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A new stochastic cellular automata model for traffic flow simulation with drivers’ behavior prediction
Accepted Manuscript Title: A New Stochastic Cellular Automata Model for Traffic FlowSimulation with Drivers’ Behavior Prediction Author: Marcelo Zamith Regina C´elia P. Leal-Toledo Esteban Clua Elson M. Toledo Guilherme V.P. Magalhandnbsp;aes PII: DOI: Reference:
S1877-7503(15)00043-5 http://dx.doi.org/doi:10.1016/j.jocs.2015.04.005 JOCS 348
To appear in:
Please cite this article as: Marcelo Zamith, Regina C´elia P. Leal-Toledo, Esteban Clua, Elson M. Toledo, Guilherme V.P. Magalhandnbsp;aes, A New Stochastic Cellular Automata Model for Traffic FlowSimulation with Drivers’ Behavior Prediction, Journal of Computational Science (2015), http://dx.doi.org/10.1016/j.jocs.2015.04.005 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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A New Stochastic Cellular Automata Model for Traffic Flow Simulation with Drivers’ Behavior Prediction Marcelo Zamith1 , Regina C´elia P. Leal-Toledo2 , Esteban Clua2 , Elson M. Toledo3 , and Guilherme V.P. Magalh˜aes2 1
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UFRRJ. [email protected] 2 Federal Fluminense University. leal,[email protected],[email protected] 3 LNCC and Federal University of Juiz de Fora [email protected]
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Abstract In this work we introduce a novel, flexible and robust traffic flow cellular automata model. Our proposal includes two important stages that make possible the consideration of different profiles of drivers’ behavior in a simple way. We first consider the motion expectation of vehicles that are in front of each driver. Secondly, we define how a specific vehicle decides to get around, considering the foreground traffic configuration. Our model uses stochastic rules for both situations, using the Probability Density Function of the Beta Distribution to model three drivers’ behavior, adjusting different parameters of the Beta distribution for each one. Keywords: Cellular automata, Traffic flow modeling, Drivers’ behaviors, Stochastic model, Numerical simulation
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Highlights
A novel, flexible and robust traffic flow cellular automata model is presented. Drivers’ behavior are easily treated by one specific Probability Density Function. The model reproduces results compatible with theoretical and measured data.
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Introduction
Vehicles traffic is becoming one of the largest problems found in big cities and highways. While a large amount of resources has being spent and invested in this field, the number of vehicles is still expanding and traffic problems remain. In this sense, traffic flow models have been of 1
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fundamental importance in order to understand the mechanisms of traffic jams formation, helping to improve the design of traffic networks and the definition of more efficient transportation systems. A great number of theoretical and computational models have been proposed to study traffic flow and microscopic models, as those based on Cellular Automata (CA), have been widely used existing today an extensive amount of research addressing various numerical techniques for models of this type. Microscopic models typically focus their attention on the behavior of individual vehicles, the road topology and on the influence from neighborhood vehicles. Despite its simplicity, CA is able to represent the different stages of traffic flow, including the metastability region beeing able to capture some essential features observed in realistic traffic, such as density waves and the influence of drivers’ behavior in the flux. The first use of stochastic cellular automata for traffic flow was introduced by Nagel and Schreckenberg [1], referred as the NaSch’s model. Although this method cannot represent the synchronized flow region, it has been widely used and it became the basis for several developments and improvements, and many others CA models were proposed trying more realistic traffic representation. Among these we find the so called “slow to start” or “stop and go” models. These models try to emulate the drivers’ inertia to represent the synchronized flow. They use a combination of more than one random variable and normally need two distinct initial conditions: one for free flow, where the vehicles must be distributed uniformly by the CA lattice, and another for jammed flow, represented by vehicles in an initial jammed state [2, 3, 4]. Some others models embody anticipation rules in order to take into account the drivers’ movement in the next instant of time. Several anticipation models have been proposed and the concept adopted in those methods is that every driver considers that its leading vehicle, the one immediately ahead of it, will move with the same velocity as in the previous time frame [5, 6, 7] while others include two set of rules in order to better represent the complexity of traffic dynamics, as seen in brake-light models [8, 9, 10]. More recently several CA models, using the above described principles were proposed for modeling specific situations such as traffic flow behavior in a roundabout [11], on intersections, urban or not [12, 13, 14], while others try to model the influence of driver’s behavior on traffic flow. Over the past decades there has been a considerable development in modeling human behavior in traffic which have been adopted in some CA models. Usually these models define a set of rules to capture drivers’ reactions in the traffic in a better way [6, 15, 16, 12] or are used with other techniques as car-following, agents, among others [17, 18]. Recently Saifuzzaman and Zheng [19] presented a literature review with specific focus on the latest advances in carfollowing models to include human behavior. In this work we present a novel model that makes possible the consideration of different drivers’ behaviors profiles more efficiently, enhancing the results of our previous model that needed an iterative process to implement the anticipation rules and used for its random variable a normal probability density function. The new technique here presented is composed by two stages: first we infer the car’s motion expectation of the vehicle ahead; secondly, we define how this car decides to get around, considering the traffic configuration ahead of it. Uncoupling these two steps allows us to consider the influence of different drivers’ behaviors in traffic, as defined in Fancher et al.[20]. The model implements stochastic rules for defining the movement expectation of cars that are ahead, considering also the adjustment of their velocities. To do this, it uses a Probability Density Function (PDF) of the Beta Distribution configured in such way that three different drivers’ behavior are obtained only by changing parameters of the Beta distribution. Our proposal differs from others due to its capacity to model behaviors of individual drivers using only the PDF Beta. 2
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This work is organized as follows: Section 2 presents our proposed model. In Section 3 we show how different drivers’ profiles and behaviors are modeled and in Section 4 we present our conclusions and some future works are proposed.
