Queuing Models $-$ Single$-$Server Queuing Model $($M$/$M$/1)$

Example $2-$ Calls arrive at a hotel switchboard at an average rate of $2$ calls per minute according to a Poisson distribution. The average time to handle each call is $20$ seconds, with negative exponentially distributed service times. There is only one switchboard operator at the current time. The manager wants answers to the following questions:

$($a$)$ What is the probability that the operator is busy?

$($b$)$ What is the average amount of time that a customer waits before reaching the operator?

$($c$)$ What is the average time that a customer spends in the system $($i$.$e$.$ waiting for the operator and being served$)?$

$($d$)$ What is the average number of calls waiting to be answered?

$($e$)$ What is the average number of calls in the system $($that is; on hold and in$-$service$)?$