Summary Currently, in countries where automobiles are moving on the right, drivers are required to drive in the rightmost 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 subsession, 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.
Transcript
Summary
Currently, in countries where automobiles are moving on the right, drivers are required to drive in therightmost lane. To overtake another vehicle, they move one lane to the left, pass, and return to theirformer travel lane. How well does this rule function? Is there any better alternative? What do we meanby better performance? In this paper, we will build mathematical models to examine all these questionsby focusing mainly on two significant measurements of performance: efficiency and safety. After a seriesof qualitative and quantitative comparisons, in the end, we draw a conclusion that there is indeedanother superior rule, which allows drivers to overtake at most three vehicles at once before returningback 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, weprovide an essential proposal on velocity distribution of freeway vehicles, which is bestapproximated by the Normal Distribution. Then, we discuss two distinct scenarios, which are lightand heavy traffics. Finally, we introduce our proposed rule that permit drivers to pass at most threevehicles.
● In session 2, we present a fast but intuitive way to model the overtaking mechanism as a specialspring block. Based on the normal distribution property of velocity, we select the mostrepresentative velocity data to simulate one overtaking process. Finally, we officially justify our ownsolution to the overtaking problem from one aspect and explore the interrelationship between thespring effect and the intelligent system.
● In session 3, we present three methodologies to explain our proposal in other ways, namely, onemore 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 lastsubsession, we consider various determining factors on the choices of safety levels and itsinfluence.
● Session 4 summarizes our findings, including advantages and weaknesses of our models. It is alsoworthwhile to discuss the generalizability of our model in countries where automobiles travel on theleft. Finally, in “Further Thinking”, we try to integrate more factors to improve the existing model.
SAFLUX — Study on Influence of OvertakingRule 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 anothervehicle, they move one lane to the left, pass, and return to their formertravel lane. In this paper, we will build mathematical models to evalu-ate this overtaking rule and further seek alternative rules that enhancetransportation performance more. We believe performance consists oftwo main aspects: efficiency and safety. We study them by 1) exten-sively using Greenshields Model and introducing unprecedented modifiedversion, called “LYZ“ Model, 2) implementing Cellular Automaton, a two-dimensional visualizing tool to simulate pre-specified instructions. Usingthese powerful tools, we succeed in providing qualitative and quantitativecomparison between the existing and our proposed rule. Under carefulmodel and data justification, we conclude that allowing drivers to over-take at most three vehicles consecutively before returning to the originallane exceed the current rule in improving traffic performance.
When driving, some people prefer speed while others are more concerned aboutsafety. In the paper, we endeavor to give balanced solutions to maximize bothaspects.
1.1 Preliminary Assumptions
For model establishment simplicity, the audience should be aware of the follow-ing 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 makeway 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 withthe vehicle right in front of them. No vehicles in front of the front vehicleare in the drivers’ sight.
1.2 Defining Fundamental Variables
• Traffic Density (ρ) — the average number of vehicles per unit length ofroad at the segment and time specified.
• Average velocity (V) — the average velocity of vehicles of road at onesegment and time specified.
• Traffic Flux (Q) — the average number of vehicles going by one segmentof the road per unit time. It can be expressed as the product of trafficdensity and average velocity. i.e., Q = P * V
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• Cutoff density (ρc) — We define a specific value of density before whichthe average velocity of vehicles will keep as the set speed limit. This is theturning density point from light traffic to heavy traffic. We will furtherquantify ρc in our later section.
As demonstrated earlier, performance includes two factors: efficiency andsafety. After careful research, we believe that efficiency can be mathematicallybest measured by traffic flux, and safety by speed variance, which requires morebackground information from readers, thus shall be defined later.
1.3 Basic Greenshields Model
The Greenshields traffic model is the most fundamental model of uninterruptedtraffic flow that predicts and explains the trends observed in real traffic flows.Greenshields Model assumes that speed and density are linearly correlated. Thisrelationship is expressed mathematically and graphically below.
