Wireless Ad Hoc and Sensor Networks

442 Wireless Ad Hoc and Sensor NetworksTheorem 9.3.4 and Theorem 9.3.5 state that the proposed backoff selectionscheme ensures convergence of the actual value to its target of outgoingtraffic in both (1) the ideal case, in which the throughput dynamicsare known, and (2) the general case, In which the dynamics are estimatedby an adaptive scheme. In the latter case, the estimation error is boundedby a known value, ε N . The proofs guarantee stability in the sense ofLyapunov (Jagannathan 2006, Zawodniok and Jagannathan 2006) as thebackoff interval selection scheme is posed as a feedback dynamic system.9.3.2.1 Adaptive Backoff Interval SelectionDifferentiate Equation 9.11 to geṫ−2R() t = B⋅TVR ()[ t VR ̇ () t ⋅TVR() t −VR()t ⋅Ti i i i i̇VR i( t )](9.13)To transform the differential equation into the discrete-time domain,Euler's formula is used:−2R( k+ 1) − R( k) = B⋅TVR ( k)i i i× ⎡⎣ ( VRi( k + 1) −VRi( k))TVRi( k) − VRi( k)( TVRi( k + 1) −TVRi( k))⎤ ⎦(9.14)After applying Equation 9.11, we can transform Equation 9.14 to getR ( k+ 1) = [ R ( k) VR ( k+ 1) VR ( k )] +R ( k)[ 1− TVR ( k+ 1) TVR( k)]i i i i i ii(9.15)Now, define α i ( k ) = 1− TVR i ( k+ 1) TVR i ( k), β i ( k ) =R i ( k) VR i ( k), andvi( k ) =VRi( k+ 1) = 1 BOi( k+ 1). The variable α i describes a variation ofbackoff intervals of flows at the neighboring nodes from the time instant kto k + 1. This variation is caused because of congestion resulting fromtraffic and fading channels. Because this information is not availablelocally, it is considered an unknown parameter and thus estimated by thealgorithm. The parameter β i is the ratio between the actual and the usedvirtual rate at time instant k and can be easily calculated. The term ν i isthe backoff interval that needs to be calculated for the node under consideration.Now, Equation 9.15 can be written asR( k+ 1) =R( k) α ( k ) + β ( k) v( k)i i i i i(9.16)Equation 9.16 indicates that the achieved rate at the instant k + 1depends on the variations of backoff intervals in the neighboring nodes.Now, select the backoff interval asv( k) = ( ( k)) − 1β [ f ( k) − R ( k)ˆ α ( k) + κ e ( k)]i i i ij i v i(9.17)