The Cellular Automata Model
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In this section we present our robust and stable cellular automata model for traffic simulation [21]. We first divide the road into discrete cells and define the movement of each vehicle (Vi ) by a set of simple update rules. To represent multiple vehicle types, a multicell representation was adopted allowing different sizes of vehicles, being this metric defined by the number of occupied cells (li ). In our metric, a standard vehicle occupies 5 cells of a regular discretized road where each individual cell size corresponds to 1.5 meters. This discretization gives the model a comfortable steering, allowing the driver to assume different acceleration policies. Each cell can only be occupied by one vehicle. The highway is considered to have periodic boundary condition, i.e., the position X is the same position as X + L where L is the total length of the circuit. The variable (vit ) denotes the velocity of the ith vehicle at time t, given in cells per unit time and xti denotes the spatial position. The distance between two vehicles at time t is represented by dti , and the maximum speed allowed is given by vmax . Our proposal for the driving strategy is defined by two stages: one is related to the definition of the effective distance that each vehicle can move and another is related to the definition of its effective velocity. In the first stage we consider that at small distances drivers adjust their velocity in such a way that safe driving is always possible, defining a safety distance (ds ). We also consider possible that the driver can infer the distance to the upfront vehicle i and can move considering a movement tendency for the vehicle Vi+1 . In the second stage we define the velocity and the effective movement of each vehicle having in mind those distances. For the definition of the safety distance (ds ), which depends on the distance between Vi and Vi+1 , we consider that vehicle i + 1 will keep the same velocity from previous time instant only in cases where this is possible, ie, if there is not any obstacle limiting its movement. To do so, we calculate the relative velocity (∆v) between Vi and Vi+1 , considering that vehicle i, Vi can have the maximum allowable acceleration rate (δv). With this considerations for obtaining ds it is also possible to guarantee that only one car will occupy a specified cell. We also consider the proposition presented by Knospe et al. [9] that evaluates, in time unities, the proximity between two successive vehicles. When they are considered close to each other, an additional safety distance is included. This process uses an empirical parameter (h) for determining the time that separates two vehicles. For both situations a random variable α is used in the definition of the safety distance. For the definition of the effective distance (dis ) between the vehicles Vi and Vi+1 , it is considered that vi+1 keeps its motion, and that it may accelerate if possible. The definition of this acceleration depends on the random variable α, as stated in Eq. 1: { t [ t ] } t dt+1 is = max di + min vi+1 + int (δv × (1 − α)) , di+1 − ds , 0
(1)
where int(x) defines the integer value closer to x. With this definition, the vehicle i + 1, Vi+1 can maintain its velocity, if α = 1 and can accelerate if 0 ≤ α < 1. It can also decrease its velocity in case there is some vehicle, immediately in front of it, restricting its movement (dti+1 ). Besides that, the value of ds is also considered for obtaining the effective distance, which will be reduced when necessary. 3
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At the stage of the velocity adjustment, the model allows the usage of several acceleration policies depending on the random variable α and on the defined maximum acceleration (δv). The desired velocity is expressed by: (2)
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[ ] vit+1 = min vit + int (δv × (1 − α)) , vmax .
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Therefore, if α = 1, the vehicle will keep its velocity and if 0 ≤ α < 1 the vehicle will accelerate up to its maximum allowed velocity. If there is enough space, the vehicle will move following the desired velocity. Otherwise it will reset its velocity to adapt it to the available space for its movement. The update rules consist of the following two stages algorithm: Definition of the safety distance
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dti = (xti+1 − li+1 ) − xti dti+1 = (xti+2 − li+2 ) − xti+1 α = random variable t ∆v = (vit + δv) − vi+1 ds = 0 if ∆v > 0 1 th = dti ; ds min = th ∆v δd = |dt − dt | i i+1 if δd ≤ int(δv × (1 − α)) d = int(δv × (1 − α)) s if t ≤ int(h × α) h d = d + int(d s s smin × α)
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Stage 1:
Definition of effective distance
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t+1 ] ] ] [[ t [ t t d is = max di + min vi+1 + int (δv × (1 − α)) , di+1 − ds , 0
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Definition of the effective velocity and movement
vit+1 = min [vit + int (δv × (1 − α)) , vmax ] t+1 > dt+1 i is if vt+1 t+1 v is = dis t+1 xt+1 = xti + vis i
(5)
In this algorithm the random variable α is modeled by a continuous Probability Density Function (PDF) and its choices are implemented based on a Monte Carlo rejection process, as described in Zamith et al. [22]. This model is explicit in time and there is no dependency among vehicles’ movement. Due to this, its implementation can perform the velocity definitions and cars’ positioning updates in parallel, taking advantage of multi-core or multi-thread hardware architectures. This parallel version may be implemented through sets of vehicles that are independently distributed using separated threads. 4
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Simulation Results of the Cellular Automata Model
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Preliminarily, we test the proposed model with a uniform PDF, where the random variable α varies uniformly between 0 and 1. We adopt a circular highway, under periodic boundary condition composed by 10, 000 cells with 1.5 m each. The simulations were executed using highway densities ranging from 0.01 up to 0.99, and evolving for 10,000 time units. The results for the first 1,000 units were discarded, since we were not concerned with transient effects in this simulation. For these tests we used vmax = 25 cells/s and δv = 5 cells/s, which is equivalent to consider vmax = 5 cells/s and δv = 1 cells/s for typical discretization used in CA models for traffic, with cells equivalent to 7.5 m.
