Working in the fall of 2000 and spring of 2001, Tonya and Nancy studied the effects of the railroad crossing at England St. (Rt 54) on automobile traffic at the intersection of England St and U.S. Rt. 1. In particular, they sought to determine the liklihood of westbound traffic on England St. backing all the way up to this intersection while waiting for a train to pass. The map below shows the railroad crossing on the left, and the major intersection with route 1 on the right. Click on the map to view an animation simulating the traffic backing up toward the intersection during a three minute train crossing at rush hour. In the movie, the gate is shown as a red bar when down (3 minutes), and as a pair of green lights when it is raised. Note how traffic continues to back up after the gate is raised. The numerical scale counts the number of cars that back up. In this example, the line gets to 83 cars in length.
Tonya and Nancy developed a mathematical model for the length of the queue of cars that forms. The model depends on the values of three parameters: T, lambda, and d. T denotes the time that the gate remains closed (measured in minutes); lambda is the mean traffic density (measured in cars/minute (we assume cars arrive according to a Poisson process with mean lambda); and d is the "delay time", a concept that comes into play after the crossing gate goes up and the cars begin to move. The delay time d is the time it takes for a car to begin moving after the car immediately preceeding it begins to move. We assume that d is constant, and measure it in minutes. Observation places the value of d near 0.02 minutes. During the evening rush hour (4:45 - 5:45 pm), the value of lambda is slightly above 14 cars/minute in summer 2001. With a projected growth in traffic of 4 percent annually, lambda is expected to reach 19 cars/minute by 2008.
With the values of these parameters fixed, the model gives the probability distribution of the number of cars N that will back up. The main results follow (for a derivation of these formulae, see The UMAP Journal, Vol 22.2, Summer 2001):
Since approximately 100 cars will fit in the corridor between the railroad tracks and Rt. 1, we are concerned with the probability that the number of cars N that back up exceeds 100. These probabilities are summarized in the bar chart below for several values of T and lambda. We look here only at gate times T between 2.5 and 3 minutes. Traffic density lambda ranges from its current value (near 14 cars/minute) to its expected value in 2008 (near 19 cars/minute).
Note that in 2008 (lambda = 19), a 3 minute train crossing during rush hour will yield roughly a 30% chance that westbound traffic will back all the way up to Rt. 1 . This is certainly cause for concern. The model predicts that such saturated traffic conditions will be a routine occurence if traffic density rises as projected.