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Team #70260

Page 1 of 17

The End of Traffic: A Probabilistic Model of

Combating Congestion with Self Driving Cars

Abstract

In order to analyze the possible uses of self-driving and cooperative

cars in the effort to minimize congestion on busy roads this paper creates

a probabilistic model of how AV’s can mitigate congestion caused by the

cascading nature of deceleration. Since there are many different causes of

traffic, some of which have little to no overlap of possible solutions, we

decided to center our work around the largest cause of congestion on busy

streets: the deceleration of cars entering or exiting lanes.

Our model first seeks to analyze the way in which the deceleration of

a car causes the driver behind them to decelerate at a faster rate. Combining research of the risk-adverse biological impulses of humans with

Cascade Control Systems, commonly discussed in reference to logic systems, we describe how deceleration of a car affects the cars behind it. We

looked at these effects in terms of the rate of deceleration, the period of

deceleration, the distance between the decelerating car and a chosen car

behind it, initial velocity and a constant c used to capture humans’ tendency to decelerate more then necessary.

We begin to discuss the utilization of AV’s to counteract this cascade

by creating 6 different possible formations of AV’s (3 formations at 10%

market penetration and 3 formations at 50%). Upon creation, we evaluate

under what conditions the AV’s could completely mitigate the affects of

the cascade by not having to decelerate at all, while still managing to not

collide with the car in front of it. If the AV is able to maintain its initial

velocity the chain of cars behind it will be unaffected by the deceleration

ahead and have no reason to slow down. In our model, we evaluate the

efficacy of different platoon sizes - considering the number and position of

AV’s as well as human driven cars - under varying conditions.

After creating and evaluating these different formations, we go on to

create a joint PDF which provides the probability of different ways of exiting a lane as a function of time, deceleration and c (where the marginal

density of c describes how much drivers are likely to over-decelerate). Using our analysis of AV’s ability to handle different forms of deceleration

as the range of integration for our joint distribution, we develop a mechanism for determining what percent of congestion can be mitigated by

simple communications between AV’s at any market share.

While this model is built off of a number of simplifications and assumptions about the way in which drivers decelerate and the way in which

this deceleration’s ripple affect creates congestion, our model still provides

analysis and results that can be applied to traffic situations regardless of

speed limits, number of lanes, types of roads, weather conditions and

more. Our model lays the ground work for optimization and other solutions while additionally demonstrating the extreme benefits that will be

seen immediately when AV’s hit the market.

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Team #70260

Page 2 of 17

Contents

1 Introduction

2

2 Model

2.1 Assumptions . . . . . . . . . . . . . . . .

2.2 Human Traffic . . . . . . . . . . . . . . .

2.2.1 Cascade Control System in Traffic

2.2.2 Human reaction . . . . . . . . . .

2.3 AV response . . . . . . . . . . . . . . . . .

2.3.1 10% AV cars . . . . . . . . . . . .

2.3.2 50% AV cars . . . . . . . . . . . .

2.3.3 90% AV cars . . . . . . . . . . . .

2.4 Probability of Success . . . . . . . . . . .

2.5 Accidents and other neglected factors . .

2.6 Strengths and Weaknesses . . . . . . . . .

2.6.1 Strengths . . . . . . . . . . . . . .

2.6.2 Weaknesses . . . . . . . . . . . . .

3 Example: Application to

3.1 Causes of Traffic . . .

3.2 Imaginary Day . . . .

3.3 Rainy Day . . . . . . .

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Seattle

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4 Conclusion

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5 Appendix A

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6 Letter to the governor

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1

Introduction

Traffic jams and congestion on the road feel like a never ending

problem. From the immense number of drivers on the road to accidents,

construction, bad weather, and bad drivers, it can seem like there are simply too many problems to fix. However, if steps aren’t taken to combat

traffic jams and congestion the problem will only get worse which is hard

to imagine seeing as congestion already costs Americans 63.2 Billion dollars per year(Longley). We don’t have a proposition to solve every traffic

problem, but we feel confident that the introduction of AV’s for general

use can improve the quality of driving, decrease travel times and decrease

the number of deadly accidents that occur.

