Summary Currently, in countries where automobiles are moving on the right, drivers are required to drive in the right most lane. To overtake another vehicle, they move one lane to the left, pass, and return to their former travel lane. How well does this rule function? Is there any better alternative? What do we mean by better performance? In this paper, we will build mathematical models to examine all these questions by focusing mainly on two significant measurements of performance: efficiency and safety. After a series of qualitative and quantitative comparisons, in the end, we draw a conclusion that there is indeed another superior rule, which allows drivers to overtake at most three vehicles at once before returning back instead of just one vehicle. The structure of this paper is as follows. In session 1, we explain some preliminary assumptions, variables, and metrics about traffic flow for readers comprehension. We briefly illustrate the Greenshields Traffic flow model. Also, we provide an essential proposal on velocity distribution of freeway vehicles, which is best approximated by the Normal Distribution. Then, we discuss two distinct scenarios, which are light and heavy traffics. Finally, we introduce our proposed rule that permit drivers to pass at most three vehicles. In session 2, we present a fast but intuitive way to model the overtaking mechanism as a special spring block. Based on the normal distribution property of velocity, we select the most representative velocity data to simulate one overtaking process. Finally, we officially justify our own solution to the overtaking problem from one aspect and explore the interrelationship between the spring effect and the intelligent system. In session 3, we present three methodologies to explain our proposal in other ways, namely, one more application of Greenshields Model, the Cellular Automaton simulation, and the LYZ Model. All these powerful tools provide justification of our proposed rule from another view. In the last sub session, we consider various determining factors on the choices of safety levels and its influence. Session 4 summarizes our findings, including advantages and weaknesses of our models. It is also worthwhile to discuss the generalizability of our model in countries where automobiles travel on the left. Finally, in Further Thinking, we try to integrate more factors to improve the existing model.
SAFLUX Study on Influence of Overtaking Rule on Freeway Performance Team #29225 February 11, 2014 Abstract Currently, in countries where automobiles are moving on the right, drivers are required to drive in the right-most lane. To overtake another vehicle, they move one lane to the left, pass, and return to their former travel lane. In this paper, we will build mathematical models to evaluate this overtaking rule and further seek alternative rules that enhance transportation performance more. We believe performance consists of two main aspects: efficiency and safety. We study them by 1) extensively using Greenshields Model and introducing unprecedented modified version, called LYZ Model, 2) implementing Cellular Automaton, a twodimensional visualizing tool to simulate pre-specified instructions. Using these powerful tools, we succeed in providing qualitative and quantitative comparison between the existing and our proposed rule. Under careful model and data justification, we conclude that allowing drivers to overtake at most three vehicles consecutively before returning to the original lane exceed the current rule in improving traffic performance. Keywords: Freeway Performance; Greenshields Model; LYZ Model; Cellular Automaton. 1
Control #29225 Page 2 of 24 Contents 1 Introduction 3 1.1 Preliminary Assumptions...................... 3 1.2 Defining Fundamental Variables................... 3 1.3 Basic Greenshields Model...................... 4 1.4 Distribution Of Freeway Vehicles Velocity............. 5 1.5 Light vs. Heavy Traffic........................ 5 1.6 Rules To Be Considered....................... 5 2 Why Three 5 2.1 Application of Greenshield Model Under Light Traffic Situation. 5 2.2 Spring Block............................. 6 2.3 Why Three.............................. 8 2.4 Under Intelligent system....................... 11 3 Macro-view Of Proposed Rule 11 3.1 Greenshields Model Application Under Heavy Traffic....... 11 3.1.1 The Law Of Conservation.................. 11 3.1.2 Application and Re-model of Greenshields in Double-Lane Freeway............................ 