Highway Toll Plaza Design for Uncontrolled & Controlled Lane Merging

Transcription

1 For office use only T1 T2 T3 T4 Team Control Number Problem Chosen B 2017 MCM/ICM Summary Sheet For office use only F1 F2 F3 F4 Highway Toll Plaza Design for Uncontrolled & Controlled Lane Merging Summary Delays at toll stations are among the main concerns for highway efficiency. It is essential to smartly design the toll station configuration, in order to reduce latency, congestions, and accidents. The thesis mainly focuses on the situation where tool barriers are more than traffic lanes, and there is a lane merging zone (toll plaza) at the exit. We discuss two different type of lane merging systems: uncontrolled merging and controlled merging, and successfully build models to solve the Toll Plaza Design problem for both systems. For uncontrolled merging system, we develop a Cellular Automaton (CA) based on Nasch model, to simulate vehicle s behavior. With simulation results, the shape of toll plaza could be determined intuitively. To our best knowledge, the idea of designing shape according to simulation is quite novel. We have tested our method in a variety of circumstances, including different toll booth types and traffic flow intensity, and most of them received good effects. For controlled merging system, we define a cost function which combined indexes in several aspects, in order to determine the best merging pattern that minimize this function. However, the explicit expression of this function was difficult to figure out. To handle this issue, we utilize Vissim (a professional software for traffic simulation) simulation to produce these indexes, and then compute the cost function with these indexes. In order to build applicable model, we made some assumptions to simplify the situation. Therefore, we didn t concern enough about some real-life behaviour of drivers. Future works may focus on this issue and make improvement. Keywords: Cellular Automaton, Nasch model, Cost function, Vissim.

2 Team # Page 1 of 22 Contents 1 Introduction Background Our works Notations and Assumptions Terminology Symbol Description General Assumptions Overview of Our Models Uncontrolled Merging Controlled Merging Model A: Shape Design Based on Cellular Automaton Treading a Path on the Lawn: An Inspiration Cellular Automata for Simulation Setting the Rules for CA Designing Plaza Shape by Simulation Results Generalization to Complicated Circumstances Throughput: Light or Heavy Taking Financial Expense into Consideration Self-driving Cars and Mixed Types of Tollbooths Model B: Merging Pattern Design Based on an Evaluation System Lane Junctions and Average Delay Input/Output Ratio Accident Risk Comparison Between Merging Patterns Validating Model B with Vissim 13

3 Team # Page 2 of A Brief Introduction to Vissim Intuitive Observation of Traffic Condition Data Acquired From Simulation Discussion and Conclusion Strengths Weaknesses Sensitivity Analysis Appendices 19 Appendix A Source Code 19 Appendix B Latency Distribution 22

4 Team # Page 3 of 22 1 Introduction 1.1 Background Most people who often drive on highways have encountered toll station congestions. When reaching the station, you not only have to slow down the car in advance, but also to spend extra time paying tolls. That would delay the speed of traffic flow to a certain extent. Although it s impossible to eliminate toll stations, we can greatly reduce their negative effects by improving design of toll stations. Considering its slow down effect, most toll stations are designed to have more tollbooths than traffic lanes. There is also a toll plaza following the booths, where egress lanes merge into traffic lanes. The following picture could exactly illustrate that kind of configuration. Figure 1: A toll station 1.2 Our works It is true that this configuration could reduce delayed time on average, but the process of merging lanes also increases the possibility of traffic accidents, which may in turn cause heavier congestions. Our works balance the throughput, expense and accident prevention, and give out an optimal design of the shape, size and merging pattern of toll station plazas. We develop two different models to deal with different ways of merging, namely, uncontrolled merging and controlled merging. In both models, we firstly assume that all vehicles are of the same type, and all tollbooths have the same working rate. After that, we adjust our design to more complicated circumstances, for example, whether the

