On a class of new nonlocal traffic flow models with look-ahead rules
aa r X i v : . [ n li n . C G ] J a n ON A CLASS OF NEW NONLOCAL TRAFFIC FLOW MODELS WITHLOOK-AHEAD RULES
YI SUN AND CHANGHUI TAN
Abstract.
This paper presents a new class of one-dimensional (1D) traffic models withlook-ahead rules that take into account of two effects: nonlocal slow-down effect and right-skewed non-concave asymmetry in the fundamental diagram. The proposed 1D cellularautomata models with the Arrhenius type look-ahead interactions implement stochastic rulesfor cars’ movement following the configuration of the traffic ahead of each car. In particular,we take two different look-ahead rules: one is based on the distance from the car underconsideration to the car in front of it; the other one depends on the car density ahead. Bothrules feature a novel idea of multiple moves, which plays a key role in recovering the non-concave flux in the macroscopic dynamics. Through a semi-discrete mesoscopic stochasticprocess, we derive the coarse-grained macroscopic dynamics of the CA model. We also designa numerical scheme to simulate the proposed CA models with an efficient list-based kineticMonte Carlo (KMC) algorithm. Our results show that the fluxes of the KMC simulationsagree with the coarse-grained macroscopic averaged fluxes for the different look-ahead rulesunder various parameter settings. Introduction
The study on traffic flows has received considerable effort in the past few decades. Manymathematical frameworks and models have been proposed and analyzed in the literature[4, 9, 17, 19, 23, 33, 37, 38, 43, 44, 50]. These models can be categorized by different scales.The microscopic models focus on modeling the behaviors of individual cars. Two types ofmicroscopic models are widely used in traffic flows.(i).
The agent-based models . The location of each car is traced in time, by an inter-acting ODE system. A variety of interaction rules are proposed, followed by analysis andsimulations. In the classical follow-the-leader model [12]˙ v i = κ i ( v i +1 − v i ) , the driver in the i -th car accelerates/decelerates according to the relative velocity towardsthe ( i + 1)-th car in front. In the optimal velocity (OV) model [2]˙ v i = κ i ( V i − v i ) , where V i is the optimal velocity of the i -th car, which depends on the distance toward the carin front. The OV model has several extensions. In [29,36], the optimal velocity is modeled bymultiple cars in front. In [14, 15, 41], cars behind are also taken into consideration. Ref. [54]discussed and analyzed multiple look-ahead models, which reproduce many real flow features.Besides the common fundamental properties of traffic flow, free flow and congested flow, all Mathematics Subject Classification.
Key words and phrases.
Traffic flow, cellular automata model, nonlocal macroscopic models, multiplejumps, kinetic Monte Carlo. these models above can reproduce a third fundamental property of traffic flow, the so-calledsynchronized flow [20].(ii)
The lattice models . The road is configured as a fixed lattice. Each site has values 1(car is present) or 0 (car is absent). Explicit rules for car movement on the lattice sites aredescribed to represent the traffic flow. The lattice models, also known as cellular automata (CA) models [55, 56], have been widely used to represent traffic flows. A vast literatureexists addressing various analytical and numerical techniques for models of this type [3, 5,10, 21, 22, 30, 39, 40, 42]. Compared with the agent-based models, lattice models are simplerto implement and are more amenable to numerical investigation. A major advantage oflattice models is that they allow for a systematic derivation of the coarse-grained dynamics,which typically requires some simplifying assumptions about the statistical behavior of themicroscopic model.To understand the emergent phenomena for large crowded traffic systems, a variety of macroscopic models have been proposed. Instead of working on individual cars, these modelsdescribe the evolution of the density distribution of the traffic. One classical model in 1Dwith a single-lane dynamics is the Lighthill-Whitham-Richards (LWR) model [32], ∂ t ρ + ∂ x ( ρu ) = 0 , u = u max (1 − ρ ) . (1)Here, ρ is the density of the traffic and u is the macroscopic velocity, which takes maximumvalue u max if ρ = 0, and becomes 0 if the maximum density ρ = 1 is reached. It is well-knownthat the Burgers-type nonlinearity leads to a finite-time wave breakdown for any smoothinitial configuration. This corresponds to the creation of traffic jams.The LWR model (1) has many extensions. One direction is to consider the nonlocal slow-down effect : drivers intend to slow down if heavy traffic is ahead. This would involve anonlocal interaction with a look-ahead distance a , ∂ t ρ + ∂ x ( ρu ) = 0 , u = u max (1 − ρ ) exp (cid:20) − Z a K ( y ) ρ ( x + y ) dy (cid:21) , (2)where K is a look-ahead kernel. The model was first introduced by Sopasakis and Katsoulakis(SK) in [46], with K ≡
1. Another kernel K ( r ) = 2(1 − ra ) was discussed in [8] for pedestrianflows, followed by an extensive numerical study.The wave breakdown phenomenon for the SK model (2) and related nonlocal models hasbeen studied in [25, 27]. Recently, it is shown in [28] that the nonlocal slowdown effect canhelp avoid traffic jams for a family of initial configurations.Another concern on the LWR model (1) is on its fundamental diagram . The flux F = ρu = u max ρ (1 − ρ )is a concave function of ρ with an even symmetry at ρ = 1 /
2, which does not agree withthe experimental data [23, 44]. A better fit would be a right-skewed non-concave flux, whichlooks like F = u max ρ (1 − ρ ) J , J > . (3)To take into account of the two effects, we propose a new class of macroscopic traffic modelswith the form ∂ t ρ + ∂ x ( ρu ) = 0 , u = u max (1 − ρ ) J exp (cid:20) − Z a K ( y ) ρ ( x + y ) dy (cid:21) . (4) RAFFIC FLOW MODELS 3
For J = 1, it reduces to the SK model (2). For J = 2, it has been recently introduced anddiscussed in [26]. The behavior of the solution is different from the SK model, due to thenon-concavity of the flux. Yet, the nonlocal slowdown effect could still help avoid finite-timewave breakdown for some initial configurations.It has been a very active area to study the connections between the microscopic andmacroscopic dynamics. From the microscopic models, one can let the number of cars go toinfinity, and derive a mean-field limit. The resulting mesoscopic models are characterizedby kinetic equations. The macroscopic dynamics can then be obtained through appropriatehydrodynamic limits.In this paper, we focus on the lattice models. In [46], a CA model with Arrhenius typelook-ahead interactions was proposed (see Sec. 2 for the description in detail). Through a semi-discrete mesoscopic stochastic process , the SK model (2) can be formally derived as acoarse-grained description of the CA model. Extensions to multilane and multiclass traffichave also been developed in [1,11]. An improved mesoscopic model has been discussed in [16].Two different look-ahead rules in both one-dimensional (1D) and two-dimensional (2D) CAmodels were compared in [48].A natural question that arises is whether there is a lattice model that connects our newmacroscopic