11 3.1.3 Further Math Derivation.................. 11 3.1.4 Current Traffic Rule Analysis In The Modified Greenshield Model......................... 12 3.1.5 Flow Optimization In Mathematical Interpretation.... 14 3.1.6 Extension To Multiple-Lane Freeway............ 16 3.2 Let the Data Tell........................... 16 3.2.1 Cellular Automata...................... 16 3.3 The LYZ and Speed Limit.................... 18 3.3.1 Advanced Greenshields Model................ 18 3.3.2 The LYZ Model...................... 19 3.3.3 The Role Of Speed Limit Under LYZ Model...... 20 4 Conclusion 22 4.1 Advantages.............................. 22 4.2 Weaknesses.............................. 22 4.3 Generalizability............................ 22 4.4 Further Thinking And Potential Improvement........... 23 5 Reference 24 2
Control #29225 Page 3 of 24 1 Introduction When driving, some people prefer speed while others are more concerned about safety. In the paper, we endeavor to give balanced solutions to maximize both aspects. 1.1 Preliminary Assumptions For model establishment simplicity, the audience should be aware of the following assumptions. Regarding the road: The road is straight, with infinity length and with no road slop. The motion is directed left-to-right on a one-lane roadway. Vehicles move in the positive direction of x-axis. This one-way road is closed, with no entrance or exit. Regarding drivers and vehicles: All vehicles do not break down in the movement. The length of all vehicles are 5m long. All vehicles have the same priority. In other words, drivers do not make way for others (like what we do for ambulance). Vehicles have all the aspects equal except for their velocity. Drivers strictly obey do not exceed speed limits. In a given lane, drivers only know their relative position and speed with the vehicle right in front of them. No vehicles in front of the front vehicle are in the drivers sight. 1.2 Defining Fundamental Variables Traffic Density (ρ) the average number of vehicles per unit length of road at the segment and time specified. Average velocity (V) the average velocity of vehicles of road at one segment and time specified. Traffic Flux (Q) the average number of vehicles going by one segment of the road per unit time. It can be expressed as the product of traffic density and average velocity. i.e., Q = P * V 3
Control #29225 Page 4 of 24 Cutoff density (ρ c ) We define a specific value of density before which the average velocity of vehicles will keep as the set speed limit. This is the turning density point from light traffic to heavy traffic. We will further quantify ρ c in our later section. As demonstrated earlier, performance includes two factors: efficiency and safety. After careful research, we believe that efficiency can be mathematically best measured by traffic flux, and safety by speed variance, which requires more background information from readers, thus shall be defined later. 1.3 Basic Greenshields Model The Greenshields traffic model is the most fundamental model of uninterrupted traffic flow that predicts and explains the trends observed in real traffic flows. Greenshields Model assumes that speed and density are linearly correlated. This relationship is expressed mathematically and graphically below. Mathematical representation: Graphic representation: v = A B ρ Figure 1: Basic Greenshields ρ v curve 4
Control #29225 Page 5 of 24 1.4 Distribution Of Freeway Vehicles Velocity Vehicle velocity is one of the most important factors in calculating traffic flow and influencing safety. Thus, it is tempting to obtain the vehicle velocity distribution and predict them accurately in the freeway. Extensive studies have shown that normal distribution is the best fit for freeway vehicle velocity. 1 1.5 Light vs. Heavy Traffic Light or heavy traffic can be easily sensed, but mathematical cutting line between them should be defined in serious mathematical proofs. We observe that in light traffic, adding another vehicle will not influence the traffic too much. However, as we repeat this process, the density of the traffic increase and adding another car will begin to lower the velocity of the existing cars, thus result in congestion. Based on these observations, we define ρ c. If the density of the lane is less than ρ c, vehicles can move at the fastest speed possible, i.e., the speed limit. If the density is more than ρ c, velocity starts to decrease till 0, where ρ max is reached. At ρ max, no vehicles can move the traffic jam exists. 