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1 Team # Page 1 of 2 When analyzing the performace of the Right-Lane rule, we approached this problem using both statistics and systems of differential equations. We modeled each driver as a particle, assumed each driver was only affected by desire for safety, the fear of the law, and personal desire, and assumed an infinitely long road to let the system adequately develop in time. Statistically, two main factors go into a traffic system: safety and flux. We considered the flux and safety as functions of speed and spatial densities, which we quantified by discretizing the area around a driver. We created Gaussian functions to model the speed and spatial densities over a stretch of roadway. Once we had these distributions, created simulations that model how each driver reacts to what happens around him. We considered factors such as the speed limit, desired speed, safety, and passing options. We used differential equations to describe the flow of traffic relative to nearby cars and wrote an algorithm that mimics the thought process involved in passing and merging. We ran simulations for both One Lane and Two Lane roads, showing how the capabilities of passing greatly increase the speed density and thus the flux score. Using both the spatial and speed densities, we came up with a statistically stable method for scoring the flux and safety, making sure to prioritize safety over speed. We came up with a final evaluation function that combines the two to give a general effect of the Right- Lane rule. Upon applying this method to urban and rural areas, we determined that the Right-Lane rule is beneficial to urban areas but harmful to rural areas.

2 Right-Lane Rule in a Left-Wing World Using statistical scoring and differential simulations to model traffic flow Team # February 1, 214 Abstract When describing a large system with many parameters and facets, diving into the microscopic details without a general idea about the system is sure to cause frustration. Trying to understand an overall behavior from the combination of each event or action requires a strong imagination and intesnsive computational power. Thus we first turn to a smoother, more general description of the system. We will apply statistical and probabalistic methods in the modelling of traffic flow. Inspired by the success of using such method to predict the fate of a particle or a whole system in quantum mechanics and thermodynamics, this paper presents an analogous treatment to the evaluation of traffic flow. We use the ideas of speed expecation and density expectation to predict how the Right-Lane rule affects traffic dynamics. Through calculating a safety score and flux score, we are able to evaluate the affects the Right-Lane rule has on traffic. Futhermore, we apply our general model to a microscopic simulation, wherein we consider each driver as a particle in our system and model the movements in a large system of differential equations. Through interative simulations, we can see the dynamics of each particle, and given enough time, how the the whole system behaves as a whole. 2

3 Team # Page 3 of 2 Contents 1 Introduction Prompt Assumptions Summary of Notation Optimization Models Analysis of the Problem General Approach: Gaussian Distribution Speed and Spacial Density from Gaussians Density Normal Distributions Speed Normal Distribution Simulations and Microscopic Modelling One Lane Simulations Lone Driver Multiple Drivers in One Lane Traffic Flow in Two Lanes Passing and Merging Algorithms Simulation: Six Cars in Two Lanes Evaluation: Traffic Systems with Speed and Density Expectations Flux Score Safety Score Overall Evaluation Score Conclusion: Considering the Scores of Different Systems 18

4 Team # Page 4 of 2 1 Introduction 1.1 Prompt In countries where driving automobiles on the right is the rule, multi-lane freeways often employ 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. Build and analyze a mathematical model to analyze the performance of this rule in light and heavy traffic. 1.2 Assumptions To account for all the physical limitations and uncertainties involved in traffic flow, we make several assumptions as follows: (A1) Every vehicle is a particle (A2) The road vehicles are driving on is infinitely long (A3) Each individual driver is only affected by three things: safety, the speed limit, and personal feelings 1.3 Summary of Notation u ū v v v n v m, L Φ(u) φ(u) Ψ(v) ψ(v) F (ū, v) S(ū, v) E(ū, v) V ar( ) H(x) x a b s spacial density(or density) spacial density expectation speed speed expectation stable speed number density speed limit in an area distribution of spacial density normalized distribution of spacial density distribution of speed normalized distribution of speed flux score safety score final evaluation score variance Heaviside function position fear of the law intensity of feelings safety reaction coefficient

