1134 THE BELL SYSTEM TECHNICAL JOURNAL, OCTOBER 1951 



Also, ^(0) = 1 and g(oo) = so that 



Pi{0,x) = [1 + (:^- 1)]^ = x' (14) 



Fi{<x>,x) = exp (x - l)b - (15) 



showing that in zero time no transit to another state is possible, and in 

 infinite time the equihbrium probabiHties are reached no matter what the 

 initial state has been. 



Finally, in a Markov process (cf. Feller [2], Chap. 15) the simple transi- 

 tion probabilities alone are needed since 



Pijkit, U) = P^J{t)PJk{u) 



A test for this is the Chapman-Kolomogorov equation, namely 



Pikit + u) =J2 Pij{t)Pjk{u) 

 j 



Using (12), the corresponding relation of generating functions is 



[l+{x- \)g{t + «)]^ exp h{x - 1)[1 - g{t + u)] 



= [1 + (:. - \)g{t)g{u)Ytx^h{x - 1)[1 - g{i)g{u)]', 



so the process is Markovian only if 



g{t + w) = g{t)g[u) 



which is true only for exponential holding time. 



For the next transition probability Pijk{t, u), consider first the condition 

 in which no call arrives in the whole interval / + w. As before 



'.vw = (*y,(i - g.)'-' 



where for convenience gt is written for g{t). For the next transit, however, 

 there is a difference, namely 



J-k 



'*> = (i)(t)'0-T)' 



since gt+u/gt is the conditional probability that -a, call which has lasted / 

 will last u more; Pjk{u) is the conditional probability of a transit from j 

 to k in «, given the transit i toj in /. 



The generating function for the double transition probabilities in this 

 case is, then, 



E L Piy*(/,w;0)xy = [! + (:,- \)g, + x{y - l)gt+uY (16) 

 j * 



Now suppose a single call arrives at random in interval /. As before, the 

 probability that it will occupy a trunk at time / is Q{t) = htr^(l — g(t)) 