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Figure 1: Flow-Density diagrams - Proposed model with Uniform PDF
Figure 2: Flow-Density diagrams - Real data [23] 5
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Definition of Drivers’ Behavior
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Figure 1 shows results of the fundamental diagram for the proposed model while Fig. 2 shows real data for flow-density diagram for Northbound lanes on Interstate 80 in Hayward, California [23]. For comparisons purposes in Figure 1 flow is measured in cells by seconds and density as the percentage of occupied cells in our lattice, which means that a flow of 0.4 corresponds to 1500 vehicles per hour in Fig. 2, 0.77 corresponds to 2700 vehicles per hour and also means that a density of 0.84 corresponds to an occupation of 180 vehicles in each mile with cells with 7.5 m of length. Comparing these results it is possible to observe that the proposed model fits well with real data, reproducing the three states of typical traffic flow phenomena (free flow, wide moving jams and synchronized traffic) and, even when using uniform PDF, a region of metastability appears, without any specific treatment for the representation of that region.
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Traditional cellular automata models for traffic simulation usually do not differentiate drivers’ behavior, assuming that all agents have the same characteristics. However, in real situations, drivers may present different styles and react according with his/her driving personality. In this sense, including different profiles in the simulation models may create more realistic scenarios and results, giving better understanding and traffic prediction. Fancher et al. [20] gave relevant contribution in the field of automatic navigation systems of vehicles, identifying different drivers’ behaviors. This proposal is a result of a technical cooperation between the National Highway Traffic Safety Administration (NHTSA) and the Transportation Research Institute (UMTRI) of the University of Michigan, trying to give subsidies for an operational test for intelligent Cruise Control systems (ICC). According to the authors, ICC systems require different drivers’ profiles in order to better adapt the automatic system for the vehicle owner’s profile. They identified 5 classes of behaviors, taking into consideration the tendency of a person to drive faster or slower and to stay closer or farther from the upfront vehicle. The classes are Hunter/Tailgater, Planner, Flow Conformist, Extremist and Ultraconservative. Based on this, our approach proposes to model different profiles of drivers’ behaviors using only an adequate PDF.
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Inclusion of Different Drivers’ Behavior to our Proposed Model
As it is possible to observe in Stage 2 of our proposed model shown in Section 3, the desired velocity is independent of the effective distance, although the real movement of the vehicle be limited by this distance. In this way, Stage 1 is responsible for defining the perception of the distance, considering the movement of the upfront vehicle and the proximity between them (ds ) and Stage 2 defines separately the desired velocity of the considered vehicle and its real motion. Values of α close to 1 define more conservative behaviors and values close to zero identify more aggressive drivers. Considering this, we propose to model different parameters using a suitable non-uniform PDF in order to define a random value. Here we used the Beta PDF, which depending on the adopted parameters may have most of its values close to zero or to one. The Beta PDF is defined as follows: f (x) =
Γ(a + b) a−1 x (1 − x)b−1 Γ(a)Γ(b)
(6)
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where 0 ≤ x ≤ 1 and Γ(n + 1) function depicts a factorial of a number when n is a positive integer (Γ(n + 1) = n!). Furthermore, the a and b parameters are integer values greater than zero and they also define the curve shape. Doing so, we choose at the Stage 1 of the algorithm on Section 3, parameters for the Beta PDF that define the perception of the front car distance. In Stage 2 the parameters of the Beta function must be used to define the intent velocity. Three behaviors were modeled in our contribution:
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• Hunter and Tailgater: describes the behavior of a driver that intends to stay close to the upfront vehicle and wants to go fast. In this work we call this driver daring.
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• Ultraconservative: describes the driver with the opposite profile of the daring. This driver prefer to keep a great distance to the upfront vehicle and drives slower. We also call this driver slow. • Flow Conformist: describes a driver that intends to go according with the flow and is not characterized by none of the previous profiles. It is also called a standard driver.
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In this work we use the following parameters of the Beta distribution for these profiles: Daring: for distance and velocity, a = 1 and b = 4; Slow: for distance, a = 15 and b = 1 and for velocity, a = 6 and b = 6; Standard: for distance, a = 9 and b = 5 and for velocity, a = 5 and b = 9. Figure 3 presents the curves for the Beta PDF of the different values of a and b.
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Figure 3: Beta PDF
Simulation Results with the Inclusion of Drivers’ Profile
In this section we present simulation results for the traffic flow, considering different drivers’ behavior, in order to validate our proposed method and the adopted methodology to represent them. Simulation 1 Figures 4(a) to 4(c) present the fundamental diagrams when the road is occupied each time with drivers having the profiles considered here, using the same setup described in Section 2 while Fig. 4(d) presents superposed results for these three kind of drivers. The maximum velocity for standard and daring profiles was vmax = 25 cells/s and for slow driver we adopted 7
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vmax = 22 cells/s. As it can be seen, the proposed model is able to reproduce the three states observed in real traffic flow: (i) Free-flow states, (ii) synchronized states and (iii) region with jams and scattered data, in agreement with empirical observations [23] and showing qualitative and quantitative results similar to those suggested by the literature [9, 6]. Figure 4(a) shows the flux-density graphic for a slow driver profile. All drivers in the road are conservative and trying to keep distant from the upfront vehicle, causing premature jamming. It is worth to note in the flux-density graphic presented in Fig. 4(b), for the daring profile, that the road displays more free flow than that presented in Fig. 4(a) and Fig. 4(b). This occurs because vehicles are at a higher velocity and are also considering daring drivers at vehicles ahead of them.
