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 [1] analyzed the influence of mixed traffic flow car’s speed under different traffic flow on the highway. He studied the relationship between traffic flow and traffic speed change and built the model of speed-flow. At the same time, he also analyzed the impact of medium-sized car under different ratios on highway mixed traffic flow speed, researched the relation between the ratio of large and medium-sized cars with the changes of the rate of traffic and built relationship model between speed with the speed-large and the proportion of medium-sized cars. Daganzo and Knoop [2] derived the analytic form of the capacity formula inspired by analytic upper and lower bounds derived with variational theory for a version of the problem where cars are treated as a fluid. Daganzo and Lehe [3] considered a signalized street of uniform width and blocks of various lengths. It has shown that the macroscopic fundamental diagram of an arterial street controlled by traffic signals was defined by the solution of a linear program. Sorigueraa et al. [4] examined both the macroscopic and microscopic effects of different speed limits on a traffic stream. Jamshidnejad et al. [5] proposed a frame- work to interface and integrate macroscopic flow models and microscopic emission models. Yang et al. [6] addressed the planning and optimization of intermodal hub-and-spoke (IH & S) network considering mixed uncertainties in both transportation cost and travel time. Celikoglu and Silgu [7] evaluated the performance of a dynamic approach to classify flow patterns reconstructed by a switching-mode macroscopic flow model considering a multivariate clustering method. Yang et al. [8] proposed a bi-objective hub-and-spoke (H & S) network design problem with type-2 (T2) fuzzy transportation cost and travel time described by parametric secondary possibility distributions. Those give us some inspiration about research the mixed ratio of self-driving cars and the mixed speed of flows. We will consider mainly the delay time to self-driving cars deal with signal which results to the changes of the traffic flows.

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.

In order to model the above mentioned problem, the following notations are employed (Table 1).

Since they drive on the proportion of directly affects the road traffic capacity, this article defined equilibrium and tipping point as follows.

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. [9] ).

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. [10] ).

In order to establish the model, we make the following assumptions, which have been used extensively in Gupta and Dhiman [11] , Qu et al. [12] , and Tang et al [13] .

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 Figure 1. In order to calculate the average speed of mixed traffic flow, we since the driving ratio of 10% (Chen et al. [14] ).

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 [5] [15] and Arizona from driving the test data [4] [16] [17] , on the proportion of self-drive delay time to simulate, you can get different delivery delay time under the proportion, as shown in Table 2. The computational experiments are carried out on a personal computer (Dell with Intel (R) Core (TM)i5-2450M CPU 2.50 GHZ and RAM 2.50 GB), using the Microsoft Windows 7 operating system (Cheng et al. [18] ).

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 Table 3.

According to the data in Table 3, in order to establish the average velocity v ¯ and put in the function v ¯ ( k ) .

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 Table 4.

Table 4 shows that when the driving on the proportion between 10% - 70%, the stretch of road traffic is only the self-drive road traffic to reduce, and when the self-drive delivery ratio above 70%, the stretch of road traffic significantly higher than that of only the self-drive road traffic. Results are basically consistent with the above theoretical analysis.

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 Table 4, since the driving input to a certain extent, can actually increase the road capacity. When driving on the ratio between 10% - 70%, the stretch of road traffic is lower than only the self-drive road traffic, and when the self-drive delivery ratio above 70%, the stretch of road traffic significantly higher than that of only the self-drive road traffic. It is also consistent with our vehicles in this paper the relationship between average speed ratio coefficient and mixing road.