MCM (PDF)

Team #70260

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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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Contents
1 Introduction

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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.

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2
2.1

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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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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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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

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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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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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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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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.

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