70 BELL SYSTEM TECHNICAL JOURNAL 



manner that can be understood by those who are not experts on the 

 subject. It is hoped that this article will assist the reader in under- 

 standing both what has been and will be written on the subject. As 

 Poisson ^ has said "a problem relative to games of chance and proposed 

 to an austere Jansenist by a man of the world, was the origin of the 

 calculus of probabilities," and today the reader will find that in the 

 majority of text books the subject is introduced by the solution of 

 games of chance and particularly of dice problems. This established 

 custom will be followed by the present writer who, in the course of 

 this article, will show how various fundamental trunking problems 

 can be transformed into equivalent dice problems. This being done, 

 solutions will be found to be at hand. 



Three trunking problems, each one step more complicated than 

 the preceding, will be dealt with. In order to facilitate the trans- 

 formation to the three equivalent dice problems it is desirable that 

 the basic assumptions made be as simple as possible. The asump- 

 tions made in all three problems are: 



A — During the period of time under consideration, the busy hour of 

 the day, each subscriber s line makes one call which is as likely to fall at 

 any one instant as at any other instant during the period. 



Conditions substantially approximating this assumption frequently 

 occur in practice. 



B — If a call when initiated obtains a trunk immediately it retains 

 possession of that trunk for exactly two minutes. In other words, a 

 constant holding time of two minutes duration will be assumed. 



In practice, holding times, of course, vary from a few seconds to 

 many minutes and it may at first sight seem that the assumption of 

 a constant holding time might lead to results deviating too much 

 from practice to be of value. On this point, the theory of probabilities 

 itself sheds some interesting light. As will be pointed out in the 

 following problems, the assumption of a constant holding time is 

 the equivalent of a dice problem in which a single die, or several iden- 

 tical dice are considered. The telephone problem with variable 

 holding times may be reduced to the consideration of many dice, 

 each with a different number of faces. Suppose 600 throws are made 

 with a die having 6 faces so that on the average 3^ of 600 or 100 aces 

 would be expected. With Bernoulli's formula it is easy to find the 

 probability that the number of aces which turn up shall lie between 

 75 and 125, that is to say, within 25 on each side of the average. Now 

 suppose 200 throws are made with a die having 20 faces, 200 with a 



» Poisson, Recherches Sur La Probabilite Des Jugements, 1837. 



