation of the metastable states in the flow-density relation. Thus it could be determinedby comparing the flow-density relation of the s2s-OVCA and observed ones. We have shown an inverse ultradiscretization from the s2s-OVCA (1) to an integral-differentialequation (9), which is an extension of the Newell model (10). Since the Newell model [8] andthe s2s-OVCA [9] are extended models of the OV [1] and the s2s models [12] respectively, thes2s-OVCA is interpreted as a CA-type hybrid of the OV and the s2s models.Using the features of the stationary flows observed in the numerical experiments, we havederived the flow-density relations of the stationary flow of the s2s-OVCA. The flow-density0 H. Ujino and T. Yajimarelations of the s2s-OVCA were numerically obtained [9] and then derived by use of a set ofstationary flows [15].Since the s2s-OVCA is a deterministic CA, the model is suitable for a simulation with a muchbigger system size than the length of the circuit, L = 100, in our numerical simulation. Weexpected that it would be sufficient to capture the characteristics of stationary flows of the s2s-OVCA on the circuit, which is determined by the density of cars and initial configuration. Inorder to observe finite-size effects of open boundaries, for example, we expect that the s2s-OVCAwill be a good tool.The s2s-OVCA has several types of monotonicity in its time evolution, which extend theresults shown for the n = 1 case [13]. We expect that the monotonicity determines the relaxationto the stationary flow from the initial configuration as well as the property of the stationary flowwe assume here. We hope that results on the relaxation to stationary flows and the monotonicityin the time evolution of the s2s-OVCA will be reported soon. Acknowledgments
One of the authors (HU) is grateful to K. Oguma for the previous collaboration.
References [1] Bando M., Hasebe K., Nakayama A., Shibata A., Sugiyama Y., Dynamical model of traffic congestion andnumerical simulation,
Phys. Rev. E (1995), 1035–1042.[2] Barlovic R., Santen L., Schadschneider A., Schreckenberg M., Metastable states in cellular automata fortraffic flow, Eur. Phys. J. B (1998), 793–800, cond-mat/9804170.[3] Chowdhury D., Santen L., Schadschneider A., Statistical physics of vehicular traffic and some relatedsystems, Phys. Rep. (2000), 199–329, cond-mat/0007053.[4] Fukui M., Ishibashi Y., Traffic flow in 1D cellular automaton model including cars moving with high speed,
J. Phys. Soc. Japan (1996), 1868–1870.[5] Helbing D., Traffic and related self-driven many-particle systems, Rev. Modern Phys. (2001), 1067–1141,cond-mat/0012229.[6] Helbing D., Schreckenberg M., Cellular automata simulating experimental properties of traffic flow, Phys.Rev. E (1999), R2505–R2508, cond-mat/9812300.[7] Kanai M., Isojima S., Nishinari K., Tokihiro T., Ultradiscrete optimal velocity model: a cellular-automatonmodel for traffic flos and linear instability of high-flux traffic, Phys. Rev. E (2009), 056108, 8 pages,arXiv:0902.2633.[8] Newell G.F., Nonlinear effects in the dynamics of car flowing, Operations Res. (1961), 209–229.[9] Oguma K., Ujino H., A hybrid of the optimal velocity and the slow-to-start models and its ultradiscretization, JSIAM Lett. (2009), 68–71, arXiv:0908.3377.[10] Takahashi D., Matsukidaira J., On a discrete optimal velocity model and its continuous and ultradiscreterelatives, JSIAM Lett. (2009), 1–4, arXiv:0809.1265.[11] Takahashi D., Satsuma J., A soliton cellular automaton, J. Phys. Soc. Japan (1990), 3514–3519.[12] Takayasu M., Takayasu H., 1 /f noise in a traffic model, Fractals (1993), 860–866.[13] Tian R., The mathematical solution of a cellular automaton model which simulates traffic flow with a slow-to-start effect, Discrete Appl. Math. (2009), 2904–2917.[14] Tokihiro T., Takahashi D., Matsukidaira J., Satsuma J., From soliton equations to integrable cellular auto-mata through a limiting procedure,