Distributed power optimization for spectrum-sharing femtocell networks: A fictitious game approach

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Distributed power optimization for spectrum-sharing femtocell networks:A fictitious game approach

Wei Zheng a,*, Tao Su a, Haijun Zhang a, Wei Li a, Xiaoli Chu b, Xiangming Wen a

a Beijing Key Lab of Network System Architecture and Convergence, School of Information and Communication Engineering, BeijingUniversity of Posts and Telecommunications, Beijing 100876, Chinab Department of Electronic and Electrical Engineering, The University of Sheffield, Sheffield S1 3JD, UK

a r t i c l e i n f o

Article history:Received 20 September 2012Received in revised form11 February 2013Accepted 6 March 2013

Keywords:FemtocellDistributed power controlGame theoryInterference priceMultichannel

45/$ - see front matter Crown Copyright & 20x.doi.org/10.1016/j.jnca.2013.03.004

esponding author. Tel.: þ86 13810415171ail addresses: [email protected], zhengwe

e cite this article as: Zheng W, etapproach. Journal of Network and

a b s t r a c t

Power control techniques are becoming increasingly important for a two-tier network, where a centralmacrocell is underlaid with femtocells, since cross-tier and co-tier interference severely limits networkperformance. In this paper, we propose a distributed power control scheme for the uplink transmissionof spectrum-sharing femtocell networks based on fictitious game. Each user announces a price thatreflects its sensitivity to the current interference level, and adjusts its power to maximize its utility.Power and price are updated at terminals and base stations, respectively. The scheme is proved toconverge to a unique optimal equilibrium. Furthermore, we propose a simple macrocell link protectionscheme, where a macro user can protect itself by increasing its price. Most importantly, we investigatethe power optimization scheme proposed in frequency-selective channels based on the Stackelberggame, in which each user prices its limited power allocated to subchannels. Numerical results show thatthe proposed schemes are effective in resource allocation for spectrum-sharing two-tier networks.

Crown Copyright & 2013 Published by Elsevier Ltd. All rights reserved.

1. Introduction

Recent research has shown that nearly 90% of data services and60% of voice services take place in indoor environments(Chandrasekhar et al., 2008). As one of the most promisingtechnologies for indoor wireless communications, femtocells haveattracted much attention. Femtocells are short-range, low-powerand low-cost home base stations, which are deployed by endconsumers and operate in licensed spectrum. Femtocells areconnected to the operator's network via broadband connectionssuch as digital subscriber line (DSL) or cable modem. Femtocellscan provide better indoor user experience with lower transmitpower, resulting in power saving of mobile devices. Femtocells canoffload traffic from macrocells, consequently improving the net-work coverage and capacity (Kang et al., 2011).

Femtocells combined with orthogonal frequency-division mul-tiple access (OFDMA) have been adopted by most next generationwireless networks, such as the 3GPP long-term evolution (LTE)(Ergen, 2009). It is necessary to study power optimization inmultichannel femtocell networks, e.g., how users allocate theirlimited power across the available subchannels.

In practice, there are still some problems that need to be resolvedfor massive deployments of femtocells. A two-tier femtocell networkis usually implemented with shared spectrum rather than split

13 Published by Elsevier Ltd. All r

[email protected] (W. Zheng).

al. Distributed power optiComputer Applications (201

spectrum between tiers (Kang et al., 2011). However, cross-tierand co-tier interference may greatly impair network performanceof spectrum-sharing femtocell networks (López-Pérez et al., 2009).This motivates the use of power optimization for interferencemanagement in two-tier femtocell networks. Prior research hasinvestigated the power optimization scheme for cross-tier interfer-ence. In Kang et al. (2011, 2012), the authors presented a distributedpower control algorithm for spectrum-sharing femtocell networksusing Stackelberg game, which is very effective in distributed powerallocation. In Chandrasekhar et al. (2009), the authors proposed adistributed utility-based Signal-to-Interference-plus-Noise Ratio(SINR) adaptation algorithm to alleviate cross-tier interference fromco-channel femtocells to macrocells. Jo et al. (2009) proposedinterference mitigation strategies that adjust the maximum transmitpower of femto user equipments (FUEs) to suppress the cross-tierinterference at macrocell base stations (MBSs). Yun and Shin (2011)proposed a distributed and self-organizing femtocell managementarchitecture to mitigate cross-tier interference which consists ofthree control loops. But they did not consider the co-tier inter-ference between femtocells. Mitigation of co-tier interference hasalso been investigated in previous work. In Sahin et al. (2009), theauthors proposed an interference avoidance framework betweenmacrocell and femtocell through frequency scheduling. Lee et al.(2011) devised a cooperative resource-allocation algorithm toimprove intercell fairness in femtocell networks. However, only co-tier interference has been considered in these literatures. And itrequires complete network information, including interferencechannel gains.

ights reserved.

