Analysis and Design of Cognitive Radio Networks and Distributed Radio Resource Management Algorithms - PowerPoint PPT Presentation

1 / 56
About This Presentation
Title:

Analysis and Design of Cognitive Radio Networks and Distributed Radio Resource Management Algorithms

Description:

Analysis and Design of Cognitive Radio Networks ... Permits analysis without well defined decision processes (only the goals are needed) Can be supplemented ... – PowerPoint PPT presentation

Number of Views:457
Avg rating:3.0/5.0
Slides: 57
Provided by: james176
Category:

less

Transcript and Presenter's Notes

Title: Analysis and Design of Cognitive Radio Networks and Distributed Radio Resource Management Algorithms


1
Analysis and Design of Cognitive Radio
Networksand Distributed Radio Resource
Management Algorithms
James Neel Aug 23-Sep 6, 2006
2
Presentation Schedule
  • August 23 Modeling Cognitive Radio Networks
  • August 30 Model Based Analysis of Cognitive
    Radio Networks
  • September 6 Model Based Design of Cognitive
    Radio Networks
  • () Formal Defense

3
Research in a nutshell
  • Hypothesis Applying game theory and game models
    (potential and supermodular) to the analysis of
    cognitive radio interactions
  • Provides a natural method for modeling cognitive
    radio interactions
  • Significantly speeds up and simplifies the
    analysis process (can be performed at the
    undergraduate level Senior EE)
  • Permits analysis without well defined decision
    processes (only the goals are needed)
  • Can be supplemented with traditional analysis
    techniques
  • Can provides valuable insights into how to design
    cognitive radio decision processes
  • Has wide applicability
  • Focus areas
  • Formalizing connection between game theory and
    cognitive radio
  • Collecting relevant game model analytic results
  • Filling in the gaps in the models
  • Model identification (potential games)
  • Convergence
  • Stability
  • Formalizing application methodology
  • Developing applications

4
Analyzing Cognitive Radio Networks
  • James Neel
  • August 30, 2006

5
Presentation Overview
  • Analysis Objectives
  • Analysis based on dynamical systems
  • Steady-states
  • Optimality
  • Convergence
  • Noise/Stability
  • Analysis based on game models
  • Steady-states
  • Optimality
  • Convergence
  • Noise/Stability

6
Analysis Objectives
7
Modeling Review
Dynamical System
  • The interactions in a cognitive radio network
    (levels 1-3) can be represented by the tuple ltN,
    A, ui, di,Tgt
  • A dynamical system model adequately represents
    inner-loop procedural radios
  • A myopic asynchronous repeated game adequately
    represents ontological radios and random
    procedural radios
  • Suitable for outer-loop processes
  • Not shown here, but can also handle inner-loop
  • Some differences in models
  • Most analysis carries over
  • Some differences

Game Model
8
Analysis Objectives
  • Steady state characterization
  • Steady state optimality
  • Convergence
  • Stability/Noise
  • Scalability

Steady State Characterization Is it possible
to predict behavior in the system? How many
different outcomes are possible?
Optimality Are these outcomes desirable?
Do these outcomes maximize the system target
parameters?
Convergence How do initial conditions impact
the system steady state? What processes will
lead to steady state conditions? How long
does it take to reach the steady state?
Stability/Noise How do system
variations/noise impact the system? Do the
steady states change with small
variations/noise? Is convergence affected by
system variations/noise?
Scalability As the number of devices
increases, How is the system impacted?
Do previously optimal steady states remain
optimal?
9
Dynamical Systems Analysis
10
Steady-states
  • Recall model of ltN,A,di,Tgt which we
    characterize with the evolution function d
  • Steady-state is a point where a d(a) for all t
    ?t
  • Obvious solution solve for fixed points of d.
  • For non-cooperative radios, if a is a fixed
    point under synchronous timing, then it is under
    the other three timings.
  • Works well for convex action spaces
  • Not always guaranteed to exist
  • Value of fixed point theorems
  • Not so well for finite spaces
  • Generally requires exhaustive search

11
Example steady-state solution
  • Consider Standard Interference Function

12
Optimality
  • In general we assume the existence of some design
    objective function JO??
  • The desirableness of a network state, o, is the
    value of J(o).
  • In general maximizers of J are unrelated to fixed
    points of d.