Our model for the commercial introduction of AV’s focuses on how

the AV’s will be able to reduce general traffic and road congestion by mitigating the cascading effect of deceleration through traffic. We propose the

dispersion of AV’s evenly throughout general traffic. Our model analyzed

the effects of the AV’s on traffic at 10%, 50%, and 90% market penetration

assuming that the main factor causing traffic jams is the deceleration of

cars during lane changes, merging, and at exits.

2

Team #70260

2

2.1

Page 3 of 17

Model

Assumptions

Below is a list of assumptions made to decrease the complexity of our

model in order to allow this paper to highlight some of the key benefits

of the integration of AV’s into daily traffic:

Assumption 1.1: All cars are traveling at the same velocity (what velocity they are all traveling at can be changed).

This assumption helps us determine the beginning velocity of each

car on the road, which simplifies our calculations.

Assumption 1.2: All cars have the goal of traveling at the speed limit.

This assumption specifies that all the cars will accelerate to the speed

limit if it does not impact their perceived risk of collision with the car in

front of them.

Assumption 1.3: The distance between each human cars is a constant

of 10 meters.

We are making this assumption because 10 meters is a relatively safe

distance; it provides drivers enough time to response to current road situations.

Assumption 1.4: The nth human drivers slow down at a rate of a ×

c(n−1) m/s2 , where c is a constant, a is the acceleration rate of the 1st car

entering the exit and c ≥ 1.

This assumptions states the fact that once the front car starts to decelerate, the rear car begins to decelerate at a faster rate.

Assumption 1.5: AV’s are going to be cooperative cars to some extent

allowing clear communication between cars near each other on the road.

Since we will need AV cars to be able to intelligently switch lanes we

need to give AV cars more flexibility.

2.2

Human Traffic

We’ve all encountered the same situation: we are driving on the

highway and suddenly there is a traffic jam. We are thinking that at some

place in the front there is a car accident or road construction. But when

we move forward there is nothing. All the cars seem to stop inevitably

while after a certain point, the traffic goes back as normal. This is usually

caused by the traffic flow instabilities. In this model, we introduce and

analyze the real causes of traffic and highlight one potential solution which

utilizes self driving cars. To describe one of the main causes of traffic we

must first discuss Cascade Control Systems.

2.2.1

Cascade Control System in Traffic

Definition 1.1: A cascade control system is a multiple-loop system where

the primary variable is controlled by adjusting the setpoint of a related

secondary variable controller. The secondary variable then affects the primary variable through the process (”Industrial controllers”).

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Team #70260

Page 4 of 17

In our traffic model, we treat each car and its driver as a system.

The primary variable is the car’s acceleration, while the secondary variables are the acceleration of the back car and the relative distance between

the two cars. The primary variable is therefore affected by the secondary

variables. In the next system, the third car’s primary variable is affected

by the secondary variables of the second car. When all the cars are traveling at the same velocities with constant distance between each other

the whole system will stay constant. However, once the first car starts to

decelerate, even if it is only for a small amount, the second car is affected

to decelerate eventually in order to avoid collision. In this way, it causes

a chain reaction, or cascade, that disrupts the whole system (Office).

2.2.2

Human reaction

After discussing the cascaded control system in traffic, we next analyze human reaction in traffic. This is best illustrated with a simple

example. Suppose there are many cars traveling in the same lane with

constant and equal velocities on a highway. This is the constant state

of the cascade control system. However, as one car approaches an exit

it decelerate in order to safely exit the lane. It starts to decelerate at a

rate of 5 sm2 . This action affects the tailing car (the car behind the decelerating car). It sees the first one decelerating, after a short response

time of (≈ 1.5) seconds (Green), and starts to decelerate at a rate of 6 sm2 .

Similarly, the third car sees the second car decelerating and starts to decelerate at 7 sm2 . This cascade could continue until one car slows to a stop.

After that, other cars behind it finally stop as well, causing a small traffic

jam. However, on a busy highway where there are plenty of exits and cars

traveling one next to other merging lanes consistently, according to the

cascaded control system, a small deceleration of a car will interfere the

whole system for creating miles of congestion and instability (Horn).

