DELAY CURVES FOR CALLS SERVED AT RANDOM 107 



Sit u = 0; this is as follows 



J, . . ^ , u , a (uY a{2a - 1) {uf 



(5) 



n + 1 2 (n + 1) 3! (n + 1) 



ai2oL - l)(3ce - 2) {uY _ 

 ■^ 4! (n + 1) * * * 



But this is the same* as: 



Fn(u) ;^ [1 - (1 - a)u/(n + 1)]^'^^-^ (5a) 



As a approaches unity, (5a) approaches (4). Equation (5a) has been 

 used, for large values of a, in the direct computations mentioned above. 

 It may also be noted that for a = 0, equation (la) has the solution 

 (now writing Fn{u, a) for Fn{u)) 



Fniu, 0) = <l>{u, n) - — ^ <t>(u, n-1) (6) 



where <f){u, n) is the Poisson sum 



1 + U+I+--- + 



n\) 



Finally, for completeness, note that for small values of u, the Mac- 

 Laurin series for F{u) is 



F{u) = \ — u log 



I - a 

 + |(l-„)[2-l^"log^] (7) 



-|(l-a)[l + 3.-(l-«)log^-i:g] 

 4. MOMENTS 



The /c'th moment (about the origin) of the delay density function 

 which is -F'{u) (F{u) itself is a distribution function is defined as 



Mk = / u'[-F'{u)] du, 



•^^ (8) 



= k f u'-'F(u) du, k> 0, 

 Jo 



the last by integration by parts. 



* G. W. Abrams is due credit for noticing this. 