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Figure 4: Flow-Density diagrams (a) slow driver; (b) daring driver; (c) standard driver; (d) superposed profiles These results are in accordance with the theoretical results shown in Treiber [24] and with real results shown in Kockelman and Kara [23]. Simulation 2 We present in Fig. 6 to Fig. 8 results for different drivers’ behavior in the road. In order to create this situation, we used the same initial condition in an unidirectional road where 18 equally spaced vehicles were assumed to have the same velocity vi = 0 (Fig 5). The maximum velocity for both profiles standard and daring was vmax = 25 cells/s and for slow driver we 8
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Figure 5: Vehicle positioning - Initial condition
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adopted vmax = 22 cells/s. For this simulation we included an obstacle in the road, which forces vehicles to stop.
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Figure 6: Vehicle positioning - Slow driver at t=20; 31 and 50 s
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Figure 7: Vehicle positioning - Standard driver at t=20; 31 and 50 s
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Figure 8: Vehicle positioning - Daring driver at t=20; 31 and 50 s Figures 6 to 8 show the obtained results for the three drivers’ behavior: daring, standard and slow in different time steps of motion. We can observe that the usage of the PDF Beta made possible to see these behaviors clearly and model them in a simple way.
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Conclusions
In this work we presented an anticipation cellular automata model for traffic flow where each vehicle movement is defined by the motion expectation of the upfront vehicle as well as how it decides to move taking into account traffic configuration ahead of it. To make the model more realistic we propose to simulate the heterogeneous nature of human behavior and random interactions among drivers to evaluate their effects on traffic flow. These different drivers’ behaviors are modeled through the use of an appropriate Probability Density Function. The PDF Beta proved to be an efficient form to model, with quality, those behaviors and it is used both to infer the cars motion expectation of vehicles ahead, as well to define how each car decides to get around, considering the traffic configuration. This modeling of distinct drivers’ behavior allowed us to capture its effect on traffic dynamic in order to evaluate the most important features of traffic flow phenomena. It was shown, for instance, how the fundamental diagram depends on these behaviors profiles. As it is here proposed the model 9
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allows the treatment of different vehicles sizes and several acceleration policies making possible the representation of a safe and comfortable steering. In all carried tests the model was able to reproduce a diversity of flux phases showing results compatible with theoretical and measured data. Based on the results obtained in this paper we can state that the presented model is robust, flexible and able to simulate the most complex traffic situations. Due to independence of vehicle movements, we can perform the updates in parallel, taking advantage of multi-core or multi-thread hardware architectures. The treatment here used allow us to consider roads with multiple lanes and different drivers’ profiles in change of lanes policies and heterogeneous traffic flows which will be considered in a future work.
Acknowledgments
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The authors wish to thank to FAPERJ, CNPQ and LNCC.
References
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[15] R. Mu and T. Yamamoto. An analysis on mixed traffic flow of conventional passenger cars and microcars using a cellular automata model. Procedia-Social and Behavioral Sciences, 43:457–465, 2012. [16] X. G. Li, B. Jia, Z. Y. Gao, and R. Jiang. A realistic two-lane cellular automata traffic model considering aggressive lane-changing behavior of fast vehicle. Physica A, 367:479–486, 2006. [17] E. Rodaro and O. Yeldan. A multi-lane traffic simulation model via continuous cellular automata. CoRR, abs/1302.0488, 2013. [18] R. Wang, W. Zhang, and Q. Miao. Effects of driver behavior on traffic flow at three-lane roundabouts. International Journal of Intelligent Control and Systems, 10:123–130, 2005. [19] M. Saifuzzaman and Z. Zheng. Incorporating human-factors in car-following models: A review of recent developments and research needs. Transportation Research Part C, 48:379–403, 2014. [20] P. Fancher, R. Ervin, J. Sayer, M. Hagan, S. Bogard, Z. Bareket, M. Mefford, and J. Haugen. Intelligent cruise control field operational test. Technical report, Universitv of Michigan-Transportation Research Institute, may 1998. [21] M. Zamith. A cellular automata model applied to traffic with multiple drivers profiles. PhD thesis, Federal Fluminense University, 2013. [22] M. Zamith, R. C. P. Leal-Toledo, M. Kischinhevsky, E. Clua, D. Brand˜ ao, A. A. Montenegro, and E. B. Lima. A probabilistic cellular automata model for highway traffic simulation. Procedia CS, 1(1):337–345, 2010. [23] K. M. Kockelman. Modeling traffic’s flow-density relation: accommodation of multiple flow regimes and traveler types. Transportation, 28:363–374, 2001. [24] M. Treiber and A. Kesting. Traffic flows dynamics: Data, Models and Simulation. Springer, 2013.
