940 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1957 



without regard to costs of observation, and (ii) determination of the 

 anticipated accuracy of certain methods of estimation which arise natu- 

 rally from the discussion of complete extraction. 



The method by which we attack problems (ij and (iij in this paper 

 has three stages. First we choose a small number of significant properties 

 of, or factors in, the ph^ysical system we are studying. Then we abstract 

 these properties into a mathematical model of the physical system. Fi- 

 nally, from the properties of the model, we derive results which may be 

 interpreted as answers to the two problems treated. The advantage of 

 this method is that we can use the precise, powerful apparatus of mathe- 

 matics in studying the model; its limitation is that it yields results which 

 are only as accurate as the model in describing reality. 



A method similar to the above forms the theoretical underpinning of 

 telephone traffic engineering itself. To design equipment effectively, the 

 traffic engineer needs a description of the traffic that is handled by central 

 offices. He decides what properties of the entire system of telephone 

 equipment and customers will be most useful to him in describing the 

 traffic. He then designates certain parameters to serve as mathematically 

 precise idealizations of these properties, and in terms of these parameters 

 constructs a model of the traffic, upon which he bases much of his engi- 

 neering. 



In choosing a mathematical model for a physical system, one is con- 

 fronted with two generally opposed desiderata: fidelity to the system 

 described, and mathematical simplicity. The model may involve impor- 

 tant departures from physical reality; a model that is sufficiently amena- 

 ble to mathematical analysis often results only after one has introduced 

 admittedly false assumptions, ignored certain effects and correlations, 

 and generally oversimplified the system to be stvidied. However, the 

 abstract model will be an exact and simple tool for analysis. 



We can construct a simple mathematical model for the operation of 

 a telephone central office by leaving out of consideration many impor- 

 tant facts about such systems, and by concentrating on factors most 

 relevant to operation. Since we are interested in telephone traffic and 

 in the availability of plant, it seems natural to require that a realistic 

 model take account of at least the following five significant factors: (1) 

 the demand for telephone service; (2) the rate at which requests for 

 service can be processed and connections established; (3) the lengths of 

 conversations; (4) the supply of central office equipment; and (5) the 

 manner in which the first four factors are interrelated. I'nfortmiately, 

 the mathematical complexity of such a realistic model precludes easy 

 investigation. Therefore, the model used in this paper is based only on 

 factors (1) and (3). 



