Wireless Ad Hoc and Sensor Networks

Predictive Congestion Control for Wireless Sensor Networks 445THEOREM 9.3.5 (BACKOFF SELECTION IN GENERAL CASE)Assume the hypothesis as given in Theorem 9.3.4, and let the uncertain parameterα i be estimated using Equation 9.18 with ε( k), the error in estimationconsidered bounded above such that ε( k)≤ εN, where ε N is a known constant.Then, the mean error in throughput and the estimated parameters are boundedprovided Equation 9.20 and Equation 9.20 hold.PROOF Define a Lyapunov function candidate as in Equation 9.21whose first difference is given by Equation 9.22. The first term ∆J 1 and thesecond term ∆J 2 can be obtained respectively as2 2∆J =e ( k) k + [ k e ( k)][ α̃( k) R ( k )] + [ α̃( kR ) ( k)]1 i v 2 v i i i i2+ ε ( k ) + 2[ kvei( k)] ε( k ) + 2ε( k) ei( k)− ei( k) 2i2(9.28)∆J = − [ k e ( k)][ α̃( k) R ( k)]2 2v i i i− 2[ α̃( kR ) ( k)]+ σR ( k)[ k e ( k ) + α̃( k) R ( k)]−ii22⎡⎣1−σRi( k)⎤⎦ e i( k ) ε( k )2 2 2i v i i i(9.29)2 2+ 2σR ( k)[ k e ( k)] ε( k ) + σR ( k) ε 2 ( k)i v i iFollowing Equation 9.25 and completing the squares for̃α i ( kR ) i ( k)yields⎛∆J ≤− 1 −δkvmaxei( k)⎝⎜( ) −2σk2 2 vmax1 − δkR ( k)i2vmax2εNe ( k)iδ−1 − δk2vmax⎞2εN⎠⎟−2( 1 −σRi() l ) ̃α i2σ Ri( k)( kR ) i( k)−1 − σ R ( k)i2( ke( k ) + ε( k))v i2(9.30)with d as given after Equation 9.20. Taking expectations on both sides,⎧⎪⎛E( ∆J) ≤ −E⎨1 −δkvmaxei( k)⎝⎜⎩⎪2 2( ) −2kvmax( ) −−σ R ( k)1 − δkv2iδεNei( k)−2 2max1 − δkvmax2σ Ri( k)+ 1 −σ Ri( k) α̃ i( k) Ri( k)ke v i(2( k)+ε( k))1 σ R ( k)i22⎫⎪⎬⎪⎭⎞2εN⎠⎟(9.31)