model (4).We propose a CA model with two different types of look-ahead rules. The rules feature anovel idea of multiple moves , which plays a key role in recovering the non-concave flux in themacroscopic dynamics, for any J ∈ Z + . The first rule has a slowdown factor that is basedon the distance from the car under consideration to the car in front of it. The correspondingmacroscopic dynamics is a local scalar conservation law with non-concave flux (3). The otherrule’s slowdown factor depends on the car density ahead. With such long-range interaction,we derive the target coarse-grained macroscopic dynamics (4).We also design a numerical scheme for the proposed CA models. To improve computationalefficiency, we use the kinetic Monte Carlo (KMC) algorithm [7] due to its main feature —“rejection-free”. When the dynamics of the traffic system features a finite number of distinctprocesses in configurational changes, we develop an efficient list-based KMC algorithm usinga fast search that can further improve the efficiency compared to the general KMC method.On the other hand, the Metropolis Monte Carlo (MMC) method [35] is adopted in mostof current CA models for vehicular flows and pedestrian flows. But the MMC method isa way of simulating an equilibrium distribution for a model, and trial steps are sometimesrejected because the acceptance probability is small, in particular when a system approachesthe equilibrium, or the car density is high. Therefore, we choose the KMC, which is moresuitable for simulating the time evolution of the traffic systems with the transition rates thatare associated with possible configurational changes in the system. With reasonable values ofthe model parameters (the characteristic time unit and the interaction strength), the KMCsimulations are used to predict the time evolution of 1D traffic flows. Our results showthat the rules induce nonlocal slow-down effect and right-skewed non-concave asymmetryin the fundamental diagram. Moreover, the fluxes of the KMC simulations agree with thecoarse-grained macroscopic averaged fluxes for the different look-ahead rules under variousparameter settings.The rest of the paper is organized as follows. In Sec. 2, we introduce the CA models withtwo look-ahead rules. In Sec. 3, we discuss the derivation of the macroscopic models starting Y. SUN AND C. TAN from our proposed CA models. In Sec. 4, we describe the list-based KMC algorithm and itsimplementation. In Sec. 5, we provide a series of numerical simulations in various parameterregimes for the 1D flows, and compare the microscopic and macroscopic models. Finally, westate our conclusions in Sec. 6.2.
Cellular Automata Models with Look-ahead Rules
We describe the construction of the cellular automata (CA) model for 1D traffic flow inthis section. The CA model is defined on a periodic lattice L with M evenly spaced cells, L = { , , . . . , M } . For simplicity, we assume that all cars move toward one direction on asingle-lane loop highway with no entrances or exits. The configuration at each cell i ∈ L isdefined by an index σ i : σ i = ( i, i is empty . (5)The state of the system is represented by σ = { σ i } Mi =1 , which lies in the configuration spaceΣ = { , } M .2.1. Interaction rules.
We now describe the dynamics of the CA models. The car move-ment can be represented by the transitions in the state of the system, which obey the rules ofan exclusion process [31]: two nearest-neighbor lattice cells exchange values in each transitionand cars cannot occupy the same cell. In addition, cars are only allowed to move one cellto the right in one transition. Therefore, the only possible configuration changes are of theform { σ i = 1 , σ i +1 = 0 } → { σ i = 0 , σ i +1 = 1 } . (6)The transition rate for (6) depends on spatial Arrhenius type one-sided interactions anda look-ahead feature to represent drivers’ behavior. These rules allow cars (or drivers) toperceive the traffic situation up to L cells ahead in which L is the look-ahead range parameter.The interactions between a pair of successive cars cannot be neglected if the gap betweenthem is shorter than L ; in such situations, the following car must decelerate to avoid collisionwith the leading car. This is similar to the spin-exchange Arrhenius dynamics in which thesimulation is driven based on the energy barrier a particle has to overcome in changing fromone state to another [1, 46]. During a spin-exchange between nearest-neighbor sites i and i + 1, the system will allow the index σ i at location i to exchange its index with the one at i + 1. This is interpreted as a car move from i to i + 1 with the rate given by the Arrheniusrelation: r i = ω exp (cid:0) − E b ( i ) (cid:1) , (7)where the prefactor ω = 1 /τ corresponds to the car moving frequency or speed and τ isthe characteristic time. The moving energy barrier E b ( i ) for the car located in the i -th celldescribes the slow-down effect due to the traffic in front. It is assumed to depend only on thetraffic situation up to range L ahead of the car under consideration, namely E b ( i ) dependson { σ j } i + Lj = i +2 .The energy barrier can be modeled through E b = E s + E c , where E s is the externalpotential associated with the site binding of the car, which could vary in both space andtime to account for spatial and temporal traffic situations, such as rush hour traffic, local RAFFIC FLOW MODELS 5 weather anomalies, etc. In this study, we set E s = 0. The term E c enforces the look-aheadrules (Fig. 1). We consider the two rules in [48], described as follows.( a ) cell i N v = 4 N v = 3 N v = 2 N v = 1 N v = 0 L ( b ) cell i N c = 0 N c = 1 N c = 2 N c = 2 N c = 3 L Figure 1.
Schematic representation of two look-ahead rules. (a): The rulebased on the distance N v : a car in gray with different numbers ( N v ) of vacantcells between it and the first car (in blue) ahead of it in the range L (here, L = 4). (b): The rule based on the density N c /L : a car in gray with differentnumbers ( N c ) of cars (in blue) ahead of it in the range L .The first look-ahead rule is based on the distance from the car under consideration to thecar in front of it, in other words, the number of vacant cells, N v , between these two cars (asshown in Fig. 1(a)). Therefore, the energy barrier is given by E b ( i ) = L − N v ( i ) L E , (8)where the parameter E is the car look-ahead interaction strength. Based on the formulas(7) and (8), we can see that the smaller is the value of N v , the larger is the energy barrier E b ,thus the smaller is the transition rate r . This reflects the fact that the closer is the distancebetween cars, the stronger is the slowdown factor.The second rule is based on the density of cars ahead of the car under consideration [46].This is due to the fact that in real traffic drivers usually observe not only the leading car butalso other cars ahead of the leading car. In this rule, the energy barrier is given by E b ( i ) = N c ( i ) L E , (9)where N c is the number of cars in the range L ahead of the car under consideration (as shownin Fig. 1(b)). It can be defined as N c ( i ) = i + L X j = i +1 σ j . This look-ahead rule with the formula (9) indicates that a slowdown factor is stronger whenthe forward car density is high, i.e., when the road is congested.
