On the accuracy of loss estimates

In telephone traffic studies, the observed “proportion of unsuccessful attempts over a given time interval is one of the measures commonly used to evaluate the grade of service provided by trunk groups. This paper deals with the derivation of an approximate formula for the variance of this estimate when (i) call arrivals constitute a Poisson process, (ii) service times are independent of each other and identically distributed according to a negative exponential law, and (iii) calls placed when all trunks are busy are either cancelled or sent via some alternate route (loss system). Comparison of simulation data with numerical values computed by means of this formula indicates that the latter is accurate enough for practical purposes. The observed proportion of time during which all trunks are occupied is also an estimate of the grade of service (defined as the probability that a call will be lost or overflow). It is shown here, that for relatively small loads, this estimate has a smaller variance than the observed proportion of lost or rerouted calls. However, as the load is increased, the inequality between the variances of these two estimates is reversed, the cross-over occurring in the vicinity of the point where the load (in erlangs) is equal to the number of trunks. For a given observation period, the proportion of time when all trunks are busy can be either measured exactly or estimated by “switch-counting.” In the latter case, the group is scanned at regular intervals and one observes, for each scan, whether all trunks are occupied or not. The average number of scans which indicate that all trunks are busy is an estimate of this proportion and is, a fortiori, an estimate of the probability of loss. The effect of the scanning rate on the accuracy of this estimate is investigated.