Mathematical representation:
v = A−B ∗ ρ
Graphic representation:
Figure 1: Basic Greenshields ρv curve
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1.4 Distribution Of Freeway Vehicles Velocity
Vehicle velocity is one of the most important factors in calculating traffic flowand influencing safety. Thus, it is tempting to obtain the vehicle velocity dis-tribution and predict them accurately in the freeway. Extensive studies haveshown 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 be-tween them should be defined in serious mathematical proofs. We observe thatin 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 addinganother car will begin to lower the velocity of the existing cars, thus result incongestion. Based on these observations, we define ρc. If the density of the laneis less than ρc, vehicles can move at the fastest speed possible, i.e., the speedlimit. If the density is more than ρc, velocity starts to decrease till 0, whereρmax is reached. At ρmax, 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 passinganother vehicle, they move one lane to the left, overtake, and then returnto the original lane.
• Proposed rule: Drivers drive in the right-most lane. When moving leftand passing, if they perceive that the vehicles in front of the car overtakenare 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 TrafficSituation
In a light traffic situation, ideally the average velocity will be the speed limit setin freeway, util the system reach a certain density (ρmax). Therefore, we firstmodify the Greenshield model in light traffic condition as the representation inFigure 2. We realize that when the traffic density is low, adding another vehiclewill not congest the road nor decrease the flux. Therefore, it is not necessary tostudy this situation deeply, and the following microscopic models will be appliedin a relative heavy traffic.
1The research group conducts freeway cross-section operation speed survey using laserspeed detector, analyze the data collected from various vehicles and roads combined. ThenSPSS is used to study the characteristics of the distributions and finds that normal distributiongive the most appropriate fit of the data out of the Gamma, Logistic, and Weibull distributions.
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Figure 2: Greenshields model when speed limit is applied
2.2 Spring Block
Based on previous researches, the spring model has been successfully imple-mented into different fields such as earthquake simulation. We see great pos-sibility that the spring-block type models can also be applied in the vehicleovertaking. Assume a situation where the traffic is heavy so the distance be-tween each car is about the safe distance, but this is not a jam situation whereall vehicles barely move. When a car overtakes from left lane to right, we needto consider the that that the driver of the car behind in right lane may slowdown and keep the safe distance for that overtaking vehicle. Similarly, therest 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 behindwont affect the car ahead. Also, the distance for each following car to adjustdepends on both the reaction time for each individual driver and the originaldistance between each pair cars. Therefore, we model the pattern for a singlecar overtaking into a lane of cars as a special “spring” model.
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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 ofreaction time and distance d between each car. For this derivation, we assumek is a constant and the distance d between each identical car is the same. Whenanother vehicle overtakes to the front of this lane, treating this fleet as a entityand considering the speed difference ∆v, we can model the situation where thesegroup 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 willthen be
F = n ∗m ∗ a
Since this is a special spring model, where the behavior of each spring blockonly depends on the former block, the force exerted on the first car from thefirst spring will be:
F1 = n ∗m ∗ a = k ∗∆x1
∆x1 =n ∗m ∗ a
k
where ∆x1 is the distance this car need to adjust. Following this pattern andconsidering the fact that all the vehicles are treated as an entity and share thesame acceleration, we get:
∆x2 =(n− 1) ∗m ∗ a
k
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And finally we get:
∆xn =(n− n) ∗m ∗ a
k= 0
For large n, the nth vehicle, it will not need to adjust the distance anymore,which is the extent to which the system is not disturbed. This brief proof isintended to give our audience a sense that the influence of one single overtakewill have spring effect to the following lane, while the extent of this effect, suchas how many cars will be interfered by this overtaking depends on many outsidecoefficients. And this can be connected with the friction coefficient for thisspring 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 an-other one. In this section we will justify the number of vehicles the drivers areallowed to overtake once.