Y. SUN AND C. TAN
The coarse-grained macroscopic dynamics corresponding to the two CA models above areLWR model (1) and SK model (2), respectively. The formal derivation for the latter casecan be found, for instance, in [16, 46].2.2.
A new class of models.
We propose a new class of CA models, and show their relationsto the macroscopic models with a non-concave flux (3), for any parameter J ∈ Z + .A novel discovery is that the parameter J in the macroscopic dynamics is related to thenumber of steps a car moves in a transition of the microscopic states. A car can make multiple J -moves, if the J cells in front are not occupied.The new rule only allows the following configuration change { σ i = 1 , σ i +1 = · · · = σ i + J = 0 } → { σ i = · · · = σ i + J − = 0 , σ i + J = 1 } . (10)It represents a car move from i to i + J . The transition rate is modeled similarly as (7).Since the car moves J cells, we take the following rate r i = ω J exp (cid:0) − E b ( i ) (cid:1) , (11)so that the estimated velocity is comparable among different choices of J .Note that the rule (10) only allows cars to move if no cars occupy the J cells in front.Therefore, heuristically speaking, the larger J is, the slower the cars move. Such a phe-nomenon is also shared by the macroscopic dynamics (4), which is verified in the numericalexperiments in Sec. 5.4.3. Coarse-grained macroscopic models
In this section, we perform a formal derivation of the coarse-grained macroscopic modelfor our CA model.3.1.
Semi-discrete mesoscopic models.
In a time step ∆ τ , the probability of the config-uration change P (cid:0) { σ i = 1 , σ i +1 = · · · = σ i + J = 0 } → { σ i = · · · = σ i + J − = 0 , σ i + J = 1 } (cid:1) = (∆ τ ) r i , (12)where the rate r i is given in (11).Followed from [16, 46], we define σ ( τ ) = { σ i ( τ ) } Mi =1 be a continuous-in-time stochasticprocess with a generator( Aψ )( τ ) = lim ∆ τ → E [ ψ ( σ ( τ + ∆ τ ))] − ψ ( σ ( τ ))∆ τ , (13)for any test function ψ : Σ → R , where τ is the time variable. All possible configurationchanges from σ ( τ ) to σ ( τ + ∆ τ ) obey the transition rule (12).By the definition of the generator, we have ddτ E ψ = E [ Aψ ] . (14) RAFFIC FLOW MODELS 7
In particular, let us take ψ ( σ ) = σ i . We calculate (13) explicitly, and obtain Aσ i ( τ ) = − r i ( τ ) σ i ( τ ) J Y j =1 (1 − σ i + j ( τ )) + r i − J ( τ ) σ i − J ( τ ) J Y j =1 (1 − σ i − J + j ( τ )) =: F i − J ( τ ) − F i ( τ ) , (15)where F i is defined as F i ( τ ) = r i ( τ ) σ i ( τ ) J Y j =1 (1 − σ i + j ( τ )) . Let ρ i ( τ ) = E [ σ i ( τ )] = P ( σ i ( τ ) = 1). Then, from (14) and (15), the dynamics of { ρ i } Mi =1 reads ddτ ρ i ( τ ) = E [ Aσ i ( τ )] = E [ F i − J ( τ )] − E [ F i ( τ )] . (16)Note that the right hand side of the equation is not yet a closed form of { ρ i ( τ ) } Mi =1 . We shallapproximate the term E [ F i ( τ )] and make a closure to the system.3.2. Approximations as M → ∞ . We start with a crucial assumption that helps us toobtain a closed system. It is called the propagation of chaos , which means that { σ i ( τ ) } Mi =1 are independent to each other, namely E [ σ i ( τ ) σ j ( τ )] = E [ σ i ( τ )] E [ σ j ( τ )] , ∀ i = j, t ≥ . (17)Due to the look-ahead interaction, condition (17) is not true for a system with fixed M cells.However, as the number of cells M tends to infinity, the system can become chaotic, andcondition (17) can be valid as M → ∞ .By formally assuming the chaotic condition (17), we get E [ F i ( τ )] = ρ i ( τ ) J Y j =1 (1 − ρ i + j ( τ )) E [ r i ( τ ) | σ i ( τ ) = 1 , σ i +1 ( τ ) = · · · = σ i + J ( τ ) = 0 ] . For the rest of the section, we drop the τ -dependence for simplicity.To estimate the rate r i = ω J exp( − E b ( i )), we perform a formal Taylor expansion on E b ( i )around its mean E [ E b ( i )]. e − E b ( i ) = e − E [ E b ( i )] ∞ X n =0 ( − n n ! (cid:0) E b ( i ) − E [ E b ( i )] (cid:1) n . Taking the expectation, we obtain E [ r i ] = ω J e − E [ E b ( i )] ∞ X n =2 ( − n n ! E (cid:2)(cid:0) E b ( i ) − E [ E b ( i )] (cid:1) n (cid:3)! . (18)Next, we estimate the energy barrier E b . To proceed, we first introduce the relative look-ahead distance a , defined as a = LM . (19)We assume a is a fixed positive number (0 < a ≤ M → ∞ , the look-aheaddistance L = aM would also tends to infinity.The two barriers (8) and (9) will be discussed separately. Y. SUN AND C. TAN
Recall the first energy barrier (8) E b ( i ) = L − N v ( i ) L E , where the random variable N v ( i ) takes integer values between 0 and L . Conditioned withthe configuration { σ i = 1 , σ i +1 = · · · = σ i + J = 0 } , N v ( i ) takes values in { J, J + 1 , . . . , L } .We impose the following assumptionmax N v ( i ) ≪ L = aM, (20)where the maximum is taken across all possible values of N v and all locations i = 1 , . . . , M .The assumption (20) describes the scenario that many cars are on the road, so that no twoneighboring cars have a large distance comparable to the look-ahead distance L .Under the assumption (20), we get E [ E b ( i )] = L − E [ N v ( i )] L E = (cid:18) − E [ N v ( i )] L (cid:19) E L →∞ −−−−→ E , and consequently, E b ( i ) − E [ E b ( i )] = L − N v ( i ) L E − E = − N v ( i ) L E L →∞ −−−−→ . Therefore, plug back into (18), we end up with E [ r i ] L →∞ −−−−→ ω J e − E . Next, we move to the second energy barrier (9) E b ( i ) = E L i + L X j = i +1 σ j = E L i + L X j = i + J +1 σ j , where the second equality is valid under the conditional configuration { σ i ( τ ) = 1 , σ i +1 ( τ ) = · · · = σ i + J ( τ ) = 0 } . Compute E [ E b ( i )] = E L i + L X j = i + J +1 ρ j . Then, we have E [( E b ( i ) − E [ E b ( i )]) ] = E L E i + L X j = i + J +1 ( σ j − ρ j ) ! = E L E " i + L X j = i + J +1 ( σ j − ρ j ) + 2 E L E " i + L X j = i + J +1 j − X k = i + J +1 ( σ j − ρ j )( σ k − ρ k ) . By condition (17), the cross terms E [( σ j − ρ j )( σ k − ρ k )] = E [ σ j − ρ j ] E [ σ k − ρ k ] = 0 . RAFFIC FLOW MODELS 9
Moreover, since | σ j − ρ j | ≤
1, we have E [( σ j − ρ j ) ] ≤
1. Hence, E [( E b ( i ) − E [ E b ( i )]) ] = E L i + L X j = i + J +1 E [( σ j − ρ j ) ] ≤ E ( L − J ) L L →∞ −−−−→ . Similarly, higher moments vanishes when L → ∞ : E [( E b ( i ) − E [ E b ( i )]) n ] L →∞ −−−−→ , ∀ n ≥ . Plug back into (18), we conclude with E [ r i ] L →∞ −−−−→ ω J exp − E L i + L X j = i + J +1 ρ j ! . To sum up, we achieve an approximation of E [ F i ( τ )] in terms of { ρ i ( τ ) } Mi =1 as M → ∞ E [ F i ( τ )] M →∞ −−−−−→ ω J ρ i ( τ ) J Y j =1 (1 − ρ i + j ( τ )) e − E First rule ω J ρ i ( τ ) J Y j =1 (1 − ρ i + j ( τ )) exp − E L i + L X j = i + J +1 ρ j ( τ ) ! Second rule (21)3.3.
