1130 THE BELL SYSTEM TECHNICAL JOURNAL, OCTOBER 1951 



concreteness it may be mentioned that the most important, at the moment, 

 seems to be that of setting confidence limits for the average traffic. 



The most important of the Umits to this study are those impHed by the 

 assumptions of statistical equihbrium with fixed average, and an infinite 

 number of trunks. The former Umits appHcation to periods in which, roughly 

 speaking, average traffic is neither rising nor falling; the latter is justified 

 only by the extreme mathematical difficulties produced by assuming other- 

 wise. The traffic variable is the number of busy trunks in a period of statisti- 

 cal equilibrium. For pure chance call input, the call holding time character- 

 istic is left arbitrary throughout the development, but main interest lies in 

 the two extreme cases of constant holding time and exponential holding 

 time, which are examined in detail,* For calls from a limited number of 

 sources, results are obtained only for exponential holding time. 



More precisely, if N{t) is the random variable for the number of busy 

 trunks at time t, the variable studied, the average number of calls in an 

 interval of length T, is 



M{T) =\, f N{t) dt (1) 



1 Jo 



The question is: What are the statistical properties of M{T)? 



The results given are the first four cumulants (semi-invariants) of M{T), 

 which seem to have the simplest expressions. For the convenience of the 

 reader it may be noticed that the first cumulant is the mean, the second 

 the second moment about the mean which is the variance, the third the 

 third moment about the mean, and the fourth the fourth moment about 

 the mean less three times the square of the variance. 



In all cases the mean of M{T) is the mean of N(t) and for pure chance call 

 input is called b, the average number of calls in unit average holding time, h. 



The other cumulants for pure chance call input, kn , have the general 

 expression 



k. = b "^"^7 ^^ jj dxgix) {T - x)x"-'; « = 2, 3, 4 



with 



* F. W. Rabe [6] has reported results for these two cases for relatively long averaging 

 intervals, which are verified below. I owe my interest in this problem to a report on Rabe's 

 work made by Messrs. Gibson, Hayward and Seckler in a probability colloquium at Bell 

 Telephone Laboratories initiated and directed by Roger Wilkinson. 