First, few assumptions must be raised:
• The state among vehicle B, C, and D is relatively stable, which meansthat front vehicles are faster than the rear ones. Otherwise, surpassingwill take place and break this stableness.
• Vehicles B, C, and D are keeping minimum safety distance at the momentvehicle A is ready to overtake (we will give careful explanation of thisminimum 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 Va, and Va ranges from 0th per-centile (V0, subscript denotes number of percentile) to V100, which is the max-imum speed. As depicted Figure 4, since A is faster than B and the expectedvelocity of all vehicles is V50, it is safe to infer that expected velocity of A(E(A)) is greater than V50. Since C is faster than B, the expected velocity C isthe expected velocity between V50 and V100, which is V75. Then if A is aboutto pass C, its velocity must be greater than V75, the halfway between V50 andV100. Similarly, if A wants to overtake D, its velocity has to be at least V87.5.Various studies shows that most freeways set the speed limit at V85. Thus, if Ais still fast enough to overtake D, high chance is that A has broken the speedlimit, which contradicts with our preliminary assumption that no vehicle moveabove it. At this stage, we can conclude that only in few cases can vehicle Aovertake 3 cars without moving back to former lane, and the proposed rule isreasonable enough.
Now we are measuring the validity of our prediction. Take the real data asexample, which is the same data surveyed in the research to derive the velocitydistribution. In a real freeway system, if a vehicle wants to overtake, it needsto be in a fast speed. Based on the data we collect from the normal speed
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Figure 4: A standardized normal distribution curve
distribution from one freeway in China. We observe that V50 is 85 km/h (22.22m/s), V75 = 95 km/h (26.39 m/s), V87.5 = 105 km/h (29.17m/s) and V95. Weassume a car A travels with Va = 30.56 m/s. It wants to overtake the vehiclesahead. Specifically,
Here we need to introduce a concept of safety distance for a vehicle, whichis a function of the speed (v) of this vehicle and the vehicle acceleration (A). Itcan be formulated as
S(V ) =V 2
2A+ 0.3V
Where we model A = 5m/sec2. And we calculate the original safe distancebetween Vehicle BC, CD, and DE. According to the data from former section3.1, we model the headway 2 to be 3s, and calculate a corresponding distancefrom A to the former car. This distance is 91.68 m. Here we present a charttracing the location of each vehicle starting from the process of this overtaking.A is located at position 0 originally.
2Headway measures the time it will take for the overtaking vehicle to travel from its tip tothe one of the front vehicle
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Time(s)\Location(m) CAR A CAR B CAR C CAR D CAR Et = 0 0 91.68 147.7 225.26 319.1t = 10.99 335.85 335.85 437 546 655t = 17.7 540 485 613 741 859.7t = 35.2 1074.8 Safely Surpassed 1074.8 1251.48 1394.5t = 35.89 1643.18 Safely Surpassed 1565 1794 1962
The road efficiency and this spring block effect are related. Go back to ourspring model. We believe that letting vehicles surpassing more than one car atone time will distribute more to the overall increase of freeway efficiency thanonly letting one overtake at one time. Since for each single overtake, there willbe a spring effect. And we want to reduce the spring effect. Therefore, duringthis iteration, we take into the factor of safety issue and want to minimize thespring effect. Therefore we specify that A can only get back into the lane whenthere is enough space. This length of this space should be larger than or equalto the sum of the headway distance of car A and safety distance of the vehiclethat car A just surpassed. From this simulation, at time 17.7 s when car Aovertook car B and left car B enough safety space, the distance between car Aand car C is shorter than As headway distance.
dAC(t=17.7s) = 613− 540 = 73m
dA headway distance = 91.68m
dAC < dA headway distance
→ A cannot surpass C
And therefore car A needs to keep going and overtake car C.When car Aovertook car C and left C enough safety distance (78.18 m), distance betweencar A and car D will be large enough:
DAD(t=53.89s) = 1794− 1643.18 = 150.82m > 91.68m
According to our rule, A has to get back into the lane. Therefore car Aneeds to get back into the lane first and travel until it reaches the point whenit is allowed to overtake car D. Therefore, in this very simulation, we first getthe 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 VB , VC , VD, andVE are defined in a decreasing occurrence in that normal distribution. Realizingthat car A is more possible to run into cars with median speed, we thereforeallow one car to overtake one more car, which is three cars at most at one singletime.
