 Research
 Open Access
Cooperative power control in OFDMA small cell networks
 Gaofeng Nie^{1},
 Hui Tian^{1}Email author,
 Jiazhi Ren^{1},
 Jing Wang^{2},
 Liu Liu^{2} and
 Huiling Jiang^{2}
https://doi.org/10.1186/s1363801503047
© Nie et al.; licensee Springer. 2015
 Received: 4 September 2014
 Accepted: 18 February 2015
 Published: 11 March 2015
Abstract
This paper aims at improving the system throughput of orthogonal frequency division multiple access small cell networks. Different with traditional schemes that neglect the cooperation among small cells, a scheme named as resource block exclusionbased power control (RBEBPC) is proposed by sharing the interference correlated information. RBEBPC consists of two steps that are iteratively conducted. First, based on current power allocation results, partial system resource blocks are excluded by playing the formulated cooperative coalition formation games. Second, the transmission power of each small cell is determined by solving a modified throughput maximization problem after the resource block exclusion. As the generated interference is constrained in the second step, part of the small cells transmit without full power. Thereby, the overall system interference keeps nonincreasing after RBEBPC is adopted. Simulation results indicate that about 15% system throughput gain and 13% power saving gain are obtained compared to traditional iterative water filling scheme.
Keywords
 Cooperative power control
 Orthogonal frequency division multiple access
 Small cell
 Coalition game
 Iterative water filling
1 Introduction
The wireless data traffic of cellular system is soaring. Based on the report of future mobile data forecast by Cisco, the compound annual growth rate of mobile data traffic from 2011 to 2016 will be 78% [1]. The wireless industry faces a big challenge: a 1,000fold data traffic increase in a decade [2]. The emergence of small cell releases the cost pressure on the dense cell deployment, which makes it one of the most promising solutions to the explosive wireless traffic growth. Small cell is the node whose transmission power is lower than that of the macro base station [3]. Both picocell and femtocell are included in the concept of small cell. Small cell is cheap, plug and play, and selfconfiguring, which can be easily embedded in the existing cell deployment. The deployment of small cell shortens the distance between service providers and subscribers. As a consequence, subscribers can achieve the same quality of service with lower power consumption. The shorter distance also enables the small cell to use a higher carrier frequency that is not suitable for the macro cell.
However, when small cells share the same spectrum band with macro cells, strong interference may exist. There are two categories of interference in small cell networks. The first category, which is known as crosstier interference, is the interference between small cells and macro cells. The second category, which is known as cotier interference, is the interference among small cells. Strong interference drops the advantages of small cell deployment and becomes the bottleneck of throughput improvement. Some previous works on small cell interference mitigation focus on orthogonal resource reuse from time or frequency domain. Authors of [47] propose orthogonal time domain interference mitigation schemes, e.g., almost blank subframe and the enhanced versions. Authors of [8,9] provide frequency domain interference mitigation scheme, e.g., fractional frequency reuse. The orthogonal resource reuse schemes have the common drawback that the interference is mitigated at the cost of resource utilization. Authors of [10] analyze the development trend of small cell networks and point out that small cell deployment would become denser in the future. Marginal reward would be obtained in the orthogonal resource reuse schemes due to the large quantity of weak interference sources. In addition to orthogonal resource reuse, power control is also considered to mitigate the interference. Authors of [11] provide a pricebased power control scheme to manage the first category of interference. But only one subchannel case is considered. In [12], hierarchical competition schemes for downlink power allocation in orthogonal frequency division multiple access (OFDMA) femtocell networks are proposed. A Stackelberg game model is used to manage the crosstier interference of Femtocell and macro cell. In [13], the potential game is used with a designed payoff function to manage both the cotier and crosstier interferences. Some works adopt the centralized manner to manage the power control issues in small cell networks. Authors of [14] provide centralized suboptimal algorithms to manage the downlink scheduling and power allocation issues in multicell OFDMA networks. Authors of [15] provide a binary power allocation scheme to mitigate the second category of interference but the research is limited to indoor scenario.
As described in [3], the scenario where spectrum bands used for small cells and macro cells are different is one of the scenarios remaining to be studied. Besides, the deployment of small cells are more random and flexible than conventional macro cells, which makes the interference mitigation issues among small cells rather challenging. In this paper, we resort to the cooperative game theory and convex optimization theory to deal with the interference issues in small cell networks. Our target is to establish an effective method to improve the system throughput. Considering the deployment trend [2,3], we only deal with the cotier interference of the small cells. Based on the resource structure of OFDMA small cell networks, the proposed scheme, which is called resource block exclusion based power control (RBEBPC), conducts the following two steps iteratively. 1) Small cells cooperatively exclude the disadvantage resource blocks by using the coalition formation game theory. An interference constraint is obtained based on the channel exclusion results for each small cell. 2) The transmission power of small cell is determined by solving the modified throughput maximization problems. The system total interference is nonincreasing during the conduction of RBEBPC. Besides, RBEBPC is also energy efficient, i.e., not all available power is used during the transmission.
