Single Server Systems -- I. Relations Between Some Averages

A typical but rather homely example of the systems considered here is a barber shop in which there is only one barber, the "single server." Let N -- 1 be the number of chairs provided for customers waiting for service so that the "capacity" of the system is N. When the shop is full, one customer is being served and N -- 1 are waiting. A prospective customer (a "demand" for service) arriving when the shop is full is turned away and is said to be "lost." If the shop is not full, he waits and is eventually served. The demands arrive at an average rate of a per unit time. The server would serve b customers per unit time (on the average) if he were to work steadily.it follows that the average interval between arrivals is 1 /a and the average service time is 1/6. It is assumed that the rates a and b do not change with time. The first part of the paper is concerned with several quantities of interest, including the fraction L of demands lost and the average length of the busy periods, i.e., the periods during which the server is continuously busy. The values of these quantities are expressed in terms of a, b, and two other quantities po and r. Here p 0 is the probability that the server is idle (at an instant selected at random) and r is the average duration of an idle period. Both PO and r depend upon N and upon the probability laws governing the arrivals and service times. However, only the simplest cases of this dependence are mentioned in the first part, of the paper. The results are summarized in Table I.