Energy-Efficient Resource Management Games in Wireless Networks - - PowerPoint PPT Presentation

Energy-Efficient Resource Management Games in Wireless Networks

Majed Haddad Yezekael Hayel, Piotr Wiecek and Oussama Habachi INRIA Sophia-Antipolis CEFIPRA Workshop 2014 January 15, 2014

Roadmap

• The energy efficiency framework
• Related work
• Characterization of the Stackelberg equilibrium
• Performance results
• Conclusion

1

The energy-efficiency – spectral efficiency trade-off

• Recent trends in mobile client access recognize energy efficiency as an

additional constraint for realizing efficient and sustainable computing.

• Achieving a high SINR level requires the user terminal to transmit at a high

power, which in turn results in low battery life.

1 2 3 4 5 6 7 8 9 10 5 10 15 20 25 30 35 Spectral efficiency [bps/Hz] Energy efficiency [bits/Joule] SNR = 5 dB SNR = 10 dB SNR = 15 dB SNR = 20 dB

Figure 1: The energy-efficiency – spectral efficiency trade-off for different SNR values: Very energy-efficient communications are typically not efficient spectrally.

2

The energy efficiency utility

• The energy efficiency (EE) of a communication between a transmitter and a

receiver is defined as the ratio of the the transmission benefit (in bit/s) to the cost (in Watts)1: EE = net data rate radiated power (in bit/Joule) (1)

• This definition translates formally as

un(p1, . . . , pN) = Rnf(γn) h(pn) (2) where pn is the power, Rn is the transmission rate and γn is the SINR of user n.

• 1D. J. Goodman and N. B. Mandayam, ”Power Control for Wireless Data”, in IEEE Personal Communications,
• Vol. 7, No. 2, pp. 48–54, April 2000.

3

The efficiency function f

1 2 3 4 5 6 7 8 9 10 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 γ f(γ)

Data transmission becomes error−free and the throughput grows asymptotically to a constant Data transmission results in massive errors and the throughput tends to 0

Figure 2: A typical efficiency function f representing the packet success probability as a function of received SINR.

4

The power radiated function h

• The total power consumption of the whole transmit device includes

the computation power, the circuit power (mainly consumed by the power amplifier),

• The power radiated function is given by

hn(pn) = apn + b, a ≥ 0, b ≥ 0

20 40 60 80 100 0.05 0.1 0.15 0.2 0.25 0.3 0.35 Radiated power level [Watts] Energy Efficiency [bit/Joule] b=9.6 Watts (Femto Cell) b=260 Watts (Macro Cell) b=103 Watts (Pico Cell) without circuit power, b=0

Figure 3: EE is typically a quasi-concave function of the radiated power and has a single maximum point.

5

System model

BS

PU SU2 SU4 SU1 SU3

: Signal : Interference

Figure 4: An example of an uplink communication wireless network.

6

The single carrier model

• The EE utility is given by

un(p1, . . . , pN) = Rnf(γn) pn (3) where γn = pngn

• j=n pjgj + σ2.

(4) pn and gn are resp. the power and the channel gain of transmitter n.

7

The non-cooperative (Nash) game problem for single carrier systems

• When it exists, the Nash equilibrium of this game is given by2:

pNE

n

= σ2 gn γ∗ 1 − (N − 1)γ∗, ∀n ∈ {1, ..., N} (5) where γ∗ is the positive solution of the equation xf ′(x) = f(x). (6)

• This type of equation has a positive solution if the function f is sigmoidal3.
• 2C. U. Saraydar, N. B. Mandayam and D. J. Goodman, “Pricing and Power Control in a Multicell Wireless Data

Network”, IEEE Journal on Selec. Areas in Comm., Vol. 19, No. 10, pp. 1883-1892, 2001.

• 3V. Rodriguez, “An Analytical Foundation for Resource Management in Wireless Communication”, IEEE Proc.
• f Globecom, 2003.

8

The hierarchical (Stackelberg) power control game problem

Definition 1. (Stackelberg equilibrium): A vector of actions p = ( pl, pf) is called Stackelberg equilibrium (SE) if and only if:

• pl = argmax

pl UL(pl, pf(pl)),

where ∀pl, pf(pl) = argmax

pf

UF(pl, pf), and pf = pf( pl).

9

The hierarchical (Stackelberg) equilibrium for single carrier systems (K = 1)

• There is a unique Stackelberg equilibrium (pSE

n , pSE −n) in the hierarchical

game4: pSE

n

= σ2 gn γ∗(1 + γ∗) 1 − (N − 1)β∗γ∗ − (N − 2)γ∗ (7) where γ∗ is the positive solution of the equation xf ′(x) − f(x) = 0, β∗ the positive solution of the equation: x

• 1 −

(K − 1)γ∗ 1 − (K − 2)γ∗x

• f ′(x) − f(x) = 0
• 4S. Lasaulce, Y. Hayel, R. El Azouzi, and M. Debbah, ”Introducing Hierachy in Energy Games”, IEEE

Transaction on Wireless Communication, vol. 8, no. 7, pp. 3833-3843, July 2009.

10

The multi-carrier model (K ≥ 2)

• The utility function over the K carriers is:

un(p1, ..., pN) = Rn ·

K

• k=1

f(γk

n) K

• k=1

pk

n

. where Rn and γk

n are respectively the transmission rate and the SINR of user

n over carrier k defined by γk

n =

gk

npk n

σ2 +

M

• m=1

m=n

gk

mpk m

:= pk

n

hk

n

11

The two-user case

PU SU BS BS: Base Station PU: Primary User SU: Secondary User p1, g1 p2 , g2

Figure 5: An example

• f

a cognitive radio network comprising an uplink primary communication (PU→BS) and a secondary communication.

