On the power allocation strategies in coordinated multicell networks using Stackelberg game
 Shima Haddadi^{1},
 Ashkan Oliaiee^{1},
 Hamid Behroozi^{1}Email author and
 Babak Hossein Khalaj^{1, 2}
https://doi.org/10.1186/s1363801605793
© Haddadi et al. 2016
Received: 11 May 2015
Accepted: 2 March 2016
Published: 1 April 2016
Abstract
In this paper, we study the power allocation problem in multicell OFDMA networks, where given the tradeoff between user satisfaction and profit of the service provider, maximizing the revenue of the service provider is also taken into account. Consequently, two Stackelberg games are proposed for allocating proper powers to central and celledge users. In our algorithm, assuming the fact that users agree to pay more for better QoS level, the service provider imposes optimum prices for unitpower transmitted to users as they request different levels of QoS. In addition, in order to improve system performance at celledge locations, users are divided into two groups based on their distance to the corresponding basestation (BSs): Central users and celledge users. The paper also exploits unique features of Coordinated MultiPoint Joint Transmission (CoMPJT) where coordinated BSs are clustered together statically or dynamically in order to also address the requirements of celledge users. After simulating the proposed game with static clustering, a simple dynamic clustering algorithm is introduced for intercell coordinated networks where its performance is evaluated through simulations.
Keywords
Game theory Intercell interference coordination (ICIC) Coordinated multipoint joint transmission Resource allocation problem Static and dynamic clustering Power consumption1 Introduction
1.1 Background and related works
Satisfying the growing demand for mobile services and radio communications makes future wireless networks to fully reuse resources such as available frequency bands. This increase in use of resources in turn leads to high increase in intercell interference (ICI) level. Increased interference levels degrade the system performance in serving users and due to the nearfar effects on the signaltointerferenceplusnoise ratio (SINR) level, users in celledges are more sensitive to high interference than central ones. ICI mitigation along with demand for higher datarate requires utilization of advanced technologies. Subsequently, resource management plays a major role in next generation mobile networks.
In Long Term Evolution (LTE) standard, developed by the 3rd Generation Partnership Project (3GPP) standardization body, three general approaches are proposed to overcome intercell interference effect: ICI randomization, ICI cancellation and ICI coordination (ICIC) [1–3]. Our research is based on the last approach which has lower computational complexity. The ICIC approach tries to improve the celledge performance by accepting some level of interference and thus provides more homogeneous service to users located at different regions of the network. Fractional Frequency Reuse (FFR) is an asserted idea in this context that divides available frequency bands into different groups, where each band is designed to be dedicated to certificated users with different reuse factors which correspond to their channel conditions [4]. Consequently, quality of communication of users who suffer more from interference can be improved by less reuse of their allocated subchannels.
Another promising technology to improve the conditions at celledges is coordinated multipoint joint transmission (CoMPJT) [5]. In cooperative networks, in order to eliminate the interference on celledge users, a set of coordinated BSs which are clustered together serve one celledge user simultaneously over the same resource [6]. It is wellknown that better performance can be achieved by increasing the number of cells that are grouped together (called the cluster size). However, in reality, only a limited number of BSs can coordinate the transmission [7]. This raises the question that coordinating transmission among which selection of BSs will lead to a more efficient CoMPJT implementation. In practice, cells can form clusters either statically or dynamically [8]. Static clusters are constructed according to the geographical criteria. Consequently, they are fixed for all users and are not changed over time. On the other hand, adaptive clustering applies realtime cluster reformation by perceiving the condition of each user at each timeinstant. In fact, variations in system loading or mobility of users may lead to new clustering architecture. Naturally, dynamic clustering can improve system performance noticeably at the price of higher system complexity.
Although CoMPJT can enhance the throughput of users located in celledges noticeably, it also increases the complexity, signaling overhead on backhaul network which is commonly known as X2 interface and information exchange among cells. The authors in [5] brought some practical techniques to overcome the fundamental requirements of CoMPJT. In addition, field trial results of [2, 9–11] have confirmed the advantages of applying CoMP in practical systems. So in this paper, these preliminaries are assumed to be provided. We also ignore the imposed complexity caused by backhaul requirements of CoMP and assume the channel gains between users and BSs to be properly estimated.
