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3. Queuing case study This part of the assignment requires a memo format. Answers must be typed in a word processor (such as MS Word or Google Docs). Submit also the spreadsheet model. Case Study: At an intersection where a two-way street crosses another, the first street divides to accommodate both vehicles turning left and those traveling straight through the intersection. The traffic light at the intersection has a left-turn signal, which in a two-minute light-change cycle is green for twenty seconds and then red for one minute and forty seconds. The present timing pattern allows up to eight cars in the left- turn lane to make their turn per light-change cycle. Recently, however, a new housing development in the area has begun to contribute more traffic through the intersection during peak commuting times. At such times, the average number of cars arriving during a two-minute light-change cycle increases by 50% over the previous eight. Consequently, a queue is forming in the left-turn lane that extends beyond the ten car-length size of the lane. Not only are left-turn drivers inconvenienced as they sometimes must wait through several light-change cycles, but through traffic is also being obstructed as the queue extends beyond where the left lane divides. Irate commuters have petitioned for the left- turn signal to be kept green longer in each light-change cycle, in order to shorten the queue, and they are meeting with the transportation planner next week to discuss their petition. After consulting with the transportation planner, the traffic engineer thinks the solution to this problem lies in extending the green arrow about eight seconds longer-this would allow three more cars to turn left during each light-change cycle. Because the traffic engineer is a busy person and cannot look into this matter on short notice, the matter has landed on your desk. To prepare for her meeting with the commuters, the transportation planner would like serviceable answers to the following questions: Just what is the magnitude of the problem? That is ... i. during the peak thirty minutes of the morning journey-to-work time, what is the average size of the queue? ii. On average, how many light changes must the driver at the end of the queue in each light-change cycle wait before getting through the intersection? iii. Would leaving the light green eight seconds longer resolve the problem as the traffic engineer conjectures? What other kinds of problems might this "solution" beget? Obviously, going out to the intersection is not the course of action for you to take. What you can do, however, is to run a small simulation several times, analyze the results, and then give the transportation planner some quick answers to her questions. And so you are interested in simulating the random arrival of traffic over the fifteen two-minute intervals in the peak morning commuting period and determine what happens to this traffic at the intersection. You recall from the class discussion that-as long as the independence assumption is satisfied-the arrival of traffic follows a Poisson distribution. You have no reason to suspect otherwise. Thus, on the basis of the information that, on average, twelve vehicles arrive during each two-minute period, you can use the Excel built-in function to produce samples of fifteen two-minute arrival observations each that will be distributed according to the Poisson distribution. You know that it will be difficult to size up the situation on the basis of one such sample, but then again you are pressed for time. So what you do is generate three samples-or, in other words, you simulate arrivals at the intersection of left-turning traffic for three mornings. Run the traffic simulations in Excel and then, for each "day," set up a table with a) a column for arrivals and, in addition, for each light time-length (twenty seconds and twenty-eight seconds) set up columns for b) queue sizes, and c) the number of cycles the last driver on the queue at each interval must wait. Compute the averages for these three metrics (arrivals, queue size, and \# of cycles for the last driver) for each day. Average all of these items over the three days and compare the "present situation" with the "proposed alternative." In an advisory memo to the transportation planner, present this comparison to the greatest effect in a small summary table. Upload also to