2 System model and basic assumptions
In this paper, bold lower case letters denote vectors and bold upper case letters denote matrices. The element of the ith row and the jth column of X is denoted by [X]_{ ij }. The transpose of X is X ^{ T }. \(\mathbb {R}^{M\times N}\) represents the M×N real space and represents the real line. Calligraphic upper case letters denote sets. The cardinal number of is denoted by \(\vert \mathcal {S}\vert \), and the power set of is denoted by \(\eta (\mathcal {S})\). Notation \(\left (\begin {array}{l}n\\k\end {array}\right)\)represents ‘n choose k’ and equals \(\frac {n!}{k!(nk)!}\). Notation ⇔ means ‘if and only if’. Finally, ‘s.t.’ is the abbreviation of ‘subject to’.
The wireless channel is assumed to vary in a slower pace compared to our power control scheme, i.e., the channel gain during our analysis remains constant. The channel gain between the ith UE and the jth small cell on the lth subchannel is denoted by \(G_{\textit {ij}}^{l}\). The system channel gain matrix is denoted by \(\mathbf {G}\in \mathbb {R}^{M\times N\times L}\). The channel gain matrix related to the ith UE is defined as \(\mathbf {G}_{i \cdot }^{\cdot }\in \mathbb {R}^{N\times L}\), which contains all the channel gain between the ith UE and all the small cells on all the subchannels. Similarly, the channel gain related to the jth small cell and the lth subchannel are defined as \(\mathbf {G}_{\cdot j}^{\cdot }\in \mathbb {R}^{M\times L}\) and \(\mathbf {G}_{\cdot \cdot }^{l}\in \mathbb {R}^{M\times N}\) separately.
For the ith small cell, the power allocation result is denoted by \(\mathbf {p}_{i}=[{p_{i}^{1}},{p_{i}^{2}},\dots,{p_{i}^{L}}]^{T}\in \mathbb {R}^{L\times 1}\), where \({p_{i}^{l}}\) represents the allocated power level on the lth subchannel. The system power allocation result is denoted by \(\mathbf {P}=[\mathbf {p}_{1},\mathbf {p}_{2},\dots,\mathbf {p}_{N}]\in \mathbb {R}^{L\times N}\).
3 Motivation of cooperative power control
Since P 1 is nonconcave to its domain, it is difficult to obtain the global optimal solution. Some suboptimal solutions concentrate on the noncooperative manner such as the iterative water filling (IWF) to slove P 1. Due to the neglect of possible mutual cooperation, the noncooperative suboptimal solutions to P 1 may not be efficient. To see this, a simple example of small cell network that motivates the cooperation is analyzed below.
and the corresponding throughput is (R _{1},R _{2})=(6.2668,5.6147), which is comparable to the full transmission power allocation results \(\hat {\mathbf {P}}\). The great improvements of \(\hat {\mathbf {P}}\) and \(\tilde {\mathbf {P}}\) motivate the cooperation of small cell networks where strong interference exists. Besides, for the noncooperative solutions such as IWF, the convergence is not always guaranteed [16].
In this paper, we aim at achieving such cooperation gain. Some papers seek cooperation solution to the interference issues in small cells via coalition game. Authors of [17] use the recursive core approach to cluster the small cells. Small cells first form clusters by using the recursive core approach. To mitigate the interference, a time division duplex scheduling strategy is used in the formed clusters. The scheme proposed in [18] promotes the scheme in [17] by allowing a small cell to join more than one coalitions. There are obvious differences between the schemes in [17,18], and RBEBPC which will be introduced in the next section. First, both the schemes proposed in [17,18] are distributed schemes, while RBEBPC is a centralized solution. Second, no power control is used in the schemes of [17,18], while RBEBPC uses the power control method to deal with the interference.
4 Resource block exclusionbased power control
In RBEBPC, the goal is to improve the system throughput by excluding the strong interference resource blocks and by redistributing the system interference among small cells. To conduct RBEBPC, some mutual interference correlated information, such as G and previous P, should be exchanged among small cells. Considering the large number of small cells and the backhaul ability of each small cell, the centralized cooperation structure is adopted to guarantee the efficiency of information exchange. The macro cell, which covers all the small cells in the system, acts as the centralized node to collect and forward all the necessary information. The procedure of RBEBPC is described in Algorithm 1.
In order to obtain G, in the initialization state, all small cells transmit with equal power on all available subchannels (line 3 in Algorithm 1). Then, the macro cell collects and forwards the necessary part of G to the small cells (line 4 and line 5). After information collection, small cells will conduct the cooperative power control for K rounds. In each round, small cells first report the used power levels to the macro cell (line 7). The macro cell, based on the reported power level, plays the resource block exclusion game and calculates the interference constraint based on the resource block exclusion results for each small cell (step 1). Then each small cell solves a throughput maximization problem by following the received constraint from the macro cell (step 2).
