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Traffic Control Analysis

By Ken Guinto

San Diego City College

Professor Camarena

Math 151 – Calculus with Analytic Geometry 2

Department of Mathematics

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Abstract

As our current century flourishes with advancements in mechanical science, it is the art

of a civil engineer to facilitate these new technologies for the convenience and use of all men. In

the process, it is important to monitor state traffic and continue to upgrade all the reliable

resources that are necessary to achieve maximum revenue and safety for all citizens. These

resources are bridges, dams, roads, aqueducts, and canals, which provide humans with smooth

transitions from point A to point B. Further, as the proportion between new technology and the

innovation of metropolitan cities increases, population and drivers increase too. As a result,

traffic starts to accumulate within metropolitan cities, which provide delay and inconveniences

for all drivers. During construction maintenance in highways, traffic sometimes accumulates

when drivers intend to merge from one lane to another. Moreover, in between countries, such as

the United States and Mexico’s border, traffic continues to be an inconvenience for drivers. In

the past couple of decades, research has been conducted on traffic flow problems. With a further

analysis of traffic in intersection, highways, and the Otay Mesa border, civil engineers can

provide an understanding on how traffic is produce in order for all drivers to be more effective

and patience while driving, which simultaneously stimulates production in our nation.

Objectives: To expose the major factors of traffic in order to empower all drivers’ understanding

of traffic. Thus, providing solutions for drivers in order to proper accommodate to traffic and

alleviate the nation from the discomfort of traffic within big cities.

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The Lighthill-Whitham Model

M.J. Lighthill, specialist in fluid dynamics, worked with G. B. Whitham in 1955 in order

to simulate macroscopic traffic flow (Jackson 4). This was known as the LW model. However,

their initial equation treats traffic only as a continuum (Jackson 4).

pt+qx=0

Where we have these following representations:

p(x,t) is the traffic density (Vehicles per miles)

q(x,t) is the traffic flow (Vehicles per hour)

v is the mean traffic speed (miles per hour) & as a function of p.

A different model of this equation was founded by Payne in 1971 by adding a second differential

equation including a mean speed-density relationship and introducing other terms, such as

drivers’ anticipation, traffic flow relaxation, and changes in traffic volume (Jackson 5).

This equation looks like:

( pv)t+( p (v2+c02 ))x=

p (U ( p )−v)T

and holds these representations:

v(x,t) is the space mean speed (km per hour),

T is the relaxation constant

c02 is a positive constant

U(p) is the equilibrium speed-density relationship

Analysis 1 – Jamitons

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With the Payne-Whitham model, vehicles traveling in a single-lane and uniformed road

can provide waves represent continuum traffic models. Assuming that all drivers behave under

the same laws and all unanticipated actions are eliminated, the traffic models can be viewed only

as a low traffic density model (Seibold). Clearly, not all roads are linear. Therefore, if one would

like to examine a road that curves, small perturbations would amplify and as a result, yielding to

models referred as phantom traffic jams. According to Seibold, these jams arise in the absence of

any obstacles. He states, “Vehicles are forced to brake when they run into such waves. In

analogy to other traveling waves, so called solitons, we call such traveling traffic waves

jamitons.” To further elaborate, in Pane-Whitham type of traffic models, instabilities grow into

traveling detonation waves (Seibold).

Below, there is a chart that illustrates a jamiton-induced fundamental with measurement data

*Tagged from Seibold.

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From the data of Seibold’s studies, one can state that two major causes of traffic. The first

statement is that denser traffic travels slower. The second is that all drivers require a certain time

interval in order to react to incoming traffic. However, it is still crucial to remember that

Seibold’s studies did not introduce a scenario where a traffic light creates an increase in

congestion. Thus, it transitions us to discuss how traffic lights affect traffic.

Analysis 2 - Intersections

In a continuous flow intersection, traffic is allowed to flow from one intersection to

another, if and only if there are green lights present in a linear direction. When this occurs,

vehicles are allowed to advance on their travels. Co-terminus, the perpendicular linear road has a

red light and the cars waiting for the light to turn are unable to proceed.

The image above illustrates a continuous flow intersection. Suppose in one scenario, the first

signal light is red and the second signal light is green. This scenario creates a line of vehicles

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behind the first signal light that are unable to proceed, which creates a limit on the number of

cars that are allowed to advance pass the second signal light. Simultaneously, on the

perpendicular linear road, cars are also unable to proceed because they must stop in order to

allow the cars from the second light to pass. Clearly, we can see how this creates a problem.