J
(shamelessly lifted from Matlabs logo)
13
Showing convergence with nonlinear programming
?
Left unanswered where does ? come from?
14
Stability
Attractive, but not stable
15
Lyapunovs Direct Method
Left unanswered where does L come from?
16
Analysis models appropriate for dynamical systems
  • Contraction Mappings
  • Identifiable unique steady-state
  • Everywhere convergent, bound for convergence rate
  • Lyapunov stable (??)
  • Lyapunov function distance to fixed point
  • General Convergence Theorem (Bertsekas) provides
    convergence for asynchronous timing if
    contraction mapping under synchronous timing
  • Standard Interference Function
  • Forms a pseudo-contraction mapping
  • Can be applied beyond power control
  • Markov Chains (Ergodic and Absorbing)
  • Also useful in game analysis

17
Markov Chains
  • Describes adaptations as probabilistic
    transitions between network states.
  • d is nondeterministic
  • Sources of randomness
  • Nondeterministic timing
  • Noise
  • Frequently depicted as a weighted digraph or as a
    transition matrix

18
General Insights (Stewart_94)
  • Probability of occupying a state after two
    iterations.
  • Form PP.
  • Now entry pmn in the mth row and nth column of PP
    represents the probability that system is in
    state an two iterations after being in state am.
  • Consider Pk.
  • Then entry pmn in the mth row and nth column of
    represents the probability that system is in
    state an two iterations after being in state am.

19
Steady-states of Markov chains
  • May be inaccurate to consider a Markov chain to
    have a fixed point
  • Actually ok for absorbing Markov chains
  • Stationary Distribution
  • A probability distribution such that ? such that
    ?T P ?T is said to be a stationary
    distribution for the Markov chain defined by P.
  • Limiting distribution
  • Given initial distribution ?0 and transition
    matrix P, the limiting distribution is the
    distribution that results from evaluating

20
Ergodic Markov Chain
  • Stewart_94 states that a Markov chain is
    ergodic if it is a Markov chain if it is a)
    irreducible, b) positive recurrent, and c)
    aperiodic.
  • Easier to identify rule
  • For some k Pk has only nonzero entries
  • (Convergence, steady-state) If ergodic, then
    chain has a unique limiting stationary
    distribution.

21
Absorbing Markov Chains
  • Absorbing state
  • Given a Markov chain with transition matrix P, a
    state am is said to be an absorbing state if
    pmm1.
  • Absorbing Markov Chain
  • A Markov chain is said to be an absorbing Markov
    chain if
  • it has at least one absorbing state and
  • from every state in the Markov chain there exists
    a sequence of state transitions with nonzero
    probability that leads to an absorbing state.
    These nonabsorbing states are called transient
    states.

a4
a5
a3
a1
a2
a0
22
Absorbing Markov Chain Insights (Kemeny_60 )
  • Canonical Form
  • Fundamental Matrix
  • Expected number of times that the system will
    pass through state am given that the system
    starts in state ak.
  • nkm
  • (Convergence Rate) Expected number of iterations
    before the system ends in an absorbing state
    starting in state am is given by tm where 1 is a
    ones vector
  • tN1
  • (Final distribution) Probability of ending up in
    absorbing state am given that the system started
    in ak is bkm where

23
Two-Channel DFS
Timing Random timer set to go off with
probability p0.5 at each iteration
24
Procedural Radio Analysis Models
25
Model Steady States
26
Model Convergence
27
Model Stability
28
Using Game Theory to Analyze Cognitive Radio
Networks
29
Modeling Assumptions
  • Model of ltN,A,ui,di,Tgt
  • Adaptations increase players goal
  • Asynchronous myopic repeated game model

30
Nash Equilibrium (Steady-state)
31
Nash Equilibrium Identification
  • Exhaustive Search
  • Time to find all NE can be significant
  • Only appropriate for finite action spaces
  • Not every game has an NE

32
Nash Equilibrium as a Fixed Point
  • Best Response function
  • Synchronous Best Response
  • Nash Equilibrium as a fixed point
  • Fixed point theorems can be used to establish
    existence of NE (see dissertation)
  • NE can be solved by implied system of equations