5 Team # Page 5 of 2 2 Optimization Models 2.1 Analysis of the Problem The two main factors that go into a traffic system are speed and safety. These can be analyzed and compared by assigning them a flux score and a safety score, where flux score describes how many vechicles pass through an area in time, and safety score describes the likelihood of accidents and fatalities. Let us begin with the flux score. Following the Right Lane rule, a group of drivers will reorder themselves so that given enough time on an infinitely long road, the fastest vehicle will be first, followed by the second and third, and so on. In this case, the speed expectation of the system is maximized. Considering a traffic flow where the Right Lane rule is not follwed, not everyone can drive at their maximum speeds (or stable speeds) since slower drivers can block faster drivers. However, the density expectation wil be in this case. Since both speed density and flux density affect the traffic flow, we must turn to quanitative calculations in order to come up with a flux score. We are, however, able to gain some intuition about the safety score by looking at the following data: Figure 1: Motor Vehicle Traffic Fatalities, by Year and Location, Note that only 19 percent of U.S. population lived in rural areas, but rural fatalities accounted for 55 percent of all traffic fatalities in 211. Intuitively, people tend to obey to the drive to the right rule in urban areas and drive spread out in rural areas. The advantage of the Right-Lane rule is to maximize speed and decrease density. But the rural areas with a smaller spatial density suffer a higher traffic fatalities! It seems that speeding is more likely to play a larger role in traffic accidents. If so, then maximizing speed adversly affects safety 2.2 General Approach: Gaussian Distribution We will form a macroscopic model of our system using a Gaussian functions of the form: F (x) = Ce λ(x x)2, (1) where C, λ and x are positive constants that affect the shape of the Gauassian funtion, as shown in Figure 2.

6 Team # Page 6 of 2 Figure 2: Different Shapes of Gaussian functions To determine the values of the parameters, we will use a Gaussian function normalized on our domain from to. To calculate the normalized Gaussian function, use the following formula: f(x) = F (x) F (x)dx and for the expectation of x on the domain we are interested in, we use the statistical definition: x = (2) xf(x)dx (3) Because we are also interested in the stability of our model, we will use the statistical variance defined as: V ar(x) = x Speed and Spacial Density from Gaussians Density Normal Distributions x 2 f(x)dx (4) Before going any further, we will specify the definition of density. Let density mean the spacial density a vehicle encounters while driving on a road, and let us denote it with the letter u. To gain a quantitative understanding, we will partition each single lane into blocks of the same size. Each block can fit one and only one vehicle. It is not possible to describe the spacial density by the number of vehicles occupying a set of neighboring blocks. Figure 3 shows two examples of calculating such density, where the crossed green block is our reference vehicle and the red blocks are neighboring vehicles. By observation, the minimum density is u = and the maximum is u max = 2, 5, 8 for one-lane, two-lane, and three-or-more-lane cases. Let us now describe the distribution of u by Gaussian function Φ(u) = C d e λ d(u u ) 2 (5)

7 Team # Page 7 of 2 Figure 3: Quantized Spacial Density u C d will canceled during normalization, so we set it to 1. After a series of tests, λ d = 1 is found to be reasonable with FWHM 1. To find the center of distribution u, we want to introduce the concept of number density n. The number density of a certain area is calculated by dividing the number of vehicles with the total number of blocks: n = number of vehicles number of blocks in the area For example, in a certain two-lane case shown below: (6) Figure 4: Number Density n the result is simply n = 3/1 =.3. In reality, the number of vehicles will be the number of vehicles running in the area at a certain time, and the total number of blocks will depend on how fine the lanes are partitioned. Notice the value of n will never be larger than 1. We can use it as probability reference of encountering the maximum density. In other words, we multiply n with u max to get the center of density distribution. With u = nu max the sample numerical expression of the density distribution becomes: Φ(u) = e 1(u numax)2 (7) When n =.1, and for the two-lane case u max = 5, it takes the form shown in Figure 5. Figure 5: Density Distribution Φ(u)

8 Team # Page 8 of 2 To normalize the density distribution, we use a formula with same form as (2): φ(u) = Φ(u) Φ(u)du Refering to (3) and (4), the expectation and variance of density is described by: (8) and ū = uφ(u)du (9) V ar(u) = ū Speed Normal Distribution u 2 φ(u)du (1) A larger speed limit usually means drivers will drive faster. However, not all will drive the same speed, as they are affected by both the speed limit and how they feel like driving. We divide the drivers into two classes: those with a lower comfortable speed and those with a higher comfortable speed. If we consider the center of speed distributions for these two types of driver, when the speed limit increases, we find that the center of the second class is going to shift faster than the first class. Therefore we need two terms in the speed distribution formula, denoted as Ψ(v): Ψ(v) = C s e λs(v vs (vm))2 + C f e λ f (v v f (vm))2 (11) C s and C f describe the population ratio of two types of drivers and λ s and λ f describe the width of Gaussian functions. v m is the speed limit and v s and v f are functions of v m that limit the shift rate of the center of the corresponding distributions (slower drivers and faster drivers), as seen in Figure 6. Figure 6: Separation of Gaussians So far, we have not considered the affects of spacial density. When surrounded by other vehicles, a driver tends to driver slower. We account for this phenomenon with a third term in our speed distribution: Ψ(v, ū) = C s e λs(v vs (vm))2 + C f e λ f (v v f (vm))2 + C u e λu(v vu (ū))2 (12) The third term smooths out the separation between faster and slower drivers and drags down the overall speed expectation.

