Enhancing Mathematical Understanding through an Optimization Problem of Delay at Fixed-Cycle Traffic Lights

Delay at traffic lights is a real life problem that each of us has ever encountered. Therefore it is an ideal example for the promotion of the applicability and usefulness of mathematics. As delay is to be avoided, an optimization problem arises as a function of the time period of green lights. The function to be minimized is also another interesting discussion item. Queues at fixed-cycle traffic lights are mostly studied from the point of view of an individual driver, but as the time period of green lights coincides with the time period of red lights in the perpendicular direction, the interests of the different drivers in those directions are opposite. The more advanced modeling of the general delay at an intersection depends on the arrival rates in each of the directions and the departure rate. The model is the basis to find an optimal tuning of the red and green periods at the fixed-cycle traffic lights. The Webster’s [1] and Newell’s [2] delay formula’s are studied and discussed for varying arrival rates in the perpendicular directions. Restrictive assumptions are motivated and handled. The delay model is useful in mathematical teaching as only elementary knowledge is a prerequisite. A detailed analysis offers easy progression in comprehension of the influences of each of the determining parameters. This example can go beyond regular high school mathematics and reaches a deeper beleef in the usability of mathematics. Visualizations with software (Maple, Matlab...) give an additional value to select an optimal tuning of the traffic lights. Classroom activities are presented ready for use.