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Summary
The key to indictors to evaluate a traffic rule is the traffic flow and the safety index. The traffic flow is the number of passing vehicles per unit timeand the safety index is the ratio of vehicles without accident. However, it’s impossible to maximize these two indexes at the same time. To keep the balance, we define efficiency as the ratio of the average value and the upper bound of the traffic flow as the criterion of a traffic rule. Higher efficiency reflects higher traffic flow and more space to ensure safety.
In the two-lane freeway automata model we built, two lanes are regarded as two ribbon regions and the vehicle as a small rectangle in the two regions, and a unidimensional coordinate system is established on the lane. The state of each vehicle includes coordinate, length, lane, instantaneous velocity and maximum velocity. Each vehicle travels in a certain rule (narrated in the following two paragraphs). We can get the efficiency by calculating the average value and the upper bound of the traffic flowin the rule by making use of two following rules. Because the analytical solution is difficult to be obtained, the automata model is solved by computer simulation and numerical methods.
The Keep-Right-Except-To-Pass Rule is employed by multi-lane freeways, which requires vehicles to keep right unless they are overtaking. If vehicles intend to overtake, on the premise of ensuring safety, they move to the left, pass, and return to their former lane. By computer simulation, it is shown that the efficiency is higher in light traffic; whereas the efficiency is lower in heavy traffic due to the waste of space and loss of velocity.
In order to increase the efficiency in heavy traffic, we put forward the No-Overtaking Rule. That is, vehicles are divided into high-speed and low-speed vehicles according to the critical velocity calculated by maximum velocity and minimum velocity,the overtaking lane and the main lane in the keep-right rule are used as the high-speed lane and the low-speed lane, and overtaking is not allowed. A vehicle should travel on the high-speed lane if its maximum velocity is greater than the critical velocity. By computer simulation, the efficiency of the No-Overtaking Rule is 1.5-2 times of that of the original rule in heavy traffic. Thus, we suggest an alternative: the Keep-Right-Except-To-Pass Rule is adopted in light traffic and the No-Overtaking Rule is adopted in heavy traffic.
On interchanging the high-speed lane and the low-speed lane and changing left-hand cars to the right-hand cars, the solution can be carried over in countries where driving automobiles on the left by symmetry. In an intelligent system, all velocitiesare controlled by computers. So the safety index increase greatly, then the safety distance in the model could decrease.By computer simulation, we know that the efficiency and the safety distance are negative correlation, and it can be increased in the system.
Contents
Summary......
1Introduction......
2Problem Restatement......
3Notations......
4Part I......
4.1 Model I: the Traffic Flow in the Keep-Right-Except-To-Pass Rule
4.2 Solutions to Model I
5Part II......
5.1 Model II: the Traffic Flow in the No-Overtaking Rule
5.2 Solutions of Model II
6Sensitivity Analysis......
6.1 The Influence of Vehicle Rates on Efficiency
6.2 The Influence of the Safety Distance on Efficiency
6.3 Model Changes in Other Cases
7Strengths and Weaknesses......
7.1 Strengths
7.2 Weaknesses
References......
The Keep-Right-Except-To-Pass Rule
1Introduction
A correct rule has a significant influence on promoting better traffic flow. In most countries, when a vehicle travels on multi-lane freeways, the driver must obey the keep-right-except-pass rule that requires drivers to drive in the right-most lane unless they are passing another vehicle. Considering the keep-right-except-pass rule, designers from 66 percent countries in the world make driver’s seat on the left, in order to ensure the driver can keep control of the steering wheel with his left hand, and use the right hand to finish some complex movements such as making a turn and operating dashboard in the meantime. What’smore, when driving on the right, the driver can keep an on the left if he intends to overtake. And if the driver keeps left, he needs an assistant beside him, causing the waste of human. In other words, therule broadens the driver’s horizon then avoids traffic accidents in a way
This thesis aims at developing a model that can evaluate the rationality of the rule.
2Problem Restatement
Develop a model to analyze the traffic flow in keep-right-except-pass rule and judge if it is effective in promoting better traffic flow. In order to improve the rule, we put forward our own rules and argue whether or not our solution can be carried over with changes and additional requirements in countries where driving automobiles on the left is the norm. Take traffic flow, safety, the role of under- or over-posted speed limits and the thought of the drivers into consideration, we analyze the problem in the condition of heavy and light traffic.
The model is based on the following assumptions:
(1)Keeping safe and within the speed limit, all drivers obey the keep-right-except-pass rule and endeavor to speed up in order to save time.
(2)Travelling vehicles are in good condition, if broken, there is an emergency parking strip to make the vehicle behind travel normally.
(3)Lanes are straight.
(4)Each side has only two lanes (except the emergency parking strip), the road conditions can ensure the steady travelling of vehicle.
