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