Let us suppose that customers arrive at a counter in accordance with a Poisson process of density λ. The customers are served by a single server in order of arrival. The service times are identically distributed, mutually independent, positive random variables with distribution function H(x). Suppose that after being served each customer either immediately joins the queue again with probability p or departs permanently with probability q (p + q =1). In this paper we shell determine for a stationary process the distribution of the queue size as well as the Laplace-Stieltjes transform and the first two moments of the distribution function of the total time spent in the system by a customer.
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L Takács (1963) studied this question.
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