INTRODUCTION

In the last decades, traffic dynamics has attracted much attention from physicists and mathematicians. There are a lot of models for traffic flow proposed thus far, from various viewpoints, such as macroscopic and microscopic, differential equations and cellular automata, deterministic and probabilistic, etc. (Chowdhury et al., 2000). Among these traffic models, the Cellular Automata (CA) model achieves a remarkable success although it is conceptually simpler compared with other dynamical approaches. The most famous one is NaSch model (Nagel and Schreckenberg, 1992). Later, various generalizations and extensions of the NaSch model have been proposed to simulate traffic flow, such as VDR model, TT model, CD model and so on (Chowdhury et al., 2000).

Lane reduction bottleneck, is common in road system. Near the merging point, drivers show obvious behaviors such as the aggressive lane change and lane squeeze, which affects the capacity seriously. Moreover, unreasonable traffic management measures have also magnified the adverse effects of bottleneck. Jia et al. (2003) investigated the traffic behaviors upstream of reduction bottleneck . Xue et al. (2010) studied two-lane traffic flow with partial reduction. Sheng et al. (2010) proposed a new CA model to study the temporary bottleneck induced by a special accident. However, the above research are mostly focused on the phase transition and the influence of this bottleneck on traffic flow. In this paper, using CA model, we aim to exploring the influence of different measures on traffic flux.

MODEL

As shown in Fig. 1, the road is divided into three sections: sections A, B and C. In the downstream of the merging point M (section C), the two-lane road merges into single-lane road. Thus, single-lane NaSch model is used in section C.

In sections A and B, vehicles can change lane, so the update step is usually divided into two sub-steps: in the first sub-step, cars may change lanes in parallel according to lane-changing rules and in the second sub-step, the two lanes are considered as independent single-lane NaSch models.

In section A, vehicles are not influenced by section C, consequently, they change lanes according to a symmetric rule (Chowdhury et al., 1997). if the following condition is satisfied:

(1)

Fig. 1:	Schematic illustration of the road

Here d_n = x_n+1 - x_n is the gap of the vehicle n; d_{n, other}, d_{n, back} denote the number of free cells between the nth car and its two neighbor cars on the other lane at time t, respectively; p_c is the probability of changing lane; v_maxi is the maximum velocity in section A, which is the same as of section C.

Section B is near the merging point M, it is usually deceleration zone and drivers’ behaviors are different from that in other sections. One is the control scheme of first-arrive and first-in (called Scheme 1), namely, those vehicles on both lanes which first arrive at the merging point enter the reduction road first; the other is the periodic control scheme (called Scheme 2), i.e., supposing there is a signal light at the merging point, when the light is green, vehicles on left lane can enter, while the light red, vehicles on right lane can do. We also assume that there is no randomization deceleration, for drivers do not except to be hindered here and tend to move forward. Under Scheme 1, if condition

(2)

is met, the car n on the left lane will change to right lane. If condition

(3)

is met, the car n on the right lane will change to left lane. v_max2 is the maximum velocity in section B. Under Scheme 2, the changing rules are symmetric, and here we still adopt the Eq. 3.

The boundary conditions are adopted as follows. On the left of the road system, if the hindmost vehicle position x_hindmost> v_max1, a car with velocity v_max1 is injected with probability p_in at the cell min [x_hindmost - v_max1, v_max1]. On the right of the road system, if the most leading vehicle position x_leading > L (L is the length of road), it moves without any hindrance.

SIMULATIONS AND DISCUSSION

In the simulations, section C is divided into (200xv_max1) cells, section B into L_B cells and section A into (200xv_max1 - L_B) cells. Each cell corresponds to 7.5 m and a vehicle has a length of one cell. One time step corresponds to 1s. The parameters v_max1= 5, v_max2 = 3, p = 0.3, p_c = 0.5 are selected, here p is probability of randomization deceleration. Each performance period is 1.2x10⁴ time steps, and the first 10⁴ time steps are discarded to let the transient time die out.

Fig. 2:	The flux against the length of section B with pin = 0.3

Figure 2 gives the flux of section C against the length of section B in the case of p_in = 0.3. Note that as p_in>0.24, the flux remains a constant with respect to p_in, so the flux in Fig. 2 is also the capacity of bottleneck. Different from those in condition 3 where the capacity of bottleneck is independent on L_B, considering different traffic measures leads to the variation of the capacity of bottleneck against L_B. As L_B is small, the flux under Scheme 1 is highest. As for Scheme 2, with the increase of signal period T, flux also increases in view of the overall situation. This reflects when L_B is comparatively large, vehicles move orderly with the signal, which, to some extent, reduces the frequent aggressive lane change and lane squeeze, and thus the flux has been improved.

The density distribution as p_in = 0.3 for Scheme 1 is shown in Fig. 3. One can see that p_l is always equal to p_r in this case since vehicles on both lanes which first arrive at the merging point enter the reduction road first, it is needless to move on a specific lane. With the increase of L_B, a lot of vehicles move into this deceleration zone, and thus the density increases evidently. As L_B = 50, the density is close 0.8, much higher than that of L_B = 5.

Next, we investigate the influence of the signal period T on density distribution at the fixed p_in = 0.3 and L_B = 20 (Fig. 4). In contrast to p_r, p_l first slightly decreases and then increase as approaching M and is always much smaller than p_r. In addition, the density difference between p_l and p_r first increases and then decreases with the increase of T. In section C, p_l tends to stabilize at an approximately constant value.

Fig. 3:	Density distribution near M under different LB with pin = 0.3 for Scheme 1

Fig. 4:	Density distribution near M under different T with Pin = 0.3, LB = 20 for Scheme 2

CONCLUSION

Our aim is to explore the influence of different management measures on bottleneck flux. To do so, we have proposed the first-arrive and first-in and the periodic control schemes. The numerical results show: (1) when the different traffic measures are considered, the capacity of bottleneck will vary against L_B; (2) to some extent, increasing period T is helpful for improving the bottleneck capacity under Scheme 2. The comparison of our simulation results with those in condition 3 enlightens us to obtain that the phenomenon of density inversion caused by the bottleneck may disappear due to the traffic regulations.

ACKNOWLEDGMENTS

This work was financially supported by the National Natural Science Foundation of China (10902076, 60904068, 10962002 and 11047003), the Top Young Academic Leaders of Higher Learning Institutions of Shanxi, the National Natural Science Foundation of Shanxi Province (2010011004), the National Natural Science Foundation of Zhejiang Province (Y6110502) and the Science Research Foundation for Doctors of Taiyuan University of Science and Technology (20082017).