Wireless Ad Hoc and Sensor Networks

Distributed Power Control of Wireless Cellular and Peer-to-Peer Networks 181Calculation of SIR, R i , at sender of ith link, (Rappaport 1999) yieldsR =i⎛⎜⎝⎜≠∑j iGii* pi⎞Gij pj + κ i⎟⎠⎟(5.2)where i , j ∈{ 123… , , , , n},p i is the link’s transmitter power, and κ i > 0is the thermal noise at its receiver node. For each link i, there is a lowerSIR threshold γ i . Therefore, we requireR i≥γ i(5.3)for every i = 1, 2, 3, ... n. The threshold values for all links can be takenequal to γ for convenience, reflecting a certain QoS the link has to maintainto operate properly. An upper SIR limit is also set, so that the transmitterpower of a link is minimized, which, in turn, will decrease theinterference due to its transmitter power at other receiver nodes. Therefore,we haveR()li≤ γ*i(5.4)Let ς be the set of all links, and call a link i ∈ς active or operationalduring the lth step ifR()≥γliwhere Ri( l) is the measured SIR, and A l is the set of all active links duringthe lth step. Alternatively, call a link i ∈ς inactive or new link during thelth step ifiR()< l γiiwhereis the set of all in active links during the lth step.B lp * ,FACT 5.2.1 (EXISTENCE OF A FEASIBLE POWER VECTOR)There exists a power vector > 0 which is optimal, such thatγ i < SIR < γ i* . This statement is equivalent to maintaining the SIRRi ≥ γ i i ∈Al, the set of active or operational links, and R i ≤ γ i i ∈ B l , theset of inactive or new links trying to get added during the lth step.