Traffic flow cellular automaton model with bi-directional information in an open boundary condition

With Connected Vehicle Technologies being popular, drivers not only perceive downstream traffic information but also get upstream information by routinely checking backward traffic conditions, and the backward-looking frequency or probability is usually affected by prevailing traffic conditions. Meanwhile, the bi-directional perception range of drivers is expected to significantly increase, which results in more informed and coordinated driving behaviours. So, we propose a traffic flow bi-directional CA model with two perception ranges, and perform the numerical simulations with the field data collected from a one-lane highway in Richmond, California, USA as the benchmark data. Numerical results show that the CA model can effectively reproduce the oscillation of relatively congested traffic and the traffic hysteresis phenomenon. When adjusting the backward-looking probability and the perception range, the CA model can well simulate the travel times of all vehicles, and the generation and dissolution of traffic jams under various scenarios.

Different from car-following models, CA models are conceptually simpler, and can be easily implemented on computers for numerical simulations. Since the first traffic flow CA model was proposed by Nagel and Schreckenberg (NS model) [18] in 1992, many researchers have contributed to the improvement of the NS model by incorporating several important characteristics in reality. For example, Benjamin et al. [19] looked at braking behaviours. Li et al. [32] modelled the virtual velocity of the preceding vehicle. Larraga et al. [33] described the safe distance. Chen et al. [34] incorporated the expected moving distance of the preceding vehicle. Ge et al. [35] studied the cooperative behaviour of the three nearest preceding vehicles. Knospe et al. [36] considered the driving smoothness and comfort, i.e. comfortable driving (CD). Jiang et al. [37] captured driving smoothness and comfort by considering synchronized flow, i.e. modified CD (MCD). Obviously, existing modifications to the NS model do not consider the influence of the following vehicles.
However, the extended OV models [28][29][30][31] with the bi-directional traffic information perform better than those with only forward-looking or backward-looking traffic information. Moreover, the following vehicle information helps to improve the traffic flow stability and then to provide a safer and more comfortable driving environment. Inspired by these extended OV models, Zheng et al. [38] proposed a modified MCD (MMCD) model with a honk stimulation term to represent the impact of the following vehicles, and numerical results showed the better performance of the MMCD model than other CA models without the following vehicle information. The limit of the MMCD model is that honking usually occurs in extreme conditions, e.g. spatial headway within safe distance. In reality, drivers can still get the following traffic conditions via rear mirrors, or via Connected Vehicle Technologies [39,40] which are being incorporated into future vehicles under non-hazardous conditions. Hence, previous CA models can be further improved by handling the back-looking phenomenon more realistically for normal traffic conditions. The remainder of this study is organized as follows. First, Section 2 proposes the CA model with the bi-directional traffic information. Section 3 carries out the numerical experiments with the field headway data as the benchmark data. Finally, some important conclusions are drawn.