1.6 Rules To Be Considered Current rule: Drivers should drive in the right-most lane. When passing another vehicle, they move one lane to the left, overtake, and then return to the original lane. Proposed rule: Drivers drive in the right-most lane. When moving left and passing, if they perceive that the vehicles in front of the car overtaken are close to each other, they can overtake at most three vehicles in a row. Then, they are required to the former lane. 2 Why Three 2.1 Application of Greenshield Model Under Light Traffic Situation In a light traffic situation, ideally the average velocity will be the speed limit set in freeway, util the system reach a certain density (ρ max ). Therefore, we first modify the Greenshield model in light traffic condition as the representation in Figure 2. We realize that when the traffic density is low, adding another vehicle will not congest the road nor decrease the flux. Therefore, it is not necessary to study this situation deeply, and the following microscopic models will be applied in a relative heavy traffic. 1 The research group conducts freeway cross-section operation speed survey using laser speed detector, analyze the data collected from various vehicles and roads combined. Then SPSS is used to study the characteristics of the distributions and finds that normal distribution give the most appropriate fit of the data out of the Gamma, Logistic, and Weibull distributions. 5
Control #29225 Page 6 of 24 Figure 2: Greenshields model when speed limit is applied 2.2 Spring Block Based on previous researches, the spring model has been successfully implemented into different fields such as earthquake simulation. We see great possibility that the spring-block type models can also be applied in the vehicle overtaking. Assume a situation where the traffic is heavy so the distance between each car is about the safe distance, but this is not a jam situation where all vehicles barely move. When a car overtakes from left lane to right, we need to consider the that that the driver of the car behind in right lane may slow down and keep the safe distance for that overtaking vehicle. Similarly, the rest of the cars in this lane will adjust their speeds and distances accordingly. Since this is not a bi-directional pattern, namely, the reaction of the car behind wont affect the car ahead. Also, the distance for each following car to adjust depends on both the reaction time for each individual driver and the original distance between each pair cars. Therefore, we model the pattern for a single car overtaking into a lane of cars as a special spring model. 6
Control #29225 Page 7 of 24 Figure 3: A graphical representation of our spring model In real case, the spring coefficient k(τ,d) for this model will be a function of reaction time and distance d between each car. For this derivation, we assume k is a constant and the distance d between each identical car is the same. When another vehicle overtakes to the front of this lane, treating this fleet as a entity and considering the speed difference v, we can model the situation where these group of n number of frictionless spring blocks hit a wall with relative speed v. Assume each car weights m. The total force on this group of spring block will then be F = n m a Since this is a special spring model, where the behavior of each spring block only depends on the former block, the force exerted on the first car from the first spring will be: F 1 = n m a = k x1 x1 = n m a k where x1 is the distance this car need to adjust. Following this pattern and considering the fact that all the vehicles are treated as an entity and share the same acceleration, we get: x2 = (n 1) m a k 7
Control #29225 Page 8 of 24 And finally we get: xn = (n n) m a k For large n, the n th vehicle, it will not need to adjust the distance anymore, which is the extent to which the system is not disturbed. This brief proof is intended to give our audience a sense that the influence of one single overtake will have spring effect to the following lane, while the extent of this effect, such as how many cars will be interfered by this overtaking depends on many outside coefficients. And this can be connected with the friction coefficient for this spring block model, which in our case has been assumed as 0. 