942 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1957 



fortunately, we do not take advantage of this similarity, since we make 

 the mathematically convenient but wholly unrealistic assumption that 

 the number of trunks in the model is infinite. 



The model we investigate thus depends on only two of the factors j 

 previously listed as essential to a realistic model: namely, (1) the demand 

 for service, and (3) the lengths of conversations. In view of the simplicity 

 and inaccuracy of this model, the question arises whether much is gained 

 from a detailed analysis. Such scrutiny may indeed reveal little that is 

 of great practical value to traffic engineers. It is important methodologi- 

 cally, however, to have a detailed treatment of at least one approximate 

 case. We undertake this detailed treatment largely for the insight that 

 it may give into methods which could be useful in dealing with more 

 complex and more accurate models. 



Once a designer has chosen a model and has specified the parameters 

 he would like to have measured, it is up to the statistician to invent effi- 

 cient means of measurement, by choosing, for each parameter, some 

 function of possible observations to serve as an estimate of that parame- 

 ter. One measure of efficiency that is of mostly theoretical interest is the 

 observation time required to achieve a given degree of anticipated ac- 

 curacy ; the most reafistic measure of efficiency is in terms of dollars and 

 man-hours. It may often be more efficient, in the sense of the latter 

 measure, to spread observation over enough more time to compensate 

 for the inability of an intrinsically cheaper method of measurement to 

 extract all of the information present in a fixed time of observation. For 

 example, periodic scanning of switches in a telephone exchange is usually 

 less costly than continuous observation. As a result, telephone traffic 

 measurement is usually carried out by averaging sequences of instan- 

 taneous periodic observations of the number of calls present, rather than 

 by continuous time averaging, although it can be shown that continuous 

 observation is more efficient at extracting information. Thus statistical 

 efficiency, which may be expensive in terms of measuring equipment, 

 can be exchanged for observation time, which may be less costly. This 

 exchange brings about a reduction in cost without impairing accuracy. 



Our concern in this paper is with the less practical problems of com- 

 plete extraction, and of the anticipated accuracy of estimation methods 

 based on complete extraction. Let us consider how our mathematical 

 model can shed light on these problems. A mathematical model may or 

 may not be a faithful description of the behavior of real telephone sys- 

 tems. Nevertheless random numbers, with or without modern computing 

 machines, enable one to make experiments and observations on physical 

 situations which approximate, arbitrarily closely, any mathematical 

 model. Thus we can speak meaningfully of events in the model, and of 