Joint multicell coordinated scheduling and multipleinput multipleoutput (MIMO) techniques are also wellknown schemes applied in LTE [12]. A multicell cooperative zeroforcing beamforming (ZFBF) scheme is proposed in [13] in which all BSs use CoMP transmission to maximize the energy efficiency (i.e., the average datarate per unit power). A locationaware multicell cooperation scheme for downlink transmission in orthogonal frequencydivision multiple access (OFDMA)based networks is proposed in [14] which only uses CoMP transmission to serve users with low SINR level while users with acceptable SINR conditions are served by a single BS.
The efficiency of a wireless cellular network mostly depends on allocating schemes for valuable resources such as power and bandwidth. Providing a minimum quality of service (QoS) level for all users, minimizing power consumption and maximizing the difference between the benefits and costs of the network for one user result in a typical choice of metrics used to allocate resources among different users. It should be noted that in this paper, we choose the requested SINR of each user as the QoS index.
Despite optimality of centralized techniques in resource allocation, finding the analytic formulation and in turn the optimum point is prohibitively difficult due to the nonconvexity of the associated optimization problem [16, 17]. Furthemore, the resource allocation problem in CoMP is proved to be NPcomplete [18]. Game theory which covers such scenarios is a powerful tool in providing suboptimal solutions to model and analyze resource allocation problems in a distributed way.
In recent years, various noncooperative games are proposed in the literature, trying to maximize the utility of selfish users [19–23]. Also, [24] presents a resource allocation based on cooperative game. But, in addition to the user satisfaction, profit of the service provider is another important factor that should be considered in the network. Hierarchical games such as Stackelberg or leaderfollower game can properly cover the pricingbased interactions between users and the service provider [25]. Namely, the service provider determines differential prices per unit of transmitting powerlevel for each user, proportional to the requested service of user, in order to maximize its revenue from the network. On the other hand, users (which are also called followers) adjust their power levels to get maximum services from the network with the least imposed costs. In [26], the authors try to model the pricingbased interactions between secondary and primary users in a timedivision multiple access (TDMA) cooperative cognitiveradio networks using Stackelberg game. Furthermore, [27] presents a Stackelberg game to model the power allocation of codedivision multiple access (CDMA) cognitiveradio network in uplink side. However, key issues such as power consumption and QoS of users are addressed in none of these references. Another Stackelberg game proposed for the same network in the downlink side [28] also does not taken QoS into account.
Various algorithms for static clustering were already presented in earlier works [29–31]. However, due to their limited performance gain, dynamic clustering has attracted more attention in recent years [32–35]. For example, the work in [32] is based on the graph theory with the aim of maximizing the sumcapacity. Indeed, each BS will be associated with the cluster which has the maximum joint capacity rate with other BSs belonging to that cluster. In [33] cells are clustered dynamically according to whether perfect channel state information (CSI) is available or not. In [34], a cochannel intercluster interference canceler is proposed for MIMO networks by performing linear processing combined with multiuser beamforming. Also, the authors in [35] introduce overlapping clustering. Requiring high level of computational complexity is the major drawback of the aforementioned algorithms. In addition, reference signal receiving power (RSRP) is used in [36, 37] to select cooperating BSs. The authors in [37] propose a resources allocation algorithm based on noncooperative game considering the throughput of users and their QoS to evaluate the system performance.
1.2 Motivations, results, and paper organization
To the best of our knowledge, the main challenge of future networks is the ability to satisfy the growing and high quality demands of users for various kind of services even in celledges while their revenue is also maximized. Consequently, the first novelty of our approach compared with aforementioned studies is that we allow users to request different QoS levels due to their interest for different services and of course their affordable payment. Since CoMPJT is a key technology to cope with high quality demands of celledge users and no revenue maximization game model that incorporates such characteristics in coordinated multicell networks is yet applied, the main motivation in this paper is proposition of a Stackelberg game for OFDMA multicell cellular networks using a combination of CoMPJT technologies and FFR technique. We also compare static and dynamic clustering and analyze their effects in terms of system performance. Subsequently, a simple dynamic clustering algorithm is proposed in order to verify the effects of clustering model in system performance.
Our earlier results show that the proposed power allocation algorithm can properly satisfy even high quality demands of users near to BSs. In contrast, the expanded version of our game to apply CoMPJT demonstrates that static BS selection is only capable of affording high QoS level for celledge users in special locations. Finally, by proposing a simple dynamic clustering method, the performance of system in celledges can be noticeably improved.