The necessary signaling overhead among the small cells, the serving UEs, and the macro cell in RBEBPC scheme can be estimated as follows. In the Basic information collection phase, if the quantification of a subchannel gain of a certain small cell needs α bits, each UE should feedback α N L bits to the access small cell to indicate the interfering channel gain on all the subchannels. Then, each small cell forwards the α N L bits to the macro cell via the backhaul link. Each small cell receives α N L bits information to indicate the channel gains correlated to the interfering UEs in the system. In the Cooperation based power control phase, each small cell forwards β L bits to indicate the previous power level in step 1, where β is the necessary bits to quantify the power level on a subchannel. Each small cell receives NL bits that indicate the resource block exclusion results and receives γ(L+1) bits that indicate the suffered interference on each subchannel and the generated total interference, where γ is the necessary bits to quantify the interference level on a subchannel. So in Algorithm 1, the signaling overhead between a UE and the serving small cell is α N L bits. The signaling overhead between a small cell and the macro cell is 2α N L+K(β L+N L+γ(L+1)) bits. As for the typical wireless channel, the time scale for change of path amplitude is several hundreds of millisecond [19]. In longterm evolution (LTE), the transmission time interval is 1 ms, which is much smaller than the time of path amplitude change. So in the extreme case, the backhaul link between the small cell and the macro cell should satisfy the data rate max{α N L,β L,N L+γ(L+1)} kbps. For the typical configuration α=β=γ=64, N=10, L=50, the backhaul link data rate should satisfy 0.032 Gbps. If the backhaul links are implemented by using the standardized passive optical network (PON) [20] systems whose downstream is about 2.5 Gbps, the signaling overhead of RBEBPC is acceptable in practice.
In the following subsections, the resource block exclusion, interference calculation, and power optimization in RBEBPC are described in detail.
4.1 Resource block exclusion
For the jth small cell, the lth subchannel is available if and only if [A]_{ lj }=1. Otherwise, [A]_{ lj }=0 and the jth small cell does not use the lth subchannel.
In Figure 2, the total available system resource blocks can be partitioned into \(\mathcal {C}=\{\mathcal {N}_{1},\mathcal {N}_{2},\dots,\mathcal {N}_{L}\}\), where \(\vert \mathcal {C}\vert =L\) and \(\mathcal {N}_{l}\subseteq \mathcal {N}\) denotes the set of small cells for which the lth subchannel is available (i.e., \(\mathcal {N}_{l}=\{j\in \mathcal {N}[\mathbf {A}]_{\textit {lj}}=1\}\)). Given a subchannel, the allocated power is not transferable, i.e., the power cannot be shared among small cells in an arbitrary ratio. For the subchannels with strong interference, the system throughput will improve if some small cells mute the subchannels. In this subsection, we resort to the coalition game to determine the subchannels that need to be muted. Note that the system interference on the lth subchannel is only determined by the small cells in \(\mathcal {N}_{l}\). Besides, the system interferences of different subchannels are independent once P is given. So in the following analysis, we only consider the lth subchannel and the correlated small cells in \(\mathcal {N}_{l}\).
Definition 1.
Let \(\mathcal {N}=\{1,2,\dots,N\}\) be a set of fixed players called the grand coalition. Nonempty subsets of are called coalitions. A collection (in the grand coalition ) is any family \(\mathcal {D}=\{\mathcal {D}_{1},\mathcal {D}_{2},\dots,\mathcal {D}_{s}\}\) of mutually disjoint coalition, and s is called its size. If additionally \(\mathop \cup \limits _{t=1}^{s}\mathcal {D}_{t}=\mathcal {N}\), the collection is called a partition of .
Note that for ∅, we have v(∅)=0. Based on Equation 13, the establishment of coalition (\(\vert \mathcal {E}\vert \ge 2\)) has two effects. The first effect is the reduction of transmission resource blocks for small cells in . Before the formulation of , small cells in can have at most \(\vert \mathcal {E}\vert \) transmission resource blocks (i.e., each small cell owns a transmission resource block) and at least two transmission resource blocks (i.e., is established based on two existing coalitions). However, when is formulated, small cells in only have one transmission resource block. The second effect is the reduction of interference of the remaining transmission resource block. Since the small cells in mute in a cooperative manner, the interference of small cells in is canceled.
Some properties of \(\mathcal {G}^{l}\) are summarized as follows.
Definition 2.
The form of a coalition game \((\mathcal {N},v)\) is the characteristic form, if and only if the value of a coalition \(\mathcal {E}\in \eta (\mathcal {N})\) solely depends on the players in and is in independent with how the players in \(\mathcal {N}\backslash \mathcal {E}\) are structured.
Property 1.
The form of game \(\mathcal {G}^{l}\) is the characteristic form.
Proof.
This property directly follows by the definition of payoff function (Equation 13).
From Property 1, we can see that the value of the formulated payoff function is only sensitive to the players in the given coalition. In small cell networks, small cells are coupled with each other via the effect of interference. In (Equation 13), the effect of small cells beyond the given coalition is treated as a whole and is independent with their partition structure. By using Equation 13, we can obtain the value of a coalition once the coalition is given.
