Wireless Ad Hoc and Sensor Networks

270 Wireless Ad Hoc and Sensor Networks6.9.2 Cost FunctionThe proposed cost function includes both the cost of queue occupancyand the cost of transmitting a given burst of data. The latter is thenapproximated with a quadratic form to obtain an optimal control law,which can be derived using the standard Riccati equation. Moreover, thissolution is proven to be stable and will converge.6.9.2.1 Queue Occupancy CostConsider a known, ideal queue utilization, q ideal , which renders desiredsystem performance in terms of throughput and delay. The buffer cost insuch a case can be selected as a quadratic function of error between idealand actual queue utilization:2Bq ( ( k )) = γ ⋅( q( k) −qideal )ii(6.41)where g is a scaling factor and qi( k)is the current queue utilization. Insuch a case, the cost of queue utilization is the lowest for qi( k)=qideal, andincreases when the utilization is lower or higher than desired.To get a quadratic cost function as in the Riccatti equation, the statevariable is substituted by transiting queue utilization by a constantq idealx(k)=q(k)qi i − ideal(6.42)Now, the state equation can be rewritten asx ( k+ 1) = x ( k ) +u( k ) +w ( k)i i i i(6.43)and the cost function is expressed asBx ( ( k )) = γ ⋅( x( k))ii2(6.44)The selection of the parameter gamma will influence the convergenceof the queue utilization dynamics to a target value. The bigger the gammaparameter, the more effort (cost) will be spent on converging to the targetqueue utilization and corresponding performance level. However, thiswill render higher cost for transmissions, thus consuming more energythan in the case of a lower gamma.6.9.2.2 Transmission CostThe wireless nodes consume energy to transmit packets. This energy dependson transmission power needed for successful transmission and durationof transmission. The selection of the modulation rate adds complexity tothis term, because duration of transmission will vary with modulation