A Sufficient Set of Statistics for a Simple Telephone Exchange Model

F o u r i m p o r t a n t k i n d s of theoretical problems arise in the m e a s u r e m e n t of telephone traffic. These a r e : (1) t h e choice of a m a t h e m a t i c a l model, containing p a r a m e t e r s characteristic of the traffic, to serve as a description; (2) t h e devising of efficient m e t h o d s of e s t i m a t i n g t h e p a r a m e t e r s ; (3) t h e d e t e r m i n a t i o n of t h e anticipated accuracy of m e a s u r e m e n t s ; a n d (4) t h e assessment of actual accuracy, a f t e r m e a s u r e m e n t s h a v e been made. T h e present p a p e r deals with aspects of t h e second and t h i r d kinds of problem, for t h e simplest a n d least realistic m a t h e m a t i c a l model of telephone traffic. Specifically, for this model, we t r e a t t h e problems of (i) complete extraction of the information from a given observation period, 939 95(> THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1 9 5 7 without regard to costs of observation, and (ii) determination of the anticipated accuracy of certain methods of estimation which arise naturally from the discussion of complete extraction. T h e method by which we attack problems (i) and (ii) in this paper has three stages. First we choose a small number of significant properties of, or factors in, the physical system we are studying. Then we abstract these properties into a mathematical model of the physical system. Finally, from the properties of the model, we derive results which m a y be interpreted as answers to the two problems treated.