2.3 Why Three The above section depicts the situation when one individual vehicle passing another one. In this section we will justify the number of vehicles the drivers are allowed to overtake once. First, few assumptions must be raised: The state among vehicle B, C, and D is relatively stable, which means that front vehicles are faster than the rear ones. Otherwise, surpassing will take place and break this stableness. Vehicles B, C, and D are keeping minimum safety distance at the moment vehicle A is ready to overtake (we will give careful explanation of this minimum safety distance later in the real scenario part). Please recall the normal distribution of vehicle velocity in the freeway. Suppose vehicle A is moving with velocity V a, and V a ranges from 0 th percentile (V 0, subscript denotes number of percentile) to V 100, which is the maximum speed. As depicted Figure 4, since A is faster than B and the expected velocity of all vehicles is V 50, it is safe to infer that expected velocity of A (E(A)) is greater than V 50. Since C is faster than B, the expected velocity C is the expected velocity between V 50 and V 100, which is V 75. Then if A is about to pass C, its velocity must be greater than V 75, the halfway between V 50 and V 100. Similarly, if A wants to overtake D, its velocity has to be at least V 87.5. Various studies shows that most freeways set the speed limit at V 85. Thus, if A is still fast enough to overtake D, high chance is that A has broken the speed limit, which contradicts with our preliminary assumption that no vehicle move above it. At this stage, we can conclude that only in few cases can vehicle A overtake 3 cars without moving back to former lane, and the proposed rule is reasonable enough. Now we are measuring the validity of our prediction. Take the real data as example, which is the same data surveyed in the research to derive the velocity distribution. In a real freeway system, if a vehicle wants to overtake, it needs to be in a fast speed. Based on the data we collect from the normal speed = 0 8
Control #29225 Page 9 of 24 Figure 4: A standardized normal distribution curve distribution from one freeway in China. We observe that V 50 is 85 km/h (22.22 m/s), V 75 = 95 km/h (26.39 m/s), V 87.5 = 105 km/h (29.17m/s) and V 95. We assume a car A travels with V a = 30.56 m/s. It wants to overtake the vehicles ahead. Specifically, V B = 22.22m/sV C = 26.39m/sV D = 29.17m/sV E = 30.56m/s Here we need to introduce a concept of safety distance for a vehicle, which is a function of the speed (v) of this vehicle and the vehicle acceleration (A). It can be formulated as S(V ) = V 2 2A + 0.3V Where we model A = 5m/sec 2. And we calculate the original safe distance between Vehicle BC, CD, and DE. According to the data from former section 3.1, we model the headway 2 to be 3s, and calculate a corresponding distance from A to the former car. This distance is 91.68 m. Here we present a chart tracing the location of each vehicle starting from the process of this overtaking. A is located at position 0 originally. 2 Headway measures the time it will take for the overtaking vehicle to travel from its tip to the one of the front vehicle 9
Control #29225 Page 10 of 24 Time(s)\Location(m) CAR A CAR B CAR C CAR D CAR E t = 0 0 91.68 147.7 225.26 319.1 t = 10.99 335.85 335.85 437 546 655 t = 17.7 540 485 613 741 859.7 t = 35.2 1074.8 Safely Surpassed 1074.8 1251.48 1394.5 t = 35.89 1643.18 Safely Surpassed 1565 1794 1962 The road efficiency and this spring block effect are related. Go back to our spring model. We believe that letting vehicles surpassing more than one car at one time will distribute more to the overall increase of freeway efficiency than only letting one overtake at one time. Since for each single overtake, there will be a spring effect. And we want to reduce the spring effect. Therefore, during this iteration, we take into the factor of safety issue and want to minimize the spring effect. Therefore we specify that A can only get back into the lane when there is enough space. This length of this space should be larger than or equal to the sum of the headway distance of car A and safety distance of the vehicle that car A just surpassed. From this simulation, at time 17.7 s when car A overtook car B and left car B enough safety space, the distance between car A and car C is shorter than As headway distance. d AC(t=17.7s) = 613 540 = 73m d A headway distance = 91.68m d AC < d A headway distance A cannot surpass C And therefore car A needs to keep going and overtake car C.When car A overtook car C and left C enough safety distance (78.18 m), distance between car A and car D will be large enough: D AD(t=53.89s) = 