2014MCM竞赛论文

analyze the performance of this rule in light and heavy traffic. In the paper, we developed four mathematical models to solve the problem that we find according to the topic, and then get the scientific and reliable conclusion from them. Finally, we designed some new arithmetic on the basis of our conclusion to effectively promote the traffic flow and safety in multi-lane freeways.

In light traffic, through in-depth analysis and demonstration, we developed a model of Queuing Theory to perform the Keep-Right-Except-To-Pass Rule in light traffic. This model shows the relationship among traffic flow, speed and traffic density directly. So according to the model, we get the conclusion that the traffic rule can effectively promote the traffic flow and safety.

In heavy traffic, through the analysis of traffic condition in both single-lane and multi-lane, we developed a model of Enhanced Cellular Automation to perform the Keep-Right-Except-To-Pass Rule in heavy traffic. In the process of solving this model, we use MATLAB, and finally prove that the principle cannot distinctly promote the traffic flow in heavy traffic, while it can only ensure the safety to some extent.

By using VISSIM Simulation System, we propose an effective arithmetic. This arithmetic is also proposed by analyzing the influence of the Keep-Right-Except-To-Pass Rule on traffic flow in both light and heavy traffic. The arithmetic can effectively promote the traffic capacity; moreover we proved that it is also suitable for the countries where driving automobiles on the left is the norm. In this paper, we meticulously explained the using characters of the arithmetic under this kind of condition. Furthermore, in order to make it be suitable under the condition of intelligent system, we do scientific and reasonable adjustment of it. We also developed the model of HRRN on the basis of Cellular Automation. At the end of the paper, we rigorously analyze the sensitivity of the models and objectively evaluate the solution.

Contents

1. Introduction (1)

1.1 Restatement of the Problem (1)

1.2 Significance of Solving this Problem (1)

1.3 Theory Knowledge Introduction (2)

1.3.1 Queuing Theory (2)

1.3.2 Traffic Flow Simulation Theory (2)

1.3.3 Dynamics Lateral Model Differential Equation Theory (2)

1.3.4 Cellular Automata (CA) (3)

2. Analysis of the problem (3)

3. Assumptions, symbols and terminologies (4)

3.1Assumptions (4)

3.2 Symbols (4)

3.3 Terminology (5)

3.3.1 Queuing Theory (5)

3.3.2Cellular Automation Theory (CA) (5)

3.3.3 Traffic Flow Model (5)

3.3.4 State generator of the signal (5)

3.3.5 Intelligent Transportation System (ITS) (5)

3.3.6 VISSIM Software System (6)

4. Keep-Right-Except-To-Pass Rule in Light Traffic (6)

4.1 Backgrounds (6)

4.2 Queuing Theory................................................................................. .6 4.2.1 Elementary Analysis (6)

4.2.2 Models (7)

4.2.3Two –lane and Three –lane Models (8)

4.3Collection of Data (9)

4.4 Analysis of the Queuing Theory Model (10)

4.5 Verification and Improvement of the Model (10)

4.6Further Discussion about Flow and Rate (11)

5. Keep-Right-Except-To-Pass Rule in Heavy Traffic (12)

5.1Backgrounds (12)

5.2 Tradeoffs between Traffic Flow and Safety (12)

5.2.1 Analysis (12)

5.5 Models (14)

5.5.1Analusis of the Initial Model (14)

5.5.2 Cellular Automation Theory (CA) (14)

5.5.3 CA model and Studies (16)

5.5.4 The Arithmetic Model of MATLAB (17)

5.5Numerical Simulation of the Mode (20)

5.6Scientific Analysis of the Model (21)

6. The Keep-Left-Except-To-Pass Rule (22)

6.1Backgrounds (22)

6.2Improved Models (22)

7. Under the Control of Intelligence Traffic System (24)

7.1Analysis and Establishment (24)

7.1.1Analysis of the Model (25)

7.1.2The Establishment of Models (25)

7.2Model’s Solution and Testing (25)

7.3 Sensitivity Analysis and Evaluation (26)

8.Evaluation of the Model (27)

8.1Strengths (27)

8.2Weaknesses (28)

9. Future Work (32)

9.1 Generalizability of the Models (34)

9.2 The prospects of the Models’ Development (35)

10. References (35)

The Keep-Right-Except-To-Pass Rule

Abstract

The object of this study is the Keep-Right-Except-To-Pass Rule, and our target is to analyze the performance of this rule in light and heavy traffic. In the paper, we developed four mathematical models to solve the problem that we find according to the topic, and then get the scientific and reliable conclusion from them. Finally, we designed some new arithmetic on the basis of our conclusion to effectively promote the traffic flow and safety in multi-lane freeways.

