For Whom The Booth Tolls

rucksackbulgeAI and Robotics

Dec 1, 2013 (3 years and 11 months ago)

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For Whom The Booth Tolls

Brian Camley

Pascal Getreuer

Brad Klingenberg

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:

What about merging?

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