374 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1952 



Equation (7) becomes: 



ix<o\X) dx — 



Letting 



and noting that 



Px<o(a^) dx = c^ 



" • 1 c— 1 1 ni—x 



I -\- X ^<r^ 1 --^— C — n 



Px>o(x) dx = 



c—l -• ni— 



n=0 i 



rfx 



i - x^ I -^ 



n=l ^ 



(8) 



(ix 



= 2/j 7v = ^ e "'^ as before 



n=0 



c-1 



— nr 



ne = 



71 = i 



K - cb 



i + y.\{K-'y:^-^^ + ^ 



Px>o(x) dx = e V 



c I - e-" 



y 



dy 



'-'r)['' cl 



rc 

 IK - cb 



(9) 



dy 



The moment generating function of this pair of equations is the sum 

 of their separate m.g.f.'s: 



AL 



i{a) = [ Py<oiy)e''" dy + [ Py^y) dy e"" dy 



J-r Jo 



rc + A' + c - 2 -^ + ^ — e- + ' 



(10) 



1 - e-*- ^ ' 1 - e^^ 



+ aiK + rc - e~'Ke~°") 



rc(l + a)" 

 Neglecting terms of order higher than «-, 



1 



(11) 

 = ^Ux^ a{h' - rK) + ~ ('rctnh('"') - 2^ (.rc - rK + 26') | 



Now the number of Type I calls present in an observation is a vari- 

 able^ — with average "a" and a Poisson distribution. Similarly the num- 

 ber of Type II calls is a A-ariable, independent of the number of Type I 



T 

 calls, with an average of "a -^" or ''arc'' and a Poisson distribution. Ac- 