Given a partition \(\mathcal {P}=\{\mathcal {P}_{1},\mathcal {P}_{2},\dots,\mathcal {P}_{s}\}\) of \(\mathcal {N}_{l}\), we can find a unique vector \(\mathbf {v}=[v(\mathcal {P}_{1}),v(\mathcal {P}_{2}),\dots,v(\mathcal {P}_{s})]^{T}\in \mathbb {R}^{s\times 1}\) to represent the value of each coalition in . Besides, since the coalition value of \(\mathcal {G}^{l}\) is the maximum achievable throughput by a single member small cell, game \(\mathcal {G}^{l}\) has a transferable utility, i.e., the achievable throughput can be arbitrarily portioned among the small cells of a coalition (for example, via the proper choice of coding strategy [18]).
Definition 3.
For the transferable utility coalition game with supperadditivity, establishing a bigger coalition is always beneficial.
Property 2.
Due to the implied tradeoff between interference and throughput in (Equation 13), the formulated game \(\mathcal {G}^{l}\) is not supperadditive and the grand coalition is not always formed; thus, disjoint independent coalitions will form in the network.
The proof of Property 2 can be found in the Appendix. The payoff function (Equation 13) shows that in the formulated coalition, only one player transmits and the rest in the coalition keep silent. Once a coalition is formulated, the available number of resource blocks reduces but the quality of the remaining resource block improves. Due to the random deployment of small cell and the fluctuation of the wireless channel quality, small cells may not be prone to form the grand coalition. In smallscale area where strong interference exists, small cells form a coalition with a higher possibility since the payoff of the coalition is probably higher if the interference is removed. The main obstacle of the grand coalition construction lies in (Equation 13) because the reduction of the available resource blocks is the cost of coalition construction. In practice, the construction of grand coalition needs harsh conditions. In the case that the interferences among small cells are so strong that if more than one small cell share the subchannel the total throughput of the subchannel reduces, the grand coalition will form.
Definition 4.
A comparison relation ⊳ is defined for two collections \(\mathcal {D}=\{\mathcal {D}_{1},\mathcal {D}_{2},\dots,\mathcal {D}_{s}\}\) and \(\mathcal {F}=\{\mathcal {F}_{1},\mathcal {F}_{2},\dots,\mathcal {F}_{w}\}\) that satisfy \(\mathop \cup \limits _{m=1}^{s}\mathcal {D}_{m}=\mathop \cup \limits _{n=1}^{w}\mathcal {F}_{n}=\mathcal {H}\subseteq \mathcal {N}\). Thus, \(\mathcal {D}\triangleright \mathcal {F}\) means that the way partitions is preferred to the way partitions .
Definition 5.
Definitions 4 and 5 provide a preference among partitions. Definition 5 indicates that ‘social welfare’ (the total throughput) is considered as the baseline. So the defined preference is coordinate with the target of system throughput improvement. We can use the exhaustive search method to obtain the maximum throughput partition structure of \(\mathcal {G}^{l}\), in which a rather large search space should be considered. For the player set \(\mathcal {N}_{l}\), the number of possible partition, which is given by a value known as Bell number, grows sharply with the number of players in \(\mathcal {N}_{l}\). For example, the Bell number of \(\mathcal {N}_{l}\) is 115,975 when \(\vert \mathcal {N}_{l}\vert =10\). However, by following some simple rules, we can obtain a stable practical partition structure of \(\mathcal {G}^{l}\).
For any two collections \(\mathcal {D}=\{\mathcal {D}_{1},\mathcal {D}_{2},\dots,\mathcal {D}_{s}\}\), \(\mathcal {F}=\{\mathcal {F}_{1},\mathcal {F}_{2},\dots,\mathcal {F}_{w}\}\) and the grand coalition that satisfy \((\mathop \cup \limits _{m=1}^{s}\mathcal {D}_{m})\cup (\mathop \cup \limits _{n=1}^{w}\mathcal {F}_{n})=\mathcal {N}\) and \((\mathop \cup \limits _{m=1}^{s}\mathcal {D}_{m})\cap (\mathop \cup \limits _{n=1}^{w}\mathcal {F}_{n})=\emptyset \), we define two operation rules to construct the stable partition structure.
Definition 6.
Definition 7.
Note that the above operation rules of Definitions 6 and 7 use ⊳ ‘locally’, by focusing on the coalitions that take part and result from the merge operation and split operation. Algorithm 2 summarizes the procedure that uses the mergesplit rule to obtain the stable partition structure of \(\mathcal {G}^{l}\). Due to Property 1, the conduction of the split rule in Algorithm 2 can be transformed into the merge rule, if the coalition remains to be splitted is treated as a smaller grand coalition. The split operation is equivalent to the merge operation in the smaller grand coalition as long as we treat the effect of the players beyond the smaller grand coalition as a whole.
It is difficult to directly describe the complexity of Algorithm 2, because the termination condition of Algorithm 2 depends on the specific state of the small cell network. But we can estimate the complexity of Algorithm 2 in some extreme cases. Since the split operation can be treated as a special kind of merge operation, we only analyze the complexity of merge operation here. In the worst case where no coalition with size bigger than two is formed, the potential number of possible coalitions that should be considered is \(\zeta _{1}=2^{\vert \mathcal {N}_{l}\vert }\vert \mathcal {N}_{l}\vert 1=\sum \limits _{k=2}^{\vert \mathcal {N}_{l}\vert }\begin {pmatrix}\vert \mathcal {N}_{l}\vert \\k\end {pmatrix}\). While in the best case where the grand coalition is formed, the potential number of possible coalitions that should be considered is \(\zeta _{2}=\vert \mathcal {N}_{l}\vert 1\). So for each merge or split operation, the number of coalitions that should be considered is between ζ _{1} and ζ _{2}.