9 Team # Page 9 of 2 Now we want to figure out the constants. First we concentrate on the first two terms, for simplicity, assume λ s = λ f =.1, v s = vm.9, and v f = v m. We will, however, use real data to determine the population ratio. According to NHTSA[2] Age Total Percentage 57% 5% 42% 27% 24% 13% 3% Figure 7: Percentage of Drivers Who Tend to Drive Above Speed Limit From Figure 7, we can roughly estimate C s =.7 and C f =.3. Thus, the sample numerical speed distribution without the spacial density term will be Ψ(v) =.7e.1(v v.9 m )2 +.3e.1(v vm)2 (13) Let us now consider the spacial density. Because we consider it is a general effect on all drivers, we set C u = 1. Because the slower drivers have a more dominant effect on the system, the center of the third term distribution will be set with the reference to the center of slower drivers. Thus we define the following relation v u = (1 ū 1 )vs (v m ) = (1 ū 1 )v.9 m (14) where ū never exceeds 8 by its definition. An increase in density expectation will results 1% decrease in the center of the density weight on the speed distribution. This is reasonable if we consider the density on a road hardly exceeds 2 except during traffic jams. Finally the λ u term is treated with statistical method while considering it to be the total variation of both faster and slower drivers: 1 = ( 1 ) λ u λ 2 + ( 1 ) s λ 2 (15) s With the previous assumption λ s = λ f =.1, we obtain λ u.7. Combining the result we get from the speed densities and spacial densiites, we obtain a sample numerical speed distribution formula: Ψ(v) =.7e.1(v v.9 m )2 +.3e.1(v vm)2 + e.7(v (1 ū)v.9 m )2 (16) With speed limit v m = 8 units and density expectation ū =.5. We normalize the function according to the following formula: ψ(v) = which yields the result we see in Figure 8. Ψ(v, ū) Ψ(v, ū)dv, (17) With these distributions defined, we can calculate the expectation of speed by and the speed variation by v = vψ(v, ū)dv (18) vm V ar(v) = v 2 + v 2 ψ(v, ū)dv. (19)

10 Team # Page 1 of 2 Figure 8: Change from Ψ(v) to Ψ(v, ū) 3 Simulations and Microscopic Modelling Now that we have equations describing the overall behavior of our system, let us turn to microscopic models and simulations. We will use our statisticaly conclusions to determine the desired speed of each individual driver compared to the speed limit. 3.1 One Lane Simulations Lone Driver We will first analyze how we drive when we are the only ones on the road. While driving, we are aware of several factors, such as road conditions and construction, personal safety and feelings, pedestrians, speed limits and posted signage, etc. For simplicity, let us assume that we are driving on a well paved open highway. This allows us to rule out road conditions, construction, and pedestrians. Thus, we are only considering safety, the speed limit, and our own feelings (that is, how fast or slow we want to drive). Since there are no other drivers on the road, personal saftey is not a concern here there are no other drivers to collide with. Thus our driving will be controlled entirely by the speed limit, our personal feelings, and the awareness or weight we give to each of them. Every driver can be modeled as a combination between the two forces: a fear of the law that pulls us towards the speed limit and the intensity of our feelings which pulls us toward our desired speed. We will resolve in some middle ground between the two that depends on the weights we give to each attracting force. We can model this phenomenon with the differential equation v = a(l v) + b(v v) (2) where L is the posted speed limit, V is how fast we feel like driving, a is our attention or fear of the law, and b is the intensity of our feelings. This equation produces a direction field shown in Figure 9. As we can see, no matter our initial speed, we will always converge to a stable speed. The stable point for (2) can be deteremined by the following equation: v = Using the parameters from Figure 9, we see al + bv a + b. (21)

