376 BELL SYSTFAf TECHNICAL JOURNAL 



Similarly thf prohahilit)- tli;it cxartly two switrb counts will he contributed 

 by such a call is 



i 2i 



p2 = P{>i) - P(>2i) = e'^ - e"^ . 

 Likewise, 



p^ = P(>2/,) - POM) = e~^ - e~'\ 



_ ( »-l)t _ IM 



p„ = P\>(u - \)i] - P(>ui) = c ' - e i . 



If now there have been m such calls observed on switch count number 1, 

 we shall need to add m variables of the type 



( u — 1 )r u i ui i u i 



fill) = e ^~ - e~^ = e~'^ (e^ - l) = ce~^, 



where u may take all values from 1 to r + 1- The exact addition of these 

 variables when m is more than a small number, say 3 or 4, becomes quite 

 complex. However, in such cases (which may be the rule) we revert to 

 the method of combining their individual moments to obtain the moments 

 (and parameters) of the resultant distribution. We find for a single 



variable, 



'■+1 ' 1 r .,..»! 



(3)- 





(4) 



The factors shown in brackets in equations (3) and (4) will approach unity 

 very closely in any practical applications of the present type; they will 

 therefore be omitted in the subsequent analysis. 



The mean and standard deviation of the sum of m such variables are 

 readily determined, of course, as^ 



■* It is interesting to note that if the first switch count had been omitted so that onl\- m 

 could have been estimated from the average of the switch counts from M 2 onward, we 



~mg-m 



might have assumed a Poisson distribution for ;;; , that is p„, = — i , and thereby have ob- 



tained an estimate of the switch counts contributed bj- calls from the preceding period 

 as follows, 



-' "' (->^ 



sm = -. , (,j ; 



1 — e j 



Ymie J + 1) ^^,^ 



'rn ~ ,■ . (0 ) 



1 - e 