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2.4 Under Intelligent system
Both of our rule and the current rule rely upon human judgment for compli-ance. Psychological factors play an important role here. Take a real-life exam-ple. Sometimes when a new car is passing from behind, we will subconsciouslybrake even we already have kept a very safe distance with it. This situation isinevitable in the freeway traffic. However, when the speed is under control of anintelligent 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 withrespect to time equals the dierence in ow rate or ux in and out of that segment.
∂ρ(x, t)
∂t+∂q(x, t)
∂x= 0
3.1.2 Application and Re-model of Greenshields in Double-Lane Free-way
The Greenshields theory treats freeway as a single entity. In our case, sinceour focus will be mainly on the interactions between the passing lane and theright lane, it is reasonable for us to apply the Greenshields model into twodifferent freeway. Because the Greenshields defines values of the constants Aand B through field observations for each fixed position on the freeway, thecross-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 inthe GreenShield Model.Since in this case we are considering a heavy traffic, wechoose 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 becausefor both of the lane, the largest amount of vehicles that can be fitted into thesetwo 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 onehand, 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 desirableto 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
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Figure 5: Modified Greenshields model for a two lane freeway
this new density line whose corresponding flux is the sum of flux1 and flux2times a coefficient α, usually α ¡ 1. Since what we only care is the sum of fluxon the two lanes, there is indeed one line in between the original two lines.It connects ρmax and the midpoint of both lines x-intercepts. To see whetherthis line passes the above test, it suffices to pick the maximum of the sum offlux1 and flux2, which is the sum of maximum flux1 and flux2. If it couldrepresent the maximum, it can represent all the other combinations. Sincea + b = 2c, ρc(v) = −2ρmax/(a + b) ∗ v + ρmax. Since as the maximum fluxoccur at midpoints, maxflux1 = maxflux2 = ρmax/2, maxsum = ρmax/2 ∗2 = maxsum. Similarly, Fluxcmax = −2ρmax/(a + b) ∗ (a + b)/4 + ρmax =1/2ρmax. Thus, we find Fluxcmax = 1/2maxsum, which means ρc(v) is a goodrepresentation of ρa and ρb.
3.1.4 Current Traffic Rule Analysis In The Modified GreenshieldModel
The existing rule requires drivers to drive in the right-most lane unless theyare passing another vehicle. In this two-lane freeway case, we assume that theoverall density of the right lane is supposed to be much heavier than the leftlane, since after a passing, driver will be obligated to get back. Therefore, wemodify the state of each lane as the following graphic representation Figure 7.Intuition indicates that the black line represents the passing line, since it haslarger range of average velocity.
Our goal is to optimize the sum of the flux from both of the cross-section
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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 thecurrent traffic rule, this sum of flux is not maximized because we over-use the
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left lane while the right lane is too vacant. However in our proposal, if onevehicle is allowed to surpass multiple vehicles at one time, the density of thefast lane will increase, which will release the burden in the slow lane. Thisdynamic interchange can be represented in the following graph Figure 8. Wecan see that both ρ and V are closer to the midpoint, which means both lanesare 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 middlecurve equals to half of the maximum possible combination of the red and blackshaded area, the middle curve thus contains all the possible combinations of halfof the areas combination between red and black. And therefore we could usethis specific line as a mathematical model to interpret the dynamic behavior forthat interchanging system. From now this traffic physical system problem canbe purely transferred into a math model, represented by the following graphicrepresentation (Figure 9).
