Wireless Ad Hoc and Sensor Networks

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Distributed Power Control of Wireless Cellular and Peer-to-Peer Networks 203as the vector of powers of active links andPn() l = ( PN+ 1( l), PN+ 2(),..., l PN+ i( l), PN+ i+1 ( l), …, PN+M( l))T(5.30)the vector of powers of inactive links.Also we havePa( l+ 1) = [ R( l) + v( l)] I( l+1)(5.31)Substituting vl () =− Kxl () +η into Equation 5.31 results inPa( l+ 1) = [( I− K) R( l) + ( Kγ+ ηI)] I( l+1),(5.32)where K, P, R, η, γ are defined in Equation 5.19. Moreover, Rl () =Il ( +1)and ϕ() l = < β()l from Theorem 5.2.6.Il ()Therefore Equation 5.31 yieldsPl ()Il ()Pa( 1) = ( I− K) ϕ() 0 P( 0) + ( Kγ + ηI) ϕ( 0) I( 0)Pa( 2) = ( I− K) ϕ( 1) P() 1 + ( Kγ + ηI) ϕ()().1 I 1(5.33)(5.34)Substituting P a () 1 in P a ( 2)yieldsPa( 2) = ( I− K) 2 ϕ( 1) ϕ( 0) P( 0) + ( Kγ + ηI)[( I−K) ϕ( 1)ϕ( 0) I( 0) + ϕ( 1) I( 1)](5.35)Consequently, we can write Pa( l)asll−1P() l = ( I−K) ϕ( l−1) ϕ( l− 2) ⋯ϕ( 0) P( 0) + ( Kγ + ηI)[(I−K) ϕ( l−1)al−1ϕ( l− 2) ⋯ ϕ( 0) I( 0) + ( I−K) ϕ(l−1) ϕ( l− 2) ⋯ϕ( 1) I( 1)] + ⋯+ ϕ( l−1) I( l−1)].(5.36)From Equation 5.36, we can clearly see that the Pa( l)value converges tolim ()*Pal →D,l →∞which is a constant. (5.37)Recall, that the SIR of the ith link at the lth power update is given byRl ( + 1) = Rl ( )[ I− K]+ Kγ + η(5.38)
204 Wireless Ad Hoc and Sensor NetworksNow, at each power control scheme iteration, we haveR() 1 = R()[ 0 I− K]+ Kγ+ η(5.39)R( 2) = R( 1)[ I− K]+ Kγ+ η(5.40)Substituting the value of R(1) in R(2) to get,2R( 2) = R( 0)( I− K) + ( I− K)( Kγ + ηI) + ( Kγ + ηI)(5.41)Similarly, the value of SIR at the lth time instant is given byl l lRl () = R( )( I− K) + [( I− K) − 1 −20 + ( I− K) + ⋯+ 1](Kγ + ηI)(5.42)Taking limits, on both sides to getlim η ()*Rl = γ I+ = ,l →∞ KR(5.43)because | I− K|
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18. Chaos in Automatic Control, edi
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CRC PressTaylor & Francis Group6000
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Table of Contents1 Background on Ne
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3 Congestion Control in ATM Network
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5.4.2 Distributed Power Controller
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7.3 Adaptive and Distributed Fair S
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9.2.2 Overview.....................
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the number of connections in each l
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y Maheswaran Thiagarajan, and tirel
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2 Wireless Ad Hoc and Sensor Networ
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10 Wireless Ad Hoc and Sensor Netwo
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2BackgroundIn this chapter, we prov
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Background 51u(k)x n (k + 1)x n (k)
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Background 53with x( 0)being the in
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Background 55with tr(.) the matrix
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Background 57nGiven a function ft (
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Background 592.3.2 Lyapunov Stabili
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Background 61as well as Jagannathan
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Background 63The subsequent example
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Background 65Evaluating this along
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Background 67Now, select the feedba
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Background 69is SISL if and only if
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Background 71L 1 (x)L(x, t)L 0 (x)0
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Background 73for some R > 0 such th
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Background 75Evaluating the first d
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Background 77Section 2.4Problem 2.4
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4Admission Controller Design for Hi
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7Distributed Fair Scheduling in Wir
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Distributed Fair Scheduling in Wire
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8Optimized Energy and Delay-Based R
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Optimized Energy and Delay-Based Ro
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9Predictive Congestion Control for
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Predictive Congestion Control for W
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10Adaptive and Probabilistic Power
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Adaptive and Probabilistic Power Co
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