Question $1$ Barber Shop $($Part I$).($PLEASE SOLVE AND SHOW WORK AND FORMULA USED$)$

Alex owns a one$-$chair barber shop. At Barber College, they told Alex that his customers would exhibit a Poisson arrival distribution $($equivalently inter$-$arrival time is exponentially distributed$),$ and that he would provide an exponential service distribution. His market survey data indicate that customers arrive at an average rate of $2$ customers per hour. It will take Alex an average of $20$ minutes to give a haircut. Based on these pieces of information, find the followings: $($Assume that the waiting area for customers is inside the shop.$)$

$($a$)$ The average utilization of Alex$$s time. For what percentage of time is Alex idle? $($b$)$ The average number of customers waiting.

$($c$)$ The average time a customer is in the shop.

$($Part II$).$ Alex now is considering the addition of a second chair $($a second barber$).$ Customers would be selected for a haircut on a first come first served basis from those waiting in a single line. Alex has assumed that both barbers would take an average of $20$ minutes to give a haircut, and that the business would remain unchanged with customers arriving at an average rate of $2$ per hour. Find the following information to help Alex decide if a second chair should be added:

$($a$)$ The average number of customers waiting. $($b$)$ The average time a customer is in the shop.