Consider a small pharmacy which administers vaccines in a separate area of the store. Patients arrive at a rate of 30 per hour according to a Poisson process. All patients first need to check in and sign the necessary paperwork. The service time at this step is exponentially distributed with a mean of 4 minutes and there are 3 people working at the check in desk. The site offers two different types of vaccines. 80% of the patients prefer vaccine A. Vaccine A administration takes on average 8 minutes (service time exponentially distributed) and there are 4 stations where it can be administered. 20% of the patients prefer vaccine B. Vaccine B administration takes on average 5 minutes (service time exponentially distributed) and there is 1 station where it can be administered. After receiving their shots, all patients need to proceed to a room where they will be monitored for adverse effects on average for 15 minutes (exponentially distributed). The monitoring room has 10 seats. For the monitoring room, assume that any time spent waiting to get a seat does not count towards the 15-minute monitoring time.

1. a. What are the average throughput times (Ws) at the check-in desk, vaccine A stations, vaccine B station, and waiting area, respectively?
2. b. On average, what is the total number of patients waiting to get a seat at the monitoring area?
3. c. Now just focus on the check-in desk and the vaccine administration stations. Assume that patients incur a cost of $20 per hour they spend at the pharmacy. If the hourly cost of adding one more staff member is $15, at which step (check in, vaccine A, vaccine B) would you add an additional staff member to achieve the biggest benefit, i.e., the reduction in patient waiting cost per hour minus the hourly cost of a staff member? Does adding an additional server at that step justify the cost (i.e., $15 per hour)?