At a large international airport, a currency exchange bank with only one teller is open 24 hours a day, 7 days a week. Suppose that at some time t = 0, the bank is free of customers and new customers arrive at random times T1, T1 + T2, T1 + T2 + T3,

. . . , where T1, T2, T3, . . . are identically distributed and independent random variables with E(Ti) = 1/λ. When the teller is free, the service time of a customer entering the bank begins upon arrival. Otherwise, the customer joins the queue and waits to be served on a first-come, first-served basis. The customer leaves the bank after being served. The service time of the ith new customer is Si, where S1, S2, S3, . . . are identically distributed and independent random variables with E(Si) = 1/μ. Therefore, new customers arrive at the rate λ, and while the teller is busy, they are served at the rate μ. Show that if λ < μ, that is, if new customers are served at a higher rate than they arrive, then with probability 1, eventually, for some period, the bank will be empty of customers again.