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Suppose that a test customer in an queueing system can get service only if he has access to the server and a separate event has occurred. All other customers only require access to the server. The time until the event occurs is assumed to be an exponentially distributed random variable, if the test customer reaches the server before occurs, he must then return to the back of the queue. At any time, however, the test customer is allowed to give up his place in the queue and join the back of the queue. The test customer represents a computational task that depends upon the results of an associated task. The test customer's mean delay until service is derived assuming that he always maintains his position in the queue until he reaches the server. Conditions are given for which this "move-along" policy is optimal, i.e., minimizes the test customer's mean delay until service. A condition is also given for which the move-along policy is not optimal.