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Characterization of the Stackelberg equilibrium: The follower’s power allocation vector

Prop 1. Given the power allocation vector p1 of the leader, the best-response

• f the follower is given by5:

pk

2(p1) =

   γ∗(σ2 + gk

1pk 1)

gk

2

, for k = L2(p1), 0, for all k = L2(p1) (8) with L2(p1) = argmax

k

• hk

2(pk 1) and γ∗ is the unique (positive) solution of the first

• rder equation:

x f ′(x) = f(x) (9)

• Equation (9) has a unique solution if the efficiency function f(·) is sigmoidal.
• 5F. Meshkati, M. Chiang, H. V. Poor and S. C. Schwartz, “A game-theoretic approach to energy-efficient power

control in multi-carrier CDMA systems”, IEEE Journal on Selected Areas in Communications, Vol. 24, No. 6,

• pp. 1115–1129, June 2006.

13

Characterization of the Stackelberg equilibrium: The leader’s power allocation vector for K = 2

Prop 2. At the Stackelberg equilibrium, the leader and the follower transmit

• ver distinct carriers if and only if:

1 1 + γ∗ < g1

2

g2

2

< 1 + γ∗. (10) where γ∗ is the unique (positive) solution of the first order Equation (9).

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Characterization of the Stackelberg equilibrium: The leader’s power allocation vector for K = 2

Prop 3. At the Stackelberg equilibrium, when the channel gains of the follower satisfy (10), the power control vector p1 for which the leader’s utility is maximized is unique and is given by

• pk

1 =

   σ2γ∗ gk

1

, for k = k, 0, for all k = k (11) where γ∗ is the unique (positive) solution of the first order Equation (9) and k denotes the ”best” carrier of the leader, i.e. k = argmax

k

gk

1.

15

Stackelberg equilibrium regions

[SU,PU] [SU,PU]

[SU,PU]

[SU,PU]

[PU,SU]

[PU,SU]

[PU,SU] [PU,SU]

g21/g22 g11/g12 1/(1-γ*) 1+γ* 1-γ* 1-γ* 1/(1-γ*) 1/(1+γ*) 1

[*,PU SU] [PU SU,*]

[SU,PU] [PU,SU] [PU,SU] [PU,SU]

Figure 6: Stackelberg equilibrium regions for the case of two users and two carriers. The point [SU, P U] means that the PU transmits over the second carrier and the SU over the first one. Contrary to the result obtained in the non-cooperative game in Meshkati’s paper, the proposed Stackelberg model always admits an equilibrium.

16

EE as function of the SNR

Figure 7: Energy efficiency at the equilibrium as function of the SNR for different schemes.

17

Learning Algorithm

• Determining the equilibrium strategy of both users requires the knowledge
• f several information that can not be observed in a realistic scenario.
• We propose a learning-based algorithm that allow users to determine their

strategies on-the-fly.

• The PU maintains the state-value function q(g, p) as a lookup table, which

determines the optimal action to chooses in the current time slot: q(gt−1, pt−1) ← βtq(gt−1, pt−1) (12) +(1 − βt)(u1 + γq(gt, pt)),

• The SU chooses its action following state-value function Q(g, p):

Q(gt−1, pt−1) ← αtQ(gt−1, pt−1) (13) +(1 − αt)(u2 + γQ(gt, pt)),

18

Learning the Stackelberg equilibrium

Figure 8: The energy efficiency at the Stackelberg equilibrium for both PU and SU.

19

The probability of no coordination for K ≥ 2

Prop 4. The probability that there is no coordination between the players is bounded above by (1 + γ∗)B(1 + γ∗, k) ∼ O(K−(1+γ∗)) where B denotes the Beta function, which is the exact probability of no coordination in the simultaneous-move version of the model.

2 3 4 5 6 7 8 9 10 0.02 0.04 0.06 0.08 0.1 0.12 Number of carriers Probability of no coordination Simulations Analytical upper−bound

Figure 9: The probability of no-coordination between the leader and the follower as function of the the number of carriers.

20

EE as function of the number of carriers for K ≥ 2

2 3 4 5 6 7 8 9 10 11 12 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 Number of Carriers

Average Energy Efficiency at equilibrium [Mbits/Joule]

Stackelberg PU Stackelberg SU Sense PU Sense SU Nash PU Nash SU

Figure 10: Energy efficiency at the equilibrium as function of the number of carriers.

21

The sum spectral efficiency for K ≥ 2

Prop 5. The sum spectral efficiency in case there is a coordination between the users is strictly bigger than log(1 + γ∗)(1 − (1 + γ∗)B(1 + γ∗, K)) which is equal to the spectral efficiency in the simultaneous-move game.

2 4 6 8 10 12 14 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 Number of carriers Sum spectral efficiency [bps/Hz] Simulated SE Theoretical lower−bound

Figure 11: Sum spectral efficiency as function of the the number of carriers.

22

Conclusion

• We have analyzed the impact of hierarchically allocating transmit power in

energy efficiency wireless networks.

• Contrary to the result obtained in the non-cooperative game in Meshkati’s

paper, the proposed Stackelberg model always admits an equilibrium.

• We have shown that, under some assumptions, introducing a certain degree
• f hierarchy in a multi-carrier system induces a natural coordination pattern.
• For implementation purposes in a cognitive radio network, the secondary

user (follower) has only to reliably sense the spectral environment and not the primary user’s transmit power and then decides to transmit only on the best carrier left idle by the primary user (leader).

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Last Slide Thank you....

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