The proposed CA model
Empirically, drivers not only look ahead continuously to avoid collisions with front vehicles, but also look backwards with a certain frequency so as to prevent rear-end accidents. Here, the backwardlooking behaviours are realized by Connected Vehicle Technologies, rearview mirror, or honking communication. Moreover, the backward-looking frequency or probability is positively correlated to traffic density. The proposed CA model starts by formulating the impact of the following vehicles. The gap and relative velocity between two successive vehicles are taken into account to determine whether following vehicles pose a potential threat to their predecessors. If the gap is smaller than the security gap (i.e. ga p saf e ), and the velocity of vehicle n exceeds that of its front vehicle n−1, then vehicle n is a potential threat to vehicle n−1, denoted as rs n (t) = 1. Otherwise, it does not, that is, rs n (t) = 0. So, this rule is expressed as follows.  where d n (t) = x n−1 (t) − x n (t) − 1 is the gap between two successive vehicles n and n−1.
Meanwhile, only if the nearest following vehicle is in the back perception range of the driver (i.e. L b ) is it meaningful for the driver to judge the gap and relative velocity of the follower. So, if there is a vehicle that is a potential threat to the driver in his/her backward perception range, the driver would receive the threat stimulation from the follower. Otherwise, there is no threat stimulation. This rule is expressed as follows (see Fig. 1).
where s n (t) = 1 indicates that driver n receives the potential threat from the follower n+1 in his/her  Similarly, each driver can only obtain the gap with and the braking light of the nearest front vehicle in the front perception range L f (see Fig. 1), and then adjust his/her driving behaviour. If a cellular between x n (t) and x n (t) + L f is occupied by the vehicle n−1, whose braking light is on (i.e. b n−1 (t) = 1), vehicle n may brake accordingly. In this condition, vehicle n receives the stimulation of the brake light within the front perception range. Otherwise, there is no braking stimulation to vehicle n from downstream. The rule is given as follows.
where b n−1 (t) is the brake light state of vehicle n−1 at time step t (i.e. b n−1 (t) = 0, when on; b n−1 (t) = 1, otherwise), and bs n (t) denotes whether vehicle n receives the brake light effect in response to the nearest vehicle within the front perception range at time step t (bs n (t) = 1 if applied, and bs n (t) = 0, otherwise). Then, a backward-looking probability p lb is introduced to describe the possible backwardlooking frequency in response to traffic density. Fig. 2 describes the relationship between the backward-looking probability versus the traffic density. In simple terms, the probability is relatively large when in congested traffic, but relatively small when in free flow. It is formulated as p lb = min(w · ρ, 1), where w = 1/ρ c is a critical spacing, i.e. the inverse of a traffic density threshold ρ c . When 0 < ρ ≤ ρ c , the probability increases linearly as the traffic density increases. Once the traffic density reaches ρ c , p lb is always 1, which implies that drivers are very careful and look backwards at each time step. The selection of the piecewise linear curve is to reflect the general trend regarding the relationship between backward-looking probability and traffic density.
Finally, the proposed CA model with the bidirectional traffic information is summarized as follows.
Rule 1: Judge the potential threat between two successive vehicles.
Rule 2: Determine the potential influence within the back perception range.
Rule 3: The brake light effect from the front perception range.
Rule 4: Determine the backward-looking probability. where lb n (t) denotes whether driver n looks backwards at time step t (lb n (t) = 1 if dthey o; otherwise, lb n (t) = 0). Rule 6: Acceleration.
where d ef f is the expected velocity of the preceding vehicle in the next time step. When a driver observes the safety threats from the following vehicle in the back perception range rather than the preceding vehicle in the front perception range, they take an effective acceleration value of d ef f n (t). The driver will choose the acceleration value of d n (t) that is not larger than then v n (t + 1) = min(d ef f n (t), v n (t + 1)); else v n (t + 1) = min(d n (t), v n (t + 1)).
If driver n perceives the safety threats through backward looking, he/she should have a small deceleration to avoid influencing the acceleration or the current speed of the following vehicle n + 1. Thus, the velocity of vehicle n becomes min(d ef f n (t), v n (t + 1)) after braking. Otherwise, the updated velocity of vehicle n is min(d n (t), v n (t + 1)).
where rand( ) returns a random number between 0 and 1, and p is the randomization probability.
Rule 9: Determination of brake light state.
That is, vehicle n switches on the brake light if its speed decreases, and switches it off if it increases. Otherwise, the braking light state is unchanged.
Obviously, rules 1, 2, 4, 5 and 7 describe the backward-looking behaviours, i.e. the responses of a driver to the potential threat from upstream. Rule 3 is based on the brake light effect from downstream. Rule 6 involves the bi-directional traffic information. Thus, the proposed CA model takes into account the bi-directional information that can be observed in real traffic, and may lead to a better description of traffic flow dynamics.

Numerical experiments
To validate how well the proposed CA model can reproduce the driving processes, we select the San Pablo Dam Road Data recorded on a single-lane road in Richmond, California (http://www.ce.ber keley.edu/∼daganzo/spdr.html) as the benchmark data. The data collection process is described as follows. From about 6:45 AM to 9:00 AM on Tuesday, 18 November 1997, eight human observers were positioned on one side of the road at equal distances from each other with a traffic light at the end of the road (see Fig. 3(a)). Each observer clicked a key on a laptop each time a vehicle passed them. The data sets therefore consisted of the arrival times of all vehicles passing the observers. In this way, we could capture the travel times between the observers for each vehicle. Usually, the traffic congestion was caused by the traffic light, and propagated backwards.
Then, the experimental site is modelled by cellular automaton with the open boundary condition (see Fig. 3(b). The entire single-lane road is divided into 815 cellulars (i.e. D = 815) of length 7.5 m, which can either be empty or occupied by just one vehicle at each time step. Meanwhile, the time is scattered into time steps of one second. Based on the arrival times for all vehicles at observer '1' and at the traffic light at the end of the

Analysis of flux-density diagrams
During the numerical simulations, we record the traffic flux and density at each time step and then depict the flux-density diagrams in Fig. 4(a, c, e, g). It is obviously known that vehicles are distributed sparsely in the road and that the traffic state is free flow at the beginning (i.e. at about 6:45 AM). As more and more vehicles enter, the traffic state has some random oscillation and the traffic flux fluctuates around a certain value. In particular, when the vehicle density is in the range (0.3, 0.4), the traffic flux value is scattered randomly. That is also to say, the flux and density do not have a one-to-one relationship. Through auto-covariance [41] of the traffic flow time series (see Fig. 4(b, d, f, h)), it is easily seen that the traffic flow time series has no long-range correlation, which validates the random volatility of the traffic flow when the vehicle density is relative large.
Moreover, Fig. 4(a, c, e, g) show that the fluxdensity diagram has two branches due to the change in direction of vehicle density in the open boundary condition. As the vehicles are fed into the road by the actual time headways, the vehicle density gradually grows, and results in traffic flux increase and traffic flux fluctuation in a certain region. This can be depicted by the upper branch of the flux-density diagram. Later, the vehicle density is relatively large and begins to decrease, that is, the traffic flow evolution in the next stage starts from the traffic jam state. In this condition, the flux-density diagram goes along the lower branch rather than returning to the upper branch. In conclusion, the proposed CA model in the open boundary condition is able to reproduce the hysteresis phenomenon.

