966 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1957 



abilities as 



and let 



f{u) = [ h{x) dx, 



so that the average holding-time, h, is given by 



h = j f{u) du. 

 Jo 



Riordan" has given the following formula for P,„{t, x) : 



P,M, x) = [1 + (.r - \)g{t)]"' exp {{x - l)ah[l - git)]], (1) 

 with 



git) = ^ fin) du. 



For exponential holding-time density, this formula had alread}' been 

 derived (as the solution of a differential equation) by Palm." 



In private communication, J. Riordan has suggested that his proof of 

 (1) is incomplete. We therefore give a new proof of (1). 



We seek the generating function of N{t), conditional on the event 

 JSfiO) = m. We obtain it by first computing the joint generating function 

 of 7^(0) and A^(^) ; that is, 



^{/^V^'M- (2) 



The desired conditional generating function is then the coefficient of 

 I/'" in (2), divided by the probability that A^(0) = m. 



To obtain a formula for (2), we exhaust the interval {— --^ , 0) by 

 division into a countable set of disjoint intervals, /„ , the n^ having 

 length Tn > 0. Let S„ be the sum of the first n lengths, Tj . Let ^„(0, 

 f or ^ > — »S'„_i , be the number of those calls which arrive in /« and are 

 still in progress at /. And let ri{t) be the number of calls arriving during 

 (0, t), t > 0, and still in existence at /. Then 



NiO) = Z UO), (3) 



«>i 



Nit) = vit) + E Ut), t > 0. (-i) 



"SI 



Since calls arri\-ing during disjoint intervals are independent, we know 



