Wireless Ad Hoc and Sensor Networks

Congestion Control in ATM Networks and the Internet 119or Equation 3.68 holds. According to a standard Lyapunov extensiontheorem (Jagannathan 2006), this demonstrates that the buffer occupancyerror and the error in weight estimates are UUB.COROLLARY 3.4.3Let the desired buffer length x d be finite and the NN traffic reconstruction errorbound ε N and the disturbance bound d M be known constants. Take the trafficrate input for Equation 3.37 as Equation 3.43 with weight tuning provided byEquation 3.60. Then the packet losses ck ( ) and the NN weight estimatesWk ˆ ( ) are globally UUB, provided the conditions hold in Equation 3.61 throughEquation 3.63.The next result describes the behavior of the closed-loop queue lengtherror system, in the ideal case of no traffic modeling error in the networkdynamics, with no disturbances present.THEOREM 3.4.4Let the desired buffer length x d be finite and the network traffic estimation errorbound ε N and the disturbance bound d M be equal to zero. Let the source rate forEquation 3.37 be provided by Equation 3.43 with weight tuning given byWk ˆ ( ) Wk ˆ ( ) ( xk ( )) eT+ 1 = + αϕ ( k + 1 ),(3.69)where α > 0 is a constant learning rate parameter or adaptation gain. Thenthe buffer occupancy error ek ( ) approaches asymptotically to zero, andthe weight estimates are bounded provided the condition in Equation 3.61holds andk vmax < 1 η(3.70)where η is given by1η =2( 1−α|(())|)ϕ xk(3.71)PROOFFollow the steps as in the previous theorem.COROLLARY 3.4.5Let the desired buffer length x d be finite, and the network traffic estimation errorbound and the disturbance bound be equal to zero. Let the source rate forε Nd M