Question $1$ Barber Shop $($Part I$).$Alex owns a one$-$chair barber shop. At Barber College, they told Alex that his customers wouldexhibit a Poisson arrival distribution $($equivalently inter$-$arrival time is exponentially distributed$),$and that he would provide an exponential service distribution. His market survey data indicatethat 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 t he f ollowings: $($ Assume thatthe waiting area for customers is inside the shop.$)($a$)$ The average utilization of Alexs 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$).$ Customerswould 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, andthat 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.