In light traffic, through in-depth analysis and demonstration, we developed a model of Queuing Theory to perform the Keep-Right-Except-To-Pass Rule in light traffic. This model shows the relationship among traffic flow, speed and traffic density directly. So according to the model, we get the conclusion that the traffic rule can effectively promote the traffic flow and safety.

In heavy traffic, through the analysis of traffic condition in both single-lane and multi-lane, we developed a model of Enhanced Cellular Automation to perform the Keep-Right-Except-To-Pass Rule in heavy traffic. In the process of solving this model, we use MATLAB, and finally prove that the principle cannot distinctly promote the traffic flow in heavy traffic, while it can only ensure the safety to some extent.

By using VISSIM Simulation System, we propose an effective arithmetic. This arithmetic is also proposed by analyzing the influence of the Keep-Right-Except-To-Pass Rule on traffic flow in both light and heavy traffic. The arithmetic can effectively promote the traffic capacity; moreover we proved that it is also suitable for the countries where driving automobiles on the left is the norm. In this paper, we meticulously explained the using characters of the arithmetic under this kind of condition. Furthermore, in order to make it be suitable under the condition of intelligent system, we do scientific and reasonable adjustment of it. We also developed the model of HRRN on the basis of Cellular Automation. At the end of the paper, we rigorously analyze the sensitivity of the models and objectively evaluate the solution.

Key words：Queuing Theory，Mathematical modeling, Cellular Automation Theory, keep-right-except-to-pass rule, traffic flow

1. Introduction

1.1 Restatement of the Problem

We know that in many countries such as USA, China and most other countries except for Great Britain, Australia, and some former British colonies where driving automobiles on the right is the rule, multi-lane freeways often adopt a rule that requires drivers to drive in the right-most lane unless they are passing another vehicle, in which case they move one lane to the left, pass, and return to their former travel lane. However, does this rule can effectively promote the traffic flow[1] and safety whether in light or heavy traffic? We suppose that the Keep-Right-Except-To-Pass

Rule is the optimal rule, so we develop a mathematical model and use many data to demonstrate our hypothesis.

If our hypothesis is right, then does this model suitable in countries where driving automobiles on the left is the norm or we need some additional requirements? If vehicle transportation on the same roadway was fully under the control of an intelligent system, to what extent would this change the results of your earlier analysis? We analyze these problems step by step and finally give our own models and methods.

1.2 Significance of Solving this Problem

With the development of our society, there are more and more cars, so more and more traffic problems occur in our daily life. So through in-depth studies about Keep-Right (or Keep-left) traffic rules, we can find many potential problems in it, thus we can avoid many traffic accidents and promote greater traffic flow. Furthermore, we can estimate the development of the transportation in the future, and find a more suitable traffic rule in future traffic system.

1.3 Theory Knowledge Introduction

1.3.1 Queuing Theory

As early as a few decades ago, queuing theory has already been widely applied to solve traffic flow problems. In any traffic sites, people must understand: 1) the distribution of vehicles after entering the queuing system[2]. 2) the source of cars is limited or not. 3) The distribution of service time in each traffic system. One of typical system in Queuing Theory is the rule of traffic flow enter or leave the parking garage. If cars come to the parking garage in random, and follow the rules of first come first service to enter or leave the garage, then the time that the vehicle required to come into or out of garage form a similar distribution of exponential function.

If the vehicle arrive a place randomly, then as for the time interval between two continuous arrived vehicles can be described as Poisson distribution. From a simple Poisson distribution model— Poisson distribution of vehicles only come from single lane – we can get the basic knowledge of queuing theory. In order to better understand Queuing Theory, knowing the waiting time (w) that cars spent before they got the service and the whole time needed to form a real queue (v) are as important as knowing expected number of the queue and the average length of the queue. Nowadays, the application of queuing theory is mainly manifested in following aspects: 1. study the relationship between departure intervals of two buses and length of queues.2.study vehicles queuing in parking places.3.study the relationship between vehicles queuing [3] and delaying before the intersection.

