Question $4$:

A maintenance worker repairs machines that break down at an average rate of $4$ per hour. The breakdown follows Poisson distribution. The time to repair a machine follows exponential distribution with a mean of $12$ minutes. The maintenance worker is paid $$9$ per hour. It is estimated that the cost of machine waiting time in line is $$3$ per hour.

A$)$ Determine the proportion of time that the maintenance worker is busy

B$)$ What is the average number of machines in the queue

C$)$ What fraction of time do you expect to have more than three machines in the system

D$)$ How long will the machine have to wait before serving by the maintenance worker

E$)$ Assume that another maintenance worker has been interviewed. He demands $$12$ per hour and services at an average rate of $7$ machines per hour. Show the cost analysis. Which repairman would you recommend and why?