THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 509 



Using this in (1.5), it is found that 



(a -h k - au + (u - l)~\ GM = (1.8) 



Hence 



1 dGM^^^J^ ^j_g^ 



Gk(u) du I — u 



and, by easy integrations, 



Gk{u) = ce"" (1 - ur\ (1.10) 



with c an arbitrary constant, which is clearly identical with Gk(0) = 

 M(.)(0). 



Expansion of the right-hand side of (1.10) shows that 



il/a,(m) = Ma)(0) Z "^ •^. , "" ■„ = Ma,(0)a-.(m), (1.11) 



j=o \ J / {m - j)l 



if 

 <jo{m) = a'/ml and, a,(w) = ^ ( •^- ~ ) y-^ ^ri (1-12) 



The notation ak(m) is copied from Xyquist; the functions are closely 

 related to the ^^^"^ used by Kosten; indeed akim) = e'ipm'''' ■ They have 

 the generating function 



00 



Qkiu) = 53 (TkMu" = e""(l — u)~'' (1.13) 



from which a number of recurrences are found readily. Thus 

 Qkiu) = (1 - u)gk+Xu) 



u -^ — = augkiu) + kugk+i(u) 

 du 



= -agk-iiu) + (a - k)gk(u) + kgk-i(u) 



(the last by use of the first) imply 



ckim) = ak+iim) — (Tk+iim — 1) 



m(Tk(m) = ackim — 1) + k<jk+i(m — 1) 



= - a<jk-i{m) + (a - k)<Tk{m) + k<Xk+i(ni) 