1.3.2 Traffic Flow Simulation Theory

Road traffic simulation model is to abstract the road traffic system based on the characteristics of simulation technology. It involves the basic elements of road traffic system and interactions between these factors and their change process. The characteristics of the road traffic system determines the complexity of the simulation mode, nevertheless, it become extraordinarily complex because the microscopic

simulation[4] model detailed description of the various elements in the system. The simulation study of traffic flow in expressway can realize dynamic virtual representation of traffic running state, and provide traffic management and traffic planning personnel with an effective experimental platform. Using traffic flow simulation model to make simulation experiments, and then through the analysis, comparison and evaluation of the simulation output result to obtain the parameters of the traffic flow. Finally we can use the parameters to provide gist of decision and technical support for the traffic management and control, traffic scheme[5] comparison and effect evaluation.

1.3.3 Dynamics Lateral Model Differential Equation Theory

When using the computer simulation, under normal circumstances we need to give a scientific mathematical model, so it is necessary to master a certain method to set up mathematical model. In dynamic systems, in most cases you can use the differential equation to represent the dynamic characteristics of a system, and can also simplify the system into state equation and difference equation model by using the differential equation. In real engineering, the system usually be divided into two types, one is the continuous system[6] in which the mathematical model is usually the high-order differential equation; The other is the discrete system, and its mathematical model is difference equation.

The vehicle dynamics model is made up by many sub-models, and there are many complex interactions between them. Under a certain sections, some sub-models may change more quickly, while others are relatively slowly. Accordingly, there are fast varying component and the slow varying component in the integral of differential equation to describe the process. If there is a large gap between fast-change sub-models and low-change sub-models[7], then it is called rigid equation in math. In lateral linear and nonlinear model, because the ratio of two characteristic values of matrix in the differential equation (Jacobi) has massive swings in the case of wheel speed change, so we can believe that the change of speed lead to the inconformity in the change of speed in Lateral model differential equation.

1.3.4 Cellular Automata (CA)

Cellular automata is a kind of local dynamic model in the form of spatial-temporal dispersions, and a typical method to research complex systems, particularly suitable for time and space dynamic simulation study. Cellular automata is not determined by strictly defined physical equation or function, but in the CA model, each cell that spreads in Lattice Grid has a limited discrete state, following a same reaction rules and occurring synchronous renewal according the same partial rule. A large number of cells through simple interaction constitute the evolution of the dynamic system. The characteristics of the CA model [8]: time, space, status are all scattered, each variable has limited multiple states, at the same time the rules of the state change in time and space are partial. It is composed of a series of rules which used to make model construction. As long as the models meet these rules then they can be called

automata model. Therefore, Cellular Automata is an ageneraldesignation of a kind of model, or a framework of a method.

The main advantages of CA model:

(1)The model is simple, particularly is easy to implement on a computer.

(2)Can represent the various complex phenomena in traffic, and reflect the traffic flow characteristics. People in the process of simulation by investigating the state change of cellular automata, can not only get cars‘ speed, displacement, and time headway to describe the micro-characteristic[9] of traffic flow but also get the parameters of average velocity, density and flow rate to present macro-characteristic of traffic flow.

(3)Can represent the Traffic Flow Modeling in one lane, multi-lane and road network; as well as Modeling of motor vehicles and non-motor vehicles.

2. Analysis of the problem

In the study, several crucial issues should be considered, and each of them contains many factors. After the comprehensive analysis about Problem A, we find the following questions.

(1)Analyzing the performance of the rule in TITLE in light and heavy traffic through examine tradeoffs between traffic flow and safety，the role of under- or over-posted speed limits (that is, speed limits that are too low or too high), and/or other factors that may not be explicitly called out in this problem statement.

(2)How to find alternatives that might promote greater traffic flow, safety, and/or other factors that we deem important and take rational analysis of them based on our quantitatively analysis about the keep-right-except-to-pass rule?s effe ct on traffic flow and safety.

(3)How to modify our mathematical models to make it suitable for countries where driving automobiles on the left is the norm.

(4)How to modify our mathematical models to make it suitable for vehicle transportation on the same roadway was fully under the control of an intelligent system.

