On the Accuracy of Loss Estimates

In this paper, we shall consider the simplest type of loss systems, namely full availability groups with Poisson inputs, negative exponential 1139 1140 THE BELL SYSTEM TECHNICAL JOURNAL, JULY-AUG. 1965 service times, and cancellation or rerouting of calls finding all trunks occupied. Under these assumptions, we shall obtain an approximate expression for the variance of the measured call congestion, the latter being defined here as the proportion of calls which either are lost or overflow to some alternate group during a given time interval. In the derivation of this expression, use is made of the classical formula for the propagation of errors, whose computation requires the evaluation of the first- and second-order moments of the joint distribution of the number of offered and the number of overflow calls. Since the marginal means and variances of this distribution are known (cf. Ref. 1), the emphasis is placed here on the derivation of the covariance. Computed values of the variance of the measured call congestion are shown to be in good agreement with simulation results (cf. Figs. 1-5).* Charts giving the variance of this ratio for group sizes up to 50 and offered loads (in erlangs) per trunk of 0.1 to 10, are reproduced in Figs. 6-8.* For a given observation period, the measured call congestion and the observed proportion of time when all servers are busy -- here called measured time congestion -- provide us with two estimates of the probability that a call will either be lost or overflow to some alternate route.