Wireless Ad Hoc and Sensor Networks

Admission Controller Design for High-Speed Networks 155where W max and V max are the maximum bounds on the unknown weights.Then the error in weights during estimation is given bỹW ( k) = W − Wk ˆ ( ), Ṽk ( ) = V−Vkˆ ( ) , ̃Z ( k) = Z−Zkˆ( ) , (4.9)Wwhere Z = ⎡ and .⎣ ⎢ 0 ⎤ ⎡, ˆWˆ0 ⎤Z = ⎢ ⎥0 V ⎦⎥⎢ Vˆ⎥⎣ 0 ⎦FACT 4.3.1The activation functions are bounded by known positive values so thatϕ2( xk ( )) ≤ ϕ2maxand ϕ̃2( xk ( )) ≤ ϕ̃2max.Using the NN traffic rate estimate, the closed-loop buffer occupancydynamics becomeTek ( + ) = kek ( ) + e( k) + W ( k) ̃ϕ ( k) + ε( k) + dk ( ), (4.10)1 v i2where the traffic flow modeling error is defined byTei( k) = W ̃ ( k) ̃ϕ 2 ( x( k))(4.11)4.3.2 Weight Updates for Guaranteed EstimationIt is required to demonstrate that the performance criterion in terms ofcell losses, c(k), and transmission delay monitored through buffer occupancyestimation error, ek ( ), is suitably small and that the NN weightsWk ˆ ( ) and Vk ˆ ( ) remain bounded. In the following theorem, a discrete-timeweight-tuning algorithm based on the error in buffer occupancy is given,which guarantees that both the error in buffer occupancy and weightestimates are bounded.THEOREM 4.3.1 (GENERAL CASE IN PRESENCE OF NETWORKDISTURBANCES)Let the desired buffer length, x d , be finite. Also, let the traffic modeling errorbound, ε N , obtained from target CLR and the disturbance bound, d M , be knownconstants. Let the NN weight tuning be provided byˆ ( ) ˆ ( ) ˆTVk+ = Vk− αϕ( k)[ˆ y( k) + Bkek ( )] , (4.12)1 1 1 1 1vˆ ( ) ˆ ( ) ˆTWk+ 1 = Wk+ αϕ(ke ) ( k+1), (4.13)2 2