Obviously, the four above-mentioned questions are the crucial part of the study, so they play an important role in solving these issues. First of all, some necessary assumptions must be considered. Firstly, whether motorcycle types (such as truck, car, motorcycle and so on) have impact on traffic flow and safety? Secondly, it is necessary to notice the different influences of multi-lanes (two-lane or three-lane) in the keep-right-except-to-pass rule? Thirdly, is it necessary to establish a model to describe velocity (too low or too high) influence on traffic and safety quantitatively? The data is the most reliable evidence to verify the theory, hence, how to get reliable data we need has come into question. In the process of establishing model, many other factors should be taken into consideration, so in order to make the model more practical in our daily life, appropriate improvement mast be made in different conditions (the keep-right-except-to pass rule, the keep-left rule and intelligent system)

3. Assumptions, symbols and terminologies

3.1Assumptions

3.1.1 Unidirectional lane is closed that is to say cars can either drive into or out the lane.

3.1.2 Regardless of motorcycle type in the whole experiment.

3.1.3 Ignore the driver's personal factors (such as drive skills, psychological quality, and moral quality).

3.1.4 The driving direction of vehicles is always consistent

3.1.5 There are no traffic accidents

3.1.6 Ignore the influence of weather conditions and the influence of Carioles Acceleration and earth magnetic field on cars.

3.2 Symbols

q ——estimated traffic flow be used to confirm the direction

x ——the number of car that driven to the opposite direction y ——net overtaking number of overtaking (the number of cars that surpass the target car)

a t —— the time that the car used to drive to backward

w t ——the time that the car used to drive to forward

t ——the average time that the car used to drive to forward

l —— the length of the whole lane

s u —— the car‘s average speed in the trial lane

c n —— the number of lanes

nl —— the length of the trial lane

v —— average speed

d —— number of tim

e o

f changin

g driving lane p —— the density of traffic flow

t d —— the Simulation Step Time, determining the change speed of the state of cellular automata

t n —— the simulation step number, determining the time that lasting time of the simulation

p f ——the probability of cars to enter the trial lanes, determining the density of traffic flow.

3.3 Terminology

3.3.1 Queuing Theory

It also called the theory of random service system. The content of its research has the following three parts:

A：Status. It studies the regularity of the probability of queuing system. The status contains instantaneous state and steady state.

B：Optimization. It contains the optimization in static and active state. The former optimization refers to the optimal design. The latter optimization refers to the optimal implement based on the existing queuing system.

C：Statistical inference. It refers to a method to estimate which model is suitable for a given queue, in order to take further study based on the Queuing Theory.

3.3.2Cellular Automation Theory (CA)

It is a dynamical system that is discrete in both space and time. Each cell that spreads in Lattice Grid has a limited discrete state, following a same reaction rules and occurring synchronous renewal according the same partial rule.

Traffic Flow Simulation Theory

3.3.3 Traffic Flow Model

Road traffic simulation model is to abstract the road traffic system based on the characteristics of simulation technology. Traffic simulator is a microscopic traffic flow simulation model; it includes car-following model and Lane-change model.

3.3.4 State generator of the signal

It refers to a control model of signal, and is based on simulation step to get the information from Traffic simulator, determining the state of the signal in next simulation time, at the same time, sending the information back to the Traffic simulator.

3.3.5 Intelligent Transportation System (ITS)

It is a Real-time, accurate and efficient integrated transportation and management system that combined advanced information technology, communication technology, sensor technology, control technology and computer technology together, and then to be used in the whole transport management system to solve the problem in a larger scale.

3.3.6 VISSIM Software System

It refers to the microscopic traffic flow simulation software system that designed for the PTV Company in Germany to be used in transportation systems for a series of operation analysis. It is an effective tool that has the functions of analyzing, evaluating optimizing the transportation network and designing scheme comparison, so it is an effective tool for the analysis of many traffic problems.

4. Performance of the Keep-Right-Except-To-Pass Rule in Light Traffic

4.1 Backgrounds

To analyze the performance of The Keep-Right-Except-To-Pass Rule in light traffic, we must abstract the actual road conditions, and then to develop reasonable mathematical models and to deeply analyze the performance in the traffic. In light traffic, the density of the trial lane is small, so almost all the cars are driving on the

right lane (also called the slow lane) under The Keep-Right-Except-To-Pass Rule. If we want to test whether the rule can promote the traffic flow, then we must analysis each of the lane.

