Wireless Ad Hoc and Sensor Networks

188 Wireless Ad Hoc and Sensor Networks5.2.3.2 Bounded Power OvershootThe following theorem proves that the transmitter power of active linksis finite and can only increase in small increments while accommodatingthe new links that are trying to gain admission into the network.THEOREM 5.2.4For any fixed η i ∈( 1, ∞),we have Pi( l+ 1) ≤ β( l) Pi(),l where β( l)is a positivenumber for every l ∈{ 0, 1, 2 , …}and every i ∈ A l under the DPC/SSCD orDPC/optimal updating scheme.PROOF By definition, i ∈ A l implies that Ri() l ≥ γ i,usingPi( l+ 1) = Ri( l+1)Ii( l), where Ii() Pi() ll = ; substituting in Pl we getRi() li ( ) Ri( l+1)+ 1 = . PlR l i ( ), pi()i( l+ 1)=[( 1− ki) Ri( l) + kiγi + ηi]. Pl i( ). This further implies that Pl whereR () li( + 1) ≤ β( lPl ) i(),β( l)iγiηiis a positive number given by β() l = ( 1− ki) + ki Rwhere >() l+ R () lγi iminRi() l < ( γ i + ηi).This clearly shows that the overshoots of the DPC/ALPscheme are bounded by β . The value of β lies betweenβmin< β < βmaxwhere is slightly larger than 1. Therefore, the powers of active linkscan only increase smoothly to accommodate the new links that are poweringin the channel.β minη i ∈ ∞THEOREM 5.2.5 (NONACTIVE LINK SIR INCREASES)For any fixed ( 1, ), we have Ri() l ≤ Ri( l+1)for every l ∈{ 012… , , , } andevery i ∈ B l , in the set of fully admissible inactive links under the DPC/SSCDor DPC/optimal scheme.PROOF Applying the proposed DPC vi() l =− kx i i()l +ηiinto SIR Equation5.10 to get Ri( l+ 1) = Ri( l) − kx i i()l + η i = Ri() l + ki( γ i − Ri()) l + ηi.Thevalue ( γ i − R i ( l))is a positive number for a nonactive link in the set offully admissible inactive links. Therefore, this implies that R( l+ 1) ≥R( l).iiTHEOREM 5.2.6 (NONACTIVE LINK INTERFERENCE DECREASES)For any fixed η ∈( 1, ∞),we have Ii( l+ 1) ≤ Ii( l)for every l ∈{ 0, 1, 2 , …}andevery i ∈ B l , the set of inactive links which are fully admissible, under the DPC/SSCD or DPC/optimal scheme.PROOF From Theorem 5.2.5, for nonactive links we have Ri( l+ 1) ≥Ri( l).Pi( l+1)Pi() lThis can equivalently be represented as ≥ . From Theorem 5.2.4, weIi( l+ 1)Ii() lhave Pi( l+ 1) ≤ β( l) Pi()l for all inactive links. Using this condition in the previoustheorem results in Ri( l+ 1) ≥Ri( l)because β( lPlI ) i() i() l≥ PlI i() i( l+1)⇒ Ii( l+ 1) ≤β( l) Ii().lTHEOREM 5.2.7 (FINITE ADMISSION TIME)For i ∈ , each inactive link becomes active if at all admitted in a finite time.B l