The following generalization of the classical ballot theorem is given: Suppose that an urn containsncards marked with nonnegative integers whose sum isk≦n. All thencards are drawn without replacement from the urn. The probability that forr= 1, 2, ···,nthe sum of the firstrnumbers drawn is less than r is 1 —k/n. By using the ballot theorem and its generalization the author findsGn(x) the probability that a busy period consists of servingncustomers and its length is ≦xfor single-server queues when either the inter-arrival times or the service times have an exponential distribution. Finally, the author gives the general solution of the classical ballot problem as well as an application of it in the theory of Bernoulli trials.