The remainder of this paper is organized as follows: The proposed architecture for the system model is described in Section 2. Section 3 focuses on the power allocation strategies of users near to the center of cells (central users) and the extension of the proposed game to apply COMP JT for celledge users is provided in Section 4. Both proposed power allocation algorithms are simulated in hypothetical static clusters in Section 5. Presented results show how CoMPJT helps to improve celledge performance. In Section 6, a dynamic clustering algorithm with the aim of performance improvement by imposing lowlevel of complexity to the system is introduced. Simulation results for the proposed method of clustering are presented in Section 6.3 in which performance improvement in comparison with the static clustering is evaluated. Finally, Section 7 concludes the paper.
2 Proposed system architecture
Meanwhile, in this paper, we intend to provide dynamic SINR thresholds for all users. In order to reach this goal, we consider a minimum SINR threshold γ _{ th } for all users in the network. The requested SINR threshold for user i in central region will be α _{ i,m } γ _{ th } and in celledge region will be α _{ i,c } γ _{ th }, where the coefficient α _{ i,m } will be determined by central user i in cell m and the coefficient α _{ i,c } will be assigned by celledge user i in cluster c.
By sectorizing each cell, indeed, for central users we are facing to singlecell scenario of resource allocation while multicell scenario should be applied for celledge users. In our investigation, first we focus on resource allocation problem in central region with singlecell approach, then try to exploit COMP JT improvements using multicell scenario.
3 Power allocation strategy for central users
In this section, we focus on serving users located in central region. After defining the required parameters, the proposed Stackelberg game formulation in noncooperative framework is determined. Eventually, convergence and uniqueness of the Nash equilibrium point for this proposed game is evaluated.
3.1 Problem definitions
System parameters in central region
Parameter  Definition 

N _{ int }  Number of users located randomly in central region of each cell 
M  Number of cells 
\({\sigma _{0}^{2}}\)  Additive noise power 
\(h_{i,l_{i}}^{m,j}\)  Channel gain between user i of cell m on channel l _{ i } and BS of cell j 
\(p_{i,l_{i}}^{m,j}\)  Transmitted power from BS of cell j to user i of cell m on channel l _{ i } 
\(p_{i,l_{i}}^{m,m}\)  Transmitted power to user i of cell m on channel l _{ i } from BS of its cell 
\(\gamma _{i,l_{i}}^{m}\)  SINR of user i in cell m on channel l _{ i } 
γ _{ th }  Minimum target SINR of users 
p _{ max }  Maximum transmitting power 
where \(I_{I.N.}^{i,m}\) refers to the summation of noise power and the received interference power at user i in cell m on channel l _{ i }, caused by users of other cells using the same channel.
3.2 Stackelberg game model for central users

\(\mathcal {N}=\{1,\ldots,n\}\) denotes the set of players or decision makers. Due to our Stackelberg game model in central region, this set consists of central users and BSs.

\(\mathcal {A}=\{A_{1},\ldots,A_{n}\}\) determines the set of possible actions that each player can choose as its strategy. This set is referred to as the strategy space. Each Stackelberg game consists of users strategies as well as BSs strategies.

The main part of each game is the set of utility functions which is defined according to player’s preferences. In each step of the game, each player chooses the strategy from its strategy space which maximizes its objectives. In our model, each BS wants to earn higher income from users by consuming as lowlevel of power as possible, while each user is interested in gaining higher level of SINR or equivalently higher datarate while paying less to the service provider.
Stackelberg games can be solved by using backward induction. In fact, first each user considering fixed unitprices for power transmitted from BSs, maximizes its utility function and choose its optimum power from the user’s strategy space. Then, each BS, considering required transmitting powers, maximizes its revenue to determine the best price for each user’s unitpower.
where \({\lambda _{i}^{m}}\) and \({\varphi _{i}^{m}}\) are respectively the unitprice and the subscription cost that BS of cell m imposes to user i located in this cell. The first term of (3) is proportional to the received rate at user i of cell m and the second term is the imposed cost of serving BS to such user. The third term and also the coefficient of the rate in the first term are adopted to ensure the SINR of user i to be more than its demand. Indeed, the objectives of user i are considered as follows: 1) Ensuring that its received SINR is larger than or equal to its target SINR, 2) maximizing its data sumrate and 3) paying the lowest price to the service provider. Also, in (3), α _{ i,m } is the factor that shows the willing of i ^{ t h } user to pay more in lieu of getting better quality of services.
