In a queue with unlimited waiting space, arrivals are Poisson (parameter ) and service times are exponentially

In a queue with unlimited waiting space, arrivals are Poisson (parameter λ) and service times are exponentially distributed (parameter μ). However, the server waits until K people are present before beginning service on the first customer;

thereafter, he services one at a time until all K units, and all subsequent arrivals, are serviced. The server is then “idle” until K new arrivals have occurred.

(a) Define an appropriate state space, draw the transition diagram, and set up the balance equations.

(b) In terms of the limiting probabilities, what is the average time a customer spends in queue?

(c) What conditions on λ and μ are necessary?