Suppose jobs arrive at a single$-$machine workstation at a rate of $20$ per hour and the average process time is $2\frac{1}{2}$ minutes.

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

$($b$)$ Suppose that interarrival and process times are exponential,

i$.$ What is the average time a job spends at the station $($i$.$e$.,$ waiting plus process time$)?$

ii$.$ What is the average number of jobs at the station?

iii. What is the long$-$run probability of finding more than three jobs at the station?

$($c$)$ Now suppose process times are not exponential, but instead have a mean of $2\frac{1}{2}$ minutes and a standard deviation of $5$ minutes

i$.$ What is the average time a job spends at the station?

ii$.$ What is the average number of jobs at the station?

iii. What is the average number of jobs in the queue?