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Please cite this article as: Marcelo Zamith, Regina C´elia P. Leal-Toledo, Esteban Clua, Elson M. Toledo, Guilherme V.P. Magalhandnbsp;aes, A New Stochastic Cellular Automata Model for Traffic FlowSimulation with Drivers’ Behavior Prediction, Journal of Computational Science (2015), http://dx.doi.org/10.1016/j.jocs.2015.04.005 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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A New Stochastic Cellular Automata Model for Traffic Flow Simulation with Drivers’ Behavior Prediction Marcelo Zamith1 , Regina C´elia P. Leal-Toledo2 , Esteban Clua2 , Elson M. Toledo3 , and Guilherme V.P. Magalh˜aes2 1
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UFRRJ. [email protected] 2 Federal Fluminense University. leal,[email protected],[email protected] 3 LNCC and Federal University of Juiz de Fora [email protected]
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Abstract In this work we introduce a novel, flexible and robust traffic flow cellular automata model. Our proposal includes two important stages that make possible the consideration of different profiles of drivers’ behavior in a simple way. We first consider the motion expectation of vehicles that are in front of each driver. Secondly, we define how a specific vehicle decides to get around, considering the foreground traffic configuration. Our model uses stochastic rules for both situations, using the Probability Density Function of the Beta Distribution to model three drivers’ behavior, adjusting different parameters of the Beta distribution for each one. Keywords: Cellular automata, Traffic flow modeling, Drivers’ behaviors, Stochastic model, Numerical simulation
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Highlights
A novel, flexible and robust traffic flow cellular automata model is presented. Drivers’ behavior are easily treated by one specific Probability Density Function. The model reproduces results compatible with theoretical and measured data.
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Introduction
Vehicles traffic is becoming one of the largest problems found in big cities and highways. While a large amount of resources has being spent and invested in this field, the number of vehicles is still expanding and traffic problems remain. In this sense, traffic flow models have been of 1
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Stochastic Cellular automata for traffic flow simulation
Zamith, Leal-Toledo, Clua, Toledo, Magalh˜ aes
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fundamental importance in order to understand the mechanisms of traffic jams formation, helping to improve the design of traffic networks and the definition of more efficient transportation systems. A great number of theoretical and computational models have been proposed to study traffic flow and microscopic models, as those based on Cellular Automata (CA), have been widely used existing today an extensive amount of research addressing various numerical techniques for models of this type. Microscopic models typically focus their attention on the behavior of individual vehicles, the road topology and on the influence from neighborhood vehicles. Despite its simplicity, CA is able to represent the different stages of traffic flow, including the metastability region beeing able to capture some essential features observed in realistic traffic, such as density waves and the influence of drivers’ behavior in the flux. The first use of stochastic cellular automata for traffic flow was introduced by Nagel and Schreckenberg [1], referred as the NaSch’s model. Although this method cannot represent the synchronized flow region, it has been widely used and it became the basis for several developments and improvements, and many others CA models were proposed trying more realistic traffic representation. Among these we find the so called “slow to start” or “stop and go” models. These models try to emulate the drivers’ inertia to represent the synchronized flow. They use a combination of more than one random variable and normally need two distinct initial conditions: one for free flow, where the vehicles must be distributed uniformly by the CA lattice, and another for jammed flow, represented by vehicles in an initial jammed state [2, 3, 4]. Some others models embody anticipation rules in order to take into account the drivers’ movement in the next instant of time. Several anticipation models have been proposed and the concept adopted in those methods is that every driver considers that its leading vehicle, the one immediately ahead of it, will move with the same velocity as in the previous time frame [5, 6, 7] while others include two set of rules in order to better represent the complexity of traffic dynamics, as seen in brake-light models [8, 9, 10]. More recently several CA models, using the above described principles were proposed for modeling specific situations such as traffic flow behavior in a roundabout [11], on intersections, urban or not [12, 13, 14], while others try to model the influence of driver’s behavior on traffic flow. Over the past decades there has been a considerable development in modeling human behavior in traffic which have been adopted in some CA models. Usually these models define a set of rules to capture drivers’ reactions in the traffic in a better way [6, 15, 16, 12] or are used with other techniques as car-following, agents, among others [17, 18]. Recently Saifuzzaman and Zheng [19] presented a literature review with specific focus on the latest advances in carfollowing models to include human behavior. In this work we present a novel model that makes possible the consideration of different drivers’ behaviors profiles more efficiently, enhancing the results of our previous model that needed an iterative process to implement the anticipation rules and used for its random variable a normal probability density function. The new technique here presented is composed by two stages: first we infer the car’s motion expectation of the vehicle ahead; secondly, we define how this car decides to get around, considering the traffic configuration ahead of it. Uncoupling these two steps allows us to consider the influence of different drivers’ behaviors in traffic, as defined in Fancher et al.[20]. The model implements stochastic rules for defining the movement expectation of cars that are ahead, considering also the adjustment of their velocities. To do this, it uses a Probability Density Function (PDF) of the Beta Distribution configured in such way that three different drivers’ behavior are obtained only by changing parameters of the Beta distribution. Our proposal differs from others due to its capacity to model behaviors of individual drivers using only the PDF Beta. 2
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Stochastic Cellular automata for traffic flow simulation
Zamith, Leal-Toledo, Clua, Toledo, Magalh˜ aes
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This work is organized as follows: Section 2 presents our proposed model. In Section 3 we show how different drivers’ profiles and behaviors are modeled and in Section 4 we present our conclusions and some future works are proposed.
