FLUCTUATIONS OF TELEPHONE TRAFFIC 971 



are independent, because no call which contributes to N{t) can survive 

 until {t -\- k), with probability 1. 

 Using (11), we see that 



w 



/•OO 



(/) = 4 / cos 27r/r/?(r) dr 

 Jo 



= 46 / cos 27r/rgr(T) dr 

 Jo 



= 4a / cos 27r/r / / h{u) du dy dr (12) 



vQ J J Jy 



= —. I sin 27r/r / h{u) du dr 

 irj Jo Jt 



r ,'« 



= ^Tf^ 1 - / COS 27r/r/l(T) C?7 



Equation (12) expresses the mean square of the frequency spectrum of 

 the fluctuations of the traffic away from the average in terms of the call- 

 ing-rate and the cosine transform of the holding-time density, h{i(). 

 The calling-rate appears only as a factor, and so does not affect the shape 

 of w{f). The function iv(j) is what Doob^ (p. 522) calls the "spectral 

 density function (real form)." 



V EXAMPLE 1. N(t) MARKOVIAN 



Let the frequency h{u) be negative exponential, so that 



h{u) = ^ e-'\ (13) 



where /) is the mean holding-time. It is shown in Riordan" p. 1134, 

 that N{t) is Markovian if and only if h{u) has the form (13). From page 

 523 of Doob^ we know that the covariance function of a real, stationary 

 Markov process (wide sense) has the form 



R{t) = R{0)c~"\ a constant. (14) 



Under the assumption (13), the covariance of N(t) is 



Rir) = bgir) = ^^ f^ f h{u) du dy 



= he 



-Tlh 



