Sunset Airlines is concerned about the level of service for its check$-$in procedure. Currently, a single clerk spends an average of three minutes per passenger checking luggage and issuing boarding passes. Passengers arrive at the rate of $15$ per hour. Make all the assumptions of M$/$M$/1$ model.

Calculating system performance measures $(10$ points$)$

$($a$)$ What is the utilization of the clerk?

$($b$)$ What is the average number of passengers in service?

$($c$)$ What is the average number of passengers in the waiting line?$$

$($d$)$ What is the average waiting time in line of a passenger?

$($e$)$ What is the average time spent in the system of a passenger?

$($f$)$ What is the average number of passengers in the system?

$($g$)$ What is the probability of having more than $1$ passenger in the waiting line?

$($h$)$ What service capacity would be required if you want the average time spent in the system of a passenger to be $6$ minutes?

$($i$)$ If the clerk is replaced by a self$-$serve ATM with a constant service time of three minutes, what would be the average waiting time in line of a passenger?