Research ArticleStackelberg Game Based Power Control with Outage ProbabilityConstraints for Cognitive Radio Networks

Helin Yang1 Xianzhong Xie1 and Athanasios V Vasilakos2

1Chongqing Key Laboratory of Mobile Communications Technology and Institute of Personal CommunicationChongqing University of Posts amp Telecommunications Chongqing 400065 China2Department of Computer Science Electrical and Space Engineering Lulea University of Technology 93187 Skelleftea Sweden

Correspondence should be addressed to Helin Yang yhelincqupt163com

Received 16 June 2015 Revised 20 September 2015 Accepted 22 October 2015

Academic Editor Kijun Han

Copyright copy 2015 Helin Yang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper firstly investigates the problem of uplink power control in cognitive radio networks (CRNs) with multiple primary users(PUs) and multiple second users (SUs) considering channel outage constraints and interference power constraints where PUs andSUs compete with each other to maximize their utilities We formulate a Stackelberg game to model this hierarchical competitionwhere PUs and SUs are considered to be leaders and followers respectively We theoretically prove the existence and uniquenessof robust Stackelberg equilibrium for the noncooperative approach Then we apply the Lagrange dual decomposition method tosolve this problem and an efficient iterative algorithm is proposed to search the Stackelberg equilibrium Simulation results showthat the proposed algorithm improves the performance compared with those proportionate game schemes

1 Introduction

The high energy consumption and exponential growth inwireless communication networks face serious challenges tothe design of more energy efficiency and spectrum efficiencygreen communications that should deal with the scarcity ofradio resources A promising approaches technique calledcognitive radio networks (CRNs) is proposed as a key designto improve spectral efficiency of the LTE-advanced standardGame theory is an effective tool for resource allocation inmultiuser communication systems which has been appliedto CRNs and [1] summarized the advance of game theory forCRNs Recently Stackelberg game has been formulated forresource allocation which addresses the relationship betweenPUs and SUs in [2ndash9] In these two-tier femtocell networksthe PUswere served by themobile operators whereas the SUsare deployed by indoor users for their own interests Theirdifferent service requirements and design objectivesmotivateus to adopt the framework of hierarchical game to the powerallocation problem with the leader-follower structure whichis actually a multiple-leader multiple-follower Stackelberggame Specifically the PUs compete with each other in

a noncooperative manner to maximize their utilities all thetime anticipating the response of the followersThis subgameis referred to as the upper subgame while after the PUsapply their strategies the SUs update their power allocationstrategies in response to the PUsrsquo strategies The SUs alsocompete with each other in a noncooperative manner tomaximize their own utility function Moreover this subgameis referred to as the lower subgame

Reference [2] presented a joint pricing and power alloca-tion scheme for CRNs with Stackelberg game PUs and SUscan benefit from the channel sharing model by achievingthe Stackelberg equilibrium (SE) In [3 4] a Stackelberggame is used to study the joint utility maximization of thePUs and the SUs with a maximum tolerable interferencepower constraint (IPC) at the primary base station (PBS)However the algorithm is suboptimal because it does notconsider how to control the IPC among PBS Therefore [5]proposed an optimal price-based power algorithm for thePBS and SUs maximize their revenues by Stackelberg gamewith IPC To maximize usersrsquo energy efficiency of CRNs anew green power control scheme was studied in [6] wherePUs and SUs aim at maximizing their energy efficiency with

Hindawi Publishing CorporationInternational Journal of Distributed Sensor NetworksVolume 2015 Article ID 604915 9 pageshttpdxdoiorg1011552015604915

2 International Journal of Distributed Sensor Networks

h1h1

h2

hL

h11

h21

h22

h22h1K

SUi

SU2

SU1

PU2

PU1

PUK

PBSSBS

h2

h2K

hLK

hK

hL2

hL1

Figure 1 System model

Stackelberg game In [7] the authors formulated a Stackelberggame model to maximize the payoff of both SUs and PUs byjointly optimizing transmission powers of SUs and subbandallocations of SUs Moreover in [8] the authors analyzedthe cooperation interaction between the PUs and SUs trans-mitters from a Stackelberg game theoretic perspective withsecrecy constraints Moreover [9] proposed a game theoreticframework to model the interactions among multiple PUsand multiple SUs under different system parameter settingsand under system perturbation

However the above papers aim to maximize usersrsquo rev-enue without considering the outage probability of multi-PUs and multi-SUs It is clear that both PUs and SUs need toupdate their transmission power frequently to maintain theirSignal to Interference plus Noise Ratio (SINR) level (usuallyreferred to as QoS) due to channel fading and interferencesfrom other users when they transmit on the same channelTherefore in CRNs the fundamental performance traits ofmultiple SUs power control should be investigated with mul-tiple PUs also the issues ofmultiple PUs power control shouldbe studied in the presence of multiple SUs in fading channelsand interferences In the game problem formulation theoutage probabilities of both PUs and SUs represent their ownutility below the required target SINR used to guarantee theirdesired QoS Hence the channel outage probabilities withrespect to both PUs and SUs transmissions should be requiredin the multiple usersrsquo game framework if usersrsquo SINR fallsbelow a certain threshold

In this paper we proposed a new Stackelberg game tomodel the hierarchical competition between PUs and SUs inCRNs with global interference and outage constraints Wetheoretically prove the existence and uniqueness of robustStackelberg equilibrium for the noncooperative approachThen we employ the Lagrange dual decomposition methodto solve the game problem by decomposing the game prob-lem into independent suboptimal solution Furthermore wedevelop an efficient iterative algorithm to converge the SE

2 System Model

We consider a CRN composed of a single PBS with a set ofPUs K fl 1 119870 having priority to use a set of channelsN fl 1 119873 and a secondary base station (SBS) with aset of SUs L fl 1 119871 allowed to share 119873 channels fromthe PBS shown in Figure 1We define the channel gains of the

PU 119896 and the SU 119897 on the channel 119899 as ℎ119899119896119896and ℎ119899119897119897 respectively

The channel gain between the 119896th PU transmitter and the SU119897 receiver is denoted by ℎ119899

119896119897 the channel gain from the 119897th SU

transmitter to the 119896th PU receiver is ℎ119899119897119896 The transmit power

of the 119896th PU and the 119897th SU on the channel 119899 is denoted by119901119899

119896and 119901119899

119897 We assume each user can transmit simultaneously

over multiple channels and one channel can be served onlyby one PU but can be used by multiple SUs

The SINR at the 119896th PU receiver on the 119899th channel canbe defined as

120574119899

119896=

119901119899

119896ℎ119899

119896119896

sum119871

119897=1120588119899119897119896119901119899119897ℎ119899119897119896+ 1205752

=119901119899

119896ℎ119899

119896119896

119868 (p119899minus119896) (1)

where if the PU 119896 and the SU 119897 transmit on the same channel119899 120588119899119897119896= 1 otherwise 120588119899

119897119896= 0 1205752 is variance of AWGN and

the interference is given by 119868(p119899minus119896) = sum119871

119897=1120588119899

119897119896119901119899

119897ℎ119899

119897119896+ 1205752

The SINR at the 119897th SU receiver on the channel 119899 can beexpressed by

120574119899

119897=

119901119899

119897ℎ119899

119897119897

sum119871

119895=1119895 =119897120588119899119897119895119901119899119895ℎ119899119897119895+ 119901119899119896ℎ119899119896119897+ 1205752

=119901119899

119897ℎ119899

119897119897

119868 (p119899minus119897) (2)

where 119868(p119899minus119897) = sum119871

119895=1119895 =119897120588119899

119897119895119901119899

119895ℎ119899

119897119895+ 119901119899

119896ℎ119899

119896119897+ 1205752

The required target SINR threshold 120574119899

119906of the user 119906

(PUs or SUs) which is not satisfied is defined as the outageprobability on the channel 119899 The outage probability Pr(120574119899

119906lt

120574119899

119906) of the user 119906 on the channel 119899 can be expressed as [10]

Pr (120574119899119906lt 120574119899

119906) = 1 minusprod

119894 =119906

(1 + 120588119899

119894119906

120574119899

119906119901119899

119894ℎ119899

119894119906

119901119899119906ℎ119899119906119906

)

minus1

(3)

Therefore the above equation shows the outage prob-ability of one user in the presence of multiple SUs andorPUs Accordingly the goal of each PU 119896 is to transmit withthe minimum target SINR 120574

119899

119896on the channel 119899 with the

prescribed limited channel outage 120585119899119896constraint

Pr (120574119899119896lt 120574119899

119896) lt 120585119899

119896 forall119896 forall119899 (4)

where (4) means the outage constraint of the PU 119896 on thechannel 119899 that guarantees its required QoS Similarly wedefine every SUrsquos minimum target SINR 120574119899

119897 and the channel

outage 120585119899119897constraint of each SU 119897 on the channel 119899 is

Pr (120574119899119897lt 120574119899

119897) lt 120585119899

119897 forall119897 forall119899 (5)

For each PU 119896 taking (3) into (4) we can get

119871

prod119897

(1 + 120588119899

119897119896

120574119899

119896119901119899

119897ℎ119899

119897119896

119901119899119896ℎ119899119896119896

) lt1

1 minus 120585119899119896

forall119896 forall119899 (6)

After rewriting (6) we can obtain

119871

sum119897

log(1 + 120588119899119897119896

120574119899

119896119901119899

119897ℎ119899

119897119896

119901119899119896ℎ119899119896119896

) le log( 1

1 minus 120585119899119896

) forall119896 forall119899 (7)

International Journal of Distributed Sensor Networks 3

Similarly taking (3) into (5) rewrite (5) as follows

119871+1

sum119895 =119897

log(1 + 120588119899119895119897

120574119899

119897119901119899

119895ℎ119899

119895119897

119901119899119897ℎ119899119897119897

) le log( 1

1 minus 120585119899119897

) forall119897 forall119899 (8)

In addition the global aggregate interference from allSUs to each channel should not be larger than the maximuminterference threshold 119879119899

119896to ensure the SUsrsquo transmission

would not cause unendurable interference on every channelof each PU Mathematically this can be written as

119871

sum119897=1

120588119899

119897119896119901119899

119897ℎ119899

119897119896lt 119879119899

119896 forall119896 forall119899 (9)

3 Stackelberg Game Theoretic Approach

The PUs (leaders) price the SUs (followers) to control theinterference power made by the SUs under the IPC EachPU will offer a suitable price to maximize its revenue byselling resource to SUs Based on the interference priceprovided by PUs each SU will adjust its transmission powerto maximize its revenue PUs have higher priority than SUswe use Stackelberg game to model the strategy between thePUs and SUs

Then as the leaders PUswillmaximize their utility (SINRperformance plus the payment from the SUs occupying thechannels) The utility function of the PU 119896 is as follows

119880119896(p119896 pminus119896 119908119896) =

119873

sum119899=1

120588119899

119896120574119899

119896+ 119908119896

119873

sum119899=1

119871

sum119897=1

120588119899

119897119896119901119899

119897ℎ119899

119897119896 (10)

where if the PU 119896 transmit on the channel 119899 120588119899119896= 1

otherwise 120588119899119896= 0 p

119896= (1199011

119896 1199012

119896 119901

119873

119896) is the transmission

power vector over the transmitted channels of the PU 119896 pminus119896

represent the transmit power vector of all users except the PU119896 119908119896denotes that the PU 119896 charges the price over all SUs if

the SUs transmit on the channel of the PUHence the revenue utility optimization problem for the

PU 119896 is as follows

P1 Max 119880119896(p119896 pminus119896 119908119896)

st (8) (10) 119901119899

119896ge 0

119873

sum119899=1

120588119899

119896119901119899

119896le 119875

max119896

(11)

where 119875max119896

is the maximum transmission power of the PU 119896Then for SUs (followers) we set the revenue utility of the

119897th SU with two parts the first one is the income from theSINR achieved from the PUs The second one is the paymentfor the PUs Then the revenue utility function of the 119897th SUis as follows

119880119897(p119897 pminus119897w119897) =

119873

sum119899=1

120588119899

119897120574119899

119897minus

119873

sum119899=1

119870

sum119896=1

120588119899

119897119896119908119896119901119899

119897ℎ119899

119897119896 (12)

where if the SU 119897 transmit on the channel 119899 120588119899119897= 1 otherwise

120588119899

119897= 0 p

119897= (1199011

119897 1199012

119897 119901

119873

119897) is the transmission power vector

over the transmitted channels of the SU 119897 pminus119897represent the

transmission power vector of all users except the SU 119897 w119897=

(1199081 1199082 119908

119870) denotes that the SU 119897 pays the price vector

for all PUs if the SU 119897 does not transmit on the channel of thePU 119896 119908

119896= 0

Hence the optimization problem for the SU 119897 is as follows

P2 Max 119880119897(p119897 pminus119897w119897)

st (9) (10) 119901119899

119897ge 0

119873

sum119899=1

120588119899

119897119901119899

119897le 119875

max119897

(13)

where 119875max119897

is the maximum transmission power of the 119897thSU

4 Solution of the Proposed Stackelberg Game

The optimization problems P1 and P2 taken together are theproposed Stackelberg game problemwith several constraintsThe goal of the proposed Stackelberg game is to achieve theSE in which point both PUs and SUs have no incentive todeviate [3] so the distributive algorithms convergence tothe SE is difficult Thus we depart the game problem intosuboptimal independent solution and employ an iterativealgorithm to achieve the SE

41 Global Efficiency of the RSE In noncooperative gamesthe existence and uniqueness of equilibrium are not alwaysachieved [11] due to multiple playersrsquo competition Hence forthemultifollower subgamewe should study the existence anduniqueness of the global SG response to the leadersrsquo priceIn particular a variational equilibrium (VE) [12] is appliedto analyze the global SG for our case This is because a VEis more stable than any other generalized Nash equilibriumunder parameter uncertainty [13] Particularly a number ofSUs aim to achieve their QoS requirement through applyingthe resource from PUs in cognitive radio networks VE isregarded as an appropriate solution

For a market fixed price at the PUs all the SUs aim tomaximize their own utility by buying the resource throughPUsThus we formulate a new objective function for all SUsthe new utility is the sum utilities of all SUs and can beexpressed as follows

119865

Sum =119871

sum119897=1

119873

sum119899=1

120588119899

119897120574119899

119897minus

119871

sum119897=1

119873

sum119899=1

119870

sum119896=1

120588119899

119897119896119908119896119901119899

119897ℎ119899

119897119896 (14)

Therefore in order to achieve the actable outcome ofthe proposed SG our goal is guarantying the existence anduniqueness of the SGwhenmaximizing (14)The correspond-ing Lagrangian function and the KTT conditions for the SU119897 expressed in and the KTT conditions for the SU 119897 are givenby

nablap119897119880119897 (p119897 pminus119897w119897) minus nablap119897 (119873

sum119899=1

120588119899

119897119901119899

119897minus 119875

max119897

)120592119897

minus nablap119897 (119873

sum119899=1

120596119899

119897(

119871+1

sum119895 =119897

log(1 + 120588119899119895119897

120574119899

119897119901119899

119895ℎ119899

119895119897

119901119899119897ℎ119899119897119897

)

4 International Journal of Distributed Sensor Networks

minus log( 1

1 minus 120585119899119897

))) minus nablap119897

119873

sum119899=1

120593119899

119897(

119871

sum119897=1

120588119899

119897119896119901119899

119897ℎ119899

119897119896

minus 119879119899

119896)120592119897(

119873

sum119899=1

120588119899

119897119901119899

119897minus 119875

max119897

) = 0

119873

sum119899=1

120596119899

119897(

119871+1

sum119895 =119897

log(1 + 120588119899119895119897

120574119899

119897119901119899

119895ℎ119899

119895119897

119901119899119897ℎ119899119897119897

) minus log( 1

1 minus 120585119899119897

))

= 0

119873

sum119899=1

120593119899

119897(

119871

sum119897=1

120588119899

119897119896119901119899

119897ℎ119899

119897119896minus 119879119899

119896) = 0

(15)

We note that the robust followersrsquo game shows a jointlyconvex generalized SE problem therefore the solution of theSE problem with constraints in (13) is a variational inequalityVI(P F) where P is the set of joint convexity It is importantto determine a vector 119911lowast isin P sub 119877119899 such that ⟨F(zlowast) zminuszlowast⟩ ge0 for all 119911 isin P and F(p) = minus(nabla

119901119897(p119897))119871

119897=1[13] Then the

solution of VI(P F) is a variational SEIn this paper we only focus on the power control in cog-

nitive radio networks by assuming the channel assignmenthas already been done Then we can divide the variationalinequality VI(P F) into 119873 subproblems each subproblemdenotesVI(P

119899 F119899) on the subchannel 119899 and they are indepen-

dentTherefore on the subchannel 119899 the KKT conditions canbe expressed as [12]

F119899(p) + 120592

119899nabla119901(

119871

sum119897=1

120588119899

119897119901119899

119897minus

119871

sum119897=1

120588119899

119897119875max119897

)

+ 120596119899nabla119901(

119871

sum119897=1

(

119871+1

sum119895 =119897

log(1 + 120588119899119895119897

120574119899

119897119901119899

119895ℎ119899

119895119897

119901119899119897ℎ119899119897119897

))

minus

119871

sum119897=1

log( 1

1 minus 120585119899119897

)) + 120593119899nabla119901(

119871

sum119897=1

120588119899

119897119896119901119899

119897ℎ119899

119897119896minus 119879119899

119896)

(16)

Now from the definition of [12] we have

F1=

[[[[[[[[[[[[[

[

1205881

1119896(119908119896ℎ1

1119896minus

1

119868 (1199011minus1))

1205881

2119896(119908119896ℎ1

2119896minus

1

119868 (1199011minus2))

1205881

119871119896(119888119896ℎ1

119871119896minus

1

119868 (1199011minus119871))

]]]]]]]]]]]]]

]

F119899=

[[[[[[[[[[[[[

[

120588119899

1119896(119888119896ℎ119899

1119896minus

1

119868 (119901119899minus1))

120588119899

2119896(119888119896ℎ119899

2119896minus

1

119868 (119901119899minus2))

120588119899

119871119896(119888119896ℎ119899

119871119896minus

1

119868 (119901119899minus119871))

]]]]]]]]]]]]]

]

(17)

Therefore the Jacobian of F119899is

J1=

[[[[[[[

[

1205881

1119896ℎ1

11198960 sdot sdot sdot 0

0 1205881

2119896ℎ1

2119896sdot sdot sdot 0

0

0 0 sdot sdot sdot 1205881

119871119896ℎ1

119871119896

]]]]]]]

]

J119899=

[[[[[[

[

120588119899

1119896ℎ119899

11198960 sdot sdot sdot 0

0 120588119899

2119896ℎ119899

2119896sdot sdot sdot 0

0

0 0 sdot sdot sdot 120588119899

119871119896ℎ119899

119871119896

]]]]]]

]

(18)

Each F119899J119899is a diagonal matrix and all the diagonal

elements are positive Therefore F119899J119899is positive definition

on P119899 and so F

119899is strictly monotone Hence the global

SG problem admits a unique global variational equilibriumsolution [12] Due to the jointly convex nature of the globalSE problem the variational equilibrium is the unique globalmaximizer of (14) [12] which completes the proof in theliterature [12]

42 Solution of the Optimization for SUs (Followers) For theP2 the utility function of each SU is a concave function of 119901119899

119897

and the constraints are all linear so a partial Lagrange dualdecompositionmethod (LDDM) [4] for the problems is used

For the problem in (13) the corresponding Lagrangianfunction for the SU 119897 on the subchannel 119899 can be expressedas

L119897(p119897w119896 120596119897 120592119897120593119897) =

119873

sum119899=1

120588119899

119897120574119899

119897minus

119873

sum119899=1

119870

sum119896=1

120588119899

119897119896119908119896119901119899

119897ℎ119899

119897119896

minus

119873

sum119899=1

120596119899

119897(

119871+1

sum119895 =119897

log(1 + 120588119899119895119897

120574119899

119897119901119899

119895ℎ119899

119895119897

119901119899119897ℎ119899119897119897

)

minus log( 1

1 minus 120585119899119897

)) minus 120592119897(

119873

sum119899=1

120588119899

119897119901119899

119897minus 119875119898

119897) minus

119873

sum119899=1

120593119899

119897

sdot (

119871

sum119897=1

120588119899

119897119896119901119899

119897ℎ119899

119897119896minus 119879119899

119896)

(19)

International Journal of Distributed Sensor Networks 5

where 120596119899119897 120592119897 and 120593119899

119897are the nonnegative dual variables of the

constraints in (13)We decompose the optimization problem into 119873 inde-

pendent subproblems Then on the subchannel 119899 taking theKarush-Kuhn-Tucker (KKT) condition [4]

120597L119897

120597119901119899119897

=ℎ119899

119897119897

119868 (p119899minus119897)minus 119908119896ℎ119899

119897119896

+ 120596119899

119897

119871+1

sum119895 =119897

(120588119899

119895119897

120574119899

119897119901119899

119895ℎ119899

119895119897

119901119899119897(119901119899119897ℎ119899119897119897+ 120574119899

119897119901119899119895ℎ119899119895119897) ln 2

)

minus 120592119897minus 120593119896ℎ119899

119897119896

(20)

Simply in (20) we assume that the 119895th user causing theinterference power 119901119899

119895ℎ119899

119895119897to the SU 119897 on the subchannel 119899 can

be denoted as the average interference power except the SU 119897119866119899

119897= 119864[sum

119871+1

119895=1119895 =119897120588119899

119895119897119901119899

119895ℎ119899

119895119897] Hence we rewrite (20) as follows

120597L119897

120597119901119899119897

=ℎ119899

119897119897

119868 (p119899minus119897)minus 119908119896ℎ119899

119897119896+ 120596119899

119897

120574119899

119897119866119899

119897(sum119871+1

119895 =119897120588119899

119895119897)

119901119899119897(119901119899119897ℎ119899119897119897+ 120574119899

119897119866119899119897) ln 2

minus 120592119897minus 120593119896ℎ119899

119897119896

(21)

Set (21) to zero and get the optimal transmission power ofthe SU 119897 if it transmits on the channel 119899

119901119899

119897

lowast

=

minus120574119899

119897119866119899

119897+ radic4120596119899

119897ℎ119899119897119897120574119899

119897119866119899119897(sum119871+1

119895 =119897120588119899119895119897) 119883 ln 2 + 120574119899

119897119866119899119897

2ℎ119899119897119897

(22)

where119883119899119897= (119908119896+ 120593119899

119897)ℎ119899

119897119896+ 120592119897minus ℎ119899

119897119897119868(p119899minus119897)

The transmission power of the SU 119897 is zero if theinterference price for it is larger than payoff threshold 119876119899

119897on

the channel 119899 Then setting 119901119899119897

lowast

= 0 we can get the payoffthreshold of the SU 119897 if it transmits on the channel 119899

119876119899

119897=1

ℎ119899119897119896

(4120596119899

119897ℎ119899

119897119897120574119899

119897119866119899

119897(sum119871+1

119895 =119897120588119899

119895119897)

((120574119899

119897119866119899119897)2

minus 120574119899

119897119866119899119897) ln 2

minus 120592119897minus 120593119899

119897ℎ119899

119897119896

+ℎ119899

119897119897

119868 (p119899minus119897))

(23)

From (23) if the price 119908119896gt 119876119899

119897 the price is above the

payoff threshold of the SU 119897 and it will stop transmitting onthe channel 119899 without buying the interference power