The Cellular Automata Model
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In this section we present our robust and stable cellular automata model for traffic simulation [21]. We first divide the road into discrete cells and define the movement of each vehicle (Vi ) by a set of simple update rules. To represent multiple vehicle types, a multicell representation was adopted allowing different sizes of vehicles, being this metric defined by the number of occupied cells (li ). In our metric, a standard vehicle occupies 5 cells of a regular discretized road where each individual cell size corresponds to 1.5 meters. This discretization gives the model a comfortable steering, allowing the driver to assume different acceleration policies. Each cell can only be occupied by one vehicle. The highway is considered to have periodic boundary condition, i.e., the position X is the same position as X + L where L is the total length of the circuit. The variable (vit ) denotes the velocity of the ith vehicle at time t, given in cells per unit time and xti denotes the spatial position. The distance between two vehicles at time t is represented by dti , and the maximum speed allowed is given by vmax . Our proposal for the driving strategy is defined by two stages: one is related to the definition of the effective distance that each vehicle can move and another is related to the definition of its effective velocity. In the first stage we consider that at small distances drivers adjust their velocity in such a way that safe driving is always possible, defining a safety distance (ds ). We also consider possible that the driver can infer the distance to the upfront vehicle i and can move considering a movement tendency for the vehicle Vi+1 . In the second stage we define the velocity and the effective movement of each vehicle having in mind those distances. For the definition of the safety distance (ds ), which depends on the distance between Vi and Vi+1 , we consider that vehicle i + 1 will keep the same velocity from previous time instant only in cases where this is possible, ie, if there is not any obstacle limiting its movement. To do so, we calculate the relative velocity (∆v) between Vi and Vi+1 , considering that vehicle i, Vi can have the maximum allowable acceleration rate (δv). With this considerations for obtaining ds it is also possible to guarantee that only one car will occupy a specified cell. We also consider the proposition presented by Knospe et al. [9] that evaluates, in time unities, the proximity between two successive vehicles. When they are considered close to each other, an additional safety distance is included. This process uses an empirical parameter (h) for determining the time that separates two vehicles. For both situations a random variable α is used in the definition of the safety distance. For the definition of the effective distance (dis ) between the vehicles Vi and Vi+1 , it is considered that vi+1 keeps its motion, and that it may accelerate if possible. The definition of this acceleration depends on the random variable α, as stated in Eq. 1: { t [ t ] } t dt+1 is = max di + min vi+1 + int (δv × (1 − α)) , di+1 − ds , 0
(1)
where int(x) defines the integer value closer to x. With this definition, the vehicle i + 1, Vi+1 can maintain its velocity, if α = 1 and can accelerate if 0 ≤ α < 1. It can also decrease its velocity in case there is some vehicle, immediately in front of it, restricting its movement (dti+1 ). Besides that, the value of ds is also considered for obtaining the effective distance, which will be reduced when necessary. 3
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Stochastic Cellular automata for traffic flow simulation
Zamith, Leal-Toledo, Clua, Toledo, Magalh˜ aes
At the stage of the velocity adjustment, the model allows the usage of several acceleration policies depending on the random variable α and on the defined maximum acceleration (δv). The desired velocity is expressed by: (2)
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[ ] vit+1 = min vit + int (δv × (1 − α)) , vmax .
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Therefore, if α = 1, the vehicle will keep its velocity and if 0 ≤ α < 1 the vehicle will accelerate up to its maximum allowed velocity. If there is enough space, the vehicle will move following the desired velocity. Otherwise it will reset its velocity to adapt it to the available space for its movement. The update rules consist of the following two stages algorithm: Definition of the safety distance
(3)
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dti = (xti+1 − li+1 ) − xti dti+1 = (xti+2 − li+2 ) − xti+1 α = random variable t ∆v = (vit + δv) − vi+1 ds = 0 if ∆v > 0 1 th = dti ; ds min = th ∆v δd = |dt − dt | i i+1 if δd ≤ int(δv × (1 − α)) d = int(δv × (1 − α)) s if t ≤ int(h × α) h d = d + int(d s s smin × α)
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Definition of effective distance
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t+1 ] ] ] [[ t [ t t d is = max di + min vi+1 + int (δv × (1 − α)) , di+1 − ds , 0
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Definition of the effective velocity and movement
vit+1 = min [vit + int (δv × (1 − α)) , vmax ] t+1 > dt+1 i is if vt+1 t+1 v is = dis t+1 xt+1 = xti + vis i
(5)
In this algorithm the random variable α is modeled by a continuous Probability Density Function (PDF) and its choices are implemented based on a Monte Carlo rejection process, as described in Zamith et al. [22]. This model is explicit in time and there is no dependency among vehicles’ movement. Due to this, its implementation can perform the velocity definitions and cars’ positioning updates in parallel, taking advantage of multi-core or multi-thread hardware architectures. This parallel version may be implemented through sets of vehicles that are independently distributed using separated threads. 4
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Simulation Results of the Cellular Automata Model
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Preliminarily, we test the proposed model with a uniform PDF, where the random variable α varies uniformly between 0 and 1. We adopt a circular highway, under periodic boundary condition composed by 10, 000 cells with 1.5 m each. The simulations were executed using highway densities ranging from 0.01 up to 0.99, and evolving for 10,000 time units. The results for the first 1,000 units were discarded, since we were not concerned with transient effects in this simulation. For these tests we used vmax = 25 cells/s and δv = 5 cells/s, which is equivalent to consider vmax = 5 cells/s and δv = 1 cells/s for typical discretization used in CA models for traffic, with cells equivalent to 7.5 m.
