Research on phase diagram of traffic flow in a single loop roundabout under open boundary condition

The cellular automation traffic flow model of roundabout with inner-roundabout-lane, injection-roundabout-lane and extraction-roundabout-lane is used to study the traffic flow of single loop roundabout under open boundary conditions. Through computer simulation, the relationships between the entrance probability, exit probability, vehicle braking probability and system flow, the average speed of each lane and system in the open boundary conditions are studied.Besides, the space-time diagram of vehicles on the injection-roundabout-lane with different entrance exit probabilities and the system phase diagram are established. In the system phase diagram, the critical entry probability and exit probability values of each part of the phase diagram under different braking probability are given. The results show that the system traffic flow can be divided into three phases: the phase in which the system flow changes with the entrance probability, the phase in which the system flow changes with the exit probability, and the phase in which the system flow has nothing to do with the entrance probability and exit probability. This will provide some guidance for road management.


Introduction
In recent years, with more and more private cars, the increasing traffic flow can not meet the existing road traffic conditions. The resulting traffic congestion has become one of the most important and urgent problems in the process of smart city construction. These traffic jams not only attract people's attention, but also attract many research experts and scholars at home and abroad to analyse and study them. Among them, the most popular way to solve the problem of road traffic jam is to improve the scientific traffic planning and management mode. Because this can not only make full use of the existing transportation resources, but also improve the use efficiency of the existing facilities to the greatest extent. Therefore, these scholars pay more attention to the scientific theoretical research of traffic flow, and use the obtained traffic flow theoretical research results to guide the road planning and design, traffic flow control and management of actual traffic roads, and have achieved many satisfactory results of actual traffic flow control.
R. Marzoug et al. [1] used cellular automation model to simulate two-way intersections under open boundary conditions, and studied the traffic characteristics and probability of traffic accidents at twoway intersections. H. Echab et al. [2] studied a non signalized T-intersection under open boundary conditions using cellular automatic model. Among them, two different priority rules are introduced in the intersection to eliminate the congestion. The phase diagram and its variation with the left rear vehicle ratio are studied. In addition, the spatial, temporal and density distributions are also studied. R. Jorge L zapotecatl et al. [3] proposed a new method to realize self-organization and coordination of intersection traffic by using simple sensors based on cellular automation model. Xianyan Kuang and  [4] analysed and studied the impact of the length of waiting area on traffic flow by establishing a cellular automation model of intersection with non motor vehicle waiting area. Zhongbao Dai et al. [5] based on the optimization model of NaSch and BML coupling model, introduced the slow start rule, established the intersection cellular automation traffic flow model, and studied the intersection traffic flow model. Based on NaSch model, Dingqiang Yuan et al. [6] established a cellular automation model considering U-turn of vehicles at T-junction, and analysed the relationship between the position of U-turn lane and the flow of each lane. Kezhao Bai et al. [7] established a cellular automation traffic flow model of plane roundabout under open boundary conditions, and analysed the influence of boundary conditions on system flow and system phase diagram. Changsheng Zhu et al. [8] analysed and studied the effects of entrance probability, exit probability, maximum vehicle speed and braking probability on system flow based on cellular automation traffic flow Nash model under open boundary conditions, and studied the phase diagram of Nash model. Qilang Li et al. [9] established the traffic flow model of low-speed intersection composed of two single lanes, constructed the system phase diagram by using the local occupancy probability method, and gave the flow expression of each part of the phase diagram.
However, there are few studies on the traffic flow phase diagram at the single loop roundabout at home and abroad. Therefore, based on the work of the above scholars, this paper takes the single loop intersection as the prototype to establish the cellular automation model under the open boundary conditions, and studies the effects of the boundary conditions and braking probability on the traffic flow, the average speed of each lane and the system through computer simulation.Furthermore, the system phase diagram of the single loop roundabout is fully constructed and discussed.  represents the distance between the -n th vehicle and the vehicle in front. The model includes the following four steps to update rules.

Model and evolution rules
Step 1 Acceleration: Step For multiple vehicles about to reach the merging points, the vehicles that arrive at the merging points first have priority; If the time is equal, vehicles close to the merging points have priority; If the time and distance required for multiple vehicles to reach the merging points are the same, vehicles in the inner-roundabout-lane can take priority.
 Rules for merging vehicles at diverging points at intersections. For a non right turning vehicle approaching or at the diverging points, if its speed at the current time step is 0, in order to effectively avoid the deadlock state and simulate the situation that the vehicle changes out of the ring intersection when it is blocked at the intersection, it will turn right and drive into the out of the ring lane according to a certain turning probability t P .

Numerical simulation results and analysis
We assume that i L represents the total number of discrete cells in the -     . At this time, the average speeds of these lanes are small, indicating that they are in a relatively congested state. However, the average speed of the extraction-roundabout-lane is still large, so the extraction-roundabout-lane is more conducive to vehicle traffic than other lanes, which indicates that when  is the same, the unblocked state of each lane is also different. significantly when  is greater than the critical value, and it leads to congestion on the whole injection-roundabout-lane. This is due to the fact that the speed of the vehicle in the system is low and has been saturated with the increase of  .  The system phase diagram shown in Figure 7 can be made through the influence on the entrance probability, exit probability, system flow and the average speed. Among them, zone Ⅰrepresents the phase whose flow is independent of the change of  and determined with the determination of  , and the system flow can be improved by increasing  in this region. Zone Ⅱ represents the phase whose flow is independent of the change of  and determined with the determination of  , and the system flow can be improved by increasing  in this region. Zone Ⅲrepresents the phase whose flow is independent of the changes of  and  . At the same time, as can be seen from Figure 7, When d P becomes larger, the area of regionⅠof the phase diagram becomes larger and the area of region Ⅱ decreases. This shows that when d P becomes larger, each lane is more likely to be blocked, which will make  have a more significant impact on the system flow.

Conclusion
Single loop roundabout is a common non signal control road intersection in daily life. It is loved by people because it can reduce the waiting time caused by signal lights. However, when the traffic flow is large, the congestion problem has been perplexing us. Through computer simulation, it is concluded that both  and  affect the system flow within a certain range, and the critical value of  at the intersection is small, which shows that the single loop roundabout is more suitable for the intersection with small traffic flow to a certain extent, so as to ensure the smoothness of the road. Simultaneously, the phase diagram of the system is obtained through analysis and simulation, and change the specific values of  and  as needed to adjust the system flow and provide reference for actual traffic control,