1、For office use only T1 _ T2 _ T3 _ T4 _ Team Control Number 25142 Problem Chosen A For office use only F1 _ F2 _ F3 _ F4 _ 2014 Mathematical Contest in Modeling (MCM/ICM) Summary Sheet Freeway Traffic Model Based on Cellular Automata and Monte-Carlo Method Summary Based on Cellular Automata and Mont
2、e-Carlo method, we build a model to discuss the influence of the “Keep right except to pass ” rule. First we break down the process of vehicle movement and establish corresponding sub-models, inflow model for car-generation, vehicle-following model for one vehicle following another, and overtaking m
3、odel for one vehicle passing another. Then we design rules to simulate the movement of vehicles in sub-models. We further discuss rules for our model to adapt to the keep-right situation, the unrestricted situation, and the situation where transportation is controlled by intelligent system. We also
4、design a formula to evaluate the danger index of the road. We simulate the traffic on two-lane freeway (two lanes per direction, four lanes in total), and three-lane freeway (three lanes per direction, six lanes in total) via computer and analyze the data. We record the average velocity, overtaking
5、rate, road density and danger index and assess the performance of the keep-right rule by comparison with the unrestricted rule. We vary the upper speed limitations to analyze the sensitivity of the model and see the impacts of different upper speed limits. Left-hand traffic is also discussed. Based
6、on our analysis, we come up with a new rule combining the two existing rules (the keep-right rule and the unrestricted rule) for an intelligent system to achieve better performance. Team # 25142 Page 2 of 34 Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7、4 1.1 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2 The Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.1 Design of Cellular Automata . . . . . . . . . . . .
8、 . . . . . . . . . 6 2.2 Inow Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.3 Vehicle-Following Model . . . . . . . . . . . . . . . . . . . . . . . . 8 2.4 Overtaking Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.4.1 Overtaking Probability . . . . . . . .
9、 . . . . . . . . . . . . . 10 2.4.2 Overtaking Condition . . . . . . . . . . . . . . . . . . . . . 10 2.4.3 Danger Index . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.5 Two Sets of Rules for CA Model . . . . . . . . . . . . . . . . . . . . 13 2.5.1 Keep Right Except to Pass Rule . . . . .
10、 . . . . . . . . . . . 13 2.5.2 Unrestricted Rule . . . . . . . . . . . . . . . . . . . . . . . . 14 3 Supplementary Analysis on the Model . . . . . . . . . . . . . . . . . . 14 3.1 Design of the Acceleration and Deceleration Probability Distribu- tions . . . . . . . . . . . . . . . . . . . . . . .
11、. . . . . . . . . . . . . 14 3.2 Design to Avoid Collision . . . . . . . . . . . . . . . . . . . . . . . 14 4 Model Implementation with Computer . . . . . . . . . . . . . . . . . . 15 5 Data Analysis and Model Validation . . . . . . . . . . . . . . . . . . . 16 5.1 Average Velocity . . . . . . . . .
12、 . . . . . . . . . . . . . . . . . . . 16Team # 25142 Page 3 of 34 5.2 Average Velocity of Fast Cars . . . . . . . . . . . . . . . . . . . . . 17 5.3 Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 5.4 Overtaking Rate . . . . . . . . . . . . . . . . . . . . . . . . . .
13、 . . . 19 5.5 Danger Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 6 Sensitivity Evaluation of the Model under Different Speed Limitations 20 7 Driving on the Left . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 8 Transportation under Intelligent System . . . .
14、. . . . . . . . . . . . . 21 8.1 New Rule for Intelligent System . . . . . . . . . . . . . . . . . . . . 21 8.2 Adaption of the Model . . . . . . . . . . . . . . . . . . . . . . . . . 22 8.3 Result of Intelligent System . . . . . . . . . . . . . . . . . . . . . . 23 9 Conclusions . . . . . . . . . .
