Question $1$ Barber Shop $($Part I$).$

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 f o llowings: $($A ssume 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.