Only when both the first and second signal lights are green is the traffic able to flow continuous.

According to the Great Britain’s Ministry of Transportation, the maximum seconds that a signal

light is allowed to be green is 70 seconds and the minimum is 35 seconds. When a light is green

at its maximum, it is referred as a Max-Out or Maximum Green. When a light is green at its

minimum, it is referred to as a Minimum Green. Moreover, when there is a Max-Out, the flow

rate of vehicles increases. However, the probability that a Max-Out is present is lower than that

of a Minimum Green. Therefore, a Minimum Green is one factor to a buildup of traffic within an

intersection.

*The chart below represents the probability of a Max-Out with respect to the Total Flow Rate

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Analysis 3 – Peak Hours

During a certain period of day, the linkage between home and place of work create a jam

of cars that try vigorously to maneuver through each other, but instead create a big contribution

to traffic. This period of the day is called peak hours. Great Britain states, “Every morning, and

in the reverse direction every evening, great tidal flows of movement take place between the

residential areas and the places of work” (36). It needs to be understood that unless the hours of

work for all citizens is adjusted for different hours within all twenty-four hours of a day, peak

hours are inevitable. Altering the hours of work for all citizens, however, will create an even

more inconvenience than traffic itself. Therefore, when in traffic, drivers should remember that it

is never as bad as it seems. If drivers learn to anticipate the amount of time that is required to go

from point A to point B, (anticipating the amount of traffic, car maintenances, displacement, etc.)

than there could be less inconvenience within the time of travel.

*The chart below represents the volume of vehicles that are present at a given time interval due

to peak hours.

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Analysis 4 – Speed Limits

Speed limits are another factor that affects traffic. For example, a high speed limit will

decrease the time of travel, but simultaneously puts the driver at risk of severe injury or death

due to the high impact if there is contact with any obstacles on the road. However, the limit of a

speed limit is determined by such factors: Environmental Capacity, State Minimum, and

Landscape.

*This chart illustrates the number of vehicles that would travel in 1 hour under the following

conditions:

1. A maximum speed limit of 90 and a minimum at 50.

2. 2 Lanes with no speed limit

3. 3 Lanes with a speed limit of 60.

5:00 - 5:15 PM

5:15 - 5:30 PM

5:30 - 5:45 PM

5:45 - 6:00 PM

0200400600800

100012001400

Time Interval

Volume (Ve-hicles)

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Analysis 5 – Borders

The amount of drivers in America are increasing and as a result, creating traffic. The

ministries of transportation from Great Britain have published the “Traffic in Towns” textbook,

which contains studies on the attempts of past engineers that have tried to permanently remove

traffic, as well as provided theoretical approaches, based on statistics, to ease traffic. Great

Britain agrees when they state, “American experience shows that as vehicles increase in numbers

the usage per vehicle also tends to increase” (Great Britain 27). We can infer that if there were

fewer cars on the roads, there would also be less traffic. However, in order to reduce the number

of cars, we would first need to determine what factors increases the use of private vehicles. Great

Britain argues that “the growth in the number of private cars would seem to depend primarily

01000

20003000

40005000

60000

102030405060708090

2 Lanes, No Speed Limit3 Lanes, Speed Limit 60

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upon the growth of incomes, but it is also dependent upon the price of cars, insurance rates, [and]

taxation levels” (26). Unexpectedly, the amount of income that individuals allocate plays a part

in the amount of traffic that is formed at a given time because there is a correlation between

income and cost of maintaining a car. In the United States, a large majority of our population

comes from Latino communities, which yields a lot of tourism from the U.S. and Mexico, a

developing nation. In the Otay Mesa U.S. and Mexico’s border, there often seems to be traffic

for incoming vehicles headed to the U.S. However, if we study closely, we can determine that

the busiest hours of traffic are throughout the middle of the day. With this statistic, we can

provide drivers with appropriate hours of traveling from Mexico to the U.S.

*The chart below yields the following days, respectively: Tuesday, Wednesday, and Thursday.

