Wireless Ad Hoc and Sensor Networks

Admission Controller Design for High-Speed Networks 1734.8 ProblemsSection 4.5Problem 4.5.1: Convergence analysis. Prove the admission controlscheme analytically by using Lyapunov stability analysis.Section 4.6Problem 4.6.1: Performance evaluation. Compare the performanceof the admission controller developed using Lyapunov stabilitywith that of the conventional rule based system.Problem 4.6.2: Performance evaluation. Evaluate the performance ofthe admission controller developed using Lyapunov stability for theend to end network presented in Chapter 3 by using MPEG traffic.APPENDIX 4.APROOFDefine the Lyapunov function candidateT 1 T TJ = e ( k) e( k) + tr[ Ṽ ( kVk ) ̃ 1( )] + trW [ ̃ ( k)̃W ( k)],αα1 2The first difference is given by∆J = ∆J 1 + ∆J2,(4.A.1)(4.A.2)TT∆J1 = e ( k+ 1) e( k+ 1) −e ( k) e( k),(4.A.3)1 TT∆J2= tr[ W̃ ( k+ 1) W̃ ( k+1)−W̃ ( k) W̃( k)]α21 TT+ trV [ ( k ) Vk ( ) V ( kVk ) ( )],α̃ + 1 ̃ + 1 − ̃ ̃1(4.A.4)Considering the input and hidden-layer weight updates (Equation 4.13),and using these in Equation 4.A.4 and combining with Equation 4.A.3,one obtains:⎡TTαϕ 2 ˆ2 k 2 k∆J ≤−( 1 −αϕ2 ˆ( )ˆ ϕ( ) δ( k)⎤2 ( k)ˆ ϕ2( k) )× ⎢e( k)−αϕˆ T⎥⎣⎢1 − 2 2 ( k)ˆ ϕ2( k)⎦⎥T