Coarse-grained PDE models.
We rescale the lattice L into a fixed interval Ω = [0 , h = 1 /M . The i -th cell is rescaled to the interval [( i − h, ih ].Define the macroscopic density ρ : Ω × R + → R , where ρ ( x, t ) = ρ i ( τ ) , with x = ih, t = τ h. A coarse-grained model can be obtained by formally letting h →
0. Under our setup, theparameters are scaled as 1 ≤ J ≤ N v | {z } O (1) ≪ L ≤ M | {z } O ( h − ) , where L and M goes to infinity with a fixed ratio a = L/M .The flux in (21) as h → F ( x, t ) := J · lim h → E [ F i ( τ )] = ω ρ ( x, t )(1 − ρ ( x, t )) J e − E , (22) ω ρ ( x, t )(1 − ρ ( x, t )) J exp (cid:18) − E Z x + ax ρ ( y, t ) dy (cid:19) . (23)For the first rule, the macroscopic flux (22) depends locally on the density ρ ; while for thesecond rule, the flux (23) is nonlocal in ρ .The dynamics of ρ in (16) becomes the following scaler conservation law: ∂ t ρ ( x, t ) = 1 h ddτ ρ i ( τ ) = E F i − J ( τ ) − E F i ( τ ) h h → −−−−→ lim h → F ( x − J h, t ) − F ( x, t ) J h = − ∂ x ( F ( x, t )) . We end up with the following coarse-grained PDE models: (i). For the first look-ahead rule , ∂ t ρ + ∂ x (cid:0) ω ρ (1 − ρ ) J e − E (cid:1) = 0 . (24)This is the LWR type local model with flux (3) and u max = ω e − E .(ii). For the second look-ahead rule , ∂ t ρ + ∂ x (cid:18) ω ρ (1 − ρ ) J exp (cid:18) − E Z x + ax ρ ( y, t ) dy (cid:19)(cid:19) = 0 . (25)This is indeed our proposed macroscopic model (4) with a nonlocal look-ahead interactionkernel K ≡ E .If the relative look-ahead distance a = 0, the equation (25) becomes the local dynamics(24). On the other hand, if we consider the periodic domain (loop highway) and set a = 1,namely L = M , the interaction becomes global. By conservation of mass, the averaged cardensity Z x +1 x ρ ( y, t ) dy = ¯ ρ is a constant for any x and t . Equation (25) again reduces to the local dynamics (24), witha different u max = ω e − ¯ ρE , which also depends on the constant ¯ ρ .4. The Kinetic Monte Carlo Method
To investigate the evolution of the nonlocal traffic system, we apply the kinetic Monte Carlo(KMC) method [7] to the microscopic CA model with look-ahead interactions. The reason tochoose the KMC instead of the Metropolis Monte Carlo (MMC) method [35] is that trial stepsin the MMC are sometimes rejected because the acceptance probability is small, in particularwhen a system approaches the equilibrium, or the density of cars is high. A main featureof the KMC algorithm is that it is “rejection-free”. In each step, the transition rates for allpossible changes from the current configuration are calculated and then a new configurationis chosen with a probability proportional to the rate of the corresponding transition. Theother feature of the KMC method is its capability of providing a more accurate descriptionof the real-time evolution of a traffic system in terms of these transition rates since the KMCmethod is more suitable for simulating the non-equilibrium system.We emphasize that although the KMC algorithm was presented in [48], there is a majorupdate due to the multiple jumps J introduced in the current work. To make the presentationself-contained, we include the details of the algorithm here again. The KMC algorithm is builton the assumption that the model features N independent Poisson processes (correspondingto N moving cars on the lattice) with transition rates r i in (11) that sum up to give thetotal rate R = P Ni =1 r i . In simulations with a finite number of distinct processes, it is moreefficient to consider the groups of events according to their rates [6, 45, 47]. This can be doneby forming lists of the same kinds of events according to the values of N v in (8) of the firstlook-ahead rule or the values of N c in (9) of the second look-ahead rule. Therefore, we canput the total N events into ( L + 1) lists, labeled by l = 0 , . . . , L . All processes in the l -th listhave the same rate r l . We denote the number of processes in this list by n l , which is calledthe multiplicity , and we have N = P Ll =0 n l . To each list, we assign a partial rate, R l = n l r l ,and a relative probability, P l = R l /R . Then the total rate is given by R = P Ll =0 n l r l . A RAFFIC FLOW MODELS 11 fast list-based KMC algorithm at each KMC step based on the grouping of events is givenas follows.
