Modeling the Optimization of a Toll Booth Plaza By Liam Connell, Erin Hoover and Zach Schutzman Figure 1. Plot of k values and outputs from Erlang’s Formula Figure 2. Selection of queuing simulation in Excel for the Delaware Memorial Bridge. Introduction Assumptions Methods Conclusion Many highways across the globe require tolls at certain intervals to pay for necessary infrastructure and road repair. Whenever building a toll booth plaza, a main concern is optimizing the number of toll booths so that it is as convenient as possible for the driver without wasting money on unnecessary booths. It is then desirable to minimize annoyance by limiting the amount of traffic disruption caused by the toll plazas by optimizing the number of toll booths. In our model, we define optimal as the number of toll booths that gives the proportion of people waiting longer than a specified time is less than a certain percentage, which are both defined and calculated in the model. Important Assumptions made in our model: 1.The number of toll booths allowed ranges from a minimum equal to the number of lanes to an infinite maximum. The Infinite Toll Plaza is ruled out as a solution based on its inefficiency rather than the impossibility of constructing such a plaza. 2.We assume that the toll plaza is the only cause for traffic congestion. Reduced speed from weather effects, road construction, car accidents, and bad driving are not considered in the choice of the number of tolls. All drivers operate as if they had full knowledge of the choices of every other driver on the road. 3.Every toll is a standard cash toll in the middle of a freeway. Options like EZ Pass, high-speed tolls, or the Massachusetts Turnpike system of tolls at on- and off-ramps are not considered in the main analysis. 4.The arrival rate of cars at the plaza follows a random Poisson distribution, and the interarrival time follows an exponential distribution. 5.The rule governing for which toll a driver queues is that if all lines contain the same number of cars, a line will be chosen at random. If not all lines have the same number of cars, a line will be chosen at random from among those with the shortest lines. The Basic idea of our model is then to use Erlang’s formula, based on the assumptions of arrival and interarrival time distributions, and Mathematica to optimize a plaza by finding the number of booths needed to create a negligible chance of large lines forming. We then use this number to test the data from real life situations and to verify our model. Figure 1 is the graphical representation of our model. The x axis of Figure 1 is the number of cars in the system not currently being served. In other words, it is all the cars in all the lines. The y axis is the percent chance that there will be this number of people in the lines. Imagine we are working at a Maine Tollbooth. Our citizens are easygoing, so we can allow up to 10 lines of 3 or more per day on average. Assuming that the traffic intensity 10.7 and the service time is 15 seconds, that makes the arrival rate 42.8 cars per minute, or 61,632 cars per day. If we only want 10 occurrences of a long line, we need the probability of it occurring to be roughly 0.016%. We choose the number of tollbooths that gives the closest statistic: 14 tollbooths, with a chance of occurrence of 0.014% or 8.63 per day. This is just one scenario which exemplifies how to calculate the optimal number of toll booths based upon our model. By using the probabilities of the formation of a line of certain length, service rate, and arrival rate we can determine the optimal number of toll booths for a plaza which minimizes the wait time and makes the experience easier for a driver. We can then use this number of toll booth to further examine the applicability of our model to real life situations and verify our model using previously collected data. Our model fits the reality of the situation, based upon the realistic results of 28 minute wait time at a bridge toll booth during rush hour, however it shows that more toll booths may be required in some circumstances to make the experience better for the customers. Our model’s greatest strength is its flexibility. For any given threshold line length and probability, we can find the optimal number of tolls in a plaza for that freeway’s traffic volume. The model allows for quick adjustment of any of the parameters to determine the effect of possibly increasing or decreasing the number of tolls, or relaxing or strengthening the line criteria. The other major strength of our model is how easily the queuing portion generalizes to other cases. If we substitute cash registers and exits for tolls and merging, we can use our model to determine the optimal number of checkout counters at a shop. Similarly, we could model security lines at an airport, entrance lines at a concert, or service calls at an IT help center. The weaknesses of our model largely stem from the lack of data on toll plazas available. We made a lot of assumptions and estimations about arrival rates and service rates which may or may not hold in the real-life scenarios. We also made several simplifications, such as assuming that the mean service rate held across all vehicles and booths and that there are no other sources of traffic congestion aside from the toll itself. Testing with real-life Data Once the optimal number of toll booths is determined for a certain waiting line, we can use a queueing simulation in Excel and real data to test if our model is appropriate. A queue in Excel is created based on the distribution of the arrival and departure, mean service, and mean arrival rates as well as queuing theory in Excel. To verify our model, we used data from the Delaware Memorial Bridge for 2 toll booths under rush hour traffic conditions. We then calculate the arrival rate to be approximately 14 cars per minute with a service rate of 4 cars per minute. Putting these values into the queue, we can determine the average wait time for n cars. We then run multiple simulations to get the average wait time for multiple trials, which we determined to be approximately 28 minutes for our parameters (1000 cars, 2 toll booths). Figure 2 shows a selection of the queue we ran for the Delaware Memorial Bridge, showing data for 30 cars. This figure also shows the data and set up necessary to run a queue in Excel. The waiting time of 28 minutes for the Delaware Memorial Bridge, while realistic, is not optimal. A 28 minute wait time is frustrating and inconvenient for many drivers, so the number of toll booths at the Delaware Memorial Bridge need to be increased to better service their customers.