Customer arrivals at an ATM are a random Poisson variable, and there is an average time of $5$ minutes between each arrival. Moreover, the time spent by a customer at the counter is an exponential random variable with an average time of $3$ minutes. Using the minute as a unit of time, answer the following questions:

$($a$)$ What is the probability that a client occupies the counter for less than $3$ minutes?

$($b$)$ What is the probability that there are at least two clients in the queue?

$($c$)$ If the bank installs a second counter, what will be the average time spent by a client in the queue?

$($d$)$ If the bank installs a second counter, what will be the average number of unoccupied counters?

$($e$)$ Suppose the bank is willing to install a second counter only when clients will have to wait at least $6$ minutes on average before being able to access a counter. What should the arrival rate of clients be to justify the installation of a second counter?