THE THEORY OF PROBABHJTIES 71 



10 face die, 100 with a 5 fac-e die and finally 100 with a 2 face die. 

 These 600 throws would also give on the average 100 aces. Using 

 Poisson's generalization of the Bernoulli formula' we can calculate 

 the probability that these (iOO throws with various kinds of dice shall 

 gi\e a number of aces lying between 75 and 125. This probability 

 will be greater than in the case of the 600 throws with the die 

 with a constant number of faces, i.e., the chance that the result will 

 come outside the range 75 to 125 is less. 



The thought is at once suggested that for the same total volume of 

 traffic and a\-erage holding time, fewer calls would be lost when the 

 holding time is not constant.'* The above theory was tested in practice 

 a few \ears ago by the engineers of the American Telephone and Tele- 

 graph Company, who made pen register records of hundreds of thou- 

 sands of actual calls as handled by groups of machine switching 

 trunks at Newark, New Jersey. A pen register was made which 

 operated as follows: Each trunk in the group was represented by a 

 pen. These pens were mounted side by side and each was controlled 

 by a magnet in such a manner that when the trunk was busy the pen 

 made a mark on a wide strip of paper driven at constant speed under 

 the pens. There was thus obtained a record showing when each call 

 originated and when it was concluded. An artificial record was now 

 made showing what would have happened if each call had lasted for 

 the average holding time as determined from the original record. 

 Some 100,000 calls w^ere analyzed in this manner and it w^as found that 

 with a group of trunks of a size to carry the calls of the original record 

 with only a small loss, 30 per cent, more calls would have been lost if 

 the traffic had been as shown by the artificial record. It should be 

 borne in mind, however, that a 30 per cent, change in a probability 

 of the order of one in one hundred, considering the values we are deal- 

 ing with, is practically negligible. 



C — // a trunk is not obtained immediately the calling subscriber 

 waits for two minutes and then withdraws his call. If while waiting 

 a trunk becomes idle he takes it and converses for the interval of time 

 remaining before his two minutes are up. 



This assumption, although artificial, simplifies materially the analy- 

 sis of the problems. Just what happens in practice to every call 



'' This result is here reached by assuming that each subscriber originates one call 

 per hour. The conclusions are the same, however, even when this is not true, pro- 

 vided the term "holding time" is understood to mean the aggregate of all the talking 

 times of the subscriber in an hour. 



It may also be mentioned in passing that for a fixed volume of traffic, the dis- 

 crepancy decreases as the number of subscribers who originate that traffic increases; 

 that is, it is less when the group is composed of a large number of relatively idle 

 lines than when it is composed of a small number of very busy ones. 



