Wireless Ad Hoc and Sensor Networks

Congestion Control in ATM Networks and the Internet 115Substituting the buffer error dynamics Equation 3.42 in Equation 3.47yields( v )2 2∆J ≤− 1−k max |( e k)|.(3.48)The closed-loop system is globally asymptotically stable.COROLLARY 3.4.1Let the desired buffer length x d be finite and the network disturbancebound, d M be equal to zero. Let the source rate for Equation 3.37 be provided byEquation 3.41, then the packet losses ek ( ) approach zero asymptotically.REMARK 1This theorem shows that in the absence of bounded disturbing traffic patterns,the error in buffer length ek ( ) and packet losses converges to zero asymptotically,for any initial condition. The rate of convergence and transmission delays andnetwork utilization depends upon the gain matrix k v .THEOREM 3.4.2 (GENERAL CASE)Let the desired buffer length x d be finite and the disturbance bound d M be a knownconstant. Take the control input for Equation 3.37 as Equation 3.41 with anestimate of network traffic such that the approximation error, ̃f (), ⋅ is boundedabove by f , then the error in buffer length ek ( ) is GUUB, providedM0 < k < Iv(3.49)PROOF Let us consider the following Lyapunov function candidate,Equation 3.46, and substitute Equation 3.42 in the first difference Equation3.47 to getTT∆J = ( k e+ f̃+ d) ( k e+ f̃+ d) −e ( k) e( k).vv(3.50)∆J ≤ 0 if and only if( k | e| + f̃+ d ) < | e|vmaxM(3.51)or| e|>f +M dM( 1− k )vmax(3.52)