Definition 8.
A partition \(\mathcal {P}=\{\mathcal {P}_{1},\mathcal {P}_{2},\dots,\mathcal {P}_{s}\}\) of the grand coalition is \(\mathbb {D}_{c}\) stable, if no players in are interested in leaving through any operation (not necessary merge or split) to form a partition different with . A partition \(\mathcal {Q}=\{\mathcal {Q}_{1},\mathcal {Q}_{2},\dots,\mathcal {Q}_{w}\}\) of the grand coalition is \(\mathbb {D}_{\textit {hp}}\) stable if no coalition has the incentive to split or merge.
Property 3.
The partition structure of \(\mathcal {G}^{l}\) obtained by Algorithm 2 is \(\mathbb {D}_{\textit {hp}}\) stable. The existence of \(\mathbb {D}_{c}\) stable partition in game \(\mathcal {G}^{l}\) is not always guaranteed. If the \(\mathbb {D}_{c}\) stable partition of \(\mathcal {G}^{l}\) exists, the partition structure obtained by Algorithm 2 is \(\mathbb {D}_{c}\) stable.
The proof of Property 3 can be found in the Appendix. Property 3 indicates that the final outcome of Algorithm 2 is stable. The outcome of Algorithm 2 can not be improved by any merge or split operation. Generally speaking, the optimal partition results can be found by using the exhaustive method. But the number of possible cases (the Bell number) that should be considered in the exhaustive method is too huge to manage. The outcome of Algorithm 2 may not be globally optimal but it is at least stable.
4.2 Interference calculation
The generated interference depends on three factors: the previous allocated power vector, the interference channel gain and the channel availability. If the subchannel is unavailable for some small cells, the interference to the small cell on this subchannel has no contribution to the generated interference. Based on the system channel gain matrix G, previous system power allocation results P and system current mapping matrix A, the macro cell calculates the generated interference vector \(\bar {\mathbf {I}}=[\bar {I}_{1},\bar {I}_{2},\dots,\bar {I}_{N}]^{T}\in \mathbb {R}^{N\times 1}\) and the suffered interference vector \(\tilde {\mathbf {I}}_{j}=[\tilde {I}_{j}^{1},\tilde {I}_{j}^{2},\dots,\tilde {I}_{j}^{L}]^{T}\in \mathbb {R}^{L\times 1}\) (\(\forall j\in \mathcal {M}\)) after the subchannel exclusion operation. The two interference correlated vectors are delivered to the small cells to be used in the power optimization procedure.
4.3 Power optimization
where \(\mathcal {L}_{j}\) is the set of available subchannels for the jth small cell (i.e., \(\mathcal {L}_{j}=\{l\in \mathcal {L}[\mathbf {A}]_{\textit {lj}}=1\}\)). The constraint of (Equation 22) is used to redistribute the total generated interference on all the available subchannels. The formulated P 2 is a concave problem and the proof can be found in the Appendix.
It is easy to verify that P 3 is a convex optimization problem. By using the KKT conditions, constraints (Equation 28) can be neglected, i.e., we can first solve P 3 without constraints (Equation 28) and then using (Equation 28) to examine the correctness of the solution. So P 3 can be efficiently solved by using the Newton Method, which is a method suitable for the convex optimization problem without constraints [22]. In some cases, we can obtain the close form optimal solution (λ ^{∗},μ ^{∗}) to P 3.
Case 1.
Case 2.
When the optimal (λ ^{∗},μ ^{∗}) is obtained, we can use Equation 26 and solve \(\frac {\partial L(\lambda ^{*},\mu ^{*},\mathbf {p}_{j})}{\partial {p_{j}^{l}}}=0\) to obtain the optimal \( {p}_{j}^{l*}\). Due to the elimination of θ, we must verify the correctness of the power results. If \( p_{j}^{l*}\ge 0\) for all \(l\in \mathcal {L}_{j}\), we can conclude that \(\mathbf {p}_{j}^{*}\) is the solution to P 2. If some elements of \(\mathbf {p}_{j}^{*}\) are negative, we must remove these subchannels with the minimum power and solve P 3 again until a solution \({p}_{j}^{l*}\ge 0\) for all \(l\in \mathcal {L}_{j}\) is found.
The power optimization procedure is summarized in Algorithm 3. It is difficult to analyze the complexity of Algorithm 3 directly. But the number of iteration times of the Newton Method of Algorithm 3 can be estimated in some extreme cases. In the best cases where the optimal value can be achieved by (Equation 29) or (Equation 30), the iteration time of the Newton Method is one. In the worst case where the optimal value is obtained without using these close form equations, the maximum number of iterations is bounded by \(\frac {g(\lambda ^{0},\mu ^{0})g(\lambda ^{*},\mu ^{*})}{\tau }+6\) [22], where λ ^{0} and μ ^{0} separately represent the initialized value of λ and μ, respectively, and τ is the maximum reduction value of function g(λ,μ) during the iterations.
