Solve the following two problems. You must show your full work to earn full credit for both problems. Submissions that do not use this template would be penalized.

Question $1$:

Consider two departments in a company which have very similar photocopying needs and are located in adjacent office space. Both departments are getting ready to replace the aging photocopiers in their separate copying rooms. The options available to them are:

Option $1$: Buy two new model $x$ copiers and operate them in separate, independent copying rooms each dedicated to one department.

Option $2$: Buy two model $x$ copiers and operate them in a joint copying room, which serves both departments.

Option $3$: Buy one new model Y copier with the exact same features as the model X$,$ but with twice the speed and operate it in a joint copying room.

During discussions between the two departments, some concerns have been expressed about the responsiveness of a busy copying room serving twice as many customers. You are asked to apply waiting time analysis to provide additional input into the purchasing decision.

In your analysis, assume that each department generates copying jobs at a rate of $9$ per hour, and that the time to complete a job averages $6$ minutes on the model $x$ and $3$ minutes on the model $Y.$ You may assume that the coefficient of variation of the time between arrivals and the service times are both $1,$ i$.$e$.,C{V}_a=1$ and $C{V}_p=1.$

$($a$)$ Evaluate the average flow time of jobs under these three options? $($Flow time includes waiting time as well as processing time, i$.$e$.,$ flow time $=$ waiting time $+$ processing time$)$

$($b$)$ What other factors you would consider in determining between these three options.