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Figure 1: Flow-Density diagrams - Proposed model with Uniform PDF
Figure 2: Flow-Density diagrams - Real data [23] 5
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Stochastic Cellular automata for traffic flow simulation
Zamith, Leal-Toledo, Clua, Toledo, Magalh˜ aes
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Definition of Drivers’ Behavior
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Figure 1 shows results of the fundamental diagram for the proposed model while Fig. 2 shows real data for flow-density diagram for Northbound lanes on Interstate 80 in Hayward, California [23]. For comparisons purposes in Figure 1 flow is measured in cells by seconds and density as the percentage of occupied cells in our lattice, which means that a flow of 0.4 corresponds to 1500 vehicles per hour in Fig. 2, 0.77 corresponds to 2700 vehicles per hour and also means that a density of 0.84 corresponds to an occupation of 180 vehicles in each mile with cells with 7.5 m of length. Comparing these results it is possible to observe that the proposed model fits well with real data, reproducing the three states of typical traffic flow phenomena (free flow, wide moving jams and synchronized traffic) and, even when using uniform PDF, a region of metastability appears, without any specific treatment for the representation of that region.
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Traditional cellular automata models for traffic simulation usually do not differentiate drivers’ behavior, assuming that all agents have the same characteristics. However, in real situations, drivers may present different styles and react according with his/her driving personality. In this sense, including different profiles in the simulation models may create more realistic scenarios and results, giving better understanding and traffic prediction. Fancher et al. [20] gave relevant contribution in the field of automatic navigation systems of vehicles, identifying different drivers’ behaviors. This proposal is a result of a technical cooperation between the National Highway Traffic Safety Administration (NHTSA) and the Transportation Research Institute (UMTRI) of the University of Michigan, trying to give subsidies for an operational test for intelligent Cruise Control systems (ICC). According to the authors, ICC systems require different drivers’ profiles in order to better adapt the automatic system for the vehicle owner’s profile. They identified 5 classes of behaviors, taking into consideration the tendency of a person to drive faster or slower and to stay closer or farther from the upfront vehicle. The classes are Hunter/Tailgater, Planner, Flow Conformist, Extremist and Ultraconservative. Based on this, our approach proposes to model different profiles of drivers’ behaviors using only an adequate PDF.
4.1
Inclusion of Different Drivers’ Behavior to our Proposed Model
As it is possible to observe in Stage 2 of our proposed model shown in Section 3, the desired velocity is independent of the effective distance, although the real movement of the vehicle be limited by this distance. In this way, Stage 1 is responsible for defining the perception of the distance, considering the movement of the upfront vehicle and the proximity between them (ds ) and Stage 2 defines separately the desired velocity of the considered vehicle and its real motion. Values of α close to 1 define more conservative behaviors and values close to zero identify more aggressive drivers. Considering this, we propose to model different parameters using a suitable non-uniform PDF in order to define a random value. Here we used the Beta PDF, which depending on the adopted parameters may have most of its values close to zero or to one. The Beta PDF is defined as follows: f (x) =
Γ(a + b) a−1 x (1 − x)b−1 Γ(a)Γ(b)
(6)
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where 0 ≤ x ≤ 1 and Γ(n + 1) function depicts a factorial of a number when n is a positive integer (Γ(n + 1) = n!). Furthermore, the a and b parameters are integer values greater than zero and they also define the curve shape. Doing so, we choose at the Stage 1 of the algorithm on Section 3, parameters for the Beta PDF that define the perception of the front car distance. In Stage 2 the parameters of the Beta function must be used to define the intent velocity. Three behaviors were modeled in our contribution:
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• Hunter and Tailgater: describes the behavior of a driver that intends to stay close to the upfront vehicle and wants to go fast. In this work we call this driver daring.
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• Ultraconservative: describes the driver with the opposite profile of the daring. This driver prefer to keep a great distance to the upfront vehicle and drives slower. We also call this driver slow. • Flow Conformist: describes a driver that intends to go according with the flow and is not characterized by none of the previous profiles. It is also called a standard driver.
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In this work we use the following parameters of the Beta distribution for these profiles: Daring: for distance and velocity, a = 1 and b = 4; Slow: for distance, a = 15 and b = 1 and for velocity, a = 6 and b = 6; Standard: for distance, a = 9 and b = 5 and for velocity, a = 5 and b = 9. Figure 3 presents the curves for the Beta PDF of the different values of a and b.
4.2
Figure 3: Beta PDF
Simulation Results with the Inclusion of Drivers’ Profile
In this section we present simulation results for the traffic flow, considering different drivers’ behavior, in order to validate our proposed method and the adopted methodology to represent them. Simulation 1 Figures 4(a) to 4(c) present the fundamental diagrams when the road is occupied each time with drivers having the profiles considered here, using the same setup described in Section 2 while Fig. 4(d) presents superposed results for these three kind of drivers. The maximum velocity for standard and daring profiles was vmax = 25 cells/s and for slow driver we adopted 7
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vmax = 22 cells/s. As it can be seen, the proposed model is able to reproduce the three states observed in real traffic flow: (i) Free-flow states, (ii) synchronized states and (iii) region with jams and scattered data, in agreement with empirical observations [23] and showing qualitative and quantitative results similar to those suggested by the literature [9, 6]. Figure 4(a) shows the flux-density graphic for a slow driver profile. All drivers in the road are conservative and trying to keep distant from the upfront vehicle, causing premature jamming. It is worth to note in the flux-density graphic presented in Fig. 4(b), for the daring profile, that the road displays more free flow than that presented in Fig. 4(a) and Fig. 4(b). This occurs because vehicles are at a higher velocity and are also considering daring drivers at vehicles ahead of them.
