THE INTERCONNECTION OF TELEPHONE SYSTEMS 563 



are to be given all values such that 



rt> 0, tn < (v - r). 

 t=i 



For computnig purposes, note that the function F satisfies the finite 

 difference equation 



F(g — 1, X, y — r) = P(x + y — r, a) 



+ [1 - P(.v, a)-]F{g - 2, X, y - r) 



This difference equation becomes obvious if one considers the change 

 which takes place in the value of P2 when the number of subgroups 

 in a graded multiple is increased from (g — 1) to g. 



Appendix II 



Mathematical Theory of Graded Multiple with 

 Unequal Subgroup Loads 



If we deal with a single stage (or simple) graded multiple having g 

 subgroups of " X " individual trunks each, and " y " common trunks; 

 and if to each subgroup, " ni " for instance, is submitted a particular 

 load, " flm," in average simultaneous calls which are originated at 

 random according to the Poisson Distribution Law requirements; and 

 if these calls are moreover of a constant holding time obeying the 

 " lost calls held " assumption; and if they are at all times so arranged 

 on the graded trunks that " no-holes-in-the-multiple " exist; then the 

 proportion of calls not obtaining immediate service over the multiple 

 taken as a whole may be approximated by: 



a 

 p ^ a,P, + a ^P^ + • • • + a,P, _ Si''"'-^"' 



Ol + O2 + • • • + flff ^ 



m=l 



where 



Pm = P{x + y, a„,) + E -7 — r — P(g — 1, ai, 02, • • • , a,„-i, 



r=Q /x -\- r 



dm+ii • ■ •> Ofli X, y r), 



in which: 



<» rt «a—a„ 



P{x + y, a,n) = L ^^ 



