968 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1957 



Eix"'"'] = exp L(.r -!))-{ /(") diX 

 = exp \ah{x - 1)[1 - git)]}. 



(7) 



Now 



/»00 



^ aT„Qn = a I f{u) du = ah, 



71 gl Jo 



-. T 



E aT„Q„/C = a E / "/('' + ^ + ^"^'n-i) du, 



/(?/) du = ahg{t). 



Therefore the infinite product is convergent, and 

 ^{/<VM =exp{aM^— 1)[1 -^(0] 



+ E aTnQn{y[l + Or - l)K„] - 1)} (s) 



= exp {ah{{x - 1)[1 - g{f)] + (// - 1) + y{x - 1)^(0)}- 



Thus the generating function of the joint distribution of N{0) and A'"(/) 

 is independent of the division of ( — -^ , 0) into intervals /„ . By letting 

 X approach 1 in (8) and finding the coefficient of i/" in the resulting limit, 

 we find that 



—ah/ i\'n 



pv{N(0) =m} =^_^. (()) 



ml 



The coefficient of i/' in (8) itself is 



—ah/ 1, "\ "' 



' ^^""^ [1 + (.r - \)g{()r exp {{x - l)aMl - git)]}, 



ml 



and so using (9) we find that the rec^uired conditional generating function 

 of N{t), given N{0) = m, is given by Riordan's formula (1). 



Ill THE AUTOCORRELATION 



In terms of N{t) one can define various stochastic integrals which 

 will be characteristic of the process. A simple one which has been ex- 

 tensively treated in connection with estimating the average traffic is 



A/ = 1 f Nit) dt, 



J Jo 