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Figure 4: Flow-Density diagrams (a) slow driver; (b) daring driver; (c) standard driver; (d) superposed profiles These results are in accordance with the theoretical results shown in Treiber [24] and with real results shown in Kockelman and Kara [23]. Simulation 2 We present in Fig. 6 to Fig. 8 results for different drivers’ behavior in the road. In order to create this situation, we used the same initial condition in an unidirectional road where 18 equally spaced vehicles were assumed to have the same velocity vi = 0 (Fig 5). The maximum velocity for both profiles standard and daring was vmax = 25 cells/s and for slow driver we 8
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Figure 5: Vehicle positioning - Initial condition
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adopted vmax = 22 cells/s. For this simulation we included an obstacle in the road, which forces vehicles to stop.
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Figure 6: Vehicle positioning - Slow driver at t=20; 31 and 50 s
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Figure 7: Vehicle positioning - Standard driver at t=20; 31 and 50 s
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Figure 8: Vehicle positioning - Daring driver at t=20; 31 and 50 s Figures 6 to 8 show the obtained results for the three drivers’ behavior: daring, standard and slow in different time steps of motion. We can observe that the usage of the PDF Beta made possible to see these behaviors clearly and model them in a simple way.
5
Conclusions
In this work we presented an anticipation cellular automata model for traffic flow where each vehicle movement is defined by the motion expectation of the upfront vehicle as well as how it decides to move taking into account traffic configuration ahead of it. To make the model more realistic we propose to simulate the heterogeneous nature of human behavior and random interactions among drivers to evaluate their effects on traffic flow. These different drivers’ behaviors are modeled through the use of an appropriate Probability Density Function. The PDF Beta proved to be an efficient form to model, with quality, those behaviors and it is used both to infer the cars motion expectation of vehicles ahead, as well to define how each car decides to get around, considering the traffic configuration. This modeling of distinct drivers’ behavior allowed us to capture its effect on traffic dynamic in order to evaluate the most important features of traffic flow phenomena. It was shown, for instance, how the fundamental diagram depends on these behaviors profiles. As it is here proposed the model 9
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Stochastic Cellular automata for traffic flow simulation
Zamith, Leal-Toledo, Clua, Toledo, Magalh˜ aes
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allows the treatment of different vehicles sizes and several acceleration policies making possible the representation of a safe and comfortable steering. In all carried tests the model was able to reproduce a diversity of flux phases showing results compatible with theoretical and measured data. Based on the results obtained in this paper we can state that the presented model is robust, flexible and able to simulate the most complex traffic situations. Due to independence of vehicle movements, we can perform the updates in parallel, taking advantage of multi-core or multi-thread hardware architectures. The treatment here used allow us to consider roads with multiple lanes and different drivers’ profiles in change of lanes policies and heterogeneous traffic flows which will be considered in a future work.
Acknowledgments
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The authors wish to thank to FAPERJ, CNPQ and LNCC.
References
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[15] R. Mu and T. Yamamoto. An analysis on mixed traffic flow of conventional passenger cars and microcars using a cellular automata model. Procedia-Social and Behavioral Sciences, 43:457–465, 2012. [16] X. G. Li, B. Jia, Z. Y. Gao, and R. Jiang. A realistic two-lane cellular automata traffic model considering aggressive lane-changing behavior of fast vehicle. Physica A, 367:479–486, 2006. [17] E. Rodaro and O. Yeldan. A multi-lane traffic simulation model via continuous cellular automata. CoRR, abs/1302.0488, 2013. [18] R. Wang, W. Zhang, and Q. Miao. Effects of driver behavior on traffic flow at three-lane roundabouts. International Journal of Intelligent Control and Systems, 10:123–130, 2005. [19] M. Saifuzzaman and Z. Zheng. Incorporating human-factors in car-following models: A review of recent developments and research needs. Transportation Research Part C, 48:379–403, 2014. [20] P. Fancher, R. Ervin, J. Sayer, M. Hagan, S. Bogard, Z. Bareket, M. Mefford, and J. Haugen. Intelligent cruise control field operational test. Technical report, Universitv of Michigan-Transportation Research Institute, may 1998. [21] M. Zamith. A cellular automata model applied to traffic with multiple drivers profiles. PhD thesis, Federal Fluminense University, 2013. [22] M. Zamith, R. C. P. Leal-Toledo, M. Kischinhevsky, E. Clua, D. Brand˜ ao, A. A. Montenegro, and E. B. Lima. A probabilistic cellular automata model for highway traffic simulation. Procedia CS, 1(1):337–345, 2010. [23] K. M. Kockelman. Modeling traffic’s flow-density relation: accommodation of multiple flow regimes and traveler types. Transportation, 28:363–374, 2001. [24] M. Treiber and A. Kesting. Traffic flows dynamics: Data, Models and Simulation. Springer, 2013.
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