OFDM

1
OFDM

• James Neel
• July 26, 2005

2
Overview

• Models
• Case Studies
• Realistic Numbers
• A Theoretical Conjecture

3
General System Model

• Single access point
• Equal receive power at access point
• Distributed channel assignment

4
Basic Game Model

• N - set of decision making links
• C - set of available channels
• Action Set
• 2C
• fC(k) link throughput given k users of channel
• Utility
• Some function of throughput

5
General Congestion Game

• Components
• A finite set of actors (players), N 1,2,,n
• A set of facilities, F1,2,,g
• A set of payoffs, cf k where f ? F and k is the
  number of users of facility f
• Action Sets
• Ai 2F

• Utility Functions
• Non wireless apps
• Club games
• Traffic routes
• Exact Potential
• For verification of EPF, see potential game
  writeup

6
What Game Theory Buys Us

• Potential Function
• Steady-state identification (Maximizers of V(a))
• Convergence for better response
• Genetic algorithms
• Trial and error improvement
• Greedy
• Stable (Asymptotic)
• Only local, numerous fixed points

7
Case Study 1 Throughput Maximization

• Utility
• Leads to possible over-provisioning
• All channels occupied by all radios if cf gt0
• Possible first mover advantage if not all cf gt0
• Definite over provisioning if cf gt0 and

(Specifically a tragedy of the commons)
8
Case Study 2 Channel Limited Throughput
Maximization

• Action Set Constrained to L or less channels
• Utility
• Possible over-provisioning
• Grab up to L channels
• Same conditions for over-provisioning as before
• Judicious choice of L could combat this problem

9
Case Study 3 Costly Channels

• Utility
• General implications
• Steady state very sensitive to Ki
• No channels used to all channels used

Power/complexity cost
10
Case Study 4 Hard Limited Throughput

• Utility
• General implications
• Steady state very sensitive to initial conditions
• Starting above Ti no improvement path below Ti

(Generalized Ordinal Potential)
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Case Study 5 Saturated Throughput

• Utility
• General implications
• Steady state less sensitive to initial conditions
• Back to possible overprovision

(Ordinal Potential)
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Case Study 6Saturated Costly Throughput

• Utility
• Potential??
• NE exists in mixed strategies

13
Case Study 7Costly Hard Limited Throughput

• Utility
• Potential? Should be one.

14
Case Study 8 Costly Channel Limited Throughput
Maximization

• Action Set Constrained to L or less channels

15
Comments on Case Studies

• All readily implementable
• Hard limited throughput, saturated throughput,
  costly channels, and costly limited channels may
  be most realistic scenarios.
• Costly limited channels most powerful for a
  network designer to work with
• Could assign different classes of users different
  values of L
• Orthogonal frequencies between different users is
  not be a good assumption unless synchronized.

16
Some Realistic Numbers

• 802.11a (MAC wouldnt permit this operation, but
  does give hints towards appropriate modulations)
• cf(0) 0
• cf(1) 18/4 B (64 QAM, r 3/4)
• cf(2) 1/2 B (BPSK, EbN0 0 dB)
• B log2(1 S/N) (total channel capacity B)
• Spec says r 1/2 , but youre better off without
  coding
• Noise isnt really random
• cf(3) 0 (Approximately, theoretical
  capacity/link 0.13 B which means in practice
  should be 0 unless massive repetition coding.
  Radios could be programmed to believe that its
  identically 0)

17
Real Number Implications

• For Costly Channels, a Ki gt 1/2 B will prevent
  double channel occupancy.
• Since cf(k) ? 0 for k gt 2, throughput potentials
  can be rewritten as

Suspect this will be useful for hard limited
throughput
18
Random Theoretical Conjecture

• Games that are Iterated Elimination of Dominated
  Strategies (IEDS) solvable clearly have a
  generalized ordinal potential (otherwise there
  would be improvement cycles that couldnt be
  eliminated.)
• Conjecture
• IEDS solvable games also have a generalized
  synchronous potential, i.e., V such that when all
  players simultaneously play a better response, V
  increases.
• If so, game would also have a generalized
  asynchronous potential by Bertsekass general
  convergence theorem