In this utility function, the first term denotes the receiving costs of BS m from all users in the same cell. In order to keep the required powerlevel of each user as low as possible, we add the second term to each BS’s objectives. In this case, there will be no incentive for the service provider to provide a QoS better than what is required. Therefore, the power consumption could reach to the minimum possible level.
3.2.1 Nash equilibrium of proposed game

Main concepts of Nash equilibrium
Generally, the main goal of any hierarchical game is achieving the Nash equilibrium point which is defined as the profile of optimal actions for users. On the other hand, it is the point that there is no incentive for each player to deviate from its choice to have extra profile [38]. Mathematically, in our game, a strategy set is defined as a Nash equilibrium point if we have:$$ {}\begin{array}{cc} {U_{i}^{m}}\left(p_{i,l_{i}}^{\ast}\right)\!\geq\! {U_{i}^{m}}\left(p_{i,l_{i}}\right) & \forall\left(p_{i,l_{i}}\right),i=1,\ldots,N\\ U_{BS}^{\,m}\left(\boldsymbol{\lambda}^{\ast},\boldsymbol{P}^{\ast}\right)\!\geq\! U_{BS}^{\,m}\left(\boldsymbol{\lambda},\boldsymbol{P}\right) & \forall\left(\boldsymbol{\lambda},\boldsymbol{P}\right),m\,=\,1,\ldots,M \end{array} $$(6)where \(\boldsymbol {P}^{\ast }=\left [p_{1,l_{1}}^{\ast },p_{2,l_{2}}^{\ast },\ldots,p_{N,l_{N}}^{\ast }\right ]\) and similarly \(\boldsymbol {\lambda }^{\ast }=\left [\lambda _{1}^{\ast },\lambda _{2}^{\ast },\ldots,\lambda _{N}^{\ast }\right ]\).
In order to solve a Stackelberg game, Nash equilibrium of the subgame should be obtained by calculating the best responses of players using backward induction. Namely, the leader first predicts the response of the follower which is considered to be rational and then chooses a strategy which maximizes its payoff. Sequentially, the follower chooses the anticipated response to the observed strategy of the leader.
Note 1: In principles of game theory, according to “Kakutani Fixed Point Theorem”, a game should meet two following conditions to have a pure Nash equilibrium point [38]: 1.
The strategy space should be nonempty, closed, bounded and also a convex set.
 2.
The utility function should be not only continuous in strategy space but also be a concave function.
Remark 1. A set is said to be convex if and only if with the assumption of 0≤θ≤1 and having x _{1} and x _{2} as points in predetermined set, y=θ x _{1}+(1−θ)x _{2} belongs to the set too [39]. Also a function f(x) is concave if it satisfies :$$ \frac{\partial^{2}f(x)}{\partial x^{2}}\leq0 $$(7)  1.

Existence of a Nash equilibrium point in the proposed Stackelberg game
According to system’s specified parameters, the strategy space of the proposed game would be the following set: [0,p _{ max }]. It is obvious that this set is nonempty, closed and bounded. Hence, for satisfying the first condition of Note 1, the convexity of the strategy set should be proved. Assuming p _{1} and p _{2} as two allocated powers from the strategy space [0,p _{ max }], we have:$$ 0\leq\theta p_{1}+(1\theta)p_{2}\leq p_{max}. $$(8)So, the strategy space is convex.
Based on Note 1, in addition to the abovementioned condition, concavity of utility functions with respect to their own strategies should exist. By calculating the secondorder partial derivative of (3) with respect to \(p_{i,l_{i}}^{m,m}\), we obtain:$$ \frac{\partial^{2}{U_{i}^{m}}}{\partial \left(p_{i,l_{i}}^{m,m}\right)^{2}}=\frac{\alpha_{i,m}^{2}h_{i,l_{i}}^{m,m}}{I_{I.N.}^{i,m} \left(1+\gamma_{i,l_{i}}^{m}\right)^{2}}. $$(9)Since (9) is negative, the utility function of each central user is always strictly concave.
For utility function of BSs, expressed in (4), to be concave, we need to have:$$ {\lambda_{i}^{m}}<\frac{6\alpha_{i,m}^{3}\gamma_{th} h_{i,l_{i}}^{m,m}}{I_{I.N.}^{i,m}(1+4\alpha_{i,m}\gamma_{th}+4\alpha_{i,m}^{2}\gamma_{th}^{2})} +\frac{h_{i,l_{i}}^{m,m}}{I_{I.N.}^{i,m}\alpha_{i,m}\gamma_{th}}, $$(10)Satisfying (10) indicates that each proposed game has at least one pure Nash equilibrium point.

