Wireless Ad Hoc and Sensor Networks

Congestion Control in ATM Networks and the Internet 117where W max is the maximum bound on the unknown weights. Then, theerror in the weights during estimation is given bỹW ( k) = W − Wk ˆ ( )(3.56)FACT 3.4.1Because the buffer size is finite, the activation functions are bounded by knownpositive values so that |(())| ϕ xk ≤ ϕmaxand |(())| ϕ̃ xk ≤ ϕ̃max.The traffic rate uk ( ) is∑1uk− (3.57)T x W T( ) = ( d − ˆ ( k ) ϕ( x ( k )) + k v e ( k )) uk ( − RTT)and the closed-loop buffer occupancy dynamics becomeek ( + 1) = kek ( ) + e( k) + ε( k) + dk ( )vi(3.58)where the traffic flow modeling error is defined byTe( k) = W ̃ ( k) ϕ( x( k))i(3.59)3.4.2.3 Weight Updates for Guaranteed QoSIt is required to demonstrate that the performance criterion in terms ofpacket losses c(k), transmission delay, and network utilization monitoredthrough error in buffer occupancy ek ( ), is suitably small and that the NNweights Wk ˆ ( ), remain bounded for, the traffic rate uk ( ) is bounded andfinite.THEOREM 3.4.3 (TRAFFIC RATE CONTROLLER DESIGN)Let 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 traffic rateinput for Equation 3.37 as Equation 3.47 with weight tuning provided byˆ ( ) ˆ ( ) ( ( ))TWk+ 1 = Wk+ αϕ xk e ( k+ 1 ) −Γ | I−αϕ( xk ( )) ϕ T ( xk ( ))| Wk ˆ ( )(3.60)with Γ>0 a design parameter. Then the error in buffer length ek ( ) andthe NN weight estimates Wk ˆ ( ) are UUB, with the bounds specifically