78 B^LL SYSTEM TECHNICAL JOURNAL 



and first selectors to be such that the calls are distributed uniformly 

 to the first selectors, meaning that if at any instant C calls exist, C/4 

 of them are on first selectors having access to the 10 trunks of sub- 

 group Gi, C/4 are on first selectors having access to the 10 trunks 

 of sub-group G2 and so on. With a constant holding time such as 

 assumed this result could be secured by a device common to all line 

 switches which would route the first call to the first sub-group, the 

 second call to the second sub-group, etc. 



X will, as before, be interested in the calls falling in the two minutes 

 preceding him. By hypothesis y^ of these will have been distributed 

 to first selectors having access to the same sub-group of second selec- 

 tors as the first selector seized by X. Finally, the probability is % 

 that one of these calls wants the level in which X is interested. The 

 equivalent dice problem is therefore: 



1st. 1725 throws are made with a 30 face die and the number of 



aces which turn up are noted. Let this number be C. 

 2nd. C/4 throws are made with a 3 face die. 



What is the probability that this sequence of throws results in at 

 least 10 aces? This probability is not that of getting at least 10 aces 

 if 1725 throws are made with a die having 30 X 3 = 90 faces. We 

 must write separately the formula for each of the two steps of the 

 problem, then multiply them together and finally sum the product 

 for all values of C/4 from 10 up. If this is done, again ignoring the 

 restriction on the upper limit of C, the answer will come out 0.01. 

 Note that whereas in Case 1 the average volume of traffic carried 

 by a sub-group of 10 trunks was 4.13, in this case, with the same 

 probability of failure, it is 1725 (1/30) (1/4) (1/3) - 4.79. 



Case 3 



In conclusion, a third and very interesting case will be mentioned. 

 A distribution of calls collectively at random would be an appropriate 

 name, and its nature may be described as follows: 



Number each first selector and a corresponding card; shufifle the 

 cards and deal out, for example, 37 of them. The distribution under 

 consideration is such that when 37 calls exist the probability that 

 they occupy a specified set of 37 selectors is equal to the probability 

 that the cards dealt have the corresponding numbers. This dis- 

 tribution of calls would be measurably secured by arranging the line 

 switch multiple so that the trunks to the first selectors appear so far 

 as possible in a different order before every line switch. This case of 

 distribution differs from that of Case 1. In Case 1, if the first call 



