(29) 



(30) 



1140 THE BELL SYSTEM TECHNICAL JOURNAL, OCTOBER 1951 



ElM'iT)] = ^' + ^h'h + ^hh + ^k\ 



and 



h = E[{M{T) - bf] - ZE[{M{T) - bY\ 



= ^-^, [dxg{x){T-x)x' 

 It is a tempting surmise that 



kn = * ""-^^^^ [ dx g{x)(T - X) X"-' 



but this has not been proved. Note that for g{x) — l,kn = h, the cumulant 

 of the Poisson, as it should. 



For the two cases of chief interest, constant and exponential holding 

 times, the function g{x), in average holding time units (that is, x = t/h) is 

 given by 



c.h.t. g{x) = \ — X X < \ 



= x> 1 



e.h.t. g{x) = e-* 



and the results are as follows: 



Constant Holding Time 

 Cumulant T < 1 T > 1 



k2 b(l - T/3) bT-\\ - l/Sr) 



h 6(1 - T/2) bT-\\ - \/2T) 



ki h{\ - 3T/5) bT-\l - 3/5T) 



Exponential Holding Time 

 ki 2bT-\T - 1 + e-^] 



h 6bT-^[T - 2 + (r + 2)e-T] 



ki 12bT-'[2T - 6+ (T+ 4T + 6)e-^] 



It may be worth noting that, if the surmise is correct, for constant holding 

 time 



r *. _ 1 1 



T < 1 





