Wireless Pers Commun DOI 10.1007/s11277-016-3927-z

Energy-Efficient Uplink Resource Allocation Based on Game Theory in Cognitive Small Cell Networks Ya-Nan Jia1 • Dian-Wu Yue2

 Springer Science+Business Media New York 2016

Abstract In this paper, resource allocation for energy-efficient uplink communications in cognitive small cell networks is studied. We formulate the entire network energy efficiency (EE) maximization problem for the joint allocation of small cell base stations (SCBSs), spectrum resources, and transmission power in an open access mode. Since the master optimization problem belong to integer combinatorial fractional program and is essentially NP-hard, we develop a low-complexity alternative as a suboptimal solution by decomposing the master optimization issue into two sub-problems: selection of SCBSs and spectrum resources, and power allocation. The selection sub-problem of SCBSs and spectrum resources for cognitive small cell users (CSCUs) is modeled as a potential game from the viewpoints of reducing system interference and improving received signal strength. We formulate the power allocation sub-problem as a non-cooperative game which can be solved in a distributed fashion. But it is also a non-convex optimization problem in fractional form to solve the optimal power strategies based on maximizing the EE on a specific channel. We transform the nonlinear fractional programming issue into an equivalent parametric programming in subtractive form. For obtaining a social optimum power Nash equilibrium (NE), we propose a novel price-based double-loop iteration algorithm to get the transmission power strategies, which take the form of water-filling structure among different CSCUs over a same channel. Simulation results show that the proposed algorithms can converge to NE, and polish up the EE of the overall system significantly. Keywords Cognitive small cell networks  Distributed resource allocation  Energy efficiency  Game theory  Green communication  Nash equilibrium

& Dian-Wu Yue [email protected]; [email protected] Ya-Nan Jia [email protected] 1

Spreadtrum Communications Co., Ltd., Shanghai, China

2

College of Information Science and Technology, Dalian Maritime University, Dalian, China

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1 Introduction In recent years, there is a growing demand for high-speed but energy saving wireless communications in our daily life,such as the scenario of massive Machine Type Communications (mMTC)in the fifth generation (5G). New technology and network architecture are needed to cope with overwhelming demand of network capacity anytime and anywhere in the big data age. Therefore, Device to Device (D2D) communication, millimeter wave communication, massive MIMO, and small cell networks (SCNs) are studied to further improve spectrum efficiency and network throughput. However, none of these technological advances can carry a forecast large system capacity increase alone without a substantial network densification. SCNs are founded on the idea of massive network densification by deploying small cell base stations (SCBSs) which are humanly-portable, plug-and-play, low-cost and low-power devices [1–3]. For further reusing spectrum resources and enhancing system capacity, cognitive radio (CR) [4] theory which was proposed first by Dr. Mitola is applied to SCNs [5]. The concept of cognitive small cell networks (CSCNs) [6] which draw more attention in 5G communications comes from the combination of CR technology and SCNs. The CSCNs can not only make full use of spectrum resources and improve system throughput, but also increase EE through effectively decreasing path loss. When using the CR technology, CSCNs should sense their external communication environment in real time, and rationally adjust transmission parameters according to historical experience for optimizing system performance [7, 8]. When cognitive small cell users (CSCUs) want to communicate over licensed channels, they should rationally choose available spectrum as theirs channels to avoid impairing primary users’ (PUs) normal communications and seek to underlay, overlay, or interweave their signals with other CSCUs [9–11]. Although CSCNs have many advantages, they pose many new challenges to an optimal system design when plenty of SCBSs are randomly deployed in a same area [12]. The densification increases same-layer and cross-layer interference in heterogeneous networks (Het-Nets) [13, 14]. The mMTC devices will consume more power in noisy environments to satisfy pre-specified quality of service (QoS) requirements, and more energy will be wasted if no reasonable methods are used to prevent vicious competition among users who only improve their own link capacity without taking into account the interference to other competitors. So how to maximize the system EE of mMTC scenario has become a hotspot in CSCNs [15–17]. In CSCNs, SCBSs can operate in open access, closed access, or mixed access mode, and CSCUs build links with the corresponding SCBSs on dedicated channels or shared channels [18]. So a dynamic resource management scheme [9, 19] is needed to reduce cochannel interference, especially on the Industrial, Scientific and Medical (ISM) frequency band. Although centralized control and manual intervention can effectively reduce the whole system interference and increase the entire system EE, they are highly inefficient in dense CSCNs. Thus, self-organization and distributed resource allocation are of paramount importance for the successful deployment of CSCNs [3]. In this regard, as a mathematical tool to analyze strategic interactions among players, game theory is a natural paradigm to solve these problems [20–24]. More available frequency bands are helpful for reducing cochannel interference when plenty of cognitive users (CUs), coexist in a same geographical area. In conventional spectrum allocation models, the CUs always communicate over nonoverlapped channels and rarely use partially overlapped channel resources, such as in [25] which only discussed the allocation algorithm based on independent channel resources.

