At a tool service center, machinists arrive according to a Poisson process with a rate of $2$ per hour. The service time is exponentially distributed with mean $20$ minutes. The hourly wage paid to the attendant at the service center is $$15$ per hour and the hourly cost of a machinist away from his work is $$40.$

$($a$)(6$ points$)$ Calculate the average number of machinists in the system at any given time.

$($b$)(6$ points$)$ Calculate the average time a machinist spends waiting for service.

$($c$)(6$ points$)$ Calculate the total cost of operating the system for an $8-$hour day.

$($d$)(7$ points$)$ Calculate the cost of the system if there were two attendants working together as a team $($in effect acting together as a single server$),$ each paid $$15$ per hour, and the mean service time of the new system is $15$ minutes.