Calculating Nash equilibrium point of the proposed game
According to the discussed principle of Stackelberg game, in order to obtain the unique Nash equilibrium point for the proposed games, the transmitting power for each user should be optimized using (11), considering unit prices as fixed parameters:$$ \frac{\partial{U_{i}^{m}}}{{\partial}p_{i,l_{i}}^{m,m}}=0. $$(11)In maximizing procedure of (3), two cases should be considered, \(\gamma _{i,l_{i}}^{m}<\alpha _{i,m}\gamma _{th}\) and \(\gamma _{i,l_{i}}^{m}\geq \alpha _{i,m}\gamma _{th}\). In the first case, the objective function of user i defined in (3) will be always nonpositive, so the maximum utility that is equal to zero will be occurred when user’s power becomes zero as well. Now, considering \(\gamma _{i,l_{i}}^{m}\geq \alpha _{i,m}\gamma _{th}\), the optimal power allocation strategy for central user i will be obtained as the following:$$ p_{i,l_{i}}^{m,m}\left({\lambda_{i}^{m}}\right)^{\;(k+1)}= \frac{\alpha_{i,m}^{2}}{\lambda_{i}^{m\;(k+1)} {\frac{h_{i,l_{i}}^{m,m}}{I_{I.N.}^{i,m\;(k)}\alpha_{i,m}\gamma_{th}}}} \frac{I_{I.N.}^{i,m\;(k)}}{h_{i,l_{i}}^{m,m}}\;{\left{\vphantom{\frac{1}{2}}}\right.}_{0}^{P_{max}} $$(12)In the next step, we substitute (12) in BS objective functions for central users in (4). Then, by maximizing the utility function of the BS, optimum unitprice of user i in central region of cell m is calculated as:$$ {\small{\begin{aligned}{} \lambda_{i}^{m\;(k+1)}=\frac{4\alpha_{i,m}^{3}\gamma_{th}h_{i,l_{i}}^{m,m}}{I_{I.N.}^{i,m\;(k)} \left(1+4\alpha_{i,m}\gamma_{th}+4\alpha_{i,m}^{2}\gamma_{th}^{2}\right)} +\frac{h_{i,l_{i}}^{m,m}}{\alpha_{i,m}\gamma_{th}I_{I.N.}^{i,m\;(k)}}, \end{aligned}}} $$(13)Indeed, as we expected, not only the values of \({\varphi _{i}^{m}}\) is constant during all iterations and have no dependance on user’s locations, but also the service provider considers higher unitprices and subscription costs for central users requesting higher levels of QoS.
Furthermore, it is obvious that (13) satisfies the required condition of having unique Nash equilibrium point, provided in (10). Moreover, by substituting (13) in (12), it can be easily proved that obtained SINR of user i is always more than its target with a small deviation

Uniqueness of obtained Nash equilibrium point
In [40], for an utility based iterative algorithm determined as \(p_{i}^{(k+1)}=f_{i}(p_{i}^{(k)})\) to converge to a unique Nash equilibrium point, the author has pointed that the updating function f _{ i } should be:
Positive. i.e. \(f_{i}(p_{i}^{(k)})\geq 0\).

Monotone which means \(p_{i_{1}}^{(k)}\geq p_{i_{2}}^{(k)}\Rightarrow f_{i}\left (p_{i_{1}}^{(k)}\right)\geq f_{i}(p_{i_{2}}^{(k)})\).

Scalable in the sense that \(\rho f_{i}\left (p_{i}^{(k)}\right)\geq f_{i}\left (\rho p_{i}^{(k)}\right)\;\forall \rho \geq 1\).
Considering proposed game and iterative power allocation calculated in (12), first condition for convergence of our algorithm will be satisfied if:$$ \lambda_{i}^{m\;(k+1)}\leq\left(\alpha_{i,m}^{2}+{\frac{1}{\alpha_{i,m}\gamma_{th}}}\right)\frac{h_{i,l_{i}}^{m,m}}{I_{I.N.}^{i,m\;(k)}} $$(14)According to (13), Satisfying (14) coincides to \(1+4\alpha _{i,m}^{2}\gamma _{th}^{2}\geq 0\) which is evident.
