On Overflow Processes of Trunk Groups with Poisson Inputs and Exponential Service Times

When telephone networks are engineered, the loads offered to the alternate routes cannot usually be assumed to be random (Poissonian) and, in these cases, it is of considerable practical interest to determine the characteristics of the traffic overflowing the trunk groups. In this paper we shall be mainly concerned with the interoverfiow distribution, the term used here to designate the distribution of the time intervals separating consecutive epochs at which calls find all trunks busy (overflow). We shall first show that the distribution of the nonbusy periods of a group of c trunks is identical to the interoverfiow distribution of a group of c -- I trunks and then obtain new recurrence formulas for the Laplace transforms of the interoverfiow distribution of a single trunk group under the assumptions that: (a) the load submitted to the group is random; (b) the service times are independent of each other and are 383 384 T H E B E L L SYSTEM T E C H N I C A L J O U R N A L , MARCH 1963 all distributed according to the same negative exponential law; and (c) the requests which are placed when all the trunks are busy are either canceled or sent via some alternate route. As we shall see, these formulas are much simpler than the expressions obtained by C. Palm (cf Ref. 1, pp. 25-2G, and Ref. 2, pp. 36-40) and are well suited to the computation of the moments. Then, under the same three assumptions, we shall also obtain the generating function of the probability distribution of the number of consecutive successful calls or, in other words, of the number of calls which are placed during a time interval whose end points coincide with two consecutive overflows.