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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 - PowerPoint PPT Presentation
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For Whom The Booth Tolls
Brian CamleyPascal 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 andanalytic 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