2. Beijing Jiaotong University, Beijing 100044, China
In real traffic, the differences of traffic rules have their own specific characteristics, which lead to individual pattern of traffic diagram. Research on traffic rules is a hotspot in the microscopic traffic flow theory field, where traffic flow is regarded as a complex multiparticle system. The microscopic description of traffic reflects the behavior of single vehicles in traffic flow. Then, the statistical properties of the traffic system may be derived by inferring the interactions between vehicles [1].
There are two main types of microscopic traffic flow models: carfollowing model and cellular automaton model. The carfollowing model was first proposed by Pipes in 1953. When the speed of the preceding car is higher than that of the following car, the latter will accelerates; conversely, the following car will decelerates [2]. Then, Chandler et al. [3], Newell [4], Bando et al. [5], Helbing et al. [6] and Jiang et al. [7] put forward their own models. However, carfollowing models could not simulate lanechanging behavior of vehicles which exists in real traffic frequently. In contrast, the cellular automaton model can avoid the aforementioned shortcomings, so it has taken great strides from the 1990s. Cremer and Ludwing were the first to apply the cellular automaton model to transportation research [8]. Then, Nagel and Schreckenberg proposed the classic NaSch model to simulate the singlelane freeway, which is the simplest model to simulate real traffic.
Based on NaSch [9] model as well as their observations of different traffic characteristics, many scholars put forward their own models, including the cruise control model in which the car traveling at maximum velocity is free from randomization [10], TT model in which the car accelerates with certain probability when the vehicle speed is 0 and there is only one empty cell in front of the car [11], the BJH model [12], VE model [13], VDR model [14] and FI model [15], amongst many others[16][22].
In the single lane model, a car merely follows its preceding vehicle, which is inconsistent with real traffic. Thus, scholars studied multilane traffic with lanechanging rules. Rickert et al., first proposed a series of lane changing rules [23] to extend the NaSch model. Then, Chowdhury et al. [24], Pedersen et al. [25], Daoudia et al. [26], Lv et al. [27], [28], and Li et al. [29], [30], established their own multilane models. Furthermore, Simon and Gutowitz [31] studied bidirectional traffic.
Though highway traffic rules differ from country to country, lanechanging rules and speedlimit rules are generally consistent. Up to now, there are few systematic comparisons of these two traffic patterns or analysis of their influence on traffic flow. In this paper, we do this work based on cellular automaton model.
Ⅱ. TRAFFIC RULES MODELINGAlong the traveling direction of the vehicle on the twolane highway, the left lane is lane1 and the other is lane2. The following information are different physical processes of the vehicle under different traffic rules. Under lane changing rule (RL rule), vehicles are not allowed to occupy lane2 for a long time. This lane is used as overtaking. Unlike RL rule, if the speedlimit rule (SC rule) is adopted, both of two lanes are carriageways. Lane1 generally is used for faster vehicles, and lane2 is reserved for slower vehicles. Vehicles traveling on the road are limited to certain speed intervals. When the vehicle speed is not in the corresponding speedlimit interval, the vehicle has to adjust its speed or move to the other lane.
In the modeling process of multilane traffic, each time step is typically divided into two parts: in the first step the vehicle on the road operates with an update regulation which is similar to the model of NaSch. The specific update procedures are: acceleration, deceleration, randomization, and location updating.
1) Acceleration:
2) Deceleration:
3) Randomization: Under a probability
4) Vehicle updating:
Within the second step, the vehicle changes lane according to the lanechanging rules, in which the conditions of this intention are consistent with [23].
1) The lane changing mechanism under RL rule: In case the vehicle
is in lane1 and meets the lane changing conditions: if
2) The lane changing mechanism under SC rule: suppose the speed
limit on lane1 is
Twolane highway traffic is simulated based on the different traffic
rules. Suppose the length of each cell is 5.5 m, the length of road
recorded as
$ \begin{align} {{q}}=k\times\bar v\end{align} $  (1) 
$ \begin{align} k=\frac{N}{L}\end{align} $  (2) 
$ \begin{align} \overline v =\frac{1}{T}\sum\limits_{{\rm{t}}=1}^T {\frac{1}{N}\sum\limits_i^N{{v_i}}}(t)\end{align} $  (3) 
$ \begin{align} {v_{sd}}=\sqrt {\frac{1}{N}\sum\limits_{i=1}^N {({v_i}}\bar v{)^2}}\end{align} $  (4) 
$ \begin{align} c=\sum\limits_{t=1}^T{{c_t}}\end{align} $  (5) 
Fig. 1 is the fundamental and speeddensity relation diagram of RL
rule and SC rule. The traffic is in free flow and the interactions
between vehicles on the road are very weak; thus, the average speed
of vehicles on the road slightly decreases when the density
increases from 0 to
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Fig. 1 The fundamental diagram: (a) and (c) are the flowdensity plane and the speeddensity plane; (b) and (d) are the flowdensity plane on first and second lane and the speeddensity plane on left and right lane (the passing rate is 0.15) 
At the same lane changing rate and density, the lane changing
times in the RL rule, due to the forced lane changing, is almost
equal and greater than what we obtain under SC operation rule
(Fig. 2 (b)). The maximum times of lane changing is at the point
of congested flow (RL rule (
In this paper, we introduce the concept of lane changing probability introduced from the three phase traffic flow theory to explain the relationship mentioned above. First, we present the notion of lane changing probability [32].
$ \begin{align} {p_p}=\frac{{{n_p}}}{{{N_p}\times{p_c}}}\end{align} $  (6) 
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Fig. 2 (a) The flowdensity plane on left and right lane; (b) the times of lane changing; (c) the speed population deviation 
where
The lane changing probability reaches the maximum
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Fig. 3 The probability of lane changing 
Certainly, the research of lane changing rule will bring about road safety issues, and as a potential security risk, frequently lane changing often results in traffic safety hazard. In traffic engineering, researchers always aim to maximize the traffic flow, and at the same time ensure the security by reducing the potential risk of lanechanging times. Therefore, compared with RL rule, SC rule performs better.
Ⅳ. CONCLUSIONIn this paper, two different traffic rules are respectively constructed to simulate the highway traffic flow. It is concluded that: the traffic flow and the average vehicle speed under the speedlimit rules and overtaking rules show few differences when they are having the same density. However, there are significant differences on different lanes under the congestion state; the lanechanging probability can preferably indicate the real conditions of the road, and the change of the lanechanging times will have some influence on the traffic flow; the speed limit rule can reduce lanechanging motivations and thus ensures the traffic safety on the highway effectively.
In further researches, firstly, the lanechanging probabilities and quantitative characteristics of mixed traffic flow are needed to be studied in detail to determine the numerical features of specific phase transition. Then, the metastable of the traffic flow under two different traffic rules needs to be further discussed. Finally, there are still many defects in this model which need to be improved, for example, the vehicle length, the number of lanes, and the rules of lanechanging, and honk effect, need to be further defined.
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