Wireless Ad Hoc and Sensor Networks

Distributed Power Control and Rate Adaptation 273Now substituting a new state of Equation 6.41 in Equation 6.49, andincluding it in the last Nth iteration, a corresponding outgoing trafficcomponent w n, we get2 2J ( z( k )) =Q ( z( k )) +R ( u( k )) +J 1( z(k+ 1))k i k i k i k+ i(6.51)Applying the DP approach to Equation 6.42 we haveJ (( z N)) = Q ( z( N))N N iJ (()) z k = min E{ Q ( z()k ) 2 2+ R ( u( k )) +J ( z( k ) +u( k))}k2u kk ik i k+ 1 i i(6.52)First, we expand the one before the last iteration2JN−1(( z N− 1)) = min E{ QN−1( zi( N− 1)) + RN−1( u i ( N − 1))uN−12+Q ( z ( N −1) +u ( N −1)) }N−1N i i22= Q ( z ( N−1)) + min ER { ( u( N −1)) +Q ( z( N−1))iuN−1N−1i N i22+Qz ( N−1) u( N −1) +Q ( u ( N −1)) 2 }u NN i i N i(6.53)The minimization of Equation 6.44 with respect to i( − 1)is performedby differentiating Equation 6.44 and equating it to zero which yields( )ui * ( N− 1) =−zi( N−1)⋅ QN QN + RN−1(6.54)By substitutingu i( N− 1)in Equation (6.44) with Equation 6.45 we get,22J (( z N− 1)) = Q ( z ( N− 1)) + R ( z ( N−1)) (− Q /[ Q + R ])N−1 N−1i N−1iN N N−1222+ Q ( z ( N−1)) ( 1 − Q /[ Q + R ]) = G ( z ( N−1))N i N N N−1N−1i2(6.5)where2G = Q + Q R −R Q ( R Q )N−1( ) +2N−1 N−1 N N−1N−1 N N−1N= Q + Q [ ( )]N 1−QN QN −RN−1(6.56)