11 Team # Page 11 of 2 Figure 9: Direction field for (1) with parameters L = 35, V = 45, a =.4218, and b =.7922, where a and b are randomly determined using a Gaussian generator. v =.4218(35) (45) which corresponds with the stability we see in the direction field Multiple Drivers in One Lane = , (22) Deriving System of Equations Let us now add another driver to our open road. Upon encountering this new driver, we are suddenly plauged with a new concern: personal safety. Having two drivers on the same road allows for possible collisions. The best way to avoid collisions is to maintain a safe distance, so we will slow down. Mathematically, this acts as a reaction force that pulls us away from our stable speed. If the driver in front of us applies his brakes, we will react according to some minimal safe distance between us. That is, if the distance between us falls below some fixed minimum distance, we will slow down. The intensity or weight of the reaction will depend on each individual driver. This adds a new element to our differential equation, namely ẍ i = a(l x i ) + b(v x i ) s(x i 1 x i D), (23) where x (distance) is the time integral of velocity, s is the intensity of our reaction, and D is some minimum safe distance. Here we have indexed x to indicate the relation between cars that are next to each other. (Note: the minimum safe distance is commonly said to be 1 car length per every ten miles per hour you are travelling. Thus, D = (carlength)/1 x i. We can select units such that this simplifies to D = x i.) This new safety term acts as a balancing force, maintaining the distance D between us and the driver in front of us. However, this has the adverse effect of also pulling us forward

12 Team # Page 12 of 2 x 1 v 1 x 2 v 2. x Ṅ v N when the driver speeds up. Since we only want this function to work one way, we will multiply it by another function to turn it off when we don t need it. Thus, our differential equation becomes ẍ i = a(l x i ) + b(v x i ) s(x i 1 x i x i ) H( x i + x i x i 1 ), (24) where H is the Heaviside or unit step function defined as { 1 x H(x) = x < Remember that the lead car is not affected by this term. There is no psychological pressure from cars behind to driver faster; thus the lead car still behaves according to (2). We will generalize this model first by converting (23) into a system of differential equations. If we let v i = x i, then v i = ẍ i = a(l v i )+b(v v i ) s(x i 1 x i v i ) H(v i +x i x i 1 ). Thus, a system of N cars can be written as a 2N 2N matrix = 1 a 1 b 1 1 s 2 H s 2 H a 2 b 2 s 2 H a n b n x 1 v 1 x 2 v 2. x N v N + (25) a 1 L + b 1 V 1 a 2 L + b 2 V 2. a n L + b N V N (26) where we have indexed a, b, s, and V, but there is no need to index L because the speed limit does not change per each driver. We have only written H for the heaviside function to avoid redundancy. While this matrix may look overwhelming, the patterns along each diagonal allow it to be easily built. Simulation: Starting From a Stoplight Our initial assumptions about driving led us to only consider three ideas: the speed limit, personal feelings, and concern for safety. Using these assumptions, we derived a system of differential equations to describe how each driver s position and velocity change in time. Let us now use this system to simulate traffic flow in one lane. Imagine that six cars are stopped at a stoplight; their velocities are zero and each car is within some small distance from the car in front (1 units in this example). The posted speed limit is 8, and each driver feels like driving within some Gaussian distribution of the speed limit, according to the previous section. The parameters a, b, and s were created using a Gaussian generator, where extra weight was applied to the safety coefficient in order to prioritize safety and avoid collisions. The system is integrated with a time step of.1 through 24 interations with a custom built 4th Order Runge-Kutta algorithm. So the light turns green and within the first five hundred iterations, we observe very interseting behavior. As we see in Figure 1a, most drivers quickly accelerate as they are drawn towards their stable speed but then forced to decelerate and perhas brake upon approaching the driver in front of them. Red breaks away and quickly arrives at his stable speed. Magenta starts off very fast and approaches farily close to Blue before backing off, indicating that his attraction to speed is higher than his caution. Eventually, The last five,

