for whom the booth tolls brian camley pascal getreuer brad klingenberg
Post on 21-Dec-2015
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For Whom The Booth Tolls
Brian CamleyPascal Getreuer
Brad 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 M|M|n 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:
What about merging?
This characterizes the first half of the toll plaza!
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Merging
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Simple Models
We need to simply model individual cars to show how they merge…
Cellular automata!
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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”
c
J
Hysteresis effect not in theory
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Analytic and Computational
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Empirical One-Lane Data
Empirical data from Chowdhury, et al.
Our computational andanalytic 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
Minimum at n = 4
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For Three Lanes
Minimum at n = 6
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Higher n is left as an exercise for the reader
• It’s not always this simple - optimal n becomes dependent on arrival rate
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Maximum at n = L + 1
The case n = L
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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
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Strengths and Weaknesses
Strengths:• Consistency• Simplicity• Flexibility
Weaknesses:• No closed form• Computation time