33
Significance of NE for CRNs
  • Why not if and only if?
  • Consider a self-motivated game with a local
    maximum and a hill-climbing algorithm.
  • For many decision rules, NE do capture all fixed
    points (see dissertation)
  • Identifies steady-states for all intelligent
    decision rules with the same goal.
  • Implies a mechanism for policy design while
    accommodating differing implementations
  • Verify goals result in desired performance
  • Verify radios act intelligently

34
Pareto efficiency (optimality)
  • Formal definition An action vector a is Pareto
    efficient if there exists no other action vector
    a, such that every radios valuation of the
    network is at least as good and at least one
    radio assigns a higher valuation
  • Informal definition An action tuple is Pareto
    efficient if some radios must be hurt in order to
    improve the payoff of other radios.
  • Important note
  • Like design objective function, unrelated to
    fixed points (NE)
  • Inferior to evaluating design objective function
    (see dissertation)

35
Example Games
Legend
Pareto Efficient
NE
NE PE
a2
b2
a1
1,1
-5,5
b1
-1,-1
5,-5
36
Paths and Convergence
  • Path Monderer_96
  • A path in ? is a sequence ? (a0, a1,) such
    that for every k ? 1 there exists a unique player
    such that the strategy combinations (ak-1, ak)
    differs in exactly one coordinate.
  • Equivalently, a path is a sequence of unilateral
    deviations. When discussing paths, we make use of
    the following conventions.
  • Each element of ? is called a step.
  • a0 is referred to as the initial or starting
    point of ?.
  • Assuming ? is finite with m steps, am is called
    the terminal point or ending point of ? and say
    that ? has length m.
  • Cycle Voorneveld_96
  • A finite path ? (a0, a1,,ak) where ak a0

37
Improvement Paths
  • Improvement Path
  • A path ? (a0, a1,) where for all k?1,
    ui(ak)gtui(ak-1) where i is the unique deviator at
    k
  • Improvement Cycle
  • An improvement path that is also a cycle

38
Convergence Properties
  • Finite Improvement Property (FIP)
  • All improvement paths in a game are finite
  • Weak Finite Improvement Property (weak FIP)
  • From every action tuple, there exists an
    improvement path that terminates in an NE.
  • FIP implies weak FIP
  • FIP implies lack of improvement cycles
  • Weak FIP implies existence of an NE

39
Examples
40
Implications of FIP and weak FIP
  • Unless the game model of a CRN has weak FIP, then
    no autonomously rational decision rule can be
    guaranteed to converge from all initial states
    under random and round-robin timing (Theorem 4.10
    in dissertation).
  • If the game model of a CRN has FIP, then ALL
    autonomously rational decision rules are
    guaranteed to converge from all initial states
    under random and round-robin timing.
  • And asynchronous timings, but not immediate from
    definition
  • More insights possible by considering more
    refined classes of decision rules and timings

41
Decision Rules
42
Absorbing Markov Chains and Improvement Paths
  • Sources of randomness
  • Timing (Random, Asynchronous)
  • Decision rule (random decision rule)
  • Corrupted observations (not assumed yet)
  • An NE is an absorbing state for autonomously
    rational decision rules.
  • Weak FIP implies that the game is an absorbing
    Markov chain as long as the NE terminating
    improvement path always has a nonzero probability
    of being implemented.
  • This then allows us to characterize
  • convergence rate,
  • probability of ending up in a particular NE,
  • expected number of times a particular transient
    state will be visited

43
Connecting Markov models, improvement paths, and
decision rules
  • Suppose we need the path ? (a0, a1,am) for
    convergence by weak FIP.
  • Must get right sequence of players and right
    sequence of adaptations.
  • Friedman Random Better Response
  • Random or Asynchronous
  • Every sequence of players have a chance to occur
  • Random decision rule means that all improvements
    have a chance to be chosen
  • Synchronous not guaranteed
  • My random better response (chance of choosing
    same action)
  • Because of chance to choose same action, every
    sequence of players can result from every
    decision timing.
  • Because of random choice, every improvement path
    has a chance of occurring

44
Convergence Results (Finite Games)
  • If a decision rule converges under round-robin,
    random, or synchronous timing, then it also
    converges under asynchronous timing.
  • Random better responses converge for the most
    decision timings and the most surveyed game
    conditions.
  • Implies that non-deterministic procedural
    cognitive radio implementations are a good
    approach if you dont know much about the network.