List-based KMC algorithm:
Step 1: Generate a uniform random number, ξ ∈ (0 ,
1) and decide which process will takeplace by choosing the list index s such that s − X l =0 R l R < ξ ≤ s X l =0 R l R (26)Step 2: Select a car for the realization of the process s . This can be done with the help ofa list of coordinates for each kind of event, and an integer random number ξ in the range[1 , n s ]; ξ is generated and the corresponding car/event from the list is selected.Step 3: Check if there are enough vacant cells ahead of the selected car. If “Yes” (i.e., N v ≥ J ), perform Steps 4–6 for total J times so that the selected car can make J movesbefore continuing to the next KMC step. If “No” (i.e., N v < J ), perform only Step 5 fortotal J times so that the selected car will stay in its current cell for J transition time periodsbefore continuing to the next KMC step.Step 4: Perform the selected event (the car move to the next cell) leading to a newconfiguration.Step 5: Use R and another random number ξ ∈ (0 ,
1) to decide the time it takes for thatevent to occur (the transition time), i.e., the nonuniform time step ∆ t = − log( ξ ) /R .Step 6: Update the multiplicity n l , relative rates R l , total rate R and any data structurethat may have changed due to this move. (cid:3) In summary, the following parameters need to be given for the KMC simulations witheither look-ahead rule: (i) the characteristic time τ ; (ii) the car interaction strength E ; (iii)the look-ahead parameter L ; and (iv) the multiple move parameter J (1 ≤ J ≤ L ).5. Numerical experiments
We next investigate 1D nonlocal traffic flows in various parameter regimes with the nu-merical method presented in the previous section. We start by calibrating some KMC modelparameters with respect to well-known quantities from real traffic data.5.1.
Calibration and validity by the red light traffic problem.
Following [46, 48], weset the actual physical length of each cell to 22 feet ( ≈ . ≈ ≈ . τ cell = 22 feet60 miles / h = 1 cell × ×
240 cells = 14 s . (27)We calibrate the parameters τ and E by simulating a free-flow regime where all cars areexpected to drive at their desired speed that is set to 60 miles per hour ( ≈ . τ = 0 . ω = 4s − . In fact, dueto the inherent stochasticity in the simulations, sometimes cars may move faster or slower than the speed limit. We also mention that other values of τ and E may be chosen to adjustour model for considering different standards in other regions or countries. Time [s] D i s t an c e [ c e ll s ] (a)
60 m/h-10 m/h
Time [s] D i s t an c e [ c e ll s ] (b)
60 m/h-10 m/h
Figure 2.
Calibration of the interaction strength E permitting the desiredcar speed of 60 miles per hour ( ≈ . ≈ −
10 miles per hour ( ≈ − . ≈ M = 240 cells)and set the look-ahead parameter of L = 4 and the multiple move parameterof J = 2 for both look-ahead rules. The initial condition corresponds to a redlight traffic problem, i.e., bumper-to-bumper cars up to 0 .
125 miles ( ≈
201 m= 30 cells) and no cars after that, so a total of 30 cars in each simulation.The running time is up to 240 s. (a) Car traces in a simulation with the firstlook-ahead rule (8) and the interaction strength E = 4 .
5. (b): Car traces ina simulation with the second look-ahead rule (9) and E = 6 . . ≈ i = 30 cells) is turned from red to green at the initial time and the“bumper to bumper” traffic wave is released. The initial condition is given by σ i = ( ≤ i ≤ , ≤ i ≤ M. (28)The highway distance is set to 1 mile ( ≈ M = 240 cells. Then the averaged cardensity ¯ ρ = 30 /
240 = 12 . E such that the velocity of an upstream front can be approximately −
10 milesper hour ( ≈ − . L = 4 and the multiple move parameter J = 2 for both look-aheadrules, the calibrated value of the interaction strength is E = 4 . E = 6 . Numerical comparisons for different interaction strengths.
First, we analyze theeffects of the interaction strength E on the traffic flow and identify the range of significanceof parameters. In the following we take a fixed look-ahead distance of L = 4 and the multiple RAFFIC FLOW MODELS 13 move parameter J = 2 and make a series of numerical tests for different car densities withvarious values of parameters E . We show the fundamental diagrams of the density-flow,density-velocity and flow-velocity relationships and compare the results of two look-aheadrules (8) and (9). For these results we take a random car distribution at the initial timeon a loop highway of ≈ .
17 miles ( ≈ M = 1000 cells) and observe the behavior oftraffic flows as the averaged car density ¯ ρ increases incrementally from ¯ ρ = 0 .
01 to ¯ ρ = 0 . h F i in number of cars per hour and theensemble-averaged velocity of all cars h v i in cells per second.In Figs. 3(a) and (b), we plot the fundamental diagrams on the averaged fluxes h F i againstthe averaged density ¯ ρ of the first look-ahead rule (8) with E = 0 to 6 . E = 0 to 8 .
0, respectively. They all share certain characteristics:a nearly linear increase of the flow at low averaged densities (which corresponds to thefree-flow regime), a single maximum of the flow reached a critical density ¯ ρ crit , and a right-skewed asymmetry (namely ¯ ρ crit < / ρ crit and the maximum value of the flow h F i tend to decrease with increasing E because thelarger is the interaction strength the stronger is the interaction to slow down the cars.Figs. 3(c) and (d) show the fundamental diagrams of the density-velocity relationship fortwo look-ahead rules, respectively. In the free-flow regime the ensemble-averaged velocity h v i decreases approximately linearly from the maximum speed of 4 cells per second ( ≈ . ρ increases and the chance of interaction between cars gets higher. As E increases, when ¯ ρ is larger than the critical point ¯ ρ crit , the average velocity drops down to zeroand the density-velocity curve is negative exponential. This linear relationship follows theGreenshields model [13] and the negative exponential relationship belongs to the Underwoodmodel [51].Figs. 3(e) and (f) show the fundamental diagrams of the flow-velocity relationship for twolook-ahead rules, respectively, which plot the ensemble-averaged velocity h v i versus the flow h F i . For the case of the interaction strength E = 0 (shown as green “+” signs), the flow h F i reaches its maximum ≈ h v i is ata critical value h v i crit ≈ . ≈ . . E increases,the maximum value of the flow decreases and the critical value h v i crit increases and becomeshigher than 2 cells per second. The results compare favorably with observed data in [53].We remark that for a small look-ahead distance of L = 4, both rules produce similarresults as shown in Fig. 3. Indeed, the macroscopic model (24) is very close to (25) witha small a = LM = 0 . E = 4 . E = 6 . ▽ ” signs) clearly display that the regionof free-flow persists up to the density of approximately ¯ ρ crit = 0 .
