A pharmacy has started a special clinic where two pharmacists are giving flu shots to customers. Customers interested in receiving a shot visit the clinic following a Poisson process with mean rate of 8 customers/hour. For each customer, it takes a pharmacist, on average, 12 minutes to first complete paperwork and then administer a shot. The service times are exponentially distributed.

First, suppose that each customer is randomly assigned to one of the two pharmacists on arrival. Before they can receive their shot, a customer needs to wait until the designated pharmacist finishes processing all previous customers allocated to them (even if the other pharmacist is waiting idly).

1.) How long does an average customer spend in the clinic (in minutes) (i.e., the total time including waiting in line, filling up paperwork, and receiving a shot)? If your answer is not an integer, provide at least three decimal places, e.g., 7.500.

Hint: Remember that each pharmacist is assigned half of the arriving customers!

2.) How many customers are in the clinic (either waiting in line or receiving the shot) on average? If your answer is not an integer, provide at least three decimal places, e.g., 7.500.

Next, suppose that all customers wait in a single line on arriving to the clinic. As soon as a pharmacist is done processing a customer, they start processing the next customer waiting in the line (i.e., the customer who has been waiting in line the longest).

3.) How long does an average customer spend in the clinic (in minutes) (i.e., the total time including waiting in line, filling up paperwork, and receiving a shot) under this pooled system?If your answer is not an integer, provide at least three decimal places, e.g., 7.500.

4.) How many customers are in the clinic (either waiting in line or receiving the shot) on average under the pooled system?If your answer is not an integer, provide at least three decimal places, e.g., 7.500.

Finally, suppose that all customers wait in a single line on arriving to the clinic. However, the two pharmacists are now working in tandem; i.e., one of them is administering a dose on a customer while the other is completing the paperwork. As a result, it now takes the pharmacists only 6 minutes, on average, to process a patient, even though the service times are still exponentially distributed.

5.) How long does an average customer spend in the clinic (in minutes) (i.e., the total time including waiting in line, filling up paperwork, and receiving a shot) under this "in-tandem" system?If your answer is not an integer, provide at least three decimal places, e.g., 7.500.