1. Traditional Approaches to Modeling and Analysis

2. Outline Concepts: Models Dynamical Systems Model Fixed Points
Optimality
Convergence
Stability
Models
Contraction Mappings
Markov chains
Standard Interference Function

3. while noting the relevant timing model
Basic Model
Dynamical system
A system whose change in state is a function of the current state and time
Autonomous system
Not a function of time
OK for synchronous timing
Characteristic function
Evolution function
First step in analysis of dynamical system
Describes state as function of time & initial state.
For simplicity
while noting the relevant timing model

4. Connection to Cognitive Radio Model
g = d/  t
Assumption of a known decision rule obviates need to solve for evolution function.
Reflects innermost loop of the OODA loop
Useful for deterministic procedural radios
(generally discrete time for our purposes)

5. Example: ([Yates_95]) Power control applications
Defines a discrete time evolution function as a function of each radio’s observed SINR, j , each radio’s target SINR and the current transmit power
Applications
Fixed assignment - each mobile is assigned to a particular base station
Minimum power assignment - each mobile is assigned to the base station in the network where its SINR is maximized
Macro diversity - all base stations in the network combine the signals of the mobiles
Limited diversity - a subset of the base stations combine the signals of the mobiles
Multiple connection reception - the target SINR must be maintained at a number of base stations.

6. Applicable analysis models & techniques
Markov models
Absorbing & ergodic chains
Standard Interference Function
Can be applied beyond power control
Contraction mappings
Lyapunov Stability

7. Differences between assumptions of dynamical system and CRN model
Goals of secondary importance
Technically not needed
Not appropriate for ontological radios
May not be a closed form expression for decision rule and thus no evolution function
Really only know that radio will “intelligently” – work towards its goal
Unwieldy for random procedural radios
Possible to model as Markov chain, but requires empirical work or very detailed analysis to discover transition probabilities

8. Steady-states
Recall model of <N,A,{di},T> 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

9. Fixed Point Definition
Given a mapping a point
is said to be a fixed point of f if
In 2-D fixed points for f can be found by evaluating where
and intersect. 1
How much information do we need to have to know that a function has a fixed point/Nash equilibrium?
f(x) x 1

10. Visualizing Fixed Point Existence
Consider continuous
X compact, convex
Fixed Point must exist 1
f(x) x 1

11. Convex Sets Definition Convex Set
Let S  n. S is said to be convex if for all x, y  S,
the point w = x + (1- )y is in S for all  [0,1].
Equivalent expression
A set S is convex if for all possible pairs of points, x, y, drawn
from S the line segments joining x, y is also in S.
Not Convex
Convex
Convex x y

12. Compact Sets Definition Compact Set
A bounded set S is compact if there is no point xS such that
the limit of a sequence formed entirely from elements in S is x.
Equivalent – closed and bounded
Compact sets
Non-compact sets
Any closed finite interval [0,1]
(0,1]
Closed n-Ball
(A filled sphere)
[0,)
Closed Disk
(Note, mathematically a disk is just a ball)

13. Continuous Function Definition Continuous Function
A function f: XY is continuous if for all x0X the following
three conditions hold:
f(x0)  Y
Note being differentiable at x0 implies continuity at x0, but
continuity does not imply differentiability
A continuous but not differentiable function

14. Visualizing Fixed Point Existence
Consider continuous
X not compact, convex or X compact, not convex
Fixed point need not exist 1
f(x) x 1

15. Brouwer’s Fixed Point Theorem
Let f :X X be a continuous function from a non-empty compact convex set X  n, then there is some x*X such that f(x*) = x*.
(Note originally written as f :B B where B = {x  n : ||x||1} [the unit n-ball])

16. Visualizing Fixed Point Existence
Consider f :X X as an upper semi-continuous correspondence
X compact, convex 1
f(x) x 1

17. Kakutani’s Fixed Point Theorem
Let f :X X be a upper semi-continuous convex valued correspondence from a non-empty compact convex set X  n, then there is some x*X such that x*  f(x*)

18. Example steady-state solution
Consider Standard Interference Function

19. Optimality
In general we assume the existence of some design objective function J:A
The desirableness of a network state, a, is the value of J(a).
In general maximizers of J are unrelated to fixed points of d.
Figure from Fig 2.6 in I. Akbar, “Statistical Analysis of Wireless Systems Using Markov Models,” PhD Dissertation, Virginia Tech, January 2007

20. Identification of Optimality
If J is differentiable, then optimal point must either lie on a boundary or be at a point where the gradient is the zero vector

21. Convergent Sequence
A sequence {pn} in a Euclidean space X with point pX such that for every >0, there is an integer N such that nN implies dX(pn,p)< 
This can be equivalently written as or

22. Example Convergent Sequence
Given , choose N=1/ , p=0 1
Establish convergence by applying definition
Necessitates knowledge of p.

