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We study the traffic characteristics on a single-lane highway with a slowdown section using the deterministic cellular automaton (CA) model. Based on the theoretical analysis, the relationships among local mean densities, velocities, traffic fluxes, and global densities are derived. The results show that two critical densities exist in the evolutionary process of traffic state, and they are significant demarcation points for traffic phase transition. Furthermore, the changing laws of the two critical densities with different length of limit section are also investigated. It is shown that only one critical density appears if a highway is not slowdown section; nevertheless, with the growing length of slowdown section, one critical density separates into two critical densities; if the entire highway is slowdown section, they finally merge into one. The contrastive analysis proves that the analytical results are consistent with the numerical ones.

Traffic flow is a kind of self-driven many-particle system of strongly interacting vehicles, and it has complexly dynamic properties [

On one hand, as one kind of flow-conserving bottlenecks, slowdown section reduces the local road capacity due to speed limits, which will lead to different traffic states or properties up to a certain point, such as traffic jam. Nagai et al. [

On the other hand, the dynamic slowdown sections resulting from variable speed limits are treated as one of the road-based optimization measures of traffic flow, which can increase the efficiency and stability of traffic flow when the infrastructure and the traffic demand are fixed [

For the above-mentioned research results, we learn that the study of traffic phase transition induced by slowdown section applying the analytical method is relatively less, and the theoretical relationships of the basic parameters of traffic flow and their changing rules are still not very clear. Therefore, we attempt to use the analytical method to deduce the theoretical relationships and their changing laws; furthermore, how the slowdown section influences the critical densities is also studied. This contribution is organized as follows: the new model will be presented in Section

The CA model we are using is a modified Nagel-Schreckenberg model (one car occupies 7 cells instead of one) in the deterministic limit [

Schematic illustration of the traffic model for single-lane highway with a section of slowdown.

In this paper, the rules of the CA model for a parallel update are as follows.

In the following simulations, we take

The schematic diagram of local mean densities and position.

Figure

The assumed configuration of slowdown section is displayed in Figure

The parameter

When the ratio of jam queue

Since the traffic flow of slowdown section is equivalent to that of the normal one, that is,

If

When the ratio of jam queue

According to the simulations, we know that

or

When substituting

In addition, one may pay more attention to the critical densities of the traffic phase transition. When the global density reaches the first transition point, the queue starts to appear; when it comes to the second one, the queue extends to the full road. Therefore, we separately define the two transition points as the first critical density

After achieving the expressions of critical densities, a comprehensive analysis about the changing rules of local mean densities can be given.

(a) If

(b) If

(c) If

The theoretical relationships between the local densities and the global density are shown in Figure

The theoretical relationships of local densities against global density.

The simulation relationships of local densities against global density.

Figure

(a) If

or

When the global density is low, the number of vehicles is relatively less, and the mutual influences among vehicles are very weak and they can move freely; thus the local velocities

(b) If

Under this circumstance, the jam has formed, and the flux reaches saturation, but the vehicles can still move at highest velocity

(c) If

When the density

In summary, we get the theoretical density-flow relation and velocity-density relation, as shown in Figures

Theoretical curve of density-flux.

Theoretical curves of local velocities against global density.

Figure

The numerical simulations curve of density-flux.

Figure

Numerical simulations curve of local velocities against global density.

In (

Figure

The total ratio of jam and slowdown section. (a)

Figure

Plots of total ratio against the density for various proportions of slowdown section.

Figure

Plots of flux against density for different ratio of slowdown section.

Through applying the analytical method and the numerical simulation, we deduced out the theoretical relationships of local mean densities, local mean velocities, and flux against global density

Additionally, the changing laws of two critical densities with different length of slowdown section suggest that there is only one critical density when the ratio

In the present paper we have devoted ourselves to studying the situation under deterministic rule. To study the cases where the randomization is considered will be future work.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This research was funded by the National Basic Research Program of China (2012CB725406) and the National Natural Science Foundation of China (71131001 and 71390332).