A single server service station has facilities for a maximum of $4$ customers.

Once these these $4$ places are filled newly arriving customers do not wait. The

arrival rate for units is $24$ per hour and the arrival process is approximately

Poisson. Service times are exponential with a mean of $3$ minutes.

$($i$)$ Draw the rate diagram that describes this process

$($ii$)$ What are the steady$-$state probabilities for this process

$($iii$)$ What is the averge time per in minutes per hour that the sever is idle?

$($iv$)$ The average profit per customer is $$80.$ What is the lost profit per hour

due to the finite capacity?.

$($v$)$ What is the average waiting time, before service, of an arrival?

Please write the answer to these question and please provide the diagram.