13 Team # Page 13 of 2 (a) Drivers velocities in time (b) Drivers distances in time Figure 1: Six cars in one lane starting from a stop light. cars reach the same stable speed. This indicates that Blue, Magenta, Black, and Cyan want to drive faster but are stuck in traffic behind Green. Notice in the last few hundred iterations of Figure 1b that each driver in traffic maintains a consistent distance between himself and the car in front of him, indicating that they are stuck in a spacially dense clump. 3.2 Traffic Flow in Two Lanes Passing and Merging Algorithms As we just saw, a one lane scenario is very prone to traffic. If any driver s stable speed is less than that of the drivers behind him, he will cause all of them to slow down and match his speed while maintaining minimum distance. These drivers are sitting in their car wishing they could simply drive around him and be done with it. That can be accomplished with the addition of a left lane and the simple rule guiding this entire paper: drive on the right except to pass. In a one lane system, the driver is only concerned with himself and the driver directly in front of him. His level of awareness rises with the introduction of a new lane and passing opportunity. He begins by asking himself the question: Is the driver in front of me within my range? A driver does not consider cars that are too far in front of him; he waits until they are within some sight distance before asking the next question: Is his stable speed less than mine? If both of these answers are true, then the driver s passing gate has opened. (Note that this algorithm analyzes the stable speed of the driver rather than his current speed. This allows us to avoid unnecessary lane changes that may occur as drivers are speeding up or slowing down. While this is not technically accurate, it has the affect of causing the second car in the clump to pass first, followed by the third, and so on. Effectively, within a very short number of iterations, each car will change lanes and make the pass, similar to what we see in real life.) When a driver s passing gate opens, he becomes constantly aware of what is happening in the left lane. With each iteration, he checks within some safe distance around him to

14 Team # Page 14 of 2 ensure there are no cars hiding. He compares the speed of the car in the front left to the speed of the car directly in front of him. If both of these tests pass, he changes lanes and begins passing. However, if one of them fails, we immediately close the loop and wait until the next iteration before asking the questions again. This method prioritizes safety so no collisions can occur during a lane change. Once in the left lane, a driver s merging gate is always open. That is, he is always looking for a way back into the right lane. He asks a similar set of questions, and can even pass multiple cars in one trip if all of their stable speeds are lower than his. The combination of these two algorithms allows for fluid lane changes and optimal speed densities, as we will see in the next simulation Simulation: Six Cars in Two Lanes Figure 11: Initial positions of each driver on a two lane road Figure 11 presents us with an image of a two lane road, with drivers placed in random positions. They are given characteristics that result in varying stable speeds. Those speeds in order from largets to smallest are Magenta, Red, Cyan, Green, Black, and Blue. We expect that enough passing will occur over a certain number of iterations to allow the cars to reorder themselves according to their stable speeds. In order to graphically see each driver s initial position, we keep the passing and merging gates closed for the first 1 iterations. Figure 12: Full simulation of six drivers over two lanes. Right Lane drivers are plotted with solid lines. Left Lane drivers are plotted with dashed lines.

15 Team # Page 15 of 2 Figure 12 leads us to the following observations. As soon as the passing gate opens, Black merges into the right lane in front of Red and Magenta pulls into the left. Red s passing gate is open, so rather than slowing down, he merges left and begins passing Black. Between 1 and 75 iterations, Magenta follows close behind Cyan, stuck in left lane traffic. As soon as Cyan overtakes Green, he merges into the right lane, activating Green s safety reaction while allowing Magenta to accelerate to stable speed and continue. At 13 iterations, Red has passed Black, so he merges back into the right lane, activating Black s safety reaction. This puts him within Green s sight. Although Green s stable speed is higher than Black s, he cannot change lanes because Magenta is driving in his blindspot. Magenta actually has the highest stable speed of all the drivers, so he will continue to drive in the left lane until he passes everyone. This simulation shows that the Right-Lane-Rule allows for faster drivers to overtake slower drivers, thus increasing the flux density as described before. It is imperative that drivers merge back into the right lane upon a successful pass. Otherwise, slower drivers will cause traffic in the left lane, lowering the speed density and increasing the spacial density. 4 Evaluation: Traffic Systems with Speed and Density Expectations 4.1 Flux Score This score measures the transportation ability of a traffic system. It estimates the number of vehicles passing a reference point per unit of time. We call this value flux. The relation between flux and speed expectation v and density expectation ū are intuitively defined by F (ū, v) = g ū v, (27) where g is a positive constant for scaling purpose. The plot takes the form of Figure 13. If we imagine water flowing through a pipe, the flux of water is just the speed of flow multiplied by the density of water. In the water case, the flux has unit of weight (lb, kg and so on), however, since we ū is unitless (remember we quantized the road that vehicles passing by), the flux has unit of speed(mph, m/s and so on). In other words, ū is a numerical factor of speed expectation. If we set g =.1, a sample numerical formula of the flux score is obtained: F (ū, v) =.1ū v (28) We can also calculate the variance from the error propagation of ū and v: 4.2 Safety Score V ar(f ) = ( V ar(u) ū 2 + V ar(v) v 2 )F (ū, v) (29) In this section we are going to come up the a safety score for a traffic system with only speed and density expectation v and ū. Instead of deducting a formula, we claim the safety score to be:

