Problem $1\mathrm{.}1$ Consider a barbershop with two chairs, two barbers, and no room for customers to wait. The system's state is the number of customers in the shop: $0,1,$ or $2.$ Six customers arrive per hour. If there is an empty chair when a customer arrives, they enter the shop and begin a haircut. Otherwise, the customer leaves. Each haircut takes an average of $15$ minutes, and its duration has an exponential distribution. $($a$)$ Draw the transition rate diagram and provide the $\backslash ($ Q $\backslash )$ matrix. $($b$)$ Calculate the stationary distribution of the number of customers in the shop. $($c$)$ Find the long$-$term ratio of customers who get a haircut and those who leave.