(5)The initial velocity is the maximum.
(6)When moving one lane to another, vehicles keep the uniform velocity.
(7)The accelerated velocity is infinite, which means the velocity saltation exists. In fact, this assumption makes a little difference on the result.
(8)Vehicles are divided into light vehicles, mid-size vehicles and heavy vehicles, whose length makes difference.
3Notations
Table 1Notations and Description
Notation / Description / Unit/ Average traffic flow in a period of time / /min
/ The upper bound of average traffic flow in a period of time / /min
/ The efficiency of the traffic flow in a certain rule in a period of time / (none)
/ The length of the certain section of a freeway / m
/ The total amount of vehicles passing in a period of time / (none)
/ The number of every vehicle / (none)
/ The vertical position of the th vehicle / (none)
/ The coordinate of the th vehicle on freeway / m
/ The length of the th vehicle / m
/ The instantaneous velocity of the th vehicle / m/s
/ The maximum velocity of the th vehicle / m/s
/ The distance between theth vehicle and theth vehicle / m
/ Safety distance / m
/ Influence distance / m
/ An interval between each two circulation processes run by computer / s
/ The time lasting from the time when the first vehicle passing through the certain cross section / min
/ The time of lane changes / s
Here are four points to supplement in Table 1:
1. ∈, ∈, and the other belong to .
2. The maximum velocity of the th vehicle is the minimum among the maximum velocity of the vehicle per se,the speed limit and the maximum velocity under the drivers’ expectation.
3. The relations of somenotations are listed:
(1)
(2)
(3)
4. Compared with the heavy vehicles, the velocity permitted of light vehicles is higher but the weight decreases. That is to say, we can unify the safety distance as regardless of the size.
4Part I
4.1 Model I: the Traffic Flow in the Keep-Right-Except-To-Pass Rule
4.1.1 The Evaluation Criterion of the Model
We consider the performance of the traffic flow in different traffic. In a certain condition including some factors related to traffic flow, such as the trade-offs between traffic flow and safety, the role of under- or over-posted speed limits, the rate of different vehicles, the thought of the drivers, the vehicle conditions and the average interval between a vehicle enters a certain section of a freeway and the next one does, we defineas the ratio of the average traffic flowand its upper boundto evaluate the rationality of the model. In other words, we can compare the traffic flow by comparingunder different rules. The larger is, the better the rule is.
4.1.2 The Representing Method of “Two Lanes”
In order to simplify the model, we regard two lanes(the right one is called main lane and the left one is called overtaking lane) as two ribbon regions and regard a section of lanes whose length is(equals to 1000m) as two rectangles whose length is. Taking the top of the line segment for origin, we establish a unidimensional coordinate whose positive direction is opposite to the direction of the traffic flow. The coordinate of the bottom of the rectangle is 1000(m). We regard the th vehicle as a small rectangle lying on the two rectangles or across them. The coordinate of the vehicle front isand the coordinate of the vehicle rear is, which means the length of the small rectangle is. The small rectangle moves upwards at the velocity of,as the following Figure 1. (In fact,,we reduce the difference value of them in order to be clear.)
4.1.3 Making Rule for Travelling Vehicle According to the Assumptions
Taking the th vehicle for example, we can list 3 cases as follows:
1.. The th vehicle is on the right lane, which is influenced by the th,th, th vehicles. Their relationship is as follows
(4)
The th vehicle is in front of the th vehicle. The th vehicle is in front of the th vehicle but on another lane. The th vehicle is behind the th vehicle but on another lane. Besides, means that the vehicle occupies two lanes in the same time. The positions of th,th, th vehicles relative to the th vehicle and the velocity of them will decide whether the driver of the th vehicle can move to the overtaking lane successfully. The schematic diagram is Figure 2. The shadow region indicates the region where the th vehicle passes. Based on the assumption (6), the shape of shadow region is a polygon.
There are three cases when the th vehicle travelling.
(1). The distance between theth vehicle and the th vehicle (simplified into two vehicles in (2)(3)) is shorter than safety distance. The th vehicle slows down for safety (We can make an assumption ). In fact, this is an important reason why the efficiency reduces.
(2). The distance between the two vehicles is moderate. If (the coefficient of is determined to be greater than 1 ,which means the difference between the velocity of the th vehicle and the th vehicle should be big enough), the vehicle should move to the overtaking lane when they meet the following requirements.
(i)While the vehicle is moving to the left lane, it can’t make traffic collision with the th vehicle, that is to say, the small rectangle whose length is can’t overlap the left part of the shadow region after s. So here comes an inequation
(5)
(ii)After the vehicle moves to the left lane, it must still keep the safety distance with the th vehicle. Because it travels on a different lane with the th vehicle, thesafety distance can change into .So here comes an inequation
(6)
(iii)After the vehicle moves to the left lane, it must keep the safety distance with the th vehicle. Because it travels on the same lane with the th vehicle, thesafety distance is still .So we can get an inequation
(7)
In conclusion, when, the vehicle can move tothe overtaking lane if and only if
(8)
While the vehicle is moving to the overtaking lane,.