Also, calculated updating function has monotonicity decreasing prosperity if its first deviation respect to imposed interference becomes nonnegative that requires exactly the mentioned obvious equation.
Eventually, by multiplying the power level of all users by factor ρ≥1 we have:$$ \forall1\leq i\leq N_{int}\:,\:1\leq m\leq M\qquad I_{I.N.}^{i,m}(\boldsymbol{P})\leq I_{I.N.}^{i,m}(\rho\boldsymbol{P}) $$(15)where \(\boldsymbol {P}=\{p_{i,l_{i}}^{m,j}{\left {\vphantom {\frac {1^{2}}{2\frac {1}{2}}}}\right.}\forall 1\leq i\leq N_{int}\:,\:1\leq m,j\leq M\}\). In this case if we define:$$ \mu_{i,m}=\frac{\alpha_{i,m}^{2}\left(1+4\alpha_{i,m}\gamma_{th}+4\alpha_{i,m}^{2}\gamma_{th}^{2}\right)}{4\alpha_{i,m}^{3}\gamma_{th}h_{i,l_{i}}^{m,m}}\frac{1}{h_{i,l_{i}}^{m,m}} $$(16)Positivity of transmitting power leads to μ _{ i,m }≥0, so we have:$$ \mu_{i,m}I_{I.N.}^{i,m}\left(\rho\boldsymbol{P}\right)\leq\mu_{i,m}I_{I.N.}^{i,m}(\boldsymbol{P})\leq\rho\mu_{i,m}I_{I.N.}^{i,m}(\boldsymbol{P}) $$(17)By satisfying the third condition as well, the uniqueness of obtained Nash equilibrium point is ensured.

4 Power allocation problem for celledge users
As it is mentioned, COMP JT is designed to be applied in celledge region to improve communication quality in celledges in which two or more coordinated BSs are employed to provide service for one celledge user, simultaneously. Indeed, despite central region, we are facing to multicell approach of power allocation problem for celledge users. Therefore, in this section, it is assumed that coordinated cells are clustered together statically. So after describing the system parameters of celledge region and the manner of using CoMPJT technique, the proposed Stackelberg game of central region is extended applying COMP JT technique and finally after evaluating the existence and convergence of extended game, the allocated power to celledge users is calculated.
4.1 Problem specific definitions
System parameters in celledge region
Parameter  Definition 

N _{ out }  Number of users located randomly in celledge region of each cell 
C  Number of clusters 
B  Number of cells belonging to each cluster 
\({\sigma _{0}^{2}}\)  Additive noise power 
\(h_{i,l_{i}}^{{c,c,b}}\)  Channel gain between user i of cluster c on channel l _{ i } and b ^{ t h } BS of its cluster 
\(h_{i,l_{i}}^{{c,j,b}}\)  Channel gain between user i of cluster c on channel l _{ i } and b ^{ t h } BS of cluster j 
\(p_{i,l_{i}}^{{c,c,b}}\)  Transmitted power from BS b of cluster c to its i ^{ t h } user on channel l _{ i } 
\(p_{i,l_{i}}^{{c,j,b}}\)  Transmitted power to user i of cluster c on channel l _{ i } from b ^{ t h } BS cluster j 
\(\gamma _{i,l_{i}}^{c,b}\)  SINR of user i in cluster c on channel l _{ i } received from b ^{ t h } BS of its cluster 
\(\gamma _{i,l_{i}}^{c}\)  Total SINR of user i in cluster c on channel l _{ i } 
γ _{ th }  Minimum target SINR of users 
p _{ max }  Maximum affordable transmitting power 
Similar to central users, celledge ones are also allowed to request their own QoS level from the network. α _{ i,c } is a factor assumed to be determined by celledge user i belonging to cluster c. Indeed, \(\beta _{i}^{c,b}\alpha _{i,c}\gamma _{th}\) denotes the part of target SINR of user i that should be provided by b ^{ t h } BS belonging to cluster c.
4.2 Stackelberg game model for celledge users
where \(\lambda _{i}^{c,b}\) and \(\varphi _{i}^{c,b}\) are the unitprice and the subscription imposed by BS b of cluster c to user i in celledge.
Since each BS of one cluster serves all of the corresponding celledge users and the number of users in celledges of B cells belonging to cluster c is assumed to be N _{ out }, the total number of users receiving services from each BS in a cluster will be B×N _{ out }.
