We consider a waiting system with a single server and finite capacity c = 3, in which the customers arrive according to a Poisson process with rate λ and the service times are independent exponential...

We consider a waiting system with a single server and finite capacity c = 3, in which the customers arrive according to a Poisson process with rate λ and the service times are independent exponential random variables with parameter μ = 2λ. When the system is full, a fair coin is tossed to determine whether the second or third customer will be the next one to be served,

(a) Write the balance equations for this system, and calculate the limiting probabilities

.

(b) Calculate the average time that an arriving customer who finds (exactly) one customer in the system will spend in it.

(c) Calculate the average time that a customer who enters the system will spend in it.