$(8+6+6=20$ points$)$

Question $1$: $(8$ points$)$ Queueing $($arrival and service$)$

Copy machine investment: We are trying to estimate the number of copy machine$($s$)$ we should install in the basement of State Farm Hall of Business. Copy machines should be available $24/7.$ Students may walk up at any time of the day or night for copying.

The interarrival time for the student's arrival at the copy machine$($s$)$ is described by the following discrete probability distribution:

$\backslash$table$[[$Interarrival time $($in minutes$),$Probability$],[1,0\mathrm{.}2],[2,0\mathrm{.}2],[3,0\mathrm{.}2],[4,0\mathrm{.}25],[5,0\mathrm{.}15]]$

Based on past data, it can be assumed that the amount of time a student needs to use the copy machine is described by a discrete probability distribution listed below:$\backslash$table$[[$Copying time $($in minutes$),$Probability$],[1,0\mathrm{.}30],[2,0\mathrm{.}20],[3,0\mathrm{.}25],[4,0\mathrm{.}25]]$

Students do not like waiting at the copy machine for their turn to use them. Students don't depart from the queue until they use the machine. Service policy is $-$ first$-$come, first$-$serve $($whoever arrives first at the machine prioritizes using the machine$).$ Suppose that students' time is valued at $$10$ per hour, i$.$e$.,$ the student could have used the time wasted waiting to do some other work at that wage rate. Each copy machine can be leased at a rate of $$10$ per day. The cost of the leasing copy machine is ultimately charged to the common student services. Simulate the process for $30$ students to determine the best of the two choices described below. Your cost analysis is just as important as the simulation. Use only the random numbers provided. Given the above information, answer the following questions.

$1$Assume that the university leases one copy machine. What is the total cost of

operating the system per day? $($this cost includes the cost of time wasted by the

students waiting at the copy machine plus the cost of leasing the copy machine$).$

Assume that the university leases two copy machines. What is the total cost of

operating the system per day?

Which choice is best and why?

Please provide your work so I can follow along and I will up vote. Thank you!