Distributed Power Control for Time

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Distributed Power Control for Time Varying
Wireless Networks Optimality and Convergence
Andrea Goldsmith Tim Holliday Nick Bambos Peter
Glynn Stanford University
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Ad-hoc Wireless Networks
Pi
Gii
Gij
Pi

• Each node generates independent data.
• Source-destination pairs are chosen at random.
• Topology is dynamic
• Different link SIRs based on channel gains Gij
• Power control used to maintain a target Ri value

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Power Control for Fixed Channels

• Seminal work by Foschini/Miljanic 1993
• Assume each node has an SIR constraint
• Write the set of constraints in matrix form

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Optimality and Stability

• Then if rFlt1 ? a unique solution
• P is the global optimal solution
• Iterative power control algorithms P?P

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Convergence Iterations
P2
Feasible Region
Iterative Algorithm
P
P1
These results are for fixed channels
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What about random channels?

• When the channel is random, power control cannot
  maintain Ri w.p. 1 or 0.
• Thus, the optimality criterion for power control
  must be redefined.
• For a given criteria, its not clear that a
  well-behaved power control algorithm exists.
• Moreover, its not clear that a distributed
  version of this power control exists.

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F-M with a Random Channel

• Assume the matrices G(k) and F(k) are positive,
  irreducible, and stationary ergodic processes
• We show that P(k) is a geometrically ergodic
  Markov chain (with appropriate conditions on F
  and u)

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Power Stability of an Ad-hoc Network

• Convergence of the F-M Algorithm is determined by
  the statistics of F(k) and u(k)
• The Lyapunov exponent for a product of random
  matrices is defined as

• If lF lt 0 and Elog(1u(k))lt ?
• Then P(k)?P(?) and EP(?)lt ?

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SIR as a Random Process

• Since P(k) is a stochastic process, R(k) will be
  random as well
• Ri(k) converges to a stationary random variable
• Its distribution is very hard to compute

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Performance in a Random Channel

• We can define system performance in terms of the
  distribution of Ri
• Average SIR
• Outage Probability
• We show that
• This doesnt tell us anything about the power
  consumption
• The Pis can deviate significantly from their mean

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Adaptation in a Random Environment

• Power control based on channel prediction
• Potentially provides better algorithm behavior
• Need interferers channel gains (nondistributed)
• Pointless if the channel is i.i.d.
• Power must be constantly updated.
• Can we develop a power control algorithm for
  random channels with better properties
• Satisfies some form of optimality
• Powers converge to fixed values
• Permits distributed power control
• Appropriate in an i.i.d. channel

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Consider A New SIR Constraint

• ? Original constraint

? Multiply out and take expectations
? Matrix form
Same form as SIR constraint in F-M for fixed
channels
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New Criterion for Optimality

• If rFlt1 then there exists a global optimal
  solution
• For the SIR constraint
• Is this constraint appropriate?
• How do we find P in a distributed manor?

ERi?gi
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Stochastic Approximation

• Re-write the optimality conditions as
• Goal find an iterative solution for P g(P)0
• We only see samples of F and u
• This is a natural setting for the application of
  the Robbins-Monro algorithm

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Robbins-Monro algorithm

• Where ek is an estimate of g(P(k))

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Convergence and Optimality

• If the following conditions hold
• supRe r r is an eigenvalue of F) lt 1
• The noise terms ek satisfy
• The stochastic approximation iterations converge
  to the optimal power allocation

For most cases of interest, conditions hold
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Optimal Distributed Control

• The components of the centralized algorithm are
  path-wise identical to
• The above distributed algorithm satisfies all of
  the properties of the centralized version

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Example i.i.d. Fading Channel

• Suppose the network consists of 3 nodes
• Each link in the network is an independent
  exponential random variable
• Note that rF.33 so we should expect this network
  to be fairly stable

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PICTURE
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Example Correlated Fading

• The i.i.d. channel is essentially the worst case
  scenario for the F-M algorithm
• Define a channel matrix with memory M as
• where Giid is independent exponential
• In the following example M20, which is a
  substantial amount of memory

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PICTURE
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Conclusion

• Power control designed for fixed channels
  performs poorly in random channels
• We propose an on-line distributed power control
  algorithm for random channels
• The algorithms meets mean SIR constraints and our
  optimality criterion
• Algorithm can incorporate (distributed) admission
  control with active link protection