Tri$-$Cities Bank has a single drive$-$in teller window. On Friday mornings, customers arrive at the drive$-$in window randomly, following a Poisson distribution at an average rate of $30$ per hour. Service at the drive$-$in window is provided at a rate of $40$ customers per hour and follows an exponential distribution.

$($a$)$ Using Kendall notation, how to describe the queueing system at the drive$-$in teller window?

$($b$)$ How many customers would you expect to arrive in a $10-$minute interval?

$($c$)$ What is the probability that the drive$-$in window is empty?

$($d$)$ On average, how many cars wait for service?

$($e$)$ On average, what is the total length of time a customer spends in the system?

$($f$)$ Suppose the arrival rate does not change, what service rate would be required to reduce the average total time in the system to $2$ minutes?