43 Solution of the Optimization for PUs (Leaders) In orderto maximize its own utility each PU needs to adaptivelyoffer an interference price to SUs based on transmit powerresponse of the SUs P1 can be decomposed into twosubproblems fix 119908

119896to get the optimal transmission power

of each PU 119896 and then search the optimal 119908119896 The optimal

transmission power of the PU 119896 can be applied by the previousLDDM

Thus for P1 in (11) the corresponding Lagrangian func-tion can be given as

L119896(p119896 119908119896120582119896 120583119896 ^119896) =

119873

sum119899=1

120588119899

119896120574119899

119896+ 119908119896

119873

sum119899=1

119871

sum119897=1

120588119899

119897119896119901119899

119897ℎ119899

119897119896

minus

119873

sum119899=1

120582119899

119896(

119871

sum119897

log(1 + 120588119899119897119896

120574119899

119896119901119899

119897ℎ119899

119897119896

119901119899119896ℎ119899119896119896

)

minus log( 1

1 minus 120585119899119896

)) minus 120583119896(

119873

sum119899=1

120588119899

119896119901119899

119896minus 119875119898

119896) minus

119873

sum119899=1

]119899119896

sdot (

119871

sum119897=1

120588119899

119897119896119901119899

119897ℎ119899

119897119896minus 119879119899

119896)

(24)

where 120582119899119896 120583119896 and ]119899

119896are the dual variables of the constraints

in (11)Similarly in (24) we assume the 119897th SU causing the inter-

ference ℎ119899119897119896119901119899

119897to the PU 119896 on the channel 119899 can be denoted

as the average interference power 119866119899119896= 119864[sum

119871

119897=1120588119899

119897119896ℎ119899

119897119896119901119899

119897]

According to the KKT conditions we obtain the optimaltransmission power of the PU 119896 if it transmits on the channel119899

119901119899

119896

lowast

=1

2ℎ119899119896119896

(minus120574119899

119896119866119899

119896

+ radic4120582119899

119896ℎ119899

119896119896120574119899

119896119866119899

119896(sum119871

119897=1120588119899

119897119896)

(120583119896minus ℎ119899119896119896119868 (p119899minus119896)) ln 2

+ 120574119899

119896119866119899119896)

(25)

Since L119896(p119896 119908119896120582119896 120583119896 ^119896) is a stepwise function with

breakpoints at119876119899119897for the SU 119897 we should discuss the existence

of the optimal price119908119896first So we divide (24) with respect to

119908119896with two parts on each channel 119899 we have L

119901119896(119901119899

119897) =

120574119899

119896(119901119899

119897) and L

119901119896(119908119896) = (119908

119896minus ]119896)119901119899

119897ℎ119897119896 From (22) it can

be easily observed that 120574119899119896(119901119899

119897) is a concave function of 119908

119896

Therefore we only need to discuss the situation ofL119901119896(119908119896)

For the SU 119897 we first sort119876119899119897(119899 = 1 119873) in ascending order

and have 119873 intervals (0 1198761119897)(1198761

119897 1198762

119897) (119876

119873minus1

119897 119876119873

119897) where

1198761

119897lt 1198762

119897lt sdot sdot sdot lt 119876

119873

119897 Note if the SU 119897 is not allocated on the

channel 119899 (119876119899minus1119897 119876119899

119897)must be taken out of the order We take

(0 1198761

119897) for an example When 119908

119896rarr 0 we can derive that

120597L119901119896(119908119896)

120597119908119896

1003816100381610038161003816100381610038161003816100381610038161003816119908119896rarr0

gt 119901119899

119897ℎ119897119896gt 0 (26)

Taking the second derivative of L119901119896(119908119896) with respect to 119908

119896

is1205972L119901119896

1205971199082119896

= minusℎ119899

119897119897

2

sdot radic120596119899

119897120574119899

119897119866119899

119897(sum119871+1

119895 =119897120588119899

119895119897)

119883119899119897ln 2

+ 120574119899

119897119866119899119897(120596119899

119897ℎ119899

119897119896120574119899

119897119866119899

119897(sum119871+1

119895 =119897120588119899

119895119897)

(119883119899119897)2 ln 2

)

lt 0

(27)

6 International Journal of Distributed Sensor Networks

(01) Initialization set 119902 = 0 set initial 119888119896(119902) and p

119896(119902) for 119896 isin 119870 Set initial p

119897(119902) for 119897 isin 119871

Set 120591 where 120591 is positive and sufficiently small(02) For each SU 119897

(03) Use SM to find the optimal step sizes 120572lowast 120573lowast and 120579lowast and update 120596119899119897 120592119897and 120593

119896according to (28) respectively

(04) For the given 119888119896(119902) and p

119896(119902) of all PUs each SU 119897 responds with its transmit power vector plowast

119897(119902 + 1) according to (22)

(05) If 119876119899119897lt 119888119896(119902) the SU 119897 stops transmitting on the channel 119899 of the PU 119896

(06) End(07) For each PU 119896

(08) Use SM to find the optimal 120594lowast and 120599lowast and update 120582119899119896and 120583

119896by (29)

(09) For the responded plowast119897(119902) of all SUs each PU 119896 updates its transmit power vector as plowast

119896(119902 + 1) according to (25)

(10) Each PU 119896 updates its price by the solution of119888119896(119902 + 1) = argmax

119888119896

119880119896(p119896(119902 + 1) p

minus119896(119902 + 1))

(11) End(12) For each PU 119896 if plowast

119896(119902 + 1) minus p

119896(119902) le 120591 or 119902 gt 102 stop the algorithm Otherwise 119902 = 119902 + 1

repeat steps (02) and (11) until the condition is satisfied

Algorithm 1 Iterative algorithm for reaching the SE

L119901119896(119908119896) is a concave function whether (120597L

119901119896120597119908119896)|119908119896rarr119876

1

119897

gt 0 or (120597L119901119896120597119908119896)|119908119896rarr119876

1

119897

lt 0 except at the nondifferen-tiable point 1198761

119897

Through the above analysis L119896(p119896 119908119896120582119896 120583119896 ^119896) is a

concave function with respect to 119908119896except at 119876119899

119897 The

ellipsoid method [4] can be employed to solve the convexoptimization in each interval

44 Iterative Algorithm to Find the SE For the above dis-cussion we propose an iterative algorithm to search the SEDue to the fact that computation of the dual variables isa complicated task the subgradient method (SM) [14] isapplied to obtain the global optimum SE of this problemThen dual variables are updated as follows

120596119899

119897(119905 + 1) = (120596

119899

119897(119905) + 120572(

119871+1

sum119895 =119897

log(1 + 120588119899119895119897

120574119899

119897119901119899

119895ℎ119899

119895119897

119901119899119897ℎ119899119897119897

)

minus log( 1

1 minus 120585119899119897

)))

+

120592119897(119905 + 1) = (120592

119897(119905) + 120573(

119873

sum119899=1

120588119899

119897119901119899

119897minus 119901

max119897))

+

120593119899

119897(119905 + 1) = (120593

119899

119897(119905) + 120579(

119871

sum119897=1

120588119899

119897119896119901119899

119897ℎ119899

119897119896minus 119879119899

119896))

+

(28)

120582119899

119896(119905 + 1) = (120582

119899

119896(119905) + 120599(

119871

sum119897

log(1 + 120588119899119897119896

120574119899

119896119901119899

119897ℎ119899

119897119896

119901119899119896ℎ119899119896119896

)

minus log( 1

1 minus 120585119899119896

)))

+

120583119896(119905 + 1) = (120583

119896(119905) + 120594(

119873

sum119899=1

120588119899

119896119901119899

119896minus 119875

max119896

))

+

(29)

where 119905 is the iteration index and 120572 gt 0 120573 gt 0 120579 gt 0 120599 gt0 and 120594 gt 0 are sufficiently small The SM guarantees theconvergence of the above optimal dual variables if the stepsizes are chosen by following the step size policy [14]

We then design the iterative algorithm to achieve the SEshown in Algorithm 1

5 Simulation Results and Their Analysis

In this section several numerical examples are presented toevaluate the performances of the proposed SG by comparingthe optimal price-based SG considering global interferencein [5] and the nominal SG without considering globalinterference and outage constraints The cell radius of 500mwith the PBS centered at the original CRN The simulationparameters are as follows 1205752 = 10minus12W 119870 = 3 119871 = 5 and119873 = 10 The SINR threshold of PUs and SUs is set as 7 dBand 4 dB respectively All PUs and SUs deploy the maximumpower 119875max

119896= 100mWand 119875max

119897= 50mWThe channel gain

in this system is ℎ119894119895= 119889minus4

119894119895 with 119889

119894119895being their corresponding

distance The outage probability thresholds of both PUs andSUs are 0001 We set the interference-to-noise ratio (INR) as1198791205752Firstly we illustrate the convergence of the proposed

algorithm for achieving an SE of the proposed game FromFigure 2 the three PUs and five SUs iteratively update theirutilities and obviously converge to the SE The proposedalgorithm converges quickly in terms of PUs only about tentimes In addition due to the larger number of SUs theconvergence of the SUs is slower than that of the PUs

51 Impact of INR In this subsection we set the numberof PUs and SUs as 3 and 5 and INR changes from minus20 dBto 20 dB which means that IPC changes from 10

minus14W to10minus10WWe then consider the sum rate of PUs and SUs for the

three solutionswith different tolerant interference constraintsshown in Figures 3 and 4 For the performance of sum rate

International Journal of Distributed Sensor Networks 7

PU-1PU-2PU-3

SU-1SU-2SU-3

SU-4SU-5

12

13

14

15

16

Util

ities

of P

Us

1

2

3

4

5

6

7

Util

ities

of S

Us

50 15 20 2510

Iteration

50 15 20 2510

Iteration

Figure 2 The convergence of utility of PUs (leaders) setting priceof PUs and utility of SUs (followers)

of SUs Figure 3 shows the nominal SG scheme outperformsother two schemes when the interference temperature levelis stringent but is inferior to the two schemes when it isloose The proposed SG performs the worst because ofthe demand of satisfying the outage probability constraintOnce the interference constraints are loose enough to benot active accordingly our proposed solution works betterthan the others For the leaders the sum rate of the nominalSG performs the worst because of not including the globalinterference constraints This is because the performance ofPUs may be degraded with the increases of the interference

Figure 5 presents the outage probability of the systemwith different INR It is observed that the proposed schemeachieves much lower outage probability than other schemesin particular the performance gap becomes larger with theincrease of INR This is because our proposed algorithmworks best by considering the outage probability constraintsof users which prevents the outage events well

52 Impact of Different Number of SUs In this subsection theINR is set to be 10 dB The number of SUs changes from 2 to20 All the other simulation parameters are the same as thebeginning part of this section

For the PUs from Figure 6 the sum rate of the SGsolution performs the worst because of not including theglobal interference constraints So the PUs may refuse tosell more spectrum resource because of protecting their own

Our proposed SGOptimal price-based SG in [5]Nominal SG

50 10 15 20minus5minus15 minus10minus20

INR T1205752 (dB)

25

26

27

28

29

30

31

32

33

34

Sum

rate

of t

he P

Us (

bits

sH

z)Figure 3 Sum rate of PUs versus INR

26

24

22

20

18

16

14

12

10minus20 minus15 minus10 minus5 0 5 10 15 20

Our proposed SGOptimal price-based SG in [5]Nominal SG

INR T1205752 (dB)

Sum

rate

of t

he S

Us (

bits

sH

z)

Figure 4 Sum rate of SUs versus INR

communication QoS The proposed scheme outperformsmost in terms of sum rate of SUs because the algorithm allowsmore SUs to share their radio resource so that it increases PUsrsquoutilities with considering the channel uncertainty and globalinterference constraints

Figure 7 shows the sum rate of SUs versus the numberof SUs The sum rate of SUs performance of our proposedalgorithm works better than other two schemes this isbecause the proposed scheme is able to support more SUs attheBS shown in Figure 7 so thatmore SUshave opportunities

8 International Journal of Distributed Sensor Networks

Our proposed SGOptimal price-based SG in [5]Nominal SG

10minus3

10minus2

10minus1

100

Out

age p

roba

bilit

y

minus15 minus10 minus5 0 5 10 15 20minus20

INR T1205752 (dB)

Figure 5 Outage probability versus INR

Our proposed SGOptimal price-based SG in [5]Nominal SG

16

18

20

22

24

26

28

30

32

34

Sum

rate

of P

Us

4 6 82 12 14 16 18 2010

Number of SUs

Figure 6 Sum rate of PUs versus number of SUs

to transmit which increase the sum rate In addition theperformance gap between the proposed scheme and theother algorithms increases when the network grows largerby which it can be concluded that the proposed scheme ismore suitable for application in larger networks Because thescheme sets the adaptive punishment parameter among allserved SUs to control their behavior more SUs can be servedat the BS thus achieving a higher sum rate

Figure 8 shows the number of outage probability com-parison versus the number of SUs for different algorithmsThe proposed algorithm is able to support more SUs than

Our proposed SGOptimal price-based SG in [5]Nominal SG

4 6 8 10 12 14 16 18 202

Number of SUs

10

15

20

25

30

35

40

45

Sum

rate

of S

Us

Figure 7 Sum rate of SUs versus number of SUs

Our proposed SGOptimal price-based SG in [5]Nominal SG

4 6 8 10 12 14 16 18 202

Number of SUs

10minus4

10minus3

10minus2

10minus1

100

Out

age p

roba

bilit

y

Figure 8 Outage probability versus number of SUs

other schemes This is because we develop the channelassignment scheduling scheme to decrease the probabilityof the unserved SUs so more SUs are admitted to serveat the BS without causing unendurable interference to PUsIn addition the interference among SUs is taken into therevenue utility function to void serious interference for someSUs who have bad channel condition

International Journal of Distributed Sensor Networks 9

6 Conclusion

In this paper we propose a Stackelberg game for powercontrol problem in CRNs with channel outage constraintsand global interference constraints We employ LDDM tosolve the problem by decomposing it into independentsubproblems and develop an iterative algorithm to achieveSE Simulation results show that the proposed algorithmimproves the performance compared with other game algo-rithms

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the National Nature ScienceFoundation of China (61271259 and 61301123) the SpecialFund of Chongqing Key Laboratory (CSTC) the Programfor Changjiang Scholars and Innovative Research Team inUniversity (IRT2129) and the Graduate Student ResearchInnovation Project of Chongqing University of Posts andTelecommunications (Chongqing) (CYS14143)

References

[1] K Akkarajitsakul E Hossain D Niyato and D I Kim ldquoGametheoretic approaches for multiple access in wireless networksa surveyrdquo IEEE Communications Surveys and Tutorials vol 13no 3 pp 372ndash395 2011

[2] Y Wu T Zhang and D H K Tsang ldquoJoint pricing andpower allocation for dynamic spectrum access networks withStackelberg game modelrdquo IEEE Transactions on Wireless Com-munications vol 10 no 1 pp 12ndash19 2011

[3] X Kang R Zhang and M Motani ldquoPrice-based resourceallocation for spectrum-sharing femtocell networks a stack-elberg game approachrdquo IEEE Journal on Selected Areas inCommunications vol 30 no 3 pp 538ndash549 2012

[4] R Xie F R Yu H Ji and Y Li ldquoEnergy-efficient resourceallocation for heterogeneous cognitive radio networks withfemtocellsrdquo IEEETransactions onWireless Communications vol11 no 11 pp 3910ndash3920 2012

[5] Z Wang L Jiang and C He ldquoOptimal price-based power con-trol algorithm in cognitive radio networksrdquo IEEE TransactionsonWireless Communications vol 13 no 11 pp 5909ndash5920 2014

[6] M Le Treust S Lasaulce Y Hayel and G L He ldquoGreen powercontrol in cognitive wireless networksrdquo IEEE Transactions onVehicular Technology vol 62 no 4 pp 1741ndash1754 2013

[7] Y Xiao G Bi D Niyato and L A DaSilva ldquoA hierarchicalgame theoretic framework for cognitive radio networksrdquo IEEEJournal on Selected Areas in Communications vol 30 no 10 pp2053ndash2069 2012

[8] F Gabry N Li N Schrammar M Girnyk L K Rasmussenand M Skoglund ldquoOn the optimization of the secondarytransmitterrsquos strategy in cognitive radio channels with secrecyrdquoIEEE Journal on Selected Areas in Communications vol 32 no3 pp 451ndash463 2014

[9] D Niyato E Hossain and Z Han ldquoDynamics of multiple-seller and multiple-buyer spectrum trading in cognitive radionetworks a game-theoretic modeling approachrdquo IEEE Transac-tions on Mobile Computing vol 8 no 8 pp 1009ndash1022 2009

[10] S Kandukuri and S Boyd ldquoOptimal power control ininterference-limited fading wireless channels with outage-probability specificationsrdquo IEEE Transactions on Wireless Com-munications vol 1 no 1 pp 46ndash55 2002

[11] T Basar and G J Olsder Dynamic Noncooperative GameTheory SIAM Philadelphia Pa USA 1999

[12] F Facchinei and C Kanzow ldquoGeneralized Nash equilibriumproblemsrdquo 4OR vol 5 no 3 pp 173ndash210 2007

[13] D Ardagna B Panicucci and M Passacantando ldquoA game the-oretic formulation of the service provisioning problem in cloudsystemsrdquo in Proceedings of the 20th International Conference onWorld Wide Web (WWW rsquo11) pp 177ndash186 ACM April 2011

[14] D Bertsekas W Hager and O Mangasarian Nonlinear Pro-gramming Athena Scientific Belmont Mass USA 1999

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RotatingMachinery

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Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

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Shock and Vibration

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Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

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Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

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Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Chemical EngineeringInternational Journal of Antennas and

Propagation

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Navigation and Observation

International Journal of

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DistributedSensor Networks

International Journal of

2 International Journal of Distributed Sensor Networks

h1h1

h2

hL

h11

h21

h22

h22h1K

SUi

SU2

SU1

PU2

PU1

PUK

PBSSBS

h2

h2K

hLK

hK

hL2

hL1

Figure 1 System model

Stackelberg game In [7] the authors formulated a Stackelberggame model to maximize the payoff of both SUs and PUs byjointly optimizing transmission powers of SUs and subbandallocations of SUs Moreover in [8] the authors analyzedthe cooperation interaction between the PUs and SUs trans-mitters from a Stackelberg game theoretic perspective withsecrecy constraints Moreover [9] proposed a game theoreticframework to model the interactions among multiple PUsand multiple SUs under different system parameter settingsand under system perturbation

However the above papers aim to maximize usersrsquo rev-enue without considering the outage probability of multi-PUs and multi-SUs It is clear that both PUs and SUs need toupdate their transmission power frequently to maintain theirSignal to Interference plus Noise Ratio (SINR) level (usuallyreferred to as QoS) due to channel fading and interferencesfrom other users when they transmit on the same channelTherefore in CRNs the fundamental performance traits ofmultiple SUs power control should be investigated with mul-tiple PUs also the issues ofmultiple PUs power control shouldbe studied in the presence of multiple SUs in fading channelsand interferences In the game problem formulation theoutage probabilities of both PUs and SUs represent their ownutility below the required target SINR used to guarantee theirdesired QoS Hence the channel outage probabilities withrespect to both PUs and SUs transmissions should be requiredin the multiple usersrsquo game framework if usersrsquo SINR fallsbelow a certain threshold

In this paper we proposed a new Stackelberg game tomodel the hierarchical competition between PUs and SUs inCRNs with global interference and outage constraints Wetheoretically prove the existence and uniqueness of robustStackelberg equilibrium for the noncooperative approachThen we employ the Lagrange dual decomposition methodto solve the game problem by decomposing the game prob-lem into independent suboptimal solution Furthermore wedevelop an efficient iterative algorithm to converge the SE

2 System Model

We consider a CRN composed of a single PBS with a set ofPUs K fl 1 119870 having priority to use a set of channelsN fl 1 119873 and a secondary base station (SBS) with aset of SUs L fl 1 119871 allowed to share 119873 channels fromthe PBS shown in Figure 1We define the channel gains of the

PU 119896 and the SU 119897 on the channel 119899 as ℎ119899119896119896and ℎ119899119897119897 respectively

The channel gain between the 119896th PU transmitter and the SU119897 receiver is denoted by ℎ119899

119896119897 the channel gain from the 119897th SU

transmitter to the 119896th PU receiver is ℎ119899119897119896 The transmit power

of the 119896th PU and the 119897th SU on the channel 119899 is denoted by119901119899

119896and 119901119899

119897 We assume each user can transmit simultaneously

over multiple channels and one channel can be served onlyby one PU but can be used by multiple SUs

The SINR at the 119896th PU receiver on the 119899th channel canbe defined as

120574119899

119896=

119901119899

119896ℎ119899

119896119896

sum119871

119897=1120588119899119897119896119901119899119897ℎ119899119897119896+ 1205752

=119901119899

119896ℎ119899

119896119896

119868 (p119899minus119896) (1)

where if the PU 119896 and the SU 119897 transmit on the same channel119899 120588119899119897119896= 1 otherwise 120588119899

119897119896= 0 1205752 is variance of AWGN and

the interference is given by 119868(p119899minus119896) = sum119871

119897=1120588119899

119897119896119901119899

119897ℎ119899

119897119896+ 1205752

The SINR at the 119897th SU receiver on the channel 119899 can beexpressed by

120574119899

119897=

119901119899

119897ℎ119899

119897119897

sum119871

119895=1119895 =119897120588119899119897119895119901119899119895ℎ119899119897119895+ 119901119899119896ℎ119899119896119897+ 1205752

=119901119899

119897ℎ119899

119897119897

119868 (p119899minus119897) (2)

where 119868(p119899minus119897) = sum119871

119895=1119895 =119897120588119899

119897119895119901119899

119895ℎ119899

119897119895+ 119901119899

119896ℎ119899

119896119897+ 1205752

The required target SINR threshold 120574119899

119906of the user 119906

(PUs or SUs) which is not satisfied is defined as the outageprobability on the channel 119899 The outage probability Pr(120574119899

119906lt

120574119899

119906) of the user 119906 on the channel 119899 can be expressed as [10]

Pr (120574119899119906lt 120574119899

119906) = 1 minusprod

119894 =119906

(1 + 120588119899

119894119906

120574119899

119906119901119899

119894ℎ119899

119894119906

119901119899119906ℎ119899119906119906

)

minus1

(3)

Therefore the above equation shows the outage prob-ability of one user in the presence of multiple SUs andorPUs Accordingly the goal of each PU 119896 is to transmit withthe minimum target SINR 120574

119899

119896on the channel 119899 with the

prescribed limited channel outage 120585119899119896constraint

Pr (120574119899119896lt 120574119899

119896) lt 120585119899

119896 forall119896 forall119899 (4)

where (4) means the outage constraint of the PU 119896 on thechannel 119899 that guarantees its required QoS Similarly wedefine every SUrsquos minimum target SINR 120574119899

119897 and the channel

outage 120585119899119897constraint of each SU 119897 on the channel 119899 is

Pr (120574119899119897lt 120574119899

119897) lt 120585119899

119897 forall119897 forall119899 (5)

For each PU 119896 taking (3) into (4) we can get

119871

prod119897

(1 + 120588119899

119897119896

120574119899

119896119901119899

119897ℎ119899

119897119896

119901119899119896ℎ119899119896119896

) lt1

1 minus 120585119899119896

forall119896 forall119899 (6)

After rewriting (6) we can obtain

119871

sum119897

log(1 + 120588119899119897119896

120574119899

119896119901119899

119897ℎ119899

119897119896

119901119899119896ℎ119899119896119896

) le log( 1

1 minus 120585119899119896

) forall119896 forall119899 (7)