1794 1643.18 = 150.82m > 91.68m According to our rule, A has to get back into the lane. Therefore car A needs to get back into the lane first and travel until it reaches the point when it is allowed to overtake car D. Therefore, in this very simulation, we first get the conclusion that we can allow one car at most overtake two cars at one time. However, in real cases, considering the fact that the value of V B, V C, V D, and V E are defined in a decreasing occurrence in that normal distribution. Realizing that car A is more possible to run into cars with median speed, we therefore allow one car to overtake one more car, which is three cars at most at one single time. 10
Control #29225 Page 11 of 24 2.4 Under Intelligent system Both of our rule and the current rule rely upon human judgment for compliance. Psychological factors play an important role here. Take a real-life example. Sometimes when a new car is passing from behind, we will subconsciously brake even we already have kept a very safe distance with it. This situation is inevitable in the freeway traffic. However, when the speed is under control of an intelligent system, this overcautious braking wont happen anymore. Therefore, the spring block effect will be greatly reduced. 3 Macro-view Of Proposed Rule 3.1 Greenshields Model Application Under Heavy Traffic 3.1.1 The Law Of Conservation We assume the rate of change of the number of vehicles in the segment with respect to time equals the dierence in ow rate or ux in and out of that segment. ρ(x, t) t + q(x, t) x 3.1.2 Application and Re-model of Greenshields in Double-Lane Freeway The Greenshields theory treats freeway as a single entity. In our case, since our focus will be mainly on the interactions between the passing lane and the right lane, it is reasonable for us to apply the Greenshields model into two different freeway. Because the Greenshields defines values of the constants A and B through field observations for each fixed position on the freeway, the cross-sections of the left lane will be different from the right lane. Therefore, we model that the passing lane and the right lane will have different slope in the GreenShield Model.Since in this case we are considering a heavy traffic, we choose not to consider the Greenshields model we modified for light traffic.thus, we apply for the Greenshields Model in the following graphic representation. Each line segment represents one lane. We fixed the value of max because for both of the lane, the largest amount of vehicles that can be fitted into these two lanes is the same. 3.1.3 Further Math Derivation Suppose we are given a graph of Figure 5, which consists of two lines. On one hand, however, two linear lines do not form a function. On the other hand, we would like to simplify calculation and simulation flux. Hence, it is desirable to have just one line which gives a suitable representation of the sum of flux. The term representing means that, for every point in density1 and density2, and their corresponding flux1 and flux2, there is always at least one point in = 0 11
Control #29225 Page 12 of 24 Figure 5: Modified Greenshields model for a two lane freeway this new density line whose corresponding flux is the sum of flux1 and flux2 times a coefficient α, usually α 1. Since what we only care is the sum of flux on the two lanes, there is indeed one line in between the original two lines. It connects ρ max and the midpoint of both lines x-intercepts. To see whether this line passes the above test, it suffices to pick the maximum of the sum of flux1 and flux2, which is the sum of maximum flux1 and flux2. If it could represent the maximum, it can represent all the other combinations. Since a + b = 2c, ρ c (v) = 2ρ max /(a + b) v + ρ max. Since as the maximum flux occur at midpoints, maxflux1 = maxflux2 = ρ max /2, maxsum = ρ max /2 2 = maxsum. Similarly, F luxcmax = 2ρ max /(a + b) (a + b)/4 + ρ max = 1/2ρ max. Thus, we find F luxcmax = 1/2maxsum, which means ρ c (v) is a good representation of ρ a and ρ b. 3.1.4 Current Traffic Rule Analysis In The Modified Greenshield Model The existing rule requires drivers to drive in the right-most lane unless they are passing another vehicle. In this two-lane freeway case, we assume that the overall density of the right lane is supposed to be much heavier than the left lane, since after a passing, driver will be obligated to get back. Therefore, we modify the state of each lane as the following graphic representation Figure 7. Intuition indicates that the black line represents the passing line, since it has larger range of average velocity. Our goal is to optimize the sum of the flux from both of the cross-section 12
