958 



THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1957 



Applying now the Laplace-Stieltje.s transformation to (10.1), we obtain 



dt 



= y(n -\- \)(pn+i + a<p„-i - n^(Pn + HFniO, t) 



(10.3) 



[yn + a]ip„ , 



in which we have left out finietional dependence on f and / where it is 

 unnecessary. By the boundary conditions (10.2), n^F„{0, /) = for 

 n ^ and / > 0; in (10.8) we may therefore omit this term in the region 

 / > 0. Let <p be defined by 



V 



u-A,() = E -i-VXi-, /). 



n=0 



The series is absolutely convergent for | .r | < 1, since 



|^„(f, I ^ 1, for all n. 



The following partial differential equation for cp is obtained from 

 (10.3): 



^ + [f.r + T(.r- l)]f^ = a(.r-l)^. 

 dt dx 



(10.4) 



If we integrate out the information about Z by letting ^ approach in this 

 equation, we obtain the equation derived by Palm (loc. cit.) for the gene- 

 rating function of A'^(/). Therefore our equation has a solution of the 

 same form as Palm's. For the boundary conditions (10.2), this solution is 



f 



exp 



fa[l 



-(r+7)M 



(f + t)^ 



li'x + y{x - 1)] - 



a^t 



Jl Vn 



n=0 



f + tJ 



■[f.r + y{x - l)]e-'^+^^' 



+ 7 



f + 7 



(10.5) 



Actually ^p contains more information than we want since it 3'ields the 

 joint distribution of N and Z. We may integrate out the former variable 

 by letting x approach 1 in 10.5. Then, 



^{exp (-fZ)l = exp 



fafd 



-(f+T)'?' 



) afr 



(f + 7)^ 



.t + y] 



00 



u=0 



. .r + 7 . 



is the Laplace transform of the distribution of Z for an arbitrary N{0) 

 distribution {p,,}- This result is not restricted to an interval (0, T) 



