# A Cellular Automaton Model with Random Update Rules for Urban Traffic Flow

Authors: Cheng Da; Yongsheng Qian; Junwei Zeng; Yongzhi Zhang; Dejie Xu
Journal: Journal of Advanced Transportation
 (2022)

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## Abstract

Traffic jamming can easily lead to wasting time and fuel consumption and induce traffic accidents, thus seriously affecting daily life. In this study, an urban traffic flow cellular automaton (CA) model with random update rules is proposed to analyze the influence of network size and the probabilities of the change of the motion directions of cars, from up to right (pur) and from right to up (pru) on traffic flow. Simulation results show that, as the size of the system increases, the critical density tends to decrease causing larger phase transition, and for a larger size network system, the critical density is stable. The greater the pur and pru, the greater the average velocity of vehicles, which means increase in the opportunity that vehicle change directions effectively avoids the formation of traffic jamming. By studying the operational status of urban traffic flow from the microlevel, it can provide some new ideas for alleviating urban traffic jamming.


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## Body

## 1. Introduction

With the development of the economy and the increase of the urban population, the scale of cities becomes larger and larger, resulting in increasing traffic pressure and frequent traffic jams. Traffic jamming tends to cause time wastage, increase fuel consumption, and aggravate environmental pollution. Therefore, it is necessary to conduct in-depth research on urban traffic flow characteristics to alleviate urban traffic jams. Cellular automaton (CA) models are increasingly used in simulations of complex physical systems such as fluid dynamics, driven diffusive systems, sandpiles, and chemical reactions’ fluid dynamics. In these systems, the cellular automaton models provide only some general qualitative features of the system, while in other cases, useful quantitative information can be obtained. For some problems involving complex geometries, such as simulations of fluid dynamics in porous media, cellular automata are found to be superior to other methods. Many scholars have studied Cellular automaton (CA) models in two dimensions, namely, the BML model [1–4].In the original formulation of the BML model, the streets parallel toX and Y axes allow only east-bound and north-bound traffic, respectively, and vehicles are not allowed to change their direction. So, the original BML model does not have high practical value in the simulation of actual traffic for ignoring the reality. Therefore, based on the original BML model, there are many improvements which can simulate reality more comprehensively. Nagatani studied the anisotropic effect on the dynamical jamming transition in the cellular automaton model of traffic flow, showed that the traffic-jam transition occurs at higher density of vehicles with increasing the difference between density of vehicles moving to the right and density of vehicles moving to the up, and found the phase diagram of jamming transition [5].Wang et al. extended the single-lane roadway in the BML model to two-lane roadway, which is close to the reality [6]. The model proposed by Freund and Pöschel allows vehicles to move in both directions, no restrictions to one-way streets [7]. Nagatani presented the dynamical jamming transition which occurs at higher density of cars with increasing fraction of the two-level crossings below the percolation threshold [8]. Török and Kertész carried out computer simulations to study the green wave model [9], and Regragui and Moussa presented the cellular automata models for urban road traffic, which a roundabout is designed for each intersection where four roads meet [10]. In addition, some scholars have studied the property of jamming transition [11–14].Through theory and analysis, we find that traffic jamming leads traffic network to lose its function, and there are many factors that cause traffic jamming. At present, this problem is mainly using intersection optimization, signal guidance, and other means to solve from planning, design, regulation, and control of traffic. These means have achieved certain effect, and the control measures have also been proven to regulate the operation of traffic [15, 16]. However, these methods ignore the essence of traffic flow, chemical reactions’ fluid dynamics consisting of a large number of discrete vehicles. In this study, from the essence of traffic flow, based on the previous research studies [17, 18], we propose a new urban traffic flow cellular automation model with random update rule, through microscopic research on the urban traffic operation to provide new ideas for solving traffic jamming.


