A gas station has a gasoline pump. Cars that want to charge arrive according to a Poisson process at an average rate of 15 per hour. However, if the pump is in operation, potential customers can give up (go to another gas station). In particular, if there are n cars in it, the probability that a potential customer arriving will give up is n/3 for n =1,2,3. The time needed to serve an exponential auto distribution with an average of 4 minutes.

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

(b) Develop the balance equations.

(c) Solve these equations to find the steady-state probability distribution of the number of cars at the station. Verify that this solution is the same as that given by the general solution for the birth-and-death process.

(d) Find the expected waiting time (including service) for those cars that stay.