5 Team # Page 4 of 22 traffic is busy or not, whether self-driving vehicles are joining the traffic flow, and with different proportions of human-staffed, automatic and electronic toll collection booths. 2 Notations and Assumptions 2.1 Terminology Tollbooth egress lane: the lanes that extend from tollbooths. Traffic lane: the lanes that vehicles run in main section of highway. Inside lane: the tollbooth egress lanes that extend directly to traffic lanes. Outside lane: the tollbooth egress lanes that end in a finite distance, and finally merge into inside lanes. Maximum driving speed: the highest speed allowed for vehicles on highway. Merging pattern: how the tollbooth egress lane merge into traffic lane (can be expressed by a tree). For example, 8 lanes merging into 3 has ( 7 2) = 21 different patterns, such as 3-2-3, 4-2-2, 3-4-1, etc. 2.2 Symbol Description Table 1: Notations Symbol B L v max p λ r T v ij τ ρ γ Description Number of tollbooths Number of traffic lanes Maximum driving speed Emerging probability in each time step Poisson arrival rate Pivot percentage Throughput of traffic flow Velocity of the jth vehicle on the i lane Average delay Ratio between the intensity input and output traffic flow Accident risk

6 Team # Page 5 of General Assumptions We abstract the problem condition into the following easy-to-handle assumptions. ASMP 1 The road is straight and without bend. ASMP 2 The width of each lane is only enough for passing one vehicle. ASMP 3 Despite the influence of bad weather, pedestrians and unexpected driving behaviors. ASMP 4 Driving on the right is norm, so inside lanes are on the left and outside lanes on the right. ASMP 5 The process that vehicles emerge from each tollbooth is a Poisson process, namely the amount of vehicles emerging in a period of t have distribution P (N(t + s) N(s) = n) = (λt)n e λt ( s, t) n! (λ is the same on each lane). 3 Overview of Our Models We developed two different models for this problem, with different assumptions about how vehicles would change their lanes. 3.1 Uncontrolled Merging Model A works under the cases that vehicles can change their lanes freely, without causing any accidents. In such circumstances, the main concern for designing toll plaza is its shape, which means the vehicles on outside lanes must transfer to inside lanes before reaching the boundary. Since the free merging behavior is not easy to model, we use Cellular Automaton to simulate how vehicles change lanes, and design shape of toll plaza accordingly. 3.2 Controlled Merging Model B works under the cases that the merging pattern of lanes have already been designed and fixed, so that vehicles could only follow the lanes. In such circumstances, the main concern for designing toll plaza is the merging pattern. Development of this model is based on Stochastic Process theory. We build

7 Team # Page 6 of 22 up an evaluation system for every merging pattern, and choose the best pattern by comparison. (a) Uncontrolled merging (b) Controlled merging In order to simplify our models, the assumption that all cars and tollbooths are equal should be admitted at first, before we generalize our models to more complicated circumstances. 4 Model A: Shape Design Based on Cellular Automaton 4.1 Treading a Path on the Lawn: An Inspiration Suppose that you want to build a pedestrian path on the lawn, how will you do that? A simple and valid method is to let people tread on the lawn as they will, and design the path according to their footprints. By such method, your design must be the shortest and most comfortable one, because it is fully based on the user s need. This idea also gives us an inspiration on solving this shape-design problem: why can t we let the vehicles just run on the road and change lanes freely, and design the toll plaza according to the tracks? As long as we the vehicles are able to avoid accidents and ensure that their driving could finally enter traffic lanes, their tracks could give us a wonderful guidance for designing the shape of toll plaza. 4.2 Cellular Automata for Simulation It is not possible for us to do such experiments in practical, but we can simulate them by computer programs. Previous researches such as [Wagner P et

