0 wireless foundations 0july 7, 2004 distributed optimization of power allocation in interference...

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Wireless Foundations 1July 7, 2004

Distributed Optimization of Power Allocation in Interference Channel

Raul Etkin, Abhay Parekh, and David Tse

Spectrum Sharing Group - University of California, Berkeley

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Problem: Spectrum Sharing

Can multiple heterogeneous wireless systems coexist and share spectrum in a flexible and efficient manner?

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Project Goals

Find strategies to allow multiple systems to share spectrum efficiently

Algorithms must be

– Distributed

– Simple and require minimum cooperation/communication between systems

– Robust to non-complying systems

– Fair

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Interference Channel Model

M systems. Will consider M small.

Bandwidth W

Flat fading assumption: hi,j constant over frequency

System m can use total power Pm

Define

Background noise AWGN with PSD N0

Can have 1Tx-Rx pair per system, or many of them.

1

2

M

1

2

M

Tx Rx...

.

.

.

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Distributed Power Allocation Problem

User m allocates its power with power spectral density pm(f)

Noise plus interference seen by user m at frequency f:

Rate of system m with Gaussian interference assumption:

Want to maximize in a distributed way for some utility function f(.)

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Algorithmic vs. Game Theoretic Formulations

Algorithmic formulation: systems cooperate by following a distributed power allocation algorithm.

Game Theoretic formulation: systems are selfish and want to maximize their own utility function.

– Systems do not cooperate with each other, and are not willing to follow any algorithm that does not maximize their own utility.

– Can use game theory to find “good” Nash Equilibria (N.E.).

– Depending on assumptions can have different game formulations.

Some confusion in literature: papers analyze N.E. of a static game of complete information, and apply dynamic algorithms to achieve them.

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Algorithmic Approach: Iterative Water-filling

Analyzed Iterative Water-filling Algorithm (IWFA): each system updates its power allocation performing water-filling over noise+interference seen.

– Algorithm motivated by static Gaussian interference game: water-filling is the best response of each system to the other systems’ strategies in the static game

– Advantages: distributed, simple, robust, no feedback across systems.

– Disadvantages: may not converge, may converge to different Nash equilibria (when N.E. is not unique), may converge to a configuration with large price of anarchy.

Insights gained with IWFA analysis: Algorithms should …

– take into account N0 to achieve a bounded price of anarchy when N0 0.

– take into account interference created over other systems.

– “look into the future” instead of greedily maximizing immediate rate.

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Algorithmic Approach: Other algorithms

Orthogonal power allocations have bounded price of anarchy.

– Rose and Popescu propose a (flawed) distributed orthogonalization algorithm. Each system gets 1/M of the total bandwidth.

– We modified (and fixed) algo. to assign to user m a fraction of the total bandwidth.

– Advantages: fast convergence, fair, good performance for small N0 and or large cross gains ci,j, does not require knowledge of channel gains.

– Disadvantages: requires some minimal synchronization/timing, requires constant monitoring of spectrum, needs to start all over again if a new system appears.

Future research: find other distributed algorithms with better features than the 2 analyzed. May allow for limited communication across systems to improve performance.

Question: how to allow for minimal communication among systems without making the problem trivial.

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Game Theoretic Approach

Regardless of the game formulation, need to find Nash Equilibria. Why ?

Nash equilibrium strategies are self enforcing. No need to have the regulating agency monitoring the spectrum everywhere to enforce them.

Basic assumption: there is an initial stage where systems agree on following a given N.E. If each system believes that the others will comply with the agreement, its best response is also to comply.

Can have different formulations depending on the way the game is defined and the information that each player has.

– Static / Dynamic game: 1 stage-simultaneous move / multi-stage game.

– Complete / Incomplete information: all parameters are known / some not known to players.

– Perfect / Imperfect information: players know the actions taken by other players when they choose their actions / don’t know some of the actions.

Literature on spectrum sharing only considers static games of complete information. May need more general formulation for real world problems.

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Static Game of Complete Information

Case M=2: we have complete characterization of N.E.

– If N.E. (full spread) is unique.

– If there are 1 N.E. (full spread + orthogonal) Case M > 2:

– Full spread power allocations are always N.E.

– We obtained sufficient condition for uniqueness of full spread N.E.:

(more general than Cioffi’s result)

– We obtained necessary and sufficient condition for the existence of orthogonal N.E. (nice proof involving network flows):

– Many more N.E. besides full spread and full orthogonal

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Dynamic Game of Complete Information

Stage game of static model repeated indefinitely.

Sequences of N.E. of static game are N.E. in dynamic game.

