A Sufficient Set of Statistics for a Simple Telephone Exchange Model

This paper considers a simple telephone exchange model which has an infinite number of trunks and in which the traffic depends on two parameters, the calling-rate and the mean holding-time. It is desired to estimate these parameters by observing the model continuously during a finite interval, and noting the calling-time and hang-up time of each call, insofar as these times fall within the interval. It is shown that the resulting information may, for the purpose of this estimate, be reduced without loss to four statistics These statistics are the number of calls found at the start of observation, the number of calls arriving during observation, the number of calls terminated during observation, and the average number of calls existing during the interval of observation. The joint distribution of these sufficient statistics is determined, in principle, by deriving a generating function for it. From this generating function the means, variances, covariances, and correlation coefficients are obtained. Various estimators for the parameters of the model are compared, and some of their distributions, means, and variances presented.