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Energy-Efficient Uplink Resource Allocation Based on Game…

Whereas some communication systems can run on overlapped spectrum resources, such as MT-CDMA and spread spectrum communications. Then the algorithm used in [25] will be not suitable for improving the utilization rate of idle spectrum resources when CUs run in the aforementioned systems. Although the transmission power allocation algorithms have been sufficiently discussed from the rates maximization point of view, there are also many challenges for rate-adaptive and EE. In [21, 22, 25, 26], CUs can use one or several preset fixed transmission power for data traffic. The fixed transmission power context is inefficient considering that traffic load notably varies along the time and/or space dimensions. The cumulative interference will cause to the violation of the temperature-interference constraints set in a CU when plenty of other CUs communicating over a same frequency resource at the same time, even the CUs choose the minimum transmission power strategies. For overcoming the mismatch between QoS demands of CUs and limited network resources, the power assignment scheme should be rate-adaptive. References [14, 27, 28] discussed power allocation algorithms from the viewpoint of the optimal power allocation over different idle subcarriers by an individual user, but it will be much more efficient when considering the energy allocation among different CUs on a same carrier at the same time. Literatures [29, 30] analyzed resource allocation algorithms from the EE viewpoint. In [29], downlink resource allocation problem was analyzed only in a single cell network that did not consider the inter-layer and intra-layer interference. Although [30] analyzed the resource allocation problem of multi-cell networks, the proposed algorithm was lack of self-organization ability under a centralized structure. Reference [31] analyzed the resource allocation algorithm based on minimizing energy consumption per bit information in heterogeneous networks in which a macro network and Wireless Fidelity (Wi-Fi) networks coexist, but this algorithm is also lack of self-organization ability and the entire system EE may be unoptimizable owing to the non-overlapped channels assignment scheme. Motivated by the aforementioned discussions, we propose a novel joint SCBSs, spectrum resources, and power allocation algorithm for maximizing sum EE of the overall spectrum resources in the uplink of dense CSCNs. The joint allocation optimization issue is decomposed into two sub-problems. First, the optimization allocation of SCBSs and spectrum resources is modeled as a potential game over the separate and/or overlapped channels. And a novel utility function is proposed for obtaining a determined NE given temperature-interference constraints. Second, for further reducing system interference and improving spectrum EE, we model the power allocation problem as a non-cooperative game given the selected SCBSs and spectrum resources of CSCUs. In order to solve social transmission power solutions in a distributed manner, we propose a novel price-based double-loop iterative water-filling approach. The pricing function is BS-dependent and can be determined by exchanging neighborhood information among SCBSs through backhaul networks. The power NE can not only be improved, but also be global optimization after a few iterations. The remainder of the paper is organized as follows. In Sect. 2, we introduce the system model of CSCNs. In Sect. 3, we introduce the game theory, and present the selection of SCBSs and spectrum resources and power allocation algorithm respectively. In Sect. 4, we validate the feasibility and superiority of our algorithm by simulation. Finally, we conclude the paper with a brief summary in Sect. 5.

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2 System Model and Problem Formulation We consider the uplink transmission in CSCNs that coexist with multiple Wi-Fi access points (APs) and a World Interoperability for Microwave Access (Wi-Max) base station (BS) in heterogeneous networks, and the SCBSs randomly distribute around the edge of Wi-Max network, cf. Fig. 1. As shown in Fig. 2, we number the channels of the Wi-Fi and Wi-Max systems respectively, and assume that Wi-Fi channel index 1 and Wi-Max channel index a are alignment of center frequencies at 2412 MHz [32]. In the CSCNs, we suppose that a set XNm of Nm CSCUs are associated with SCBS m 2 XM = {1, 2, …, M}, and the spectrum resources XK = {1, 2, …, K} contain two attributes which are bandwidth and center frequency. The set of all the CSCUs is P SM XN = {1, 2, …, N}, and the expressions N = M m=1Nm and XN ¼ m¼1 XNm are meaningful while K \ N shows that multiple CSCUs can share a same frequency band. In order to use the overlapped spectrum resources and calculate the interference from the others, we define a jamming operator [33] for representing the interference between different frequency bands as   8 max ðBni þ Bnm Þ=2  Dfni ;nm ; 0 > > < ; Dfni ;nm  DBni ;nm ðBni þ Bnm Þ=2 ð1Þ rðfni ; fnm Þ ¼ Bnm > > : ; Dfni ;nm \DBni ;nm ðBni þ Bnm Þ=2 where Bni and fni are signal bandwidth and center frequency respectively, and satisfy Dfni ;nm ¼jfni fnm j and DBni ;nm ¼ jBni  Bnm j=2. In the uplink communication, the received signal at SCBS m from CSCU nm on spectrum resource k 2 XK is given by

Fig. 1 CSCNs coexist with Wi-Fi and Wi-Max networks

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Energy-Efficient Uplink Resource Allocation Based on Game…

Fig. 2 Channel allocation model of Wi-Fi and Wi-Max systems

yknm ¼

qffiffiffiffiffiffiffi pknm Hnkm ;m xknm þ

qffiffiffiffiffiffi pkni xkni Hnki ;m þ

X ni 2XN k ;ni 6¼nm

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} thecochannelinterference

X

  pffiffiffiffiffiffi pnj xnj Hnj ;m r fnj ; fnm þNnm

nj 62ðXN \XN k Þ

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} thecrosschannelinterference

ð2Þ xknm ,

pknm ,

Hnkm ;m

and are the transmitted date symbol, transmission power, and channel where gain between SCBS m and CSCU nm on spectrum resource k, respectively. Nk is the number of CSCUs who communicate on the same spectrum resource k and the set of CSCUs is XN k . Nnm is the additive white Gaussian noise (AWGN) at CSCU nm with zero mean and variance c2nm , and the AWGN contains interference from PUs and the thermal noise. Hni ;m ðni 6¼ nm Þ is the interference gain between SCBS m and CSCU ni. In our CSCNs, we define a signal-to-interference plus noise ratio (SINR) to indicate the uplink quality of CSCU nm on spectrum resource k.  2   pknm Hnkm ;m  ð3Þ SINRknm ;m ¼ k Idn þ Idnm þ c2n m

m

X

k Idn , m

nl 2XN k ;nl 6¼nm

Idnm ,

X

 2   pknl Hnkl ;m 

 2   pnj Hnj ;m  r fnj ; fnm

ð4Þ

ð5Þ

nj 62ðXN \XN k Þ;nm 2XN k k where Idn and Idnm are the co-channel and cross-channel interference, respectively. And m the maximum achievable data rate of CSCU nm on spectrum resource k is