2, i.e., 240 × . ≈
50 carsper mile. These results are naturally produced by the traffic dynamics in our simulationswith the calibrated parameters τ = 0 . L = 4 and the multiplemove parameter J = 2, which agrees with observations [18, 53]. Density F l o w [ c a r s / h ] (a) E =0.0E =1.0E =2.0E =3.0E =4.5E =6.0 Density F l o w [ c a r s / h ] (b) E =0.0E =1.0E =2.5E =4.0E =6.0E =8.0 Density A v e r age v e l o c i t y [ c e ll s / s ] (c) E =0.0E =1.0E =2.0E =3.0E =4.5E =6.0 Density A v e r age v e l o c i t y [ c e ll s / s ] (d) E =0.0E =1.0E =2.5E =4.0E =6.0E =8.0 Flow [cars/h] A v e r age v e l o c i t y [ c e ll s / s ] (e) E =0.0E =1.0E =2.0E =3.0E =4.5E =6.0 Flow [cars/h] A v e r age v e l o c i t y [ c e ll s / s ] (f) E =0.0E =1.0E =2.5E =4.0E =6.0E =8.0 Figure 3.
Comparison results of the traffic flow on the one-lane highway withsix different values of the interaction strength E . In all KMC simulations, wetake the highway distance of ≈ .
17 miles ( ≈ M = 1000 cells), thelook-ahead parameter of L = 4, the multiple move parameter of J = 2 and thefinal time of 1 hour. (a)(b): Longtime averages of the density-flow relationship;(c)(d): Ensemble-averaged velocity of cars versus the density ¯ ρ ; (e)(f): Long-time averages of the flow-velocity relationship. (left panel): Results of the firstlook-ahead rule (8) with E = 0 to 6 .
0. (right panel): Results of the secondlook-ahead rule (9) with E = 0 to 8 . L is large, the two look-ahead rules produce differ-ent results in all diagrams of the density-flow, density-velocity and flow-velocity relationshipsas shown in Fig. 4 and in the following Sec. 5.3. RAFFIC FLOW MODELS 15
Density F l o w [ c a r s / h ] (a) E =0.0E =1.0E =2.0E =3.0E =4.5PDE fluxes Density F l o w [ c a r s / h ] (b) E =0.0E =1.0E =2.5E =4.0E =6.0PDE fluxes Density A v e r age v e l o c i t y [ c e ll s / s ] (c) E =0.0E =1.0E =2.0E =3.0E =4.5 Density A v e r age v e l o c i t y [ c e ll s / s ] (d) E =0.0E =1.0E =2.5E =4.0E =6.0 Flow [cars/h] A v e r age v e l o c i t y [ c e ll s / s ] (e) Flow [cars/h] A v e r age v e l o c i t y [ c e ll s / s ] (f) Figure 4.
Comparison results of the traffic flow on the one-lane highway withfive different values of the interaction strength E . In all KMC simulations, wetake the highway distance of ≈ .
17 miles ( ≈ M = 1000 cells), the look-ahead parameter of L = 1000, the multiple move parameter of J = 2 and thefinal time of 1 hour. (a)(b): Long-time averages of the density-flow relationship;(c)(d): Ensemble-averaged velocity of cars versus the density ¯ ρ ; (e)(f): Long-time averages of the flow-velocity relationship. (left panel): Results of thefirst look-ahead rule (8) with E = 0 to 4 .
5. (right panel): Results of thesecond look-ahead rule (9) with E = 0 to 6 .
0. Note that the flux of the KMCsimulation in (a) and (b) agrees with the macroscopic averaged flux (29) and(30) (shown as the dashed black curves), respectively.Take the look-ahead distance L = 1000, which is equal to the length M of the loop highway.Recall the macroscopic flux under the two rules F = ω ρ (1 − ρ ) J e − E , and F = ω ρ (1 − ρ ) J e − ¯ ρE . As the dynamics will reach the equilibrium state ρ ( x ) ≡ ¯ ρ , the longtime averaged flux shouldsatisfy h F i ( ¯ ρ ) = (cid:26) ω ¯ ρ (1 − ¯ ρ ) J e − E , First rule (29) ω ¯ ρ (1 − ¯ ρ ) J e − ¯ ρE , Second rule (30)The critical density can then be obtained through a first derivative test. For the first rule,¯ ρ crit = 11 + J , (31)which depends on J but is independent of E . For the second rule,¯ ρ crit = 2( E + J + 1) + p ( E + J + 1) − E , (32)which depends on both J and E .In Fig. 4, we fix multiple move parameter J = 2, and vary the interaction strength E .Figs. 4(a) and (b) of the density-flow relationship show that as E increases, the maximumvalue of the flow h F i of both look-ahead rules tend to decrease. For the first rule (8) with E =0 to 4 . ρ crit = , which is consistent with (31). When E ≥ .
0, the maximum value of the flow h F i becomes very low as the PDE flux decreasesexponentially with increasing E indicated in (29). For the second look-ahead rule (9) with E = 0 to 6 . ρ crit tends to decrease with increasing E , as indicated in(32). Moreover, the second rule (9) can still produce relatively larger flux at small values ofaverage density ¯ ρ than the first rule (8) does.The fundamental diagrams of the density-velocity relationship in Figs. 4(c) and (d) alsoshow differences between two look-ahead rules for the large look-ahead distance L . Fig. 4(c)for the first rule (8) shows that as E increases, the value of h v i at ¯ ρ = 0 .
01 decreasesdrastically, ranging from the full speed h v i = 4 . ≈ . E = 0 (shown as green “+” signs) to almost h v i ≈ E = 4 . E decrease from high values of h v i = 3 .
74 to 4 . ≈ . . . ρ = 0 .
01 and eventually decay to zero with the increasingdensity ¯ ρ .Figs. 4(e) and (f) of the flow-velocity relationship again show big differences between twolook-ahead rules for the large look-ahead distance L . Fig. 4(e) for the first rule (8) showsthat both the magnitude of the flow h F i and the range of h v i decrease with increasing E .For the second rule (9) shown in Fig. 4(f), only the magnitude of the flow h F i decreases as E increases, but both the range of h v i and the critical value h v i crit where the flow h F i reachesits maximum do not change too much. The value of h v i crit is around 1 . ≈
10 m/s or 22 . Numerical comparisons for different look-ahead distances.
Next, we show theeffects of the look-ahead distances L on the flows in more detail in Fig. 5. For these results,we again take a random car distribution at the initial time on a loop highway of ≈ . RAFFIC FLOW MODELS 17 miles ( ≈ M = 1000 cells) and observe the behavior of traffic flows as the averagedcar density ¯ ρ increases incrementally from ¯ ρ = 0 .
01 to ¯ ρ = 0 .
