Consider a single-server queue with Poisson arrivals and exponential service times having the following variation: Whenever a service is completed a departure occurs only with probability α. With probability 1 - α the customer, instead of leaving, joins the end of the queue. Note that a customer may be serviced more than once.
(a) Set up the balance equations and solve for the steady-state probabilities, stating conditions for it to exist.
(b) Find the expected waiting time of a customer from the time he arrives until he enters service for the ?rst time.
(c) What is the probability that a customer enters service exactly n times, n = 1, 2, .. .?
(d) What is the expected amount of time that a customer spends in service (which does not include the time he spends waiting in line)?