8 Team # Page 7 of 22 al.2005] has revealed that Cellular Automaton(CA) is a useful model for simulating traffic flow. A CA consists of a regular grid of cells, each in one of a finite number of states, such as on and off. CA is easy to conduct. When setting the rules how each cell change its status, CA could simulate the development of a whole traffic system. There are hundreds of different rules for Cellular Automata, among them Rule 184 best suits our requirements. German physicists Nagel and Schreckenberg developed a model based on this rule, which is called Nasch model in brief. There are mainly four rules in Nasch model: Acceleration, Slowing down, Randomization and Car motion, and our CA inherits most of them. 4.3 Setting the Rules for CA In our model, we assume that all vehicles can change lanes freely at any time, so we replaced the Randomization rule by two Lane changing rules. The status of whole system is updated after each time step. Here are the exact rules: 1. Lane subdivision: There are B lanes in total (tollbooth egress lane), and each lane consists of a regular grid of 1000 cells, and the size of each cell is 4m*4m. 2. Vehicles emerging: After an unit time interval, a vehicle come out from each tollbooth with probability p (p reflects the intensity of traffic flow). 3. Car motion: each cell have a status v ij, which represent the speed of vehicle on it (0 means no vehicle on it). During every time step, the vehicle move a distance of v ij forward, and transfer the status to correspond cell. 4. Acceleration: if there is no other vehicles within d ahead, the vehicle would accelerate to maximum speed, namely v ij = min {v ij + a, v max } 5. Slow down: if there is other vehicle within d ahead, the vehicle would slow down, namely v ij = min {v ij, d 1} 6. Lane changing: vehicles change lane in two cases For vehicles on outside lanes, if there is no other vehicles within v ahead and v max backward on the left lane, the vehicle will change lane to the left. For vehicles on all lanes, if the present lane meets slow down condition and one of the nearby (left or right) lane meets the first lane changing condition, the vehicle will change lane to such lane. Left lane enjoy priority over right lane.

9 Team # Page 8 of 22 It should be noticed that the vehicle emerging rule is in accordance with ASMP 5. We have the following well-known theorem: Theorem 1. If np n λ, then B(n, p n ) D P (λ) (n ) With this theorem, ASMP 5 could be be seen as the limit of "vehicle emerging" rule as time step approach to zero. 4.4 Designing Plaza Shape by Simulation Results Taking B = 8, L = 3, p = 0.35, and iterate for 200 time steps, we achieve the following gray scale image by recording how many times each cell has been passed by vehicles (just like footprints on the lawn): Figure 2: Track of vehicles In this image, each row represents a tollbooth egress lane, and the 3 upper lanes are traffic lanes. Vehicles emerge from the left size and move to the right. The gray scale of each cell indicates how many vehicles have passed on it in this period of time. By finding the pivot cell according to the percentage r (here r = 60%), we design the shape of plaza as follow: Figure 3: Plaza Shape Design Here are more simulation results by changing B and L (p = 0.35), and its not hard to design the shape similarly. (a) B=4,L=2 (b) B=6,L=3

10 Team # Page 9 of 22 (c) B=5,L=2 (d) B=7,L=3 (e) B=6,L=2 (f) B=9,L=3 (The MATLAB source code is in Appendix) 5 Generalization to Complicated Circumstances 5.1 Throughput: Light or Heavy It is reasonable to define throughput as the expectation of passing vehicles during time period t, namely T E[ B X i (t)] where {X i (t)} n i=1 are independent Poisson processes. E[X i ] = n=0 i=1 n (λt)n e λt = λt T = Bλt n! Therefore, the arrival rate λ can fully reflect the throughput of a toll station. In model A, p is the discrete version of λ, and by setting different p (B=8,L=3), we can get different shape designs. (g) p=0.1 (h) p=0.2 (i) p=0.4 (j) p=0.6

