Stream of Overflows from a Finite Queue

A finite queuing system with a recurrent arrival process and a single negative exponential server is considered. A customer who, upon his arrival, finds the system full departs never to return, i.e., he “overflows.” The process of overflows is shown to be a recurrent process and the distribution of the time between overflows is derived as a recurrence time distribution in a semi-Markov process. In the special case where the maximum number of customers allowed in the system is one, the problem is known as Palm's overflow problem.