99. Here, we use E = 2 . E = 6 . J = 2 for both rules. All curves exhibit phase transitionsbetween the free-flow phase and the jammed phase. However, Fig. 5 shows that the twolook-ahead rules produce different results in all diagrams of the density-flow, density-velocityand flow-velocity relationships when the look-ahead distance L is large.Fig. 5(a) of the density-flow relationship for the first rule (8) shows that as L increases, themaximum values of the flow h F i tend to decrease, but the value of the critical density ¯ ρ crit first decreases ( L = 2, 4, 8 and 16) and later changes to increase ( L ≥ L = 1000,the flux of the KMC simulations (shown as cyan squares) agrees with the averaged flux forthe macroscopic dynamics (29) (shown as the dashed black curve). We note that the sameKMC simulation results have also been shown as blue circles for E = 2 . ρ crit = . Following (29), themaximum flux at ¯ ρ crit = is about 3600 · e − . ≈
289 cars per hour (recall that ω = 4s − ).Moreover, for a fixed ¯ ρ , the magnitude of the flow h F i decreases with increasing L sincethe larger is the look-ahead distance the longer is the effective range of interaction betweenthe cars. For the second look-ahead rule (9) shown in Fig. 5(b), as L increases, both themaximum value of the flow h F i and the value of the critical density ¯ ρ crit tend to decrease.When L = 1000, the flux of the KMC simulations (shown as cyan squares) also matches withthe averaged flux of the PDE model (30) (shown as the dashed black curve). The same casehas also been shown as cyan squares for E = 6 . L = 100 for both rules (shown as black “ ▽ ” signs in Figs. 5(a) and (b)) are close to thecorresponding macroscopic fluxes.In Figs. 5(c) and (d), the fundamental diagrams of the density-velocity relationship alsoshow differences between the two look-ahead rules. Fig. 5(c) for the first rule (8) shows thatas L increases, the ensemble-averaged velocity h v i decreases very rapidly in the low-densityregime and eventually decays to zero with the increasing density ¯ ρ . In particular, while thedensity-velocity curve of L = 100 (shown as black “ ▽ ” signs) drops down from a high valueof h v i = 3 .
53 cells per second ( ≈ . . ρ = 0 .
01, the case of L = 1000(shown as cyan squares) starts from a low value of h v i = 0 .
66 cells per second ( ≈ . . ρ = 0 .
01. We recall that the loop highway has a length of M = 1000 cells,so ¯ ρ = 0 .
01 means that there are a total of only 10 cars on this highway of ≈ .
17 miles( ≈ h v i is so low, which indicates that the first look-aheadrule (8) is not reasonable for large look-ahead distances L . On the other hand, Fig. 5(d)shows that the second look-ahead rule (9) produces more reasonable results, even for largelook-ahead distances L . The case of L = 1000 (shown as cyan squares) gradually decreasesfrom a high value of h v i = 3 .
70 cells per second ( ≈ . . ρ = 0 . L = 2, the flow-velocity curvesfor both rules (shown as green “+” signs) reach their maxima at the critical velocity h v i crit ≈ . ≈ . . L increases in Fig. 5(e) of the firstlook-ahead rule (8), the maximum value of the flow decreases and the critical value h v i crit increases to be higher than 2 cells per second ( L = 4, 8 and 16). When L = 32, the result(shown as blue “x” signs) produces two local maxima in the flow. As L further increases,the range of the average velocity eventually becomes very small ( L = 1000, shown as cyansquares). On the other hand, Fig. 5(f) shows that for all L , the flow-velocity curve of the Density F l o w [ c a r s / h ] (a) L=2L=4L=8L=16L=32L=100L=1000PDE flux
Density F l o w [ c a r s / h ] (b) L=2L=4L=8L=16L=32L=100L=1000PDE flux
Density A v e r age v e l o c i t y [ c e ll s / s ] (c) L=2L=4L=8L=16L=32L=100L=1000
Density A v e r age v e l o c i t y [ c e ll s / s ] (d) L=2L=4L=8L=16L=32L=100L=1000
Flow [cars/h] A v e r age v e l o c i t y [ c e ll s / s ] (e) Flow [cars/h] A v e r age v e l o c i t y [ c e ll s / s ] (f) Figure 5.
Comparison results of the traffic flow on the one-lane highway withseven different values of the look-ahead distance L . In all KMC simulations,we take the highway distance of ≈ .
17 miles ( ≈ M = 1000 cells), themultiple move parameter of J = 2 and the final time 1 hour. (a)(b): Long-timeaverages of the density-flow relationship; (c)(d): Ensemble-averaged velocityof cars versus the density ¯ ρ ; (e)(f): Longtime averages of the flow-velocityrelationship. (left panel): Results of the first look-ahead rule (8) with E = 2 . E = 6 .
0. Notethat for long range interactions ( L = 1000), the flux of the KMC simulation(shown as cyan squares in (a) and (b)) agrees with the macroscopic averagedflux (29) and (30) (shown as the dashed black curve), respectively.second look-ahead rule (9) has the full range from a high speed down to zero. As L increasesfrom 2, the maximum value of the flow decreases, but the critical value h v i crit first increases RAFFIC FLOW MODELS 19 and becomes higher than 2 cells per second ( L = 2 , L ≥ Numerical comparisons for different multiple move parameters.
Finally, weshow the effects of the multiple move parameter J on the flows in Fig. 6. For these results,we still take a random car distribution at the initial time on a loop highway of ≈ .
17 miles( ≈ M = 1000 cells) and observe the behavior of traffic flows as the averaged cardensity ¯ ρ increases incrementally from ¯ ρ = 0 .
01 to ¯ ρ = 0 .
99. Here, we also use E = 2 . E = 6 . L = 1000 for both rules. Fig. 6 shows the differencesbetween two look-ahead rules with the large look-ahead distance L .Fig. 6(a) shows the density-flow relationship for the first rule (8) with J increasing from1 to 5. The fluxes match beautifully with the macroscopic averaged flux (29) (shown asthe dashed black curves). The case J = 1 corresponds to the LWR type model, where thecurve is symmetric and concave. For J ≥
2, the curves become neither convex nor concave,and have a right-skewed asymmetry. This indicates that our new model is more realistic.Moreover, for a fixed ¯ ρ , the magnitude of the flow h F i decreases with increasing J , whichverifies the heuristic argument in Sec.2. For the second look-ahead rule (9) shown in Fig. 6(b),the microscopic fluxes agree with the macroscopic averaged flux (30) very well. The criticaldensity ¯ ρ crit is located at (32) and it gets smaller as J increases. For instance, when J = 1and E = 6 . ρ crit = √ ≈ .
14 and the maximumflux is about 748 cars per hour.The fundamental diagrams of the density-velocity relationship in Figs. 6(c) and (d) alsoshow differences between two look-ahead rules (note that the ranges of Y -axis are differentin two figures). Fig. 6(c) for the first rule (8) shows that at the same ¯ ρ , the ensemble-averaged velocity h v i decreases as J increases. All cases of different J start from a low valueof h v i = 0 .