15、 . . . . . . . . . . . . . . . . . . . . . . . . 23 10 Strengths and Weaknesses . . . . . . . . . . . . . . . . . . . . . . . . . 25 10.1 Strengths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 10.2 Weakness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
16、References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26Team # 25142 Page 4 of 34 1 Introduction Today, about 65% of the worlds population live in countries with right-hand trafc and
17、35% in countries with left-hand trafc. worldstandards.eu,2013 In countries with right-hand trafc, like USA and China, regulations request driv- ing and walking keep to the right side of the road. Multi-lane freeways in these countries often employ a rule that requires drivers to drive in the right-m
18、ost 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. This rule on driving and overtaking is referred to as the ”Keep right except to pass” rule, or the keep- right rule in our paper. The rule in countries with l
19、eft-hand trafc is exactly mirror symmetrical to the keep-right rule(”Keep left except to pass”). So, whats the purpose of applying such a rule? Does the keep-right rule ameliorate the freeway trafc condition? Transportation free of the restriction of the keep-right rule (Vehicles can choose either s
20、ide for overtaking.) is referred to as obeying the unrestricted rule. How does the keep-right rule perform comparing with the unrestricted rule? Based on the Cellular Automata model and the Monte Carlo algorithm, we establish a model to simulate freeway trafc under different conditions (under the ke
21、ep-right rule or the unrestricted rule, in light trafc or in heavy trafc, 2-lane or 3-lane per direction). Our model is divided into 3 sub-models the in- ow model, the vehicle-following model and the overtaking model. The inow model employs the Poisson probability distribution for the simulation of
22、the vehicle-generation process. The vehicle-following model introduces a special probability distribution model which makes the simulation of the process of a car following another more realistic. The overtaking model simulates the over- taking behavior and denes the danger index to evaluate the saf
23、ety risk of a certain freeway. We also build an extended model for transportation under the control of an intelligent system. We implement the model in MATLAB, and obtain sufcient data. We test the average velocity, the density, the overtaking rate and the danger index, analyze their properties and
24、assess the performance of the keep-right rule by comparison with the unrestricted rule. In addition, we analyze the sensitivity of our model under different speed limits. It turns out that our model is robust. Then we come to our conclusions which consist with common sense. We also put forward a new
25、 rule for transportation under the control of an intelligent system.Team # 25142 Page 5 of 34 Table 1: Notation Symbol Meaning V current velocity of the vehicle V m maximum velocity of the vehicle V l the upper speed limit of the freeway V 0 the velocity before overtaking V 1 the velocity in the ove
26、rtaking process G the distance between a vehicle and the vehicle ahead of it G s the minimum gap required for safety consideration G 0 the minimum gap after the vehicle stops T r PIEV time(human reaction time) P o the overtaking probability P a the acceleration probability P b the deceleration proba
27、bility f the frictional force when braking d danger index in one overtaking event D danger index of the road system a the acceleration during overtaking a p the component parallel to the lane of acceleration during overtaking a d the available deceleration 1.1 Terminology Two-lane road: Two lanes on
28、 right-half of the road, four lanes in total. Three-lane road: Three lanes on right-half of the road, six lanes in total. Danger index: An index designed in our paper to evaluate the danger of the road system. Minimum safety gap: The distance between two vehicles that deemed safe enough in our model
29、. Keep-right rule:Keep right except to pass rule. Unrestricted rule:Vehicles are not restricted and can overtake others from either side. Free-driving style:When there is no vehicles nearby, drivers will not accel- erate or decelerate deliberately,but the speed will still uctuate slightly. 1.2 Assum
30、ptions The road is straight and there is no bypass.Team # 25142 Page 6 of 34 The width of one lane is only enough for one vehicle. All vehicles have the same volume. There are only two kinds of vehicles on the road(fast one and slow one). The environment and climate are good for driving. Driving on
31、the right is the norm. Pedestrians are ignored. 2 The Models 2.1 Design of Cellular Automata Large quantities of former trafc simulations Wagner P et al.2005 based on Cellular Automata (CA) indicate that CA model is a feasible and effective method to emulate trafc ow. Space, time and status are all
32、discrete in Cellu- lar Automata. For example, the model divides the road into small rectangular cells, and time is divided into small units. This feature predigests the simulation process signicantly. Besides, the status of a cell is controlled by its neighboring cells following a set of rules, whic