Control #29225 Page 13 of 24 Figure 6: A graphical illustration of treating double lanes as a single lane Figure 7: One possible scenario under current rule in the fast lane and slow lane at a fixed time and position. Clearly, under the current traffic rule, this sum of flux is not maximized because we over-use the 13
Control #29225 Page 14 of 24 left lane while the right lane is too vacant. However in our proposal, if one vehicle is allowed to surpass multiple vehicles at one time, the density of the fast lane will increase, which will release the burden in the slow lane. This dynamic interchange can be represented in the following graph Figure 8. We can see that both ρ and V are closer to the midpoint, which means both lanes are closer to the maximum flux state. Figure 8: Visualize the effect of proposed rule 3.1.5 Flow Optimization In Mathematical Interpretation Go back to our math derivation Figure 6. Since the max area under middle curve equals to half of the maximum possible combination of the red and black shaded area, the middle curve thus contains all the possible combinations of half of the areas combination between red and black. And therefore we could use this specific line as a mathematical model to interpret the dynamic behavior for that interchanging system. From now this traffic physical system problem can be purely transferred into a math model, represented by the following graphic representation (Figure 9). The rectangular area under the midpoint of this blue line represents the situation which represents the percentage of the maximum combination of the flux from both lanes. Assume the current traffic rule determines the current flux point (Point A), which is not on our ideal optimized point.(figure 10) Go back to our proposal where cars are allowed to overtake multiple cars at one time, the effect of our proposal to this dynamic system can be represented 14
Control #29225 Page 15 of 24 Figure 9: max flux point plot Figure 10: One possible scenario under current rule in the Figure 11. Basically, our more flexible rule will drive the flux point A toward its optimization point, the midpoint. For a pure optimization, ideally, A will converge to the midpoint.from here, we conclude that one of the main effects of our pro- 15
Control #29225 Page 16 of 24 Figure 11: The effect of proposed rule posal, compared with the existing rule, is the increase of the possible average flux for a traffic flow system. 3.1.6 Extension To Multiple-Lane Freeway In our model for double-lane freeway, we suggest the effectiveness of allowing vehicles to overtake multiple cars at one time. We consider this interchangeability as a very important factor to maximize the overall performance of the whole system. In addition, our modified Greenshields Model can be implemented into multiple-lane freeways. The basic principle is the same, while the actual scheme to find the linear representing curve is more time-consuming, and thus it will not be presented here. 3.2 Let the Data Tell 3.2.1 Cellular Automata Cellular automaton is a discrete model in studying complex models to visualize a set of pre-specified rules. In this chapter we are going to present Cellular Automata model to simulate multi-lane traffic conditions. Below are some screenshots of our sample runs. The goal is to give concretive and quantitative performance comparison between the current and proposed rules, and our hope is that the simulation could give us confirmation on the superiority of the proposed rule. The result is as follows. 16
Control #29225 Page 17 of 24 Figure 12: Screenshot of Cellular Automata in process, starting rough at generation 10, ending at generation 30, with road length 80, maximum velocity 8 and density at 0.2, 0.4 and 0.8 respectively Figure 13: A comparison between the ρ v curve for simulations under two rules The Blue curves represent the data collected under the current rule. And the red curves represent the data collected under our proposed rule. It is obvious that under the same speed limit precondition, our cutoff density (ρ c ) is much larger. This indicates that on average, vehicles under our rules can proceed with a faster speed. Another important observation we made is that the speed variance of our system is lower and more converged. Speed variance is defined to be the variance of all the vehicle velocity after a long period of time. At this moment, vehicle velocity stop spread randomly and enter into a relatively 17
