Wireless Ad Hoc and Sensor Networks

156 Wireless Ad Hoc and Sensor Networkswhere ˆ ( ) ˆ Ty1 k = V ( k )ˆ ϕ1( k). Then the buffer occupancy error, ek ( ), the NNweight estimates, Vk ˆ ( ) , Wk ˆ ( ) , are uniformly ultimately bounded (UUB),with the bounds specifically given by Equation 4.A.10 provided thefollowing conditions hold:(1) αϕ < 2(4.14)211,max2(2) αϕ 2 < 1 , (4.15)2 maxwhere is given byc 0(3) c < , (4.16)0 1c02k1 2= − αφ maxN1 1 2(4.17)REMARK 1Note that it is very easy to verify that the conditions Equation 4.14 throughEquation 4.16 are realizable, and hence the proof is avoided.PROOF See Appendix 4.A.REMARK 2As the buffer is of finite size and sigmoid is used in the hidden layer,conditions Equation 4.14 through Equation 4.16 easily follow.REMARK 3This theorem shows that using the parameter-tuning updates presentedin Equation 4.12 and Equation 4.13, the buffer occupancy estimation errorand error in weight estimates converge to a small subset provided thatthe design parameters are selected as Equation 4.14 through Equation 4.16.As the boundedness of buffer occupancy estimation error implies theboundedness of traffic-accumulation error, the traffic accumulated at theswitch is estimated accurately.REMARK 4The network traffic modeling error bound, ε , and the disturbances, dM,increase bounds on buffer occupancy estimation error in an interesting way.REMARK 5Large values of adaptation gain, α , forces smaller buffer occupancy estimationerrors but large weight estimation errors, which, in turn, result inpoor estimation of traffic accumulation. In contrast, a small value of αforces larger buffer occupancy and small weight estimation errors.