Vehicles arrive at the Lincoln Tunnel toll gate according to a Poisson distribution, with a mean of 90 vehicles per hour. The time for passing the gate is exponential with mean 38 seconds. Drivers complaining of the long waiting time. The municipality is willing to add an additional gate, provided that the average number of vehicles waiting in line (queue) exceeds 5. a. What is the average number of cars waiting in line? Would this justify adding a new gate? (5 points) b. If a representative of the municipality arrives unexpectedly to the gate, what is the probability she will find more than 5 cars in the system? (5 points) c. The gate is operational 24 hours per day. Cars emit 200 grams of Carbon Monoxide (CO) for every hour they spend on the road. What is the average daily emission of CO from the toll gate system? (5 points) d. A second toll gate is being considered to reduce waiting time. Two scenarios are being investigated. Scenario 1: The first gate will receive, on average, half of the incoming traffic and the second gate will receive the other half. Each gate will have its own queue of vehicles (vehicles will queue up in two separate lanes). Scenario 2: All vehicles arrive to one queue (vehicles will remain in one lane) and each gate wil receive whichever vehicle is next in line. What is the utilization in each scenario? What is the average time a vehicle waits before enteri the gate in each scenario? Which scenario would you prefer? Explain carefully your calculation (10 points) e. Suppose that the arrival rate of vehicles to the toll gate increases 5 times (i.e. to 450 vehicles per hour). What is the average waiting time in system for both Scenarios 1 and 2? (5 points)