The title of the students' scientific paper, "The Booth Tolls for Thee," was humorous, but the purpose was to help solve a serious, frustrating problem for all those who regularly find themselves trapped in tollbooth lines. The paper, written last semester by Pradeep Baliga, Adam Chandler, and Matt Mian, now Duke seniors, describes a mathematical formula that will guide design of the most efficient highway toll plaza. Their work took home top honors in a mathematical modeling contest sponsored by the Consortium for Mathematics and its Applications. And it could well win them the gratitude of the multitudes of fuming, trapped drivers.
The formula prescribes the optimal number of tollbooths for a given plaza based on the number of lanes in the highway. "We weren't advocating for the drivers or for the operators of the tollbooths," says Chandler. "We looked at the whole system and translated the entire problem into money and were able to minimize this cost function that took everybody into account."
The formula included such factors as the worth of a driver's time ($6 an hour) and the cost of operating a staffed tollbooth ($180,000 a year). The students conceptualized the problem by using mathematical models that have been developed to describe other physical phenomena: beads lined up at gates, water flowing through a pipe, and robots running a race.
They found that the optimal number of tollbooths for a given plaza can be calculated by multiplying the number of lanes in the highway by 1.65, adding 0.9, and then rounding down to the nearest whole number. Thus, seven booths should adequately serve traffic pouring in from four lanes.