Control #29225 Page 18 of 24 Figure 14: Distributions of velocity variance verse density under two rules stable state. Recall that we measure safety as speed variance in the Summary section. The link between variance and safety is strong. More variant speed usually results in more overtaking, thus more possible collisions. Besides, more overtaking increase the spring effect which is not desirable in the freeway system. Up to this point, both our data and model have indicated that our proposal provides the system not only with a larger possible flux, but also a safer condition. 3.3 The LYZ and Speed Limit At the beginning, we introduced basic Greenshields Model, which serves perfectly to the superiority of our proposed rule. It is tempting to reuse it to optimize traffic flow and safety at the same time. However, this basic model is not complicated enough to model these two factors, because as stated in the Introduction section, the max flux point lies on the midpoint of the density-velocity curve. Also, the basic model does not consider speed limit and its influence on the density, velocity, and flux. Hence, advanced Greenshields Model is needed. 3.3.1 Advanced Greenshields Model Consider a more realistic case where a freeway speed limit exists. An individual car may tend to travel at the highest speed possible, either the result of speed limits or road conditions or driver caution, call it Vmax. As stated previously, before a certain density point (ρ c ), the overall average density will keep increasing without interfering the average speed. After this ρ c, velocity tends to go down with increasing density, so we should assume that dv dρ < 0. In 18
Control #29225 Page 19 of 24 addition, there is certainly a ρ max, where vehicle is next to each other without any distance, i.e., ρ = 1/L, where L is the length of the vehicle. In this case, the velocity is zero. So, taking consideration of all these three conditions, we present a more realistic version Greenshields Model: v(ρ) = v max (1 - ρ ρ max ). Figure 15: Advanced Greenshields V-P curve 3.3.2 The LYZ Model It seems that the advanced Greenshields Model approaches to reality much more than the basic version, but, based on our careful research and practical driving experiences, we are able to establish our own model to give even more realistic model for the velocity-density curve. It is called LYZ Model. The LYZ Model actually generalizes D, the distance between each vehicle. Previously, D is defined to be a constant. However, common sense and intuition suggest that the faster one is driving, the longer distance he should keep. Thus, this intuition gives us a basic idea that D is a function of velocity and should be monotonically non-decreasing. According to elementary physic, the distance basically consists of two parts: the drivers response distance and the deceleration distance. The former is defined to be the distance travelled between the moment the driver is thinking about braking and actually braking. The latter is the distance between moving at speed V and completely stopping. Thus, the equation should be of 19
Control #29225 Page 20 of 24 form D(v) = tv + v2 2A, where t is human beings reaction time and A is the deceleration rate. In general, t ranges from 0.3 0.7 seconds, so we take the average, which is 0.5 seconds. The argument about A is that for A = 3m s, 2 ordinary human beings will very uncomfortable, and for A = 5m s, they are 2 extremely likely to be hurt. So, we take the average, 4m s. 2 Now we are ready to officially propose the LYZ model. Since ρ = 1 (L+D), we have, ρ = (5 + 0.5v + v2 8 ) 1 { Vmax, ρ ρ min V = 8 ρ 36 20, ρ min ρ ρ max Where ρ min = (5 + 0.5V max + V 2 max 8 ). Figure 16: LYZ ρv curve 3.3.3 The Role Of Speed Limit Under LYZ Model We derive the most realistic function modeling the relationship between velocity U and density V. however, please recall that the variable we really want to maximize is flux Q, the average number of vehicles going by one segment of the road per unit time. Bigger flux implies more efficient transportation. Recall that flux is the product of density and velocity, i.e., Q(ρ) = p V (ρ), we have { ρv max, ρ ρ min Q = 8ρ 36ρ2 2ρ, ρ min ρ ρ max This is the objective function we would like to maximize. Suppose we have a list of choices on speed limit. We would like to see on various speed limit levels, what the maximized flux from left to ρ min is 20