In the road traffic performance experiment, statistical analysis of the traffic flow is needed. Through the statistics and analysis of the number of cars (driven in or out of the trial lane), we can get find the rule‘s influence on the traffic flow in light flow. In the experiment, we pay more attention to the chosen of the experimental lane, because many kind of lanes (tunnels, cross-sea bridge),it is not allowed to overtake.

4.2 Queuing Theory

4.2.1 Elementary Analysis

In light traffic, the density of the right lane is the highest, while the lift is the smallest. First of all, we should establish a standard to identify ―light traffic‖. We know that the traffic flow is expedite, that is to say the car that driven into the experimental lane is smaller than the number of driven out. So we should establish a quantified model to define traffic flow.

Picture1

In light traffic, the relationship between speed (v) and density ( ) can be described as the following graph (the length of the experimental is 200km):

Picture2

The selected standard is the average ratio of the vehicles‘ number, which are from the entrance to the exit.

1N is the number of cars that driven into the experimental lane, 2N presents the

number of cars that driven into the experimental lane. The relationship between the two is:

1

2N F N =, 1121122221,T T T T N N N N N N =-=-. (1)

Analyzing the data, we can get the evaluation criteria of F in light traffic. Abstracting the case, we can get a method to calculate the traffic flow on one lane. According to the theory of traffic dynamics differential equation, we can get the functional relationship between traffic flow and the density[10] of the lane. It can be used to deeply study traffic flow under The Keep-Right-Except-To-Pass Rule.

4.2.2 Models

Treat the traffic flow as the liquid, suppose (,),(,)q x t x t ρ and (,)u x t all can establish a continuous and differentiable functions with x and t, the following are the equations according to the conservation principle.

According to the integral, when the time is t, in section [a, b], the number of cars is (,)b

x t dx

a ρ?. (,)q a t means the number of cars that passed a in ten minutes; (,)q

b t means the number of cars that passed b in ten minutes

(,)(,)(,)b d q a t q b t x t dx dt a ρ-=?. (2)

This is the integral form of traffic flow, and it does not have continuous relationship with x. Under the Analyticity Properties of q and ρ, it can be changed into the following form:

(,)(,)(,)b q a t q b t q x t dx x a ?-=-??,

(,)(,)b b d x t dx x t dx dt t a a ρρ?=???0b q dx t x a ρ??=?

??（+）.(3) We chose the section [a, b] randomly, so 0q t x ρ??=??+. This is the continuous traffic

flow equation.

When we find the function relationship between q and ρ——()q q ρ=, differential coefficient dq

d ρ is th

e known function, written as ()?ρ. We can get the conclusion:

().q dq x d x x ρρ?ρρ???==???

Then equation can be written as

()0,(),0,(,0)()dq t x t x d x f x ρρ?ρ?ρρρ???+==>-∞<<∞?????=? . (4)

F(x) represents the initial density,(,)x t ρ describe the distribution of the traffic flow in any time, and from ()q ρwe can get (,)q x t .The equation is a first order quasilinear partial differential equations:

0((),)()x t t f x ρ=, 000()(()),(0)x t f x t x x x ?=+= (5)

It is easy to prove that （4），（5）can be satisfied in （3）. From ((),)(0)x t t f x ρ=,

we can get 0d dx dt t x dt ρρρ??=+=??. We can also get 0(())dx f x dt ?=.Put （4）into

()dx dt ?ρ=, we can get equation （3）, so （4），（5）are satisfied in initial conditions (,0)()x f x ρ=.（4）and （5）have obvious geometric meaning, on the plane Oxt ,

（5）can be describe as a series of straight lines, they meet the x-axis in 0x , and it‘s slope is ()1

0k x ?-=????(the relationship between t and x). When ,f ? is fixed, 0x

change, then k will also change. （4）shows that on each of the characteristic line ()x x t =, the density of the traffic flow (,)x t ρ is a fixed number 0()f x , at the same time, in different characteristic line,(,)x t ρ change according to the change of 0x .The following chart describes the characteristic line of (3):

The choice of the time has an obvious influence on traffic loading. For example, there is a big difference of traffic between day and night. Based on this condition, we can control the traffic loading by choosing different experimental time. In the study, we choose the time on 6 o‘clock in the morning, when the road is in the light traffic condition.

