TELEPHONE TRUNKING PROBLEMS 467 



appear on the varying holding time curves. This difference in the 

 two classes of curves should occasion no surprise. In the varying 

 holding time case the quantity h has no physical significance; it is 

 merely a numerical value obtained by an algebraic process called 

 averaging. In the other case the quantity h represents a physical 

 characteristic of each and every call. 



As stated on page 464 the solution presented in this paper for the 

 case where holding times are all of constant length is not exact. It is 

 desirable therefore to have some idea of the degree of approximation 

 attained. 



Figure 4 shows a comparison between our tentative solution and 

 true values which Erlang of Copenhagen, Denmark, succeeded in 

 obtaining by a method which unfortunately becomes impracticable 

 for values of c greater than 3. Our results are shown by the solid 

 curve and Erlang's results by the small circles. We have also indi- 

 cated on the c — \, 2 and 3 constant holding time charts, Erlang's 

 points for ajc = 0.50. For Erlang's work, reference may be had to the 

 Elektrotechnische Zeitschrift, December 19, 1918, page 504. 



It may be noted that for the higher values of the ratio ajc the 

 curves are practically straight lines. They depart materially from 

 straight lines for the lower values of the ratio a[c, particularly if c 

 itself is not very large. 



Assumptions Made in Mathematical Theory 

 The mathematical theory back of the curves accompanying this 

 paper is based on the following assumptions : 



1. Calls originating independently of each other, and at random with 



reference to time, have complete access to a single group of 

 trunks. 



2. The probability of a call originating during a particular infinitesimal 



interval, dt, is practically independent of the number of trunks 

 busy or number of waiting calls at the beginning of said interval. 

 This assumption implies that the total interval of time during 

 which the calls fall at random is very large compared with the 

 average holding time per call and that the total number of 

 calls under consideration is very large compared with the 

 number of calls originating per average holding time interval. 



3. Calls are served in the order in which they originate. This restric- 



tion does not apply to the average delays obtained. 

 ■iA. The average holding time being h, the holding times of individual 

 calls vary around this average in such a way that e~^"' is the 

 probability that for a call taken at random the holding time 

 is greater than /. 



