552 BELL SYSTEM TECHNICAL JOURNAL 



/x 



and is independent of the variations in any other subgroup. 

 4. " No-holes-in-the-multiple." ^^ 



The probabiHty that a caUing source fails to obtain an idle trunk 

 immediately may be divided into two parts, corresponding to two 

 essentially different sets of circumstances under which a call may be 

 interfered with. Assume, to fix ideas, that the particular source 

 under consideration belongs within subgroup No. 1 in Figure 7 (A). 

 Failure will occur if in addition to the call originating from this source, 



1. At least X + y calls originated by subgroup No. 1 occupy trunks 



(actually or potentially) ; or, 



2. The number of calls placed on the trunks consists of x -\- r origin- 



ated from subgroup No. 1 ; at least x + 1 calls originated from 

 each of .y of the other subgroups; and, moreover, that said 5 

 other groups have, collectively, placed at least sx -{- y — r 

 calls. 5 is a number which may have any value from 1 to 

 (g — 1), inclusive. 



This classification of the circumstances under which the call under 

 consideration may be delayed gives, for the desired probability when 

 " holes-in-the-multiple " are not possible, the equation 



P = Pi + P2, 



where 



a. Pi = P{x + y, a) 



V—l / .-.x+rp-a \ 



c. P{x + y, a) is the Poisson expansion L —jr- j 



t=x+v \ / 1 I 



d. F{g - \,x,y - r) 



.|(->)c.-n. + >,.):.-^S£(zS)' 



where S indicates that in the product 



TT -, — . — , ri, r-i, r^ • • • r, 

 t=i \ /x + rt I 



^^ Imagine that some method of transferring calls from common to individual 

 trunks eliminates the possibility of "holes-in-the-multiple." 



