_{1}

^{*}

The measurements on actual traffic have revealed the existence of meta-stable states with high flow. Such nonlinear phenomena have not been observed in the classic Nagel-Schreckenberg traffic flow model. Here we just add a constraint to the classic model by introducing a velocity-dependent randomization. Two typical randomization strategies are adopted in this paper. It is shown that the Matthew effect is a necessary condition to induce traffic meta-stable states, thus shedding a light on the prerequisites for the emergence of hysteresis loop in the fundamental diagrams.

In the past decades, a lot of attention has been devoted to the study of traffic flow. Since the seminal work of Nagel and Schreckenberg in the early 1990s [

In recent years, the cellular automata models have been extended to investigate the meta-stable states in traffic systems [

The study of meta-stable states is of great practical significance. On the smooth road, using cruise constant velocity can keep the vehicles running at a constant velocity and reduce fuel consumption. Based on this fact, D. Chowdhury et al. proposed a cruise-control limit model, which successfully reproduced the meta-stable states in traffic system [

For the sake of completeness, let us briefly recall the evolution rules of the classic Nagel-Schreckenberg model. This set of rules describes the principles that the vehicles must follow when driving on a one-dimensional ring road. The road is divided into a series of cells. Each cell is either empty or occupied by just one vehicle with a discrete velocity v i ( t ) ∈ [ 0 , v max ] . Here v max is the velocity limit of the vehicles. The density ρ of the road is defined as the ratio of the number N of vehicles to the length L of the road, i.e., ρ = N / L .

The configurations of the vehicles are updated in parallel according to the following four rules. R1: Acceleration, v i ( t + 1 / 3 ) = min ( v i ( t ) + 1 , v max ) ; R2: Braking, v i ( t + 2 / 3 ) = min ( v i ( t + 1 / 3 ) , d i ( t ) ) ; R3: Randomization (with probability p), v i ( t + 1 ) = max ( v i ( t + 2 / 3 ) − 1 , 0 ) ; R4: Location updating, x i ( t + 1 ) = x i ( t ) + v i ( t + 1 ) . Here x i ( t ) and v i ( t ) are the position and the velocity of ith vehicle at time t. The parameter d i ( t ) is the empty cells between vehicle i and the nearest neighbor vehicle i + 1 in front of it.

Although these rules seem simple, they can simulate some complex traffic phenomena such as the free flow and ghostly congestion. Rule 1 characterizes a driver’s trait to drive as fast as possible without exceeding the maximum velocity limit. Rule 2 is designed to avoid collisions between vehicles. Rule 3 requires drivers to slow down randomly and change their visual angle, so as to effectively alleviate visual fatigue. The randomization is crucial for the spontaneous emergence of traffic jams.

In the classic Nagel-Schreckenberg traffic flow model, the meta-stable states will not occur, because the indiscriminate randomization makes the homogeneous structure of the system difficult to maintain. In this paper, we do not modify the evolution rules of Nagel-Schreckenberg model, but add a specific function to control the random slowing probability of each vehicle. The determination of random slowing probability is placed before the acceleration step. R0: p i ( t ) = p 0 g ( v i ( t ) ) . Here g ( x ) is a bounded sine or cosine function.

For simplicity, only one type of vehicles is considered in this paper and therefore the same maximum velocity v max = 5 is applied to all vehicles. For a realistic description of highway traffic, the length of a cell is set to 7.5 m, which is interpreted as the length of one vehicle plus the average gap between two adjacent vehicles in a jam.

First, we show the differences of fundamental diagrams for three different control strategies, as shown in

The cosine control strategy mainly limits the low-velocity vehicles, but has no limit to the vehicles with maximum velocity. At the low density, only vehicles with the cosine control strategy can drive at maximum velocity, as shown in

Under the control of the sinusoidal law, the vehicles traveling at maximum velocity are most restricted, while the stationary vehicle can start at any time as long as there is free space in front of it. Therefore, in the middle and high density areas, the traffic flow under the control of sine function is significantly higher than the other two strategies, as shown in

On one hand, the cosine control strategy requires that the vehicles running at the maximum velocity do not slow down randomly, which corresponds to the phenomenon of “rich get richer” in life. On the other hand, it demands the stationary vehicles to slow start, which is equivalent to the phenomenon of “poor get poorer” in life. In this sense, the cosine control strategy is equivalent to Matthew effects, which is conducive to the separation of phases and the emergence of meta-stable states.

In

In this section, we show the effect of a minor perturbation on the traffic system.

For the random initial configuration, at the same traffic density there are already traffic congestions in the lane. After being disturbed, the chain reaction of subsequent vehicles leads to new traffic congestions. The new traffic congestions here relieve the traffic pressure elsewhere, so the width of the early traffic congestions in the system becomes narrow as shown in

For the sine control strategy, at the same traffic density some small traffic congestions are randomly scattered on the lane, as shown in

In this paper, the main factors and limiting conditions of meta-stable and hysteretic phenomena are explored. Although the Nagel-Schreckenberg model itself cannot reflect the meta-stable and hysteretic phenomena found in real traffic, it can capture more complex traffic phenomena with a little modification. For any traffic flow model, it is a challenge to describe the possibility of hysteresis loop. Our study generalizes some of the previous results and extends the possibility of

meta-stable states in traffic systems to a general criterion. Only when the Matthew effect is embedded in the evolution rules, the meta-stable state will appear as scheduled. Our results will pave the way for the research with the same dynamic background in the real traffic systems.

This work is supported by the National Natural Science Foundation of China under Grant Nos. 61563054 and 10562020, the Natural Science Foundation of Guangxi under Grant No. 2019GXNSFAA245023, the open fund of Guangxi Colleges and Universities Key Laboratory of Complex System Optimization and Big Data Processing under Grant No. 2015CSOBDP0101, Initiation Fund of Doctoral Research of Yulin Normal University under Grant No. G20150003.

The author declares no conflicts of interest regarding the publication of this paper.

Zhu, L.H. (2020) Criterion for the Emergence of Meta-Stable States in Traffic Systems. Journal of Applied Mathematics and Physics, 8, 976-982. https://doi.org/10.4236/jamp.2020.86076