## 2. A CA Model with Random Update Rule

In the ChSch model, during a certain period of travel timeT, all signals are green for the east-bound vehicles at the intersection and, at the same time, are red for the north-bound vehicles in the intersection. In the next time steps, all signals turn red for the east-bound vehicles and green for north-bound vehicles. So, the vehicle in the ChSch is not allowed vehicles to change moving direction in the intersection. All the above are far from reality. In order to overcome these disadvantages, based on the ChSch model and the model with random update rule proposed by Benyoussef [18], we present an urban traffic flow CA model with random update rule. Firstly, define an L × L road matrix, in which each cell (i, j) (1 ≤ i ≤ L, 1 ≤ j ≤ L) has three states, occupied by an up-moving car, occupied by a right-moving, and unoccupied. pur and pru are the probabilities of the change of the motion directions of cars, from up to right (pur) and from right to up (pru). ρx and ρy are the density of cars moving to the right and the density of cars moving upwards, respectively. And ρ is density of all vehicles on the road satisfying the equation ρx = ρy = ρ/2. In the initial state, vehicles are randomly distributed on the road at a given initial density ρ (0≤ρ ≤ 1). The parameter Line-right is the length of right-moving vehicles queue in the intersection, and the parameter Line-up is the length of up-moving vehicles queue in the intersection.At each discrete time stepT, the vehicles on the road are updated according to the following rules.(1)
Step 1: acceleration,vn⟶minvn+1,vmax, where vn is the vehicle speed and vmax is the maximum speed(2)
Step 2: deceleration (due to other vehicles or signal light)Case 1: if the signal is red, thenvn⟶minvn,dn,sn, dn denotes the distance between vehicle n and its preceding vehicle on the current lane, sn is the number of empty cells in the former of vehicle to the intersection, and the length of queues at the intersection are recorded using Line-right (or Line-up) and sorted in the descending order.Case 2: if the signal light is green, vehicles at the intersection are updated by the order of queue length from large to small and change moving direction with probabilitypur (or pru).(3)
Step 3 (randomization): if probabilitypr (0 ≤ pr ≤ 1), then vn=maxvn−1,0. The velocity of vehicle n is decreased randomly with probability pr. pr is the random deceleration probability and is identical for all the vehicles and does not change during the updating.(4)
Step 4 (vehicle movement): each vehicle is moved forward so that, for the right-moving vehicles,rightn⟶rightn+vrn, where rightn and vrn denote the position and speed of vehicle n at time step t, respectively, while for the upward vehicles, upn⟶upn+vnn, where upn and vnn denote the position and speed of vehicle n at time t, respectively.


## 3. Simulation and Analysis

Similar to the ChSch model, this study supposing that the traffic lights are placed at the intersection with cycleT, and two phases are set in it. In the first phase, the up-moving vehicles are allowed to pass, and right-moving vehicles are prohibited to pass, and the second phase is just opposite to the first phase. The system size L = 20 – 400. And the simulation is carried out under periodic boundary conditions.


### 3.1. Analysis of Simulation Results under Different Network Scales

Firstly, whenpur = pru = 0, we simulate the relationship between the average velocity and the initial density of the road network in different scales, as shown in Figure 1.Figure 1
The relationship between average velocity and density of the road network in different scales.As can be seen in Figure1, the simulation has two different states. When the density is less than the critical density, the vehicles in the network travel at the maximum velocity 1. Once the density is greater than the critical density, the maximum of velocity is changed to 0, which indicates all vehicles in the network are jammed. There is a phase transition from the free-flowing dynamical phase to the jammed phase at the critical density. As the network scale becomes larger, the critical density ρc tends to be lower causing a sharper phase transition. For larger network, the critical density is in a stable state (L ≥ 300).Figure2 shows the relationship between the average velocity and density of the network when different p = pur = pr is selected. The global average velocity of the network decreases with the density increases. In the case of p = 1, when ρ ≤ 0.34, the average velocity of the vehicle in the network is the maximum velocity 1, which means traffic flow is free, but when ρ > 0.34, the average velocity decreases linearly with the density increases. In the case of p ≠ 1, the average velocity decreases as the density increases, at point of which the vehicles in the network cannot travel freely. The average velocity increases as the probability p increases. Thus, the growth of p effectively avoids the traffic jamming because the vehicle in the intersection changes it direction quickly, freeing up space for other up-moving vehicles or vehicles behind it.Figure 2
The relation between average velocity and density whenp = pur = pru and L = 100.


