110 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1953 



Then the ratio 



Ui^iLmT' = (1 - af^'U,/k\ 



(16) 

 = (1 - afMu/k\ = R, 



is the ratio of these moments to those for order of arrival service; the 

 last relation is a definition. In the same way the ratio 



might be considered, but to avoid fractions the following somewhat odd 

 change of variables seems convenient: 



{'t'')gkLjk = VjkLj+k,o (17) 



where go = gi = 1 and 



g^k = (2 - aY'-\3 - 2af'-' . . . (k + 1 - ka) 



gu+i = (2 - afiS - 2af'-' . . . (/c + 1 - kaf 



Notice that 



g2k+i(g2k)~^ = g2k{g2k-i)~^ = (2 - a)(3 - 2a) • • • (/c + 1 - /ca) = Dk+i 



the last being a definition, again. 

 Since 



(1 — a)Lkfi — kLk-1,0 

 it follows from (15) that 



0' + 1 - JodPjk - i(l - (x)pj-i,k , ^ 



(18) 

 = (gk/gk-i)[(j + l)pj+i,k-i - i(l - oi)pj,k-i] 



By taking differences of this equation and writing 



Qok = Pok 



qik = pik — pok = Apok 



qu = P2k — 2pik + pok = A^Pok 



Qjk = Aqj-i.k = A%k 



a somewhat simpler recurrence relation is found to be as follows 



0' 4- 1 - ja)qjk = (gk/Qk-i) 



(19) 

 L/agy_i.jfc_i + 0' H- 1 + joc)qj,k-i + 0" + ^)qj+i,k-i] 