4.2.3Two –lane and Three –lane Models

In Queuing Theory, if ()N t represent that at the time t , the number of the car in the specific traffic system, so {(),0}N t t ≥is a particular random process. If ―Birth ‖ means that the car drives into a specific highway in a particular time, the n ―Death ‖ is the car leaving ，as for many queuing processes,{(),0}N t t ≥is a particular Stochastic Process －Birth and Death Process. Generally, it is difficult to get the distribution of ()N t , which represents that ()P{N(t)n}(n 0,1

,2,)=== n P t , so it is usually try to find the distribution when the system keep balance, marked n p ,n 0,1,2,= to find the balance distribution under the light traffic, considering the possible state of system n . Assumption in a period time ,record the time of entrance state n and leaving state n , because the ―come ‖and ―leave ‖is alternative ,the two numbers is equal or the discrepancy is 1. On average, we can think they are equal. When the system became balance ，for any state n ，in unit time, the entrance time 0

equals the leaving time, this is the theory ―in =out ‖ under the condition of statistic balance .According to this theory, we can get equations:

0 1100p p μλ=

1

()2200111p p p μλλμ+=+ (6) ??

n ()1111n n n n n n n p p p μλλμ++--+=+ (7)

From the above, we can get: 0101p p λμ=, 10112111010222211(p p )o p p p p λλλλμλμμμμμ=+-==, (8)

101110111111(p p )n n n n n n n n n n n n n n n n p p p p λλλλλμλμμμμμμ-+--++++=+-== .(9) If 10

11n n n n n C λλλμμμ-+= , n 0,1,2,= then the distribution of the steady state:

0=n n p c p .By the probability distribution requirements

01n n p ∞==∑, so 01[1]p 1

n n C ∞=+=∑,then 0111n n p C

∞==+∑. Attention: Only the serial 1n

n C

∞=∑is convergent ，that is 1n n C

∞=<∞∑.We can get the probability distribution of the

steady state from above .Thurs, in light traffic situations ,even there are several overtaking situations, they will not waste too much time, we can regard that the car drive fluently .So we can conclude that the rules are helpful in improving the traffic flow .

4.3Collection of Data

As for multi –lane and security coefficient date, we use the intelligent vehicle detection based on video technology. With the rapid development of intelligent transportation, vehicle detection based on video technology has become a famous research subject all over the world .The vehicle detection based on video technology got rapid development because of the advantages of wide detection range and it can provide the comprehensive traffic information . Intelligent transportation system requires real-time detection, in the process of testing in order to reduce the amount of calculation and improve the computing speed of the system, many algorithms are

adopted the way of setting detecting area, that is, in the video image just set the part of the detection area, it will ensure the accuracy of detection to improve the operation speed of the system.

The reported average annual daily traffic volumes on selected Interstate highways are given in Exhibit 8-18. Most of these high-volume freeways are found in the largest metropolitan areas. Daily traffic volumes on these heavily used highways exceed200, 000 veh/day. Exhibit 8-19 contains a sample of the maximum reported hourly one-way volumes and the average volumes per lane on rural and urban freeways in the United States. Most volumes in this table exceed 2,000 veh/ h/ln, with several freeways featuring average lane volumes of more than 2,400 veh/ h/ln. The highest reported lane volumes on selected freeways are given in Exhibit 8-20.Freeway capacity analysis procedures of this manual use a rate of flow of 2,400pc/h/ln for freeways with free-flow speeds of 70 to 75 mi/h and 2,300 pc /h/ln for freeways with free-flow speeds of 65 mi/h as the capacity under base conditions. Exhibit 8-19 contains observations of values higher than this standard, but these are the maximums reported on a given freeway and are not expected to be achieved on most other freeway segments [11].

Two-lane, two-way rural highways in the United States and Canada rarely operate at volumes approaching capacity, and thus the observation of capacity operations for such highways in the field is difficult. A sampling of high-volume observations is given in Exhibit 8-22, but it is emphasized that none may be taken to represent capacity for the facilities shown. Observations on two-lane, two-way rural highways in Europe have been reported at far higher volumes. Volumes of more than 2, 700 veh per hour have been observed in Denmark, more than 2,800 in France, more than 3,000 in Japan, and more than 2,450 in Norway. Some of these volumes have contained significant numbers of trucks, some as high as 30 percent of the traffic stream.

