THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 445 



By choosing x -]- y large compared with the submitted load a a "lost 

 calls held" situation or infinite-overflow-trunks result can be approached 

 as closely as desired. 



Kendrick suggested solving the series of simultaneous equations (7) by 

 determinants, and also by a method of continued fractions. However 

 little of this numerical work was actually undertaken until several years 

 later. 



Early in 1935 Miss E. V. Wyckoff of Bell Telephone Laboratories be- 

 came interested in the solution of the (x -\- 1)(^/ + 1) lost calls cleared 

 simultaneous equations leading to all terms in the /(m, n) distribution. 

 She devised an order of substituting one equation in the next which pro- 

 vided an entirely practical and relatively rapid means for the numerical 

 solution of almost any set of these equations. By this method a con- 

 siderable number of /(m, n) distributions on x, y type multiples with 

 varying load levels were calculated. 



From the complete m, n matrix of probabilities, one easily obtains the 

 distribution 9m{n) of overflow calls when exactly m are present on the 

 lower group of x trunks; or by summing on m, the d{n) distribution with- 

 out regard to m, is realized. A number of other procedures for obtaining 

 the/(m, n) values have been proposed. All involve lengthy computations, 

 very tedious for solution by desk calculating machines, and most do not 

 have the ready checks of the WyckofT-method available at regular points 

 through the calculations. 



In 1937 Kosten^ gave the following expression for /(m, n) : 



/(», n) = (- l)V.fe) i (i) M^- "f^'l., (8) 



i=0 



(Pi^l{x)ipi(x) 



where 



(po{x) = 



x^—a 



a e 



xl 



; and for i > 0, 



;=o \ J / (.^• - J)i 



These equations, too, are laborious to calculate if the load and num- 

 1 K^rs of trunks are not small. It would, of course, be possible to program a 

 modern automatic computer to do this work with considerable rapidity. 



The corresponding application of the statistical equilibrium equations 

 to the graded multiple problem was visualized by Kendrick who, how- 

 ever, went only so far as to write out the equation for the three-trunk 



