A Sufficient Set of Statistics for a Simple 

 Telephone Exchange Model 



By V. E. BENES 



(Manuscript received October 17, 1956) 



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 co- 

 efficients are obtained. Various estimators for the parameters of the model 

 are compared, and some of their distributions, means, and variances pre- 

 sented. 



I THEORETICAL PROBLEMS AND METHODS OF TRAFFIC MEASUREMENT 



Four important kinds of theoretical problems arise in the measurement 

 of telephone traffic. These are: (1) the choice of a mathematical model, 

 containing parameters characteristic of the traffic, to serve as a descrip- 

 tion; (2) the devising of efficient methods of estimating the parameters; 

 (3) the determination of the anticipated accuracy of measurements; 

 and (4) the assessment of actual accuracy, after measurements have 

 been made. 



The present paper deals with aspects of the second and third kinds of 

 problem, for the simplest and least realistic mathematical model of tele- 

 phone traffic. Specifically, for this model, we treat the problems of (i) 

 complete extraction of the information fiom a given observation period, 



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