For Whom The Booth Tolls Brian Camley Pascal

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: This characterizes the first half of the toll plaza! What about merging?

Merging

Simple Models We need to simply model individual cars to show 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” J Hysteresis effect not in theory c

Analytic and Computational

Empirical One-Lane Data Our computational and analytic results Empirical data from Chowdhury, et al.

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

The case n = L Maximum at n = L + 1

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