5 Simulations and discussions
Simulation parameters
Parameters  Value 

N  10 
M  10 
F2 (MHz)  1.25/5 
L  6/25 
d (m)  100 to 160 
(R(m),r(m))  (15,10), (20,15), (25,20), (20,10) 
System carrier frequency (GHz)  3.5 
Noise power density (dBm/Hz)  174 
P0 (dBm)  14/20 
Path loss model  Table A.2.1.1.23 in [23] 
5.1 Track of RBEBPC
In this subsection, the system performance of RBEBPC based on the topology of Figure 5 is analyzed. For convenience, the value of K in Algorithm 1 equals 10 and the number of subchannels is L=6. The maximum power of each small cell is 14 dBm. The distance pair (R,r) equals (20,10) and bandwidth F2 equals 1.25 M.
Resource exclusion results, k=1
l  \(\boldsymbol {\mathcal {P}^{*}}\) 


1  {1}{2}{4}{5}{8}{9}{3,6}{7,10}  {1,2,4,5,8,9,3,10} 
2  {2} {4} {5} {8} {1,9} {3,6} {7,10}  {2,4,5,8,1,3,10} 
3  {1} {2} {3} {4} {5} {9} {10} {6,7,8}  {1,2,3,4,5,9,10,8} 
4  {5} {7} {8} {10} {1,9} {2,4} {3,6}  {5,7,8,10,1,4,6} 
5  {4} {8} {9} {1,5} {3,6} {2} {7,10}  {4,8,9,1,3,2,10} 
6  {1} {4} {5} {8} {9} {10} {2,7} {3,6}  {1,4,5,8,9,10,2,3} 
Resource exclusion results, k=2
l  \(\boldsymbol {\mathcal {P}^{*}}\) 


1  {1}{2}{3}{4}{5}{8}{9}{10}  {1,2,3,4,5,8,9,10} 
2  {1} {2} {3} {4} {5} {8} {10}  {1,2,3,4,5,8,10} 
3  {1} {2} {3} {4} {5} {8} {9} {10}  {1,2,3,4,5,8,9,10} 
4  {1} {4} {5} {6} {7} {8} {10}  {1,4,5,6,7,8,10} 
5  {1} {2} {3} {4} {8} {9} {10}  {1,2,3,4,8,9,10} 
6  {1} {3} {4} {5} {9} {10} {2,8}  {1,3,4,5,9,10, 8} 
Power allocation results of IWF (Watt)
l =1  l =2  l =3  l =4  l =5  l =6  Total  

SC1  0.0043  0.0041  0.0046  0.0045  0.0042  0.0034  0.0251 
SC2  0.0025  0.0065  0.0077  0  0.0034  0.0050  0.0251 
SC3  0.0044  0.0033  0.0046  0.0037  0.0047  0.0044  0.0251 
SC4  0.0042  0.0040  0.0041  0.0043  0.0043  0.0042  0.0251 
SC5  0.0049  0.0049  0.0049  0.0052  0  0.0052  0.0251 
SC6  0.0041  0.0092  0.0094  0.0025  0  0  0.0251 
SC7  0.0041  0.0131  0  0.0080  0  0  0.0251 
SC8  0.0050  0.0003  0.0050  0.0050  0.0048  0.0050  0.0251 
SC9  0.0039  0.0043  0.0047  0.0029  0.0049  0.0044  0.0251 
SC10  0.0043  0.0036  0.0044  0.0043  0.0043  0.0044  0.0251 
Power Allocation Results of RBEBPC (Watt)
l =1  l =2  l =3  l =4  l =5  l =6  Total  

SC1  0.0041  0.0044  0.0043  0.0045  0.0041  0.0038  0.0251 
SC2  0.0026  0.0046  0.0028  0  0.0124  0  0.0224 
SC3  0.0055  0.0026  0.0056  0  0.0069  0.0046  0.0251 
SC4  0.0042  0.0040  0.0041  0.0043  0.0042  0.0043  0.0251 
SC5  0.0032  0.0046  0.0077  0.0052  0  0.0044  0.0251 
SC6  0  0  0  0.0042  0  0  0.0042 
SC7  0  0  0  0.0042  0  0  0.0042 
SC8  0.0042  0.003  0.0045  0.0040  0.0040  0.0047  0.0251 
SC9  0.0110  0  0.0062  0  0.0042  0.0021  0.02344 
SC10  0.0042  0.0041  0.0042  0.0042  0.0041  0.0042  0.0251 
Compare Table 4 to Table 5 and we can see that RBEBPC is energysaving, i.e., not all small cells transmit with the total available power. The small cells can be divided into three groups based on the two tables. The first group (including small cells 2, 6, 7, and 9) consists of the small cells whose total transmit power and available resource blocks vary greatly in RBEBPC compared to IWF. The small cells belong to the first group follow the constraint (Equation 22) and transmit without full power. The second group (including small cell 3) consists of the small cells that use all the available power but different resource blocks compared to IWF. Small cells in the second group are influenced by the resource block exclusion operation. The disadvantage resource blocks are excluded. The third group (including small cell s1, 4, 5, 8, and 10) consists of the small cells that only have different power on all the subchannels compared with IWF. In the third group of small cells, the power allocation difference among RBEBPC and IWF comes from the other small cells because all the small cells in the system are influenced by the interference. The power allocation variation of the other two groups leads to the power allocation result of the third group.
5.2 Average performance of RBEBPC
In this subsection, we will provide the average performance of RBEBPC. The used system parameters are P0=20 dBm, F2=5 MHz, L=25. The proposed overlapping coalition formation (OCF) scheme in [18] is also simulated as the comparison scheme. Note that when we conduct the OCF, the power limitation of coalition formation is removed, i.e., all the small cells in the simulation areas have the potentials to form coalition.
6 Conclusions
This paper focuses on the system throughput improvement via cooperative power control in OFDMA small cell networks. Based on the system resource structure, a cooperative power control scheme named as RBEBPC is proposed. Two steps are iteratively conducted in RBEBPC. First, the small cells play L independent coalition formation games to cooperatively determine the exclusion of resource blocks. Interference constraints are calculated based on the exclusion results for each small cell. Second, each small cell solves a modified throughput maximization problem to determine the power level on each subchannel. By following the interference constraints, the system total interference is nonincreasing and part of the small cells transmit without full power. Simulation results show that both system throughput and energy consumption are improved in RBEBPC compared to the traditional IWF scheme. However, the conduction of RBEBPC relies on the backhaul link ability between the small cell and the macro cell. The performance improvement is based on the performance sacrifice of partial small cells, which may lead to the unfairness of the network. Both signaling overhead optimization and small cell fairness will be considered in our future study.
7 Appendix
7.1 Proof of Property 2
Proof.
Consider two coalitions, \(\mathcal {E}_{1}\subseteq \mathcal {N}_{l}\), \(\mathcal {E}_{2}\subseteq \mathcal {N}_{l}\) and \(\mathcal {E}_{1}\cap \mathcal {E}_{2}=\emptyset \), which satisfy a) slightly interference with each other such that \(\tilde {R}(\mathcal {E}_{1}\cup \mathcal {E}_{2})\le R(\mathcal {E}_{1}\cup \mathcal {E}_{2})=R(\mathcal {E}_{1})+R(\mathcal {E}_{2})\) and b) strong interference within each coalition such that \(\tilde {R}(\mathcal {E}_{1})> R(\mathcal {E}_{1})\) and \(\tilde {R}(\mathcal {E}_{2})>R(\mathcal {E}_{2})\). In this case, \(v(\mathcal {E}_{1}\cup \mathcal {E}_{2})=0<\tilde {R}(\mathcal {E}_{1})+\tilde {R}(\mathcal {E}_{2})=v(\mathcal {E}_{1})+v(\mathcal {E}_{2})\), which means game \(\mathcal {G}^{l}\) is not supperadditive.
7.2 Proof of Property 3
Proof.
The \(\mathbb {D}_{\textit {hp}}\) stability of partition structure obtained by Algorithm 2 follows by the termination condition of Algorithm 2. The necessary and sufficient conditions for \(\mathbb {D}_{c}\) stability of a partition can be found in [24]. Since the partition structure of \(\mathcal {G}^{l}\) is based on location and wireless channel state which are all random variables, these conditions are not always satisfied. But if \(\mathbb {D}_{c}\) stable partition of \(\mathcal {G}^{l}\) exists, the mergesplit rule converges to this partition because the optimal \(\mathbb {D}_{c}\) stable partition is the unique outcome of any arbitrary mergesplit rule operation [24].
7.3 Concavity Proof of P 2
Proof.
where \({\delta _{j}^{l}}=\frac {W}{L\ln 2}\frac {1}{({p_{j}^{l}}+\frac {\sigma ^{2}+\tilde {I}_{j}^{l}}{G_{\textit {jj}}^{l}})^{2}},\forall l\in \mathcal {L}_{j}\). Note that when we calculate the Hessian, the interference on each subchannel is processed as a constant. Since the Hessian of the target function of P 2 is a diagonal matrix and each element in the diagonal is negative, the Hessian is negative semidefinite. Based on the second order conditions [22], we conclude that the target function of P 2 is a concave function. So P 2 is a concave optimization problem.
Declarations
Acknowledgements
This work was supported by the ‘Radio Resource Management for Small Cell Deployment’ Project of DOCOMO Beijing Communications Laboratories Co., Ltd, the National Natural Science Foundation of China (Grant No.61471060) and the Funds for Creative Research Groups of China (Grant No.61421061).
Authors’ Affiliations
References
 Cisco Visual Networking Index: Global Mobile Data Traffic Forecast Update, 2011–2016. http://www.cisco.com/en/US/solutions/collateral/ns341/ns525/ns537/ns705/ns827/white_paper_c11520862.html.
