(24) 



TELEPHONE TRAFFIC TIME AVERAGES 1137 



Finally, the third order correlation turns out to be 

 ipiu, V, w) = E[NiO)N(u)N(u + v)Niu + v + w)] 

 = b^ + b^lgM + g(v) + g(w) 



+ g(u + v) -{- g(v +w) + g(u-{-v-\- w)] 

 + ^%(w -i-v) + g{v + w) + 2g{u + v + w)] 

 + b^lgWgM + g(u + v)g(v + w) 

 + gWgi'i^ + v + w)] -{- bg{u + V + w) 



As will appear, the arrangement of terms in (22), (23) and (24) corresponds 

 to the expansion of ordinary moments in terms of cumulants (semi-in- 

 variants); e.g. (24) corresponds to ma = b^ -{- 6b^k2 + 4bkz -+- 3kl + k4 

 with ki the i'th cumulant (for the Poisson of mean b, ki = b). 



4. Moments 



Moments are obtained from these results by integrations. As already 



noted, equation (2), the first moment is b for any holding time distribution. 



Since there are two ways of ordering the epochs t, «, the second moment is 



ElM\T)] = j^^l dtj^ du<p{t-u) 



= b'+^^j\tf^dug{t-u) (25) 



=^b'^-^±j\xg{x){T-'x) 



The last step is by the formula for reversing the order of integration indi- 

 cated by 



T 



I dt I du = I du / dt 



Jo Jo Jo Ju 



The variance or second central moment, which is also th^ second cumulant 

 ^2 , is then 



Var [M{T)] = E[{M{T) - bf 

 = ElM\T)] - b' 



^^l' dxg(xKT-x) 



(26) 