11 Team # Page 10 of Taking Financial Expense into Consideration For the same group of parameters B,L,p, the simulation results are similar, but with different financial budgets, we can make different design. In normal sense, the financial expense of building the plaza is proportional to its area, namely, the sum of every lane s length. In our model, the lengths are determined by percentage r, so by changing it, we can moderate the shape of plaza according to expected cost. For example, still taking B=8, L=3 and p=0.35, here are the results when r = 40%, 60%, 80% (yellow area stand for the toll plaza): Figure 4: r=40% Figure 5: r=60% Figure 6: r=80% 5.3 Self-driving Cars and Mixed Types of Tollbooths Nowadays, rapid development of self-driving and Electronic Toll Collection(ETC) technology promotes corresponding improvement in traffic system facilities. In many toll stations, in addition to traditional manual charges, there are automatic tolls and electronic toll devices, which greatly improve the rate of charge. In order to deal with that cases, we adjust our model and set different emerging rate p on different tollbooths. For example, suppose B=8,L=3, and 8 tollbooths are divided into 3 types:

12 Team # Page 11 of 22 The 1st and 2nd are electronic booths and only work for self-driving vehicles, where p = 0.5. The 3rd, 4th, 5th, and 6th are automatic booths working for human drove vehicles, where p = 0.3. The 7th and 8th are human-staffed booths, which work at a low rate p = 0.1. The simulation result and design are as follows (r = 60%): Figure 7: Design for mixed condition 6 Model B: Merging Pattern Design Based on an E- valuation System In this model, We build up a evaluation system for merging patterns, and then compare different patterns to choose the best one according to this evaluation system. We consider three major indexes: average delay (τ), input/output ratio (ρ) and accident risk (γ). 6.1 Lane Junctions and Average Delay Consider a two-to-one lane junction, where two egress lanes merge into one. Previous researches such as [Jacob Tsao et al.1997] has modeled and analyzed the distribution of delay during lane changing based on ASMP 5 (Poisson process), and we apply these analysis to our stochastic model. When a vehicle arrive at the junction, if we ignore the width of road, its distance to the vehicle ahead is a random variable d j Exp( λ v j ), because P (d j > d) = P (N( d v j ) = 0) = e λd v j Suppose that the latency t j = α d j, and v j U(0, v max ) is independent to d j. When v j is given, distribution of t j could be computed, and the average delay is a double expectation. τ = E[t j ] = E[E[t j v j ]]

13 Team # Page 12 of 22 Figure 8: Junctions Similar analysis could be applied to three-to-one junctions and even k-to-1 junctions. However, explicit expression of the average delay is hard to compute, so we don t show it here. 6.2 Input/Output Ratio When the traffic flow is light, average delay is a good indicator of the negative influence caused by lane changes on the traffic. However, heavy traffic may cause congestion around the junctions, in which case the average delay is not enough to reflect the influence. Therefore, we add the ratio ρ = outputflow to inputflow estimating this influence. 6.3 Accident Risk In a k-to-1 junction, suppose in each time step δt, each input lane have possibility p of vehicle arrival, and if more than one vehicles arrive simultaneously, each of them has possibility p c of driving ahead carelessly (ignoring other the existence vehicles). Unfortunately, if more than two cars ignore each other, then a crash is likely to occur. So we define the accident risk as γ k i=2 i j=2 ( k i )( i j ) p i (1 p) k i p j c(1 p c ) i j (Derivative of this formula borrowed some idea from [Zhang Ning.2005]) 6.4 Comparison Between Merging Patterns Cost function is the combination of the indexes above, which we aim to minimize. Considering that ρ and γ would have cumulative effect on traffic condition,