Analaysis 6 – The Work Zone

On time of construction, the rate of traffic flow reduces due to it safety protocols, such as

providing a toll road for incoming vehicles to proceed to their destination. Co-terminus, this

provides a sweep distance from the work zone and allows space for engineers to work. One

possible solution to the decreased of traffic flow is to reduce the number of hours in the work

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zone, which eliminates the need for a toll road. However, if the hours of work zone are reduce,

there needs to be an increase on the quality of work and materials used, which yields a higher

national budget cut. When this occurs, it is referred to as accelerated construction. Further, when

the time of construction is high, but the budget is at a constant, it is referred to as traditional

construction. From the chart below, we can argue that since 2009, accelerated construction has

been increasing. Therefore, we can conclude that traffic will reduce due to the accelerated

construction, but at the cost of our nation’s budget. In addition, it is important for drivers to

decide if it is worth tolerating the inconvenience of traffic for a few minutes and in exchange

keep our national budget high or to eliminate as much traffic as possible at any cost.

*Traditional Construction and Accelerated Construction with respects to the National Budget in

Millions.

*Tagged from the U.S. Department of Transportation.

Convergence rates of a stable Payne-Whitham Model

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Convergence rates of the Payne-Whitham Model can be determined by comparing other

forms of the PW model. When the PW model is stable, at time T 0=500T for two different lanes

in a given road, 2N and N, yields this following equation (Jin 13).

(U ¿¿ i2 N) fori=12 N 2 N cell¿ & ¿ (U ¿¿ iN) fori=1

N N Cells ¿

A difference vector, ¿ ,between these two solution is define as (Jin 13)

e i2 N −N=1

2¿ + U 2 i

2 N) - U iNi=1, …, N,

And the relative error with respect to L1 , L2 ,∨L∞ normally can be computed as (Jin 13)

∈2 N −N = || e2 N −N∨¿

Finally, the convergence rate r is obtained by comparing the two relative errors: (Jin 13)

r = log2(∈2N −N

∈4 N−2 N )

Here U can be p or v (Jin 13).

*Convergence rates for the implicit method (Jin 14).

p 128-64 Rate255-128 Rate

512-256 Rate

1024-512

1.95E-01 0.79

1.12E-01 0.88

6.12E-02 0.93

3.20E-02

2.57E- 0.64 1.65E- 0.76 9.78E- 0.85 5.42E-

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01 01 02 025.48E-

01 0.374.24E-

01 0.562.88E-

01 0.731.74E-

01

v 128-64 Rate256-128 Rate

512-256 Rate

1024-512

4.21E-05 0.78

2.45E-05 0.87

1.34E-05 0.93

7.04E-06

5.61E-05 0.62

3.65E-05 0.74

2.19E-05 0.84

1.22E-05

1.30E-04 0.35

1.02E-04 0.55

6.98E-05 0.72

4.25E-05

Conclusion

Despite our nation’s current status of advanced technology that would imply faster

transitions for road traveling, it is essential that drivers accept the presence of traffic as a

sacrifice for urban development and safety. Although the various factors that affect traffic have

been analyzed and targeted for accommodation – such as, state population, peak hours,

geographical landscaping, and road structures – to yield efficient driving, drivers should keep in

mind that traffic is almost inevitable. Further, it is crucial to remember that other factors, non-

traffic related, can create more discomfort than the frustration of traffic. A simple example is

when the quality of life of a nation is infiltrated by changing the hours of work of all citizens in

order to eliminate peak hours. Moreover, the best way to execute a safe driving experience for all

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drivers and non-drivers is to be prepared to deal with traffic as soon as they leave their homes.

Otherwise, traffic will continue to create an unwanted discomfort and make life more difficult.

Although for some drivers, patience is difficult to gain, with this analysis of traffic, drivers

should be more informed about what affects traffic. Further, for all civil engineers, the Payne-

Whitham model should be a starting point towards analyzing traffic. With the Payne-Whitham

model, many simulations can be articulated and use to anticipate traffic. However, the Payne-

Whitham model is limited due to its perfect error. Yet, one should always remember that traffic

can never be anticipated precisely. A benefit that all drivers should have gained through this

analysis is a deeper understanding of what affects traffic to accommodate for it and as a result,

create a safer driving experience.

Annotated Bibliography

Great Britain. Ministry of Transport. Traffic in Towns; a study of the long term problems of

traffic in urban areas. London, H. M. Stationery Off., 1963.