International Journal of Distributed Sensor Networks 3

Similarly taking (3) into (5) rewrite (5) as follows

119871+1

sum119895 =119897

log(1 + 120588119899119895119897

120574119899

119897119901119899

119895ℎ119899

119895119897

119901119899119897ℎ119899119897119897

) le log( 1

1 minus 120585119899119897

) forall119897 forall119899 (8)

In addition the global aggregate interference from allSUs to each channel should not be larger than the maximuminterference threshold 119879119899

119896to ensure the SUsrsquo transmission

would not cause unendurable interference on every channelof each PU Mathematically this can be written as

119871

sum119897=1

120588119899

119897119896119901119899

119897ℎ119899

119897119896lt 119879119899

119896 forall119896 forall119899 (9)

3 Stackelberg Game Theoretic Approach

The PUs (leaders) price the SUs (followers) to control theinterference power made by the SUs under the IPC EachPU will offer a suitable price to maximize its revenue byselling resource to SUs Based on the interference priceprovided by PUs each SU will adjust its transmission powerto maximize its revenue PUs have higher priority than SUswe use Stackelberg game to model the strategy between thePUs and SUs

Then as the leaders PUswillmaximize their utility (SINRperformance plus the payment from the SUs occupying thechannels) The utility function of the PU 119896 is as follows

119880119896(p119896 pminus119896 119908119896) =

119873

sum119899=1

120588119899

119896120574119899

119896+ 119908119896

119873

sum119899=1

119871

sum119897=1

120588119899

119897119896119901119899

119897ℎ119899

119897119896 (10)

where if the PU 119896 transmit on the channel 119899 120588119899119896= 1

otherwise 120588119899119896= 0 p

119896= (1199011

119896 1199012

119896 119901

119873

119896) is the transmission

power vector over the transmitted channels of the PU 119896 pminus119896

represent the transmit power vector of all users except the PU119896 119908119896denotes that the PU 119896 charges the price over all SUs if

the SUs transmit on the channel of the PUHence the revenue utility optimization problem for the

PU 119896 is as follows

P1 Max 119880119896(p119896 pminus119896 119908119896)

st (8) (10) 119901119899

119896ge 0

119873

sum119899=1

120588119899

119896119901119899

119896le 119875

max119896

(11)

where 119875max119896

is the maximum transmission power of the PU 119896Then for SUs (followers) we set the revenue utility of the

119897th SU with two parts the first one is the income from theSINR achieved from the PUs The second one is the paymentfor the PUs Then the revenue utility function of the 119897th SUis as follows

119880119897(p119897 pminus119897w119897) =

119873

sum119899=1

120588119899

119897120574119899

119897minus

119873

sum119899=1

119870

sum119896=1

120588119899

119897119896119908119896119901119899

119897ℎ119899

119897119896 (12)

where if the SU 119897 transmit on the channel 119899 120588119899119897= 1 otherwise

120588119899

119897= 0 p

119897= (1199011

119897 1199012

119897 119901

119873

119897) is the transmission power vector

over the transmitted channels of the SU 119897 pminus119897represent the

transmission power vector of all users except the SU 119897 w119897=

(1199081 1199082 119908

119870) denotes that the SU 119897 pays the price vector

for all PUs if the SU 119897 does not transmit on the channel of thePU 119896 119908

119896= 0

Hence the optimization problem for the SU 119897 is as follows

P2 Max 119880119897(p119897 pminus119897w119897)

st (9) (10) 119901119899

119897ge 0

119873

sum119899=1

120588119899

119897119901119899

119897le 119875

max119897

(13)

where 119875max119897

is the maximum transmission power of the 119897thSU

4 Solution of the Proposed Stackelberg Game

The optimization problems P1 and P2 taken together are theproposed Stackelberg game problemwith several constraintsThe goal of the proposed Stackelberg game is to achieve theSE in which point both PUs and SUs have no incentive todeviate [3] so the distributive algorithms convergence tothe SE is difficult Thus we depart the game problem intosuboptimal independent solution and employ an iterativealgorithm to achieve the SE

41 Global Efficiency of the RSE In noncooperative gamesthe existence and uniqueness of equilibrium are not alwaysachieved [11] due to multiple playersrsquo competition Hence forthemultifollower subgamewe should study the existence anduniqueness of the global SG response to the leadersrsquo priceIn particular a variational equilibrium (VE) [12] is appliedto analyze the global SG for our case This is because a VEis more stable than any other generalized Nash equilibriumunder parameter uncertainty [13] Particularly a number ofSUs aim to achieve their QoS requirement through applyingthe resource from PUs in cognitive radio networks VE isregarded as an appropriate solution

For a market fixed price at the PUs all the SUs aim tomaximize their own utility by buying the resource throughPUsThus we formulate a new objective function for all SUsthe new utility is the sum utilities of all SUs and can beexpressed as follows

119865

Sum =119871

sum119897=1

119873

sum119899=1

120588119899

119897120574119899

119897minus

119871

sum119897=1

119873

sum119899=1

119870

sum119896=1

120588119899

119897119896119908119896119901119899

119897ℎ119899

119897119896 (14)

Therefore in order to achieve the actable outcome ofthe proposed SG our goal is guarantying the existence anduniqueness of the SGwhenmaximizing (14)The correspond-ing Lagrangian function and the KTT conditions for the SU119897 expressed in and the KTT conditions for the SU 119897 are givenby

nablap119897119880119897 (p119897 pminus119897w119897) minus nablap119897 (119873

sum119899=1

120588119899

119897119901119899

119897minus 119875

max119897

)120592119897

minus nablap119897 (119873

sum119899=1

120596119899

119897(

119871+1

sum119895 =119897

log(1 + 120588119899119895119897

120574119899

119897119901119899

119895ℎ119899

119895119897

119901119899119897ℎ119899119897119897

)

4 International Journal of Distributed Sensor Networks

minus log( 1

1 minus 120585119899119897

))) minus nablap119897

119873

sum119899=1

120593119899

119897(

119871

sum119897=1

120588119899

119897119896119901119899

119897ℎ119899

119897119896

minus 119879119899

119896)120592119897(

119873

sum119899=1

120588119899

119897119901119899

119897minus 119875

max119897

) = 0

119873

sum119899=1

120596119899

119897(

119871+1

sum119895 =119897

log(1 + 120588119899119895119897

120574119899

119897119901119899

119895ℎ119899

119895119897

119901119899119897ℎ119899119897119897

) minus log( 1

1 minus 120585119899119897

))

= 0

119873

sum119899=1

120593119899

119897(

119871

sum119897=1

120588119899

119897119896119901119899

119897ℎ119899

119897119896minus 119879119899

119896) = 0

(15)

We note that the robust followersrsquo game shows a jointlyconvex generalized SE problem therefore the solution of theSE problem with constraints in (13) is a variational inequalityVI(P F) where P is the set of joint convexity It is importantto determine a vector 119911lowast isin P sub 119877119899 such that ⟨F(zlowast) zminuszlowast⟩ ge0 for all 119911 isin P and F(p) = minus(nabla

119901119897(p119897))119871

119897=1[13] Then the

solution of VI(P F) is a variational SEIn this paper we only focus on the power control in cog-

nitive radio networks by assuming the channel assignmenthas already been done Then we can divide the variationalinequality VI(P F) into 119873 subproblems each subproblemdenotesVI(P

119899 F119899) on the subchannel 119899 and they are indepen-

dentTherefore on the subchannel 119899 the KKT conditions canbe expressed as [12]

F119899(p) + 120592

119899nabla119901(

119871

sum119897=1

120588119899

119897119901119899

119897minus

119871

sum119897=1

120588119899

119897119875max119897

)

+ 120596119899nabla119901(

119871

sum119897=1

(

119871+1

sum119895 =119897

log(1 + 120588119899119895119897

120574119899

119897119901119899

119895ℎ119899

119895119897

119901119899119897ℎ119899119897119897

))

minus

119871

sum119897=1

log( 1

1 minus 120585119899119897

)) + 120593119899nabla119901(

119871

sum119897=1

120588119899

119897119896119901119899

119897ℎ119899

119897119896minus 119879119899

119896)

(16)

Now from the definition of [12] we have

F1=

[[[[[[[[[[[[[

[

1205881

1119896(119908119896ℎ1

1119896minus

1

119868 (1199011minus1))

1205881

2119896(119908119896ℎ1

2119896minus

1

119868 (1199011minus2))

1205881

119871119896(119888119896ℎ1

119871119896minus

1

119868 (1199011minus119871))

]]]]]]]]]]]]]

]

F119899=

[[[[[[[[[[[[[

[

120588119899

1119896(119888119896ℎ119899

1119896minus

1

119868 (119901119899minus1))

120588119899

2119896(119888119896ℎ119899

2119896minus

1

119868 (119901119899minus2))

120588119899

119871119896(119888119896ℎ119899

119871119896minus

1

119868 (119901119899minus119871))

]]]]]]]]]]]]]

]

(17)

Therefore the Jacobian of F119899is

J1=

[[[[[[[

[

1205881

1119896ℎ1

11198960 sdot sdot sdot 0

0 1205881

2119896ℎ1

2119896sdot sdot sdot 0

0

0 0 sdot sdot sdot 1205881

119871119896ℎ1

119871119896

]]]]]]]

]

J119899=

[[[[[[

[

120588119899

1119896ℎ119899

11198960 sdot sdot sdot 0

0 120588119899

2119896ℎ119899

2119896sdot sdot sdot 0

0

0 0 sdot sdot sdot 120588119899

119871119896ℎ119899

119871119896

]]]]]]

]

(18)

Each F119899J119899is a diagonal matrix and all the diagonal

elements are positive Therefore F119899J119899is positive definition

on P119899 and so F

119899is strictly monotone Hence the global

SG problem admits a unique global variational equilibriumsolution [12] Due to the jointly convex nature of the globalSE problem the variational equilibrium is the unique globalmaximizer of (14) [12] which completes the proof in theliterature [12]

42 Solution of the Optimization for SUs (Followers) For theP2 the utility function of each SU is a concave function of 119901119899

119897

and the constraints are all linear so a partial Lagrange dualdecompositionmethod (LDDM) [4] for the problems is used

For the problem in (13) the corresponding Lagrangianfunction for the SU 119897 on the subchannel 119899 can be expressedas

L119897(p119897w119896 120596119897 120592119897120593119897) =

119873

sum119899=1

120588119899

119897120574119899

119897minus

119873

sum119899=1

119870

sum119896=1

120588119899

119897119896119908119896119901119899

119897ℎ119899

119897119896

minus

119873

sum119899=1

120596119899

119897(

119871+1

sum119895 =119897

log(1 + 120588119899119895119897

120574119899

119897119901119899

119895ℎ119899

119895119897

119901119899119897ℎ119899119897119897

)

minus log( 1

1 minus 120585119899119897

)) minus 120592119897(

119873

sum119899=1

120588119899

119897119901119899

119897minus 119875119898

119897) minus

119873

sum119899=1

120593119899

119897

sdot (

119871

sum119897=1

120588119899

119897119896119901119899

119897ℎ119899

119897119896minus 119879119899

119896)

(19)

International Journal of Distributed Sensor Networks 5

where 120596119899119897 120592119897 and 120593119899

119897are the nonnegative dual variables of the

constraints in (13)We decompose the optimization problem into 119873 inde-

pendent subproblems Then on the subchannel 119899 taking theKarush-Kuhn-Tucker (KKT) condition [4]

120597L119897

120597119901119899119897

=ℎ119899

119897119897

119868 (p119899minus119897)minus 119908119896ℎ119899

119897119896

+ 120596119899

119897

119871+1

sum119895 =119897

(120588119899

119895119897

120574119899

119897119901119899

119895ℎ119899

119895119897

119901119899119897(119901119899119897ℎ119899119897119897+ 120574119899

119897119901119899119895ℎ119899119895119897) ln 2

)

minus 120592119897minus 120593119896ℎ119899

119897119896

(20)

Simply in (20) we assume that the 119895th user causing theinterference power 119901119899

119895ℎ119899

119895119897to the SU 119897 on the subchannel 119899 can

be denoted as the average interference power except the SU 119897119866119899

119897= 119864[sum

119871+1

119895=1119895 =119897120588119899

119895119897119901119899

119895ℎ119899

119895119897] Hence we rewrite (20) as follows

120597L119897

120597119901119899119897

=ℎ119899

119897119897

119868 (p119899minus119897)minus 119908119896ℎ119899

119897119896+ 120596119899

119897

120574119899

119897119866119899

119897(sum119871+1

119895 =119897120588119899

119895119897)

119901119899119897(119901119899119897ℎ119899119897119897+ 120574119899

119897119866119899119897) ln 2

minus 120592119897minus 120593119896ℎ119899

119897119896

(21)

Set (21) to zero and get the optimal transmission power ofthe SU 119897 if it transmits on the channel 119899

119901119899

119897

lowast

=

minus120574119899

119897119866119899

119897+ radic4120596119899

119897ℎ119899119897119897120574119899

119897119866119899119897(sum119871+1

119895 =119897120588119899119895119897) 119883 ln 2 + 120574119899

119897119866119899119897

2ℎ119899119897119897

(22)

where119883119899119897= (119908119896+ 120593119899

119897)ℎ119899

119897119896+ 120592119897minus ℎ119899

119897119897119868(p119899minus119897)

The transmission power of the SU 119897 is zero if theinterference price for it is larger than payoff threshold 119876119899

119897on

the channel 119899 Then setting 119901119899119897

lowast

= 0 we can get the payoffthreshold of the SU 119897 if it transmits on the channel 119899

119876119899

119897=1

ℎ119899119897119896

(4120596119899

119897ℎ119899

119897119897120574119899

119897119866119899

119897(sum119871+1

119895 =119897120588119899

119895119897)

((120574119899

119897119866119899119897)2

minus 120574119899

119897119866119899119897) ln 2

minus 120592119897minus 120593119899

119897ℎ119899

119897119896

+ℎ119899

119897119897

119868 (p119899minus119897))

(23)

From (23) if the price 119908119896gt 119876119899

119897 the price is above the

payoff threshold of the SU 119897 and it will stop transmitting onthe channel 119899 without buying the interference power

43 Solution of the Optimization for PUs (Leaders) In orderto maximize its own utility each PU needs to adaptivelyoffer an interference price to SUs based on transmit powerresponse of the SUs P1 can be decomposed into twosubproblems fix 119908

119896to get the optimal transmission power

of each PU 119896 and then search the optimal 119908119896 The optimal

transmission power of the PU 119896 can be applied by the previousLDDM

Thus for P1 in (11) the corresponding Lagrangian func-tion can be given as

L119896(p119896 119908119896120582119896 120583119896 ^119896) =

119873

sum119899=1

120588119899

119896120574119899

119896+ 119908119896

119873

sum119899=1

119871

sum119897=1

120588119899

119897119896119901119899

119897ℎ119899

119897119896

minus

119873

sum119899=1

120582119899

119896(

119871

sum119897

log(1 + 120588119899119897119896

120574119899

119896119901119899

119897ℎ119899

119897119896

119901119899119896ℎ119899119896119896

)

minus log( 1

1 minus 120585119899119896

)) minus 120583119896(

119873

sum119899=1

120588119899

119896119901119899

119896minus 119875119898

119896) minus

119873

sum119899=1

]119899119896

sdot (

119871

sum119897=1

120588119899

119897119896119901119899

119897ℎ119899

119897119896minus 119879119899

119896)

(24)

where 120582119899119896 120583119896 and ]119899

119896are the dual variables of the constraints

in (11)Similarly in (24) we assume the 119897th SU causing the inter-

ference ℎ119899119897119896119901119899

119897to the PU 119896 on the channel 119899 can be denoted

as the average interference power 119866119899119896= 119864[sum

119871

119897=1120588119899

119897119896ℎ119899

119897119896119901119899

119897]

According to the KKT conditions we obtain the optimaltransmission power of the PU 119896 if it transmits on the channel119899

119901119899

119896

lowast

=1

2ℎ119899119896119896

(minus120574119899

119896119866119899

119896

+ radic4120582119899

119896ℎ119899

119896119896120574119899

119896119866119899

119896(sum119871

119897=1120588119899

119897119896)

(120583119896minus ℎ119899119896119896119868 (p119899minus119896)) ln 2

+ 120574119899

119896119866119899119896)

(25)

Since L119896(p119896 119908119896120582119896 120583119896 ^119896) is a stepwise function with

breakpoints at119876119899119897for the SU 119897 we should discuss the existence

of the optimal price119908119896first So we divide (24) with respect to

119908119896with two parts on each channel 119899 we have L

119901119896(119901119899

119897) =

120574119899

119896(119901119899

119897) and L

119901119896(119908119896) = (119908

119896minus ]119896)119901119899

119897ℎ119897119896 From (22) it can

be easily observed that 120574119899119896(119901119899

119897) is a concave function of 119908

119896

Therefore we only need to discuss the situation ofL119901119896(119908119896)

For the SU 119897 we first sort119876119899119897(119899 = 1 119873) in ascending order

and have 119873 intervals (0 1198761119897)(1198761

119897 1198762

119897) (119876

119873minus1

119897 119876119873

119897) where

1198761

119897lt 1198762

119897lt sdot sdot sdot lt 119876

119873

119897 Note if the SU 119897 is not allocated on the

channel 119899 (119876119899minus1119897 119876119899

119897)must be taken out of the order We take

(0 1198761

119897) for an example When 119908

119896rarr 0 we can derive that

120597L119901119896(119908119896)

120597119908119896

1003816100381610038161003816100381610038161003816100381610038161003816119908119896rarr0

gt 119901119899

119897ℎ119897119896gt 0 (26)

Taking the second derivative of L119901119896(119908119896) with respect to 119908

119896

is1205972L119901119896

1205971199082119896

= minusℎ119899

119897119897

2

sdot radic120596119899

119897120574119899

119897119866119899

119897(sum119871+1

119895 =119897120588119899

119895119897)

119883119899119897ln 2

+ 120574119899

119897119866119899119897(120596119899

119897ℎ119899

119897119896120574119899

119897119866119899

119897(sum119871+1

119895 =119897120588119899

119895119897)

(119883119899119897)2 ln 2

)

lt 0

(27)

6 International Journal of Distributed Sensor Networks

(01) Initialization set 119902 = 0 set initial 119888119896(119902) and p

119896(119902) for 119896 isin 119870 Set initial p

119897(119902) for 119897 isin 119871

Set 120591 where 120591 is positive and sufficiently small(02) For each SU 119897

(03) Use SM to find the optimal step sizes 120572lowast 120573lowast and 120579lowast and update 120596119899119897 120592119897and 120593

119896according to (28) respectively

(04) For the given 119888119896(119902) and p

119896(119902) of all PUs each SU 119897 responds with its transmit power vector plowast

119897(119902 + 1) according to (22)

(05) If 119876119899119897lt 119888119896(119902) the SU 119897 stops transmitting on the channel 119899 of the PU 119896

(06) End(07) For each PU 119896

(08) Use SM to find the optimal 120594lowast and 120599lowast and update 120582119899119896and 120583

119896by (29)

(09) For the responded plowast119897(119902) of all SUs each PU 119896 updates its transmit power vector as plowast

119896(119902 + 1) according to (25)

(10) Each PU 119896 updates its price by the solution of119888119896(119902 + 1) = argmax

119888119896

119880119896(p119896(119902 + 1) p

minus119896(119902 + 1))

(11) End(12) For each PU 119896 if plowast

119896(119902 + 1) minus p

119896(119902) le 120591 or 119902 gt 102 stop the algorithm Otherwise 119902 = 119902 + 1

repeat steps (02) and (11) until the condition is satisfied

Algorithm 1 Iterative algorithm for reaching the SE

L119901119896(119908119896) is a concave function whether (120597L

119901119896120597119908119896)|119908119896rarr119876

1

119897

gt 0 or (120597L119901119896120597119908119896)|119908119896rarr119876

1

119897

lt 0 except at the nondifferen-tiable point 1198761

119897

Through the above analysis L119896(p119896 119908119896120582119896 120583119896 ^119896) is a

concave function with respect to 119908119896except at 119876119899

119897 The

ellipsoid method [4] can be employed to solve the convexoptimization in each interval

44 Iterative Algorithm to Find the SE For the above dis-cussion we propose an iterative algorithm to search the SEDue to the fact that computation of the dual variables isa complicated task the subgradient method (SM) [14] isapplied to obtain the global optimum SE of this problemThen dual variables are updated as follows

120596119899

119897(119905 + 1) = (120596

119899

119897(119905) + 120572(

119871+1

sum119895 =119897

log(1 + 120588119899119895119897

120574119899

119897119901119899

119895ℎ119899

119895119897

119901119899119897ℎ119899119897119897

)

minus log( 1

1 minus 120585119899119897

)))

+

120592119897(119905 + 1) = (120592

119897(119905) + 120573(

119873

sum119899=1

120588119899

119897119901119899

119897minus 119901

max119897))

+

120593119899

119897(119905 + 1) = (120593

119899

119897(119905) + 120579(

119871

sum119897=1

120588119899

119897119896119901119899

119897ℎ119899

119897119896minus 119879119899

119896))

+

(28)

120582119899

119896(119905 + 1) = (120582

119899

119896(119905) + 120599(

119871

sum119897

log(1 + 120588119899119897119896

120574119899

119896119901119899

119897ℎ119899

119897119896

119901119899119896ℎ119899119896119896

)

minus log( 1

1 minus 120585119899119896

)))

+

120583119896(119905 + 1) = (120583

119896(119905) + 120594(

119873

sum119899=1

120588119899

119896119901119899

119896minus 119875

max119896

))

+

(29)

where 119905 is the iteration index and 120572 gt 0 120573 gt 0 120579 gt 0 120599 gt0 and 120594 gt 0 are sufficiently small The SM guarantees theconvergence of the above optimal dual variables if the stepsizes are chosen by following the step size policy [14]

We then design the iterative algorithm to achieve the SEshown in Algorithm 1

5 Simulation Results and Their Analysis

In this section several numerical examples are presented toevaluate the performances of the proposed SG by comparingthe optimal price-based SG considering global interferencein [5] and the nominal SG without considering globalinterference and outage constraints The cell radius of 500mwith the PBS centered at the original CRN The simulationparameters are as follows 1205752 = 10minus12W 119870 = 3 119871 = 5 and119873 = 10 The SINR threshold of PUs and SUs is set as 7 dBand 4 dB respectively All PUs and SUs deploy the maximumpower 119875max

119896= 100mWand 119875max

119897= 50mWThe channel gain

in this system is ℎ119894119895= 119889minus4

119894119895 with 119889

119894119895being their corresponding

distance The outage probability thresholds of both PUs andSUs are 0001 We set the interference-to-noise ratio (INR) as1198791205752Firstly we illustrate the convergence of the proposed

algorithm for achieving an SE of the proposed game FromFigure 2 the three PUs and five SUs iteratively update theirutilities and obviously converge to the SE The proposedalgorithm converges quickly in terms of PUs only about tentimes In addition due to the larger number of SUs theconvergence of the SUs is slower than that of the PUs

51 Impact of INR In this subsection we set the numberof PUs and SUs as 3 and 5 and INR changes from minus20 dBto 20 dB which means that IPC changes from 10

minus14W to10minus10WWe then consider the sum rate of PUs and SUs for the

three solutionswith different tolerant interference constraintsshown in Figures 3 and 4 For the performance of sum rate

International Journal of Distributed Sensor Networks 7

PU-1PU-2PU-3

SU-1SU-2SU-3

SU-4SU-5

12

13

14

15

16

Util

ities

of P

Us

1

2

3

4

5

6

7

Util

ities

of S

Us

50 15 20 2510

Iteration

50 15 20 2510

Iteration

Figure 2 The convergence of utility of PUs (leaders) setting priceof PUs and utility of SUs (followers)

of SUs Figure 3 shows the nominal SG scheme outperformsother two schemes when the interference temperature levelis stringent but is inferior to the two schemes when it isloose The proposed SG performs the worst because ofthe demand of satisfying the outage probability constraintOnce the interference constraints are loose enough to benot active accordingly our proposed solution works betterthan the others For the leaders the sum rate of the nominalSG performs the worst because of not including the globalinterference constraints This is because the performance ofPUs may be degraded with the increases of the interference

Figure 5 presents the outage probability of the systemwith different INR It is observed that the proposed schemeachieves much lower outage probability than other schemesin particular the performance gap becomes larger with theincrease of INR This is because our proposed algorithmworks best by considering the outage probability constraintsof users which prevents the outage events well

52 Impact of Different Number of SUs In this subsection theINR is set to be 10 dB The number of SUs changes from 2 to20 All the other simulation parameters are the same as thebeginning part of this section

For the PUs from Figure 6 the sum rate of the SGsolution performs the worst because of not including theglobal interference constraints So the PUs may refuse tosell more spectrum resource because of protecting their own

Our proposed SGOptimal price-based SG in [5]Nominal SG

50 10 15 20minus5minus15 minus10minus20

INR T1205752 (dB)

25

26

27

28

29

30

31

32

33

34

Sum

rate

of t

he P

Us (

bits

sH

z)Figure 3 Sum rate of PUs versus INR

26

24

22

20

18

16

14

12

10minus20 minus15 minus10 minus5 0 5 10 15 20

Our proposed SGOptimal price-based SG in [5]Nominal SG

INR T1205752 (dB)

Sum

rate

of t

he S

Us (

bits

sH

z)

Figure 4 Sum rate of SUs versus INR

communication QoS The proposed scheme outperformsmost in terms of sum rate of SUs because the algorithm allowsmore SUs to share their radio resource so that it increases PUsrsquoutilities with considering the channel uncertainty and globalinterference constraints

Figure 7 shows the sum rate of SUs versus the numberof SUs The sum rate of SUs performance of our proposedalgorithm works better than other two schemes this isbecause the proposed scheme is able to support more SUs attheBS shown in Figure 7 so thatmore SUshave opportunities

8 International Journal of Distributed Sensor Networks

Our proposed SGOptimal price-based SG in [5]Nominal SG

10minus3

10minus2

10minus1

100

Out

age p

roba

bilit

y

minus15 minus10 minus5 0 5 10 15 20minus20

INR T1205752 (dB)

Figure 5 Outage probability versus INR

Our proposed SGOptimal price-based SG in [5]Nominal SG

16

18

20

22

24

26

28

30

32

34

Sum

rate

of P

Us

4 6 82 12 14 16 18 2010

Number of SUs

Figure 6 Sum rate of PUs versus number of SUs

to transmit which increase the sum rate In addition theperformance gap between the proposed scheme and theother algorithms increases when the network grows largerby which it can be concluded that the proposed scheme ismore suitable for application in larger networks Because thescheme sets the adaptive punishment parameter among allserved SUs to control their behavior more SUs can be servedat the BS thus achieving a higher sum rate

Figure 8 shows the number of outage probability com-parison versus the number of SUs for different algorithmsThe proposed algorithm is able to support more SUs than

Our proposed SGOptimal price-based SG in [5]Nominal SG

4 6 8 10 12 14 16 18 202

Number of SUs

10

15

20

25

30

35

40

45

Sum

rate

of S

Us

Figure 7 Sum rate of SUs versus number of SUs

Our proposed SGOptimal price-based SG in [5]Nominal SG

4 6 8 10 12 14 16 18 202

Number of SUs

10minus4

10minus3

10minus2

10minus1

100

Out

age p

roba

bilit

y

Figure 8 Outage probability versus number of SUs

other schemes This is because we develop the channelassignment scheduling scheme to decrease the probabilityof the unserved SUs so more SUs are admitted to serveat the BS without causing unendurable interference to PUsIn addition the interference among SUs is taken into therevenue utility function to void serious interference for someSUs who have bad channel condition

International Journal of Distributed Sensor Networks 9

6 Conclusion

In this paper we propose a Stackelberg game for powercontrol problem in CRNs with channel outage constraintsand global interference constraints We employ LDDM tosolve the problem by decomposing it into independentsubproblems and develop an iterative algorithm to achieveSE Simulation results show that the proposed algorithmimproves the performance compared with other game algo-rithms

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the National Nature ScienceFoundation of China (61271259 and 61301123) the SpecialFund of Chongqing Key Laboratory (CSTC) the Programfor Changjiang Scholars and Innovative Research Team inUniversity (IRT2129) and the Graduate Student ResearchInnovation Project of Chongqing University of Posts andTelecommunications (Chongqing) (CYS14143)

References

[1] K Akkarajitsakul E Hossain D Niyato and D I Kim ldquoGametheoretic approaches for multiple access in wireless networksa surveyrdquo IEEE Communications Surveys and Tutorials vol 13no 3 pp 372ndash395 2011

[2] Y Wu T Zhang and D H K Tsang ldquoJoint pricing andpower allocation for dynamic spectrum access networks withStackelberg game modelrdquo IEEE Transactions on Wireless Com-munications vol 10 no 1 pp 12ndash19 2011

[3] X Kang R Zhang and M Motani ldquoPrice-based resourceallocation for spectrum-sharing femtocell networks a stack-elberg game approachrdquo IEEE Journal on Selected Areas inCommunications vol 30 no 3 pp 538ndash549 2012

[4] R Xie F R Yu H Ji and Y Li ldquoEnergy-efficient resourceallocation for heterogeneous cognitive radio networks withfemtocellsrdquo IEEETransactions onWireless Communications vol11 no 11 pp 3910ndash3920 2012

[5] Z Wang L Jiang and C He ldquoOptimal price-based power con-trol algorithm in cognitive radio networksrdquo IEEE TransactionsonWireless Communications vol 13 no 11 pp 5909ndash5920 2014

[6] M Le Treust S Lasaulce Y Hayel and G L He ldquoGreen powercontrol in cognitive wireless networksrdquo IEEE Transactions onVehicular Technology vol 62 no 4 pp 1741ndash1754 2013

[7] Y Xiao G Bi D Niyato and L A DaSilva ldquoA hierarchicalgame theoretic framework for cognitive radio networksrdquo IEEEJournal on Selected Areas in Communications vol 30 no 10 pp2053ndash2069 2012

[8] F Gabry N Li N Schrammar M Girnyk L K Rasmussenand M Skoglund ldquoOn the optimization of the secondarytransmitterrsquos strategy in cognitive radio channels with secrecyrdquoIEEE Journal on Selected Areas in Communications vol 32 no3 pp 451ndash463 2014

[9] D Niyato E Hossain and Z Han ldquoDynamics of multiple-seller and multiple-buyer spectrum trading in cognitive radionetworks a game-theoretic modeling approachrdquo IEEE Transac-tions on Mobile Computing vol 8 no 8 pp 1009ndash1022 2009

[10] S Kandukuri and S Boyd ldquoOptimal power control ininterference-limited fading wireless channels with outage-probability specificationsrdquo IEEE Transactions on Wireless Com-munications vol 1 no 1 pp 46ndash55 2002

[11] T Basar and G J Olsder Dynamic Noncooperative GameTheory SIAM Philadelphia Pa USA 1999

[12] F Facchinei and C Kanzow ldquoGeneralized Nash equilibriumproblemsrdquo 4OR vol 5 no 3 pp 173ndash210 2007

[13] D Ardagna B Panicucci and M Passacantando ldquoA game the-oretic formulation of the service provisioning problem in cloudsystemsrdquo in Proceedings of the 20th International Conference onWorld Wide Web (WWW rsquo11) pp 177ndash186 ACM April 2011

[14] D Bertsekas W Hager and O Mangasarian Nonlinear Pro-gramming Athena Scientific Belmont Mass USA 1999

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Navigation and Observation

International Journal of

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DistributedSensor Networks

International Journal of

International Journal of Distributed Sensor Networks 3

Similarly taking (3) into (5) rewrite (5) as follows

119871+1

sum119895 =119897

log(1 + 120588119899119895119897

120574119899

119897119901119899

119895ℎ119899

119895119897

119901119899119897ℎ119899119897119897

) le log( 1

1 minus 120585119899119897

) forall119897 forall119899 (8)

In addition the global aggregate interference from allSUs to each channel should not be larger than the maximuminterference threshold 119879119899

119896to ensure the SUsrsquo transmission

would not cause unendurable interference on every channelof each PU Mathematically this can be written as

119871

sum119897=1

120588119899

119897119896119901119899

119897ℎ119899

119897119896lt 119879119899

119896 forall119896 forall119899 (9)

3 Stackelberg Game Theoretic Approach

The PUs (leaders) price the SUs (followers) to control theinterference power made by the SUs under the IPC EachPU will offer a suitable price to maximize its revenue byselling resource to SUs Based on the interference priceprovided by PUs each SU will adjust its transmission powerto maximize its revenue PUs have higher priority than SUswe use Stackelberg game to model the strategy between thePUs and SUs

Then as the leaders PUswillmaximize their utility (SINRperformance plus the payment from the SUs occupying thechannels) The utility function of the PU 119896 is as follows

119880119896(p119896 pminus119896 119908119896) =

119873

sum119899=1

120588119899

119896120574119899

119896+ 119908119896

119873

sum119899=1

119871

sum119897=1

120588119899

119897119896119901119899

119897ℎ119899

119897119896 (10)

where if the PU 119896 transmit on the channel 119899 120588119899119896= 1

otherwise 120588119899119896= 0 p

119896= (1199011

119896 1199012

119896 119901

119873

119896) is the transmission

power vector over the transmitted channels of the PU 119896 pminus119896

represent the transmit power vector of all users except the PU119896 119908119896denotes that the PU 119896 charges the price over all SUs if

the SUs transmit on the channel of the PUHence the revenue utility optimization problem for the

PU 119896 is as follows

P1 Max 119880119896(p119896 pminus119896 119908119896)

st (8) (10) 119901119899

119896ge 0

119873

sum119899=1

120588119899

119896119901119899

119896le 119875

max119896

(11)

where 119875max119896

is the maximum transmission power of the PU 119896Then for SUs (followers) we set the revenue utility of the

119897th SU with two parts the first one is the income from theSINR achieved from the PUs The second one is the paymentfor the PUs Then the revenue utility function of the 119897th SUis as follows

119880119897(p119897 pminus119897w119897) =

119873

sum119899=1

120588119899

119897120574119899

119897minus

119873

sum119899=1

119870

sum119896=1

120588119899

119897119896119908119896119901119899

119897ℎ119899

119897119896 (12)

where if the SU 119897 transmit on the channel 119899 120588119899119897= 1 otherwise

120588119899

119897= 0 p

119897= (1199011

119897 1199012

119897 119901

119873

119897) is the transmission power vector

over the transmitted channels of the SU 119897 pminus119897represent the

transmission power vector of all users except the SU 119897 w119897=

(1199081 1199082 119908

119870) denotes that the SU 119897 pays the price vector

for all PUs if the SU 119897 does not transmit on the channel of thePU 119896 119908

119896= 0

Hence the optimization problem for the SU 119897 is as follows

P2 Max 119880119897(p119897 pminus119897w119897)

st (9) (10) 119901119899

119897ge 0

119873

sum119899=1

120588119899

119897119901119899

119897le 119875

max119897

(13)

where 119875max119897

is the maximum transmission power of the 119897thSU

4 Solution of the Proposed Stackelberg Game

The optimization problems P1 and P2 taken together are theproposed Stackelberg game problemwith several constraintsThe goal of the proposed Stackelberg game is to achieve theSE in which point both PUs and SUs have no incentive todeviate [3] so the distributive algorithms convergence tothe SE is difficult Thus we depart the game problem intosuboptimal independent solution and employ an iterativealgorithm to achieve the SE

41 Global Efficiency of the RSE In noncooperative gamesthe existence and uniqueness of equilibrium are not alwaysachieved [11] due to multiple playersrsquo competition Hence forthemultifollower subgamewe should study the existence anduniqueness of the global SG response to the leadersrsquo priceIn particular a variational equilibrium (VE) [12] is appliedto analyze the global SG for our case This is because a VEis more stable than any other generalized Nash equilibriumunder parameter uncertainty [13] Particularly a number ofSUs aim to achieve their QoS requirement through applyingthe resource from PUs in cognitive radio networks VE isregarded as an appropriate solution

For a market fixed price at the PUs all the SUs aim tomaximize their own utility by buying the resource throughPUsThus we formulate a new objective function for all SUsthe new utility is the sum utilities of all SUs and can beexpressed as follows

119865

Sum =119871

sum119897=1

119873

sum119899=1

120588119899

119897120574119899

119897minus

119871

sum119897=1

119873

sum119899=1

119870

sum119896=1

120588119899

119897119896119908119896119901119899

119897ℎ119899

119897119896 (14)

Therefore in order to achieve the actable outcome ofthe proposed SG our goal is guarantying the existence anduniqueness of the SGwhenmaximizing (14)The correspond-ing Lagrangian function and the KTT conditions for the SU119897 expressed in and the KTT conditions for the SU 119897 are givenby

nablap119897119880119897 (p119897 pminus119897w119897) minus nablap119897 (119873

sum119899=1

120588119899

119897119901119899

119897minus 119875

max119897

)120592119897

minus nablap119897 (119873

sum119899=1

120596119899

119897(

119871+1

sum119895 =119897

log(1 + 120588119899119895119897

120574119899

119897119901119899

119895ℎ119899

119895119897

119901119899119897ℎ119899119897119897

)

4 International Journal of Distributed Sensor Networks

minus log( 1

1 minus 120585119899119897

))) minus nablap119897

119873

sum119899=1

120593119899

119897(

119871

sum119897=1

120588119899

119897119896119901119899

119897ℎ119899

119897119896

minus 119879119899

119896)120592119897(

119873

sum119899=1

120588119899

119897119901119899

119897minus 119875

max119897

) = 0

119873

sum119899=1

120596119899

119897(

119871+1

sum119895 =119897

log(1 + 120588119899119895119897

120574119899

119897119901119899

119895ℎ119899

119895119897

119901119899119897ℎ119899119897119897

) minus log( 1

1 minus 120585119899119897

))

= 0

119873

sum119899=1

120593119899

119897(

119871

sum119897=1

120588119899

119897119896119901119899

119897ℎ119899

119897119896minus 119879119899

119896) = 0

(15)

We note that the robust followersrsquo game shows a jointlyconvex generalized SE problem therefore the solution of theSE problem with constraints in (13) is a variational inequalityVI(P F) where P is the set of joint convexity It is importantto determine a vector 119911lowast isin P sub 119877119899 such that ⟨F(zlowast) zminuszlowast⟩ ge0 for all 119911 isin P and F(p) = minus(nabla

119901119897(p119897))119871

119897=1[13] Then the

solution of VI(P F) is a variational SEIn this paper we only focus on the power control in cog-

nitive radio networks by assuming the channel assignmenthas already been done Then we can divide the variationalinequality VI(P F) into 119873 subproblems each subproblemdenotesVI(P

119899 F119899) on the subchannel 119899 and they are indepen-

dentTherefore on the subchannel 119899 the KKT conditions canbe expressed as [12]

F119899(p) + 120592

119899nabla119901(

119871

sum119897=1

120588119899

119897119901119899

119897minus

119871

sum119897=1

120588119899

119897119875max119897

)

+ 120596119899nabla119901(

119871

sum119897=1

(

119871+1

sum119895 =119897

log(1 + 120588119899119895119897

120574119899

119897119901119899

119895ℎ119899

119895119897

119901119899119897ℎ119899119897119897

))

minus

119871

sum119897=1

log( 1

1 minus 120585119899119897

)) + 120593119899nabla119901(

119871

sum119897=1

120588119899

119897119896119901119899

119897ℎ119899

119897119896minus 119879119899

119896)

(16)

Now from the definition of [12] we have

F1=

[[[[[[[[[[[[[

[

1205881

1119896(119908119896ℎ1

1119896minus

1

119868 (1199011minus1))

1205881

2119896(119908119896ℎ1

2119896minus

1

119868 (1199011minus2))

1205881

119871119896(119888119896ℎ1

119871119896minus

1

119868 (1199011minus119871))

]]]]]]]]]]]]]

]

F119899=

[[[[[[[[[[[[[

[

120588119899

1119896(119888119896ℎ119899

1119896minus

1

119868 (119901119899minus1))

120588119899

2119896(119888119896ℎ119899

2119896minus

1

119868 (119901119899minus2))

120588119899

119871119896(119888119896ℎ119899

119871119896minus

1

119868 (119901119899minus119871))

]]]]]]]]]]]]]

]

(17)

Therefore the Jacobian of F119899is

J1=

[[[[[[[

[

1205881

1119896ℎ1

11198960 sdot sdot sdot 0

0 1205881

2119896ℎ1

2119896sdot sdot sdot 0

0

0 0 sdot sdot sdot 1205881

119871119896ℎ1

119871119896

]]]]]]]

]

J119899=

[[[[[[

[

120588119899

1119896ℎ119899

11198960 sdot sdot sdot 0

0 120588119899

2119896ℎ119899

2119896sdot sdot sdot 0

0

0 0 sdot sdot sdot 120588119899

119871119896ℎ119899

119871119896

]]]]]]

]

(18)

Each F119899J119899is a diagonal matrix and all the diagonal

elements are positive Therefore F119899J119899is positive definition

on P119899 and so F

119899is strictly monotone Hence the global

SG problem admits a unique global variational equilibriumsolution [12] Due to the jointly convex nature of the globalSE problem the variational equilibrium is the unique globalmaximizer of (14) [12] which completes the proof in theliterature [12]

42 Solution of the Optimization for SUs (Followers) For theP2 the utility function of each SU is a concave function of 119901119899

119897

and the constraints are all linear so a partial Lagrange dualdecompositionmethod (LDDM) [4] for the problems is used

For the problem in (13) the corresponding Lagrangianfunction for the SU 119897 on the subchannel 119899 can be expressedas

L119897(p119897w119896 120596119897 120592119897120593119897) =

119873

sum119899=1

120588119899

119897120574119899

119897minus

119873

sum119899=1

119870

sum119896=1

120588119899

119897119896119908119896119901119899

119897ℎ119899

119897119896

minus

119873

sum119899=1

120596119899

119897(

119871+1

sum119895 =119897

log(1 + 120588119899119895119897

120574119899

119897119901119899

119895ℎ119899

119895119897

119901119899119897ℎ119899119897119897

)

minus log( 1

1 minus 120585119899119897

)) minus 120592119897(

119873

sum119899=1

120588119899

119897119901119899

119897minus 119875119898

119897) minus

119873

sum119899=1

120593119899

119897

sdot (

119871

sum119897=1

120588119899

119897119896119901119899

119897ℎ119899

119897119896minus 119879119899

119896)

(19)

International Journal of Distributed Sensor Networks 5

where 120596119899119897 120592119897 and 120593119899

119897are the nonnegative dual variables of the

constraints in (13)We decompose the optimization problem into 119873 inde-

pendent subproblems Then on the subchannel 119899 taking theKarush-Kuhn-Tucker (KKT) condition [4]

120597L119897

120597119901119899119897

=ℎ119899

119897119897

119868 (p119899minus119897)minus 119908119896ℎ119899

119897119896

+ 120596119899

119897

119871+1

sum119895 =119897

(120588119899

119895119897

120574119899

119897119901119899

119895ℎ119899

119895119897

119901119899119897(119901119899119897ℎ119899119897119897+ 120574119899

119897119901119899119895ℎ119899119895119897) ln 2

)

minus 120592119897minus 120593119896ℎ119899

119897119896

(20)

Simply in (20) we assume that the 119895th user causing theinterference power 119901119899

119895ℎ119899

119895119897to the SU 119897 on the subchannel 119899 can

be denoted as the average interference power except the SU 119897119866119899

119897= 119864[sum

119871+1

119895=1119895 =119897120588119899

119895119897119901119899

119895ℎ119899

119895119897] Hence we rewrite (20) as follows

120597L119897

120597119901119899119897

=ℎ119899

119897119897

119868 (p119899minus119897)minus 119908119896ℎ119899

119897119896+ 120596119899

119897

120574119899

119897119866119899

119897(sum119871+1

119895 =119897120588119899

119895119897)

119901119899119897(119901119899119897ℎ119899119897119897+ 120574119899

119897119866119899119897) ln 2

minus 120592119897minus 120593119896ℎ119899

119897119896

(21)

Set (21) to zero and get the optimal transmission power ofthe SU 119897 if it transmits on the channel 119899

119901119899

119897

lowast

=

minus120574119899

119897119866119899

119897+ radic4120596119899

119897ℎ119899119897119897120574119899

119897119866119899119897(sum119871+1

119895 =119897120588119899119895119897) 119883 ln 2 + 120574119899

119897119866119899119897

2ℎ119899119897119897

(22)

where119883119899119897= (119908119896+ 120593119899

119897)ℎ119899

119897119896+ 120592119897minus ℎ119899

119897119897119868(p119899minus119897)

The transmission power of the SU 119897 is zero if theinterference price for it is larger than payoff threshold 119876119899

119897on

the channel 119899 Then setting 119901119899119897

lowast

= 0 we can get the payoffthreshold of the SU 119897 if it transmits on the channel 119899

119876119899

119897=1

ℎ119899119897119896

(4120596119899

119897ℎ119899

119897119897120574119899

119897119866119899

119897(sum119871+1

119895 =119897120588119899

119895119897)

((120574119899

119897119866119899119897)2

minus 120574119899

119897119866119899119897) ln 2

minus 120592119897minus 120593119899

119897ℎ119899

119897119896

+ℎ119899

119897119897

119868 (p119899minus119897))

(23)

From (23) if the price 119908119896gt 119876119899

119897 the price is above the

payoff threshold of the SU 119897 and it will stop transmitting onthe channel 119899 without buying the interference power

43 Solution of the Optimization for PUs (Leaders) In orderto maximize its own utility each PU needs to adaptivelyoffer an interference price to SUs based on transmit powerresponse of the SUs P1 can be decomposed into twosubproblems fix 119908

119896to get the optimal transmission power

of each PU 119896 and then search the optimal 119908119896 The optimal

transmission power of the PU 119896 can be applied by the previousLDDM

Thus for P1 in (11) the corresponding Lagrangian func-tion can be given as

L119896(p119896 119908119896120582119896 120583119896 ^119896) =

119873

sum119899=1

120588119899

119896120574119899

119896+ 119908119896

119873

sum119899=1

119871

sum119897=1

120588119899

119897119896119901119899

119897ℎ119899

119897119896

minus

119873

sum119899=1

120582119899

119896(

119871

sum119897

log(1 + 120588119899119897119896

120574119899

119896119901119899

119897ℎ119899

119897119896

119901119899119896ℎ119899119896119896

)

minus log( 1

1 minus 120585119899119896

)) minus 120583119896(

119873

sum119899=1

120588119899

119896119901119899

119896minus 119875119898

119896) minus

119873

sum119899=1

]119899119896

sdot (

119871

sum119897=1

120588119899

119897119896119901119899

119897ℎ119899

119897119896minus 119879119899

119896)