14 Team # Page 13 of 22 we define the cost function as follow: F (τ, ρ, γ) τe (1 ρ)+γ2 γ is quadratic on the exponential, because we think that accident should be taken seriously in traffic system design. Theoretically, for every merging pattern, we can derive the cost function by calculating the three indexes. However it is too difficult to calculate and compare for us. Therefore we choose to validate that model with Vissim, which means we achieve the key indexes from simulation, instead of frustration hand computation. 7 Validating Model B with Vissim 7.1 A Brief Introduction to Vissim Vissim is a microscopic multi-modal traffic flow simulation software package developed by PTV Planung Transport Verkehr AG. With the support of multiple microsimulation libraries, Vissim gives access to realistic modeling of a variety of transport conditions. Based on CF (Car Following) model, Vissim performs quite remarkable in this relatively complicated situation, with highly parameterized depiction of practical transports, which keeps the simulation simple since all what needs to be concerned is the accuracy and rationality of the parameters, including the length and width of each lane, the velocity of every individual vehicle, the safe distance between two vehicles, the deceleration when road merging occurs ahead, etc. 7.2 Intuitive Observation of Traffic Condition Using Vissim, we can observe vivid traffic condition, either light or heavy. Figure 9: Light traffic condition Figure 10: Heavy traffic condition

15 Team # Page 14 of Data Acquired From Simulation Simulation seconds: 600s, Vehicle velocity: 15km/h, assume that accident risks are the same. The major indexes of 4 different patterns under light and heavy traffic are listed in the following table, and the cost function is clearly computed in it. Table 2: Indexes from simulation Merging pattern Mode heavy light heavy light heavy light heavy light Input Output Ratio ρ 85.03% 95.21% 84.16% 95.21% 84.16% 95.21% 82.56% 95.21% Average delayτ 20.51s s 20.96s s 24.51s s 29.96s s Cost function By comparing the cost function, we find that 233 merging pattern is the best, among the four patterns. From the table we can also infer that 4-to-1 junction is not inspired by our evaluation system, if it could be replaced by 2-to-1 and 3-to-1 junctions. 8 Discussion and Conclusion 8.1 Strengths Intuitive design based on real situation Using Cellular Automata to simulate the driving behavior of vehicles out of the toll station, the Monte Carlo method is used to simulate the process of changing lane spontaneously. This approach is more in line with the real situation, and ensure better results for our design. The use of grayscale images also make it intuitive to see the distribution of vehicles in long term. Effective simulation The strategy of parallel updating is adopted, which can effectively simulate the influence of local changes (such as deceleration) on the overall traffic flow. Combining theory with simulation In our second model, we built an evaluation system to distinguish different merging patterns. Although it s too difficult to compute the result directly, the system can guide us to deal with the data we got from simulation results. In that way, our results would be more convincable.

16 Team # Page 15 of Weaknesses Regardless of some driving behavior The cellular automata are based on the Nasch model and simple assumptions on merging paths that do not cover all driving behavior in the real driving situation, nor can it cope with unexpected conditions outside the criterion Simplification put it away from real cases Not considering the different size, speed, and acceleration of the vehicles, and in actual highway, different lanes have different functions and driving rules. We assumed many things to be equal, which put our model distance away from real cases. 8.3 Sensitivity Analysis We take several factors in consideration, including traffic density, toll booth type, accident occurrence. Among them, toll booth type have a direct impact on the car s density when the traffic volume is large enough. According to the cellular automaton simulation results, the traffic density directly affects the fan-out area to set the shape (please refer to data shown in section 5). Generally speaking, results of our model indicated that traffic flow changes are stable and sensitive. Our first model did not take the accident conditions into account when designing the guidelines, and therefore did not design emergency response lanes and other elements, thus is not good at dealing with traffic accidents. Our second model take the cost function as F (τ, ρ, γ) = τe (1 ρ)+γ2, so it is obvious that the result is most sensitive to the accident risk γ, less sensitive to input/output ratio ρ, and least sensitive to average delay τ. Especially, when γ is large or ρ is small, the model is unstable. That is quite reasonable, because when congestion or accident rate is high, the traffic system is likely to be paralyzed.

