FLUCTUATIONS OF TELEPHONE TRAFFIC 967 



that 7]{t) is independent of all the ^'s, and that ^n{t) is independent of 

 ^y(r) if Ji 9^ j. Of course, ^„{t) and ^nir) are not independent. It follows 

 that if the infinite product converges, then for / > 



^^{/%-^''^l = E\x''"} ni?h/"%-'"^'^}. (5) 



We now compute the terms of the product. If a call originates in in- 

 terval /„ , it still exists at with probability 



Qn =~ [ " fiu + Sn-l) dU = ^ f " /(m) du. 

 -f n •'0 1 n •'S„_i 



Hence if k calls arrived in /„ , the probability that m of them are still 

 in progress at is 



pr{^n(0) = m I A- calls arrive in /„} 



M QZd - Qn)'-"\ m ^ k. 



Similarly, if a call originates in /„ and exists at 0, it also exists at ? > 

 with probability 



T 



Kn = {QnTnV f ^ fin + t + Sn-i) du. 

 Jo 



Therefore 



Eix^"'''^ I ^„(0) = m and A- calls arrive in /„} 



= [1 + {x - l)Knr, 

 and so 



- {1 + (^[1 + {x - 1)K„] - 1)Q,}' 



k 



= a . 



The number of calls arriving during /„ has a Poisson distribution with 

 mean aT„ ; hence 



£;{2/«"'V«^'^} = exp {aT.ia - 1)} 



(6) 



= exp {aTnQn{y{l + {x - 1)K„] - 1)}. 



By reasoning like that leading to (6), it can be shown that 



