Based on the idea of infinitesimal analysis, we establish the basic model of relation between speed and flow. Since putting a certain amount of self-driving car will affect the average speed of mixed traffic flow, we choose the proportion of self-driving car to be a variable, denoted by k. Based on the least square method, we find two critical values of k that are 38.63% and 68.26%. When k < 38.63%, the self-driving cars have a negative influence to the traffic. When 38.63% < k < 68.26%, they have a positive influence to the traffic. When k > 68.26%, they have significant improvement to the traffic capacity of the road.
In order to indicate the origin of self-driving, cooperating cars, the following background is worth mentioning.
As we all known that the traffic capacity is limited in many areas of the United States owing to the number of lanes of roads. However, self-driving, cooperating cars have been provided a solution to increase the capacity of highways without increasing the number of lanes of roads. The self-driving, cooperating cars are popular with its advantages, for example, faster speed, faster acceleration, and shorter baking distance.
Lu [
We use the infinitesimal analysis method to establish the model of relationship between flow and velocity, and combine with the least square method to calculate driving the delivery ratio as a function of the average speed of mixed traffic flow. Using least square method can easily find the function relationship between the variables, and make the error between the predicted value and the actual value of sum of squares to a minimum. Therefore, it is reasonable for us to find the best function relation between the variables with the least squares approximation.
The rest of this paper is organized as follows. In Section 2, we describe the problem in detail. In Section 3, we establish the model and derive the solution based on the least square method. Finally, some conclusions are made in Section 4.
Traffic capacity is limited the designed capacity of the road networks. Once over the capacity of the road networks, drivers would experience long delays during peak traffic hours. Self-driving, cooperating cars have been proposed as a solution to increase capacity of highway without increasing number of lanes. Our model is required to solve the influence of traffic capacity about traffic lane number, peak or average, self-drive and percentage of self-driving. Further, to achieve a certain number, the number of self-drive capacity has a significant effect for roads, whether should set lanes for self-drive and corresponding policy change. By analysis, we proposed to decompose the problem into three sub- problems:
1) Sure the driving cars with the self-drive cars in different sections of mixed traffic flow speed.
2) Built a model to deal with the impact of traffic lane number, peak or average model analysis, the driving and percentage of self-drive influence on traffic capacity.
3) Reaching a certain number, the number of self-drive consideration for self-drive lanes and pipe policy tendency, and find out under different number of lanes, the number of lanes and self-drive best delivery ratio.
In order to model the above mentioned problem, the following notations are employed (
Since they drive on the proportion of directly affects the road traffic capacity, this article defined equilibrium and tipping point as follows.
Symbol | Definition | Units |
---|---|---|
The average velocity of mixed traffic flow | mile/hour | |
The vehicle by the total distance of a road | mile | |
Some sections of the vehicle by the total time | hour | |
The safe distance from driving | mile | |
The length of self-driving, cooperating cars | mile | |
The proportion of self-driving, cooperating cars | mile | |
Through a section of distance | mile | |
Through a road the total time of driving cars | hour | |
Through a road the total time of self-driving cars | hour | |
Under the influence of the driving cars, the self-drive delay time | hour | |
The proportion of self-driving | ||
A road traffic flow | hour | |
Some sections of the vehicle density | ||
A road block coefficient | ||
The average speed of driving cars | mile/hour | |
The average speed of self-driving cars | mile/hour | |
The average speed of zero flow | mile/hour |
Tipping point: The proportion of the self-driving, cooperating cars when self- driving cars on road traffic capacity has the largest inhibition effect (Chen et al. [
Equilibrium: The proportion of the self-driving, cooperating cars when self- driving cars and driving cars and driving cars go to the Nash equilibrium point (Chen et al. [
In order to establish the model, we make the following assumptions, which have been used extensively in Gupta and Dhiman [
Assumption 1. Self-driving cars evenly distributed in the driving cars.
Assumption 2. All the driving and the self-drive conductor are equal.
Assumption 3. Ignore to slope impact on the traffic of flow.
We divide the establishing process of the model into the following two steps.
