Wireless Ad Hoc and Sensor Networks

Distributed Power Control of Wireless Cellular and Peer-to-Peer Networks 201Defining xi() l = ei()l for each link, the SIR update equation for the entirenetwork can be represented in the vector form as Rl ( + 1) = Rl ( ) + vl (),where vl () =− Kxl () +η,with Rl (), vl (),K, xl (),γ , η represented as⎡R1()l ⎤⎢ ⎥⎢R2()l ⎥⎢ ⎥Rl () = ⎢.⎥ ,⎢ ⎥⎢.⎥⎢ ⎥⎢⎣Ri()l⎥⎦⎡v1()l ⎤⎢ ⎥⎢v2()l ⎥⎢ ⎥vl () = ⎢.⎥ ,⎢ ⎥⎢.⎥⎢ ⎥⎢⎣vi()l⎥⎦⎡k⎢ 1 000..0⎤⎥⎢0 k20 0 .. 0⎥⎢⎥K = ⎢. . . . . . . ⎥ ,⎢⎥⎢. . . . . . . ⎥⎢⎥⎢⎣0000..k⎥i ⎦⎡x1()l ⎤⎢ ⎥⎢x2()l ⎥⎢ ⎥xl () = ⎢.⎥ ,⎢ ⎥⎢.⎥⎢ ⎥⎢⎣xi()l⎥⎦⎡γ1 ⎤⎢ ⎥⎢γ⎥2⎢ ⎥γ = ⎢.⎥ ,⎢ ⎥⎢.⎥⎢ ⎥⎢ ⎥⎣γi ⎦⎡η1⎤⎢ ⎥⎢η2⎥⎢ ⎥η = ⎢.⎥⎢ ⎥⎢.⎥⎢ ⎥⎢⎣η⎥i ⎦(5.19)Using the SSCD or optimal power control approach presented in thispaper, the active links are always maintained active. Inactive links, aftersome time, get added into the network if they are admissible. They remainactive until the end of their transmission.We now consider a group of N + M links, such that originally the onesin the setA0 = {, 1 2, 3, … , N−1, N}(5.20)are active, whereas the ones in the setB0 = { N+ 1, N+ 2, N+ 3, … , N+ M− 1, N+M}(5.21)are inactive new links. We are mainly interested in whether the new linkswill eventually become active.THEOREM 5.3.1 (TOTALLY INADMISSIBLE NEW LINKS)Given that the network operates under the state-space-based or optimal powercontrol scheme, ifAl =A0 ≠φ and Bl = B0 ≠φ(5.22)