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People arrive at an automatic teller machine (ATM) according to a Poisson process with intensity λ. The service time required at the ATM is constant, a seconds. Unfortunately, this machine does not allow for any waiting customers (i.e., no queue is allowed), which means that persons who arrive while the ATM is busy have to leave. When the a seconds of a customer have elapsed, the ATM is free to serve again, and so on. Suppose that the ATM is free at time zero, and let Tn be the time of the arrival of the nth customer. Find the distribution of Tn, and compute E Tn and Var Tn. Remark. Customers arriving (and leaving) while the ATM is busy thus do not affect the service time.

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