4.4 Analysis of the Queuing Theory Model

（1）In different lanes ,the flow is different ,the type of the car is different ,and the flow is changing all the time;

（2）When cars overtaking ,they need change the lane and will back to the previous lane after they do it, it will waste much time and effect the maximum traffic capacity ; （3）If it is light traffic ,we should limit the minimum speed .

4.5 Verification and Improvement of the Model

In this question ,under the light traffic condition ,the right lane will be crowded，we decided to use the queuing theory to solve this problem .Queuing is the normal phenomenon in our lives and it requires the number of service is larger than the capacity of service system. In this question, that is, arrival customer cannot get service im mediately, so they should queue. The line?s busy in telephone station ,bus station ,wharfs and other transportation junctions ?crowd ,and broken machines, all of them problems ?solving are about queuing phenomenon . This picture is a schematic diagram of our experiment:

Schematic diagram of queuing theory

According to the light traffic model ,the flow in middle and inner lane account for 90% among all lanes ，in test 1,in order to ensure cars get though the cross section at exit ,all crowded cars need to change the lanes once or twice，the twice one account for 35%among them, except the waste time ,the deceleration and cars ?waiting time will rise up . Except flow ,different type of cars is one of the reasons which make the difference of the highway real transportation capacity ,according to the ―The Keep-Right-Except-To-Pass Rule‖ ，the speed is increasing in outside lane ,middle lane and inner lane .Though the model we established, wecan analysis that the rule can help improve the flow under light traffic situation. As for the security, just think that if drivers drive cars casually without a rule, the accident will be increasing definitely.

4.6Further Discussion about Flow and Rate

In order to seeing the influence of ―The Keep-Right-Except-To-Pass Rule ‖ clearly ，adopted the ― floating car method ‖record the flow at a specific distance in a street. The method has two parts :(1) using float car to record the speed and time, the float car drives as fast as the car flow, this method need not accuracy equipment, so we cannot get the accuracy information either .This method can also be divided into two parts: people in the car record the speed and time or using the speed indicator;(2)This method can record speed and flow at the same time ,it is suitable in the fluent street and the highway without automatic testing instrument . This method

is based on observation cars drive on the road back and forth, the equations are as follows:

q ()/()x y t t αω=++ (10) /t t y q ω=-

(11)

/s u l t = (12)

Fix the time before testing ,the car should drive depends this time ,it allows stop at the side of the road .According to the national urban transport committee ,at the main road 19min/km ，at second road 6min/km ，it usually allows return and back 12～16times ，then we can get the result .Additional, the turn of the vehicle will affect the result, so, in this survey we should avoid the main entrance and exit.

5. Keep-Right-Except-To-Pass Rule in Heavy Traffic

5.1Backgrounds

In heavy traffic, through careful observation and experiments, we can find that if we limit cars in a certain section of the road, then the high density of traffic can reduce the speed of the car. So from this we can know that in high traffic, The Keep-Right-Except-To-Pass Rule cannot greater promote the traffic flow. However, this is just a intuitionistic and shallow conclusion that we got according to our daily experience. So we need further scientific experiments to prove that and get a reliable conclusion. The following picture shows the performance of The Keep-Right-Except-To-Pass Rule. In heavy traffic, all the lanes may be occupied by cars, so even the cars in the left lanes have to follow this rule, in other words, the drivers are trying to drive into the right lane.

Picture3 the performance of The Keep-Right-Except-To-Pass Rule

At the same time, in heavy traffic, we need to limit the speed of cars to ensure the safety. We know that the condition of the road is more complex than in light traffic, so if we was asked to follow the Keep-Right-Except-To-Pass Rule all the time (regardless of the road condition), then the probability of traffic jam will rise, and it will also influence the safety. In order to solve this problem deeply and precisely, we should develop an appropriate mathematical model to show the performance of The Keep-Right-Except-To-Pass Rule, and compare the traffic flow with safety.