### 3.2. Analysis of Simulation Results under Differentp

Whenpur ≠ pru, the average velocity and density under different p are shown as follows.The results in Figure3 show that the three kind of average velocity decrease as the density increases, regardless of global (Figure 3(a)), upward-moving (Figure 3(b)), or right-moving (Figure 3(c)) situations. Besides, the average velocities of upward and right-moving vehicles change with different values of pur. After further analysis, the average velocity of right-moving vehicles increases as pur increases; because the vehicle in the intersection quickly move to right, it gives road space for other vertical vehicles or vehicles behind it. However, the average velocity of vehicles upward does not change with pur.Figure 3
Whenpru = 1 and L = 100, (a) the relationship between global average velocity and density, (b) the relationship between average velocity of upward vehicles and density, and (c) the relationship between average velocity of right-moving vehicles and density, with different values of pur.
(a)(b)(c)In order to have a better understand of the change of velocity when the density is fixed, we need to study how the average speed is affected bypur and pru. Due to the symmetry of the road network, as shown in Figures 4 and 5, we study the corresponding changes of the average velocity vu of upward vehicles (Figures 4(a) and 5(a)) and the average velocity vr of right-moving vehicles (Figures 4(b) and 5(b)) when pur and pru are different. Here, in Figure 4, the density is low with ρ = 0.3, and in Figure 5, the density is high with ρ = 0.7.Figure 4
Whenρ = 0.3 and L = 100, (a) the relationship between the average velocity of upward vehicles and pur and (b) the relationship between the average velocity of right-moving vehicles and pur, with different pru.
(a)(b)Figure 5
Whenρ = 0.7 and L = 100, (a) the relationship between the average velocity of upward vehicles and pur and (b) the relationship between the average velocity of right-moving vehicles and pur, with different pru.
(a)(b)In the case ofpru = 0 and vu=1, as shown in Figure 4(a), there are only upward vehicles traveling freely, since the road flow is free. However, in the interference phase pru = 0 and vu≠1, it is different, as shown in Figure 5(a). If the right-moving vehicles change to move upward, it will affect the velocity of upward-moving vehicles and then restrict the average velocity of all upward vehicles. When the value of pur is low, the average velocity of upward vehicles increases as pru gradually increases, and as pur increases, the right-moving vehicles can cause traffic jamming of through vehicles. The average velocity of upward vehicle is reduced more obviously when the value is pru is lower. In fact, for low pru values, the longer the right-moving vehicles remain in their original direction, more serious jamming and longer vehicles’ queue are.At low density, the average velocity of upward vehicles increases aspru increases when pur is constant, for upward vehicles often change moving direction and avoids jams appearing. Since the traffic jamming is easier to appear at a low pru value, the increase in the velocity of upward vehicle is exhibited when the value of pur is low. At high density, the average velocity of upward vehicles remains the same as at the low density, except that when pru = 0.1 and the pur is low, the average velocity of upward vehicles is obviously decreased. When pru > 0.5, the average velocity of upward vehicles will decrease as pur increases. In fact, at high density, due to the obvious decrease in the average velocity, almost cells are occupied by the vehicles and the vehicle is unable to enter next cell and is unable to leave in time, so traffic jamming appears. This also shows that, under the average velocity of upward vehicles decreases, pur increases and then the number of right-moving vehicles increases because these vehicles will line up in the intersection. Therefore, upward vehicles wait inside the intersection and cannot cross.


## 3.1. Analysis of Simulation Results under Different Network Scales

Firstly, whenpur = pru = 0, we simulate the relationship between the average velocity and the initial density of the road network in different scales, as shown in Figure 1.Figure 1
The relationship between average velocity and density of the road network in different scales.As can be seen in Figure1, the simulation has two different states. When the density is less than the critical density, the vehicles in the network travel at the maximum velocity 1. Once the density is greater than the critical density, the maximum of velocity is changed to 0, which indicates all vehicles in the network are jammed. There is a phase transition from the free-flowing dynamical phase to the jammed phase at the critical density. As the network scale becomes larger, the critical density ρc tends to be lower causing a sharper phase transition. For larger network, the critical density is in a stable state (L ≥ 300).Figure2 shows the relationship between the average velocity and density of the network when different p = pur = pr is selected. The global average velocity of the network decreases with the density increases. In the case of p = 1, when ρ ≤ 0.34, the average velocity of the vehicle in the network is the maximum velocity 1, which means traffic flow is free, but when ρ > 0.34, the average velocity decreases linearly with the density increases. In the case of p ≠ 1, the average velocity decreases as the density increases, at point of which the vehicles in the network cannot travel freely. The average velocity increases as the probability p increases. Thus, the growth of p effectively avoids the traffic jamming because the vehicle in the intersection changes it direction quickly, freeing up space for other up-moving vehicles or vehicles behind it.Figure 2
The relation between average velocity and density whenp = pur = pru and L = 100.