Fitting analysis of travel time
This section aims to study how well the proposed CA model can simulate the vehicle movements in the single-lane road given the actual input data, and why there is different fitting accuracy about travel times with the different perception ranges and looking backward probability set in the CA model. Here, the arrival times at observer '1' and observer '8' for all vehicles are used to calculate the actual travel times (ATTs) for all vehicles that pass the entire road. The entry and departure times for each vehicle during simulation are utilized to compute the simulated travel times. The numerical results in Fig. 5 show that the actual and simulated travel times for all vehicles are close to each other. That is, the proposed CA model with bi-directional traffic information is able to reproduce the vehicle movements at a high level of accuracy, although the perception range and looking backward probability are set as different values.
Moreover, Fig. 6 illustrates the average relative errors (AREs) for travel time (see equation (1)  Theoretically, when L increases, the driver is able to notice the braking behaviours of more distant vehicles, which would cause a larger probability of braking. Meanwhile, when a driver looks backwards at some time steps, the security gap ga p saf e is a critical factor in the proper choice of driving operations. When L is larger than ga p saf e + 1, and the nearest follower is not in the range [0, ga p saf e + 1] but in the perception range, there is no safety threat from the follower according to rules 1 and 2. Thus, as L increases, the brake light effect from downstream is strengthened and causes more braking behaviours, but the potential threat from upstream is only reinforced to some extent and results in greater acceleration and less deceleration. The two contradicting impacts lead to the more stable and homogeneous traffic flow.
However, the numerical results in Fig. 6 show that ARE drops a little for w = 1 but grows for w = 5, as L increases from 1 to 5. That is, the numerical results do not agree well with the theoretical analyses. The reason can be analysed as follows. The data set collected in 1997 by Daganzo does not involve Connected Vehicle Technologies. In that period, drivers could not obtain and deal with bi-directional information in the long perception range, and could only perform driving operations with back and front vehicle information in the short range. In other words, the proposed CA model with a larger L does not accord with the reality in that period. However, with the development of Connected Vehicle Technologies, drivers can easily receive bi-directional information in the long perception range. From this perspective, the proposed CA model would give some enlightenment for the development of Connected Vehicle Technologies so as to reach a safe and comfortable traffic state.

Efficiency of congestion dissolution
According to the outflow condition, vehicles near the end of the road would depart only if the traffic light turns green. This would result in periodical traffic jams at the end of the road, which meanwhile propagate backwards at a certain speed. The spatial-temporal velocity diagrams in Figs. 7(a-c) illustrate the small time headway between two successive vehicles in the downstream section during the morning peak period (about 7:10 AM to 7:18 AM). This in fact results from the large inflow rate and the relatively stable outflow rate determined by the signal cycle. Once the inflow rate exceeds the outflow rate, the traffic jams at the traffic light would propagate backwards and the propagating speed depends on the difference between inflow and outflow rates. So, Figs. 7(a-c) show the formulation and propagation of traffic jams.
On the other hand, Figs. 7(d-f) illustrate that the time headway between two successive vehicles becomes relatively large at the end of morning peak period (about 8:41 AM to 8:49 AM). This results from the inflow rate dropping to be smaller than the unchanged outflow rate. In this condition, traffic jams dissolve gradually and the direction of the shock wave becomes forwards. Moreover, when L is fixed at 1, the dissolution efficiency of traffic jams becomes higher as the backwardlooking probability (represented by w) increases (see Figs. 7(d, e)). The reason for this can be analysed as follows. Usually, drivers look backwards to keep a safe and comfortable driving state, and the larger backward-looking probability benefits the traffic congestion dissolution. However, when w is fixed at 5, such dissolution efficiency does not continue to improve and even performs worse, although L grows from 1 to 5. The reason for this can be seen in the theoretical analyses in Section 3.2.

Conclusions
This study proposes a CA model with two important factors, that is, the bi-directional perception range and the backward-looking probability of drivers. Based on the data set recorded by Daganzo's research team, the simulation experiments are performed by the proposed CA model under the open boundary condition. Then, the numerical results are used to analyse the characteristics of the flux-density diagrams and the impact of the above two important factors on the fitting accuracy of travel time and the dissolution efficiency of traffic jams. Finally, we draw some important findings as follows. 1) The proposed CA model under the open boundary condition is able to reproduce the scattered points in the fluxdensity plane and the hysteresis phenomenon.
2) The backward-looking probability has a positive effect in the fitting of actual travel times, while the bi-directional perception range does not.
3) The proposed CA model is able to reproduce the generation of traffic jams and the backwardpropagating shock wave at the beginning of the morning peak hours, and also the dissolution of traffic jams and the forward-propagating shock wave once the inflow rate falls below the outflow rate. 4) The backward-looking probability produces a positive effect in the dissolution of traffic congestion, but the bi-directional perception range does not.