Wireless Ad Hoc and Sensor Networks

154 Wireless Ad Hoc and Sensor Networkswhere ϕ2( k)is the vector of hidden-layer activation functions, ϕ1( k)is avector linear function, and |( ε k)|≤ εNwith the bounding constant, ε N ,known. For suitable approximation properties, it is necessary to select alarge enough number of hidden-layer neurons. It is not known how tocompute this number for general multilayer NN. Typically, the numberof hidden-layer neurons is selected by a trial-and-error procedure.4.3.1 Estimator StructureDefining the NN traffic estimate in the buffer occupancy estimator byˆ( ( )) ˆ TTf x k = W ( k) ϕ ( V ϕ ( x( k)))2 1(4.7)with Wk ˆ ( ) and Vk ˆ ( ) being the current NN weights, the next step is todetermine the weight updates so that the performance of the closed-loopbuffer occupancy estimation error dynamics at the switch is guaranteed.The structure of the estimator is shown in Figure 4.3, where the currentand past values of buffer occupancy are used as inputs to the NN modelso that an accurate estimate of the network traffic accumulation isobtained. The current and past traffic estimates are used to derive thebandwidth equivalent.Let W and V be the unknown target NN weights required for theapproximation to hold in Equation 4.5 and assume that they are boundedso thatW ≤W , V ≤Vmaxmax. (4.8)d(k)u(k)Buffer dynamicsx(k+1)e(k+1)+−Estimatorx(k + 1)z −1f (x(k))f (x(k − 1))+−∆f(x(k))1/tBandwidthequivalent∆Bw(k)FIGURE 4.3Neural network bandwidth estimator structure.