TELEPHONE TRAFFIC TIME AVERAGES 1135 



and the conditional probability that it will also occupy a trunk at time 



t -\- uis 



I ff{t + u-x)dx-^Q{l) 

 t Jo 



(17) 



/ Jo 

 or 



^^^^^ - m,uX s.y. 



The corresponding generating function, with x and y the indicators of calls 

 at / and / + u, resp. is 



1 - eW + Qim - R(t, u)]x + Q{t)R(t, u)xy 

 or 



1 + (:*: - 1)(1 - g(t))h/t + x{y - iMu) - g(t + u)]h/t 



The generating function for c calls in this interval is this expression raised 

 to the c'th power, since calls arrive independently; and since c calls arrive 

 with probabihty e~°'\aty/c\, the generating function for calls arriving in 

 this interval is 



Z [1 + (^ - i)e + <y - WRYe-'^'iaD'/ci 



= expb[{x - 1)(1 - g{t)) + x(y - l){giu) - git + u))] 



For brevity Q and R have been written for Q(t) and R{t, u). 

 Finally the generating function for calls arriving in /, / + «, is 



exp h{y - 1)(1 - g{u)) (18) 



Hence 

 Z Z Ay.(/, u)x^y' = [1 + (x - \)g{t) + x{y - \)g{i + u)V 



i k 



• exp blix - 1)(1 - git)) + (y - 1)(1 - giu)) (19) 



+ xiy - Digiu) - git + u))] 



By similar argument, the generating function for triple transition proba- 

 bihties is 



j:i:i:PiMhu,v)x'yV 



j k t 



= [1 + (:*; - \)gt + xiy - l)gt+u + xyiz - l)gt+u+J* 



. exp biix - 1)(1 - gt) -h (y - 1)(1 - gu) + (20) 

 (2 - 1) (1 - g.) 4- xiy - Digu - gt+u) + 



yiz - Digv - g«+r)+ xyiz - l)(^u+r - gt+u+v)) 



