1142 THE BELL SYSTEM TECHNICAL JOURNAL, OCTOBER 1951 



and for exponential holding time 



k^ = b "-^^^^ [(n - 2)! r - (» - 1)! + e-' {T + a)-'] 

 where in the last term {T + a:)"~^ is a symboHc expression or shorthand for 



(r + ay-' = Z f" ~ ^) r-^- «„ 



\ w / 

 and«« = (w + 1)!; e.g. 



(r + ay = T' + 6^ + 1ST + 24 



For small values of T, the two cases coalesce (e~' ^ 1 — x) and at T = 

 approach b as they should. For large values of T, and constant holding 

 time, 



kn^b/T-\ {n= 2,3,4); 



for exponential holding time 



kn^n\b/T''-\ (n= 2,3,4). 



For w = 2, these results agree with Rabe [6]. 



As T increases, for either holding time, the cumulants are progressively 

 smaller, and the approximation of the distribution of M{T) by a normal 

 curve (which has all cumulants, except the first and second, zero) improves. 

 This is what follows from the central limit theorem if the subdivision of T 

 into a large number of intervals results in mutually independent random 

 variables (cf. Rice [7] 3.9). 



Figure 1 shows a comparison of the variances (^2) for the two holding 

 time cases. Figure 2 shows a comparison of the cumulants ^2 , h and ki for 

 constant holding time, and Fig. 3 shows the same thing for exponential 

 holding time. 



5. Finite Sources — Exponential Holding Time 

 The generating function for transitional probabilities for N subscribers, 

 each originating calls independently with probabihty X, and for exponential 

 holding time, as given by Jensen (l.c.) is as follows: 



Piit. ^) = [1 + qi{x - 1)]»[1 + qo{x - l)]^-» (31) 



with 



?■ = /> + 9 " 



p = l-q = \/{\+y) 



7= 1/h 