Step 1 The mixed speed of self-driving cars and driving cars
Based on our assumption that all the self-drive uniform distribution in the self-drive, as shown in
v ¯ = s sum t sum (1)
s sum = ∑ i = 0 9 ( ∫ t i − 1 t i d s i d t + i ( l 1 + l 2 ) ) (2)
t sum = T 1 + T 2 + k * T 3 (3)
T 1 = ∑ i = 0 8 ( ∫ t i − 1 t i d s i d t + i ( l 1 + l 2 ) ) v 1 (4)
T 2 = s + 9 ( l 1 + l 2 ) v 2 (5)
According to (1)-(5), we can get the average speed of mixed traffic flow as
follows.
v ¯ = ∑ i = 0 9 ( ∫ t i − 1 t i d s i d t + i ( l 1 + l 2 ) ) 9 s + 36 ( l 1 + l 2 ) v 1 + s + 9 ( l 1 + l 2 ) v 2 + k * T 3 (6)
Step 2 Speed-flow model
In the continuous traffic flow, average speed, the density of flow and between the following relations:
Q = ρ * v ¯ (7)
We based on the hypothesis that density has a linear relation with the flow, can derive the average speed and flow rate of the quadratic parabola relationship model:
Q = k j ( v ¯ − v ¯ 2 v 0 ) (8)
Among from them, the for block coefficient, the average of speed at the time of zero flow. In theory, when the traffic flow to maximum, the average speed of traffic flow half of vitamin to zero flow speed, while the maximum flow is traffic capacity.
Step 1 Establish automatic driving vehicle delay time
Because of self-drive obstacle information processing, with the percentage of self-drive on different, delay would happen. According to the driving on the proportion of 50% had the greatest influence of traffic capacity [
Step 2 The driving with the self-drive mixed average speed
A safe distance from l 1 = 0.06 miles, conductor l 2 = 0.0031 miles, the distance by vitamin s = 200 miles, the self-drive miles per hour, the speed of the driving miles per hour, the speed of the delay time of self-drive, in (6) available in the average speed of mixed traffic flow under different delivery ratio, the results are shown in
According to the data in
Proportion | 10% | 20% | 30% | 40% | 50% | 60% | 70% | 80% | 90% | 100% |
---|---|---|---|---|---|---|---|---|---|---|
0.3 | 0.8 | 1.2 | 1.6 | 1.85 | 1.6 | 1.2 | 0.9 | 0.6 | 0.1 |
Proportion | 10% | 20% | 30% | 40% | 50% | 60% | 70% | 80% | 90% | 100% |
---|---|---|---|---|---|---|---|---|---|---|
39.62 | 38.47 | 37.75 | 37.05 | 36.88 | 38.47 | 40.83 | 43.02 | 45.46 | 49.38 |
Using the MATLAB software, first of all, the relationship between the scatterplot, found that increasing its change trend to decline after the first, in line with the basic facts.
Next, by adopting the idea of least square method, average speed can be on different on the proportion of mixed traffic flow with the quadratic curve fitting function of v ¯ ( k ) = a k 2 + b k + c , the average speed of mixed traffic flow and the function relation between driving the proportion is the function:
v ¯ = 0.3259 k 2 − 2.5180 k + 41.9942 (9)
When the driving on the tipping point k 1 = 38.63 % , at this point the drive to prevent traffic capacity of the largest. When driving on the proportion of equilibrium k 2 = 68.26 % , the point where the driving influence on traffic capacity. When the delivery ratio k < k 1 , self-drive for road traffic capacity has obvious inhibiting effect. When put in the proportion of k 1 < k < k 2 , self-drive for road traffic capacity and some improvement. When dropping ratio k 1 > k 2 , consider setting up self-drive lanes.