The rectangular area under the midpoint of this blue line represents thesituation which represents the percentage of the maximum combination of theflux from both lanes. Assume the current traffic rule determines the currentflux 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 atone time, the effect of our proposal to this dynamic system can be represented
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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 opti-
mization point, the midpoint. For a pure optimization, ideally, A will convergeto the midpoint.From here, we conclude that one of the main effects of our pro-
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Figure 11: The effect of proposed rule
posal, compared with the existing rule, is the increase of the possible averageflux 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 allowingvehicles to overtake multiple cars at one time. We consider this interchangeabil-ity as a very important factor to maximize the overall performance of the wholesystem. In addition, our modified Greenshields Model can be implemented intomultiple-lane freeways. The basic principle is the same, while the actual schemeto find the linear representing curve is more time-consuming, and thus it willnot 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 visual-ize a set of pre-specified rules. In this chapter we are going to present Cellu-lar Automata model to simulate multi-lane traffic conditions. Below are somescreenshots of our sample runs.
The goal is to give concretive and quantitative performance comparison be-tween the current and proposed rules, and our hope is that the simulation couldgive us confirmation on the superiority of the proposed rule. The result is asfollows.
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Figure 12: Screenshot of Cellular Automata in process, starting rough at generation10, ending at generation 30, with road length 80, maximum velocity 8 and density at0.2, 0.4 and 0.8 respectively
Figure 13: A comparison between the ρv curve for simulations under two rules
The Blue curves represent the data collected under the current rule. And thered curves represent the data collected under our proposed rule. It is obviousthat under the same speed limit precondition, our cutoff density (ρc) is muchlarger. This indicates that on average, vehicles under our rules can proceedwith a faster speed. Another important observation we made is that the speedvariance of our system is lower and more converged. Speed variance is definedto be the variance of all the vehicle velocity after a long period of time. Atthis moment, vehicle velocity stop spread randomly and enter into a relatively
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Figure 14: Distributions of velocity variance verse density under two rules
stable state. Recall that we measure safety as speed variance in the “Sum-mary” section. The link between variance and safety is strong. More variantspeed usually results in more overtaking, thus more possible collisions. Besides,more overtaking increase the spring effect which is not desirable in the freewaysystem. Up to this point, both our data and model have indicated that ourproposal provides the system not only with a larger possible flux, but also asafer condition.
3.3 The “LYZ” and Speed Limit
At the beginning, we introduced basic Greenshields Model, which serves per-fectly to the superiority of our proposed rule. It is tempting to reuse it to opti-mize traffic flow and safety at the same time. However, this basic model is notcomplicated enough to model these two factors, because as stated in the “Intro-duction” section, the max flux point lies on the midpoint of the density-velocitycurve. Also, the basic model does not consider speed limit and its influence onthe 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 indi-vidual car may tend to travel at the highest speed possible, either the resultof speed limits or road conditions or driver caution, call it Vmax. As statedpreviously, before a certain density point (ρc), the overall average density willkeep increasing without interfering the average speed. After this ρc, velocitytends to go down with increasing density, so we should assume that dv
dρ < 0. In
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addition, there is certainly a ρmax, where vehicle is next to each other withoutany 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, wepresent a more realistic version Greenshields Model:
v(ρ) = vmax(1 -ρ
ρmax).