17 Team # Page 16 of 22 Letter to New Jersey Turnpike Authority To whom it may concern, I am on behalf of us three enthusiastic undergraduates that apply ourselves to solving traffic problems such as annoying congestion and unfortunate accidents with advanced mathematical methods, which we exactly major in. The day before, we happened to hitch a ride from Jersey City to Linden via New Jersey Turnpike, bumping into a badly congested traffic especially at exits of toll plazas when every vehicle trying to leave the jam as fast as possible. That spoiled us a lot. So we decided to do something, instead of leaving it alone. (a) New Jersey Turnpike (b) A typical toll plaza We analyzed every possible reason that could lead to the frequent congestion on the Turnpike with the help of the Internet. Apparently the causes are multiple, but one of them, the design of toll barrier did capture our attraction. Since the Turnpike is a major thoroughfare providing access to various localities in New Jersey, as well as Delaware, Pennsylvania, and New York, vehicle flow rate is quite colossal. This means that toll barriers are really indispensable, but how to choose the size and shape is indeed interesting matter. Generally, toll plazas usually look similar with the above picture a lot, it is a good choice but not the best, actually our analysis proves that this shape will hamper the traffic flowing to some extent. To avoid this, we ve devoted an advanced method to simulating this situation, whose main principle is letting all vehicles freely choose lanes with spontaneity, of course each one moving with specific order (just same as practical driving). After testing enough times, we only need to draw a proper contour as the toll plaza boundary. The following picture is one of simulation results, where the blue polyline is exactly the boundary.

18 Team # Page 17 of 22 Design of toll plaza This method realizes two essential goals: Less congestion More cost-effective Needless to say it is really a practical and convenient approach since this simulation doesn t cost much time and isn t sensitive to parameters, which means that the method we speak highly of is accessible for all situations! To conclude, I would like to acknowledge that it is a great pleasure that we can have an opportunity to depict our method with pride. I sincerely hope that our studies can be helpful in the highway arrangement. Look forward to your favorable reply! Cordially, Team #56510

19 Team # Page 18 of 22 References [1] Cellular Automaton, Wikipedia. [2] Nagel-Schrechenberg model, Wikipedia. [3] Wagner P, Nagel K, Wolf D E. Realistic multi-lane traffic rules for cellular automata[j]. Physica A: Statistical Mechanics and its Applications, 1997,234(3): [4] Models and Simulations of Traffic System Based on the Theory of Cellular Automaton. Science Press. [5] Jacob Tsao, Randolph W Hall, Indrajit Chatterjee. Analytical Models for Vehicle/Gap Distribution on Automated Highway Systems. Transportation Science, /97/ [6] Sabastiaan Vermeulen. Modeling road traffic flow with queueing theory. Bachelor Project, Universiteit van Amsterdam, [7] Zhang Ning. The Semi-Markov model for capacity of highway with merging two lanes into one lane. System Engineering-theory & Practice, 2005,25(7): [8] Lin, W. and H. Liu (2010) Enhancing Realism in Modeling Merge Junctions in Models for System Optimal Dynamic Traffic Assignment. IEEE Transactions on Intelligent Transportation Systems, 11(4), [9] Sheldon M. Ross (1996) Stochastic Processes, Second Edition. UC Berkley.

20 Team # Page 19 of 22 Appendices Appendix A Source Code Matlab source for Cellular Automata: clc; close all; %GUI setup plotbutton=uicontrol( style, pushbutton, string, Run,... fontsize,12, position,[100,400,50,20], callback, run=1; ); erasebutton=uicontrol( style, pushbutton, string, Stop,... fontsize,12, position,[200,400,50,20], callback, freeze=1; ); plazabutton=uicontrol( style, pushbutton, string, Plaza,... fontsize,12, position,[300,400,50,20], callback, design=1; ); quitbutton=uicontrol( style, pushbutton, string, Quit,... fontsize,12, position,[400,400,50,20], callback, stop=1;close; ); number = uicontrol( style, text, string, 1, fontsize,... 12, position,[20,400,50,20]); %CA setup n = 30; vmax = 3; L = 3; B = 8; cells = zeros(b, n); z = cells; v = cells; destiny = zeros(b, n + vmax); p = 0.35; r = 0.4; run = 0;%wait for start stop = 0;%state of quit freeze = 0;%state of stop design = 0;%state of design imh=imshow(cells); set(imh, erasemode, none ) axis equal axis tight while (stop == 0) if run == 1 for i = 1 : B %open border if rand <= p z(i, 1) = 1; v(i, 1) = 1; destiny(i, 1) = destiny(i, 1) + 1; for j = 1 : n if cells(i, j) == 1 %outside line if i > L for k1 = j + 1 : n if cells(i - 1, k1) ==1 break; for k2 = j : -1 : 1 if cells(i - 1, k2) ==1 break;

