$(22$ points. This problem does not require the use of Excel. However, the use of Excel is recommended for the calculations.$)$

AUI's shop has a single waiting line and a single server. Customers $($mainly students$)$ arrive at the shop at a rate of $23\mathrm{.}5$ customers per hour, and are served at a rate of $30\mathrm{.}4$ customers per hour. We assume Poisson arrivals and exponential service times.

$($a$)(3$ points$)$ Is this waiting line system stable? Justify your answer.

$($b$)(3$ points$)$ Determine the probability that $20$ customers or less arrive to the shop in an hour $($Hint: you may need to use Excel here, for convenience!$).$ Round the probability value to $3$ digits.

GBU $5306(01)-$ Final Exam

$6$

$($c$)(3$ points$)$ Determine the probability that a customer is served in less than $2$ minutes. Round the probability value to $3$ digits.

$($d$)(3$ points$)$ Determine the probability that no customers are in the shop. Round the probability value to $3$ digits.

$($e$)(3$ points$)$ Determine the average number of customers waiting for service. Round the value to $3$ digits.

$($f$)(3$ points$)$ Determine the average waiting time $($in hours$).$ Round the value to $3$ digits.

$($g$)(4$ points$)$ In order to decrease the customer waiting time, AUI considers to hire parttime $($PT$)$ help to the cashier working at the shop. The PT worker would assist the students in packing their purchases, which would result in an estimated increase of $5$ customers per hour in the service rate. The cost to the University of hiring the PT worker would be $1500$ Dirhams per week. The University estimates that there would be gains of $500$ Dirhams per week in business for every minute of waiting time saved. Should AUI hire the PT worker? Justify your answer by showing your calculations.

WITHOUT EXCEL OLEASE