Wireless Ad Hoc and Sensor Networks

Admission Controller Design for High-Speed Networks 175In general ∆J ≤ 0 in a compact set as long as Equation 4.14 throughEquation 4.16 are satisfied and Equation 4.A.10 holds. According to astandard Lyapunov extension theorem (Jagannathan 2006), this demonstratesthat the cell losses and the error in weight estimates are UUB. Toshow the weights are bounded, consider the dynamics relative to theerrors in the weight estimates are given by:̃TWk xk xk Wk ̃ T( + 1) = ( 1− αϕ 2 2( ( )) ϕ2( ( )) ) ( ) + e( kk ) v T TB 1(4.A.11)̃TVk xk xk Vk ̃ T( + 1) = ( 1− αϕ 1 1( ( )) ϕ1( ( )) ) ( ) + e ( kk ) v T TB 1(4.A.12)where the buffer occupancy estimation error is considered bounded.Applying the persistence of excitation condition (Jagannathan 2006) andnoting that the traffic-modeling error and disturbance are bounded, theboundedness of ̃W ( k) and Ṽk ( ) implies that the weight estimates Wk ˆ ( )and Vk ˆ ( ) are bounded.