Distributed Power Allocation Game for Uplink OFDM Systems

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Distributed Power AllocationGame for Uplink OFDM Systems

Gaoning He, Sophie Gault - MOTOROLA LABSMérouane Debbah - SUPÉLEC

Eitan Altman - INRIA INSTITUTE

Motorola confidential proprietaryMOTOROLA and the Stylized M Logo are registered in the US Patent & Trademark Office. All other product or service names are the property of their respective owners. © Motorola, Inc. 2005

2

Outline

1. System model

2. Research background

3. Distributed power allocation (DPA) game


3

System model

• System settings– The uplink of OFDM system in a single cell– K users (Tx) are communicating with BS (Rx) using N subcarriers– Transmitters (Tx) have partial (or incomplete) channel state information (CSI).


4

System model

• A simplified model (MAC)– K=2, N=1.– gk(t) is the uplink channel gain of user k.– Assume gk(t) has finite discrete channel states, following a certain distribution.

• Different channel state information (CSI) levels at Tx:– Complete CSI– Partial CSI– Statistical CSI

MT1MT2

1g t

BS

2g t


5

System model

• Individual rationality → optimization problem with selfish objective– Channel gain matrix , where each element is gk, i

– Selfish objective of user k (average capacity)

– Average power constraint of user k

,1

N

k i ki

p P

GE

, ,2

21, ,

1,

log 1N

k i k ik K

ij i j i

j j k

p gC

p g

GE

K NG R

, 0, k ip i


6

Research background

• Complete CSI at Tx → The waterfilling game [1]– Optimization problem for (selfish) user 1 (which is similar for user 2):

– Results in an interesting equilibrium (individual rationality=social rationality):

• Statistical CSI at Tx → Uniform power allocation (in the time domain)

1 21 1 2

1 2

1 1 2 1, 2 2( , )

2 1 2 2

, 1 1 2 1 1 1 2

( , )log 1

( , )

, & , 0

g gp g g

g g

p g g gmax

p g g g

s.t. p g g P p g g

E

E

22

1 1 2 1 1 21 1

21

2 1 2 2 2 12 2

, , when

, , when

p g g g gg

p g g g gg

1 2

1 2

22

, 1 1 2 11 1

21

, 2 2 1 22 2

g g

g g

g g Pg

g g Pg

E

E

1 2 where and

are decided from


7

Research background

• Partial CSI at Tx → Distributed power allocation (DPA) game– The optimization problem for user 1, maximizing its own average capacity C1

(and similarly for user 2):

– From Lagrangian duality, user 1’s best strategy p1(.) is given by equation:

– Note that user 2’s strategy p2(.) is unknown to user 1, and user 1’s strategy can not depend on the particular realization of g2.

1 21 1

1

1 1 11 , 2 2( )

2 2 2

1 1 1 1 1

( )log 1

( )

& 0

g gp g

g

p g gmaxC max

p g g

s.t. p g P p g

E

E

2

12

1 1 1 2 2 2 1

1g

g

p g g p g g

E

11 1 1 1 g p g P E,where is chosen such that


8

DPA game

• A non-cooperative game model

– Player set: – Action set: – Utility (payoff) function set:

• Intuitively, u1,u2 are defined as the average capacity of user 1 and 2

, ,k kk ku

K KK Pu

1,2 , 2.K K Kwhere

1 2 1 1 1 1 2 2 2 1 2 2, , , , , .p a p a p a p a PuPu Pu Puwhere

1 2 1 1 2 2 1 2, , ( , ) ( , ).u u u uPuPu PuPudenoted by and


9

DPA game

• What the game brings us?– Each user can decide his “best strategy” relying on the assumption of rationality

of the other user.– Results in Nash equilibrium (NE): a “stable” situation, where both (selfish) users

are satisfied with the choices of their strategies , if given any other strategies

– The social performance of NE might be inefficient.

• Equilibrium analysis [2]-[5]– Theorem 1: A NE exists in the game.– Theorem 2: The game has a unique equilibrium. (proofs refer to the paper)

• Can we improve the performance of NE?

1 1 2 1 1 2

2 1 2 2 1 2

, ,

, ,

u u

u u

P P P P

P P P P

1 2, Pu Pu 1 2,PuPu


10

DPA game

• Improve the performance of NE for the symmetric case– Assume the channel state gk (k=1,2) can be a1 and a2 with probability ρ1 and ρ2

respectively.– Assume both users coordinate to the same strategy set, i.e.

More precisely, . – Assume both users have the same maximum power constraint, i.e. – Then the problem goes to find the optimal strategy set that maximizes the

average capacity for user k.– The best strategy is given by:

1 2 Pu Pu P1 2( ) ( ) ( ), 1, 2p a p a p a

P

1 2,p a p a 2 1 1 2

2

0, , P

a a

2 1 1 21

,0 , P

a a

1 2P P P


11

Numerical results

a1 a2 ρ1 ρ2 P σ2

Left 1 3 0.5 0.5 1 0.1

Right 1 10 0.5 0.5 1 0.1Table of parameters used in the simulation


12

Numerical results


13

Conclusions and future works

• We study the power allocation problem in the uplink of OFDM system with partial knowledge at the transmitters. Most of the preview works on power allocation game was achieved under assumption of complete CSI at the transmitters.

• We introduced a theoretic approach to provide a natural setting for the construction of efficient distributed algorithms.

• We establish the existence and uniqueness of Nash equilibrium• For the symmetric case, we found the efficiency of NE can be improved by

forcing all users coordinating to the same optimal strategy set.• Future work will consider this problem to the case of MIMO and correlated

equilibrium.


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Bibliography

• [1] L. Lai and H. E. Gamal, “The Water-Filling Game in Fading Multiple Access Channels,” IEEE Trans. on Information Theory, Dec. 2005.

• [2] G. Debreu, “A Social Equilibrium Existence Theorem,” in Proc. Nat. Acad. Science, vol. 38, pp. 886893, 1952.

• [3] K. Fan, “Fixed Point and Minimax Theorems in Locally Convex Topological Linear Spaces,” in Proc. Nat. Acad. Sciences, vol. 38, 1952, pp. 121126.

• [4] I. L. Glicksberg, “A Further Generalization of the Kakutani Fixed Point Theorem with Application to Nash Equilibrium Points,” in Proc. Amer. Math. Soc., vol. 3, 1952, pp. 170174.

• [5] D. Fudenberg and J. Tirole, Game Theory. MIT Press, Cambridge, MASS, London, England, 1991.


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Thank you !Questions?

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Distributed Power Allocation Game for Uplink OFDM Systems