4.3 Nash equilibrium of the extended game
4.3.1 Existence of Nash equilibrium point in proposed Stackelberg game
(25) is apparently always negative. Consequently, it is induced that the utility functions of each celledge user is strictly concave.
4.3.2 Calculating available Nash equilibrium point
which is exactly the same as imposed subscription to central users and obviously has all properties described for \({\varphi _{i}^{m}}\) in Section 3.2.1. Furthermore, it can be easily seen that the utility of the user is more than its target with a small deviation.
4.3.3 Uniqueness of obtained Nash equilibrium point
According to (28), satisfying (32) results in \(\alpha _{i,c}^{2}+4\beta _{i}^{c,b}\alpha _{i,c}^{4}\gamma _{th}^{2}\geq 0\) which is always true.
In order to satisfy the second condition, we show that updating functions are monotone, i.e., their first derivative with respect to imposed interference should become nonnegative which conclude exactly the same as (32), and these are always satisfied.
Thus, the third condition is easily met. As a result, not only proposed algorithm has at least one pure Nash equilibrium point but it is also unique.
5 Simulation results of proposed power allocation in static clusters
where \(d_{i,l_{i}}^{m,j}\) denotes the distance of i ^{ t h } user of cell m to the BS of cell j. Path loss exponent n is set to 3.6 and the value of constant A is fixed to 7.75×10^{−3}. Furthermore, noise variance and P _{ max } are considered to be 5×10^{−15} and 1.5^{ w } respectively for all users. Also γ _{ th } is chosen to be 5, while each user requests its target SINR by determining α from 1 to 2 which is set randomly in our simulations.
5.1 Algorithm convergence and power consumption
Meanwhile, low steady state level of power consumptions associated with users in the network is the second thing that can be educed from Fig. 3. As we observed after the algorithm converged, the level of powers allocated to users are in mW range.
5.2 Affording the target QoS of users
Also, it can be elicited from Fig. 6a that the sum total level of power transmitted from 3 BSs to one celledge user is decreased in comparison with the needed transmitted power shown in Fig. 5a. The reason is that the signal sent from BSs of two closest neighbor cells are exchanged from interfering signal to useful ones. So the required power to provide the target SINR of the user is reduced.
5.3 The effect of user’s location in system performance
Power level consumption of three BSs in each cluster for all three users
User 1  User2  User 3  

BS 1  0.00876  0.0382  1.173×10^{−6} 
BS 2  0.00879  4.3499×10^{−5}  1.1858×10^{−6} 
BS 3  0.00884  0.0382  0.0138 
5.4 The effect of user’s demand for QoS in imposed unitprice
5.5 The effect of number of users in power consumption
Finally it should be noted that since we consider an OFDMA cellular network where frequency bands are orthogonal and intracall interference is ignored, increasing the number of users in central region or in celledge will not cause any changes on last results. In other words, in our architecture the number of users in each associated frequency band of central and celledge regions are equal to the number of cells and the number of clusters respectively, which are both constant. Consequently, the number of users in each cell will not have any effect on the number of users interfering with each other and in turn the level of needed powers will not change as well.
5.6 Algorithm complexity
6 A dynamic clustering algorithm for intercell coordinated networks
So far, we presented the resource allocation algorithm for a coordinated network which is statically clustered for celledge users. With respect to obtained results, despite being so helpful for users located near to center of predetermined clusters, static clustering cannot improve the system performance for all celledge users. In order to overcome this limitation, here we propose a dynamic clustering algorithm which enables the network to serve more celledge users with better available clusters. Indeed, the main goal in defining such dynamic algorithm is to show that conspicuous improvement in system performance can be achieved even by applying simple dynamic clusters.
In this section after introducing the considered system model, our proposed dynamic clustering algorithm is described. Finally simulation results for the proposed method of clustering are presented in which achieved improvement in system performance is evaluated.
6.1 Proposed architecture for dynamic clustering model
6.2 Proposed dynamic clustering algorithm
Hence, the collected information in the network will be presented as a list of pairs in the form of [C _{ d }],N _{ d }] in which N _{ d } is the number of events that C _{ d } has been requested to be formed by users. Undoubtedly, a cluster observed more in users’ reports has got the higher potential to improve the system performance. But due to the presumption of having disjointed clusters, in some cases, simultaneous forming of clusters with the maximum number of requests might be impossible. So the decision must be taken in a way that the final clusters are not only the maximum requested ones, but there must be no common cells between them.
in which X _{ i }={x _{ i,1},x _{ i,2},…} is the set of clusters which can be formed together. Indeed, the weights assigned to each available cluster in (38) equals to the corresponding element in vector N which determines the number of requests by all users for each cluster. Under the second constraint in (38), the sets containing clusters with common cells are discarded. Obviously, the network will benefit more from forming the set of clusters with higher weights. So, among all possible sets, the one that maximizes the objective function of (38) under the mentioned constraints will be determined as the most requested disjoint clusters.
 1.
Step 1: Specify the size of the cluster by setting the parameter B and consider the null set of C _{opt}.
 2.
Step 2: Generate all available clusters with size B, or equivalently C _{ j } ∀1≤j≤D _{ max }, and the vector N containing the demands number associated to each cluster. Also, obtain the matrix Y according to (42).
 3.Step 3: Define the square symmetric matrix X of size D _{ max }×D _{ max } as follows:$$ x_{i,j}=\left\{\begin{array}{ll} 1 & i\neq j,C_{i}\bigcap C_{j}=\text{\"{\i}\textquestiondown}\frac{1}{2}\\ 1 & i=j\\ 0 & \text{o.w.} \end{array}\right.i,j=1,\ldots,D_{max} $$(39)
It means that if there exists a cell which belongs to both C _{ i } and C _{ j } clusters, the corresponding element of these clusters will be set to zero and it will be 1 if there is no common cell between them.
 4.
Step 4: Search for the maximum value in vector N. Find the cluster for which this element is assigned to as the first CoMP cluster and add the index of cluster to C _{opt}. Then, set its corresponding element in vector N to zero.
 5.
Step 5: Choose the C _{ d } cluster corresponding to N _{ d } which is the biggest element in modified vector N as the candidate for next CoMP cluster and set the element of N _{ d } in vector N to zero.
 6.Step 6: Select the candidate cluster as the best one if (40) is satisfied and add its index to C _{opt}. Otherwise, go back to step 5.$$ x_{d,j}=1\qquad\forall j\in C_{\text{opt}} $$(40)
 7.
Step 7: Stop the algorithm if all elements in vector N are equal to zero. Otherwise, go back to step 5.
Finally, it should be noted that the best CoMP clusters can be obtained from the columns of Y matrix which their numbers exist in C _{opt}. Simulation results show that this proposed algorithm completely satisfies our purposes.
6.3 Simulating the dynamic clustering algorithm
In this section the effect of considering the position of users in opting the serving BSs is evaluated through clustering the cells by the proposed dynamic method. In other words, the resource allocation strategy presented in Section 4.2 is applied among users and the role of dynamic clustering in improving the system performance is clearly demonstrated through different simulations.
6.3.1 Cells clustering based on proposed dynamic algorithm
6.3.2 The effect of clustering method in system performance
It should be noted again that although it is possible to find a user with better situation by making the static clusters in use, but number of this type of users will be less than the ones who benefit more from dynamic CoMP clustering. As the result, dynamic clustering method will be able to improve the performance of network and despite imposing higher level of complexity, is preferred to static clustering approach.
6.3.3 The effect of clustering method in power consumption
The interference imposed to the specified user
Clustering method  Interference level 

Statically  19.3095×10^{−14} 
Dynamically  4.291×10^{−14} 
6.3.4 The effect of increasing number of users in power consumption
7 Conclusions
Since in conventional cellular networks, performance of nearedge users may be degraded due to strong interference, CoMPJT is recently introduced as a promising technology to address such issues. In this paper, we proposed a new power allocation algorithm based on game theory for multicell networks. In our model, users are separated according to their distances from BSs of cells they are located in. In order to improve the condition of users in celledges, lower channel reuse factors are assigned to them through clustering and applying CoMPJT technology. As a result, dynamic target SINR levels are adopted for different users and by applying a Stackelberg game model, utility of users and revenue of the service providers are concurrently optimized.
Due to random channel association model used in this paper, the utility of users does not directly depend on allocated frequency channels. Therefore, one direction for future work is to provide a game theory based approach for joint channel assignment and power allocation in coordinated multicell networks. User mobility is also another issue that can be addressed in future extensions of this work.
Declarations
Acknowledgment
The authors would like to thank the anonymous reviewers for their helpful comments. This research was in part supported by a grant from IPM and also partially supported by Iran National Science Foundation (INSF) under contract No. 92/32575.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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