This source is composed of reports form the Steering and Working Group. The authors of this source intended to have their work read by not only those working on the field of transportation, but to the public as well. The authors believed that public understanding ofr traffic will be the best factor towards finding a solution. This source contains lessons learned from various cities that have tried to ease the tension of traffic as well as theoretical approach towards solving the problem of too much traffic. Thus, I will draw from this source in my analysis. However, this source also contains an abundant amount of reports, which I will not include on my paper.

Hyperborder : the contemporary U.S.-Mexico border and its future.  New York : Princeton

Architectural Press, ©2008.  

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This source contains statistics and a short history of Mexico’s and United States’ border. I will use the information of this source to touch upon the environment, use of vehicles, and economic status of Mexico to support my claims. However, this source also contains information about narcotics and how they contribute to the safety of Mexico’s citizens. I will not use this information on my paper.

Jackson, Amy. “Estimation of Parameters in Traffic Flow Models Using Data

Assimilation.” University of Reading. September (2011): 1-67. PDF file.

Amy Jackson studies in The University of Reading, and she provides an excellent source that effectively introduces the Payne-Whitham model. In her dissertation, targeted to the Deparment of Mathematics and Statistics, she also provides an overview on how less traffic can provide a benefit for all drivers. This source is a key element on my paper. A weakness in this dissertation is that Jackson also includes data about Adjoint Models and Implementation of the Minimisation, which I will not touch upon.

Jin, W.L., and Zhang, H.M. “Solving the Payne-Whitham Traffic Flow Model as a Hyperbolic

system of Conservation Laws with Relaxation.” University of California, Davis. (2001):

1-24. PDF file.

Jin and Zhang have studied in the University of California, Davis. Similar to Amy Jackson’s dissertation, Jin and Zhang have provided an abstract on the Payne-Whitham Model. In addition, they have shown the PW model as a hyperbolic system of conservation laws with relaxation. In my paper, I mix the ideas of Amy Jackson and these authors to create a vivid explanation of the Payne-Whitham Models. In their paper, however, an irrelevant section would be the Kerner-Konhauser model of speed-density to flow-density relations.

Leibbrand, Kurt, 1914-1985. Transportation and town planning. Cambridge, Mass. : M.I.T.

Press, 1970, ©1964.

The author of this source has wide experience as a transportation planner and remarkable knowledge of the history of urbanization. In his work, the reader can access a reliable approach to town planning. Leibbrand states that a town entirely free of traffic and citizens adapting to traffic easily is impossible. This source contains a section titled Traffic Counts and Estimates, as well as another titled Road Networks, which contain relevant information to

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my topic. As a result, I will include information from these two sections. Hence, this source also contains irrelevant information, such as public transportation units. That section will not be included on my paper.

Muir Wood, David, 1949. Civil engineering : a very short introduction. Oxford : Oxford

University Press, 2012. 

The author, David Muir Wood, is Emeritus Professor of Civil Engineering at the University of Bristol and Professor of Geotechnical Engineering, in the University of Dundee. Further, he is a Fellow of the Royal Academy of Engineering and of the Royal Society of Edinburgh. In his published short introduction, David describes the role of the civil engineer. Further, he outlined challenges that he predicts will arise within the industry of civil engineering. Although Wood’s short story only provides a quick overview on civil engineering, I will use this source to provide an accurate introduction about civil engineering in my paper.

Seibold, Benjamin. “Traffic Modeling – Phantom Traffic Jams and Traveling

Jamitons.” National Science Foundation. (2009): Web. 10 Apr. 2015.

Benjamin Seibold has lead a research group from students in Temple University, Massachusetts Institute of Technology, University of Alberta, McGill, and KAUST. On their research, they have discussed the functions of jamitons, phantom traffic jams, and properties of jamitons that have been produced from applying the Payne-Whitham model into various simulations. In my paper, I have used their results and data to support my analysis. However, I did not touched upon their discussion of circular roads.

Wright, Paul H. Transportation engineering : planning and design. New York : J. Wiley, c1998.  

The authors of this source are professors from Georgia Institute of Technology, University of Technology, Loughborough, and Vanderbilt University. The authors have split their textbooks into six parts. The first three parts deal with planning and other non-engineering aspects of transportation. On the other hand, the remaining parts focus on the design of land, air and water transportation facilities. Part two of this multimodal work contains information on traffic control devices and procedures, which I will use on my paper to examine research already conducted on traffic control. However, this book also contains irrelevant information that does not support my topic, such as evaluation of transportation plans, airport configuration, and more others that I will not use.

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