12. Exponential queueing systems in which the state is the number of customers in the system are known as birth and death queueing systems. For such a system, let λn denote the rate at which a new customer joins the system when it is in state n, and let μn denote the rate at which there is a departure from the system when in state n.

(a) Give the quantities λn and μn for the M/M/1 queue with finite capacity N.

(b) Write down the balance equations.

(c) Show how the balance equations can be reduced to the set of equations

λnPn = μn+1Pn+1, n 0

(d) Give a direct argument for the preceding equations.

(e) Solve the preceding equations, and in doing so, give the condition that is needed for there to be a solution.

(f) What is the average arrival rate λa ?

(g) What is the average amount of time that a customer spends in the system?