STATISTICS FOR A SIMPLE TELEPHONE EXCHANGE MODEL 



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making measurements and observations on the model. The mathematical 

 model elucidates our problems in the following ways: (1) it enables us 

 to state precisely what information is provided by observation; (2) it 

 enables us to explain what we mean by complete extraction of informa- 

 tion; and (3) it enables us to derive results about the anticipated ac- 

 curacy of measurements in the model. These results will have approxi- 

 mately true analogues in physical situations to which the model is 

 applicable. 



The calls existing during the observation interval (0, T) fall into four 

 categories: (i) those which exist at 0, and terminate before T; (ii) those 

 which fall entirely within (0, T) ; (iii) those which exist at and last 

 beyond T; and (iv) those which begin within (0, T) and last beyond T. 

 For calls of category (i), we assume that we observe the hang-up time 

 of each call; for category (ii), we observe the matching calhng-time and 

 hang-up time of each conversation ; for category (iii) , we observe simply 

 the number of such calls; and for category (iv), we observe the caUing- 

 times. Table I summarizes the kinds of calls and the information ob- 

 served about each. 



What we mean by the complete extraction of information is made 

 precise by the statistical concept of sufficiency. By a statistic we mean 

 any function of the observations, and by an estimator we mean 

 a statistic which has been chosen to serve as an estimate of a particular 

 parameter. Roughly and generally, a set S of statistics is sufficient for a 

 set P of parameters when S contains all the information in the original 

 data that was relevant to parameters in P. If S is sufficient for P, there 

 is a set E of estimators for parameters in P, such that the estimators in 

 E depend only on statistics from S, and such that an estimator from E 

 does at least as well as any other estimator we might choose for the same 

 parameter. Thus we incur no loss in reducing the original data (of speci- 

 fied form) to the set »S of statistics. It remains to state what it means for 

 S to contain all the relevant information. We do this in terms of our 

 model. 



The mathematical model we are adopting contains two distribution 



Table I — • Information Observed 



