1136 the bell system technical journal, october 1951 



3. Expected Correlations 



Correlation expectations, like E[N{t)N(u)] in equation (3), are needed 

 for evaluation of the moments of M(T). They may be determined from the 

 transition probability generating functions, if it is agreed, as a matter only 

 of convenience, that the time epochs /, u, v, etc. are in that order {t < u < 

 V < •••)• Since, on the assumption of statistical equilibrium, the call 

 probabilities at the first epoch /, are independent of its value, as already 

 noticed, this value may be taken as zero without loss of generahty. 



Thus for the second moment it is sufficient to determine 



^{u) = E[N{0)N(u)] = E ipi T^JPiM (21) 



with p, = Pr[N(0) = i] = e-^b^/il 

 Write 



Guix, y) = J2 Pi^' Z PiM)y 

 By (12), this is the same as 



G^{x, y) = exp ^»[:*: - 1 + y - 1 + (x - l)(y - V)g{u)] 

 or 



Hu{x, y) = Gu{x + 1, y + 1) = exp 6(:x: + y + xyg{u)) 

 and 



<p{u) = 



'.v=o (22) 



dxdy 

 = b' + bg(u) 



In the same way the second order correlation expectation, that is 

 ifiu, v) = E[N{0)N{u)N(u + v)], 

 is obtained from 



GuAx, y, z) = Z /'i^* Z Z PiJkiu, v)yz' 

 and 

 Hu,v(x, y, z) = Gu.vix + 1, y + 1, 2 + 1) 



= exp6(x + y + z + xyg(u) + yzg{v) + x{y + l)zg(u + v)) 

 Hence 



^(«, v) = b' + b'[g(u) + g(v) + g{u + v)] + bg(u + v) (23) 



