Wireless Ad Hoc and Sensor Networks

Admission Controller Design for High-Speed Networks 163Once the admission controller is expressed as a state equation (Equation4.30), its performance and overall stability can be analyzed in conjunctionwith the congestion controller using hybrid system theory. Here, a nonsmoothLyapunov function will be used along the system trajectoryndefined by Equation 4.30. The continuous state space, X ⊆ R , is partitionedinto a finite number of connected open regions, Ω q ,where X =∪qΩ q, q∈ Q = N. Note that the regions, Ω q , are not required tobe disjoint. In each region a controller, u is designed by any technique. Itis assumed that the controller gives a satisfactory performance only inregion Ω q for all q∈ Q.As each closed-loop system is stable,xk ( + 1) = f( xk ( ), uk ( )) is stable for all x ∈Ω q,a standard Lyapunov functionVq( x, t):X x R→R exists in Ω q. These Lyapunov functions can be+combined to produce a nonsmooth Lyapunov function, which globallyrepresents the dynamics of the hybrid system. Here, the congestion controlleralong with Equation 4.27 becomes a local control law and the rulesused for admission control switch the controller.DEFINITION 4.5.1 (ANTSAKLIS ET AL. 1995, RAMADGE AND WONHAM 1989)A nonsmooth Lyapunov function, Vxk ( ( ), k), is assumed to be left continuouson k ∈[ k0 , ∞)and differentiable almost everywhere on ( t O , ∞)excepton the set T s = { k0, k1, ..., } when switches occur. The constraint ∆V < 0 isreplaced by a stronger condition: the nonincreasing condition.Vxk ( ( ), k ) < Vxk ( ( ), k ) if k > ki i j j i j(4.33)THEOREM 4.5.1 (ADMISSION CONTROL SCHEME STABILITY)Given the hybrid system H whose dynamics are governed by Equation 4.1 andEquation 4.30, if its controller is selected as Equation 4.6 (Jagannathan 2002) tothe buffer dynamics and (4.31) for congestion control, and the Lyapunov functionsin each region have the same values on the boundaries, Ω q , then the originof the discrete-state space of the hybrid system is asymptotically stable.The next section details the simulation results.4.6 Simulation ResultsThis section discusses the network model, parameters, and constants usedin the simulations to test the effectiveness of the proposed algorithm. Thetraffic sources used in the simulation are also discussed and the simulationresults are explained. In our admission control example, we drive thesources independently with both voice (ON/OFF) and video (MPEG)data. We use fixed packets of length 53 bytes referred to here as cells.