Queuing system question: To cope with the high demand during high school summer break, DMV office is restructuring the process of taking written test and issuing driving permit. This process includes the following steps

A. Verify document, fill/sign forms, take vision test, and process payment (the process time of each staff/server is 6 minutes in average with exponential distribution). Almost all applicants have document/form ready, pass the vision test, and have enough money to pay.

B. Take written test in front of computer (each test occupies one computer for an average process time 15 minutes with exponential distribution). The passing rate of written test is 80% in average. Those applicants fail the test will be notified by computer and leave the queue immediately.

C. If passed the test, the applicants need to take photo, collect finger print, and issue temporary driving permit (the process time of each staff/server is 5 minutes in average with exponential distribution).

Assume the applicants of written test arrive in Poisson with average rate 20 per hour throughout the entire DMV office hour (8am to 5pm). You have 6 staffs available to serve the written test applicants and each staff is equally capable of handling both Step A or C. There are 7 computers available in Step B.

Q1) How would you deploy the 6 staffs to cover Task A and C to minimize the average total time (including waiting and service time) that each applicant spends in DMV office? Based on your deployment plan, what is average total time each applicant spends in DMV for this written test process?

Q2) Assume new union regulation mandates your staff to take 15 minutes union break for every 2 hours of work. What is the best arrangement to rotate the break among the 6 staff? How would this union break regulation affects the average total time each applicant spends in DMV for this written test process?

Q3) How would the service be improved if you request for 1 extra staff (total 7 staffs, with union break) to handle the written test process in the summer? (Hint: The departure rate of each M/M/C queuing system in previous step will become Poisson arrivals for next step)