## 3.2. Analysis of Simulation Results under Differentp

Whenpur ≠ pru, the average velocity and density under different p are shown as follows.The results in Figure3 show that the three kind of average velocity decrease as the density increases, regardless of global (Figure 3(a)), upward-moving (Figure 3(b)), or right-moving (Figure 3(c)) situations. Besides, the average velocities of upward and right-moving vehicles change with different values of pur. After further analysis, the average velocity of right-moving vehicles increases as pur increases; because the vehicle in the intersection quickly move to right, it gives road space for other vertical vehicles or vehicles behind it. However, the average velocity of vehicles upward does not change with pur.Figure 3
Whenpru = 1 and L = 100, (a) the relationship between global average velocity and density, (b) the relationship between average velocity of upward vehicles and density, and (c) the relationship between average velocity of right-moving vehicles and density, with different values of pur.
(a)(b)(c)In order to have a better understand of the change of velocity when the density is fixed, we need to study how the average speed is affected bypur and pru. Due to the symmetry of the road network, as shown in Figures 4 and 5, we study the corresponding changes of the average velocity vu of upward vehicles (Figures 4(a) and 5(a)) and the average velocity vr of right-moving vehicles (Figures 4(b) and 5(b)) when pur and pru are different. Here, in Figure 4, the density is low with ρ = 0.3, and in Figure 5, the density is high with ρ = 0.7.Figure 4
Whenρ = 0.3 and L = 100, (a) the relationship between the average velocity of upward vehicles and pur and (b) the relationship between the average velocity of right-moving vehicles and pur, with different pru.
(a)(b)Figure 5
Whenρ = 0.7 and L = 100, (a) the relationship between the average velocity of upward vehicles and pur and (b) the relationship between the average velocity of right-moving vehicles and pur, with different pru.
(a)(b)In the case ofpru = 0 and vu=1, as shown in Figure 4(a), there are only upward vehicles traveling freely, since the road flow is free. However, in the interference phase pru = 0 and vu≠1, it is different, as shown in Figure 5(a). If the right-moving vehicles change to move upward, it will affect the velocity of upward-moving vehicles and then restrict the average velocity of all upward vehicles. When the value of pur is low, the average velocity of upward vehicles increases as pru gradually increases, and as pur increases, the right-moving vehicles can cause traffic jamming of through vehicles. The average velocity of upward vehicle is reduced more obviously when the value is pru is lower. In fact, for low pru values, the longer the right-moving vehicles remain in their original direction, more serious jamming and longer vehicles’ queue are.At low density, the average velocity of upward vehicles increases aspru increases when pur is constant, for upward vehicles often change moving direction and avoids jams appearing. Since the traffic jamming is easier to appear at a low pru value, the increase in the velocity of upward vehicle is exhibited when the value of pur is low. At high density, the average velocity of upward vehicles remains the same as at the low density, except that when pru = 0.1 and the pur is low, the average velocity of upward vehicles is obviously decreased. When pru > 0.5, the average velocity of upward vehicles will decrease as pur increases. In fact, at high density, due to the obvious decrease in the average velocity, almost cells are occupied by the vehicles and the vehicle is unable to enter next cell and is unable to leave in time, so traffic jamming appears. This also shows that, under the average velocity of upward vehicles decreases, pur increases and then the number of right-moving vehicles increases because these vehicles will line up in the intersection. Therefore, upward vehicles wait inside the intersection and cannot cross.


## 4. Conclusion

This study proposes a new urban traffic flow cellular automation mode with random update rule and analyzes the impact of system scale and the probabilities of the change of the motion directions of cars, from up to right (pur) and from right to up (pru) on traffic flow. Through simulation, we obtain the following conclusions. When the density of traffic flow is near the critical density, there is a phase transition from the free flow phase to the completely jam phase. As the system scale increases, the critical density tends to decrease, resulting in a sharper phase transition, and for larger networks, the critical density is stable. This is because the large scale of the road network makes the distance between the intersections too large, resulting in too many vehicles queuing at the intersection, which easily leads to the traffic flow phase transition from the free state to the congestion state, causing congestion. When pur = pru, the average velocity increases as the probability of the change of the motion directions of cars increases at a constant density. This is because vehicles can adjust the driving direction at the intersection and choose the road with lower vehicle density and better driving conditions, so as to improve the average speed. And this is consistent with the conclusion in [17, 18]. When pur ≠ pru, the average velocity of right-moving vehicles increases with the increase of the probability of the change of the motion directions of cars, while there is no significant relationship between the average speed of the up-moving vehicle and the steering probability. At low density, with the constant probability of the change of the motion directions of cars, the average velocity of upward vehicles increases as pru increases. At high density, when pru ≤ 0.5, the change of average velocity of upward vehicles is the same as the change at low density, but when pru > 0.5, the average velocity of upward vehicles decreases as pur increases. And the above conclusions enrich the findings of [17, 18].Therefore, in the process of constructing urban road network, the scale of road network should be reasonably set to make the distance between intersections not too large, so as to increase the critical density required for the phase transition of traffic flow from free phase to jam phase, so as to keep the traffic flow in free flow for more time. Secondly, make full use of the regulation and evacuation function of intersections for road congestion, reasonably set the road markings, increase the conditions and probability of vehicles changing direction at the intersection, and induce vehicles to choose to travel to low density roads, so as to alleviate traffic congestion. Besides, in the design of transportation infrastructure, even if the structure and capacity are fixed, the maximum free-flowing phase of road network can be realized by adjusting the control strategy for the traffic flow. Fully understand the useful factors in the design and develop reasonable control strategy, to improve the efficiency of transportation systems and optimize their operation. The research in this study helps to better understand the self-organization of traffic jams in urban and to further explore measures to deal with traffic jamming, which has a good application prospect.



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*Source: 4607340-2022-04-28.xml*