470 



BELL SYSTEM TECHNICAL JOURNAL 



Case No. 1 — Holding Times Vary Exponentially 

 P {> t) for this case was obtained by Erlang of Copenhagen. In 

 1917 he published the formula without its proof. The following 

 deduction of his formula is therefore submitted. 



The particular call "X" considered above will be delayed if the 

 number of calls he encountered, x, is equal to or greater than the 

 number, c, of trunks in the group. Suppose that the x = c + {x — c) 

 calls encountered by "X" are handled by the trunks in the manner 

 indicated in the following Fig. 2, where m^ -\- mi -\- - - ■ mc = x — c. 



2 2 J 3 



t^ 



2 3 4 



call under consideration 



Fig. 2. 



By assumption ^A and taking h as the unit of time, the probability 

 that trunk No. 1 will be busy during an interval of time t is 



{e-'^dti){e-Hh) ••• (e~'"'idtn,,)e- 



it-h-t2 



tm,) 



Giving (/i, 1-2 •• • trnt) all positive values consistent with their sum > /, 

 we obtain (see Todhunter's "Integral Calculus," sixth edition, art. 

 276) 



^ 71 ^ • 



(|Wj) 



Therefore the compound probability that all c trunks are busy during 

 the interval t with the x calls existing at the instant under consideration 

 and that then one of the trunks becomes idle in the interval dt is 



(e- 



E 



t"^i f' 



t" 



mi m^ • • • Mr 

 where E means that we are to give Wi, mo • 



nil + Dio + • • • nic = X 



(cdt), 



}Uc all values such that 

 c. 



