Study on Origins of Reverse Lane Usage using Nagel-Schreckenberg Model

On highways with two lanes, cars are requested to run on the slow lane and are allowed to run on the fast lane for overtaking. However, on real two-lane highways, we daily observe a phenomenon called"reverse lane usage", in which the flow on the fast lane exceeds that on the slow lane. The origins of the phenomenon have not been discussed clearly. In this paper, we study a two-lane extension of the Nagel-Schreckenberg model. We employ two types of lane-changing rules: a German and a Japanese type. The German type suppresses overtaking slow cars on the fast lane through the slow lane. The Japanese type, on the contrary, allows overtaking through both lanes. If two lanes allow the same maximum speed, the suppression of overtaking through the slow lane is the key for the reverse lane usage. If the maximum speed on the fast lane is faster than that on the slow lane, the reverse lane usage is observed in the Japanese type. The effects of stochasticity in lane changing, and mixing trucks and passenger cars are also discussed.


Introduction
Congestion in vehicle traffic is one of familiar phenomena observed in highways and city streets. Congestion phenomena are also observed in flows of various granular materials such as pedestrians and logistics, and have been attracting research interests. Since 1990's traffic flow have been studied in a physics point of view through theoretical and simulation methods. [1][2][3][4][5][6][7][8] In particular, experimental studies have been conducted recently and phase transition from free flow to congestion has been confirmed. [9][10][11][12][13] Highways usually have multiple lanes, where some interesting phenomena different from those on single-lane roads are observed. Figure 1 shows the ratio of the flow on the slow lane to the total flow observed on a Japanese two-lane highway. As the total flow increases, the  14) Two lane models have been also used for studying effects of a single slow car 19) and blockades. 20) A few studies have focused on reverse lane usage however. The phenomenon has been reproduced using an extended NS model, where the lane-changing rule contains suppression of overtaking through the slow lane. 14) The phenomenon has also been reproduced using the   Let us consider the n-th car. The position, the speed and the distance to the preceding car are denoted by x n , v n , and ∆x n = x n+1 − x n , respectively. The speed is updated by the following three steps: Acceleration: Accelerate by 1 in the range up to the maximum speed v max , Deceleration: Decelerate so as not to collide with the preceding car, v n = min (v n , ∆x n − 1) .
Randomization: Decelerate by 1 with probability p, After the speeds of all cars are updated, cars run with the new value of speeds. Thus the positions are updated virtually in parallel.

Two-lane NS model
In this section, we will construct a two-lane   there is another preceding car running with speed v np , whom the headway distance to is ∆x np .
There is another car running ∆x nb behind on the adjacent lane.  Table I). It consists of that the preceding car on the same (slow) lane runs slower than the car focused on, and that the preceding car on the adjacent (fast) lane runs faster than the 5/15 preceding car on the same lane,  Table I) . It is that the speed of the preceding car on the adjacent lane is faster than that of the car focused on, We remark that d = 16 corresponds to about 100m if v max ≃ 120km/h.
Next we construct the safety condition (S) for lane changing. This condition is common to the two types. For avoiding collision, the following condition must be satisfied, where v * max denotes the maximum speed of the car running behind on the adjacent lane seen from the focused car. In all sections except Sect. 6,  We investigate the relation between the Japanese type of the lane-changing rule and the reverse lane usage in this section. In the Japanese type, as mentioned previously (Table I) 7): The safety condition consists of only the common part represented by Eq. (6).
The maximum speed on the fast lane v f max varies as a parameter. Figure 3 Even if the slow lane is clear, a car cannot move to the slow lane if the preceding car is slower than itself. The car must give room for the preceding car to return to the slow lane. As in the Japanese type, the safety condition consists of only the common part represented by Eq. (6).

9/15
For observing the effects of the suppression of overtaking through the slow lane, we perform two types of simulations: with and without the suppression. In the system without the suppression, the condition of the demand for lane changing consists of only the common parts. For this purpose, we set the same maximum speed for cars on both lanes. For the effects of the distance of vision d, we find, in Fig. 4(b), the same effects as in the Japanese type. If the distance of vision is short, d = 5, the reverse lane usage does not occur. The longer the distance of vision is, the greater the chances to change lanes are.

Mixed traffic of trucks and passenger cars
In this section, we investigate the reverse lane usage in mixed traffic of trucks (slow cars) and passenger cars (fast cars), where the maximum speed of each car depends only on whether the car is a truck or a passenger car. In this section, v * max in the safety condition (Eq. (6)) denotes the maximum speed of the car running behind on the adjacent lane seen from the focused car.
For both the Japanese and the German type, we set the maximum speeds of passenger cars to v car max = 6 and those of trucks to v truck max = 5. The ratio of passenger cars to all cars is set to 90%, 50% and 10%.    We have introduced the distance of vision d within which a car observes the speeds of the preceding cars on both lanes. We have observed that, in both the Japanese and the German type, if d is short (resp. long), the reverse lane usage does not (resp. does) occur. This is because the longer the distance of vision is, the greater the chances to change lanes are.
The effects of inhomogeneity of speed have been studied. We have prepared mixed traffic of trucks (slow cars) and passenger cars (fast cars). By naive expectation, it is guessed that