66 cells per second ( ≈ . . ρ = 0 .
01. As we pointed outin Fig. 5(c) in Sec.5.3, the low average velocity h v i at low densities indicates that the firstlook-ahead rule (8) is not reasonable for large look-ahead distances L . Here again, Fig. 6(d)shows that for large look-ahead distances L , the second look-ahead rule (9) produces morereasonable results. All curves of different cases of J gradually decrease from a high value of h v i = 3 .
74 cells per second ( ≈ . . ρ = 0 .
01 and eventually decay tozero with the increasing density ¯ ρ .Figs. 6(e) and (f) of the flow-velocity relationship show that for a fixed value of h v i , themagnitude of the flow h F i decreases with increasing J . However, the value of h v i crit wherethe flow h F i reaches its maximum does not change too much as J increases. For the firstlook-ahead rule (8) shown in Fig. 6(e), the value of h v i crit is around 0 .
25 cells per second( ≈ .
67 m/s or 3 .
75 miles/h). For the second rule (9) shown in Fig. 6(f), the value of h v i crit is around 1 . ≈
10 m/s or 22 . Conclusion
We have presented a new class of one-dimensional (1D) models to study traffic flows. Ourwork is motivated by the growing need to understand mechanisms leading to traffic jamsand develop a quantitative approach to the optimal design of transportation systems. Thecellular automata (CA) traffic models proposed here incorporate stochastic dynamics for the
Density F l o w [ c a r s / h ] (a) J=1J=2J=3J=4J=5PDE fluxes
Density F l o w [ c a r s / h ] (b) J=1J=2J=3J=4J=5PDE fluxes
Density A v e r age v e l o c i t y [ c e ll s / s ] (c) J=1J=2J=3J=4J=5
Density A v e r age v e l o c i t y [ c e ll s / s ] (d) J=1J=2J=3J=4J=5
Flow [cars/h] A v e r age v e l o c i t y [ c e ll s / s ] (e) J=1J=2J=3J=4J=5
Flow [cars/h] A v e r age v e l o c i t y [ c e ll s / s ] (f) J=1J=2J=3J=4J=5
Figure 6.
Comparison results of the traffic flow on the one-lane highway withfive different values of the multiple move parameter J . In all KMC simulations,we take the highway distance of ≈ .
17 miles ( ≈ M = 1000 cells), thelook-ahead distance of L = 1000 and the final time 1 hour. (a)(b): Long-timeaverages of the density-flow relationship; (c)(d): Ensemble-averaged velocityof cars versus the density ¯ ρ ; (e)(f): Long-time averages of the flow-velocityrelationship. (left panel): Results of the first look-ahead rule (8) with E = 2 . E = 6 .
0. Notethat for each value of J = 1 , , . . . ,
5, the flux of the KMC simulation in (a)and (b) agrees with the macroscopic averaged flux in (29) and (30) (shown asthe dashed black curves), respectively. Also, note the differences in the rangesof Y -axis between the left and right panels.movement of cars by using the Arrhenius type look-ahead rules of each car, which take into RAFFIC FLOW MODELS 21 account of the nonlocal slow-down effect. In particular, we considered two different look-ahead rules: the first one is based on the distance from the car under consideration to the carin front of it; the second one depends on the car density ahead. Both rules feature a novelidea of multiple moves, which plays a key role in recovering the right-skewed non-concaveflux in the macroscopic dynamics. Through a semi-discrete mesoscopic stochastic process,we derive the coarse-grained macroscopic dynamics of the CA model.To simulate the proposed CA models, we applied an efficient list-based KMC algorithmwith a fast search that can further improve computational efficiency. In the KMC method,the dynamics of cars is described in terms of the transition rates corresponding to possibleconfigurational changes of the system, and then the corresponding time evolution of the sys-tem can be expressed in terms of these rates. While the Metropolis Monte Carlo (MMC)method is a way of simulating an equilibrium distribution for a model, the KMC is moresuitable for simulating the time evolution of the traffic systems. Moreover, since the KMCalgorithm is “rejection-free”, we choose the KMC as one of our contributions in terms of com-putational efficiency. The KMC simulations relied on the calibration of model parameters:the characteristic time τ , the interaction strength E , the look-ahead parameter L and themultiple move parameter J . Then we used the KMC simulations to quantitatively predictthe time evolution of the traffic flows.Our numerical results show that the fluxes of the KMC simulations agree with the coarse-grained macroscopic averaged fluxes under various parameter settings. We obtained funda-mental diagrams that display several important observed traffic states. In particular, ourmodels capture the right-skewed non-concave asymmetry in the fundamental diagram of thedensity-flow relationship, which is well-known in realistic traffic measurements [23]. Com-parison of the numerical results of the two look-ahead rules shows that in long-range interac-tions limit with large look-ahead parameter L , the two rules produce different coarse-grainedmacroscopic averaged fluxes. But for small L , both rules produce similar results.Physically we do not expect that human drivers would (or even could) have a perceptionof traffic up front for many cars. Therefore, the look-ahead horizon is typically in the rangeof 50 to 150m, which corresponds to the small-to-intermediate values of L = 8 to 24 and bothrules exhibit reasonable behavior in this regime. However, with the fast development of self-driving vehicles equipped with vehicle-to-vehicle communication and a variety of techniquesto perceive their surroundings, such as radar, Lidar, sonar, odometry and GPS [49], we/carsmay “look” far ahead to reach large L . In that situation, the second look-ahead rule may bemore suitable than the first one.As one of our main goals is to compare the two look-ahead rules, we propose our CAmodels in a closed system and take the periodic boundary conditions to keep the numberof cars and the density constant in a single simulation. Therefore, we have not applied ourmodels to simulate some more complex non-stationary features, such as traffic breakdownsat bottlenecks [19]. It is possible to improve the models further in the following directions.We can include entrances and exits in the models by adding dynamical mechanisms such asadsorption/desorption. In reality, there are multi-lanes on highways and fast vehicles maychange lanes to bypass slow ones. We also need to consider different types of vehicles, suchas cars and trucks with unequal sizes and speeds. More complicated models addressing theseaspects will be explored in the future. Acknowledgment
YS is partially supported by the NSF Grants DMS-1620212, DMS-1913146 and a SCEPSCoR GEAR Award. CT is partially supported by the NSF Grant DMS-1853001.
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Department of Mathematics, University of South Carolina, 1523 Greene St., Columbia, SC29208, USA
E-mail address : [email protected] (Changhui Tan) Department of Mathematics, University of South Carolina, 1523 Greene St., Columbia, SC29208, USA
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