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Fig. 1. Singal macrocell with underlaid femtocells.

W. Zheng et al. / Journal of Network and Computer Applications ∎ (∎∎∎∎) ∎∎∎–∎∎∎2

A variety of game-theoretic approaches have been applied tointerference management in different networks (Kim et al., 2010).In Wang et al. (2009), the authors investigated power controlstrategies to maximize the utility for spectrum-sharing cognitiveradio networks (CRNs) using Stackelberg game. Huang et al.(2006) presented distributed power control algorithms for bothsignal-channel and multichannel ad hoc networks, in which powerprice and interference price are introduced to represent the dualvariables corresponding to user's total power constraint andsensitivity to the current interference level, respectively. In thispaper, we focus on femtocell networks instead of ad hoc networks.

In Kang et al. (2012) and Chandrasekhar et al. (2009), only the MBScan price uplink interference from FUEs. It is impractical to usecentralized power control, since a central controller would requirecomplete network information, including interference channel gains.Hence, we consider distributed power control in this paper.

In this paper, we propose a distributed power optimizationscheme for the uplink transmission of spectrum-sharing femtocellnetworks based on fictitious game, where not only macrocells butalso femtocells can price interference. Furthermore, in order toreduce signaling overhead, power and price are updated at term-inals and base stations (BSs), respectively. Each user can estimateother users' total prices based on the sensing results from theenergy detector. Moreover, if macro user equipment (MUE) cannotachieve its target SINR (Razaviyayn et al., 2010) with the maximumtransmit power, the MBS increases its price. For multichannelnetworks, we need consider how the users allocate their poweracross to the available subchannels. We investigate the poweroptimization scheme for multichannel femtocell networks basedon the Stackelberg game by introducing the power price, whichrepresents the dual variable corresponding to user's total powerconstraint. Under the total power constraint, each user prices itspower allocated to subchannels. Each user minimizes its loss due tothe total power constraint. In the Stackelberg game, we also studytwo pricing schemes: uniform pricing and non-uniform pricing. Inpractice, the uniform pricing scheme can be implement in the samecentralized way as that for the non-uniform pricing scheme, whichrequires large amounts of computation. However the optimizationproblem in the uniform pricing scheme has some nice propertiesthat can be explored for the implement. Based on properties, wepresent an effective distributed power price bargaining algorithm,which only requires a small amount of computation.

The rest of this paper is organized as follows. We introduce thesystem model and formulate the multichannel power optimizationproblem in Section 2. In Section 3, we present the distributedpower optimization algorithm for a single-channel network basedon fictitious game (ADP algorithm) and propose the macrocell linkprotection scheme (ADP-P algorithm). Then we formulate theStackelberg game to allocate power for multichannel networkssubject to a power constraint and propose the ADP algorithm withmacrocell link protection for multichannel femtocell networks(ADP-M algorithm) in Section 4. Numerical results are given inSection 5. Finally, we make the conclusion in Section 6.

2. System model and problem formulation

In this section, we first introduce the system model for thespectrum-sharing femtocell network. Then we formulate themultichannel power optimization problem.

2.1. System model

A two-tier femtocell network has been considered as shown inFig. 1. For analytical tractability, we ignore co-channel interferencefrom neighboring macrocell transmissions. Assuming that the

Please cite this article as: Zheng W, et al. Distributed power optigame approach. Journal of Network and Computer Applications (201

system consists of a central MBS B0 providing a macrocell coveragearea with radius Rm: Within the macrocell coverage at a distanceDf from the MBS B0, N co-channel femtocells fBig, i¼ 1⋯N, arelocated in a square grid—e.g. residential neighborhood—of areaD2grid Km

2 (Chandrasekhar et al., 2009). Each femtocell has a radiorange of Rf meters. It is assumed that femtocells and macrocell usethe same frequency bands, and there is one scheduled active userin each femtocell during each signaling slot. Each user is able totransmit over a set of K¼ f1,⋯,Kg orthogonal subchannels. Let gðkÞi,jdenote the channel gain between transmitting mobile j and BS Bi

at subchannel k. The channel gains are modeled following thechannel model in Chandrasekhar et al. (2009).

2.2. Problem formulation

Let i∈f0,1,⋯,Ng denotes the scheduled active user connected toits BS Bi. Transmit power on subchannel k of user i is pðkÞi . Thevariance of additive white Gaussian noise (AWGN) on each sub-channel is s2. The interference on subchannel k of user i is denotedby IðkÞi , where

IðkÞi ¼∑j≠ipðkÞj gðkÞi,j þs2 ð1Þ

Consequently, the SINR γðkÞi of user i at subchannel k can beexpressed as

γðkÞi ¼pðkÞi gðkÞi,i

∑j≠ipðkÞj gðkÞi,j þs2

ð2Þ

The utility function uðkÞi on subchannel k of user i is an

increasing and strictly concave function of SINR γðkÞi . For example,the utility function can be expressed as (Huang et al., 2006)

uðkÞi ¼W logðγðkÞi Þ ð3Þ

where W denotes the bandwidth of each subchannel. In this case,since a user's SINR is typically larger than 1, the utility function uðkÞ

iis approximately equal to the data rate of user i at subchannel k.The overall utility of user i is

ui ¼ ∑K

k ¼ 1uðkÞi ð4Þ

Then the overall utility of the network can be defined as

u¼∑iui ð5Þ

The problem we consider is to determine power allocation ofall users to maximize the utility summed over all users under theconstraints ∑k∈Kp

ðkÞi ≤pmax

i and pðkÞi ≥pmini .

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3. Distributed power optimization for single channel femtocellnetworks

In this section, we first investigate the distributed poweroptimization for single-channel networks based on fictitiousgame. Then, we devise a distributed power optimization algorithm(ADP algorithm) and analyze the convergence of the algorithm.Finally, we propose a macrocell link protection scheme (ADP-Palgorithm).

3.1. A game theoretic framework

The problem for signal-channel femtocell networks can bemodeled as a non-cooperative power control game. Assumingthat all users only transmit at a single subchannel and thebandwidth of the subchannel is 1. Let G¼ ½N ,fPig,fuið⋅Þg� denotesthe non-cooperative power control game (NPG), wherePi ¼ ½pmin

i ,pmaxi � is the strategy set describing the domain of

transmission power for user i and N ¼ f0,1,…,Ng refers to theplayer index set. Notice that a special case is pmin

i ¼ 0. Let p−idenotes the vector of transmit power of all users other than user i.Then this non-cooperative power control game can be expressedas

max0≤pi≤pmax

uiðpi,p−iÞ

for each user in N :

It has been proved in Huang et al. (2006) that the Nashequilibrium of G exists and is unique, although the Nash equili-brium is not efficient in general (Saraydar et al., 2002). In non-cooperative power control game, each user selfishly optimizes itsown utility and transmits at the maximum power, regardless ofthe interference imposed on other users. In order to decrease theinterference, a price should be paid by each user for the inter-ference caused (Huang et al., 2006), and the pricing function ofuser i is given by

ciðpi,p−iÞ ¼ pi∑j≠i

∂uj

∂pið6Þ

where ∂uj=∂pi denotes user j's slight increase in utility per unitdecrease in user i's transmit power. Based on the price strategy,the non-cooperative power game with interference compensationcan be expressed as

siðpi,p−iÞ ¼ uiðpi,p−iÞþciðpi,p−iÞ: ð7ÞEach user updates its transmit power based on the received

prices announced by other users and selfishly optimizes its ownutility. However, since there is no penalty for user i announcing ahigh price, each user may announce a large enough price to forceother users to transmit at the minimum power (Huang et al.,2006). To avoid this undesirable outcome, we turn to the followingfictitious power control game with compensation (FPGC)

GFPGC ¼ ½ℱP∪ℱC,fPℱPi ,PℱC

i g,fsℱPi ,sℱC

i g�where ℱP and ℱC are both include Nþ1 players. ℱP denotes thefictitious power player set, and each player i∈ℱP chooses a powerpi from the strategy set PℱP

i ¼ Pi to maximize its payoff

sℱPi ðpi,p−iÞ ¼ uiðpi,p−iÞ−pi∑j≠iμjgj,i ð8Þ

where μi ¼−∂ui=∂Ii is nonnegative. ℱC denotes the fictitious priceplayer set and each player i∈ℱC chooses a price μj from thestrategy set PℱC

i ¼ ½0,μmaxi � to maximize its payoff

sℱCi ðpi,p−iÞ ¼−ðμi−ð−∂ui=∂IiÞÞ2 ð9Þwhere μmax

i denotes the upper bound of −∂ui=∂Ii. In FPGC, eachuser is split into two fictitious players, one in ℱP controls transmit

Please cite this article as: Zheng W, et al. Distributed power optigame approach. Journal of Network and Computer Applications (201

power and the other one in ℱC controls interference price. Thismotives the following ADP algorithm.

3.2. ADP algorithm

In the ADP algorithm, each player sets its transmit power basedon the prices announced by other players. Suppose that user iupdates its power and price at time instances Ti,p and Ti,μ,respectively. User i updates its power according to

p*i ¼ arg maxpi∈Pi

sℱPi ðpi,p−i,μ−iÞ: ð10Þ

By taking the first-order partial derivative of sℱPi with respect to pi,

the power update function can be written as

p*i ¼1

∑j≠iμjgj,i

" #pmaxi

pmini

ð11Þ

where ½z�ba denotes maxfminfz,ag,bg. The interference price can beupdated according to

μ*i ¼−∂ui

∂Ii¼ 1

Iiðp*−iÞ

: ð12Þ

We can rewrite (11) and (12) as an iterative power control updateand an iterative interference price update, respectively. Theiterative power update pðtþ Þ ¼ zðpðtÞÞ can be expressed as

piðtþ Þ ¼1

∑j≠iμjðtÞgj,i

" #pmaxi

pmini

: ð13Þ

The iterative interference price update can be rewritten as

μiðtþ Þ ¼γiðtÞ

piðtÞgi,i: ð14Þ

It can be seen that to implement the updates, each player ineeds to know its channel gain gi,i, its current SINR γi, the“adjacent” channel gains gj,i, and the prices announced by otherusers. Notice that getting to know the adjacent channel gains andthe prices announced by other players is complicated and maysignificantly increase signal overhead. In order to implement theupdates more efficiently, we introduce a simply strategy that eachuser only needs to know the sum of other users' prices, withweights equal to the adjacent channel gains. This can be derivedfrom the sensing results of the energy detector. Assuming thatuser i's price update is performed at BS i, and the uplink channelgain gj,i between transmitting mobile i and BS Bj equals thedownlink channel gain g′j,i between transmitting BS Bj and mobilei (Yun and Messerschmitt, 1995). Each BS periodically broadcasts abeacon with the transmit power μipr , where pr is a fixed constantand is known by each user. Then user i can calculate ∑j≠iμjgj,iaccording to the sensing result pr∑j≠iμjg′j,i ¼ pr∑j≠iμjgj,i:

Based on the above discussion, users participate in the fictitiouspower control game, and BSs participate in the fictitious pricegame. In the rest of this section, a “player” refers to a user or a BS.

As for the convergence of the ADP algorithm, according toHuang et al. (2006), the fixed point of fictitious power controlgame (or the Nash Equilibrium given by (11)) is unique, so the ADPalgorithm globally converges to the best response from any initialstrategies.

3.3. Macrocell link protection

In this section, we propose the macrocell link protectionscheme, which is ADP algorithm with macrocell link protectionin single-channel networks (ADP-P algorithm). In ADP-P algo-rithm, the MUE can protect itself from severe interference byincreasing its price. In order to achieve the MUE's target SINR Γ0,

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we can get the smallest transmit power of MUE pmin0 ¼ Γ0I0=g0,0,

which is derived from γmin0 ¼ Γ0. Then the smallest power of user i

is given as

pmini ¼ maxf0,p⌢ ig, i¼ 0

0, i¼ 1, 2, …, N:

(ð15Þ

The maximum tolerable interference level of the MBS isdenoted by Ilim, which can be derived from pmax

0 g0,0=Ilim ¼ Γ0. If aMUE cannot achieve its target SINR with the maximum transmitpower, that is, the interference to the MBS exceeds the thresholdIlim, then the MBS increases its price by δ0, which is larger than 1.Note that the FUE that causes more interference to the MBS willpay more for its interference, in other words, the FUE will reducemore power. Then the fictitious price game can be rewritten as

sℱCi ðpi,p−iÞ ¼−ðμi−δið−∂ui=∂IiÞÞ2 ð16Þ

where δi denotes user i's parameter. Then the interference pricecan be updated according to

μiðtþ Þ ¼ δiðtÞγiðtÞ

piðtÞgi,i: ð17Þ

Algorithm 1. ADP-P algorithm

(1)

Plga

Initialize pð0Þ ¼ pmin and μð0Þ ¼ 0:

(2) Power Update: At each t∈Ti,p, user i updates its minimum

power according to (15) and updates its transmit poweraccording to

piðtþ Þ ¼1

∑j≠iμjðtÞgj,i

" #pmaxi

pmini

:

(3)

Interference Price Update: At each t∈Ti,μ, BS iupdates its priceaccording to

μiðtþ Þ ¼ δiðtÞγiðtÞ

piðtÞgi,iand periodically broadcasts a beacon with the transmit powerμipr . If γ0oΓ0, the MBS updates its price parameter accordingto

δ0ðtþ Þ ¼ δ0ðtÞ⋅Δδ :

Based on above discussion, we can get δi ¼ 1 for i¼ 1, 2…, Nand δi≥1 for i¼ 0. It means that the MBS may announce a largeenough price to force all other users to reduce their transmitpower. In order to set a suitable proportion parameter δ0, we firstimplement the ADP algorithmwith δ0¼1, and increase the value ofδ0 by a step of Δδ if a MUE cannot achieve its target SINR with themaximum transmit power and until the MUE's target SINR is met.The ADP-P algorithm is summarized in Algorithm 1.

