Wireless Ad Hoc and Sensor Networks

92 Wireless Ad Hoc and Sensor Networkswhere ˆ ( ) ˆ Ty1 k = V ( k )ˆ ϕ1( k),and Γ>0 is a design parameter. Then, the bufferoccupancy error, ek ( ), the NN weight estimates, Vk ˆ ( ), Wk ˆ ( ), are UUB,with the bounds specifically given by Equation 3.A.9 and Equation 3.A.10,provided the design parameters are selected as:(1) αϕ < 2(3.24)21 1 max ,(2) αϕ < 1(3.25)22 2 max ,(3) 0< Γ < 1,(3.26)(4) k vmax < 1 ,(3.27)σwhere σ is given byσ = β + k β ,1 12⎡ ϕ 1 ϕβ = 1+ ϕ + − a max⎣+ Γ − aa max21− a 2 ϕ 2max1 2 2 2 2 2 2 2 2 2 2⎤max⎦2( )( )ϕϕβ2 1ϕ1 2 1 1 2 1 1 2 2⎡a+ 1−⎤max Γ a max= 1+ a +⎣⎦max.22 − a 1 ϕ 1max (3.28)(3.29)(3.30)PROOFSee Appendix 3.A.3.3.3 Simulation ExamplesThis section describes the NN model, parameters, and constants used inthe simulations. The traffic sources used in the simulation are also discussedand the simulation results are explained.3.3.3.1 Neural Network (NN) ModelThe NN used in this approach is shown in Figure 3.4. Previous six valuesof the buffer occupancy were used as inputs to the first layer NN as atrade-off between approximation and computation. The output of the NN