Figure 15: Advanced Greenshields V-P curve
3.3.2 The “LYZ” Model
It seems that the advanced Greenshields Model approaches to reality much morethan the basic version, but, based on our careful research and practical drivingexperiences, we are able to establish our own model to give even more realisticmodel for the velocity-density curve. It is called “LYZ” Model. The “LYZ”Model actually generalizes D, the distance between each vehicle. Previously, Dis defined to be a constant. However, common sense and intuition suggest thatthe faster one is driving, the longer distance he should keep. Thus, this intuitiongives us a basic idea that D is a function of velocity and should be monotonicallynon-decreasing. According to elementary physic, the distance basically consistsof two parts: the drivers response distance and the deceleration distance. Theformer is defined to be the distance travelled between the moment the driver isthinking about braking and actually braking. The latter is the distance betweenmoving at speed V and completely stopping. Thus, the equation should be of
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form D(v) = tv + v2
2A , where t is human beings reaction time and A is thedeceleration rate. In general, t ranges from 0.3 0.7 seconds, so we take theaverage, which is 0.5 seconds. The argument about A is that for A = 3m
s2 ,ordinary human beings will very uncomfortable, and for A = 5m
s2 , they areextremely likely to be hurt. So, we take the average, 4m
s2 .Now we are ready to officially propose the “LYZ” model. Since ρ = 1
(L+D) ,
we have, ρ = (5 + 0.5v + v2
8 )−1
V =
{Vmax, ρ ≤ ρmin√
8ρ − 36− 20, ρmin ≤ ρ ≤ ρmax
Where ρmin = (5 + 0.5Vmax +V 2max
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 velocityU and density V. however, please recall that the variable we really want tomaximize is flux Q, the average number of vehicles going by one segment of theroad per unit time. Bigger flux implies more efficient transportation. Recallthat flux is the product of density and velocity, i.e., Q(ρ) = p ∗ V (ρ), we have
Q =
{ρVmax, ρ ≤ ρmin√
8ρ− 36ρ2 − 2ρ, ρmin ≤ ρ ≤ ρmax
This is the objective function we would like to maximize. Suppose we have alist of choices on speed limit. We would like to see on various speed limit levels,what the maximized flux from left to ρmin is
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Qmax = ρmin ∗ Vmax = (5
Vmax+ 0.5 +
Vmin8
)−1
Based on the above formula, we give out this table:
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 Qmaxdecreases. Right to Pmin. Taking the derivative of Q with respect to ρ and setit 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 from15mph to 75mph, the choice on speed limits does not affect the theoreticalmaximum flux. The maximum flux possible under “LY” model is 0.48vehicle/sec, unless the speed limit is set under 15 mph (which is an extremelystrange and rare setting). However, under each scenario, our proposed rulegives better performance.
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4 Conclusion
4.1 Advantages
• Simulation confirms the superiority of proposed rule in promotingperformance 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 Qare based on unit length and unit time interval, not on instantaneousnumber. Otherwise, continuity of traffic flow will not hold.
• Each driver has consciousness to make active change, which makes thebehavior of the system hard to predict.
• In multiple lane system, the vehicles are only not allowed to stayconsistent in the right most lanes. Regarding the situation where carscan stay in multiple lanes, this will make our derivation verytime-consuming. We therefore simplify our system in a way that thevehicle needs to go back to the former lane after passing up to 3 othercars. 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 andsimulation can be applied in countries driving in the left or right norm. Infact, the following graphs are cited from a scientific journal by Hesham Rakhaand Brent Crowther. They collected traffic data from several main freeways indifferent 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 HollandTunnel in New York City, US.Both of these graphs are in similar shapes. So, we see considerable similaritybetween right- and left- driving countries.
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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 anintuitive sense, while this hypothesis worth future research.
• In the application of Greenshields into two lanes situation, quantifiedalgorithm is preferred. It will be very useful that future researchers canrelate a time constant τ forthesystemtoreachequilibrium.
• We also believe that our model will affect the value of ρc. We believe ourproposal is worth studying for future experiments.
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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.” TransportationResearch Record. (2002): 248-262. Print.
Qingliang Zhang and Guozhu Zhao. ”The Establishment of the Traffic SlowSystem.” 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 OperatingSpeed Distribution of Highway Cross-section. Journal of East China JiaotongUniversity. Vol. 25 No.5. Oct. 2008.
Yulong Pei, Guozhu Cheng. Research on Operation Speed and Speed Limit forFreeway in China. Journal of Harbin Institute of Technology. Vol.35, No.2.Feb. 2003
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