Wireless Ad Hoc and Sensor Networks
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Wireless Ad Hoc and Sensor Networks

Wireless Ad Hoc and Sensor Networks Wireless Ad Hoc and Sensor Networks

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12.07.2015 Views

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)

274 Wireless Ad Hoc and Sensor NetworksFollowing the preceding calculations we can calculate optimal inputfor k = N−2, N−3,....,0. In such a case, an optimal law for every k is equaltowhere( )u * ( k) =−z( k)⋅ G G + Ri i N+ 1 k+1 k(6.57)G =QnG =GNk k+ 1⎛ Gk+1 ⎞1−⎝⎜G +G +Qk+ 1 k ⎠⎟k(6.58)However, because the transmission duration is unknown, it is desirableto calculate a steady-state solution by assuming an infinite flow. Thesteady-state solution is more useful in implementation, because most ofthe calculations can be performed offline before network deployment andonly limited calculations have to be done online. In such a case, Equation6.58 becomesu * ( k) =−z( k) ⋅ G ( G+R )i i k(6.59)where G is stable state solution ( k →∞)of Equation 6.49, Rk = α ⋅P 0( k)is aparameter of a cost function with P0( k)being the transmission powervalue calculated by the DPC for the next transmission.By substituting Equation 6.33 and Equation 6.41 in Equation 6.59 wecan calculate the stationary law that depends directly on the queue utilizationasu*( k ) = −G( q( k) −q+E{ w ( k)})/( G+R )i ideal i k(6.60)This control law is applied before each transmission to calculate adesired burst size u*. Next, the optimal modulation rate is selected asin Subsection 6.9.2.2, as the lowest rate that can support a given burstsize.6.9.4 Additional Conditions for Modulation SelectionTransmission power is physically limited by the node’s hardware.Hence, the rate adaptation has to exclude modulation rates that requiretransmission power higher than the maximum available value. Thisthreshold is calculated according to Equation 6.24. The maximum powerwill dictate the maximum modulation rate that can be applied for agiven channel state. If the modulation rate is reduced, then the burstsize will be reduced as well.

274 <strong>Wireless</strong> <strong>Ad</strong> <strong>Hoc</strong> <strong>and</strong> <strong>Sensor</strong> <strong>Networks</strong>Following the preceding calculations we can calculate optimal inputfor k = N−2, N−3,....,0. In such a case, an optimal law for every k is equaltowhere( )u * ( k) =−z( k)⋅ G G + Ri i N+ 1 k+1 k(6.57)G =QnG =GNk k+ 1⎛ Gk+1 ⎞1−⎝⎜G +G +Qk+ 1 k ⎠⎟k(6.58)However, because the transmission duration is unknown, it is desirableto calculate a steady-state solution by assuming an infinite flow. Thesteady-state solution is more useful in implementation, because most ofthe calculations can be performed offline before network deployment <strong>and</strong>only limited calculations have to be done online. In such a case, Equation6.58 becomesu * ( k) =−z( k) ⋅ G ( G+R )i i k(6.59)where G is stable state solution ( k →∞)of Equation 6.49, Rk = α ⋅P 0( k)is aparameter of a cost function with P0( k)being the transmission powervalue calculated by the DPC for the next transmission.By substituting Equation 6.33 <strong>and</strong> Equation 6.41 in Equation 6.59 wecan calculate the stationary law that depends directly on the queue utilizationasu*( k ) = −G( q( k) −q+E{ w ( k)})/( G+R )i ideal i k(6.60)This control law is applied before each transmission to calculate adesired burst size u*. Next, the optimal modulation rate is selected asin Subsection 6.9.2.2, as the lowest rate that can support a given burstsize.6.9.4 <strong>Ad</strong>ditional Conditions for Modulation SelectionTransmission power is physically limited by the node’s hardware.Hence, the rate adaptation has to exclude modulation rates that requiretransmission power higher than the maximum available value. Thisthreshold is calculated according to Equation 6.24. The maximum powerwill dictate the maximum modulation rate that can be applied for agiven channel state. If the modulation rate is reduced, then the burstsize will be reduced as well.

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