ð6Þ Rknm pknm ; pknm ¼ Bnm log2 1 þ SINRknm ;m ;



where pknm ¼ pk1 ; . . .; pknm 1 ; pknm þ1 ; . . .; pkN k is the transmission power of its opponents, Bnm is the spectrum bandwidth. For designing an energy-efficient resource allocation algorithm, the total power consumption of communication devices should be considered in

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the optimization objective function. So the total power consumption of CSCU nm is given by

ð7Þ PEnkm pknm ¼ pknm þ PEnm ; where pknm is the transmission power on spectrum resource k, PEnm is the static power consumption. In the CSCNs, the uplink communications with public access are interference-limited since the aggressive full frequency reuse and an increasing density of SCBSs deployments. So a flexible resource allocation scheme is needed to reducing harmful interference. We model the resource allocation context from the viewpoint of optimizing the overall system EE under QoS requirements of CSCUs as well as transmission power constraints. And the EE function is defined as the ratio of the total system throughput to the total power consumption for all k 2 XK, the optimization EE problem is given in the following.

P P k k k k k2Xk nm 2XN k xnm Rnm pnm ; pnm

maximize EE = P P pknm k ð8Þ pkn k2X n 2X PEn k

subject to

pknm

m

Nk

m

m

2 Wnm

where xknm denotes the weight assigned to CSCU nm on spectrum k, which may be interpreted in different ways (e.g., priority or fairness factor), Wnm is a corresponding strategy space. Since the optimization resource allocation problem in (8) belongs to integer combinational fractional program and is essentially NP-hard to solve especially for getting optimal transmission power. We formulate this non-convex optimization issue as the joint allocation of SCBSs, spectrum resources, and transmission power, and propose a low-complexity alternative for suboptimal solution by decomposing the master optimization issue into two sub-problems: selection of SCBSs and spectrum resources and power allocation. The more detailed analysis is given in the following sections.

3 Resource Allocation Based on Game Theory Due to mutual interference among different CSCUs, the transmission strategies of a single user will affect the overall system performance. For solving the maximization system EE problem in a distributed fashion, we individually optimize single CSCU’s strategies. The application background of these multi-objective optimization problems is consistent with game theory. Therefore, game theory can be adopted to solve the optimal allocation of communication resources in CSCNs.

3.1 SCBS Selection and Spectrum Allocation

  For a given p ¼ p1 ; . . .; pk ; . . .; pK , where pk ¼ pk1 ; . . .; pknm 1 ; pknm þ1 ; . . .; pkN k , we can know that the denominator part in (8) is constant and the issue of maximizing system EE simplifies to solve the optimal assignment of SCBSs and spectrum resources among CSCUs in an open access mode. Since players (e.g., CSCUs) with different property generally own to different authorities and pursue diverse goals, so fully cooperative

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Energy-Efficient Uplink Resource Allocation Based on Game…

behaviors can not be taken for granted or it is too expensive, and only just a cooperation can bring performance improvement. Therefore, a key problem in CSCNs is how to stimulate the cooperation among selfish CSCUs to improve the system performance. Based on these, we model the scheme of choosing SCBSs and spectrum resources simultaneously by CSCUs as a potential game, and propose a new altruistic utility function as incentives of reducing system interference and improving signal strength to select optimal strategies. And the mathematical model is defined as D E ð9Þ G1 ¼ XN ; fF nm gnm 2XN ; fUnm gnm 2XN ; where XN is the set of CSCUs (decision makers), F nm  XK is the set of admissible strategies of CSCU nm, the utility function Unm satisfies F ! R, and F ¼ F 1      F N is the Cartesian product of all the available strategy sets of the corresponding CSCUs. In our potential game, we introduce interference constraints gðfnm ; f nm Þ  P0 ; fnm 2 F nm to avoid harmful interference to other CSCUs, where f nm ¼ ðf1 ; . . .; fnm 1 ; fnm þ1 ; . . .; fN Þ is the strategy tuple of the opponents of CSCU nm, and P0 is the interference threshold set by CSCUs according to their QoS requirements. Each CSCU selects appropriate action strategies to maximize its own utility function, and the mathematical model is given by   maximize Unm fnm ; f nm fnm ð10Þ   subject to fnm 2 F nm ; g fnm ; f nm  P0 As shown in (10), the NE problem is a set of coupled optimization problems [24]. By analyzing the outcome of this game, we know that CSCUs can independently choose their strategies based on their opponents’ strategies. The NE exists when the strategy tuple f satisfies (11), and it is pure strategy NE if the strategies are determined in game G1 . 0

0 Unm ðf Þ  Unm fnm ; f nm ; 8nm 2 XN ; fnm 2 F nm ð11Þ In what follows, we propose a new utility function which can not only fully display the channel quality, such as users’ collaboration and fairness, but also can be formulated as an exact potential game, and the mathematical expression is given by  2   ð12Þ Unm fnm ; f nm ¼ Idnm  Ionm þ pnm Hnm ;m  ; 8nm 2 XN ; Idnm ,

X

 2 pni Hni ;m  rðfni ; fnm Þ;

8nm 2 XN ;

ð13Þ

 2 pnm Hnm ;i  rðfnm ; fni Þ;

8nm 2 XN ;

ð14Þ

ni 2XN ;ni 6¼nm

Ionm ,

X ni 2XN ;ni 6¼nm

where Idnm is the interference of CSCU nm from others, and Idnm has a same attribute with  2 gðfnm ; f nm Þ, Ionm is the interference of CSCU nm to others, pnm Hnm ;m  indicates the signal strength received by SCBS m. Therefore, the physical meaning of (12) is that our utility function contains three different kinds of energy factors of CSCU nm in the uplink. We define game G1 as an exact potential game if there is an exact potential function 0 P : F ! R; 8nm 2 XN ; fnm ; fnm 2 F nm that satisfies the following relationship