(24)

where 120582119899119896 120583119896 and ]119899

119896are the dual variables of the constraints

in (11)Similarly in (24) we assume the 119897th SU causing the inter-

ference ℎ119899119897119896119901119899

119897to the PU 119896 on the channel 119899 can be denoted

as the average interference power 119866119899119896= 119864[sum

119871

119897=1120588119899

119897119896ℎ119899

119897119896119901119899

119897]

According to the KKT conditions we obtain the optimaltransmission power of the PU 119896 if it transmits on the channel119899

119901119899

119896

lowast

=1

2ℎ119899119896119896

(minus120574119899

119896119866119899

119896

+ radic4120582119899

119896ℎ119899

119896119896120574119899

119896119866119899

119896(sum119871

119897=1120588119899

119897119896)

(120583119896minus ℎ119899119896119896119868 (p119899minus119896)) ln 2

+ 120574119899

119896119866119899119896)

(25)

Since L119896(p119896 119908119896120582119896 120583119896 ^119896) is a stepwise function with

breakpoints at119876119899119897for the SU 119897 we should discuss the existence

of the optimal price119908119896first So we divide (24) with respect to

119908119896with two parts on each channel 119899 we have L

119901119896(119901119899

119897) =

120574119899

119896(119901119899

119897) and L

119901119896(119908119896) = (119908

119896minus ]119896)119901119899

119897ℎ119897119896 From (22) it can

be easily observed that 120574119899119896(119901119899

119897) is a concave function of 119908

119896

Therefore we only need to discuss the situation ofL119901119896(119908119896)

For the SU 119897 we first sort119876119899119897(119899 = 1 119873) in ascending order

and have 119873 intervals (0 1198761119897)(1198761

119897 1198762

119897) (119876

119873minus1

119897 119876119873

119897) where

1198761

119897lt 1198762

119897lt sdot sdot sdot lt 119876

119873

119897 Note if the SU 119897 is not allocated on the

channel 119899 (119876119899minus1119897 119876119899

119897)must be taken out of the order We take

(0 1198761

119897) for an example When 119908

119896rarr 0 we can derive that

120597L119901119896(119908119896)

120597119908119896

1003816100381610038161003816100381610038161003816100381610038161003816119908119896rarr0

gt 119901119899

119897ℎ119897119896gt 0 (26)

Taking the second derivative of L119901119896(119908119896) with respect to 119908

119896

is1205972L119901119896

1205971199082119896

= minusℎ119899

119897119897

2

sdot radic120596119899

119897120574119899

119897119866119899

119897(sum119871+1

119895 =119897120588119899

119895119897)

119883119899119897ln 2

+ 120574119899

119897119866119899119897(120596119899

119897ℎ119899

119897119896120574119899

119897119866119899

119897(sum119871+1

119895 =119897120588119899

119895119897)

(119883119899119897)2 ln 2

)

lt 0

(27)

6 International Journal of Distributed Sensor Networks

(01) Initialization set 119902 = 0 set initial 119888119896(119902) and p

119896(119902) for 119896 isin 119870 Set initial p

119897(119902) for 119897 isin 119871

Set 120591 where 120591 is positive and sufficiently small(02) For each SU 119897

(03) Use SM to find the optimal step sizes 120572lowast 120573lowast and 120579lowast and update 120596119899119897 120592119897and 120593

119896according to (28) respectively

(04) For the given 119888119896(119902) and p

119896(119902) of all PUs each SU 119897 responds with its transmit power vector plowast

119897(119902 + 1) according to (22)

(05) If 119876119899119897lt 119888119896(119902) the SU 119897 stops transmitting on the channel 119899 of the PU 119896

(06) End(07) For each PU 119896

(08) Use SM to find the optimal 120594lowast and 120599lowast and update 120582119899119896and 120583

119896by (29)

(09) For the responded plowast119897(119902) of all SUs each PU 119896 updates its transmit power vector as plowast

119896(119902 + 1) according to (25)

(10) Each PU 119896 updates its price by the solution of119888119896(119902 + 1) = argmax

119888119896

119880119896(p119896(119902 + 1) p

minus119896(119902 + 1))

(11) End(12) For each PU 119896 if plowast

119896(119902 + 1) minus p

119896(119902) le 120591 or 119902 gt 102 stop the algorithm Otherwise 119902 = 119902 + 1

repeat steps (02) and (11) until the condition is satisfied

Algorithm 1 Iterative algorithm for reaching the SE

L119901119896(119908119896) is a concave function whether (120597L

119901119896120597119908119896)|119908119896rarr119876

1

119897

gt 0 or (120597L119901119896120597119908119896)|119908119896rarr119876

1

119897

lt 0 except at the nondifferen-tiable point 1198761

119897

Through the above analysis L119896(p119896 119908119896120582119896 120583119896 ^119896) is a

concave function with respect to 119908119896except at 119876119899

119897 The

ellipsoid method [4] can be employed to solve the convexoptimization in each interval

44 Iterative Algorithm to Find the SE For the above dis-cussion we propose an iterative algorithm to search the SEDue to the fact that computation of the dual variables isa complicated task the subgradient method (SM) [14] isapplied to obtain the global optimum SE of this problemThen dual variables are updated as follows

120596119899

119897(119905 + 1) = (120596

119899

119897(119905) + 120572(

119871+1

sum119895 =119897

log(1 + 120588119899119895119897

120574119899

119897119901119899

119895ℎ119899

119895119897

119901119899119897ℎ119899119897119897

)

minus log( 1

1 minus 120585119899119897

)))

+

120592119897(119905 + 1) = (120592

119897(119905) + 120573(

119873

sum119899=1

120588119899

119897119901119899

119897minus 119901

max119897))

+

120593119899

119897(119905 + 1) = (120593

119899

119897(119905) + 120579(

119871

sum119897=1

120588119899

119897119896119901119899

119897ℎ119899

119897119896minus 119879119899

119896))

+

(28)

120582119899

119896(119905 + 1) = (120582

119899

119896(119905) + 120599(

119871

sum119897

log(1 + 120588119899119897119896

120574119899

119896119901119899

119897ℎ119899

119897119896

119901119899119896ℎ119899119896119896

)

minus log( 1

1 minus 120585119899119896

)))

+

120583119896(119905 + 1) = (120583

119896(119905) + 120594(

119873

sum119899=1

120588119899

119896119901119899

119896minus 119875

max119896

))

+

(29)

where 119905 is the iteration index and 120572 gt 0 120573 gt 0 120579 gt 0 120599 gt0 and 120594 gt 0 are sufficiently small The SM guarantees theconvergence of the above optimal dual variables if the stepsizes are chosen by following the step size policy [14]

We then design the iterative algorithm to achieve the SEshown in Algorithm 1

5 Simulation Results and Their Analysis

In this section several numerical examples are presented toevaluate the performances of the proposed SG by comparingthe optimal price-based SG considering global interferencein [5] and the nominal SG without considering globalinterference and outage constraints The cell radius of 500mwith the PBS centered at the original CRN The simulationparameters are as follows 1205752 = 10minus12W 119870 = 3 119871 = 5 and119873 = 10 The SINR threshold of PUs and SUs is set as 7 dBand 4 dB respectively All PUs and SUs deploy the maximumpower 119875max

119896= 100mWand 119875max

119897= 50mWThe channel gain

in this system is ℎ119894119895= 119889minus4

119894119895 with 119889

119894119895being their corresponding

distance The outage probability thresholds of both PUs andSUs are 0001 We set the interference-to-noise ratio (INR) as1198791205752Firstly we illustrate the convergence of the proposed

algorithm for achieving an SE of the proposed game FromFigure 2 the three PUs and five SUs iteratively update theirutilities and obviously converge to the SE The proposedalgorithm converges quickly in terms of PUs only about tentimes In addition due to the larger number of SUs theconvergence of the SUs is slower than that of the PUs

51 Impact of INR In this subsection we set the numberof PUs and SUs as 3 and 5 and INR changes from minus20 dBto 20 dB which means that IPC changes from 10

minus14W to10minus10WWe then consider the sum rate of PUs and SUs for the

three solutionswith different tolerant interference constraintsshown in Figures 3 and 4 For the performance of sum rate

International Journal of Distributed Sensor Networks 7

PU-1PU-2PU-3

SU-1SU-2SU-3

SU-4SU-5

12

13

14

15

16

Util

ities

of P

Us

1

2

3

4

5

6

7

Util

ities

of S

Us

50 15 20 2510

Iteration

50 15 20 2510

Iteration

Figure 2 The convergence of utility of PUs (leaders) setting priceof PUs and utility of SUs (followers)

of SUs Figure 3 shows the nominal SG scheme outperformsother two schemes when the interference temperature levelis stringent but is inferior to the two schemes when it isloose The proposed SG performs the worst because ofthe demand of satisfying the outage probability constraintOnce the interference constraints are loose enough to benot active accordingly our proposed solution works betterthan the others For the leaders the sum rate of the nominalSG performs the worst because of not including the globalinterference constraints This is because the performance ofPUs may be degraded with the increases of the interference

Figure 5 presents the outage probability of the systemwith different INR It is observed that the proposed schemeachieves much lower outage probability than other schemesin particular the performance gap becomes larger with theincrease of INR This is because our proposed algorithmworks best by considering the outage probability constraintsof users which prevents the outage events well

52 Impact of Different Number of SUs In this subsection theINR is set to be 10 dB The number of SUs changes from 2 to20 All the other simulation parameters are the same as thebeginning part of this section

For the PUs from Figure 6 the sum rate of the SGsolution performs the worst because of not including theglobal interference constraints So the PUs may refuse tosell more spectrum resource because of protecting their own

Our proposed SGOptimal price-based SG in [5]Nominal SG

50 10 15 20minus5minus15 minus10minus20

INR T1205752 (dB)

25

26

27

28

29

30

31

32

33

34

Sum

rate

of t

he P

Us (

bits

sH

z)Figure 3 Sum rate of PUs versus INR

26

24

22

20

18

16

14

12

10minus20 minus15 minus10 minus5 0 5 10 15 20

Our proposed SGOptimal price-based SG in [5]Nominal SG

INR T1205752 (dB)

Sum

rate

of t

he S

Us (

bits

sH

z)

Figure 4 Sum rate of SUs versus INR

communication QoS The proposed scheme outperformsmost in terms of sum rate of SUs because the algorithm allowsmore SUs to share their radio resource so that it increases PUsrsquoutilities with considering the channel uncertainty and globalinterference constraints

Figure 7 shows the sum rate of SUs versus the numberof SUs The sum rate of SUs performance of our proposedalgorithm works better than other two schemes this isbecause the proposed scheme is able to support more SUs attheBS shown in Figure 7 so thatmore SUshave opportunities

8 International Journal of Distributed Sensor Networks

Our proposed SGOptimal price-based SG in [5]Nominal SG

10minus3

10minus2

10minus1

100

Out

age p

roba

bilit

y

minus15 minus10 minus5 0 5 10 15 20minus20

INR T1205752 (dB)

Figure 5 Outage probability versus INR

Our proposed SGOptimal price-based SG in [5]Nominal SG

16

18

20

22

24

26

28

30

32

34

Sum

rate

of P

Us

4 6 82 12 14 16 18 2010

Number of SUs

Figure 6 Sum rate of PUs versus number of SUs

to transmit which increase the sum rate In addition theperformance gap between the proposed scheme and theother algorithms increases when the network grows largerby which it can be concluded that the proposed scheme ismore suitable for application in larger networks Because thescheme sets the adaptive punishment parameter among allserved SUs to control their behavior more SUs can be servedat the BS thus achieving a higher sum rate

Figure 8 shows the number of outage probability com-parison versus the number of SUs for different algorithmsThe proposed algorithm is able to support more SUs than

Our proposed SGOptimal price-based SG in [5]Nominal SG

4 6 8 10 12 14 16 18 202

Number of SUs

10

15

20

25

30

35

40

45

Sum

rate

of S

Us

Figure 7 Sum rate of SUs versus number of SUs

Our proposed SGOptimal price-based SG in [5]Nominal SG

4 6 8 10 12 14 16 18 202

Number of SUs

10minus4

10minus3

10minus2

10minus1

100

Out

age p

roba

bilit

y

Figure 8 Outage probability versus number of SUs

other schemes This is because we develop the channelassignment scheduling scheme to decrease the probabilityof the unserved SUs so more SUs are admitted to serveat the BS without causing unendurable interference to PUsIn addition the interference among SUs is taken into therevenue utility function to void serious interference for someSUs who have bad channel condition

International Journal of Distributed Sensor Networks 9

6 Conclusion

In this paper we propose a Stackelberg game for powercontrol problem in CRNs with channel outage constraintsand global interference constraints We employ LDDM tosolve the problem by decomposing it into independentsubproblems and develop an iterative algorithm to achieveSE Simulation results show that the proposed algorithmimproves the performance compared with other game algo-rithms

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the National Nature ScienceFoundation of China (61271259 and 61301123) the SpecialFund of Chongqing Key Laboratory (CSTC) the Programfor Changjiang Scholars and Innovative Research Team inUniversity (IRT2129) and the Graduate Student ResearchInnovation Project of Chongqing University of Posts andTelecommunications (Chongqing) (CYS14143)

References

[1] K Akkarajitsakul E Hossain D Niyato and D I Kim ldquoGametheoretic approaches for multiple access in wireless networksa surveyrdquo IEEE Communications Surveys and Tutorials vol 13no 3 pp 372ndash395 2011

[2] Y Wu T Zhang and D H K Tsang ldquoJoint pricing andpower allocation for dynamic spectrum access networks withStackelberg game modelrdquo IEEE Transactions on Wireless Com-munications vol 10 no 1 pp 12ndash19 2011

[3] X Kang R Zhang and M Motani ldquoPrice-based resourceallocation for spectrum-sharing femtocell networks a stack-elberg game approachrdquo IEEE Journal on Selected Areas inCommunications vol 30 no 3 pp 538ndash549 2012

[4] R Xie F R Yu H Ji and Y Li ldquoEnergy-efficient resourceallocation for heterogeneous cognitive radio networks withfemtocellsrdquo IEEETransactions onWireless Communications vol11 no 11 pp 3910ndash3920 2012

[5] Z Wang L Jiang and C He ldquoOptimal price-based power con-trol algorithm in cognitive radio networksrdquo IEEE TransactionsonWireless Communications vol 13 no 11 pp 5909ndash5920 2014

[6] M Le Treust S Lasaulce Y Hayel and G L He ldquoGreen powercontrol in cognitive wireless networksrdquo IEEE Transactions onVehicular Technology vol 62 no 4 pp 1741ndash1754 2013

[7] Y Xiao G Bi D Niyato and L A DaSilva ldquoA hierarchicalgame theoretic framework for cognitive radio networksrdquo IEEEJournal on Selected Areas in Communications vol 30 no 10 pp2053ndash2069 2012

[8] F Gabry N Li N Schrammar M Girnyk L K Rasmussenand M Skoglund ldquoOn the optimization of the secondarytransmitterrsquos strategy in cognitive radio channels with secrecyrdquoIEEE Journal on Selected Areas in Communications vol 32 no3 pp 451ndash463 2014

[9] D Niyato E Hossain and Z Han ldquoDynamics of multiple-seller and multiple-buyer spectrum trading in cognitive radionetworks a game-theoretic modeling approachrdquo IEEE Transac-tions on Mobile Computing vol 8 no 8 pp 1009ndash1022 2009

[10] S Kandukuri and S Boyd ldquoOptimal power control ininterference-limited fading wireless channels with outage-probability specificationsrdquo IEEE Transactions on Wireless Com-munications vol 1 no 1 pp 46ndash55 2002

[11] T Basar and G J Olsder Dynamic Noncooperative GameTheory SIAM Philadelphia Pa USA 1999

[12] F Facchinei and C Kanzow ldquoGeneralized Nash equilibriumproblemsrdquo 4OR vol 5 no 3 pp 173ndash210 2007

[13] D Ardagna B Panicucci and M Passacantando ldquoA game the-oretic formulation of the service provisioning problem in cloudsystemsrdquo in Proceedings of the 20th International Conference onWorld Wide Web (WWW rsquo11) pp 177ndash186 ACM April 2011

[14] D Bertsekas W Hager and O Mangasarian Nonlinear Pro-gramming Athena Scientific Belmont Mass USA 1999

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Active and Passive Electronic Components

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International Journal of

RotatingMachinery

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Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

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Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

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Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

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Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Chemical EngineeringInternational Journal of Antennas and

Propagation

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Navigation and Observation

International Journal of

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DistributedSensor Networks

International Journal of

4 International Journal of Distributed Sensor Networks

minus log( 1

1 minus 120585119899119897

))) minus nablap119897

119873

sum119899=1

120593119899

119897(

119871

sum119897=1

120588119899

119897119896119901119899

119897ℎ119899

119897119896

minus 119879119899

119896)120592119897(

119873

sum119899=1

120588119899

119897119901119899

119897minus 119875

max119897

) = 0

119873

sum119899=1

120596119899

119897(

119871+1

sum119895 =119897

log(1 + 120588119899119895119897

120574119899

119897119901119899

119895ℎ119899

119895119897

119901119899119897ℎ119899119897119897

) minus log( 1

1 minus 120585119899119897

))

= 0

119873

sum119899=1

120593119899

119897(

119871

sum119897=1

120588119899

119897119896119901119899

119897ℎ119899

119897119896minus 119879119899

119896) = 0

(15)

We note that the robust followersrsquo game shows a jointlyconvex generalized SE problem therefore the solution of theSE problem with constraints in (13) is a variational inequalityVI(P F) where P is the set of joint convexity It is importantto determine a vector 119911lowast isin P sub 119877119899 such that ⟨F(zlowast) zminuszlowast⟩ ge0 for all 119911 isin P and F(p) = minus(nabla

119901119897(p119897))119871

119897=1[13] Then the

solution of VI(P F) is a variational SEIn this paper we only focus on the power control in cog-

nitive radio networks by assuming the channel assignmenthas already been done Then we can divide the variationalinequality VI(P F) into 119873 subproblems each subproblemdenotesVI(P

119899 F119899) on the subchannel 119899 and they are indepen-

dentTherefore on the subchannel 119899 the KKT conditions canbe expressed as [12]

F119899(p) + 120592

119899nabla119901(

119871

sum119897=1

120588119899

119897119901119899

119897minus

119871

sum119897=1

120588119899

119897119875max119897

)

+ 120596119899nabla119901(

119871

sum119897=1

(

119871+1

sum119895 =119897

log(1 + 120588119899119895119897

120574119899

119897119901119899

119895ℎ119899

119895119897

119901119899119897ℎ119899119897119897

))

minus

119871

sum119897=1

log( 1

1 minus 120585119899119897

)) + 120593119899nabla119901(

119871

sum119897=1

120588119899

119897119896119901119899

119897ℎ119899

119897119896minus 119879119899

119896)

(16)

Now from the definition of [12] we have

F1=

[[[[[[[[[[[[[

[

1205881

1119896(119908119896ℎ1

1119896minus

1

119868 (1199011minus1))

1205881

2119896(119908119896ℎ1

2119896minus

1

119868 (1199011minus2))

1205881

119871119896(119888119896ℎ1

119871119896minus

1

119868 (1199011minus119871))

]]]]]]]]]]]]]

]

F119899=

[[[[[[[[[[[[[

[

120588119899

1119896(119888119896ℎ119899

1119896minus

1

119868 (119901119899minus1))

120588119899

2119896(119888119896ℎ119899

2119896minus

1

119868 (119901119899minus2))

120588119899

119871119896(119888119896ℎ119899

119871119896minus

1

119868 (119901119899minus119871))

]]]]]]]]]]]]]

]

(17)

Therefore the Jacobian of F119899is

J1=

[[[[[[[

[

1205881

1119896ℎ1

11198960 sdot sdot sdot 0

0 1205881

2119896ℎ1

2119896sdot sdot sdot 0

0

0 0 sdot sdot sdot 1205881

119871119896ℎ1

119871119896

]]]]]]]

]

J119899=

[[[[[[

[

120588119899

1119896ℎ119899

11198960 sdot sdot sdot 0

0 120588119899

2119896ℎ119899

2119896sdot sdot sdot 0

0

0 0 sdot sdot sdot 120588119899

119871119896ℎ119899

119871119896

]]]]]]

]

(18)

Each F119899J119899is a diagonal matrix and all the diagonal

elements are positive Therefore F119899J119899is positive definition

on P119899 and so F

119899is strictly monotone Hence the global

SG problem admits a unique global variational equilibriumsolution [12] Due to the jointly convex nature of the globalSE problem the variational equilibrium is the unique globalmaximizer of (14) [12] which completes the proof in theliterature [12]

42 Solution of the Optimization for SUs (Followers) For theP2 the utility function of each SU is a concave function of 119901119899

119897

and the constraints are all linear so a partial Lagrange dualdecompositionmethod (LDDM) [4] for the problems is used

For the problem in (13) the corresponding Lagrangianfunction for the SU 119897 on the subchannel 119899 can be expressedas

L119897(p119897w119896 120596119897 120592119897120593119897) =

119873

sum119899=1

120588119899

119897120574119899

119897minus

119873

sum119899=1

119870

sum119896=1

120588119899

119897119896119908119896119901119899

119897ℎ119899

119897119896

minus

119873

sum119899=1

120596119899

119897(

119871+1

sum119895 =119897

log(1 + 120588119899119895119897

120574119899

119897119901119899

119895ℎ119899

119895119897

119901119899119897ℎ119899119897119897

)

minus log( 1

1 minus 120585119899119897

)) minus 120592119897(

119873

sum119899=1

120588119899

119897119901119899

119897minus 119875119898

119897) minus

119873

sum119899=1

120593119899

119897

sdot (

119871

sum119897=1

120588119899

119897119896119901119899

119897ℎ119899

119897119896minus 119879119899

119896)

(19)

International Journal of Distributed Sensor Networks 5

where 120596119899119897 120592119897 and 120593119899

119897are the nonnegative dual variables of the

constraints in (13)We decompose the optimization problem into 119873 inde-

pendent subproblems Then on the subchannel 119899 taking theKarush-Kuhn-Tucker (KKT) condition [4]

120597L119897

120597119901119899119897

=ℎ119899

119897119897

119868 (p119899minus119897)minus 119908119896ℎ119899

119897119896

+ 120596119899

119897

119871+1

sum119895 =119897

(120588119899

119895119897

120574119899

119897119901119899

119895ℎ119899

119895119897

119901119899119897(119901119899119897ℎ119899119897119897+ 120574119899

119897119901119899119895ℎ119899119895119897) ln 2

)

minus 120592119897minus 120593119896ℎ119899

119897119896

(20)

Simply in (20) we assume that the 119895th user causing theinterference power 119901119899

119895ℎ119899

119895119897to the SU 119897 on the subchannel 119899 can

be denoted as the average interference power except the SU 119897119866119899

119897= 119864[sum

119871+1

119895=1119895 =119897120588119899

119895119897119901119899

119895ℎ119899

119895119897] Hence we rewrite (20) as follows

120597L119897

120597119901119899119897

=ℎ119899

119897119897

119868 (p119899minus119897)minus 119908119896ℎ119899

119897119896+ 120596119899

119897

120574119899

119897119866119899

119897(sum119871+1

119895 =119897120588119899

119895119897)

119901119899119897(119901119899119897ℎ119899119897119897+ 120574119899

119897119866119899119897) ln 2

minus 120592119897minus 120593119896ℎ119899

119897119896

(21)

Set (21) to zero and get the optimal transmission power ofthe SU 119897 if it transmits on the channel 119899

119901119899

119897

lowast

=

minus120574119899

119897119866119899

119897+ radic4120596119899

119897ℎ119899119897119897120574119899

119897119866119899119897(sum119871+1

119895 =119897120588119899119895119897) 119883 ln 2 + 120574119899

119897119866119899119897

2ℎ119899119897119897

(22)

where119883119899119897= (119908119896+ 120593119899

119897)ℎ119899

119897119896+ 120592119897minus ℎ119899

119897119897119868(p119899minus119897)

The transmission power of the SU 119897 is zero if theinterference price for it is larger than payoff threshold 119876119899

119897on

the channel 119899 Then setting 119901119899119897

lowast

= 0 we can get the payoffthreshold of the SU 119897 if it transmits on the channel 119899

119876119899

119897=1

ℎ119899119897119896

(4120596119899

119897ℎ119899

119897119897120574119899

119897119866119899

119897(sum119871+1

119895 =119897120588119899

119895119897)

((120574119899

119897119866119899119897)2

minus 120574119899

119897119866119899119897) ln 2

minus 120592119897minus 120593119899

119897ℎ119899

119897119896

+ℎ119899

119897119897

119868 (p119899minus119897))

(23)

From (23) if the price 119908119896gt 119876119899

119897 the price is above the

payoff threshold of the SU 119897 and it will stop transmitting onthe channel 119899 without buying the interference power

43 Solution of the Optimization for PUs (Leaders) In orderto maximize its own utility each PU needs to adaptivelyoffer an interference price to SUs based on transmit powerresponse of the SUs P1 can be decomposed into twosubproblems fix 119908

119896to get the optimal transmission power

of each PU 119896 and then search the optimal 119908119896 The optimal

transmission power of the PU 119896 can be applied by the previousLDDM

Thus for P1 in (11) the corresponding Lagrangian func-tion can be given as

L119896(p119896 119908119896120582119896 120583119896 ^119896) =

119873

sum119899=1

120588119899

119896120574119899

119896+ 119908119896

119873

sum119899=1

119871

sum119897=1

120588119899

119897119896119901119899

119897ℎ119899

119897119896

minus

119873

sum119899=1

120582119899

119896(

119871

sum119897

log(1 + 120588119899119897119896

120574119899

119896119901119899

119897ℎ119899

119897119896

119901119899119896ℎ119899119896119896

)

minus log( 1

1 minus 120585119899119896

)) minus 120583119896(

119873

sum119899=1

120588119899

119896119901119899

119896minus 119875119898

119896) minus

119873

sum119899=1

]119899119896

sdot (

119871

sum119897=1

120588119899

119897119896119901119899

119897ℎ119899

119897119896minus 119879119899

119896)

(24)

where 120582119899119896 120583119896 and ]119899

119896are the dual variables of the constraints

in (11)Similarly in (24) we assume the 119897th SU causing the inter-

ference ℎ119899119897119896119901119899

119897to the PU 119896 on the channel 119899 can be denoted

as the average interference power 119866119899119896= 119864[sum

119871

119897=1120588119899

119897119896ℎ119899

119897119896119901119899

119897]

According to the KKT conditions we obtain the optimaltransmission power of the PU 119896 if it transmits on the channel119899

119901119899

119896

lowast

=1

2ℎ119899119896119896

(minus120574119899

119896119866119899

119896

+ radic4120582119899

119896ℎ119899

119896119896120574119899

119896119866119899

119896(sum119871

119897=1120588119899

119897119896)

(120583119896minus ℎ119899119896119896119868 (p119899minus119896)) ln 2

+ 120574119899

119896119866119899119896)

(25)

Since L119896(p119896 119908119896120582119896 120583119896 ^119896) is a stepwise function with

breakpoints at119876119899119897for the SU 119897 we should discuss the existence

of the optimal price119908119896first So we divide (24) with respect to

119908119896with two parts on each channel 119899 we have L

119901119896(119901119899

119897) =

120574119899

119896(119901119899

119897) and L

119901119896(119908119896) = (119908

119896minus ]119896)119901119899

119897ℎ119897119896 From (22) it can

be easily observed that 120574119899119896(119901119899

119897) is a concave function of 119908

119896

Therefore we only need to discuss the situation ofL119901119896(119908119896)

For the SU 119897 we first sort119876119899119897(119899 = 1 119873) in ascending order

and have 119873 intervals (0 1198761119897)(1198761

119897 1198762

119897) (119876

119873minus1

119897 119876119873

119897) where

1198761

119897lt 1198762

119897lt sdot sdot sdot lt 119876

119873

119897 Note if the SU 119897 is not allocated on the

channel 119899 (119876119899minus1119897 119876119899

119897)must be taken out of the order We take

(0 1198761

119897) for an example When 119908

119896rarr 0 we can derive that

120597L119901119896(119908119896)

120597119908119896

1003816100381610038161003816100381610038161003816100381610038161003816119908119896rarr0

gt 119901119899

119897ℎ119897119896gt 0 (26)

Taking the second derivative of L119901119896(119908119896) with respect to 119908

119896

is1205972L119901119896

1205971199082119896

= minusℎ119899

119897119897

2

sdot radic120596119899

119897120574119899

119897119866119899

119897(sum119871+1

119895 =119897120588119899

119895119897)

119883119899119897ln 2

+ 120574119899

119897119866119899119897(120596119899

119897ℎ119899

119897119896120574119899

119897119866119899

119897(sum119871+1

119895 =119897120588119899

119895119897)

(119883119899119897)2 ln 2

)

lt 0

(27)

6 International Journal of Distributed Sensor Networks

(01) Initialization set 119902 = 0 set initial 119888119896(119902) and p

119896(119902) for 119896 isin 119870 Set initial p

119897(119902) for 119897 isin 119871

Set 120591 where 120591 is positive and sufficiently small(02) For each SU 119897

(03) Use SM to find the optimal step sizes 120572lowast 120573lowast and 120579lowast and update 120596119899119897 120592119897and 120593

119896according to (28) respectively

(04) For the given 119888119896(119902) and p

119896(119902) of all PUs each SU 119897 responds with its transmit power vector plowast

119897(119902 + 1) according to (22)

(05) If 119876119899119897lt 119888119896(119902) the SU 119897 stops transmitting on the channel 119899 of the PU 119896

(06) End(07) For each PU 119896

(08) Use SM to find the optimal 120594lowast and 120599lowast and update 120582119899119896and 120583

119896by (29)

(09) For the responded plowast119897(119902) of all SUs each PU 119896 updates its transmit power vector as plowast

119896(119902 + 1) according to (25)

(10) Each PU 119896 updates its price by the solution of119888119896(119902 + 1) = argmax

119888119896

119880119896(p119896(119902 + 1) p

minus119896(119902 + 1))

(11) End(12) For each PU 119896 if plowast

119896(119902 + 1) minus p

119896(119902) le 120591 or 119902 gt 102 stop the algorithm Otherwise 119902 = 119902 + 1

repeat steps (02) and (11) until the condition is satisfied

Algorithm 1 Iterative algorithm for reaching the SE

L119901119896(119908119896) is a concave function whether (120597L

119901119896120597119908119896)|119908119896rarr119876

1

119897

gt 0 or (120597L119901119896120597119908119896)|119908119896rarr119876

1

119897

lt 0 except at the nondifferen-tiable point 1198761

119897

Through the above analysis L119896(p119896 119908119896120582119896 120583119896 ^119896) is a

concave function with respect to 119908119896except at 119876119899

119897 The

ellipsoid method [4] can be employed to solve the convexoptimization in each interval

44 Iterative Algorithm to Find the SE For the above dis-cussion we propose an iterative algorithm to search the SEDue to the fact that computation of the dual variables isa complicated task the subgradient method (SM) [14] isapplied to obtain the global optimum SE of this problemThen dual variables are updated as follows

120596119899

119897(119905 + 1) = (120596

119899

119897(119905) + 120572(

119871+1

sum119895 =119897

log(1 + 120588119899119895119897

120574119899

119897119901119899

119895ℎ119899

119895119897

119901119899119897ℎ119899119897119897

)

minus log( 1

1 minus 120585119899119897

)))

+

120592119897(119905 + 1) = (120592

119897(119905) + 120573(

119873

sum119899=1

120588119899

119897119901119899

119897minus 119901

max119897))

+

120593119899

119897(119905 + 1) = (120593

119899

119897(119905) + 120579(

119871

sum119897=1

120588119899

119897119896119901119899

119897ℎ119899

119897119896minus 119879119899

119896))

+

(28)

120582119899

119896(119905 + 1) = (120582

119899

119896(119905) + 120599(

119871

sum119897

log(1 + 120588119899119897119896

120574119899

119896119901119899

119897ℎ119899

119897119896

119901119899119896ℎ119899119896119896

)

minus log( 1

1 minus 120585119899119896

)))

+

120583119896(119905 + 1) = (120583

119896(119905) + 120594(

119873

sum119899=1

120588119899

119896119901119899

119896minus 119875

max119896

))

+

(29)

where 119905 is the iteration index and 120572 gt 0 120573 gt 0 120579 gt 0 120599 gt0 and 120594 gt 0 are sufficiently small The SM guarantees theconvergence of the above optimal dual variables if the stepsizes are chosen by following the step size policy [14]

We then design the iterative algorithm to achieve the SEshown in Algorithm 1

5 Simulation Results and Their Analysis

In this section several numerical examples are presented toevaluate the performances of the proposed SG by comparingthe optimal price-based SG considering global interferencein [5] and the nominal SG without considering globalinterference and outage constraints The cell radius of 500mwith the PBS centered at the original CRN The simulationparameters are as follows 1205752 = 10minus12W 119870 = 3 119871 = 5 and119873 = 10 The SINR threshold of PUs and SUs is set as 7 dBand 4 dB respectively All PUs and SUs deploy the maximumpower 119875max

119896= 100mWand 119875max

119897= 50mWThe channel gain

in this system is ℎ119894119895= 119889minus4

119894119895 with 119889

119894119895being their corresponding

distance The outage probability thresholds of both PUs andSUs are 0001 We set the interference-to-noise ratio (INR) as1198791205752Firstly we illustrate the convergence of the proposed

algorithm for achieving an SE of the proposed game FromFigure 2 the three PUs and five SUs iteratively update theirutilities and obviously converge to the SE The proposedalgorithm converges quickly in terms of PUs only about tentimes In addition due to the larger number of SUs theconvergence of the SUs is slower than that of the PUs

51 Impact of INR In this subsection we set the numberof PUs and SUs as 3 and 5 and INR changes from minus20 dBto 20 dB which means that IPC changes from 10

minus14W to10minus10WWe then consider the sum rate of PUs and SUs for the

three solutionswith different tolerant interference constraintsshown in Figures 3 and 4 For the performance of sum rate

International Journal of Distributed Sensor Networks 7

PU-1PU-2PU-3

SU-1SU-2SU-3

SU-4SU-5

12

13

14

15

16

Util

ities

of P

Us

1

2

3

4

5

6

7

Util

ities

of S

Us

50 15 20 2510

Iteration

50 15 20 2510

Iteration

Figure 2 The convergence of utility of PUs (leaders) setting priceof PUs and utility of SUs (followers)

of SUs Figure 3 shows the nominal SG scheme outperformsother two schemes when the interference temperature levelis stringent but is inferior to the two schemes when it isloose The proposed SG performs the worst because ofthe demand of satisfying the outage probability constraintOnce the interference constraints are loose enough to benot active accordingly our proposed solution works betterthan the others For the leaders the sum rate of the nominalSG performs the worst because of not including the globalinterference constraints This is because the performance ofPUs may be degraded with the increases of the interference

Figure 5 presents the outage probability of the systemwith different INR It is observed that the proposed schemeachieves much lower outage probability than other schemesin particular the performance gap becomes larger with theincrease of INR This is because our proposed algorithmworks best by considering the outage probability constraintsof users which prevents the outage events well

52 Impact of Different Number of SUs In this subsection theINR is set to be 10 dB The number of SUs changes from 2 to20 All the other simulation parameters are the same as thebeginning part of this section

For the PUs from Figure 6 the sum rate of the SGsolution performs the worst because of not including theglobal interference constraints So the PUs may refuse tosell more spectrum resource because of protecting their own

Our proposed SGOptimal price-based SG in [5]Nominal SG

50 10 15 20minus5minus15 minus10minus20

INR T1205752 (dB)

25

26

27

28

29

30

31

32

33

34

Sum

rate

of t

he P

Us (

bits

sH

z)Figure 3 Sum rate of PUs versus INR

26

24

22

20

18

16

14

12

10minus20 minus15 minus10 minus5 0 5 10 15 20

Our proposed SGOptimal price-based SG in [5]Nominal SG

INR T1205752 (dB)

Sum

rate

of t

he S

Us (

bits

sH

z)

Figure 4 Sum rate of SUs versus INR

communication QoS The proposed scheme outperformsmost in terms of sum rate of SUs because the algorithm allowsmore SUs to share their radio resource so that it increases PUsrsquoutilities with considering the channel uncertainty and globalinterference constraints

Figure 7 shows the sum rate of SUs versus the numberof SUs The sum rate of SUs performance of our proposedalgorithm works better than other two schemes this isbecause the proposed scheme is able to support more SUs attheBS shown in Figure 7 so thatmore SUshave opportunities

8 International Journal of Distributed Sensor Networks

Our proposed SGOptimal price-based SG in [5]Nominal SG

10minus3

10minus2

10minus1

100

Out

age p

roba

bilit

y

minus15 minus10 minus5 0 5 10 15 20minus20

INR T1205752 (dB)

Figure 5 Outage probability versus INR

Our proposed SGOptimal price-based SG in [5]Nominal SG

16

18

20

22

24

26

28

30

32

34

Sum

rate

of P

Us

4 6 82 12 14 16 18 2010

Number of SUs

Figure 6 Sum rate of PUs versus number of SUs

to transmit which increase the sum rate In addition theperformance gap between the proposed scheme and theother algorithms increases when the network grows largerby which it can be concluded that the proposed scheme ismore suitable for application in larger networks Because thescheme sets the adaptive punishment parameter among allserved SUs to control their behavior more SUs can be servedat the BS thus achieving a higher sum rate

Figure 8 shows the number of outage probability com-parison versus the number of SUs for different algorithmsThe proposed algorithm is able to support more SUs than

Our proposed SGOptimal price-based SG in [5]Nominal SG

4 6 8 10 12 14 16 18 202

Number of SUs

10

15

20

25

30

35

40

45

Sum

rate

of S

Us

Figure 7 Sum rate of SUs versus number of SUs

Our proposed SGOptimal price-based SG in [5]Nominal SG

4 6 8 10 12 14 16 18 202

Number of SUs

10minus4

10minus3

10minus2

10minus1

100

Out

age p

roba

bilit

y

Figure 8 Outage probability versus number of SUs

other schemes This is because we develop the channelassignment scheduling scheme to decrease the probabilityof the unserved SUs so more SUs are admitted to serveat the BS without causing unendurable interference to PUsIn addition the interference among SUs is taken into therevenue utility function to void serious interference for someSUs who have bad channel condition

International Journal of Distributed Sensor Networks 9

6 Conclusion

In this paper we propose a Stackelberg game for powercontrol problem in CRNs with channel outage constraintsand global interference constraints We employ LDDM tosolve the problem by decomposing it into independentsubproblems and develop an iterative algorithm to achieveSE Simulation results show that the proposed algorithmimproves the performance compared with other game algo-rithms

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the National Nature ScienceFoundation of China (61271259 and 61301123) the SpecialFund of Chongqing Key Laboratory (CSTC) the Programfor Changjiang Scholars and Innovative Research Team inUniversity (IRT2129) and the Graduate Student ResearchInnovation Project of Chongqing University of Posts andTelecommunications (Chongqing) (CYS14143)

References

[1] K Akkarajitsakul E Hossain D Niyato and D I Kim ldquoGametheoretic approaches for multiple access in wireless networksa surveyrdquo IEEE Communications Surveys and Tutorials vol 13no 3 pp 372ndash395 2011

[2] Y Wu T Zhang and D H K Tsang ldquoJoint pricing andpower allocation for dynamic spectrum access networks withStackelberg game modelrdquo IEEE Transactions on Wireless Com-munications vol 10 no 1 pp 12ndash19 2011

[3] X Kang R Zhang and M Motani ldquoPrice-based resourceallocation for spectrum-sharing femtocell networks a stack-elberg game approachrdquo IEEE Journal on Selected Areas inCommunications vol 30 no 3 pp 538ndash549 2012

[4] R Xie F R Yu H Ji and Y Li ldquoEnergy-efficient resourceallocation for heterogeneous cognitive radio networks withfemtocellsrdquo IEEETransactions onWireless Communications vol11 no 11 pp 3910ndash3920 2012

[5] Z Wang L Jiang and C He ldquoOptimal price-based power con-trol algorithm in cognitive radio networksrdquo IEEE TransactionsonWireless Communications vol 13 no 11 pp 5909ndash5920 2014

[6] M Le Treust S Lasaulce Y Hayel and G L He ldquoGreen powercontrol in cognitive wireless networksrdquo IEEE Transactions onVehicular Technology vol 62 no 4 pp 1741ndash1754 2013

[7] Y Xiao G Bi D Niyato and L A DaSilva ldquoA hierarchicalgame theoretic framework for cognitive radio networksrdquo IEEEJournal on Selected Areas in Communications vol 30 no 10 pp2053ndash2069 2012

[8] F Gabry N Li N Schrammar M Girnyk L K Rasmussenand M Skoglund ldquoOn the optimization of the secondarytransmitterrsquos strategy in cognitive radio channels with secrecyrdquoIEEE Journal on Selected Areas in Communications vol 32 no3 pp 451ndash463 2014

[9] D Niyato E Hossain and Z Han ldquoDynamics of multiple-seller and multiple-buyer spectrum trading in cognitive radionetworks a game-theoretic modeling approachrdquo IEEE Transac-tions on Mobile Computing vol 8 no 8 pp 1009ndash1022 2009

[10] S Kandukuri and S Boyd ldquoOptimal power control ininterference-limited fading wireless channels with outage-probability specificationsrdquo IEEE Transactions on Wireless Com-munications vol 1 no 1 pp 46ndash55 2002

[11] T Basar and G J Olsder Dynamic Noncooperative GameTheory SIAM Philadelphia Pa USA 1999

[12] F Facchinei and C Kanzow ldquoGeneralized Nash equilibriumproblemsrdquo 4OR vol 5 no 3 pp 173ndash210 2007

[13] D Ardagna B Panicucci and M Passacantando ldquoA game the-oretic formulation of the service provisioning problem in cloudsystemsrdquo in Proceedings of the 20th International Conference onWorld Wide Web (WWW rsquo11) pp 177ndash186 ACM April 2011

[14] D Bertsekas W Hager and O Mangasarian Nonlinear Pro-gramming Athena Scientific Belmont Mass USA 1999

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of Distributed Sensor Networks 5

where 120596119899119897 120592119897 and 120593119899

119897are the nonnegative dual variables of the

constraints in (13)We decompose the optimization problem into 119873 inde-

pendent subproblems Then on the subchannel 119899 taking theKarush-Kuhn-Tucker (KKT) condition [4]

120597L119897

120597119901119899119897

=ℎ119899

119897119897

119868 (p119899minus119897)minus 119908119896ℎ119899

119897119896

+ 120596119899

119897

119871+1

sum119895 =119897

(120588119899

119895119897

120574119899

119897119901119899

119895ℎ119899

119895119897

119901119899119897(119901119899119897ℎ119899119897119897+ 120574119899

119897119901119899119895ℎ119899119895119897) ln 2

)

minus 120592119897minus 120593119896ℎ119899

119897119896

(20)

Simply in (20) we assume that the 119895th user causing theinterference power 119901119899

119895ℎ119899

119895119897to the SU 119897 on the subchannel 119899 can

be denoted as the average interference power except the SU 119897119866119899

119897= 119864[sum

119871+1

119895=1119895 =119897120588119899

119895119897119901119899

119895ℎ119899

119895119897] Hence we rewrite (20) as follows

120597L119897

120597119901119899119897

=ℎ119899

119897119897

119868 (p119899minus119897)minus 119908119896ℎ119899

119897119896+ 120596119899

119897

120574119899

119897119866119899

119897(sum119871+1

119895 =119897120588119899

119895119897)

119901119899119897(119901119899119897ℎ119899119897119897+ 120574119899

119897119866119899119897) ln 2

minus 120592119897minus 120593119896ℎ119899

119897119896

(21)

Set (21) to zero and get the optimal transmission power ofthe SU 119897 if it transmits on the channel 119899

119901119899

119897

lowast

=

minus120574119899

119897119866119899

119897+ radic4120596119899

119897ℎ119899119897119897120574119899

119897119866119899119897(sum119871+1

119895 =119897120588119899119895119897) 119883 ln 2 + 120574119899

119897119866119899119897

2ℎ119899119897119897

(22)

where119883119899119897= (119908119896+ 120593119899

119897)ℎ119899

119897119896+ 120592119897minus ℎ119899

119897119897119868(p119899minus119897)

The transmission power of the SU 119897 is zero if theinterference price for it is larger than payoff threshold 119876119899

119897on

the channel 119899 Then setting 119901119899119897

lowast

= 0 we can get the payoffthreshold of the SU 119897 if it transmits on the channel 119899

119876119899

119897=1

ℎ119899119897119896

(4120596119899

119897ℎ119899

119897119897120574119899

119897119866119899

119897(sum119871+1

119895 =119897120588119899

119895119897)

((120574119899

119897119866119899119897)2

minus 120574119899

119897119866119899119897) ln 2

minus 120592119897minus 120593119899

119897ℎ119899

119897119896

+ℎ119899

119897119897

119868 (p119899minus119897))

(23)

From (23) if the price 119908119896gt 119876119899

119897 the price is above the

payoff threshold of the SU 119897 and it will stop transmitting onthe channel 119899 without buying the interference power

43 Solution of the Optimization for PUs (Leaders) In orderto maximize its own utility each PU needs to adaptivelyoffer an interference price to SUs based on transmit powerresponse of the SUs P1 can be decomposed into twosubproblems fix 119908

119896to get the optimal transmission power

of each PU 119896 and then search the optimal 119908119896 The optimal

transmission power of the PU 119896 can be applied by the previousLDDM

Thus for P1 in (11) the corresponding Lagrangian func-tion can be given as

L119896(p119896 119908119896120582119896 120583119896 ^119896) =

119873

sum119899=1

120588119899

119896120574119899

119896+ 119908119896

119873

sum119899=1

119871

sum119897=1

120588119899

119897119896119901119899

119897ℎ119899

119897119896

minus

119873

sum119899=1

120582119899

119896(

119871

sum119897

log(1 + 120588119899119897119896

120574119899

119896119901119899

119897ℎ119899

119897119896

119901119899119896ℎ119899119896119896

)

minus log( 1

1 minus 120585119899119896

)) minus 120583119896(

119873

sum119899=1

120588119899

119896119901119899

119896minus 119875119898

119896) minus

119873

sum119899=1

]119899119896

sdot (

119871

sum119897=1

120588119899

119897119896119901119899

119897ℎ119899

119897119896minus 119879119899

119896)

(24)

where 120582119899119896 120583119896 and ]119899

119896are the dual variables of the constraints

in (11)Similarly in (24) we assume the 119897th SU causing the inter-

ference ℎ119899119897119896119901119899

119897to the PU 119896 on the channel 119899 can be denoted

as the average interference power 119866119899119896= 119864[sum

119871

119897=1120588119899

119897119896ℎ119899

119897119896119901119899

119897]

According to the KKT conditions we obtain the optimaltransmission power of the PU 119896 if it transmits on the channel119899

119901119899

119896

lowast

=1

2ℎ119899119896119896

(minus120574119899

119896119866119899

119896

+ radic4120582119899

119896ℎ119899

119896119896120574119899

119896119866119899

119896(sum119871

119897=1120588119899

119897119896)

(120583119896minus ℎ119899119896119896119868 (p119899minus119896)) ln 2

+ 120574119899

119896119866119899119896)

(25)

Since L119896(p119896 119908119896120582119896 120583119896 ^119896) is a stepwise function with

breakpoints at119876119899119897for the SU 119897 we should discuss the existence

of the optimal price119908119896first So we divide (24) with respect to

119908119896with two parts on each channel 119899 we have L

119901119896(119901119899

119897) =

120574119899

119896(119901119899

119897) and L

119901119896(119908119896) = (119908

119896minus ]119896)119901119899

119897ℎ119897119896 From (22) it can

be easily observed that 120574119899119896(119901119899

119897) is a concave function of 119908

119896

Therefore we only need to discuss the situation ofL119901119896(119908119896)

For the SU 119897 we first sort119876119899119897(119899 = 1 119873) in ascending order

and have 119873 intervals (0 1198761119897)(1198761

119897 1198762

119897) (119876

119873minus1

119897 119876119873

119897) where

1198761

119897lt 1198762

119897lt sdot sdot sdot lt 119876

119873

119897 Note if the SU 119897 is not allocated on the

channel 119899 (119876119899minus1119897 119876119899

119897)must be taken out of the order We take

(0 1198761

119897) for an example When 119908

119896rarr 0 we can derive that

120597L119901119896(119908119896)

120597119908119896

1003816100381610038161003816100381610038161003816100381610038161003816119908119896rarr0

gt 119901119899

119897ℎ119897119896gt 0 (26)

Taking the second derivative of L119901119896(119908119896) with respect to 119908

119896

is1205972L119901119896

1205971199082119896

= minusℎ119899

119897119897

2

sdot radic120596119899

119897120574119899

119897119866119899

119897(sum119871+1

119895 =119897120588119899

119895119897)

119883119899119897ln 2

+ 120574119899

119897119866119899119897(120596119899

119897ℎ119899

119897119896120574119899

119897119866119899

119897(sum119871+1

119895 =119897120588119899

119895119897)

(119883119899119897)2 ln 2

)

lt 0

(27)

6 International Journal of Distributed Sensor Networks

(01) Initialization set 119902 = 0 set initial 119888119896(119902) and p

119896(119902) for 119896 isin 119870 Set initial p

119897(119902) for 119897 isin 119871

Set 120591 where 120591 is positive and sufficiently small(02) For each SU 119897

(03) Use SM to find the optimal step sizes 120572lowast 120573lowast and 120579lowast and update 120596119899119897 120592119897and 120593

119896according to (28) respectively

(04) For the given 119888119896(119902) and p

119896(119902) of all PUs each SU 119897 responds with its transmit power vector plowast

119897(119902 + 1) according to (22)

(05) If 119876119899119897lt 119888119896(119902) the SU 119897 stops transmitting on the channel 119899 of the PU 119896

(06) End(07) For each PU 119896

(08) Use SM to find the optimal 120594lowast and 120599lowast and update 120582119899119896and 120583

119896by (29)

(09) For the responded plowast119897(119902) of all SUs each PU 119896 updates its transmit power vector as plowast

119896(119902 + 1) according to (25)

(10) Each PU 119896 updates its price by the solution of119888119896(119902 + 1) = argmax

119888119896

119880119896(p119896(119902 + 1) p

minus119896(119902 + 1))

(11) End(12) For each PU 119896 if plowast

119896(119902 + 1) minus p

119896(119902) le 120591 or 119902 gt 102 stop the algorithm Otherwise 119902 = 119902 + 1

repeat steps (02) and (11) until the condition is satisfied

Algorithm 1 Iterative algorithm for reaching the SE

L119901119896(119908119896) is a concave function whether (120597L

119901119896120597119908119896)|119908119896rarr119876

1

119897

gt 0 or (120597L119901119896120597119908119896)|119908119896rarr119876

1

119897

lt 0 except at the nondifferen-tiable point 1198761

119897

Through the above analysis L119896(p119896 119908119896120582119896 120583119896 ^119896) is a

concave function with respect to 119908119896except at 119876119899

119897 The

ellipsoid method [4] can be employed to solve the convexoptimization in each interval

44 Iterative Algorithm to Find the SE For the above dis-cussion we propose an iterative algorithm to search the SEDue to the fact that computation of the dual variables isa complicated task the subgradient method (SM) [14] isapplied to obtain the global optimum SE of this problemThen dual variables are updated as follows

120596119899

119897(119905 + 1) = (120596

119899

119897(119905) + 120572(

119871+1

sum119895 =119897

log(1 + 120588119899119895119897

120574119899

119897119901119899

119895ℎ119899

119895119897

119901119899119897ℎ119899119897119897

)

minus log( 1

1 minus 120585119899119897

)))

+

120592119897(119905 + 1) = (120592

119897(119905) + 120573(

119873

sum119899=1

120588119899

119897119901119899

119897minus 119901

max119897))

+

120593119899

119897(119905 + 1) = (120593

119899

119897(119905) + 120579(

119871

sum119897=1

120588119899

119897119896119901119899

119897ℎ119899

119897119896minus 119879119899

119896))

+

(28)

120582119899

119896(119905 + 1) = (120582

119899

119896(119905) + 120599(

119871

sum119897

log(1 + 120588119899119897119896

120574119899

119896119901119899

119897ℎ119899

119897119896

119901119899119896ℎ119899119896119896

)

minus log( 1

1 minus 120585119899119896

)))

+

120583119896(119905 + 1) = (120583

119896(119905) + 120594(

119873

sum119899=1

120588119899

119896119901119899

119896minus 119875

max119896

))

+

(29)

where 119905 is the iteration index and 120572 gt 0 120573 gt 0 120579 gt 0 120599 gt0 and 120594 gt 0 are sufficiently small The SM guarantees theconvergence of the above optimal dual variables if the stepsizes are chosen by following the step size policy [14]

We then design the iterative algorithm to achieve the SEshown in Algorithm 1

5 Simulation Results and Their Analysis

In this section several numerical examples are presented toevaluate the performances of the proposed SG by comparingthe optimal price-based SG considering global interferencein [5] and the nominal SG without considering globalinterference and outage constraints The cell radius of 500mwith the PBS centered at the original CRN The simulationparameters are as follows 1205752 = 10minus12W 119870 = 3 119871 = 5 and119873 = 10 The SINR threshold of PUs and SUs is set as 7 dBand 4 dB respectively All PUs and SUs deploy the maximumpower 119875max

119896= 100mWand 119875max

119897= 50mWThe channel gain

in this system is ℎ119894119895= 119889minus4

119894119895 with 119889

119894119895being their corresponding

distance The outage probability thresholds of both PUs andSUs are 0001 We set the interference-to-noise ratio (INR) as1198791205752Firstly we illustrate the convergence of the proposed

algorithm for achieving an SE of the proposed game FromFigure 2 the three PUs and five SUs iteratively update theirutilities and obviously converge to the SE The proposedalgorithm converges quickly in terms of PUs only about tentimes In addition due to the larger number of SUs theconvergence of the SUs is slower than that of the PUs

51 Impact of INR In this subsection we set the numberof PUs and SUs as 3 and 5 and INR changes from minus20 dBto 20 dB which means that IPC changes from 10

minus14W to10minus10WWe then consider the sum rate of PUs and SUs for the

three solutionswith different tolerant interference constraintsshown in Figures 3 and 4 For the performance of sum rate

International Journal of Distributed Sensor Networks 7

PU-1PU-2PU-3

SU-1SU-2SU-3

SU-4SU-5

12

13

14

15

16

Util

ities

of P

Us

1

2

3

4

5

6

7

Util

ities

of S

Us

50 15 20 2510

Iteration

50 15 20 2510

Iteration

Figure 2 The convergence of utility of PUs (leaders) setting priceof PUs and utility of SUs (followers)

of SUs Figure 3 shows the nominal SG scheme outperformsother two schemes when the interference temperature levelis stringent but is inferior to the two schemes when it isloose The proposed SG performs the worst because ofthe demand of satisfying the outage probability constraintOnce the interference constraints are loose enough to benot active accordingly our proposed solution works betterthan the others For the leaders the sum rate of the nominalSG performs the worst because of not including the globalinterference constraints This is because the performance ofPUs may be degraded with the increases of the interference

Figure 5 presents the outage probability of the systemwith different INR It is observed that the proposed schemeachieves much lower outage probability than other schemesin particular the performance gap becomes larger with theincrease of INR This is because our proposed algorithmworks best by considering the outage probability constraintsof users which prevents the outage events well

52 Impact of Different Number of SUs In this subsection theINR is set to be 10 dB The number of SUs changes from 2 to20 All the other simulation parameters are the same as thebeginning part of this section

For the PUs from Figure 6 the sum rate of the SGsolution performs the worst because of not including theglobal interference constraints So the PUs may refuse tosell more spectrum resource because of protecting their own

Our proposed SGOptimal price-based SG in [5]Nominal SG

50 10 15 20minus5minus15 minus10minus20

INR T1205752 (dB)

25

26

27

28

29

30

31

32

33

34

Sum

rate

of t

he P

Us (

bits

sH

z)Figure 3 Sum rate of PUs versus INR

26

24

22

20

18

16

14

12

10minus20 minus15 minus10 minus5 0 5 10 15 20

Our proposed SGOptimal price-based SG in [5]Nominal SG

INR T1205752 (dB)

Sum

rate

of t

he S

Us (

bits

sH

z)

Figure 4 Sum rate of SUs versus INR

communication QoS The proposed scheme outperformsmost in terms of sum rate of SUs because the algorithm allowsmore SUs to share their radio resource so that it increases PUsrsquoutilities with considering the channel uncertainty and globalinterference constraints

Figure 7 shows the sum rate of SUs versus the numberof SUs The sum rate of SUs performance of our proposedalgorithm works better than other two schemes this isbecause the proposed scheme is able to support more SUs attheBS shown in Figure 7 so thatmore SUshave opportunities

8 International Journal of Distributed Sensor Networks

Our proposed SGOptimal price-based SG in [5]Nominal SG

10minus3

10minus2

10minus1

100

Out

age p

roba

bilit

y

minus15 minus10 minus5 0 5 10 15 20minus20

INR T1205752 (dB)

Figure 5 Outage probability versus INR

Our proposed SGOptimal price-based SG in [5]Nominal SG

16

18

20

22

24

26

28

30

32

34

Sum

rate

of P

Us

4 6 82 12 14 16 18 2010

Number of SUs

Figure 6 Sum rate of PUs versus number of SUs

to transmit which increase the sum rate In addition theperformance gap between the proposed scheme and theother algorithms increases when the network grows largerby which it can be concluded that the proposed scheme ismore suitable for application in larger networks Because thescheme sets the adaptive punishment parameter among allserved SUs to control their behavior more SUs can be servedat the BS thus achieving a higher sum rate

Figure 8 shows the number of outage probability com-parison versus the number of SUs for different algorithmsThe proposed algorithm is able to support more SUs than

Our proposed SGOptimal price-based SG in [5]Nominal SG

4 6 8 10 12 14 16 18 202

Number of SUs

10

15

20

25

30

35

40

45

Sum

rate

of S

Us

Figure 7 Sum rate of SUs versus number of SUs

Our proposed SGOptimal price-based SG in [5]Nominal SG

4 6 8 10 12 14 16 18 202

Number of SUs

10minus4

10minus3

10minus2

10minus1

100

Out

age p

roba

bilit

y

Figure 8 Outage probability versus number of SUs

other schemes This is because we develop the channelassignment scheduling scheme to decrease the probabilityof the unserved SUs so more SUs are admitted to serveat the BS without causing unendurable interference to PUsIn addition the interference among SUs is taken into therevenue utility function to void serious interference for someSUs who have bad channel condition

International Journal of Distributed Sensor Networks 9

6 Conclusion

In this paper we propose a Stackelberg game for powercontrol problem in CRNs with channel outage constraintsand global interference constraints We employ LDDM tosolve the problem by decomposing it into independentsubproblems and develop an iterative algorithm to achieveSE Simulation results show that the proposed algorithmimproves the performance compared with other game algo-rithms

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the National Nature ScienceFoundation of China (61271259 and 61301123) the SpecialFund of Chongqing Key Laboratory (CSTC) the Programfor Changjiang Scholars and Innovative Research Team inUniversity (IRT2129) and the Graduate Student ResearchInnovation Project of Chongqing University of Posts andTelecommunications (Chongqing) (CYS14143)

References

[1] K Akkarajitsakul E Hossain D Niyato and D I Kim ldquoGametheoretic approaches for multiple access in wireless networksa surveyrdquo IEEE Communications Surveys and Tutorials vol 13no 3 pp 372ndash395 2011

[2] Y Wu T Zhang and D H K Tsang ldquoJoint pricing andpower allocation for dynamic spectrum access networks withStackelberg game modelrdquo IEEE Transactions on Wireless Com-munications vol 10 no 1 pp 12ndash19 2011

[3] X Kang R Zhang and M Motani ldquoPrice-based resourceallocation for spectrum-sharing femtocell networks a stack-elberg game approachrdquo IEEE Journal on Selected Areas inCommunications vol 30 no 3 pp 538ndash549 2012

[4] R Xie F R Yu H Ji and Y Li ldquoEnergy-efficient resourceallocation for heterogeneous cognitive radio networks withfemtocellsrdquo IEEETransactions onWireless Communications vol11 no 11 pp 3910ndash3920 2012

[5] Z Wang L Jiang and C He ldquoOptimal price-based power con-trol algorithm in cognitive radio networksrdquo IEEE TransactionsonWireless Communications vol 13 no 11 pp 5909ndash5920 2014

[6] M Le Treust S Lasaulce Y Hayel and G L He ldquoGreen powercontrol in cognitive wireless networksrdquo IEEE Transactions onVehicular Technology vol 62 no 4 pp 1741ndash1754 2013

[7] Y Xiao G Bi D Niyato and L A DaSilva ldquoA hierarchicalgame theoretic framework for cognitive radio networksrdquo IEEEJournal on Selected Areas in Communications vol 30 no 10 pp2053ndash2069 2012

[8] F Gabry N Li N Schrammar M Girnyk L K Rasmussenand M Skoglund ldquoOn the optimization of the secondarytransmitterrsquos strategy in cognitive radio channels with secrecyrdquoIEEE Journal on Selected Areas in Communications vol 32 no3 pp 451ndash463 2014

[9] D Niyato E Hossain and Z Han ldquoDynamics of multiple-seller and multiple-buyer spectrum trading in cognitive radionetworks a game-theoretic modeling approachrdquo IEEE Transac-tions on Mobile Computing vol 8 no 8 pp 1009ndash1022 2009

[10] S Kandukuri and S Boyd ldquoOptimal power control ininterference-limited fading wireless channels with outage-probability specificationsrdquo IEEE Transactions on Wireless Com-munications vol 1 no 1 pp 46ndash55 2002

[11] T Basar and G J Olsder Dynamic Noncooperative GameTheory SIAM Philadelphia Pa USA 1999

[12] F Facchinei and C Kanzow ldquoGeneralized Nash equilibriumproblemsrdquo 4OR vol 5 no 3 pp 173ndash210 2007

[13] D Ardagna B Panicucci and M Passacantando ldquoA game the-oretic formulation of the service provisioning problem in cloudsystemsrdquo in Proceedings of the 20th International Conference onWorld Wide Web (WWW rsquo11) pp 177ndash186 ACM April 2011

[14] D Bertsekas W Hager and O Mangasarian Nonlinear Pro-gramming Athena Scientific Belmont Mass USA 1999

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

6 International Journal of Distributed Sensor Networks

(01) Initialization set 119902 = 0 set initial 119888119896(119902) and p

119896(119902) for 119896 isin 119870 Set initial p

119897(119902) for 119897 isin 119871

Set 120591 where 120591 is positive and sufficiently small(02) For each SU 119897

(03) Use SM to find the optimal step sizes 120572lowast 120573lowast and 120579lowast and update 120596119899119897 120592119897and 120593

119896according to (28) respectively

(04) For the given 119888119896(119902) and p

119896(119902) of all PUs each SU 119897 responds with its transmit power vector plowast

119897(119902 + 1) according to (22)

(05) If 119876119899119897lt 119888119896(119902) the SU 119897 stops transmitting on the channel 119899 of the PU 119896

(06) End(07) For each PU 119896

(08) Use SM to find the optimal 120594lowast and 120599lowast and update 120582119899119896and 120583

119896by (29)

(09) For the responded plowast119897(119902) of all SUs each PU 119896 updates its transmit power vector as plowast

119896(119902 + 1) according to (25)

(10) Each PU 119896 updates its price by the solution of119888119896(119902 + 1) = argmax

119888119896

119880119896(p119896(119902 + 1) p

minus119896(119902 + 1))

(11) End(12) For each PU 119896 if plowast

119896(119902 + 1) minus p

119896(119902) le 120591 or 119902 gt 102 stop the algorithm Otherwise 119902 = 119902 + 1

repeat steps (02) and (11) until the condition is satisfied

Algorithm 1 Iterative algorithm for reaching the SE

L119901119896(119908119896) is a concave function whether (120597L

119901119896120597119908119896)|119908119896rarr119876

1

119897

gt 0 or (120597L119901119896120597119908119896)|119908119896rarr119876

1

119897

lt 0 except at the nondifferen-tiable point 1198761

119897

Through the above analysis L119896(p119896 119908119896120582119896 120583119896 ^119896) is a

concave function with respect to 119908119896except at 119876119899

119897 The

ellipsoid method [4] can be employed to solve the convexoptimization in each interval

44 Iterative Algorithm to Find the SE For the above dis-cussion we propose an iterative algorithm to search the SEDue to the fact that computation of the dual variables isa complicated task the subgradient method (SM) [14] isapplied to obtain the global optimum SE of this problemThen dual variables are updated as follows

120596119899

119897(119905 + 1) = (120596

119899

119897(119905) + 120572(

119871+1

sum119895 =119897

log(1 + 120588119899119895119897

120574119899

119897119901119899

119895ℎ119899

119895119897

119901119899119897ℎ119899119897119897

)

minus log( 1

1 minus 120585119899119897

)))

+

120592119897(119905 + 1) = (120592

119897(119905) + 120573(

119873

sum119899=1

120588119899

119897119901119899

119897minus 119901

max119897))

+

120593119899

119897(119905 + 1) = (120593

119899

119897(119905) + 120579(

119871

sum119897=1

120588119899

119897119896119901119899

119897ℎ119899

119897119896minus 119879119899

119896))

+

(28)

120582119899

119896(119905 + 1) = (120582

119899

119896(119905) + 120599(

119871

sum119897

log(1 + 120588119899119897119896

120574119899

119896119901119899

119897ℎ119899

119897119896

119901119899119896ℎ119899119896119896

)

minus log( 1

1 minus 120585119899119896

)))

+

120583119896(119905 + 1) = (120583

119896(119905) + 120594(

119873

sum119899=1

120588119899

119896119901119899

119896minus 119875

max119896

))

+

(29)

where 119905 is the iteration index and 120572 gt 0 120573 gt 0 120579 gt 0 120599 gt0 and 120594 gt 0 are sufficiently small The SM guarantees theconvergence of the above optimal dual variables if the stepsizes are chosen by following the step size policy [14]

We then design the iterative algorithm to achieve the SEshown in Algorithm 1

5 Simulation Results and Their Analysis

In this section several numerical examples are presented toevaluate the performances of the proposed SG by comparingthe optimal price-based SG considering global interferencein [5] and the nominal SG without considering globalinterference and outage constraints The cell radius of 500mwith the PBS centered at the original CRN The simulationparameters are as follows 1205752 = 10minus12W 119870 = 3 119871 = 5 and119873 = 10 The SINR threshold of PUs and SUs is set as 7 dBand 4 dB respectively All PUs and SUs deploy the maximumpower 119875max

119896= 100mWand 119875max

119897= 50mWThe channel gain

in this system is ℎ119894119895= 119889minus4

119894119895 with 119889

119894119895being their corresponding

distance The outage probability thresholds of both PUs andSUs are 0001 We set the interference-to-noise ratio (INR) as1198791205752Firstly we illustrate the convergence of the proposed

algorithm for achieving an SE of the proposed game FromFigure 2 the three PUs and five SUs iteratively update theirutilities and obviously converge to the SE The proposedalgorithm converges quickly in terms of PUs only about tentimes In addition due to the larger number of SUs theconvergence of the SUs is slower than that of the PUs

51 Impact of INR In this subsection we set the numberof PUs and SUs as 3 and 5 and INR changes from minus20 dBto 20 dB which means that IPC changes from 10

minus14W to10minus10WWe then consider the sum rate of PUs and SUs for the

three solutionswith different tolerant interference constraintsshown in Figures 3 and 4 For the performance of sum rate

International Journal of Distributed Sensor Networks 7

PU-1PU-2PU-3

SU-1SU-2SU-3

SU-4SU-5

12

13

14

15

16

Util

ities

of P

Us

1

2

3

4

5

6

7

Util

ities

of S

Us

50 15 20 2510

Iteration

50 15 20 2510

Iteration

Figure 2 The convergence of utility of PUs (leaders) setting priceof PUs and utility of SUs (followers)

of SUs Figure 3 shows the nominal SG scheme outperformsother two schemes when the interference temperature levelis stringent but is inferior to the two schemes when it isloose The proposed SG performs the worst because ofthe demand of satisfying the outage probability constraintOnce the interference constraints are loose enough to benot active accordingly our proposed solution works betterthan the others For the leaders the sum rate of the nominalSG performs the worst because of not including the globalinterference constraints This is because the performance ofPUs may be degraded with the increases of the interference

Figure 5 presents the outage probability of the systemwith different INR It is observed that the proposed schemeachieves much lower outage probability than other schemesin particular the performance gap becomes larger with theincrease of INR This is because our proposed algorithmworks best by considering the outage probability constraintsof users which prevents the outage events well

52 Impact of Different Number of SUs In this subsection theINR is set to be 10 dB The number of SUs changes from 2 to20 All the other simulation parameters are the same as thebeginning part of this section

For the PUs from Figure 6 the sum rate of the SGsolution performs the worst because of not including theglobal interference constraints So the PUs may refuse tosell more spectrum resource because of protecting their own

Our proposed SGOptimal price-based SG in [5]Nominal SG

50 10 15 20minus5minus15 minus10minus20

INR T1205752 (dB)

25

26

27

28

29

30

31

32

33

34

Sum

rate

of t

he P

Us (

bits

sH

z)Figure 3 Sum rate of PUs versus INR

26

24

22

20

18

16

14

12

10minus20 minus15 minus10 minus5 0 5 10 15 20

Our proposed SGOptimal price-based SG in [5]Nominal SG

INR T1205752 (dB)

Sum

rate

of t

he S

Us (

bits

sH

z)

Figure 4 Sum rate of SUs versus INR

communication QoS The proposed scheme outperformsmost in terms of sum rate of SUs because the algorithm allowsmore SUs to share their radio resource so that it increases PUsrsquoutilities with considering the channel uncertainty and globalinterference constraints

