952 



THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1957 



Table II 



1. e-fz 



2. e-f^ 



3. y^ 



4. e~f«i 



5. y^e~^^ 



In E[X] 



-f T + 



r'T fHl - e-(f+^>r)' 



S" + 7 (r+7)' 



.r f2(l - e-(r + r))' 



2aTC(y - 1) + ar(l - C)iy^ - 1) 

 2arC(e-f«^ - 1) + aT{l - C)(e-f/r - 1) 



V f + V 



[e-!-(+r) _!]_,- 



(■- 



j.yi 



r + r. 



By substitution, and by either letting the appropriate power series 

 variables — > 1, or letting f -^ 0, or both, we can obtain from (9.2) the 

 generating function of any combination of linear functions of the basic 

 random variables 7i, N{T), A, H, and Z. Some of the generating func- 

 tions thereby obtained are listed in Table II, in which the entries all 

 refer to an interval (0, T) of equilibrium. 



Since, for eciuilibrium (0, T), the generating functions are all exponen- 

 tials, it has been convenient to make Table II a table of logarithms of 

 expectations, with random variables X on the left, and functions In 

 E{X} on the right. C as a function of r is plotted in Fig. 2. 



Entry 1 of Table II is actually the cumulant generating function of 

 Z for ecjuilibrium (0, T) ; similarly, Entry 2 is that of il/, and depends 

 only on the average traffic b and the ratio r. The form of the general 

 cumulant of M is 



A-, 



n{n — 1) 



2^n 



i 



{T - .r)x"-V dr. 



This result coincides with a special case (exponential holding-time) of 

 a conjecture of Riordan. ' This conjecture was first established (for a 

 general holding-time distribution) in unpublished work of S. P. Lloyd. 

 The cumulant generating function permits investigation of asymptotic 

 properties. We prove in Section X that the standardized variable 



,1/2 



V = (77726)"-= (iM - b) 

 = {:r/2hf- {M - b) 

 is asymptotically normally distributed with mean and variance 1. 