Control #29225 Page 21 of 24 5 Q max = ρ min V max = ( + 0.5 + V min V max Based on the above formula, we give out this table: 8 ) 1 V(mph) V(m/s) Q(vehicle/sec) 15 6.71 0.48 20 8.95 0.46 25 11.18 0.43 30 13.41 0.39 35 15.65 0.36 40 17.88 0.33 45 20.12 0.31 50 22.35 0.28 55 24.59 0.26 60 26.82 0.24 65 29.06 0.23 70 31.29 0.21 75 33.53 0.21 Figure 17: LYZ Model, Q-P curve with P on x-axis We can see a clear pattern that as the speed limit increase, the Q max decreases. Right to P min. Taking the derivative of Q with respect to ρ and set it equal to 0, we have dq dρ = 0, ρ = 0.076 vehicle/m, V = 6.325m/sec, Q = 0.48vehicle/sec From this calculation, we can easily see that ranging from 15mph to 75mph, the choice on speed limits does not affect the theoretical maximum flux. The maximum flux possible under LY model is 0.48 vehicle/sec, unless the speed limit is set under 15 mph (which is an extremely strange and rare setting). However, under each scenario, our proposed rule gives better performance. 21
Control #29225 Page 22 of 24 4 Conclusion 4.1 Advantages Simulation confirms the superiority of proposed rule in promoting performance of freeway. We recommend that the rule be implemented. Wide investigation of relevant aspects. Establishing unprecedented model. 4.2 Weaknesses In real case the flow of cars is not identical to the fluid dynamic. Vehicles have lengths and keep distance with each other. Thus, P and Q are based on unit length and unit time interval, not on instantaneous number. Otherwise, continuity of traffic flow will not hold. Each driver has consciousness to make active change, which makes the behavior of the system hard to predict. In multiple lane system, the vehicles are only not allowed to stay consistent in the right most lanes. Regarding the situation where cars can stay in multiple lanes, this will make our derivation very time-consuming. We therefore simplify our system in a way that the vehicle needs to go back to the former lane after passing up to 3 other cars. Change of travelling lane permanently is not considered. 4.3 Generalizability Since our model is designed to analyze the straight proportion of the road, where exits and entrances are not considered, we believe that our model and simulation can be applied in countries driving in the left or right norm. In fact, the following graphs are cited from a scientific journal by Hesham Rakha and Brent Crowther. They collected traffic data from several main freeways in different countries and they fitted those data with their traffic stream model. Here we cite a pair of them, representing the Arterial Road in UK and Holland Tunnel in New York City, US. Both of these graphs are in similar shapes. So, we see considerable similarity between right- and left- driving countries. 22
Control #29225 Page 23 of 24 Figure 18: Traffic data collected in Holland Tunnel (US) And Arterial Road.(UK) 4.4 Further Thinking And Potential Improvement The Spring Block Model is only intended to give our audience an intuitive sense, while this hypothesis worth future research. In the application of Greenshields into two lanes situation, quantified algorithm is preferred. It will be very useful that future researchers can relate a time constant τ f orthesystemtoreachequilibrium. We also believe that our model will affect the value of ρ c. We believe our proposal is worth studying for future experiments. 23
Control #29225 Page 24 of 24 5 Reference Klar, A, J M. Greenberg, and M Rascle. Congestion on Multilane Highways. Siam Journal on Applied Mathematics. 63.3 (2003): 818-833. Print. Childress, Stephen. Notes on Trac ow. New York City: NewYork Univerisity, 2005. 25 Mar. 2005. Web. Rakha, Hesham, and Brent Crowther. Comparison of Greenshields, Pipes, and Van Aerde Car-Following and Traffic Stream Models. Transportation Research Record. (2002): 248-262. Print. Cellular Automaton. Wikipedia. Wikimedia Foundation, 02 Sept. 2014. Web. 10 Feb. 2014. Qingliang Zhang and Guozhu Zhao. The Establishment of the Traffic Slow System. Joumal of Handan Polytechnic College 3rd ser. 22.6 (2009): n. pag. Web. 6 Feb. 2014. Hong-min Zhou, Yu-cheng Ma, Jun Wang, Ying Yan. Study on Operating Speed Distribution of Highway Cross-section. Journal of East China Jiaotong University. Vol. 25 No.5. Oct. 2008. Yulong Pei, Guozhu Cheng. Research on Operation Speed and Speed Limit for Freeway in China. Journal of Harbin Institute of Technology. Vol.35, No.2. Feb. 2003 24