5.2 Tradeoffs between Traffic Flow and Safety

5.2.1 Backgrounds

In 1967, GWYnn first began to study the relationship between traffic flow and safety. He studied the relationship between traffic accidents and traffic flow (per hour) when he evaluated the safety of the four-lane road in New Jersey. He believed that when, the traffic volume is small, the accident rate is high; however, when the traffic volume reaches a certain value, it will get the lowest casualty ratio of accident. In other words, the relationship between accident rate and traffic flow can be shown as a U-bend. Following GWYnn.Ceder(1982), Franrzeskaki(1987), Pendelton(1989) and Martin(2002)made further studies of the issue and enhance the conclusion. Cederetal(1982)and Martin (2002)have studied the relationship between traffic accidents (single car accident and multi vehicle accident) and traffic flow (per hour), and proved that the relationship between them can also be shown as a U-bend. Figure 1.1 shows the relationship between them. In single car accident, traffic volume is smaller, accident rat is higher. And with the increase of traffic volume, the accident rate reduces gradually.

Picture4

Garber and Golob T. F(2001) studied relationships among traffic accidents, the state of traffic flow, weather condition and lighting condition. In different condition of weather and lighting, they divided into traffic flow parameters into speed, speed difference, flow and flow difference, and analyze them in different kind of combinations, and provided theoretical basis for the daily safety management of highway.

Nicholas J.，Garber and Angeka A.Ehrhart studied the relationship among speed, traffic flow, the shape of the lane and different kind of traffic accidents, and developed different kinds of models in different traffic flow. They proved that with the increase of the speed discretely, the accident rate will always rise in any

conditions of traffic flow. And the influence of traffic flow and average speed on accident rate will change according to the change of lane types.

Canadian scientist Dominique Lord et al. (2005) got the relationship between traffic accident and traffic density as well as V/C. They are showed in Figure 1.2.Cirillstudied the condition of highway and rural way in both daytime and night. He proved the conclusion of Solomon, and perfected the curve which showed the relationship between speed discrete size and accident rate. It is showed in figure 1.3.Through the picture, we can see that when the accident rate is lowest, the speed discrete size is bigger than 0 km/h，the speed become biger, the discrete accident rate become higher.

Picture 1.4 showed that，the accident rate increases with the increase of speed standard deviation. We can ignore the time error, when the number of the overtaking is small. But when the number of the overtaking increase, the time error will play a more important role on it. At the same time, the speed of cars will be reduced, and safety will become lower correspondingly[11], so the keep-right rule is far from suitable in this condition.

5.5 Models

5.5.1Analusis of the Initial Model

Queuing Theory is also called the theory of random service system，and is designed to solved this kind of problems. We mainly using queuing theory to get the Preliminary statistics and inference of inward and outward flow, the length of the queues and waiting time in trial section. Later, when we describe the performance of the Keep-Right-Except-To-Pass Rule light and heavy traffic, we used SPSS software to analyze the two sets of data and find the significant difference between them, at last, we explain the difference by using Queuing Theory.

In the study of the road performance in high traffic，we find that the effect of the Keep-Right-Except-To-Pass Rule is not significant. So a new program (based on the

initial model) must be put forward to solve the problem. Our final target is to develop a suitable model to greater promote traffic flow and safety.

Picture 5

5.5.2 Cellular Automation Theory (CA)

（1）Definition

A：Cellular Automation Theory is a kind of dynamical system which space and time are all discrete.

B：Each cell that spreads in Lattice Grid has a limited discrete state, following a same reaction rules and occurring synchronous renewal according the same partial rule. A large number of cell through simple interaction, constituting the evolution of the dynamic system.

C：The characteristics of the CA model: time, space, status are all scattered, each variable has limited multiple states, at the same time the rules of the state change in time and space are partial. It is composed of a series of rules which used to make model construction. As long as the models meet these rules then they can be called automata model. Therefore, Cellular Automata is a general designation of a kind of model, or a framework of a method.

（2）The neighbor of Cellular and boundary conditions

A: Boundary conditions

Theoretically，the space of the cellular can be expended infinitely, and this is good for studies and inference, the in practice, we cannot realize the ideal condition. So, to sum up, there are three kinds of boundary conditions-- Periodic Boundary, Constant Boundary and Reflective Boundary). Something that needs to be particularly point out is that in practice, we can combined the three kinds of Boundary conditions together. For example, in the two-dimensional space, the Upper and Lower boundaries usd Reflective Boundary, and the Right and Left boundaries use Periodic Boundary. The Cellular is far from neighbor only shows the Static components. In order to introduce dynamic into the system, the evolution rules must be added. In Cellular Automation, these rules are define d in partial space, that is to say, at next period of time a Cellar‘s state can determine its own state and the state of the neighboring Cellular. See the following picture: