Wireless Ad Hoc and Sensor Networks

Congestion Control in ATM Networks and the Internet 85SourceNetwork switchSSSu(k)MultiplexerMx d− +AdaptivealgorithmBufferS rx(k + 1)Z −1Z −1x dFIGURE 3.2Network model with adaptation.discrete-time nonlinear systems to be controlled, as shown in Figure 3.2,given in the following form:xk ( + 1) = f( xk ( )) + Tuk ( ) + dk ( )(3.5)nwith state xk ( )∈R being the buffer length (or occupancy) at timeinstant k, T being the sampling instant, f () ⋅ being the nonlinear trafficnaccumulation or buffer occupancy, and control uk ( )∈R being the trafficrate calculated by feedback, so that the sources modify their transmissionrates from their intended rates. The unknown nonlinear function,( k) (f sat x k q t TfbIniSr k )() ⋅= [ ( ) − ( − ) + − ], is defined as the actual traffic flow andit is a function of current buffer occupancy xk ( ), buffer size, x d , traffic( karrival rate at the destination buffer, I ) ni , bottleneck queue levelqt ( − T fb ) (end-to-end), service capacity Sr, at a given switch with sat() ⋅applied to the function. The unknown disturbance vector, which can bean unexpected traffic burst/load or change in available bandwidth dueto the presence of a network fault, acting on the system at the instant knis dk ( ) ∈R , is assumed to be bounded by a known constant, |( dk)| ≤ d M .The state xk ( ) is a scalar if a single switch/single buffer scenario is considered,whereas it becomes a vector when multiple network switches/multiple buffers are involved, as is the case of multiple switches intandem.Given a finite buffer size x d , define the performance criterion in termsof buffer occupancy error asek ( ) = xk ( ) −xd ,(3.6)