Dynamic Spectrum Management: Optimization, game and equilibrium Tom Luo Yinyu Ye December 18, WINE 2

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Dynamic Spectrum Management Optimization, game
and equilibriumTom Luo (Yinyu Ye)December 18,
WINE 2008
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Outline

• Introduction of Dynamic Spectrum Management (DSM)
• Social Utility Optimization
• Noncooperative Nash Game
• Competitive Spectrum Economy
• Pure exchange market
• Budget Allocation
• Channel Power Production
• The objective is to apply algorithmic
  game/equilibrium theory to solving real and
  challenging problems

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Dynamic Spectrum Management

• Communication system
• DSL, cognitive radio, cellular networks, cable TV
  networks,
• Multiple users (each has a utility function)
  access multiple channels/tones
• 2/3 allocated spectrum is not being used at any
  given times
• An efficient spectrum management scheme is
  needed

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Spectrum Allocation Problem

• Model
• Each user i has a physical power demand
• Each channel/tone j has a power supply
• maximize system efficiency and utilization

power allocation
D1
D2
D3
. . .
channel
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Shannon Utility Function
xij the power allocation to user i on channel
j x-bari power allocations to all users other
than i ?ij the crosstalk ratio to user i on
channel j aikj the interference ratio from user
k on channel j They may time varying and
stochastic

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Spectrum Management Models

• From the optimization perspective, the dynamic
  spectrum management problem can be formulated as
• 1. Social utility maximization
• May not optimize individual utilities
  simultaneously
• Generally hard to achieve
• 2. Noncooperative Nash game
• May not achieve social economic efficiency
• 3. Competitive economy market
• Price mechanism proposed to achieve social
  economic efficiency and individual optimality

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1. Social Utility Maximization

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Social Utility Maximization- a two user and two
channel example
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Difficulty of the problem

• Even in the two user case, the problem is
  NP-hard.
• No constant approximation algorithm even for one
  channel and multiple users.
• Problems under the Frequency Division Multiple
  Access (FMDA) policy can be solved efficiently
• Luo and Zhang 2007

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2. Noncooperative Nash Game

• Model
• Each user maximize its own utility under a
  physical power demand constraint
• Ciofi and Yu 2002, etc.

D1
D2
D3
power allocation
. . .
channel
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Individual rationality

The basic game assumes that there is no limit on
power supply for each channel. IWF iterative
water filling algorithm converges in certain cases
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Spectrum Nash Game- the same toy example
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Results on the problem

• No bound on price of anarchy
• Can be solved as finding solution of a linear
  complementarity problem, so that its PPAD hard
  in general
• There is a FPTAS under symmetric interference
  condition
• There is a polynomial time algorithm under
  symmetric and strong weak interference condition
• Key to the proofs the LCP matrix is symmetric
• Luo and Pang 2006, Xie, Armbruster, and Y 2008

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3. Competitive Spectrum Market

• The problem was first formulated by Leon Walras
  in 1874, and later studied by Arrow, Debreu, and
  Fisher, also see Brainard and Scarf.
• Agents are divided into two categories seller
  and buyer.
• Buyers have a budget to buy goods and maximize
  their individual utility functions sellers sell
  their goods just for money.
• An equilibrium is an assignment of prices to all
  goods, and an allocation of goods to every buyer
  such that it is maximal for the buyer under the
  given prices and the market clears.

