924 



THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1956 



Actually, each of these terms is the same form as that of the gambler's 

 exponential gain where there is no channel 



(? = X; p(s) log [b + a.a(s)]. (3) 



a 



We will maximize (3) and interpret the results either as a typical 

 term in the general problem or as the total exponential gain in the case 

 of no communication channel. Let us designate by X the set of indices, 

 s, for which a(s) > 0, and by X' the set for which a(s) = 0. Now at the 

 desired maximum 



p(s)as 



dG 



da{s) b + a(s)ai 



log e = k for seX 



dG y-^ p{s) , , 



-1- = Z^ -, — . \\ — log e = k 



dG p(s)as , ^ J r .f 



— T-r = ^ / log e :^ /c for SfX 



da{s) b ^ ~ 



where /c is a constant. The equations yield 



k = log e, b = 



b 



ais) = pis) - 



1 -P 



1 - a- 



for seX 



as 



where p = Xxp(s), a- = ^x (1/as), and the inequalities yield 



p(s)as ^ b = 



1 -p 



for SfX 



We will see that the conditions 



(X < 1 

 p(s)as > 

 p(s)as ^ 



1 



P 



1 - a 



1 -P 

 1 - <r 



se\ 



for seX 



completely determine X. 



If we permute indices so that 



p(s)ae ^ p(s + l)as+i 



