# For Whom The Booth Tolls

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1 Δεκ 2013 (πριν από 4 χρόνια και 5 μήνες)

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For Whom The Booth Tolls

Brian Camley

Pascal Getreuer

Problem

Needless to say, we chose
problem B. (We like a challenge)

What causes traffic jams?

If there are not enough toll booths,
queues will form

If there are too many toll booths, a
traffic jam will ensue when cars merge
onto the narrower highway

Important Assumptions

We minimize wait time

Cars arrive uniformly in time (toll plazas are
not near exits or on
-
ramps)

Wait time is memoryless

Cars and their behavior are identical

Queueing Theory

We model approaching and waiting as
an M|M|n queue

Queueing Theory Results

The expected wait time for the n
-
server
queue with arrival rate

,
service

,

=

/

This shows how long a typical
car will wait
-

but how often do
they leave the tollbooths?

Queueing Theory Results

The probability that
d

cars leave in time
interval

t is:

This characterizes the first half of
the toll plaza!

Merging

Simple Models

We need to
simply

model
individual cars to show how
they merge…

Cellular
automata!

Nagel
-
Schreckenberg (NS)

Standard rules for behavior in one lane:

Each car has integer position
x

and velocity
v

NS Behavior

NS Analytic Results

Traffic flux
J

changes with density
c

in
“inverse lambda”

c

J

Hysteresis effect
not in theory

Analytic and Computational

Empirical One
-
Lane Data

Empirical data from

Chowdhury, et al.

Our computational and

analytic results

Lane Changes

Need a
simple

rule to describe merging

This is consistent with Rickert et al.’s two
-
lane algorithm

Modeling Everything

Model Consistency

Total Wait Times

For Two Lanes

Minimum
at n = 4

For Three Lanes

Minimum
at n = 6

Higher n is left as an
exercise for the reader

It’s not always this simple
-

optimal
n

becomes dependent on arrival rate

Maximum
at n = L + 1

The case n = L

Conclusions

Our model matches empirical data and
queueing theory results

Changing the service rate doesn’t change
results significantly

We have a general technique for determining
the optimum tollbooth number

n = L is suboptimal, but a local minimum

Strengths and Weaknesses

Strengths:

Consistency

Simplicity

Flexibility

Weaknesses:

No closed form

Computation time