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Market Equilibrium Condition I

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Market Equilibrium Condition II

• Physical Constraint
• The total purchase volume for good j should not
  exceed its available supply

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Market Equilibrium Condition III

• Walras Law

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Whats the budget in DSM?
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3.1 Competitive Equilibrium in Spectrum Economy
for Fixed Budget and Power Supply
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Competitive Spectrum Economy

• Model
• Each user buys channel powers under her budget
  constraint and maximize her own utility
• Price control goal
• Avoid congestion
• Improve resource utilization

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Problem Formulation

• m users, each has a budget wi
• n channels, each with power capacity sj
• Design variable
• xij Power allocation for i th user in jth
  channel
• pj Price for j th channel (Nash Equilibrium
  pj1 fixed)
• User utility (Shannon utility function )

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Competitive Equilibrium Model
Theorem A competitive equilibrium always exists
for the spectrum management problem
Y 2007 based on the Lemma of Abstract Economy
developed by Debreu 1952
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Equilibrium Properties

• Every channel has a price
• All power supply are allocated
• All budget are spent

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Weak-Interference Market

• Weak-interference environment the Shannon
  utility function of user i is
• In the weak-interference environment,
• An equilibrium can be computed in polynomial
  time.
• The competitive price equilibrium is unique.
  Moreover, if the crosstalk ratio is strictly less
  than 1, then the power allocation is also unique.
  (Y 2007)

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Two methods of solving competitive equilibrium

• Centralized
• Solving the equilibrium conditions
• Decentralized
• Iterative price-adjusting

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Competitive Equilibrium Model- the same toy
example
power allocation
power allocation
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Computational Results

• Compare competitive equilibrium and Nash
  equilibrium
• Evaluate the performance in
• Individual utility and Social utility
• In most cases, CE results in a channel allocation
  
• Have a higher social utility value
• Make more users achieve higher individual
  utilities

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3.2 Budget Allocation in Competitive Spectrum
Economy
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Lin, Tasi, and Y 2008
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Budget Allocation in Competitive Spectrum Economy

• Budget allocation aims to
• satisfy a minimum physical power demand di for
  each user i
• or
• satisfy a minimum utility value ui for each user
  i e.g., all users achieve an identical utility
  value
• Theorem Such a budget equilibrium always exists.

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Two methods of solving competitive equilibrium

• Centralized
• Solving entire optimal conditions which may be
  nonconvex
• Decentralized
• Iterative budget-adjusting

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Budgeting for demand- computational results

• Number of (budget-adjusting) iterations required
  to achieve individual power demands

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Budgeting for demand- computational results

• Number of iterations and CPU time (seconds)
  required to satisfy individual power demands in
  large scale problems, error tolerance0.05

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Budgeting for demand- CE and NE comparison
results

• General cases background noise randomly selected
  from (0,m, crosstalk ratio randomly selected
  from 0,1
• In all cases, the social utility of CE is better
  than that of NE.

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Budgeting for demand- More CE and NE comparison
results

• In special type of problems, the competitive
  equilibrium performs much better than the Nash
  equilibrium does.
• For instance, the channels being divided into two
  categories high-quality and low-quality.
• (In simulations, one half of channels with
  background noise randomly selected from the
  interval (0 0,1 and the other half of channels
  with background noise randomly selected from the
  interval 1m.)

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Budgeting for demand- More CE and NE comparison
results

• Two-tier channels
• CE with power demands v.s. NE

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Budget allocation to balance utilities -
Computational results

• Number of iterations and CPU time (seconds)
  required to balance individual utilities in large
  scale problems, difference tolerance0.05

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Budgeting to balance utilities - CE and NE
comparison results

• Two-tier channels
• CE with balanced utilities v.s. NE

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Comparison result summaries

• Compare with NE, in most cases, CE with minimum
  power demands results in power allocation
• Have a higher social utility
• Compare with NE, in most cases, CE with balanced
  utilities demands results in a power allocation
• Have a higher social utility
• Make more users have higher individual utilities
• Have a smaller gap between maximal individual
  utility and minimal individual utility
• In special type of problems, for instance, two
  tiers of channels, CE performs much better than
  NE does.

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3.3 Channel Power Production in Competitive
Spectrum Economy
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Produce power supply to increase social utility
the same toy example
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Future Work

• How to systematically adjust channel power supply
  capacity to increase social utility?
• The convergence of the iterative
  variable-adjusting method for general setting
• Real-time spectrum management vs optimal policy
  at top levels