traditional approaches to modeling and analysis

1 © Cognitive Radio Technologies, 2007

Traditional Approaches to Modeling and Analysis


Outline

Concepts:– Dynamical Systems Model– Fixed Points– Optimality– Convergence– Stability

Models– Contraction Mappings– Markov chains– Standard Interference Function


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

:d A T A ,a g a t

x

y

:jj N

d d d A A

while noting the relevant timing model


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)


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.

1ˆ jm m

j jj

p p

\1

ˆ mj kj k j

k N im mj j m

jj j

g p

p pKg p

ˆ j


Applicable analysis models & techniques

Markov models– Absorbing & ergodic chains

Standard Interference Function– Can be applied beyond power control

Contraction mappings Lyapunov Stability


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


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


Fixed Point Definition

Given a mapping a point

is said to be a fixed point of f if

*x X:f X X

* *f x x

y f x y x

1

1

0 x

f(x)

In 2-D fixed points for f can be found by evaluating where and intersect.

How much information do we need to have to know that a function has a fixed point/Nash equilibrium?


Visualizing Fixed Point Existence

Consider continuous

X compact, convex Fixed Point must

exist

1

1

0 x

f(x)

:f X X


Convex Sets

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].

Definition Convex Set

Equivalent expressionA set S is convex if for all possible pairs of points, x, y, drawnfrom S the line segments joining x, y is also in S.

Convex Convex Not Convex

x

y


Compact Sets

Definition Compact SetA bounded set S is compact if there is no point xS such that the limit of a sequence formed entirely from elements in S is x.

Any closed finite interval [0,1]

Compact sets

Equivalent – closed and bounded

Non-compact sets

(0,1]

[0,)

Closed Disk (Note, mathematically a disk is just a ball)

Closed n-Ball (A filled sphere)


Continuous Function

Definition Continuous FunctionA function f: XY is continuous if for all x0X the following three conditions hold: f(x0) Y

0

limx x

f x Y

0

0limx x

f x f x

Note being differentiable at x0 implies continuity at x0, but continuity does not imply differentiability

A continuous but not differentiable function



Consider continuous

X not compact, convex or X compact, not convex

Fixed point need not exist

1

1

0 x

f(x)

:f X X


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])



Consider f :X X as an upper semi-continuous correspondence

X compact, convex

1

1

0 x

f(x)


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*)


Example steady-state solution

Consider Standard Interference Function

1ˆ jm m

j jj

p p

\1

ˆ mj kj k j

k N im mj j m

jj j

g p

p pKg p

* *

\

ˆ jj kj k j

k N ijj

p g pKg

11 1 12 1

21 22 2*

1 2

1 2

ˆ/

ˆ/

ˆ/

n

n

n n nn n

Kg g g

g Kg

g g Kg

p


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


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

1 2

1 2

ˆ ˆ ˆnn

J a J a J aJ a a a a

a a a

0


Convergent Sequence

A sequence {pn} in a Euclidean space X with point pX such that for every >0, there is an integer N such that nN implies dX(pn,p)<

This can be equivalently written as

or np plim nn

p p


Example Convergent Sequence

Given , choose N=1/ , p=0

1/np n

10

Establish convergence by applying definitionNecessitates knowledge of p.


Convergent Sequence Properties


Cauchy Sequence

A sequence {pn} in a metric space X such that for every >0, there is an integer N such that if ,m n N ,X n md p p


Example Cauchy Sequence

Given , choose N=2/, p=0

1 /nnp n

10

Establish convergence by applying definitionNo need to know p

In k, every Cauchy sequence converges, and every convergent sequence is Cauchy


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.

n mn m s s

n mn m s s


Convergent Monotonic Sequences

Suppose is a monotonic in X. Then converges if X is bounded.

Note that also converges if X is compact.

ns ns

ns


Showing convergence with nonlinear programming

(shamelessly lifted from Matlab’s logo)

J

Left unanswered: where does come from?


Stability

x

y

x

y

Stable, but not attractive

x

y

Attractive, but not stable


Lyapunov’s Direct Method

Left unanswered: where does L come from?


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.


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

O1

O2

O3

O4

O5

O6O7

O8O9

O10

O11

O11

A(t0)

A(t1)

A(t2)

A(t3)

A(t4)

A(t5)

A(t6)

A(t7)

A(t8)

A(t8)

A(t9)


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) 1

1

0 1

1

0

A Pseudo-contractionwhich is not a contraction

* *, 0 ,td a t a d a a


General Convergence Theorem

A synchronous contraction mapping also converges asynchronously


Standard Interference Function

Conditions Suppose d:AA and d satisfies:

– Positivity: d(a)>0– Monotonicity: If a1a2, 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. 1341-1347. F. Berggren, “Power Control, Transmission Rate Control and Scheduling in Cellular Radio Systems,” PhD Dissertation Royal Institute of Technology, Stockholm, Sweden, May, 2001.


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.

1ˆ jk k

j jj

p p

jj j

jkj j j

k N

g p

g p N


Example steady-state solution

Consider Standard Interference Function

1ˆ jm m

j jj

p p

\1

ˆ mj kj k j

k N im mj j m

jj j

g p

p pKg p

* *

\

ˆ jj kj k j

k N ijj

p g pKg

11 1 12 1

21 22 2*

1 2

1 2

ˆ/

ˆ/

ˆ/

n

n

n n nn n

Kg g g

g Kg

g g Kg

p


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


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.


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

0lim T k

k

P


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.


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


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.

a0 a1 a2 a3 a4 a5


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

' ab

Q RP

0 I

1 N I Q

B NR


Two-Channel DFS

(f1,f1) (f1,f2)

(f2,f2)(f2,f1)

0.25

0.25

0.25

0.25

0.25

1

1 0.25

0.25

0.25

(f1,f1) (f1,f2)

(f2,f2)(f2,f1)

0.25

0.25

0.25

0.25

0.25

1

1 0.25

0.25

0.25

0.250.250.250.25(f2,f2)

0100(f2,f1)

0010(f1,f2)

0.250.250.250.25(f1,f1)

(f2,f2)(f2,f1)(f1,f2)(f1,f1)

0.250.250.250.25(f2,f2)

0100(f2,f1)

0010(f1,f2)

0.250.250.250.25(f1,f1)

(f2,f2)(f2,f1)(f1,f2)(f1,f1)

P =

1.50.5(f2,f2)

0.51.5(f1,f1)

(f2,f2)(f1,f1)

1.50.5(f2,f2)

0.51.5(f1,f1)

(f2,f2)(f1,f1)

N =0.50.5(f2,f2)

0.50.5(f1,f1)

(f2,f1)(f1,f2)

0.50.5(f2,f2)

0.50.5(f1,f1)

(f2,f1)(f1,f2)

B =

1

1j j

jj j

f fu a

f f

1,

\ 1j j

j j jj j

f u ad f f

f F f u a

Decision Rule

Goal

TimingRandom timer set to go off with probability p=0.5 at each iteration


Analysis Models


Model Steady States


Model Convergence


Model Stability


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


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

traditional approaches to modeling and analysis

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