For%20Whom%20The%20Booth%20Tolls

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For Whom The Booth Tolls
Brian CamleyPascal GetreuerBrad Klingenberg
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Problem
Needless to say, we chose problem B. (We like a
challenge)
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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

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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

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Queueing Theory

• We model approaching and waiting as an MMn
  queue

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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?
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Queueing Theory Results

• The probability that d cars leave in time
  interval ?t is

This characterizes the first half of the toll
plaza!
What about merging?
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Merging
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Simple Models

• We need to simply model individual cars to show
  how they merge

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Nagel-Schreckenberg (NS)

• Standard rules for behavior in one lane
• Each car has integer position x and velocity v

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NS Behavior
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NS Analytic Results

• Traffic flux J changes with density c in inverse
  lambda

J
c
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Analytic and Computational
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Empirical One-Lane Data
Empirical data from Chowdhury, et al.
Our computational and analytic results
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Lane Changes

• Need a simple rule to describe merging

This is consistent with Rickert et al.s two-lane
algorithm
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Modeling Everything
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Model Consistency
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Total Wait Times
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For Two Lanes
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For Three Lanes
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Higher n is left as an exercise for the reader

• Its not always this simple - optimal n becomes
  dependent on arrival rate

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The case n L
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Conclusions

• Our model matches empirical data and queueing
  theory results
• Changing the service rate doesnt change results
  significantly
• We have a general technique for determining the
  optimum tollbooth number
• n L is suboptimal, but a local minimum

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Strengths and Weaknesses

• Strengths
• Consistency
• Simplicity
• Flexibility
• Weaknesses
• No closed form
• Computation time