THEORIES FOR TOLL TRAFFIC EXGINEERIXG IX THE U. S. A. 499 



From (38) it is easy to see that 



Fed = L{1 - Q 



= (Load carried by the group) (1 — load on last trunk) (39) 



This is a convenient relationship since for high usage trunk study work, 

 both the loads carried (in eriangs) on the group and on the last trunk 

 will ordinarily be at hand. 



If the high usage group's load is to be split in various directions at 

 the distant point for re-offer to other groups, it would appear not un- 

 reasonable to assign a variance to each portion so as to maintain the 

 ratio expressed in eciuation (38). That is, if a carried load L is divided 

 into parts Xi , X2 • • • where L = Xi -f X2 • • • , then the associated 

 variances 71 , 72 . . • would be 



71 = Xi (1 - fc) 



y, = Xo (1 - fc) (40) 



If, however, the load offered to the group is non-random (e.g., the 

 group is an intermediate route in a multi-alternate route system), the 

 procedure is not quite so simple as in the random case just discussed. 

 Equation (32) expresses the variance Vc of the carried load on a group 

 of C paths whose 'offered traffic consists of the overflow from a first 

 group of S paths to which a random load of A eriangs has been offered. 

 Vc could of course be expressed in terms of A', V and C, and curves or 

 tables constructed for working purposes. However, such are not avail- 

 able, and in any case might be unwieldy for practical use. 



A simple alternative procedure can be used which jdelds a conserva- 

 tive (too large) estimate of carried load variance. With random load 

 offered to a divided two stage multiple of x paths followed by tj paths, a 

 positive correlation exists between the numbers m and n of calls present 

 simultaneously on the x and y paths, respectively. Then the variance 

 V-n+n of the m -\- n distribution is greater than the sum of the individual 

 variances of m and n, 



y m-\-n ^ ' m l~ ' n 



or 



Vm < y^n - Vn (41) 



Now n can be chosen arbitrarily, and if made very large, Vm+n becomes 

 the offered load variance, and F„ the overflow load variance. Both of 

 these are usually (or can be made) available. Their difference then, 

 according to (41) gives an upper limit to F,„ , the desired carried load 



