Traditional Approaches to Modeling and Analysis

1. Traditional Approaches to Modeling and Analysis

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

3. 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. 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. Example: ([Yates_95]) Power control applications

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 0

10. Visualizing Fixed Point Existence • Consider continuous • X compact, convex • Fixed Point must exist 1 f(x) x 1 0

11. 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. y x 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]. Not Convex Convex Convex

12. Closed n-Ball (A filled sphere) Closed Disk (Note, mathematically a disk is just a ball) 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] [0,)

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 0

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 0

17. Kakutani’s Fixed Point Theorem • Let f :X X be a upper semi-continuousconvex 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 0 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 0 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. Stable, but not attractive Stability 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. 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.

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. 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 Markov Chains

39. General Insights ([Stewart_94]) • Probability of occupying a state after two iterations. • Form PP. • Now entry pmnin the mth row and nth column of PPrepresents the probability that system is in state antwo 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 antwo 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. [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 Pkhas only nonzero entries (Convergence, steady-state) If ergodic, then chain has a unique limiting stationary distribution. Ergodic Markov Chain

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 a4 a3 a1 a2 a0

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 amis given by tmwhere 1 is a ones vector • t=N1 • (Final distribution) Probability of ending up in absorbing state am given that the system started in akis bkmwhere

45. Decision Rule Goal Two-Channel DFS 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