Whenvehicles don’t meet the requirements above, then .
(3). The distance between the two vehicles is so long that there is no need to overtake. Based on the assumption (1), the vehicle ought to travel at themaximum velocity ().
2.. The th vehicle is between the two lanes. Based on the assumption(6), the vehicle should keep uniform velocity while moving to the other lane. After the process lasting s,. (Because we have considered safety problem under the case 1 or 3, and case 2 is between 1 and 3,so safety problem doesn’t need to be considered.)
3.. The th vehicle is on the left lane, which is influenced by the th,th, th vehicles. Their relationship is as follows
(9)
The th vehicle is in front of the th vehicle. The th vehicle is in front of the th vehicle but on another lane. The th vehicle is behind the th vehicle but on another lane. The positions of th,th, th vehicles relative to the th vehicle and the velocity of them will decide if the driver of the th vehicle can move to the right lane successfully. The schematic diagram is Figure 3. The shadow region indicates the region where the th vehicle passes. Based on the assumption (6), the shape of shadow region is a polygon.
There are two cases when the th vehicle travelling.
(1). The distance between theth vehicle and the th vehicle (simplified into two vehicles in (2)(3)) is shorter than safety distance. The th vehicle slows down for safety. (We can make an assumption) In fact, this is an important reason why the efficiency reduces.
(2) .The distance between the two vehicles is long enough. Based on the Keep-Right-Except-To-Pass Rule, the vehicle should move to the main lane when they meet the following requirements.
(i)While the vehicle is moving to the main lane, it can’t make traffic collision with the th vehicle, that is to say, the small rectangle whose length is can’t overlap the right side of the parallelogram after s. So here comes an inequation
(10)
(ii)After the vehicle moves to the main lane, it must still keep the safety distance with the th vehicle. Because it travels on a different lane with the th vehicle, the safety distance can change into .So here comes an inequation
(11)
(iii)After the vehicle moves to the main lane, it must keep the safety distance with the th vehicle. But we change the safety distance into to let vehicles move to the main lane easier.So we can get an inequation
(12)
In conclusion, when , the vehicle can move to the main lane if and only if
(13)
While the vehicle is moving to the main lane,.
When their vehicles don’t meet the requirement above, then .
4.1.4 The Expression of
is the average traffic flow in a period of time at the origin in the unidimensional coordinate, which is defined as the ratio of the amount of vehicles that pass through the origin and the time lasting from the time when the first vehicle passing through the origin. That is
(14)
Ignoring the influence among the vehicles and the accident, we assume the velocity of each vehicle is to make the traffic flow maximum. We regard the condition as an ideal condition, which is nearly non-existent in real life except traffic capacity is strong enough. Even so we can assure the upper bound of called , which is defined as the ratio of the amount of vehicles who pass through the origin and the time lasting from the time when the first vehicle passing through the origin in the ideal condition. That is
(15)
The expression of the efficiency is
(16)
measures the traffic flow in the condition of the same traffic load. Considering the trade-offs between traffic flow and safety, we should take measures (enlarge ) to reduce the traffic flow in order to increase the safety index. In a way, can evaluate the rationality of the rule.
4.2 Solutions to Model I
There is an amount of variables whose relationship is complicated, so the analytical solution is hard to get. The model is solved by computersimulationas as follows.
4.2.1 Initialization of Vehicle Parameter
To be practical, we assume that there are three sizes of vehicles including light vehicles,mid-size vehicles and heavy vehicles. Initial parameters include the rate of each size (we set their values as, whose relationship is ), the length of the vehicle (we set their values as), the maximum velocity permitted (a random number obeying the normal distribution ,is given), the interval between the appearance of adjacent vehicles (is a random number and means the interval between the th vehicle and the th, and ,is a given parameter of the exponential distribution), , and.Parameters above are given before the computer simulation[1].
4.2.2 Simulate the Circumstance by Computer
The computer simulates the circumstance according to the rules as follows.
1. Rand a number referred in 4.2.1, the initial amount of vehicles is 0, that is, after , , .
2.For , the th vehicle obeys the rule in 4.1.3( keep changing in the travelling process). And , is an interval between each two circulation processes run by computer.
3. When the first vehicle passes through the origin, the computer calculates according to the expression in 4.1.4.
4. When keeps steady, we regard as the approximatevalueof the current efficiency.
4.2.3 Result of Circumstance Simulation
The values of parameters are given as follows: