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. 35 30 Energy efficiency [bits/Joule] SNR = 5 dB 25 SNR = 10 dB SNR = 15 dB 20 SNR = 20 dB 15 10 5 0 1 2 3 4 5 6 7 8 9 10 Spectral efficiency [bps/Hz] 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 : net data rate EE = (in bit/Joule) (1) radiated power • This definition translates formally as u n ( p 1 , . . . , p N ) = R n f ( γ n ) (2) h ( p n ) where p n is the power, R n is the transmission rate and γ n is the SINR of user n . 1 D. 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 0.9 0.8 Data transmission becomes error−free 0.7 and the throughput grows asymptotically to a constant 0.6 0.5 f( γ ) Data transmission results in massive errors 0.4 and the throughput tends to 0 0.3 0.2 0.1 0 0 1 2 3 4 5 6 7 8 9 10 γ 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 h n ( p n ) = ap n + b, a ≥ 0 , b ≥ 0 0.35 without circuit power, b=0 0.3 Energy Efficiency [bit/Joule] b=9.6 Watts (Femto Cell) 0.25 b=103 Watts (Pico Cell) 0.2 b=260 Watts (Macro Cell) 0.15 0.1 0.05 0 0 20 40 60 80 100 Radiated power level [Watts] Figure 3: EE is typically a quasi-concave function of the radiated power and has a single maximum point. 5
System model SU 2 SU 4 SU 1 BS SU 3 : Signal PU : Interference Figure 4: An example of an uplink communication wireless network. 6
The single carrier model • The EE utility is given by u n ( p 1 , . . . , p N ) = R n f ( γ n ) (3) p n where p n g n � γ n = (4) j � = n p j g j + σ 2 . p n and g n 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 by 2 : = σ 2 γ ∗ p NE 1 − ( N − 1) γ ∗ , ∀ n ∈ { 1 , ..., N } (5) n g n 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 sigmoidal 3 . 2 C. 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. 3 V. Rodriguez, “An Analytical Foundation for Resource Management in Wireless Communication”, IEEE Proc. of Globecom , 2003. 8
The hierarchical (Stackelberg) power control game problem Definition 1. ( Stackelberg equilibrium ): A vector of actions � p = ( � p l , � p f ) is called Stackelberg equilibrium (SE) if and only if: p l = arg max � p l U L ( p l , p f ( p l )) , where ∀ p l , p f ( p l ) = arg max U F ( p l , p f ) , p f and � p f = p f ( � p l ) . 9
The hierarchical (Stackelberg) equilibrium for single carrier systems ( K = 1 ) • There is a unique Stackelberg equilibrium ( p SE n , p SE − n ) in the hierarchical game 4 : = σ 2 γ ∗ (1 + γ ∗ ) p SE (7) 1 − ( N − 1) β ∗ γ ∗ − ( N − 2) γ ∗ n g n where γ ∗ is the positive solution of the equation xf ′ ( x ) − f ( x ) = 0 , β ∗ the positive solution of the equation: � � ( K − 1) γ ∗ f ′ ( x ) − f ( x ) = 0 1 − x 1 − ( K − 2) γ ∗ x 4 S. 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: K � f ( γ k R n · n ) k =1 u n ( p 1 , ..., p N ) = . K � p k n k =1 where R n and γ k n are respectively the transmission rate and the SINR of user n over carrier k defined by g k n p k n � γ k n := p k h k n = n M � σ 2 + g k m p k m m =1 m � = n 11
The two-user case p 1 , g 1 p 2 , g 2 BS PU BS: Base Station PU: Primary User SU: Secondary User SU Figure 5: An example of a cognitive radio network comprising an uplink primary communication (PU → BS) and a secondary communication. 12
Characterization of the Stackelberg equilibrium: The follower’s power allocation vector Prop 1. Given the power allocation vector p 1 of the leader, the best-response of the follower is given by 5 : γ ∗ ( σ 2 + g k 1 p k 1 ) k = L 2 ( p 1 ) , , for p k 2 ( p 1 ) = g k (8) 2 0 , for all k � = L 2 ( p 1 ) 1 ) and γ ∗ is the unique (positive) solution of the first � h k 2 ( p k with L 2 ( p 1 ) = arg max k order equation: x f ′ ( x ) = f ( x ) (9) • Equation (9) has a unique solution if the efficiency function f ( · ) is sigmoidal. 5 F. 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 over distinct carriers if and only if: 1 + γ ∗ < g 1 1 2 < 1 + γ ∗ . (10) g 2 2 where γ ∗ is the unique (positive) solution of the first order Equation (9). 14
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 � p 1 for which the leader’s utility is maximized is unique and is given by σ 2 γ ∗ k = � for , k, p k g k � 1 = (11) 1 k � = � 0 , for all k where γ ∗ is the unique (positive) solution of the first order Equation (9) and � k denotes the ”best” carrier of the leader, i.e. � g k k = arg max 1 . k 15
Stackelberg equilibrium regions g 21 /g 22 [SU,PU] [PU SU,*] 1+ γ * [SU,PU] [PU,SU] [PU,SU] 1/(1+ γ *) [PU,SU] [SU,PU] [PU,SU] 1- γ * [SU,PU] [SU,PU] [PU,SU] [PU,SU] 1/(1- γ *) [PU,SU] [*,PU SU] 1- γ * 1/(1- γ *) g 11 /g 12 1 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 of 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 ( g t − 1 , p t − 1 ) β t q ( g t − 1 , p t − 1 ) ← (12) +(1 − β t )( u 1 + γq ( g t , p t )) , • The SU chooses its action following state-value function Q ( g , p ) : Q ( g t − 1 , p t − 1 ) α t Q ( g t − 1 , p t − 1 ) ← (13) +(1 − α t )( u 2 + γQ ( g t , p t )) , 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. 0.12 Simulations Analytical upper−bound 0.1 Probability of no coordination 0.08 0.06 0.04 0.02 0 2 3 4 5 6 7 8 9 10 Number of carriers Figure 9: The probability of no-coordination between the leader and the follower as function of the the number of carriers. 20
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