For Whom The Booth Tolls Brian Camley Pascal Getreuer Brad Klingenberg.
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Transcript of For Whom The Booth Tolls Brian Camley Pascal Getreuer Brad Klingenberg.
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