Wireless Ad Hoc and Sensor Networks

182 Wireless Ad Hoc and Sensor NetworksHere, B l is the set of all inactive links during the lth step. We need a parameterη i to provide the protection margin, which is used to guarantee thata link is active during link admissions, such thatR ()≥ l γ + ηi i i(5.5)Any power control scheme would have to set the individual powers atleast to p * to satisfy SIR requirements in Equation 5.3 and Equation 5.4.This solution for power p * is optimal. A good power control schemewould set the individual powers of the links to p * so as to minimize thepower consumption.5.2.1 Power Control Scheme by BambosIf Equation 5.3 or Equation 5.4 fail, the transmitter power is updated.Therefore, each link independently increases its power when its currentSIR is below its target γ i , and decreases it otherwise. The associatedpower update can be obtained from (Bambos et al. 2000) asγ ipi()lpi( l+ 1)=R()li(5.6)where l = (1, 2, 3 ...) (see Bambos 1998, Bambos et al. 2000). Ifpi( l+ 1) > pmax,a new link is not added. If power slips under a minimumthreshold power pi( l+ 1) < pmin(the minimum power needed to form alink), then the power is assigned pi( l+ 1) = pmin.The DPC scheme updatesthe transmitter powers in steps (time slots) indexed by l = 1, 2, 3, ….5.2.2 Constrained Second-Order Power ControlIn Jantti and Kim (2000), the SIR in Equation 5.2 is defined as a set oflinear equationsAP = µ(5.7)where A = I − H and P = ( p i ), in turn H = [ h ij ] is defined as a Q x Q matrix,igijsuch that hij= γ g for i≠ j and h for In addition,iiij = 0 i= j. µ = ( γ ivi/ gii)is a vector of length Q. Because the maximum transmission power of anode is limited, the following constraint is placed on the power vector:0 ≤P≤P(5.8)