Suppose that customers are in line to receive service that is pro- vided sequentially by a server; whenever a service is completed, the next person in line enters the service facility. However, each waiting customer will only wait an exponentially distributed time with rate θ; if its service has not yet begun by this time then it will immediately depart the system. These exponential times, one for each waiting customer, are independent. In addition, the service times are independent exponential random variables with rate μ. Suppose that someone is presently being served and consider the person who is nth in line.
(a) Find Pn, the probability that this customer is eventually served.
(b) Find Wn, the conditional expected amount of time this person spends waiting in line given that she is eventually served.