970 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1957 



IV THE COVARIANCE AND SPECTRAL DENSITY 



The average value of N{t) is b = ah. One way to study the fluctuations 

 of N(t) about its average is by means of the power spectrum used in the 

 analysis of noise. (Cf. Rice.^) We resolve the difference [A'^(/) — b] into 

 sinusoidal components of non-negative frequency, and postulate a noise 

 current proportional to this difference dissipating power through a unit 

 resistance. The spectrum iv{f) is then the average power due to frequen- 

 cies in the interval (/, / + df) . 



More formally, we consider the Fourier integral 



T 



Sif, T) = [ [Nit) - b]e 



•JO 



-•lirifl 



dL 



(11) 



and we recall, for completeness, the relationship between *S(/, T) and 

 the covariance function, R{t), of \N{t) — b]. If 



,,. ,. 2|^(/, T)|' 

 r->oo 1 



then 



/*C0 



w{f) = 4 / R{t) COs27r/Vf/r, 

 Jo 



rtOO 



R(t) = wU) COS 2irfTdf. 

 Jo 



(Ci. Rice,' p. 312 ff.) 



At the same time, we have 



R{r) = E{[Nit) - b][N{t+ r) - b]\ 



= 'A(r) - b' 



= bg{r). 



Let X(t) be any stochastic process which is known to be the occupancy 

 of a telephone exchange of unlimited capacity, having a probability 

 density of holding-time, and subject to Poisson traffic. From the pre- 

 ceding result it can be seen that the covariance function of .Y(/) deter- 

 mines the distributions of the X{t) process completely, since 



dR' 

 dn 



t=0 



-rdR 

 dr' 



If the holding-times are bounded by a constant, k, then readings of 

 A^(^) taken further apart than k are uncorrelated . In fact, such values 



/(r) = f^ h{u) du = -a 