Step 3 Model validation
We will block coefficient k j , average speed v 0 at the time of zero flow and mixed flow into the average speed v ¯ of these parameters (8), about 5, 90, 405, and 520 road traffic simulation calculation, obtained under different on the proportion of road traffic. The results are as shown in
In order to further validate our conclusion, the rationality of the velocity-flow model is established and calculated under different on the proportion of road traffic, as shown in
Through the above analysis, we concluded that when the driving on the tipping
Rou | Star | End | 0 | 10% | 20% | 30% | 40% | 50% | 60% | 70% | 80% | 90% |
---|---|---|---|---|---|---|---|---|---|---|---|---|
5 | 100.93 | 101.87 | 6500 | 64,441 | 63,592 | 60,492 | 59,967 | 60, 198 | 61,016 | 65,392 | 69,439 | 74,435 |
5 | 101.87 | 103.17 | 85,000 | 82,213 | 80,689 | 78,555 | 79,139 | 77,876 | 79,768 | 87,036 | 92,954 | 94,632 |
5 | 103.17 | 103.42 | 108,000 | 107,718 | 104,230 | 102,251 | 100,918 | 98, 426 | 102,491 | 109,671 | 115,172 | 121,168 |
5 | 103.42 | 104.81 | 101,000 | 101,814 | 96,983 | 93,928 | 94,268 | 94,291 | 97,284 | 101,195 | 110,261 | 113,215 |
5 | 104.81 | 105.63 | 144,000 | 144,111 | 139,951 | 134,188 | 132,234 | 131,349 | 138,550 | 146,979 | 156,448 | 163,125 |
5 | 105.63 | 106.23 | 123,000 | 120,271 | 118,952 | 116,400 | 112,257 | 113,363 | 117,849 | 126,851 | 131,878 | 138,749 |
5 | 106.23 | 107.09 | 143,000 | 141,044 | 136,960 | 134,101 | 131,554 | 129,897 | 136,787 | 144,993 | 154,294 | 161,905 |
90 | 1.94 | 2.04 | 13,000 | 13,624 | 13,622 | 11,064 | 10,362 | 12,540 | 12,572 | 11,373 | 14,318 | 16,487 |
90 | 2.04 | 2.4 | 44,000 | 42,141 | 44,142 | 43,456 | 40,198 | 40,108 | 41,304 | 43,530 | 47,197 | 51,600 |
90 | 2.4 | 2.54 | 23,000 | 22,827 | 23,802 | 22,545 | 22,620 | 22,269 | 20,898 | 24,812 | 23,085 | 27,543 |
90 | 2.54 | 2.79 | 23,000 | 23,665 | 21,662 | 20,407 | 20,162 | 21,818 | 20,486 | 22,255 | 26,051 | 25,167 |
405 | 0 | 0.09 | 75,000 | 76,154 | 73,434 | 71,480 | 69,242 | 70,702 | 70,275 | 78,301 | 81,162 | 84,328 |
405 | 0.09 | 0.37 | 110,000 | 107,338 | 107,047 | 104,788 | 100,101 | 99,640 | 104,950 | 110,705 | 117,876 | 123,697 |
405 | 0.37 | 0.54 | 169,000 | 167,564 | 163,762 | 160,859 | 154,890 | 154,211 | 160,862 | 174,094 | 179,787 | 192,726 |
405 | 0.54 | 0.75 | 176,000 | 173,921 | 170,119 | 166,163 | 164,215 | 162,871 | 170,889 | 179,841 | 189,466 | 200,169 |
520 | 0 | 0.36 | 48,000 | 47,413 | 46,140 | 44,157 | 43,533 | 44,203 | 46,132 | 48,699 | 51,041 | 52,886 |
520 | 0.36 | 0.85 | 79,000 | 78,255 | 75,589 | 73,549 | 74,231 | 72,697 | 77,210 | 81,160 | 83,491 | 88,078 |
520 | 0.85 | 1.43 | 45,000 | 46,211 | 45,263 | 41,374 | 42,904 | 40,015 | 42,589 | 47,264 | 47,029 | 52,221 |
520 | 1.43 | 1.63 | 61,000 | 59,239 | 57,717 | 58,379 | 54,919 | 57,788 | 58,873 | 61,342 | 63,852 | 70,598 |
520 | 1.63 | 4.4 | 68,000 | 66,701 | 66,067 | 65,190 | 62,865 | 63,394 | 64,961 | 69,009 | 73,941 | 78,245 |
point k 1 = 38.63 % , it prevents traffic capacity of the largest. It influences the driving traffic capacity. When the delivery ratio k < k 1 , self-drive for road traffic capacity has obvious inhibiting effect. When put in the proportion k 1 < k < k 2 , they have a positive influence to the traffic. When dropping ratio k 1 > k 2 , they have the significant improvement to the traffic capacity of the road. In this paper, we only consider the ordinary road traffic capacity. For certain capacity, we need to continue to look for a function of flow rate, density, time between.
The authors gratefully acknowledge the very helpful comments and suggestions of the editor and the anonymous reviewers for their valuable comments and suggestions that helped improve this paper.
Ku, Z.Q. and Liao, T. (2017) The Effect of Self-Driving Car on Urban Traffic. American Journal of Computational Mathematics, 7, 149-156. https://doi.org/10.4236/ajcm.2017.72013