Question $4)(16$ points$)$ Customers arrive at the Al$-$Mawalih branch of SuperBank according to a Poisson process with rate $10$ per hour. This branch is relatively small; therefore, there is only one server attending to the customers with the capacity to serve $15$ customers per hour, each in an exponentially distributed duration.

$($a$)(4$ points$)$ What is the long$-$run fraction of time that the server is idle?

$($b$)(4$ points$)$ What is the expected time a customer spends in the queue in front of the server?

$($c$)(4$ points$)$ Suppose Majid comes to the bank and observes that there are two customers waiting in line. What is the expected duration of his stay in the bank?

$($d$)(4$ points$)$ What is the probability that there are more than $3$ customers in the bank at any point in time?