Wireless Ad Hoc and Sensor Networks

272 Wireless Ad Hoc and Sensor NetworksEnergy consumed (mW ∗ burst time)160014001200100080060040020000 1 2 3 4 5 6 7Burst size (packets)FIGURE 6.17Rate selection as a function of queue utilization.Actual vs estimated energy spent on TXActual power costEstimated (2.4 ∗ u^2)REMARK 7Because the cost of transmission CQ( ui( k))is not exact, the calculated controllaw will be suboptimal.Figure 6.17 presents an example for the set of modulations presentedin Table 6.1. In this case, a is calculated to be equal to 2.4.Now, the final cost function is expressed as2 2J ( x( k )) =Q ( x( k )) +R ( u( k )) +J 1 ( x(k+ 1))k i k i k i k+ i(6.49)where Q k = γ and Rk = αP 0( k)are parameters from Equation 6.41 andEquation 6.45. The cost function in Equation 6.49 is in a quadratic form;hence, we can calculate an optimal control law using the standard Riccattiequation (Bertsekas 1987) because of the linear nature of the buffer dynamics.Next, the proposed solution is presented.6.9.3 The Riccatti EquationFirst, we notice that the rate adaptation problem should match closely thestate of the system, x i , to the outgoing flow, w i , rather than the queueutilization, because keeping an adequate flow of the data is more importantthan keeping the queue at a certain level. Hence, we consider a newstate variable that is equal to the sum of state x i and outgoing flow w i(negative value).z( k) = x( k) + w ( k)i i i(6.50)