Can obtain new N.E. through believable threats.

– Each systems threatens to apply a punishment if the other system does not comply with a given rule.

– For threats to be believable, the punishment should be a “bad” N.E.

Analyzed punishment strategies for M=2, c1,1=c2,2=1.

– Strategy: allocate power over a BW W1 as long as the other system stays orthogonal. Otherwise spread the power over the whole BW forever (punishment).

– Questions: when is this strategy a N.E. ? When it is not, do we lose much by not being able to use this kind of strategy ?

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Punishment Strategies

Can apply punishment

strategy


strategy


strategy


strategy

Spreading better for both systems

Region where can apply punishment strategy (green) larger for high SNR, where full-spread N.E. has large price of anarchy.

In most cases where orthogonal N.E. is preferable but cannot be achieved through punishment strategies, performance loss is

small.

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Punishment Strategies: what if ci,j unknown ?

Complete information formulation assumes knowledge of {ci,j}

Can apply punishment strategies when user i does not know ci,j ?

– ci,j needed to determine the W1 that maximizes the minimum rate.

– Can set W1=W/2 and apply the new strategy: start with orthogonal W/2 P.A.. If full spread allocation results in better rate, or the other system departs from orth. P.A., apply full spread allocation forever.

– Resulting region slightly smaller than green region

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Games of Incomplete Information

Each system is randomly assigned a type when the game starts.

– Can think of type for system m as {cj,m: j m}

Each system knows its own type, but does not know the types of other systems.

Model may allow messaging across systems to inform about types, but systems may lie (cheap talk game).

Each system’s best response is computed by maximizing the expected utility over the posterior distribution of the other systems’ types, and some belief about the other systems’ strategies.

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Static Game of Incomplete Information

Analyzed simple model:

– M=2

– Binary action space {a1, a2}:

a1 = full-spread power allocation

a2 = orthogonal W/2 power allocation

– c1,1=c2,2=known constant

– ci,j ~ Exp(1)

Computed expected rates for different values of c1,1=c2,2

When multiple Bayesian N.E. exist, choose the one with largest average rates.

Compared performance to that of the equivalent game with complete information.

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Static Game of Incomplete Information (cont.)

0.5 1 1.5 2

1.4

1.6

1.8

0.5 1 1.5 2

0.6

0.8

1.2

1.4

N0=0.01

N0=0.1

0.5 1 1.5 2

0.2

0.4

0.6

N0=1

Complete information

Incomplete information

c1,1=c2,2

c1,1=c2,2

c1,1=c2,2

E[R]

E[R]

E[R]

Lack of knowledge of the other system’s type is helpful for small ci,i

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Dynamic Game of Incomplete Information

Want to analyze when communication across systems is beneficial.

Structure of Game:

– Player 1 sends a message to player 2 about its type

– Player 2 chooses an action based on the message and its own type.

– Player 1 chooses an action based on player 2’s action and its own type.

Fixed c1,1=c2,2 and chose ci,j to have binary and exponential distribution.

Limited action space to {a1,a2} as defined before

Depending on parameter choices observed situations where communication is beneficial, and situations where Player 1 has no incentive in communicating its type.

Player 1 prefers not to send message when user 2 is likely to play orthogonal and Player 1 can exploit its advantage of playing later by playing full-spread.

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Games of Incomplete Information: what’s next ?

All models analyzed so far had

– M=2

– Binary action space {a1,a2}

– 1 or 2 stage game

Results depend on parameter choices and modeling assumptions.

Changing any of these assumptions would make the models more realistic but math becomes significantly more involved.

Seems hard to obtain generalizations that are mathematically tractable and resemble practical systems.

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Conclusions and Research Direction

If we can rely on systems to follow a specified algorithm, then algorithmic approach seems viable.

– May not be possible to obtain very good performance unless explicit or implicit communication is allowed.

– If we allow for communication, then problem becomes trivial.

– Challenge: find interesting problem formulation allowing for some communication among systems.

Can obtain good performance in dynamic game of complete information through punishment strategies. Could extend approach for M>2.

Analysis of games of incomplete information is hard, and results are very dependent on model assumptions and parameter choices.

Hope to get more insight into modeling assumptions and parameter choices by analyzing the case of multiple 802.11-type systems sharing spectrum. In 802.11 case:

– Each system contains multiple transceivers, which access the medium in a TDMA fashion.

– Since each terminal transmits from time to time, it may be possible to use reciprocity to justify complete information assumption.

– New problems/challenges may arise due to the larger number of terminals in each system.

0 wireless foundations 0july 7, 2004 distributed optimization of power allocation in interference...

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