21 Team # Page 20 of 22 d1 = k1 - j; d2 = j - k2; if ((v(i, j) <= d1-1) k1 == n) &... ((d2-1 >= vmax) k2 == 1) z(i - 1, j + v(i, j)) = 1; v(i - 1, j + v(i, j)) = v(i, j); destiny(i - 1, j + 1 : j + v(i, j)) =... destiny(i - 1, j + 1 : j + v(i, j)) + 1; else for k = j + 1 : n if cells(i, k) == 1 break; d = k - j; if d - 1 > v(i, j) k == n v1 = min(v(i, j) + 1, vmax); else v1 = min(d - 1, v(i, j)); if v1 > 0 z(i, j + v1) = 1; v(i, j + v1) = v1; destiny(i, j + 1 : j + v1) =... destiny(i, j + 1 : j + v1) + 1; elseif i > 1 for k = j + 1 : n if cells(i, k) == 1 break; d = k - j; if ((d - 1) > v(i, j)) (k == n) v1 = min(v(i, j) + 1, vmax); z(i, j + v1) = 1; v(i, j + v1) = v1; destiny(i, j + 1 : j + v1) =... destiny(i, j + 1 : j + v1) + 1; else for k1 = j + 1 : n if cells(i - 1, k1) == 1 break; if k1 > k z(i - 1, j + v(i, j)) = 1; v(i - 1, j + v(i, j)) = v(i, j); destiny(i - 1, j + 1 : j + v(i, j)) =... destiny(i - 1, j + 1 : j + v(i, j)) + 1; else v1 = min(d - 1, v(i, j)); if v1 > 0 z(i, j + v1) = 1; v(i, j + v1) = v1; destiny(i, j + 1 : j + v1) =... destiny(i, j + 1 : j + v1) + 1;

22 Team # Page 21 of 22 elseif i == 1 for k = j + 1 : n if cells(i, k) == 1 break; d = k - j; if d - 1 > v(i, j) k == n v1 = min(v(i, j) + 1, vmax); else v1 = min(d - 1, v(i, j)); if v1 > 0 z(i, j + v1) = 1; v(i, j + v1) = v1; destiny(i, j + 1 : j + v1) =... destiny(i, j + 1 : j + v1) + 1; z(i, j) = 0; v(i, j) = 0; cells = z; set(imh, cdata,cells(1 : B, 1 : n)) %update image pause(0.001); stepnumber = 1 + str2double(get(number, string )); set(number, string,num2str(stepnumber)) if freeze == 1 run = 0; freeze = 0; im2 = imshow(1-1.0 * destiny / max(max(destiny))); if design == 1 run = 0; freeze = 0; plaza = zeros(b,n); for i=1:l for j=1:n plaza(i,j)=1; end end for i=l:b sum = 0; for j=1:n sum = sum + destiny(i,j); end total = sum; sum = 0; for j=1:n if(sum<r*total) plaza(i,j) = 1; end sum = sum + destiny(i,j); end end im2 = imshow(plaza);

23 Team # Page 22 of 22 drawnow Appendix B Latency Distribution From Vissim simulation, we get the distribution of vehicle s latency: (c) heavy (d) light (e) heavy (f) light (g) heavy (h) light (i) heavy (j) light