A Discrete Time, Single Server Queue from a Finite Population

One of the standard assumptions in queueing theory is that the system operates for an indefinite period of time. This assumption, however, is not valid for many practical situations where there is only a finite number of customers and each customer requires service only once. This paper studies such a finite queueing system in which all customers have the same time-dependent arrival probability function.

The method of solution allows one to “follow” each customer to record his arrival and departure times. While the study of the former is straightforward, the generating function of the departure time probabilities and the expected departure time of each customer are expressed in terms of the emptiness probabilities, which can be calculated from a recurrence relation. When the arrival probability function is independent of time, explicit results are obtained.