16 Team # Page 16 of 2 Figure 13: Flux F (ū, v) with Respect to u and v 1 S(ū, v) = ( 1 + h ū v )2 (3) With a plot according to Figure 14. Figure 14: Safety Score S(ū, v) with respect to u and v We justify this formula according to the following reasoning. First, the value of this function ranges from to 1, and it has the form of the inverse of the product of v and ū with a positive constant h for scaling. If either v or ū is, the safety score is at its maximum (either all vehicles are stopped in bumper-to-bumper traffic, or there is no one else on the

17 Team # Page 17 of 2 road). Second, when ū and v are none-zero, the increase of either of them will decrease the safety score. If there re more vehicles concentrated in a certain area who drive with higher speed, the area will certainly be less safe. Lastly, without the square, we ll find that the function S(ū, v) will have the inverse grow rate as flux score F (ū, v), and this is not what we desired. Since we want to emphasize human safety, we want to make the change of safety function plays a more dominant role, so we square the function. Setting factor h =.1, yields the sample numerical formula for safety score: and its variance: 1 S(ū, v) = ( 1 +.1ū v )2 (31) V ar(s) = 2( V ar(u) ū Overall Evaluation Score + V ar(v) v 2 )S(ū, v) (32) As mentioned before, the evaluation is simply based on the product of flux score F (u, v) and safety score S(u, v): E(u, v) = S(u, v) F (u, v) (33) Notice, the decreasing rate of safety is a squared product while the increasing rate of flux is the product of two linear functions. This indicates when n and v are large, the system is dominated by the decreasing safety, which makes the score drop significantly. However at low speed or low spacial density, the flux will weigh as the more important factor because the scaling factor h (.1 in the sample) makes hū v small compared to 1. Figure 15: Final Evaluation Score E(ū, v) with Respect to u and v

18 Team # Page 18 of 2 Lastly, we consider the robustness of the system, we calculate the variance of the final score by the statistical propagation from the variance of ū and v: V ar(e) = 3( V ar(u) ū 2 + V ar(v) v 2 )E(ū, v) (34) This is the error propagation from the variance of ū and v. There is a constant factor of 3 since both ū and v is equivalent to being risen to the power of 3 during the calculation of E(ū, v). 5 Conclusion: Considering the Scores of Different Systems Let s start with assumed data from an urban and rural area: Place Number Density n Speed Limit v m Number of Lanes Urban km/h 2 Rural km/h 3 Figure 16: Data for a Certain Urban Area and a Certain Rural Area First we work out density and speed expectations and variances: Urban ū u = V ar(u u ) =.5 v u = V ar(v u ) = Rural ū r =.2948 V ar(u r ) =.322 v r = V ar(v r ) = Figure 17: Expectation and Variance of Density and Speed Then we can calculate the safety score, flux score, and the final evaluation score for both systems: Urban F u =.159 ±.5 S u =.929 ±.55 E u =.147 ±.13 Rural F r = ± S r =.4 ±.1 E r =.64 ±.27 Figure 18: Safety Score and Flux Score and Final Evaluation Score As we see, the rural area actually gets a lower evaluation score than the urban area. It s high flux score makes it suffer a much lower safety score, consistent with the data provided in the introduction section. We can improve the safety factor of rural areas by decreasing the flux. Because the Right-Lane rule actually increases the flux, it results in a lower evaluation score.

19 Team # Page 19 of 2 On the other hand, in urban areas, the flux suffers a really low score. It is possible to increase the flux by promoting the keep-right-except-to-pass rule. This will, however, undermine the safety score and cause a lower evlaution score. Thus, a balance need to be found between flux and safety. If we fix ū which relates to city population and vary v, we can obtain a plot of E( v) with fixed ū: Figure 19: Plot of E( v) with fixed ū By observation, the maximum value is above the speed limit, thus an increase in speed expectation will increase the evaluation score. Hence in urban areas, the Right-Lane rule actually benefits the overall evaluation score.

20 Team # Page 2 of 2 References [1] US Department of Transportation. Traffic Safety Facts, 211 Data. [2] US Department of Transportation. National Survey of Drinking and Driving Attitudes and Behaviors 28