Wireless Ad Hoc and Sensor Networks

Congestion Control in ATM Networks and the Internet 143from 6 to 15. Evaluate the performance of the TQ controlscheme.Problem 3.6.4: Extended topology. Use Figure 3.19 with number ofsources increased from 6 to 10, study the performance evaluationof the TQ control scheme.Appendix 3.APROOF FOR THEOREM 3.3.1Define the Lyapunov function candidateT 1 T TJ = e ( k) e( k) + tr[ Ṽ ( kVk ) ̃ 1( )] + trW [ ̃ ( k)̃W ( k)],αα1 2(3.A.1)The first difference is given by∆J = ∆J + ∆J1 2(3.A.2)TT∆J1 = e ( k+ 1) e( k+ 1) −e ( k) e( k)(3.A.3)1 TT∆J2= tr[ W̃ ( k+ 1) W̃ ( k+ 1) −W̃ ( k) W̃( k)]α21 TT+ trV [ ̃ ( k+ 1) Vk ̃ ( + 1) −V ̃ ( kVk ) ̃ ( )]α1(3.A.4)Use the buffer length error dynamics of Equation 3.20 to obtainT∆J e k kv T TT1 =− ( ) ⎡1−kv⎤⎣ ⎦ + 2 ( kve ( k )) ( ei( k ) + W ̃ϕ 2 ( k) + ε( k) + d( k))× e ( k)( e i ( k) + 2W ϕ̃( k) + 2( ε( k) + d( k))) + ( W ϕ ̃ ( k))T (3.A.5)i T T 2 T2× ( W TT̃ϕ ( k) + ( ε( k) + d( k)))( ε( k) + dk ( )) ( ε( k) + d(k))2 2