31. Consider a single-server exponential system in which ordinary customers arrive at a rate and have

31. Consider a single-server exponential system in which ordinary customers arrive at a rate λ and have service rate μ. In addition, there is a special customer who has a service rateI mage. Whenever this special customer arrives, she goes directly into service (if anyone else is in service, then this person is bumped back into queue). When the special customer is not being serviced, she spends an exponential amount of time (with meanI mage) out of the system.

(a) What is the average arrival rate of the special customer?

(b) Define an appropriate state space and set up balance equations.

(c) Find the probability that an ordinary customer is bumped n times.