$(20$ Points$)$ A service station has one gasoline pump. Cars wanting gasoline arrive according to a

Poisson process at a mean rate of $15$ per hour. However, if the pump already is being used, these

potential customers may balk $($drive on to another service station$).$ In particular, if there are $n$ cars

already at the service station, the probability that an arriving potential customer will balk is $\frac{n}{3}$ for

$n=1,2,3.$ The time required to service a car has an exponential distribution with a mean of $4$

minutes.

a$)$ Construct the rate diagram for this queueing system.

b$)$ Develop the balance equations.