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0

0

    P fnm ; f nm  P fnm ; f nm ¼ Unm fnm ; f nm  Unm fnm ; f nm

ð15Þ

when favorable and unilateral departure strategies are limited and meet a finite improvement property (FIP) criterion, there is at least one NE in the potential game [23]. Any CSCU choosing a better strategy given the others’ current strategies will necessarily lead to an improvement in the value of potential function, and the potential game will terminate to a NE in finite steps when all CSCUs adopt their best strategies sequentially. According to the rules of potential game whose utility function is Unm ðfnm ; f nm Þ, an exact potential function can be defined to be X 1  2

  P ðF nm Þ ¼ P fnm ; f nm ¼ Idnm  Ionm þ pnm Hnm ;m  ; 8nm 2 XN ð16Þ nm 2XN

2

If a single CSCU chooses the best response strategy sequentially, from (16), it shows that the increase of an individual CSCU’s utility value contributes to the increase of the overall system performance. So (16) can essentially reflect the whole system performance.

3.2 Power Allocation As shown in the aforementioned Sect. 3.1, the SCBSs and spectrum resources selection algorithm can effectively improve uplink quality and reduce system interference level. In the following, we propose a novel price-based transmission power scenario to further improve the system EE given the corresponding SCBSs and spectrum resources strategies in Sect. 3.1. In the uplink, SCBSs can exchange signaling information with other SCBSs through IP-based backhaul networks after gathering channel state information (CSI) from the serving CSCUs on a common channel. So the social optimization of the transmission power strategies can be solved in a distributed manner by each CSCU after knowing the other CSCUs’ CSI, leaving the complex and energy dissipative signaling interactions and calculations to SCBSs. So, this method can effectively reduce the complexity and energy consumption of the mMTC devices. In the following, scheme as a non-cooperation game, as  we model the power allocation n o k k , where XN k is the set of given by G2 ¼ XN k ; fP nm gnm 2X k ;k2XK ; Unm nm 2XN k ;k2XK

N

CSCUs using the same spectrum resource k, P knm is the set of transmission power strategies of CSCU nmon spectrum resource k, and Unkm is the corresponding utility function. In order to ensure the feasibility of power allocation scheme, we define P knm as 9 8 = < X k  0  pk  Pk P knm , pk 2 RN : pknm  N k P; ð17Þ nm mask ; : n 2X m

Nk

where P is the average transmitting power of a single CSCU, Pkmask is the value of maximum transmission power on spectrum resource k and defining Pmask ¼

1  Pmask ; P2mask ; . . .; PKmask to satisfy the constraints forced by Federal Communications Commission (FCC) for avoiding damaging PUs’ normal communications. In a distributed framework, the cross-channel interference value, as shown in (5), can be equivalent to AWGN after knowing the selection strategies of SCBSs and spectrum resources of CSCUs in Sect. 3.1. And the CSCUs can unilaterally maximize the utility function of the single

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Energy-Efficient Uplink Resource Allocation Based on Game…

link energy efficiency (SLEE) given others’ power allocation strategies over channel k. The mathematical model is given by



maximize Unkm ¼ EEnkm pknm ; pknm ¼ Rknm pknm ; pknm =PEnkm k pnm ; 8nm 2 XN k : ð18Þ subject to pknm 2 P knm Because of CSCUs selfishly choose their own optimal transmission power strategies without taking care of the influence to others in the distributed structure, the NE of the game G2 is not always Pareto-optimum [24, 26]. In practice, we are interested in obtaining an equilibrium solution that can maximize the EE of spectrum k, over which there are Nk CSCUs running simultaneously, but not the SLEE maximization, and the mathematical model is given by

P k k k k x R p ; p nm nm nm nm nm 2XN k

maximize EEk ðpk Þ¼ P pk k k ; 8k 2 XK : ð19Þ nm 2X k PEnm pnm N

subject to pknm 2 P knm Note that the power allocation strategies in (19) is a Pareto optimal solution on spectrum resource k. The objective function (19) is also non-convex optimal. If using brute force method to solve (19), it will lead to an exponential computational complexity and is computationally infeasible even for a small size system. Therefore, in order to obtain an efficient power allocation method, we introduce a nonlinear fractional program algorithm [34] to reduce k

the computation complexity. We define the maximum energy efficiency EE of the considered spectrum resource k as



P P k k k ~k k k k k ~ x R ; p x R p ; p p n n n n n n n n n 2X n 2X m m m m m m m m m m k Nk Nk



EE ¼ ¼ maximize ; pknm 2 P knm ; P P k p k k k k ~nm nm 2X k PEnm p nm 2X k PEnm pnm N

N

ð20Þ where p~knm is the power allocation strategy of CSCU nm when achieving the maximized k

k

EE . From [34], we know that a maximum EE exists if and only if (21) is valid.



X X k xknm Rknm pknm ; pknm  EE PEnkm pknm maximize pk

¼

X

nm 2XN k

nm 2XN k

xknm Rknm

nm 2XN k





X k p~knm ; p~knm  EE PEnkm p~knm ¼ 0; p~knm 2 P knm

ð21Þ

nm 2XN k

for Rknm ðpknm ; pknm Þ  0 and PEnkm ðpknm Þ [ 0. So it reveals that this objective function in subtractive form is equivalent with the original optimization problem whose objective function is in fractional form, moreover the optimal power allocation can be solved by an iterative algorithm. A mathematical proof of the convergence is similar to [34] and Appendix A of [29], we will not repeat the similar proof procedure. The proposed iterative algorithm for obtaining the maximized EE is summarized in Table 1.

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As shown in Table 1, we should solve the following optimization problem (22) in each iteration for a given EEk(i) over k.



X X maximize xknm Rknm pknm ; pknm  EEk ðiÞ PEnkm pknm pk nm 2XN k nm 2XN k ð22Þ k k subject to pnm 2 P nm For solving the optimization problem (22) in a distributed manner, we redefine the utility function in (18) by introducing a pricing tuple kk ¼ ðkk1 ; kk2 ; . . .kkN k Þ to incentive CSCUs to choose Pareto-optimum NE strategies, and the user-dependent pricing function is given by



maximize Unkm pknm ; pknm ¼ xknm Rknm pknm ; pknm  xknm kknm  pknm k pnm ; 8nm 2 XN k : subject to pknm 2 P knm ð23Þ

Table 1 Iterative power allocation algorithm Algorithm 1 Iterative Power Allocation Algorithm for Maximizing EE on Spectrum Resource k 1: Initialize the maximum number of iterations I max and the maximum tolerance

ϖ

,

k

2: Set iteration index i = 0 and calculate initial energy efficiency EE (0) in (19) according to initial transmission strategies. 3: repeat iterations: k

k

4:

Solve the equivalent problem in (22) for a given EE (i ) and obtain the power allocation strategy p (i ) ,

5:

if

∑

nm ∈Ω

ωnk Rnk ( pnk (i ), p-nk (i )) − EE k (i ) m

m

m

m

Nk

6:

Convergence = true

7:

 = p (i ) and EE = return p k

k

k

∑

nm ∈Ω

∑

nm ∈Ω

9:

m

m

∑

m

m

PEnkm ( pnkm (i ))

Nk

else

k

Set EE (i + 1) =

∑

nm ∈Ω

ωnk Rnk ( pnk (i ), p-nk (i )) m

Nk

∑

nm ∈Ω

10:

Nk

ωnk Rnk ( pnk (i ), p-nk (i ))

Nk

nm ∈Ω

8:

PEnkm ( pnkm (i )) ≤ ϖ , then

Nk

Convergence = false

11: end if 12: until Convergence = true or i = I max .

123

m

m

k nm

m

k nm

PE ( p (i ))

and i = i + 1

Energy-Efficient Uplink Resource Allocation Based on Game…

In the following, we introduce two Lagrange functions J k and J knm based on (22) and (23) respectively to solve the pricing factor kknm in (23) for a given parameter EEk(i) and constraint set P knm .



X X Jk ¼ xknm Rknm pknm ; pknm  EEk ðiÞ PEnkm pknm nm 2XN k

þ

X nm 2XN k

nm 2XN k

aknm pknm



X

bknm



pknm



Pkmask



0 

nm 2XN k



J knm ¼ xknm Rknm pknm ; pknm  xknm kknm  pknm



1;

X

lknm @ nm 2XN k

0

þ aknm pknm  bknm pknm  Pkmask  lknm @

X

pknm

ð24Þ

k A

N P

1 pknm  N k PA

ð25Þ

;

nm 2XN k

where aknm , bknm , and lknm are the Lagrange multipliers (nonnegative real numbers). The corresponding K.K.T. [35] conditions are given by (26) and (27) respectively. X oRknm oRkni oJ k k k ¼ x þ x  EEk ðiÞ þ aknm  bknm  lknm ¼ 0 n n m i opknm opknm n 2X ;n 6¼n opknm i m Nk i

k k k pnm  0; anm pnm ¼ 0; pknm  Pkmask  0; bknm pknm  Pkmask ¼ 0 ; 0 1 X X pknm  N k P  0; lknm @ pknm  N k PA ¼ 0; 8k 2 XK ; 8nm 2 XN k nm 2XN k

nm 2XN k

oJ knm oRknm k ¼ x  xknm kknm þ aknm  bknm  lknm ¼ 0 nm opknm opknm

pknm  0; aknm pknm ¼ 0; pknm  Pkmask  0; bknm pknm  Pkmask ¼ 0 0 1 X X pknm  N k P  0; lknm @ pknm  N k PA ¼ 0; 8k 2 XK ; nm 2XN k

ð26Þ

:

ð27Þ

8nm 2 XN k

nm 2XN k

By comparing the K.K.T conditions in (26) and (27), we can set obtaining a same optimal solution, so the pricing factor expression.

kknm

oJ k opknm

¼

oJ knm opknm

for

must be the following



X oRk 1 1 kknm pknm ¼  k xkni kni þ k  EEk ðiÞ xnm n 2X ;n 6¼n opnm xnm i m Nk i 0 1 2  2  : k  k   k  p H H B log e     X n C 2 m ni ni ;i nm ;i 1 B k k C   x ðiÞ þ EE ¼ k B  

A 2 xnm @n 2X ;n 6¼n ni k k þ pk H k  þI þ c2 i m Idni þ Idni þ c2ni Idn Nk i dni ni  ni ;i  ni i

ð28Þ

We can know that the pricing in (28) is similar to the pricing factor obtained based on maximize spectral efficiency in [26] if without taking the part of EEk(i) into account.

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Intuitively, a larger pricing factor kknm ðpknm Þ will prevent CSCU nm from taking a large transmission power strategy on k. In view of (28), CSCU nm should transmit its CSI and static power information to the corresponding SCBS m by using a signaling packet over the common channel, then this SCBS broadcasts the received CSI, static power information, the detected interference on spectrum resource k, and the corresponding transmission power strategy to other SCBSs through backhaul networks. Based on all the received broadcasting information at the SCBS, a channel status table (CST) is used to incorporate these information and facilitate the information exchange among different SCBSs. We can achieve the transmission power pknm by solving (23) given the pricing factor and CST in SCBS m, and the equilibrium solution satisfies the water-filling expression in (29). pk1 ; . . .; p^knm 1 ; p^knm þ1 ; . . .; p^kN k Þ; 8nm 2 XN k ; p^knm ¼ WFnm ð^

ð29Þ

where the water-filling operator WFnm ðÞ is defined as " ½WFnm ,

xknm Bnm log2 e lk þ xknm kknm ð^ pknm Þ



INCRnm p^knm ,

P

 INCRnm



p^knm

# k

Pmask

; 8nm 2 XN k ;

ð30Þ

0

 2 k  k  2  ^ H p  ni ;m  þ Idnm þ cnm ni 2XN k ;ni 6¼nm ni ;   Hn ;m 2 m

ð31Þ

where [x]ba = min (b, max (a, x)), a B b is the Euclidean projection [36] of x on [a, b], P k k and lk is chosen to satisfy the constraint nm 2XN k pnm  N P in (17). Setting

      T p^k ,½^ pk1 ; p^k2 ; . . .; p^kN k T , INCRk , INCR1 p^k1 ; INCR2 p^k2 . . .; INCRN k p^kNk , and wk ,½xk1 ; xk2 . . .; xkN k T , so the NE water-filling solution in (29) can be expressed as a projection with respect to the weighted Euclidean norm with weights xk1 ; xk2 . . .; xkN k of INCRk given in (32) onto the simplex P knm in (17): k

p^k ¼ ½INCRk wP kn ; 8k 2 XK : m

ð32Þ

The proof of convergence and existence of NE is consistent with [26, 36]. In practice, parallel iteration algorithm requires the system parameters to be correctly estimated for each CSCU. It is difficult to satisfy this condition for practical system. To overcome this problem, we adopt a relaxed updating scheme in each iteration, and SCBS is required to remember the most recent power strategies as well as other SCBSs’. The relaxed algorithm is more robust to occasional estimation errors and channel oscillations, but leads to certain degradation in convergence speed. Based on the characteristics of the CSCNs, we choose relaxed parallel iterative algorithm to update the transmission power strategies, and the mathematical expression of j ? 1 iteration is given by

pknm ðjÞ þ ð1  fÞWFnm p~knm ðjÞ ; 8nm 2 XN k ð33Þ pknm ðj þ 1Þ ¼ f~ where p~k ,½~ pk1 ; p~k2 ; . . .; p~kN k T is the optimal solutions obtained in Algorithm 1, f 2 [0, 1) can be interpreted as memory factor [26], the memory is longer if setting larger value of f, and this algorithm is more robust to estimation errors at the cost of slower convergence speed. So we can make a compromise between the convergence speed and the system precision by

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Energy-Efficient Uplink Resource Allocation Based on Game… Table 2 Parallel price-based iterative algorithm Algorithm 2 Parallel Price-Based Double-Loop Iterative Algorithm for Solving Game G2 k

1: Initialize the maximum number of iterations J max and the transmission power pnm = 0.1, ∀nm ∈ Ω N k . k

2: Set iteration count j = 0 and calculate energy efficiency EE (0) and pricing

λnk (0) m

according to initial transmission

strategies and CST. 3: repeat {Out loop } 4: repeat {Inner loop } Given pricing tuple

λ k , simultaneously solve the optimal issues in (23) for all CSCU nm ∈ Ω N k and obtain power

k

policies { pnm }. 5: until Convergence = true as shown in Algorithm 1. k

k

k

6: Update p ( j + 1) according to (33) and set EE ( j + 1) = EE , where EE update pricing

λ ( j + 1) k nm

according to (28) for all CSCU

k

is the optimal EE in Algorithm 1; then

nm ∈ Ω N k , j = j + 1 .

7: until Convergence = true or j = J max .

choosing a favorable f.The proposed power allocation algorithm is carried out through two nested loops in Algorithm 2, and the detailed process is summarized in Table 2.

4 Simulation Results and Analysis We incorporate the resource allocation into an operational Multiple Access Control (MAC) protocol [26] in a distributed structure. In CSCNs, it must satisfy the following features: a dedicated common channel is available to support the transmission of control packets using a pre-set transmission power, SCBSs should broadcast CSTs to others through backhaul networks, and the channel gains between any two terminals are symmetric and timeinvariant in an algorithm cycle. In the MAC protocol, we use three types of control packets for handshaking: Request-to-Send (RTS), Clear-to-Send (CTS), and Decide-to-Send (DTS). But they are different from the classic date packets switching protocol in IEEE 802.11. These control packets are not used for exclusively reserving channels, but rather to exchange the transmission parameters within CSCNs. Then we can determine the strategies based on these parameters. The control packets need to be exchanged within a certain duration, referred to as the contention window [26], which contains access window and training window, and the data will be transmitted within a data window after the convergence of the proposed algorithm. The detailed process is given in Fig. 3. According to the power allocation algorithm in Sect. 3.2, we know that this theoreticalbased algorithm (TA) can converge to NE power strategies just when the EE in the inner loop reaches the maximum value in Algorithm 1 given the selected SCBSs and spectrum resources. However, the determined power strategies obtained in Algorithm 1 are also used for the updating of the pricing in the outer loop of game G2 as shown in Algorithm 2. For simplifying these iterative procedures, we integrate the updating of the EE in Algorithm 1 and the updating of pricing in Algorithm 2 for once power updating. By using the simplified algorithm (SA), though running only once updating of the EE in Algorithm 1, the SA can converge to an equilibrium solution at last along with the game carrying out as

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Fig. 3 Flowchart of our algorithm

shown in the following simulation results. Therefore, the system convergence time and the signaling overhead can be reduced due to less time consuming and updating in Algorithm 1 when using the SA. We show that the proposed iterative approach provides comparable performance, while being much less computationally complex. In our paper, the energy-efficient resource allocation technique is simulated with MATLAB programs in CSCNs. In particular, we employ a Gaussian frequency-selective multiuser interference channel in our simulation. The CSCNs coexist with other Wi-Fi devices under the coverage of a Wi-Max base station in a 50m 50m shopping mall of the same floor, and there are M SCBSs and N CSCUs randomly distributing in it, as shown in Fig. 4. The signal transmission loss in indoor environments satisfies the log-normal shadowing model [37, 38] as given by     4pd0 d þ 10c log10 þ wdB ; ð34Þ PLðdÞ½dB ¼ 20 log10 d0 k where d0 is a reference distance for the antenna far-field and typically assumed to be 1 m in indoor environments due to scattering phenomena, d is transmission distances between devices, k is the carrier wavelength, c = 2.2 is the path loss exponent in our simulation, wdB is Gaussian random variable whose mean is 0 and standard deviation is r.

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Energy-Efficient Uplink Resource Allocation Based on Game… Fig. 4 Distribution of SCBS and CSCU

We suppose that the primary Wi-Max network will always operate and the Wi-Fi devices randomly operate. And we assumes that channel indexes 1, 5, 9, 13, and 14 are available for CSCUs as shown in Fig. 2. For convenient illustration, we renumber these five available channels from 1 to 5 according to the corresponding channel indexes. All the channels can be fully reused in our algorithm, whereas just the non-overlapped channel indexes 1, 9, and 14 can be used in [25]. At the beginning, the initial transmission power is 0.1 W, while the initial SCBSs and spectrum resources allocation strategies are randomly selected by CSCUs. The detailed simulation parameters is outlined in Table 3.

4.1 Convergence of SCBS Selection and Spectrum Allocation Algorithm Figure 5 illustrates the evolution of the SCBSs and spectrum resources selected by CSCUs based on potential game G1 , respectively. To make the figures not excessively overcrowded, we plot only the curves of 6 out of 40 CSCUs. As shown in Fig. 5a, b, the strategies of CSCUs are convergent after 3 game processes. e.g., CSCU15 and CSCU20 Table 3 Simulation parameters

Parameter

Value

Number of SCBS M

30

Number of CSCU N

40

Number of carriers K

5

Bandwidth B

22 MHz

Bandwidth efficiency

0.85

Average transmission power P

0.1 W

Maximum transmission power Pkmask

0.4 W

Static power consumption PEnm

0.1 W

Thermal noise N0

-174 ? 10lg(B) dBm

Shadowing standard deviation r

4 dB

Memory factor f

0.3, 0.6

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Fig. 5 Convergence of strategies selected by CSCUs based on potential game: a SCBS, b spectrum resource

choose SCBS14 to transmit data in the uplink simultaneously after converging to the equilibrium solutions. In order to avoiding the interference and maximizing the system performance, SCBS14 allocates two different channels 4 and 1 to CSCU15 and CSCU20 respectively. Figure 6 shows the evolution of the sum of the EE on all the spectrum resources (system EE). The system EE is consistent with the system capacity by referring to (18) when there is no power allocation in the potential game and the equilibrium value of EE improves significantly comparing with initial value. By choosing appropriate SCBSs and spectrum resources, the proposed potential game can improve the link quality of each CSCU by reducing co-channel interference and convergence within *3 iterations.

Fig. 6 EE versus the number of iterations based on potential game

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Energy-Efficient Uplink Resource Allocation Based on Game…

4.2 Convergence of Power Allocation Algorithm Based on Maximizing EE on Spectrum Resource Given the Selected SCBSs and Spectrum Resources Figure 7 illustrates the convergence of the EE in the inner loop for different outer loop respectively, e.g., 1, 2, 3, and L time iteration of the outer loop, and L & 6 represents the number of iterations reaching the NE in game G2 . In the outer loop 1, the iterative power allocation algorithm of maximizing EE in the inner loop can reach a maximum value within *3 iterations, and the initial value of EE in the outer loop 2 is the convergence value of the outer loop 1, and so on. The convergence speed in the inner loop is increased with the evolution of game G2 , and the corresponding EE is close to an optimal value just after once iteration. So it is feasible to integrate the updating process of EE in the inner loop with the updating of pricing in the outer loop as once updating of the transmission power in game G2 . Figure 8 shows the evolution of the transmission power of different CSCUs by using simplified algorithm (SA) to maximizing EE. For improving the EE, we can know that the transmission power decreases to steady state at last, and CSCU5’s transmission power comes to 0 W when the selected channel 3 is poor. Figure 9a–c illustrate the convergence of the whole EE, system capacity, and system power consumption respectively for different algorithms in game G2 . We first give a brief explanation for some symbols appeared in Fig. 9 as follows: e.g., ‘‘EE’’ denotes the algorithm proposed in our paper for maximizing EE on spectrum resource, ‘‘SE’’ denotes the algorithms used in [14, 21, 22, 26–28] mainly for maximizing spectrum efficiency on spectrum resource, ‘‘only EEk’’ denotes the proposed algorithm in which we only take the EE of the spectrum resource k as the pricing factor, and ‘‘SLEE’’ denotes the algorithm of maximizing single link energy efficiency. The convergent speed of EE-TA in Fig. 9 is slightly faster than the EE-SA, and the two algorithms converge to the same steady state within *6 iterations. The convergence speed attenuation can be ignored due to the less iteration in the inner loop when using the EE-SA. So the EE-SA is more simple and

Fig. 7 EE versus the number of iterations in inner loop for different outer loop

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Fig. 8 Convergence of transmission power for different CSCUs

Fig. 9 Performance comparison versus the number of iterations in outer loop for different algorithms and memory factors: a the whole EE, b system capacity, and c system power consumption

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efficient. In Fig. 9a, b, we observe that the proposed price-based power allocation algorithm achieves the equilibrium solutions of the EE and system capacity higher than initial values by about 85 and 4% respectively, and the initial value of EE is the equilibrium solution of potential game G1 as illustrated in Fig. 6. On the contrary, the system power consumption is reduced by about 44% comparing to initial value as shown in Fig. 9c. Although the employment of memory factor can improve system robustness, the convergence speed is reduced as shown in Fig. 9. The number of iterations achieving equilibrium state for different value of memory f = 0.3 and f = 0.6 are *10 and *19 respectively. The system will have a longer memory time when setting bigger value of f, and the updated power is more related to former allocated power strategies. So the simulation results are consistent with theoretical analysis. When we adopt the SLEE algorithm to solve the power allocation problem in a distributed architecture, CSCUs selfishly choose strategies to maximize their own single link energy efficiency without exchanging channel information with other SCBSs, and the equilibrium solution of EE is lower than the proposed price-based algorithm by about 70%. If we just use the energy efficiency EEk as pricing factor not the optimal pricing in (28), the equilibrium value of EE can increase about 27% comparing with the SLEE algorithm, but is still lower than the optimal algorithm by 34%. Therefore, we need to make a compromise between signaling overhead and system performance. Comparing the EE, system capacity, and system power consumption of the different algorithms in Fig. 9, we know that the proposed algorithm can achieve an equilibrium value of EE bigger than the algorithms used in [14, 21, 22, 26–28] by about 76%. The system capacity is just lower than the algorithm used in [26] by about 1.28%, but higher than the algorithm in which there is no price by 3%. Meanwhile, the system power consumption reduces by about 79%. Thus, our algorithm can effectively reduce system power consumption without substantially reducing system capacity in a strong interference environment. Figure 10a, b depict the performance of the EE and system power consumption versus the static power PEnm for different algorithms. We can know that the EE reduces along with the increasing of static power for the two algorithms, and the performance gain of our algorithm saturates when static power goes up tremendously. This is mainly due to that static power dominates the total power consumption, and CSCUs must improve transmission power to optimize the system performance even obtaining little EE gain.

Fig. 10 Performance comparison versus static power PEnm for the algorithms of maximizing EE and SE: a EE of the whole spectrum resources, b system power consumption

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Fig. 11 Performance comparison versus the number of iterations of the joint resource allocation for different algorithms: a the whole EE, b system capacity, and c system power consumption

Therefore, our algorithm can make a greater performance improvement when the devices have smaller static power. So it is more suitable for mobile communication networks and internet of things systems in which there are more low static power devices to communicate in uplink. Figure 11a–c show the performance of the whole EE, system capacity, and system power consumption by using joint resource allocation methods based on different power allocation algorithms, respectively. We can know that all the four algorithms can converge to equilibrium state, so the joint resource allocation algorithms based on game G1 and G2 are feasible. Although the four algorithms can achieve almost the same system capacity, our algorithm can get the largest EE and consume the least system power than other algorithms as discussed above in Fig. 9.

5 Conclusion In this paper, we proposed a new joint resource allocation algorithm for maximizing EE of the whole CSCNs based on game theory. We develop a low-complexity alternative for the suboptimal solution by decomposing the optimization problem into two sub-problems. By introducing a new interference operator that not only indicates the interference level of

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Energy-Efficient Uplink Resource Allocation Based on Game…

independent spectrum bands, but also overlapped bands. Meanwhile, we formulate a new potential function to facilitate the selection of SCBSs and spectrum resources at the same time by CSCUs in an open access model, and the potential game can be carried out in a distributed architecture. By maximizing the utility function of CSCUs respectively, this algorithm can effectively reduce the system interference level. For solving the social NE transmission power, we propose a price-based power allocation algorithm to maximize EE on the selected spectrum resource based on a non-cooperative game. The optimal power strategies of G2 can be expressed as a water-filling-like mapping among the CSCUs running over the spectrum resource. We also introduce a memory factor in the updating of transmission power to improve system robustness and reduce equipment complexity. Simulation results show that the proposed joint resource allocation algorithm can effectively improve EE and reduce system power consumption. Therefore, we may conclude that the proposed algorithm is suitable for an energy-efficient design of CSCNs and consistent with the development of green Information and Communications Technology.

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Ya-Nan Jia was born in Shangqiu, Henan, China. He received his B.E. in Electrical and Information Engineering from the PLA Information Engineering University, Zhengzhou, China, in 2010, and the M.E. degree in Information and Communication Engineering and the Ph.D. degree in Communication and Information System from Dalian Maritime University, Dalian, China, in 2012 and 2015, respectively. Since Dec. 2015, he has been a researcher at the Spreadtrum Communications Co., Ltd., Shanghai, China. His research interests include cognitive radio, optimization theory, cooperative communications, Game Theory and Small Cell Networks.

Dian-Wu Yue received the B.S. and M.S. degrees in mathematics from Nankai University, Tianjin, China, in 1986 and 1989, respectively, and the Ph.D. degree in communications and information engineering from Beijing University of Posts and Telecommunications, Beijing, China, in 1996. From 1989 to 1993, he was a Research Assistant of applied mathematics at Dalian University of Technology, Dalian, Liaoning, China. From 1996 to 2003, he was an Associate Professor of communications and information engineering at Nanjing University of Posts and Telecommunications, Nanjing, Jiangsu, China. He is currently a full Professor of communications and information engineering at Dalian Maritime University, Dalian, Liaoning, China. During 2000–2001, he was a Visiting Scholar at the University of Manitoba, Winnipeg, MB, Canada. During 2001–2002, he was a Postdoctoral Fellow at the University of Waterloo, Waterloo, ON, Canada. His current research interests include MIMO wireless systems, cooperative communications, cognitive radio, and modulation and coding.

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