The Covariance Function of a Simple Trunk Group, with Applications to Traffic Measurement

A theoretical study of sampling fluctuations in telephone traffic measurements is useful both in designing procedures for measuring traffic loads and in interpreting field observations. Hayward1 and Palm"' have given an approximate formula for the sampling error incurred when observations of the numbers of calls in existence are made at fixed intervals of time. Their formula has the disadvantage that, it is derived for a probabilistic model (of the traffic) in which there is an infinite number of available trunks. Thus there is 1 0 limit to the number of 1 calls which can be in progress at one time, and 1 0 congestion. Two im1 portant parameters, the number N of trunks in the group, and the probability ps of loss, are left out of account. For this reason the practical * This work was completed in part while the author was visiting lecturer at Dartmouth College, Fall-Winter, 1959-fK). 117 118 T H E B E L L SYSTEM T E C H N I C A L J O U R N A L , J A N U A R Y 1901 application of this model is usually restricted to large groups of trunks which are lightly loaded. In this paper we derive and study the covariance function of the simplest stochastic model of a finite group of N trunks. The sampling error in traffic measurements can be calculated exactly from the covariance. We find formulas for the magnitudes of fluctuations of observed traffic for both periodic and continuous observation. The exact formulas lead to simple approximations similar to Hayward's, which take account of the number of trunks.