Wireless Ad Hoc and Sensor Networks

Distributed Power Control of Wireless Cellular and Peer-to-Peer Networks 187Given the DPC, the issue now is whether the proposed DPC can ensurelink protection of active links during inactive link admissions. As seenbefore, a link i belonging to the set A l is considered active or an operationallink. On the other hand, link i belonging to set B l is termed asinactive. Certain links in this set of inactive set, B l , update their powersaccording to Theorem 5.2.1 and Theorem 5.2.2 and, eventually, gainadmission into the network, becoming part of active links A l . The set ofinactive links that will eventually be admitted into the channel are termedas fully admissible set of inactive links. There may be a set of links in B l thatcan never gain admission into the network because of the followingreasons: (1) the set of new links, while powering to gain admission, causesextreme interference to the existing set of active links; (2) the SIRs, Ri( l),ofthese inactive links saturate below the target value γ i with time after notgetting active. This set of links that cannot gain admission into the networkdue to channel saturation, is termed as totally inadmissible set ofinactive links.5.2.3.1 SIR Protection of an Active LinkFor any active link i, we haveR() l ≥ γ => R( l+ 1)≥γi i i i(5.17)This implies that a new link is added if and only if the new state of thesystem is stable, i.e., none of the existing links are broken. Now, using theabove SSCD/optimal DPC schemes, we prove that the active links in thenetwork continue to remain active throughout their transmission time.THEOREM 5.2.3For any fixed η i ∈( 1, ∞),for every l ∈{ 0, 1, 2 , …}and every i ∈ A l wehave Ri() l ≥ γ i ⇒ Ri( l+ 1) ≥ γ i,under the DPC/SSCD or DPC/optimal updatingscheme. Therefore i ⊂ Al⇒ i ∈ Al+1 or Al⊆ Al+1 and Bl⊆ Bl+1 forevery l ∈{ 0, 1, 2 , …}.PROOF If vi() l =− kixi()l +ηiwhere η i is the protection margin, thenusing Equation 5.13, the system error equation can be written as xi( l+ 1)=xi() l − kixi() l +ηi,or xi( l+ 1) = ( 1− ki) xi( l)+ ηiwhere 0 < k i < 1.The systemdiscussed in the preceding text is a linear time-invariant system withstable transition matrix driven by a constant, small, and boundedinput η i , which is the protection margin. This further implies that xi( l),ηiwhich is equal to ( Ri( l) −γ i)tends to at steady state. This further implieskiηithat R () l ≥ γ + , and it follows that at steady state.i i ki