THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 513 



ing the character of non-random traffic. An approximate solution of the 

 problem is offered based on this method. 



Suppose a random traffic a is offered to a straight multiple which is 

 divided into a lower Xi portion and an upper X2 portion, as follows: 



T «2 , V2 



X2. 



] OCl,Vi 



u 



From Nyquist's and Molina's work we know the mean and variance 



of the two overflows to be: 



ai = a-Ei^xiia) = a 



a"» 



•ril 





Vi = ai\ 1 — ai -\ ■ — - 



L Xi — a + ai + IJ 



a2 = a-Ei,xi+x2(0') 



V2 = aol I — a2 -\ j j : — r 



L xi + a;2 - a + 0:2 4- IJ 



Since ai and vi completely determine a and Xi , and these in turn, with 

 X2 , determine 02 and Vo , we may express 02 and V2 in terms of only ai , 

 Vi , and X2 . The overflow characteristics (0:2 and V2), are then given for a 

 non-random load (ai and Vi) offered to x trunks as was desired. 



Fig. 51 of this Appendix has been constructed by the Equivalent Ran- 

 dom method. The charts show the expected values of 0:2 and I'o when 

 ai , Vi (or vi/ai), and X2 , are given. The range of ai is only to 5 er- 

 langs, and v/a is given only from the Poisson unity relation to a peaked- 

 ness value of 2.5. Extended and more definitive curves or tables could 

 readily, of course, be constructed. 



The use of the curves can perhaps best be illustrated by the solution 

 of a familiar example. 



Example: A load of 4.5 erlangs is submitted to 10 trunks; on the "lost 

 calls cleared" basis; what is the average load passing to overflow? 



Solution: Compute the load characteristics from the first trunk when 

 4.5 erlangs of random traffic are submitted to it. These values are found 

 to be a\ = 3.G8, vi = 4.15. Now using ai and vi (or vi/ai = 4.15/3.68 = 

 1.13) as the offered load to the second trunk, read on the chart the param- 

 eters of the overflow from the second trunk, and so on. The successive 

 overflow values are given in Table XVIII. 



