The Milner Library is considering whether to add an additional printer for student use. Currently,

there is only one printer available $24/7.$ Based on historical data, the amount of time a student

needs to use the printer follows the discrete probability distribution listed below:

Historical data has provided the following discrete probability distribution for the time $($in minutes$)$

a research assistant spends assisting a student:

Assistance Time $($min$)$ Probability

$10\mathrm{.}25$

$20\mathrm{.}30$

$30\mathrm{.}20$

$40\mathrm{.}25$

Student queries arrive according to the following discrete distribution:

Interarrival Time $($min$)$ Probability

$10\mathrm{.}10$

$20\mathrm{.}20$

$30\mathrm{.}30$

$40\mathrm{.}25$

$50\mathrm{.}15$

Students value their time at $$12$ per hour, meaning that any time wasted waiting could have been

used for productive activities valued at that rate. A printer can be leased at $$20$ per day, and the

leasing cost is charged to the students in their technology fees.

Questions for Analysis:

$1.$ Assume that the library leases one printer. 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 printer and the

cost of leasing the printer.

$2.$ What is the idle time $($in $\%$ of time$)$ for the printer, and what is its utilization rate $(\%)?$

$3.$ How many printers would you consider adding? Justify your answer based on your analysis.

$4.$ The library is offered an alternative leasing plan with a cost of $$12$ per day but with a $10\%$

slower printing speed. Would this option be more cost$-$effective than the current plan?

Analyze the total cost and recommend the best option.

Simulation Requirement: Simulate at least $50$ arrivals to determine your cost analysis and justify

your recommendations.