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