For the multitudes of drivers stuck in tollbooth lines, three Duke University students have come to the rescue with a mathematical formula to guide design of the most efficient toll plaza.
The formula created by rising Duke seniors Pradeep Baliga, Adam Chandler and Matt Mian also won them top honors in the 2005 Mathematical Contest in Modeling sponsored by The Consortium for Mathematics and its Applications.
They have yet to share their solution with any highway planners, but they will present it at MathFest 2005, the annual conference of the Mathematical Association of America, on August 6, 2005.
The formula in their paper "The Booth Tolls for Thee" basically 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," explains 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."
Based on another academic paper, the students postulated that a driver's time is worth six dollars per hour. They determined from another study that; the cost of operating a manned tollbooth is $180,000 per year. However, the students drew on more than just traffic studies for their inspiration.
"Before we wrote any math we just tried to visualize physical concepts that we thought would represent this thing," explains Mian.
The students came up with three physical concepts to describe traffic moving through a toll plaza: beads lined up at gates, water flowing through a pipe and robots running a race.
Each concept, employing different types of math, contributed to their solution. The first model, an algorithm tracking beads entering and exiting lines, was the simplest and quickly gave the team a sense for how tollbooths slow traffic. The water-through-pipe concept used continuous functions, which allowed the students to apply tools of calculus to come up with an equation for the cost to the entire system that could be minimized. The racing robot concept consisted of a computer program that treated each car as a robot with instructions to move forward, sit idle or change lanes depending on the available space and factors for drivers' restlessness and attentiveness. This concept allowed the team to account for the herky-jerky motion of traffic, the shape of the plaza and bottlenecking when cars exit the plaza.
"You have to spend a lot of time going from intuitive notions about the world to some sort of formal mathematics," said team advisor, mathematics postdoctoral fellow Garrett Mitchener. "And then connect it back to reality."
Specifically, the resulting formula holds that the optimal number of tollbooths for a given plaza is the number of lanes in the highway multiplied by 1.65, plus 0.9, and rounded down to the nearest whole number. Thus, seven booths should adequately service traffic pouring in from four lanes.
The reality of toll plaza traffic, however, is not entirely captured in the students' solution, they acknowledge. For example, EZ Pass lanes, accidents and people who arrive at an exact change booth without any change are not included in their models.
Of course they'll have many opportunities to experience the implications of their theory. Baliga already had one when he was driving at night in a thunderstorm on the New Jersey Turnpike, the source of the data for the team's paper, and came to a toll plaza.
"The turnpike had three lanes , and as per our recommendations, should have five booths," he said. "It in fact had eleven booths.
"Since I've seen it from the research standpoint, and realize the need for optimization, I know the eleven [tollbooth arrangement] was excessive," he said. However, "as a driver looking to get home at ten p.m. and no longer wanting to drive in that ridiculous thunderstorm, I liked the eleven booths."