The cause of our (humans) natural tendency to slow down faster then

the car ahead of us can be attributed to our risk adverse nature. Risk

adverse means we are more willing to take accept small inconveniences

(traveling a little slower) then risk large consequences (misjudging the deceleration and crashing) even if the probability of that is small. In our

model once a driver sees the front car decelerating they press the break

to prevent from hitting the front car, even though there is still a long distance (10 meters) between the two cars. While this is a reasonable thing

to do in order to be certain you won’t get in an accident most drivers

decelerate much more then necessary creating problems. Eventually the

rapid deceleration combined with short reaction time becomes excessive

decreasing the average speed and creating other risks.

Looking back to the previous example suppose the first car decelerates at 5m/s2 . The second car, which is 10 meters behind, reacts after a

normal response time of 1.5 seconds. During the two seconds, the distance

between the two cars shorten to 7.5 meters. The second driver thinks that

it is not safe to maintain a distance of 5 meters, so he reacts quickly and

decelerates at 6 sm2 . In fact, the second driver does not need a 6 sm2 deceler-

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Team #70260

Page 5 of 17

ation rate, a 4 sm2 rate would also work since the front car would exit before

the two cars collided. However, since drivers are risk averse, they want to

make sure that everything is safe, which causes unnecessary decreasing in

speed. This chain reaction continues until all the cars in the same lane,

including any cars entering the lane miles back, are driving way below the

speed limit.

Drivers’ risk averse biological and not something easily changed. However, through the use of precise lightning fast calculations that will be

possible with autonomous (self-driving) cars and the communicative ability of of cooperating cars this cascade can be stopped before it gets to

extreme. Unlike humans, self-driving cars are driven by computers, which

will analyze current road situations, run complicated calculations and execute the best options within a fraction of a second. Self-driving cars can

easily deduce optimal options that preserve both safety and speed. In the

example above, if the second car is a self-driving car, then it will easily

calculates the optimal 4 sm2 deceleration rate and executes the command.

Besides the deceleration rate, the self- driving car can also calculate the

optimal distance it will keep from the front car. By evaluating different

variables, the self-driving car can keep the loss to the minimum which can

regulate the traffic in an entire lane. Our analysis that will be presented

in the next section goes so far as to show that even with only 10% of the

cars on the road being cooperative cars the majority of the time AV’s will

be able to maintain a safe distance from the car in front of it without

slowing down at all, effectively stopping the cascade in it’s tracks.

2.3

AV response

In terms of the three variables f (final velocity in meters/second), c

(risk) and t (time of deceleration) the deceleration of a car in front of an

AV is modeled to be:

− fm

26.8224 m

s

s

ts

∗ ctrain ,

where train is the number of cars between the initially decelerating

car and the AV including the initially decelerating car.

With this information we are able to represent the minimum distance

between an AV and the car in front of it so that the AV never gets within

two meters of the car in terms of the previously discussed variables. Using

simple physics equations and some basic algebra steps it is possible to

write:

− fm

∗ ctrain )ts

Trailing Distance = 2 − (26.8224 m

s

s

Now, in order to create a probabilistic model that the trailing distance is less then some number d we need the PDF’s (Probability Distribution Functions) for the three variables; f,c and t.

For f, because the change in velocity will always be positive, we pick

a PDF with support z > 0. Based on the assumption that most drivers

will slow down approximately 2.2352 m

(5 Miles/Hour) before exiting with

s

5

Team #70260

Page 6 of 17

some drivers slowing down more and it being unlikely that drivers slow

down more then 10 Miles/Hour we chose a Gamma distribution with a

shape parameter of 2 and a scale parameter of 1 where 1 unit in the distribution corresponds to a 2.2352 m

decrease in velocity. This gives us the

s

equation:

velocity − velocity

e 2.2352

g(velocity) = 2.2352

,

Γ(2)

where velocity is the change in velocity.