Figure 7 shows the sum rate of SUs versus the numberof SUs The sum rate of SUs performance of our proposedalgorithm works better than other two schemes this isbecause the proposed scheme is able to support more SUs attheBS shown in Figure 7 so thatmore SUshave opportunities

8 International Journal of Distributed Sensor Networks

Our proposed SGOptimal price-based SG in [5]Nominal SG

10minus3

10minus2

10minus1

100

Out

age p

roba

bilit

y

minus15 minus10 minus5 0 5 10 15 20minus20

INR T1205752 (dB)

Figure 5 Outage probability versus INR

Our proposed SGOptimal price-based SG in [5]Nominal SG

16

18

20

22

24

26

28

30

32

34

Sum

rate

of P

Us

4 6 82 12 14 16 18 2010

Number of SUs

Figure 6 Sum rate of PUs versus number of SUs

to transmit which increase the sum rate In addition theperformance gap between the proposed scheme and theother algorithms increases when the network grows largerby which it can be concluded that the proposed scheme ismore suitable for application in larger networks Because thescheme sets the adaptive punishment parameter among allserved SUs to control their behavior more SUs can be servedat the BS thus achieving a higher sum rate

Figure 8 shows the number of outage probability com-parison versus the number of SUs for different algorithmsThe proposed algorithm is able to support more SUs than

Our proposed SGOptimal price-based SG in [5]Nominal SG

4 6 8 10 12 14 16 18 202

Number of SUs

10

15

20

25

30

35

40

45

Sum

rate

of S

Us

Figure 7 Sum rate of SUs versus number of SUs

Our proposed SGOptimal price-based SG in [5]Nominal SG

4 6 8 10 12 14 16 18 202

Number of SUs

10minus4

10minus3

10minus2

10minus1

100

Out

age p

roba

bilit

y

Figure 8 Outage probability versus number of SUs

other schemes This is because we develop the channelassignment scheduling scheme to decrease the probabilityof the unserved SUs so more SUs are admitted to serveat the BS without causing unendurable interference to PUsIn addition the interference among SUs is taken into therevenue utility function to void serious interference for someSUs who have bad channel condition

International Journal of Distributed Sensor Networks 9

6 Conclusion

In this paper we propose a Stackelberg game for powercontrol problem in CRNs with channel outage constraintsand global interference constraints We employ LDDM tosolve the problem by decomposing it into independentsubproblems and develop an iterative algorithm to achieveSE Simulation results show that the proposed algorithmimproves the performance compared with other game algo-rithms

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the National Nature ScienceFoundation of China (61271259 and 61301123) the SpecialFund of Chongqing Key Laboratory (CSTC) the Programfor Changjiang Scholars and Innovative Research Team inUniversity (IRT2129) and the Graduate Student ResearchInnovation Project of Chongqing University of Posts andTelecommunications (Chongqing) (CYS14143)

References

[1] K Akkarajitsakul E Hossain D Niyato and D I Kim ldquoGametheoretic approaches for multiple access in wireless networksa surveyrdquo IEEE Communications Surveys and Tutorials vol 13no 3 pp 372ndash395 2011

[2] Y Wu T Zhang and D H K Tsang ldquoJoint pricing andpower allocation for dynamic spectrum access networks withStackelberg game modelrdquo IEEE Transactions on Wireless Com-munications vol 10 no 1 pp 12ndash19 2011

[3] X Kang R Zhang and M Motani ldquoPrice-based resourceallocation for spectrum-sharing femtocell networks a stack-elberg game approachrdquo IEEE Journal on Selected Areas inCommunications vol 30 no 3 pp 538ndash549 2012

[4] R Xie F R Yu H Ji and Y Li ldquoEnergy-efficient resourceallocation for heterogeneous cognitive radio networks withfemtocellsrdquo IEEETransactions onWireless Communications vol11 no 11 pp 3910ndash3920 2012

[5] Z Wang L Jiang and C He ldquoOptimal price-based power con-trol algorithm in cognitive radio networksrdquo IEEE TransactionsonWireless Communications vol 13 no 11 pp 5909ndash5920 2014

[6] M Le Treust S Lasaulce Y Hayel and G L He ldquoGreen powercontrol in cognitive wireless networksrdquo IEEE Transactions onVehicular Technology vol 62 no 4 pp 1741ndash1754 2013

[7] Y Xiao G Bi D Niyato and L A DaSilva ldquoA hierarchicalgame theoretic framework for cognitive radio networksrdquo IEEEJournal on Selected Areas in Communications vol 30 no 10 pp2053ndash2069 2012

[8] F Gabry N Li N Schrammar M Girnyk L K Rasmussenand M Skoglund ldquoOn the optimization of the secondarytransmitterrsquos strategy in cognitive radio channels with secrecyrdquoIEEE Journal on Selected Areas in Communications vol 32 no3 pp 451ndash463 2014

[9] D Niyato E Hossain and Z Han ldquoDynamics of multiple-seller and multiple-buyer spectrum trading in cognitive radionetworks a game-theoretic modeling approachrdquo IEEE Transac-tions on Mobile Computing vol 8 no 8 pp 1009ndash1022 2009

[10] S Kandukuri and S Boyd ldquoOptimal power control ininterference-limited fading wireless channels with outage-probability specificationsrdquo IEEE Transactions on Wireless Com-munications vol 1 no 1 pp 46ndash55 2002

[11] T Basar and G J Olsder Dynamic Noncooperative GameTheory SIAM Philadelphia Pa USA 1999

[12] F Facchinei and C Kanzow ldquoGeneralized Nash equilibriumproblemsrdquo 4OR vol 5 no 3 pp 173ndash210 2007

[13] D Ardagna B Panicucci and M Passacantando ldquoA game the-oretic formulation of the service provisioning problem in cloudsystemsrdquo in Proceedings of the 20th International Conference onWorld Wide Web (WWW rsquo11) pp 177ndash186 ACM April 2011

[14] D Bertsekas W Hager and O Mangasarian Nonlinear Pro-gramming Athena Scientific Belmont Mass USA 1999

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of Distributed Sensor Networks 7

PU-1PU-2PU-3

SU-1SU-2SU-3

SU-4SU-5

12

13

14

15

16

Util

ities

of P

Us

1

2

3

4

5

6

7

Util

ities

of S

Us

50 15 20 2510

Iteration

50 15 20 2510

Iteration

Figure 2 The convergence of utility of PUs (leaders) setting priceof PUs and utility of SUs (followers)

of SUs Figure 3 shows the nominal SG scheme outperformsother two schemes when the interference temperature levelis stringent but is inferior to the two schemes when it isloose The proposed SG performs the worst because ofthe demand of satisfying the outage probability constraintOnce the interference constraints are loose enough to benot active accordingly our proposed solution works betterthan the others For the leaders the sum rate of the nominalSG performs the worst because of not including the globalinterference constraints This is because the performance ofPUs may be degraded with the increases of the interference

Figure 5 presents the outage probability of the systemwith different INR It is observed that the proposed schemeachieves much lower outage probability than other schemesin particular the performance gap becomes larger with theincrease of INR This is because our proposed algorithmworks best by considering the outage probability constraintsof users which prevents the outage events well

52 Impact of Different Number of SUs In this subsection theINR is set to be 10 dB The number of SUs changes from 2 to20 All the other simulation parameters are the same as thebeginning part of this section

For the PUs from Figure 6 the sum rate of the SGsolution performs the worst because of not including theglobal interference constraints So the PUs may refuse tosell more spectrum resource because of protecting their own

Our proposed SGOptimal price-based SG in [5]Nominal SG

50 10 15 20minus5minus15 minus10minus20

INR T1205752 (dB)

25

26

27

28

29

30

31

32

33

34

Sum

rate

of t

he P

Us (

bits

sH

z)Figure 3 Sum rate of PUs versus INR

26

24

22

20

18

16

14

12

10minus20 minus15 minus10 minus5 0 5 10 15 20

Our proposed SGOptimal price-based SG in [5]Nominal SG

INR T1205752 (dB)

Sum

rate

of t

he S

Us (

bits

sH

z)

Figure 4 Sum rate of SUs versus INR

communication QoS The proposed scheme outperformsmost in terms of sum rate of SUs because the algorithm allowsmore SUs to share their radio resource so that it increases PUsrsquoutilities with considering the channel uncertainty and globalinterference constraints

Figure 7 shows the sum rate of SUs versus the numberof SUs The sum rate of SUs performance of our proposedalgorithm works better than other two schemes this isbecause the proposed scheme is able to support more SUs attheBS shown in Figure 7 so thatmore SUshave opportunities

8 International Journal of Distributed Sensor Networks

Our proposed SGOptimal price-based SG in [5]Nominal SG

10minus3

10minus2

10minus1

100

Out

age p

roba

bilit

y

minus15 minus10 minus5 0 5 10 15 20minus20

INR T1205752 (dB)

Figure 5 Outage probability versus INR

Our proposed SGOptimal price-based SG in [5]Nominal SG

16

18

20

22

24

26

28

30

32

34

Sum

rate

of P

Us

4 6 82 12 14 16 18 2010

Number of SUs

Figure 6 Sum rate of PUs versus number of SUs

to transmit which increase the sum rate In addition theperformance gap between the proposed scheme and theother algorithms increases when the network grows largerby which it can be concluded that the proposed scheme ismore suitable for application in larger networks Because thescheme sets the adaptive punishment parameter among allserved SUs to control their behavior more SUs can be servedat the BS thus achieving a higher sum rate

Figure 8 shows the number of outage probability com-parison versus the number of SUs for different algorithmsThe proposed algorithm is able to support more SUs than

Our proposed SGOptimal price-based SG in [5]Nominal SG

4 6 8 10 12 14 16 18 202

Number of SUs

10

15

20

25

30

35

40

45

Sum

rate

of S

Us

Figure 7 Sum rate of SUs versus number of SUs

Our proposed SGOptimal price-based SG in [5]Nominal SG

4 6 8 10 12 14 16 18 202

Number of SUs

10minus4

10minus3

10minus2

10minus1

100

Out

age p

roba

bilit

y

Figure 8 Outage probability versus number of SUs

other schemes This is because we develop the channelassignment scheduling scheme to decrease the probabilityof the unserved SUs so more SUs are admitted to serveat the BS without causing unendurable interference to PUsIn addition the interference among SUs is taken into therevenue utility function to void serious interference for someSUs who have bad channel condition

International Journal of Distributed Sensor Networks 9

6 Conclusion

In this paper we propose a Stackelberg game for powercontrol problem in CRNs with channel outage constraintsand global interference constraints We employ LDDM tosolve the problem by decomposing it into independentsubproblems and develop an iterative algorithm to achieveSE Simulation results show that the proposed algorithmimproves the performance compared with other game algo-rithms

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the National Nature ScienceFoundation of China (61271259 and 61301123) the SpecialFund of Chongqing Key Laboratory (CSTC) the Programfor Changjiang Scholars and Innovative Research Team inUniversity (IRT2129) and the Graduate Student ResearchInnovation Project of Chongqing University of Posts andTelecommunications (Chongqing) (CYS14143)

References

[1] K Akkarajitsakul E Hossain D Niyato and D I Kim ldquoGametheoretic approaches for multiple access in wireless networksa surveyrdquo IEEE Communications Surveys and Tutorials vol 13no 3 pp 372ndash395 2011

[2] Y Wu T Zhang and D H K Tsang ldquoJoint pricing andpower allocation for dynamic spectrum access networks withStackelberg game modelrdquo IEEE Transactions on Wireless Com-munications vol 10 no 1 pp 12ndash19 2011

[3] X Kang R Zhang and M Motani ldquoPrice-based resourceallocation for spectrum-sharing femtocell networks a stack-elberg game approachrdquo IEEE Journal on Selected Areas inCommunications vol 30 no 3 pp 538ndash549 2012

[4] R Xie F R Yu H Ji and Y Li ldquoEnergy-efficient resourceallocation for heterogeneous cognitive radio networks withfemtocellsrdquo IEEETransactions onWireless Communications vol11 no 11 pp 3910ndash3920 2012

[5] Z Wang L Jiang and C He ldquoOptimal price-based power con-trol algorithm in cognitive radio networksrdquo IEEE TransactionsonWireless Communications vol 13 no 11 pp 5909ndash5920 2014

[6] M Le Treust S Lasaulce Y Hayel and G L He ldquoGreen powercontrol in cognitive wireless networksrdquo IEEE Transactions onVehicular Technology vol 62 no 4 pp 1741ndash1754 2013

[7] Y Xiao G Bi D Niyato and L A DaSilva ldquoA hierarchicalgame theoretic framework for cognitive radio networksrdquo IEEEJournal on Selected Areas in Communications vol 30 no 10 pp2053ndash2069 2012

[8] F Gabry N Li N Schrammar M Girnyk L K Rasmussenand M Skoglund ldquoOn the optimization of the secondarytransmitterrsquos strategy in cognitive radio channels with secrecyrdquoIEEE Journal on Selected Areas in Communications vol 32 no3 pp 451ndash463 2014

[9] D Niyato E Hossain and Z Han ldquoDynamics of multiple-seller and multiple-buyer spectrum trading in cognitive radionetworks a game-theoretic modeling approachrdquo IEEE Transac-tions on Mobile Computing vol 8 no 8 pp 1009ndash1022 2009

[10] S Kandukuri and S Boyd ldquoOptimal power control ininterference-limited fading wireless channels with outage-probability specificationsrdquo IEEE Transactions on Wireless Com-munications vol 1 no 1 pp 46ndash55 2002

[11] T Basar and G J Olsder Dynamic Noncooperative GameTheory SIAM Philadelphia Pa USA 1999

[12] F Facchinei and C Kanzow ldquoGeneralized Nash equilibriumproblemsrdquo 4OR vol 5 no 3 pp 173ndash210 2007

[13] D Ardagna B Panicucci and M Passacantando ldquoA game the-oretic formulation of the service provisioning problem in cloudsystemsrdquo in Proceedings of the 20th International Conference onWorld Wide Web (WWW rsquo11) pp 177ndash186 ACM April 2011

[14] D Bertsekas W Hager and O Mangasarian Nonlinear Pro-gramming Athena Scientific Belmont Mass USA 1999

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

8 International Journal of Distributed Sensor Networks

Our proposed SGOptimal price-based SG in [5]Nominal SG

10minus3

10minus2

10minus1

100

Out

age p

roba

bilit

y

minus15 minus10 minus5 0 5 10 15 20minus20

INR T1205752 (dB)

Figure 5 Outage probability versus INR

Our proposed SGOptimal price-based SG in [5]Nominal SG

16

18

20

22

24

26

28

30

32

34

Sum

rate

of P

Us

4 6 82 12 14 16 18 2010

Number of SUs

Figure 6 Sum rate of PUs versus number of SUs

to transmit which increase the sum rate In addition theperformance gap between the proposed scheme and theother algorithms increases when the network grows largerby which it can be concluded that the proposed scheme ismore suitable for application in larger networks Because thescheme sets the adaptive punishment parameter among allserved SUs to control their behavior more SUs can be servedat the BS thus achieving a higher sum rate

Figure 8 shows the number of outage probability com-parison versus the number of SUs for different algorithmsThe proposed algorithm is able to support more SUs than

Our proposed SGOptimal price-based SG in [5]Nominal SG

4 6 8 10 12 14 16 18 202

Number of SUs

10

15

20

25

30

35

40

45

Sum

rate

of S

Us

Figure 7 Sum rate of SUs versus number of SUs

Our proposed SGOptimal price-based SG in [5]Nominal SG

4 6 8 10 12 14 16 18 202

Number of SUs

10minus4

10minus3

10minus2

10minus1

100

Out

age p

roba

bilit

y

Figure 8 Outage probability versus number of SUs

other schemes This is because we develop the channelassignment scheduling scheme to decrease the probabilityof the unserved SUs so more SUs are admitted to serveat the BS without causing unendurable interference to PUsIn addition the interference among SUs is taken into therevenue utility function to void serious interference for someSUs who have bad channel condition

International Journal of Distributed Sensor Networks 9

6 Conclusion

In this paper we propose a Stackelberg game for powercontrol problem in CRNs with channel outage constraintsand global interference constraints We employ LDDM tosolve the problem by decomposing it into independentsubproblems and develop an iterative algorithm to achieveSE Simulation results show that the proposed algorithmimproves the performance compared with other game algo-rithms

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the National Nature ScienceFoundation of China (61271259 and 61301123) the SpecialFund of Chongqing Key Laboratory (CSTC) the Programfor Changjiang Scholars and Innovative Research Team inUniversity (IRT2129) and the Graduate Student ResearchInnovation Project of Chongqing University of Posts andTelecommunications (Chongqing) (CYS14143)

References

[1] K Akkarajitsakul E Hossain D Niyato and D I Kim ldquoGametheoretic approaches for multiple access in wireless networksa surveyrdquo IEEE Communications Surveys and Tutorials vol 13no 3 pp 372ndash395 2011

[2] Y Wu T Zhang and D H K Tsang ldquoJoint pricing andpower allocation for dynamic spectrum access networks withStackelberg game modelrdquo IEEE Transactions on Wireless Com-munications vol 10 no 1 pp 12ndash19 2011

[3] X Kang R Zhang and M Motani ldquoPrice-based resourceallocation for spectrum-sharing femtocell networks a stack-elberg game approachrdquo IEEE Journal on Selected Areas inCommunications vol 30 no 3 pp 538ndash549 2012

[4] R Xie F R Yu H Ji and Y Li ldquoEnergy-efficient resourceallocation for heterogeneous cognitive radio networks withfemtocellsrdquo IEEETransactions onWireless Communications vol11 no 11 pp 3910ndash3920 2012

[5] Z Wang L Jiang and C He ldquoOptimal price-based power con-trol algorithm in cognitive radio networksrdquo IEEE TransactionsonWireless Communications vol 13 no 11 pp 5909ndash5920 2014

[6] M Le Treust S Lasaulce Y Hayel and G L He ldquoGreen powercontrol in cognitive wireless networksrdquo IEEE Transactions onVehicular Technology vol 62 no 4 pp 1741ndash1754 2013

[7] Y Xiao G Bi D Niyato and L A DaSilva ldquoA hierarchicalgame theoretic framework for cognitive radio networksrdquo IEEEJournal on Selected Areas in Communications vol 30 no 10 pp2053ndash2069 2012

[8] F Gabry N Li N Schrammar M Girnyk L K Rasmussenand M Skoglund ldquoOn the optimization of the secondarytransmitterrsquos strategy in cognitive radio channels with secrecyrdquoIEEE Journal on Selected Areas in Communications vol 32 no3 pp 451ndash463 2014

[9] D Niyato E Hossain and Z Han ldquoDynamics of multiple-seller and multiple-buyer spectrum trading in cognitive radionetworks a game-theoretic modeling approachrdquo IEEE Transac-tions on Mobile Computing vol 8 no 8 pp 1009ndash1022 2009

[10] S Kandukuri and S Boyd ldquoOptimal power control ininterference-limited fading wireless channels with outage-probability specificationsrdquo IEEE Transactions on Wireless Com-munications vol 1 no 1 pp 46ndash55 2002

[11] T Basar and G J Olsder Dynamic Noncooperative GameTheory SIAM Philadelphia Pa USA 1999

[12] F Facchinei and C Kanzow ldquoGeneralized Nash equilibriumproblemsrdquo 4OR vol 5 no 3 pp 173ndash210 2007

[13] D Ardagna B Panicucci and M Passacantando ldquoA game the-oretic formulation of the service provisioning problem in cloudsystemsrdquo in Proceedings of the 20th International Conference onWorld Wide Web (WWW rsquo11) pp 177ndash186 ACM April 2011

[14] D Bertsekas W Hager and O Mangasarian Nonlinear Pro-gramming Athena Scientific Belmont Mass USA 1999

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of Distributed Sensor Networks 9

6 Conclusion

In this paper we propose a Stackelberg game for powercontrol problem in CRNs with channel outage constraintsand global interference constraints We employ LDDM tosolve the problem by decomposing it into independentsubproblems and develop an iterative algorithm to achieveSE Simulation results show that the proposed algorithmimproves the performance compared with other game algo-rithms

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the National Nature ScienceFoundation of China (61271259 and 61301123) the SpecialFund of Chongqing Key Laboratory (CSTC) the Programfor Changjiang Scholars and Innovative Research Team inUniversity (IRT2129) and the Graduate Student ResearchInnovation Project of Chongqing University of Posts andTelecommunications (Chongqing) (CYS14143)

References

[1] K Akkarajitsakul E Hossain D Niyato and D I Kim ldquoGametheoretic approaches for multiple access in wireless networksa surveyrdquo IEEE Communications Surveys and Tutorials vol 13no 3 pp 372ndash395 2011

[2] Y Wu T Zhang and D H K Tsang ldquoJoint pricing andpower allocation for dynamic spectrum access networks withStackelberg game modelrdquo IEEE Transactions on Wireless Com-munications vol 10 no 1 pp 12ndash19 2011

[3] X Kang R Zhang and M Motani ldquoPrice-based resourceallocation for spectrum-sharing femtocell networks a stack-elberg game approachrdquo IEEE Journal on Selected Areas inCommunications vol 30 no 3 pp 538ndash549 2012

[4] R Xie F R Yu H Ji and Y Li ldquoEnergy-efficient resourceallocation for heterogeneous cognitive radio networks withfemtocellsrdquo IEEETransactions onWireless Communications vol11 no 11 pp 3910ndash3920 2012

[5] Z Wang L Jiang and C He ldquoOptimal price-based power con-trol algorithm in cognitive radio networksrdquo IEEE TransactionsonWireless Communications vol 13 no 11 pp 5909ndash5920 2014

[6] M Le Treust S Lasaulce Y Hayel and G L He ldquoGreen powercontrol in cognitive wireless networksrdquo IEEE Transactions onVehicular Technology vol 62 no 4 pp 1741ndash1754 2013

[7] Y Xiao G Bi D Niyato and L A DaSilva ldquoA hierarchicalgame theoretic framework for cognitive radio networksrdquo IEEEJournal on Selected Areas in Communications vol 30 no 10 pp2053ndash2069 2012

[8] F Gabry N Li N Schrammar M Girnyk L K Rasmussenand M Skoglund ldquoOn the optimization of the secondarytransmitterrsquos strategy in cognitive radio channels with secrecyrdquoIEEE Journal on Selected Areas in Communications vol 32 no3 pp 451ndash463 2014

[9] D Niyato E Hossain and Z Han ldquoDynamics of multiple-seller and multiple-buyer spectrum trading in cognitive radionetworks a game-theoretic modeling approachrdquo IEEE Transac-tions on Mobile Computing vol 8 no 8 pp 1009ndash1022 2009

[10] S Kandukuri and S Boyd ldquoOptimal power control ininterference-limited fading wireless channels with outage-probability specificationsrdquo IEEE Transactions on Wireless Com-munications vol 1 no 1 pp 46ndash55 2002

[11] T Basar and G J Olsder Dynamic Noncooperative GameTheory SIAM Philadelphia Pa USA 1999

[12] F Facchinei and C Kanzow ldquoGeneralized Nash equilibriumproblemsrdquo 4OR vol 5 no 3 pp 173ndash210 2007

[13] D Ardagna B Panicucci and M Passacantando ldquoA game the-oretic formulation of the service provisioning problem in cloudsystemsrdquo in Proceedings of the 20th International Conference onWorld Wide Web (WWW rsquo11) pp 177ndash186 ACM April 2011

[14] D Bertsekas W Hager and O Mangasarian Nonlinear Pro-gramming Athena Scientific Belmont Mass USA 1999

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of