33、h is much similar to real-life trafc where a cars movement largely depends on its neighboring cars movement. Therefore, it is rational for us to apply Cellular Automata in solving our problem. In our simulation, we divide each lane into 1000 cells. Each cell is 4 meters long in length and width and
34、has two properties, the current velocityV and the maximum velocity V m . A cell is empty when V is 0, because a car wont stop in our crash-free simulation. We consider only one direction of the freeway for simplicity. Thus, a freeway of n lanes is converted into an n*1000 matrix. In our simulation,
35、we employ two kinds of vehicles,faster ones to simulate the cars and slower ones to simulate the trucks. For each lane, the rst 6 cells are used as car-generation area, trafc ow is observed in the last 10 cells and trafc density is calculated on the basis of the last 500 cells.Our model updates once
36、 per second, while the periodT = 1s is the average reaction time of a driver. We discuss the basic processes for the CA model: Inow Process: According to the inow model that we will discuss later, assign vehicles in the vehicle-generation area.Team # 25142 Page 7 of 34 Acceleration Process: IfV G s
37、(V 0 ). ( G s (V 0 ) is the minimum gap required for safety consideration, and is to be dened later.) Specic rules will be set in the inow model, the following model and the overtaking model to simulate trafc under the Keep-Right-Except-To-Pass rule and trafc under no such restriction. 2.2 Inow Mode
38、l The inow model, or the vehicle-generation model, simulates the stochastic arrivals of vehicles at the entrance of the freeway. For each lane, the rst six cells in the cellular automata are set as vehicle-generation area. We assume that the arrival of each vehicle obeys the binomial probability dis
39、tribution. Lett s denote the sampling time interval andN denote the total of vehicle arrivals duringt s . Then N can be approximated to obey the Poisson probability distribution. Let P ts (N) be the probability ofN and we have P ts (N) = N N! e ; N 0 Witht s being one second in our implementation, w
40、e can assign the expecta- tion ofN to a range of values from 0 to 3.6. N being the total of vehicle arrivals in each second, the expectation of N, can effectively reect the trafc condition. The smaller the is, the lighter the trafc is; the greater, the heavier. Thus we are able to simulate different
41、 trafc conditions, light or heavy, by assigning cor- responding values to . After the value of is set, we get the stochastic number of vehicles entering the freeway for every second in simulation. Which lane to enter is then randomly assigned. Our model supports two kinds of vehicles of different ve
42、locity ranges, the initial speed of all vehicles are set to 20 m/s. Such practice brings simplication and doesnt weaken the result.Team # 25142 Page 8 of 34 That is because the speed of all vehicles tends to converge toward a value controlled by the trafc density and by the distribution of accelerat
43、ion proba- bility which is to be introduced later. When trafc density is low, vehicles can always accelerate freely to the maximum speed without worrying about colli- sions, so the convergence speed is near the highest speed allowed. When trafc density is high, all the lanes will be lled with vehicl
44、es, and the speed of the trafc ow is decided by the speed of the slowest vehicle on the lane, so the con- vergence speed is near the lower speed limit. The preliminary analysis on the convergence speed will be justied by later implementation of the model. The utilization of the Poisson probability d
45、istribution makes our inow mod- el close to reality and practical. Because of the convergence tendency, the same initial speed policy can yield simplication without harm to the simulation. 2.3 Vehicle-Following Model The Federal Highway Administration of the United States Department of Transportatio
46、n denes, in its Manual on Uniform Trafc Control Devices, dene drivers reaction time as PIEV time. PIEV time consists of four parts: Perception process: A driver perceives the change in driving environment. Intellection process: The driver analyzes the information about the change. Evaluation process
47、: The driver determines driving behavior based on his analysis. Volition process: The driver executes the driving behavior. We apply the PIEV process in our vehicle-following model and overtaking model. In every time cycle, we rst obtain the velocity and position of each vehicle, calculate the gap,
48、and then determine the driving behavior (whether to continue following or to change lane for overtaking). According to driving behavior, compute the acceleration and update the speed and position. Decision on driving behavior based primarily on the current gap. If the gap G is safe enough, accelerat
49、ion is feasible; otherwise, the vehicle should slow down. Here, we dene the minimum safe gapG s to beT r V . (T r stands for the PIEV time, andV is the current velocity.) We assume that decisions on driving behavior follow certain principles:Team # 25142 Page 9 of 34 When G G s , the vehicle tends to accelerate (Later we will introduce a probability model to simu
