Wireless Ad Hoc and Sensor Networks

Predictive Congestion Control for Wireless Sensor Networks 443where αˆ i ( k)is the estimate of α i ( k) , ei( k) = Ri( k) − fi( k), defined as throughputerror, and κ v is the feedback gain parameter. In this case, the throughputerror is expressed asei( k + 1) =Kvei( k ) + αi( k) Ri( k) −α ˆ i( k) Ri( k ) =Kvei( k) + ̃α i ( k) R i ( k)(9.18)where ̃αi( k) = αi( k) −αˆ i( k)is the error in estimation.The throughput error of the closed-loop system for a given link is drivenby the error in backoff intervals of the neighbors, which are typicallyunknown. If these uncertainties are properly estimated, a suitable backoffinterval is selected for the node under consideration such that a suitablerate is selected to mitigate potential congestion. If the error in uncertaintiestends to zero, Equation 9.18 reduces to ei( k+ 1) = κ v⋅ei( k). In the presenceof backoff interval variations of neighboring nodes, only a bound on theerror in backoff interval selection for the node under consideration canbe shown. In other words, the congestion control scheme will ensure thatthe actual throughput is close to its target value, but it will not guaranteeconvergence of actual backoff interval to its ideal target for all the nodes.It is very important to note that unless suitable backoff intervals areselected for all the nodes, congestion cannot be prevented.THEOREM 9.3.4 (BACKOFF SELECTION UNDER IDEAL CIRCUMSTANCES)Given the aforementioned backoff selection scheme with variable α i estimatedaccurately (no estimation error), and the backoff interval updated as in Equation9.17, then the mean estimation error of the variable α i along with the mean errorin throughput converges to zero asymptotically, if the parameter is updated asα iαˆ ( k+ 1) = αˆ ( k) + σ⋅R( k) ⋅ e ( k+1)i i i i(9.19)Provided(a) σ Ri( k ) 2 < 1 and (b) K vmax < 1 δ(9.1)where δ = 1/[ 1 −σ∗ Ri( k) ], K vmax is the maximum singular value of K v ,and σ , the adaptation gain.PROOF Define the Lyapunov function candidate2J = e 2 ( k) + σ −1α̃2 ( k)ii(9.21)whose first difference is2 2 −1 2∆J = ∆J1+ ∆J2= ei ( k+ 1) − ei ( k) + σ ⎡α̃i ( k+ 1)−α̃2 i⎣( k)⎤⎦(9.22)