05 Jul 1. (10 points)From historical data, Suzy’s Car Was
1. (10 points)From historical data, Suzy’s Car Wash estimates that dirty cars arrive at a rate of 10 per hour all day Saturday. With a crew working the wash line, Suzy figures that cars can be cleaned at a rate of one every five minutes. One car at a time is cleaned in this example of a single-server waiting line. Assuming Poisson arrivals and exponential service times, find the(a) average number of cars in line,(b) average time a car waits before it is washed,(c) average time a car spends in the service system,(d) utilization rate of the car wash,(e) probability that no cars are in the system.2. (5 points)A computer processes jobs on a first-come, first served basis in a time-sharing environment. The jobs have Poisson arrival rates, with an average of 5 minutes between arrivals. The objective in processing these jobs is that they spend no more than 8 minutes, on average, in the system. Assuming exponential service times, how fast does the computer have to process jobs (in minutes, rounded to 3 decimal places), on average, to meet this objective?3. (10 points)During peak times the entry gate at a large amusement park experiences an average arrival of 30 customers per minute, according to a Poisson distribution. The average customer requires 12 seconds to be processed through the entry gate (exponential distribution). The park’s goal is to keep the time in the system (waiting time plus time through the gate) to less than 90 seconds. What is the minimum number of gates that are necessary to meet this goal? (Note that the overall arrivals will be evenly spread among the gates, generating a separate line behind each gate.)4. (10 points)Steph Curry’s Coffee Shop decides to install an automatic coffee vending machine outside one of its stores to reduce the number of people standing in line inside. The shop charges $3.50 per cup. The service time is a constant 2.8 minutes, and the arrival rate is 15 per hour (Poisson distributed).a) What is the average waiting time in line?b) What is the average number of people in line?c) The shop decides to raise the price to $5 per cup and takes 60 seconds off the service time. However, because the coffee is now so expensive, the arrival rate drops to 10 per hour. Now what are the average wait time and the average number of people in the queue (waiting)?5. (10 points)A chemical plant stores spare parts for maintenance in a large warehouse. Throughout the working day, maintenance personnel go to the warehouse to pick up supplies needed for their jobs. The warehouse receives a request for supplies, on average, every 2 minutes (Poisson distributed). The average request requires 1.7 minutes to fill (exponential service times). Maintenance employees are paid $20 per hour, and warehouse employees are paid $12 per hour. The warehouse is open 8 hours each day. What is the optimal number of warehouse employees to hire? (Hint: Time spent by maintenance workers waiting in line for supplies should be considered an expense.) What is the total cost per hour (waiting plus warehouse employee labor cost) of your solution?6. (5 points)During the evening hours, an average of 52 customers can be found in Gihan’s Sports Bar. About 40 customers arrive each hour. Using Little’s Law, how long does each customer stay at Gihan’s?
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