For c, because we are assuming a risk averse driver, we want a density

that gives an extremely high probability of c > 1. Specifically we assume

the expected value of c to be 1.05 with the it being extremely likely that

the actual c value is within the range (1, 1.1). These conditions led us to a

Weibull distribution with a shape parameter of 100 and a scale parameter

of 1.05:

99 −( c )100

c

100 1.05

e 1.05

f (c) =

1.05

For the final variable, t, because time is obviously greater then zero

and it is possible, although unlikely, that a car did decelerate over an

extremely long period of time we looked for a PDF with support y ≥ 0.

Note that we do make it possible for the exit time to be zero because it

is possible for a car to exit without decelerating. For the shape of the

distribution we assume that it is most likely that drivers will only decelerate for a small amount of time before exiting and that the distribution

would be a decreasing function, where it is a shorter exit time is always

more likely then a longer one. These conditions led us to an exponential

distribution with a parameter of 2:

h(time) =

e−

time

4

2

,

note that time is divided by four instead of two in the exponent, this is

simply to scale the time variable we used previously to the distribution.

Because we are assuming all of these Random Variables are independent the joint distribution is simply the product of the three which

simplifies to:

j(c, velocity, time) = 0.170108c99 velocitye−0.00760449c

2.3.1

100

−0.25time−0.447387velocity

10% AV cars

With 10% market penetration AV’s would be widely dispersed across

the road. Because of this we modeled three ideal situations that a crossAV communication system would work to coordinate. The first formation

arranges the AV’s so there is one AV at the end of a line of 9 cars with

the AV tailing the last car by 10 meters. The second formation has a

line of 18 human driven vehicles tailed by two AV’s both with a following

distance of 10 meters. The final formation is is 27 human driven vehicles

tailed by three AV’s. The following plot depicts our base case, the first

formation:

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Team #70260

Page 7 of 17

The shaded region bounded by the blue and gold contains possible

combinations of final velocity, time and risk constant c that would allow

the AV to continue driving at the same velocity without slowing down

and without ever getting within 2 meters of the car in front of it. As you

will see in the next graph, switching from formation 1 to formation 2 the

number of situations formation 2 can handle without the tail car slowing

down is greater then that of formation 1:

While one may expect this pattern to continue for similar changes to

the formation (ie the change from formation 2 to formation 3), something

interesting actually happens:

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Team #70260

Page 8 of 17

While able to handle every situation in which c is extremely close to

1, formation three is less equipped to handle high c values then formation

2 or even formation 1. Upon inspection this makes sense because c is

amplified exponentially by the train size, thus an increase in c has larger

affects in formation 3 thin in formation 1 or 2 and the linear expansion of

AV’s tailing the train is unable to compensate.

2.3.2

50% AV cars

With 50% market penetration AV’s would be approximately evenly

distributed throughout traffic. Because of this we modeled three similar

formations. The first formation alternates AV and non-AV every other

car, the second alternates 2 AV’s every 2 non-AV’s and the third alternates 3 AV’s every 3 non-AV’s. Note that for any arrangement of AV’s

with longer platoons is sufficient to not have to have the tail car slow

down under any combination of the variables illustrated in the graphic.

While a notable improvement at higher c values from 10% market

share, formation 1 for 50% market share is still unable to adapt to higher

rates of deceleration, even at low c values that where manageable under

formation 2 and 3 at 10%. This highlights how increasing the number of

AV’s on the road doesn’t solve the same problems that can be solved via

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Team #70260

Page 9 of 17

communication of AV’s. The next formation of two human-driven cars

followed by 2 AV’s presents drastic improvements:

Able to adapt to any situation any of the previous formations could

this way of organizing AV’s at 50% market share demonstrates how dramatic a change a small amount of organization can make compared to

formation 1. This also highlights how, even if there are more then 2 cars

in front of 2 AV’s there is still a dramatic improvement when there is a

platoon of AV’s following a train of human driven cars of about the same

length. This fact is further validated by the next formation:

With three AV’s following three human driven cars results improve

further. This pattern continues getting better for 4, 5 more cars. Eventually high values of c can have a more dramatic impact due to the exponential affect of the length of the train, thus larger trains can become

problematic however this is not an issue when, at 50% market penetration, it is extremely unlikely that a large number of human driven cars

end up in a train without an AV nearby that is able to merge into the

lane to cut the train in half.

9

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