 I Hwang, B Song, SS Soliman, A holistic view on hyperdense heterogeneous and small cell networks. IEEE Commun. Mag. 51, 20–27 (2013).View ArticleGoogle Scholar
 TR 36.932, Scenarios and Requirements for Small Cell Enhancements for EUTRA and EUTRAN (Release 12) (2012).Google Scholar
 MI Kamel, KMF Elsayed, in Global Telecommunications Conference. Performance evaluation of a coordinated timedomain eICIC framework based on ABSF in heterogeneous LTEadvanced networks (IEEE Piscataway,New Jersey, USA, 2012), pp. 5326–5331.Google Scholar
 L Jiang, M Lei, in Personal Indoor and Mobile Radio Communications. Resource allocation for eICIC scheme in heterogeneous networks (IEEE Piscataway,New Jersey, USA, 2012), pp. 448–453.Google Scholar
 J Oh, Y Han, in Personal Indoor and Mobile Radio Communications. Cell selection for range expansion with almost blank subframe in heterogeneous networks (IEEE Piscataway,New Jersey, USA, 2012), pp. 653–657.Google Scholar
 J Pang, J Wang, D Wang, G Shen, Q Jiang, J Liu, in Wireless Communications and Networking Conference. Optimized timedomain resource partitioning for enhanced intercell interference coordination in heterogeneous networks (IEEE Piscataway,New Jersey, USA, 2012), pp. 1613–1617.Google Scholar
 S Uygungelen, G Auer, Z Bharucha, in Vehicular Technology Conference. Graphbased dynamic frequency reuse in femtocell networks (IEEE Piscataway,New Jersey, USA, 2011), pp. 1–6.Google Scholar
 W Noh, W Shin, C Shin, K Jang, Choi Hh, in Wireless Communications and Networking Conference. Distributed frequency resource control for intercell interference control in heterogeneous networks (IEEE Piscataway,New Jersey, USA, 2012), pp. 1794–1799.Google Scholar
 T Nakamura, S Nagata, A Benjebbour, Y Kishiyamaand, T Hai, S Xiaodong, Y Ning, L Nan, Trends in small cell enhancements in LTE advanced. IEEE Commun. Mag. 51, 98–105 (2013).View ArticleGoogle Scholar
 P Li, Y Zhu, in Personal Indoor and Mobile Radio Communications. Pricebased power control of femtocell networks: a Stackelberg game approach (IEEE Piscataway,New Jersey, USA, 2012), pp. 1185–1191.Google Scholar
 S Guruacharya, D Niyato, DI Kim, E Hossain, Hierarchical competition for downlink power allocation in OFDMA femtocell networks. IEEE Trans. Wireless Comm. 12, 1543–1553 (2013).View ArticleGoogle Scholar
 L Giupponi, C Ibars, in Personal Indoor and Mobile Radio Communications. Distributed interference control in OFDMAbased femtocells (IEEE Piscataway,New Jersey, USA, 2010), pp. 1201–1206.View ArticleGoogle Scholar
 L Venturino, N Prasad, X Wang, Coordinated scheduling and power allocation in downlink multicell OFDMA networks. IEEE Trans. Veh. Tech. 58, 2835–2848 (2009).View ArticleGoogle Scholar
 J Kimm, DH Cho, A joint power and subchannel allocation scheme maximizing system capacity in indoor dense mobile communication systems. IEEE Trans. Veh. Tech. 59, 4340–4353 (2010).View ArticleGoogle Scholar
 KW Shum, KK Leung, CW Sung, Convergence of iterative waterfilling algorithm for Gaussian interference channels. IEEE J. Sel. Area Comm. 25, 1091–1100 (2007).View ArticleGoogle Scholar
 F Pantisano, M Bennis, W Saad, R Verdone, M Latvaaho, in Wireless Communications and Networking Conference. Coalition formation games for femtocell interference management: a recursive core approach (IEEE Piscataway,New Jersey, USA, 2011), pp. 1161–1166.Google Scholar
 Z Zhang, L Song, Z Han, W Saad, Z Lu, in Wireless Communications and Networking Conference. Overlapping coalition formation games for cooperative interference management in small cell networks (IEEE Piscataway,New Jersey, USA, 2013), pp. 643–648.Google Scholar
 D Tse, P Viswanath, Fundamentals of Wireless Communication (Cambridge University Press, USA, 2004).Google Scholar
 CS Ranaweera, PP Iannone, KN Oikonomou, KC Reichmann, RK Sinha, Design of costoptimal passive optical networks for small cell backhaul using installed fibers. IEEE/OSA J. Opt. Commun. Netw. 5, 230–239 (2013).View ArticleGoogle Scholar
 W Saad, Z Han, M Debbah, A Hjorungnes, T Basar, Coalitional game theory for communication networks. IEEE Signal Process. Mag. 26, 77–97 (2009).View ArticleGoogle Scholar
 S Boyd, L Vandenberghe, Convex Optimization (Cambridge University Press, USA, 2004).View ArticleMATHGoogle Scholar
 3GPP, TR 36.814, Further advancements for EUTRA physical layer aspects (Release 9) (2010).Google Scholar
 KR Apt, T Radzik, in arXiv:cs/0605132v1 [cs.GT]. Stable partitions in coalitional games (New York, USA, May, 2006).Google Scholar
Copyright
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.