TELEPHONE TRAFFIC TIME AVERAGES 1133 



The corresponding generating function is 



1 - (?(/) + xQ{t) = 1 + {x - l)Q{t) 



and, since calls arriving are independent, the generating function for k 

 calls arriving is 



[1 + {x- 1)())^ 



and 



PS, x; *) = [1 + (x - l)g]«[l Jr(x- Dei' (11) 



Hence, finally by (5), 



= [1 + U - \)gV E [1 + (^ - \)Q\' '^^- 



= [1 + (:*; - l)g]*exp(a:- 1) at g 



= [1 + (:«: - DgV exp {x - 1) ah (1 - g) (12) 



The last step uses (10). 



This is the generating function for the simplest transition probabilities, 

 and is quite like Pahn's result; indeed, for exponential holding time g = f = 

 e~^'^. The probabilities themselves are obtained by expansion of the generat- 

 ing function in powers of x, or by substituting g for e~^'^ in Palm's result. 

 But they are not needed here; the generating function is most apt for deter- 

 mining the averages of interest, as will appear. 



Before going on to the other transition probabihties, it is interesting to 

 notice certain checks of equation (12). In statistical equilibrium the traffic 

 has Poisson density (Palm I.e.) that is, in the present notation 



Pr(N(t) = k) = e-^b'/kl 



where b = ah. This of course is independent of time. Then, if A^(0) has this 

 density, so should N{t) as determined from iV(0) and the transition proba- 

 bihties impHcit in (12). This is verified by 



E Pi(t,x)e-'byi\ = exp (x - l)b(l - g) E [1 + ix - ^)gV^-jr 



= exp [{x - \)b{\ -g)-b + b + {x- \)bg\ (13) 

 = exp {x — \)b. 



