TELEPHONE TRAFFIC TIME AVERAGES 1143 



It should be noticed that for / = co ^ q^ = q^ = p and 



Pi{<^,x) = [\-\-p{x- \)Y (32) 



The right hand side is the binomial generating function and, as independent 

 of i, is the generating function for the statistical equilibrium probabilities; 

 that is 



■ Pr mo = *] = (^) / q"-" 



Also the process is Markovian since 



Z ^' Z PiAt)Pik{u) = JlPiM^ + qUx - \)V [1 + qou{x - 1)]^-^' 

 k j j 



= [1 4- (.qou + quQiu — quqou)(^ — 1)]* 



[1 + (qou + qotqiu — qotqou){x — l)]^~* 



and 



qcu + qitqiu — qitqou = qi,t+u 



qou ~\~ qotqiu ~ qotqou = qo.t+u 



Here it has been convenient to indicate by the double subscript the de- 

 pendence of go and qi on a time variable. 



Moments are obtained by the process given in detail for the infinite 

 source case. For brevity it is convenient to use the binomial cumulants 

 which are as follows 



K2 = Npq 



K3 = Npqiq - p) 



Ki = Npq(\ — 6pq) 

 and the modified time variable Ti = (X + y)T. Then the results are 

 k2 = 2IT%[Ti - 1 + e-'''] 

 k, = 67T^3[ri -2+ (Ti+ 2)6-^^] 

 k, = 127T'((/c4 + kIN-')[2Ti - 6+iTl + 4^ + 6)^-^^] 



- .^^^[1 - {Tl + 2)6-'^ + e-''^]) 



These of course bear a strong resemblance to the infinite source case (ex- 

 ponential holding time), to which they converge. 