45
Impact of Noise
  • Noise impacts the mapping from actions to
    outcomes, f A?O
  • Same action tuple can lead to different outcomes
  • Most noise encountered in wireless systems is
    theoretically unbounded.
  • Implies that every outcome has a nonzero chance
    of being observed for a particular action tuple.
  • Some outcomes are more likely to be observed than
    others (and some outcomes may have a very small
    chance of occurring)

46
DFS Example
  • Consider a radio observing the spectral energy
    across the bands defined by the set C where each
    radio k is choosing its band of operation fk.
  • Noiseless observation of channel ck
  • Noisy observation
  • If radio is attempting to minimize inband
    interference, then noise can lead a radio to
    believe that a band has lower or higher
    interference than it does

47
Trembling Hand (Noise in Games)
  • Assumes players have a nonzero chance of making
    an error implementing their action.
  • Who has not accidentally handed over the wrong
    amount of cash at a restaurant?
  • Who has not accidentally written a tpyo?
  • Related to errors in observation as erroneous
    observations cause errors in implementation (from
    an outside observers perspective).

48
Noisy decision rules
  • Noisy utility

Trembling Hand
Observation Errors
49
Implications of noise
  • For random timing, Friedman shows game with
    noisy random better response is an ergodic Markov
    chain.
  • Likewise other observation based noisy decision
    rules are ergodic Markov chains
  • Unbounded noise implies chance of adapting (or
    not adapting) to any action
  • If coupled with random, synchronous, or
    asynchronous timings, then CRNs with corrupted
    observation can be modeled as ergodic Makov
    chains.
  • Not so for round-robin (violates aperiodicity)
  • Somewhat disappointing
  • No real steady-state (though unique limiting
    stationary distribution)

50
DFS Example with three access points
  • 3 access nodes, 3 channels, attempting to operate
    in band with least spectral energy.
  • Constant power
  • Link gain matrix
  • Noiseless observations
  • Random timing

3
1
2
51
Trembling Hand
  • Transition Matrix, p0.1
  • Limiting distribution

52
Noisy Best Response
  • Transition Matrix, ?(0,1) Gaussian Noise
  • Limiting stationary distributions

53
Comment on Noise and Observations
  • Cardinality of goals makes a difference for
    cognitive radios
  • Probability of making an error is a function of
    the difference in utilities
  • With ordinal preferences, utility functions are
    just useful fictions
  • Might as well assume a trembling hand
  • Unboundedness of noise implies that no state can
    be absorbing
  • NE retains significant predictive power
  • While CRN is an ergodic Markov chain, NE (and the
    adjacent states) remain most likely states to
    visit
  • Stronger prediction with less noise
  • Also stronger when network has a Lyapunov
    function
  • Exception - elusive equilibria (Hicks_04)

54
Summary
  • Skipped over a lot of stuff in the dissertation
  • Given a set of goals, an NE is a fixed point for
    all radios with those goals for all autonomously
    rational decision processes
  • Traditional engineering analysis techniques can
    be applied in a game theoretic setting
  • Markov chains to improvement paths
  • Network must have weak FIP for autonomously
    rational radios to converge
  • Weak FIP implies existence of absorbing Markov
    chain for many decision rules/timings
  • In practical system, network has a theoretically
    nonzero chance of visiting every possible state
    (ergodicity), but does have unique limiting
    stationary distribution
  • Specific distribution function of decision rules,
    goals

55
Shortcomings
  • Steady-states
  • Need not be optimal
  • Could enforce desired equilibria
  • Doesnt generally scale well
  • Identification in finite games is painful
  • Game convergence
  • Only looked at finite games
  • Arguably all DSP controlled radios have a finite
    action space, but to a casual observer, action
    space may appear infinite
  • How to identify when a game has FIP/weak FIP
    without exhaustive search?
  • Would like to be able to apply Zangwills for
    infinite action spaces
  • Noise/Stability
  • Characterizing state distribution not exactly the
    same as stability
  • Where do the functions for applying Zangwills
    Theorem and Lyapunovs Direct Method come from?
  • Solution potential games and the interference
    reducing network design framework (next week)

56
Questions?
Write a Comment
User Comments (0)
About PowerShow.com