DELAY CURVES FOR CALLS SERVED AT RANDOM 109 



and D = d/da. Hence 



• Ml = (1 - «)-' 



This is the mean delay of calls delayed and as mentioned above is the 

 same as for service in order of arrival. 



In the general case*, the following notation is convenient 



I/O* = Y^ oL'mn,k = (1 — a)~^Mk 



Ljic = E (^ + 1)(^ + 2) • • . (n + j)a''mn,k 



Using the relations 



n(n + 2) • • • (n + j) 



= n(n + 1) . • • (n +i - 1) + (j - l)n(n + 1) • • • (n +i - 2) 



+ • • • + (i - l)^Mn + 1) '" (n + j - i - 1) + "' 



+ (i - l)In, 



n(n + 1) • • • (n + i - 1) ' 



= (n + l)(n + 2) . • • (n + i) - j(n + l)(n + 2)(n + j - 1) 



with 



(j- 1)^= (i- 1)0' -2) --'(j-i), 



the summing of (12) is found to result in 



kLj,k-i = [j - {j - l)a]Lj-i,k - a[(j - l)2Ly_2.fc 



(14) 



+ (i - l)3Ly_3.. +"•-{- (j - l)iLj-i,k + ■" + U - l)!^iJ 



But this may be simplified by multiplying through by j and sub- 

 tracting from the same equation with j replaced hy j + 1 ; the result is 



(i + 1 - Jot)Ljk - fLj-i,k = kLj+i,k-i - jkLj,k-i (15) 



Notice that for j = 0, k = 1, Ui = Lw , as in (13). Notice also that 



so that 



Lyo = jLj-1,0 + «2)Ly_i.o = i!(l - «)"^' 

 *This procedure is the development of a suggestion made by S. O. Rice. 



