The medical health center is planning for setting up a vaccination facility and providing booster shots to the patients. They plan to set up2parallel vaccination stations inside the facility, and each vaccination station is managed by a nurse. For each patient, it takes a nurseexactly 3minutes to validate the patient's information and anotherexactly 2minutes to administer the shot. They expect that for each station, there will be10patients coming for booster shots per hour and patients' inter-arrival time is exponentially distributed. Suppose each station works on their own and serves their own patients.

2. What is the utilization rate of each vaccination station?
4. How long does a patient expect to wait for their booster shot?
6. On average, how many patients are inside the vaccination facility in total?
8. After learning the pooling principle, the medical health center decides to pool the queues for the two vaccination stations together. Therefore, there will be a single queue in the vaccination facility and patient can go to any vaccination station when it's available. After pooling, how long does a patient expect to wait for their booster shot?

B. Thesoccer team shirtsare manufactured at a factory in Hongkong. To produce shirts, the cloth must first be cut and folded, which requires 1 minute on average for each shirt. Three separate machines are available for this task. After the cloth is cut, the team logo is printed on the shirts. There are three printing machines, each of which can print on one shirt in 45 seconds.

Finally, the printed shirt is then reviewed and packaged by a quality control worker. On average, this takes 90 seconds per shirt, but the standard deviation of this inspection step is 2 minutes. There are five quality control employees for this task. Orders for shirts arrive at exponentially distributed intervals, with an average of 160 orders per hour.

From beginning to end, the average time to produce one shirt is 4.5 minutes.

2. Find the capacity at each resource.
4. On average, there are 8/3=2.667shirts at the cutting step and 7.21 shirts at the inspection step. How many items are on average in the printing step?
6. To reduce average wait time, the factory wants to reduce the utilization of the inspection workers to 4/7 = 57.14%. How many total workers should there be at the inspection step in order to achieve this?
8. Under the new staffing derived in(c), what is the new average waiting time (in minutes) at the inspection step?

C. The Carolina Pharmacy is offering on-campus Influenza vaccines to its patients. There are two locations (Charleston lot and Columbia). Charleston lot is for patients with appointments and Columbia is for walk-in patients. There are2nurses in Charleston lot and5nurses in Columbia. The nurses in both the locations spend on average5 minuteson each patient with standard deviation1 minute.

Suppose that there are20appointments made per hour in Charleston lot and patients show upexactlyat the scheduled times.

2. How many minutes on average does a patient spend waiting in line at theCharleston lot?
4. On average, how many patients are at theCharleston lot?
6. Assume that the patient inter-arrival times to Columbia follow an exponential distribution with an average inter-arrival time of2 min. How many minutes does a patient spend on average at theColumbia(waiting and test)?
8. At theCharleston lot, when a nurse is not busy, she spends time entering the delivered test results into the system. Suppose that on average it takes2 minutesto record a result in the system. Over a shift of12 hours, how many results can the2 nursesrecord?

D. Trader Toe's is a popular grocery store which has on average20customers arriving every hour, with a standard deviation of interarrival times of2minutes. There is one line for checkout and it is currently operated bytwoemployees. Each employee spends on average4.5minutes to checkout a customer. The checkout times follow an exponential distribution.

2. Assume that every customer coming to the store buys something and so must go through the checkout. Then, what is the average wait time that a customer has to wait in line before getting served by the checkout employees?
4. What is the average number of customers who are at the checkout desk, either waiting or currently being served?
6. If there are no customers requiring check out, the employees are sorting returned items, which there is always plenty of. Suppose that on average it takes 5 mins to sort a single item (record that it has been returned and place it in its position). How many items can the two employees sort over a 12-hour shift?
8. As a special service, the store offers free snacks for customers waiting in the checkout line (the customers who are currently served by the checkout employees are already busy with payment so they do not get free snacks.) The store manager estimates that every minute of customer waiting time costs the store 75 centsbecause of the free snack service. What is the hourly cost for the store due to this service