4. Multichannel femtocell networks

In this section, we investigate the power optimization schemeproposed in multichannel femtocell networks. Firstly, we presentthe Stackelberg game for multichannel femtocell networks anddiscuss the Stackelberg equilibrium (SE). Then, we consider twopricing schemes: non-uniform pricing, and uniform pricing.Finally, we propose the ADP algorithm with macrocell link protec-tion for multichannel femtocell networks (ADP-M algorithm).

ease cite this article as: Zheng W, et al. Distributed power optime approach. Journal of Network and Computer Applications (201

4.1. Stackelberg game

For multichannel networks, an additional consideration is howusers allocate their power across the available subchannels effi-ciently (Huang et al., 2006). Note that the total power allocated tothe available subchannels for user i cannot exceed the maximumtransmit power pmax

i of user i, i.e.,

∑K

k ¼ 1pðkÞi ≤pmax

i ð18Þ

Assuming that each user prices its limited power allocated tothe subchannels. Then a Stackelberg game is formulated to studythe joint utility maximization of users and subchannels. Stackel-berg game is a strategic game that consists of a leader andfollowers competing for certain resources (Kang et al., 2011). Weformulate the user as the leader and its subchannels as followers.The user (leader) imposes a set of prices on the transmit powerallocated to the subchannels. Then, the subchannels (followers)update their power allocation strategies to maximize the payoffbased on the power prices. The user minimizes its loss due to thetotal power constraint rather than maximizing its revenueobtained from selling the power quota to subchannels. Accord-ingly, the optimization is formulated as

Problem 1:

minπ≥0

s′iðπi,piÞ ¼ ∑K

k ¼ 1πðkÞi ðpðkÞ*i −pðkÞi Þ ð19Þ

s:t: ∑K

k ¼ 1pðkÞi ≤pmax

i ð20Þ

where πðkÞi denotes the power price and pðkÞ*i is the optimal powerallocation without the total power constraint. Problem 1 can betransformed into

Problem 2:

maxπ≥0

sLi ðπi,piÞ ¼ ∑K

k ¼ 1πðkÞi ðpðkÞi −pðkÞ*i Þ ð21Þ

s:t: ∑K

k ¼ 1pðkÞi ≤pmax

i : ð22Þ

The utility function for subchannel k of user i is defined as

sðkÞi ðpðkÞi ,pðkÞ−i ,π

ðkÞi Þ ¼ uðkÞ

i ðpðkÞi ,pðkÞ−i Þ−θ

ðkÞi pðkÞi −πðkÞi pðkÞi ð23Þ

where θðkÞi denotes the total interference price ∑j≠iμðkÞj gj,i. Then the

optimization problem for each subchannel k of user i can beformulated as

Problem 3:

maxp kð Þi

sðkÞi ðpðkÞi ,pðkÞ−i ,π

ðkÞi Þ ð24Þ

s:t: pðkÞi ≥pmini ð25Þ

The above two optimization problems form the Stackelberg game.The objective of this game is to find the SE point(s). In the Stackelberggame, a set of strategies form the SE point from which neither theleader (user) nor the followers (subchannels) have incentives todeviate. A formal definition of the SE can be found in Kang et al. (2011).

Definition 1. The point ðπ*i ,p

*i Þ is a SE for the proposed Stackelberg

game if sðkÞi ðpðkÞ*i ,pðkÞ*−i ,π

ðkÞ*i Þ≥sðkÞi ðpðkÞi ,pðkÞ*

−i ,πðkÞ*i Þ for any k, and sLi ðπ*

i ,p*i Þ

≥sLi ðπi,p*i Þ for any ðπi,piÞ with π≥0 and p≥0:

According to Kang et al. (2011), the SE can be obtained asfollows: for a given πi, Problem 3 is solved first. Then, the optimalpower price π*

i is achieved by solving Problem 2 with the bestresponse p*

i .

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Next, we discuss the SE for two pricing schemes. If the usercharges each subchannel with a different power price, then weregard this pricing scheme as non-uniform pricing. A special caseof this scheme is uniform pricing, in which the user charges eachsubchannel with the same power price.

4.2. Non-uniform pricing

Under the non-uniform pricing scheme, the user sets differentpower prices for different subchannels. The power price for subchan-nel k of user i is denoted by πðkÞi . It is observed that given interferenceprice θðkÞi and interference powers pðkÞ

−i , the utility function sðkÞi is aconcave function over pðkÞi . Thus Problem 3 is a convex optimizationproblem. For a convex optimization problem, the optimal solutionmust satisfy the Karush–Kuhn–Tucker (KKT) conditions.

Lemma 1. (KKT Conditions): Given the Lagrange multiplier λðkÞi ,the KKT conditions for Problem 3 can be written as

∂sðkÞi

∂pðkÞi

þλðkÞi ¼ 0 ð26Þ

λðkÞi ðpðkÞi −pmini Þ ¼ 0, λðkÞi ≥0 ð27Þ

By solving the KKT conditions, the optimal solution for Problem3 can be easily obtained as

pðkÞi ¼ 1

πðkÞi þθðkÞi

" #pmaxi

pmini

ð28Þ

Since the total interference price θðkÞi and the power price πðkÞi arenonnegative, Problem 2 can be rewritten as

Problem 4:

maxπ≥0

∑K

k ¼ 1πðkÞi

1

πðkÞi þθðkÞi

−1

θðkÞi

!ð29Þ

s:t: ∑K

k ¼ 1

1

πðkÞi þθðkÞi

≤pmaxi : ð30Þ

Obviously, the optimization problem is convex and the optimalsolution is given as follows.

Proposition 1. The optimal solution to Problem 4 is given by

πðkÞ*i ¼0, if ∑

K

k ¼ 1

1

θðkÞi

≤pmaxi ,

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiθ kð Þi θðkÞi þα*i

� �r−θðkÞi , otherwise

8>>>><>>>>:

ð31Þ

where α*i is the Lagrange multiplier and can be derived from thefollowing equation:

α*i ¼ h−1i ðpmaxi Þ ð32Þ

where

hiðxÞ ¼ ∑K

k ¼ 1

1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiθðkÞi ðθðkÞi þxÞ

q , and x40: ð33Þ

Proof. Please refer to Part A of the Appendix.

We relate the optimal solution of Problem 4 to that of theoriginal problem, i.e., Problem 1. Then we can obtain the optimalsolution of Problem 1 according to (31).

Theorem 3. The SE for the Stackelberg game formulated inProblems 1 and 3 is ðπ*

i ,p*i Þ, where π*

i is given by (31) and p*i is

given by (28).

Please cite this article as: Zheng W, et al. Distributed power optigame approach. Journal of Network and Computer Applications (201

Proof. As discussed above, π*i is the optimal solution to Problem

2 and p*i is the optimal solution to Problem 3. According to

Definition 1, ðπ*i ,p

*i Þ is the SE for the Stackelberg game formulated

in Problems 1 and 3.

4.3. Uniform pricing

In uniform pricing, the user sets a uniform power price for allsubchannels, i.e., πðkÞi ¼ πi, ∀k. As discussed in the non-uniformpricing scheme, the optimal power allocation for subchannel k ofuser i can be expressed as

pðkÞi ¼ 1

πiþθðkÞi

" #þ

ð34Þ

Then the optimization problem for user i is formulated asProblem 5:

maxπ≥0

∑K

k ¼ 1πi

1

πiþθðkÞi

−1

θðkÞi

!ð35Þ

s:t: ∑K

k ¼ 1

1

πiþθðkÞi

≤pmaxi : ð36Þ

Since Problem 5 is similar to Problem 4, it can be solved by thesame method for Problem 4. According to Kang et al. (2011), asimple yet effective way to update the power price is to employthe subgradient method as follows

πiðtþ Þ ¼ πiðtÞ−τ pmaxi − ∑

K

k ¼ 1pðkÞi ðtÞ

!" #þ

ð37Þ

where τ is a small step size and is non negative.

4.4. ADP-M algorithm

In practice, MUEs may simultaneous communicate with theMBS. Under the assumption that the SINR on each subchannel of aMUE is larger than its target SINR, the ADP-M algorithm withmacrocell link protection is summarized in Algorithm 2.

Algorithm 2. ADP-M algorithm

(1)

miz3),

Initialize pð0Þ ¼ pmin, μð0Þ ¼ 0 and πð0Þ ¼ 0:

(2) Power Price Update: User i increases its power price according

to (31) (non-uniform pricing) or (37) (uniform pricing).

(3) Power Update: At each t∈Ti,p, user i updates its minimum

power on subchannel k according to

pmini ¼ maxf0,ΓðkÞ

i IðkÞi =gðkÞi,i g, i¼ 0

0, i¼ 1, 2, …, N

(

and updates its transmit power on subchannel k according to

pðkÞi ðtþ Þ ¼ 1

θðkÞi ðtÞ

" #pmaxi

pmini

:

(4)

Interference Price Update: At each t∈Ti,μ, BS i updates itsinterference price on subchannel k according to

μðkÞi ðtþ Þ ¼ δðkÞi ðtÞ γðkÞi ðtÞpðkÞi ðtÞgðkÞi,i

:

(5)

If γ0oΓ0, MBS updates its price parameter on subchannel kaccording to

δðkÞ0 ðtþ Þ ¼ δðkÞ0 ðtÞ⋅Δδ :

ation for spectrum-sharing femtocell networks: A fictitioushttp://dx.doi.org/10.1016/j.jnca.2013.03.004i

Page 6

W. Zheng et al. / Journal of Network and Computer Applications ∎ (∎∎∎∎) ∎∎∎–∎∎∎6

5. Numerical result

We provide simulation results to illustrate the performanceof the proposed schemes in single-channel and multichannelfemtocell networks. Simulation parameters are in accordance withthose used in Chandrasekhar et al. (2009) and are summarized inTable 1.

Fig. 2 shows the convergence of the powers and interferenceprices for MUE and FUEs with the proposed ADP algorithmwithout macrocell link protection (ADP) and the proposed ADPalgorithm with macrocell link protection (ADP-P) in single-channel networks. Notice that in both cases ADP converges tothe optimal power allocation, and in ADP-P MUE transmits withlarger power to achieve macrocell link protection.

Fig. 3 shows the mean transmit power of users in ADP, ADP-Pand the distributed power control based SINR adaptation (DPCA)of Chandrasekhar et al. (2009) versus the number of femtocells.

Table 1Simulation parameter.

Parameter Value

Macrocellcell/Femtocell radiusðRm=Rf Þ

288/10 m

Grid size ðDgridÞ 100 mCarrier frequencyðf c,MHzÞ 2000 MHzpmax0 =pmax

i 1/0.1 WPrice parameter increment ðΔδÞ 3 dBPenetration loss of wall ðϕÞ 5 dBStep size ðτÞ 10

ADP-P

0 5 10 15 200

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

Iterations

Pow

er(W

)

FUE 1FUE 2FUE 3FUE 4MUE

0 5 10 15 20115

120

125

130

135

Iterations

Pric

e(dB

)

FUE 1FUE 2FUE 3FUE 4MUE

Fig. 2. Convegerence of the interference price and pow

Please cite this article as: Zheng W, et al. Distributed power optigame approach. Journal of Network and Computer Applications (201

Compared with DPCA, where FUEs' target SINRs are set to 20 dB,the ADP-P algorithm consumes more power due to the macrocelllink protection. The DPCA approach consumes the least power,since it only seeks to reach the target SINR, not necessarilyreaching the maximum SINR.

Fig. 4 shows the mean SINR of users in DPCA, ADP and ADP-P.Compared with DPCA, ADP-P improves the FUEs' SINRs signifi-cantly while protecting the MUE's target SINR.

Fig. 5 plots the convergence performance of ADP-M in amultichannel network with K¼4 subchannels, where frequencyselective fading across subchannels has been considered. It isobserved that under the uniform pricing, the proposed iterativepower allocation algorithm converges to the optimal powerallocation rapidly.

Fig. 6 plots the optimal power prices of FUEs at differentsubchannels vs. the total power constraint pmax

i under non-uniform pricing. It is observed that power prices decrease withthe increase of pmax

i . This is because α*i decreases with the increaseof pmax

i , which will decrease the power price according to (31).In Fig. 7, we evaluate the average spectrum efficiency of FUEs

with ADP-M algorithm under uniform pricing (ADP-MU) and non-uniform pricing (ADP-MN), as compared with ADP-P, ADP-F andfixed power allocation (Fixed) schemes. In ADP-P, the maximumpower in every subchannel of a FUE is the maximum power (i.e.,0.1 W), while in ADP-F it is the maximum power divided bythe number of subchannels (i.e., 0.025 W). In the Fixed scheme,the transmit powers in every subchannel are set to 50% (Fixed1)and 75% (Fixed2) of the maximum power of the subchannel inADP-F. It is observed from Fig. 7 that compared with ADP-P, theloss in spectrum efficiency in ADP-MU is minor, and ADP-MUoutperforms all the other schemes. This is because the uniform

ADP

0 5 10 15 20 25 300

0.02

0.04

0.06

0.08

0.1

0.12

Iterations

Pow

er(W

)

FUE 1FUE 2FUE 3FUE 4MUE

0 5 10 15 20115

120

125

130

135

Iterations

Pric

e(dB

)

FUE 1FUE 2FUE 3FUE 4MUE

er for the ADP-P (left) and ADP (right) algorithms.

mization for spectrum-sharing femtocell networks: A fictitious3), http://dx.doi.org/10.1016/j.jnca.2013.03.004i

Page 7

4 8 12 16 20 24 28 32 36-10

-5

0

5

10

15

20

25

30

Femtocells Per Cellsite

Ave

rage

Pow

er(d

Bm

)

ADP-P MUEADP MUEDPCA MUEADP-P FUEADP FUEDPCA FUE

Fig. 3. Mean transmit power for different schemes.

4 8 12 16 20 24 28 32 365

10

15

20

25

30

35

40

45

50

Femtocells Per Cellsite

Ave

rage

SIN

R(d

B)

ADP-P FUEADP FUEDPCA FUEADP-P MUEADP MUEDPCA MUE

Fig. 4. Mean SINR for different schemes.

MUE FUE

0 10 20 30 400

0.02

0.04

0.06

0.08

0.1

Iterations

Pow

er(W

)

channel 1

channel 2

channel 3

channel 4

0 10 20 30 400

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

Iterations

Pow

er(W

)

channel 1channel 2channel 3channel 4

Fig. 5. Convegerence performance of the ADP-M algorithm under uniform pricing.

0.2 0.4 0.6 0.8 10

10

20

30

40

Total power constraint of femtocell user(W)

Pow

erP

rice

channel 1channel 2channel 3channel 4

Fig. 6. Power price for femtocell user vs. total power under non-uniform pricing.

5 10 15 20 25 30 357

8

9

10

11

12

13

14

15

Femtocells Per Cellsite

Ave

rage

Spe

ctru

m E

ffici

ency

(bit/

s/H

z)

ADP-PADP-MUADP-MNADP-FFixed1Fixed2

Fig. 7. Mean spectrum efficiency for different schemes.

W. Zheng et al. / Journal of Network and Computer Applications ∎ (∎∎∎∎) ∎∎∎–∎∎∎ 7

Please cite this article as: Zheng W, et al. Distributed power optimization for spectrum-sharing femtocell networks: A fictitiousgame approach. Journal of Network and Computer Applications (2013), http://dx.doi.org/10.1016/j.jnca.2013.03.004i

Page 8

W. Zheng et al. / Journal of Network and Computer Applications ∎ (∎∎∎∎) ∎∎∎–∎∎∎8

pricing scheme maximizes the sum rate of subchannels, while thenon-uniform pricing scheme minimizes the loss in revenueof users.

6. Conclusion

In this paper, we propose ADP algorithm for the uplinktransmission of spectrum-sharing femtocell networks based onfictitious game, in which not only macro users but also femto userscan price the interference from other users. The ADP-P algorithmproposed can improve the network performance significantlywhile protecting the macro user's target SINR. Furthermore, inADP-P algorithm, power and price are updated at terminals andBSs respectively, which can reduce signaling overhead. We havealso proposed ADP-M algorithm for multichannel networks basedon the Stackelberg game and power pricing. Two pricing schemes:uniform pricing and non-uniform pricing have been investigated.Compared with non-uniform pricing scheme, uniform pricingscheme requires less computation and can achieve higher spec-trum efficiency. This is because the uniform pricing schememaximizes the sum rate of subchannels, while the non-uniformpricing scheme minimizes the loss in revenue of users. Comparedto power optimization without the total power constraint, the lossin spectrum efficiency is minor in the ADP-MU. And both ADP-MUand ADP-MN outperform other traditional schemes.

Acknowledgments

This work is supported by the National Natural Science Foun-dation of China (61101109, 61271179), the Youth Research andInnovation Project of Beijing University of Posts and Telecommu-nications, the Cobuilding Project of Beijing Municipal EducationCommission, and the National S&T Major Project of China (nos.2010ZX03003-001-01, 2011ZX03003-002-01).

Appendix A. Proof of Proposition 1

Since Problem 4 is a convex optimization problem, we can solveit by solving its dual problem. Introducing Lagrange multiplier α*i ,which is nonnegative for the total power constraint, Problem 4 canbe solved by maximizing the following expression

Lðπi,αiÞ ¼ ∑K

k ¼ 1πðkÞi

1

πðkÞi þθðkÞi

−1

θðkÞi

!

þαi pmaxi − ∑

K

k ¼ 1

1

πðkÞi þθðkÞi

!ð38Þ

Then the KKT conditions can be written as follows:

∂Lðπi,αiÞ∂πðkÞi

¼ 0, ∀k, ð39Þ

αi pmaxi − ∑

K

k ¼ 1

1

πðkÞi þθðkÞi

!¼ 0, ð40Þ

pmaxi − ∑

K

k ¼ 1

1

πðkÞi þθðkÞi

≥0, ð41Þ

Please cite this article as: Zheng W, et al. Distributed power optigame approach. Journal of Network and Computer Applications (201

αi≥0, πðkÞi ≥0, ∀k: ð42Þ

From (39), we have

θðkÞi ðθðkÞi þαiÞ−ðθðkÞi þπðkÞi Þ2 ¼ 0, ∀k, ð43Þ

From (43), we can get

πðkÞi ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiθðkÞi ðθðkÞi þαiÞ

q−θðkÞi : ð44Þ

If pmaxi −∑K

k ¼ 11=θðkÞi 40, since πðkÞi is nonnegative, we can obtain

pmaxi −∑K

k ¼ 11=ðθðkÞi þπðkÞi Þ40. From (40), we know that αi ¼ 0,

which indicates that πðkÞi ¼ 0. If pmaxi −∑K

k ¼ 11=θðkÞi o0, we have

pmaxi − ∑

K

k ¼ 1

1

πðkÞi þθðkÞi

¼ 0, ð45Þ

Substituting (44) into (45), we have the following equation

αi ¼ h−1i ðpmaxi Þ ð46Þ

where

hiðxÞ ¼ ∑K

k ¼ 1

1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiθðkÞi ðθðkÞi þxÞ

q , and x40:

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