FLUCTUATIONS OF TELEPHONE TRAFFIC 969 



the average of N(t) over an interval (0, T). The chief references in the 

 literature on M are References 3 and 5. If we consider A^(/) during an 

 interval (0, T -\- r), a measure of the coherence of N{t) during this in- 

 terval, i.e., of the extent to which A^(^) hangs together, is given by the 

 integral 



U{T, r) = ^ ^ N(t) Nit + r) dt, 



depending on values of N{t) taken r apart. When the limit ^(r) of u as 

 T approaches ^ exists, it is usually called the autocorrelation function ; 

 most statisticians, however, reserve the term "correlation" for suitably 

 normalized, dimensionless quantities. It can be shown that this limit 

 exists and is the same for almost all N{t) in the ensemble. It then coin- 

 cides with the ensemble average, i.e., 



^P{T) = lim U{T, t), almost all N(t), 



r-»oo 

 = E{N{t)N{t+ r)}. 



The function, \p, for the system we are discussing is derived by Riordan,* 

 and we reproduce his argument for ease of understanding. For equilib- 

 rium, and b = ah, we have 



E{N{t)Nit + t)\ 

 Now 



I- Pn.(r, x) 

 dx 



x=\ 



so that 



00 —bjm 



Ht) = Z'-^mlmgir) + b[\ - gir)]], 

 »i=o m I 



(10) 



= b' + bgir). 



(Cf.,5p. 1136) 



The limiting value of i/'(t) for r approaching x is the sc^uare of the 

 mean occupancy, b, and the limiting value of \j/(t) for r approaching 

 is the mean square occupancy, b' -f b, the second moment of the Pois- 

 son distribution with mean b. 