23. Convergent Sequence Properties

24. Cauchy Sequence
A sequence {pn} in a metric space X such that for every >0, there is an integer N such that if

25. Example Cauchy Sequence
Given , choose N=2/, p=0 1
Establish convergence by applying definition
No need to know p
In k, every Cauchy sequence converges, and every convergent sequence is Cauchy

26. Monotonic Sequences A sequence {sn} is monotonically increasing if .
A sequence {sn} is monotonically decreasing if
(Note: some authors use the inclusion of the equals condition to define a sequence to be respectively monotonically nondecreasing or monotonically nonincreasing.). A sequence which is either monotonically increasing or monotonically decreasing is said to be monotonic.

27. Convergent Monotonic Sequences
Suppose is a monotonic in X. Then converges if X is bounded.
Note that also converges if X is compact.

28. Showing convergence with nonlinear programming
Left unanswered: where does  come from?

29. Stability
Stable, but not attractive
Attractive, but not stable

30. Lyapunov’s Direct Method
Left unanswered: where does L come from?

31. Comments on analysis
We just covered some very general techniques for showing that a system has a fixed point (steady-state), converges, and is stable.
Could apply these to every problem independently, but can sometimes be painful (and nonobvious – where does Lyapunov function come from, convergence assumes we already know a fixed point)
My preferred approach is to analyze general models and then show that particular problems satisfy conditions of one of the general models.

32. 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

33. Contraction Mappings Every contraction is a pseudo-contraction
Every pseudo-contraction has a fixed point
Every pseudo-contraction converges at a rate of 
Every pseudo-contraction is globally asymptotically stable
Lyapunov function is distance to the fixed point)
A Pseudo-contraction
which is not a contraction

34. General Convergence Theorem
A synchronous contraction mapping also converges asynchronously

35. Standard Interference Function
Conditions
Suppose d:AA and d satisfies:
Positivity: d(a)>0
Monotonicity: If a1a2, then d(a1)d(a2)
Scalability: For all >1, d(a)>d( a)
d is a pseudo-contraction mapping [Berggren] under synchronous timing
Implies synchronous and asynchronous convergence
Implies stability
R. Yates, “A Framework for Uplink Power Control in Cellular Radio Systems,” IEEE JSAC., Vol. 13, No 7, Sep. 1995, pp
F. Berggren, “Power Control, Transmission Rate Control and Scheduling in Cellular Radio Systems,” PhD Dissertation Royal Institute of Technology, Stockholm, Sweden, May, 2001.

36. Yates’ power control applications
Target SINR algorithms
Fixed assignment - each mobile is assigned to a particular base station
Minimum power assignment - each mobile is assigned to the base station in the network where its SINR is maximized
Macro diversity - all base stations in the network combine the signals of the mobiles
Limited diversity - a subset of the base stations combine the signals of the mobiles
Multiple connection reception - the target SINR must be maintained at a number of base stations.

37. Example steady-state solution
Consider Standard Interference Function

38. 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

39. 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.

40. 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

41. 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.

42. Shortcomings in traditional techniques
Fixed point theorems provide little insight into convergence or stability
Lyapunov functions hard to identify
Contraction mappings rarely encountered
Doesn’t address nondeterministic algorithms
Genetic algorithms
Analyze one algorithm at a time – little insight into related algorithms
Not very useful for finite action spaces
No help if all you have is the cognitive radios’ goal and actions

43. Absorbing Markov Chains
Absorbing state
Given a Markov chain with transition matrix P, a state am is said to be an absorbing state if pmm=1.
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. a5 a0 a1 a2 a3 a4

44. 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
t=N1
(Final distribution) Probability of ending up in absorbing state am given that the system started in ak is bkm where

45. Two-Channel DFS Decision Rule Goal Timing
Random timer set to go off with probability
p=0.5 at each iteration

46. Analysis Models

47. Model Steady States

48. Model Convergence

49. Model Stability

50. Shortcomings in “traditional” techniques
Fixed point theorems provide little insight into convergence or stability
Lyapunov functions hard to identify
Contraction mappings rarely encountered
Doesn’t address nondeterministic algorithms
Genetic algorithms
Not very useful for finite action spaces
No help if all you have is the cognitive radios’ goal and actions

51. Comments No unified method for analyzing cognitive radio interactions
Random collection of methods for different problems
Perhaps a bit of a stretch to call it “traditional” with respect to cognitive radios
Is not suitable for analyzing radios with