Introduction

With the development of wireless communications, it becomes more and more important to make full use of scarce spectrum resource. Future wireless mobile communication system is to provide better service for users, guarantee quality of service, and support more reliable transmission. At the same time, it should also provide higher data rate and spectrum utilization. As reported in [1], 70% data services and more than 50% voice services take place in the indoor environment. Thus, it is necessary to enhance the indoor coverage and connect the data services with voice services more efficiently.

Femtocell, also named home base station, is a short-range, low-cost, and low-power wireless data access point installed by the consumers inside home or buildings for better indoor voice and data service. Femtocell is connected to the mobile operator’s network by a physical broadband connection such as digital subscriber line[1]. It can expand coverage and increase spectral efficiency. Interference management is necessary for the application of femtocell. The interference in the femtocell network includes the inter-femtocell interference among femtocells and the cross-tier interference between macrocell and femtocells.

In recent years, power control is used for mitigating the cross-tier interference. In [2], a utility-based adaptive algorithm was proposed to reduce the cross-tier interference by regulating the signal-to-interference-plus-noise ratio (SINR). In [3], the authors proposed a distributed power control method based on Stackelberg game to mitigate the interference. In [4], the authors developed an opportunistic power control algorithm to mitigate the aggregate interference from active femtocells in each cluster. However, these works only deal with the cross-tier interference. In fact, the inter-femtocell interference is serious when the femtocell is densely deployed. To resolve this problem, some inter-femtocell interference mitigation schemes are proposed. In [5], the authors jointly considered the power and sub-channel allocation to reduce the inter-femtocell interference. A distributed dynamic inter-femtocell interference mitigation scheme was studied in [6] to guarantee high SINR for each user. These works require information exchanges and increase the network overhead. A novel clustering scheme named power and coverage-aware clustering was proposed in [7]. In [8], the authors used a cluster head to determine the resource allocation within each cluster and resolved the collisions among clusters. The energy consumption of femtocell cluster head is high, and only local optimization is achieved by this method. In [9], the femtocells are divided into clusters, and the spectrum resource is allocated to each cluster. However, some femtocells cannot meet the minimum rate constraints because of the limited resource. Thus, it is necessary to allocate the bandwidth to meet the minimum rate constraint of each cluster.

In this paper, we propose a cluster-based bandwidth allocation algorithm (CBBAA) based on Nash bargaining solution (NBS) to mitigate the inter-interference in the femtocell network. The NBS was used for allocating time slots to the mobile users in cooperative relay networks [10]. In this study, we assume that the femtocells are densely deployed and divided into clusters according to a weighted interference graph. Then, the bandwidth is allocated to each cluster by NBS.

The rest of the paper is organized as follows. Section 2 establishes the network model. Section 3 constructs a weighted interference graph and develops a cluster-based bandwidth allocation algorithm using NBS. Section 4 gives the numerical results, and Section 5 concludes the paper.

Network model

We consider a two-tier cellular network, where the first tier is a macrocell and the second tier is composed of femtocells. The femtocells are randomly deployed in the coverage of the macrocell. In this study, we focus on the downlink transmission in the femtocell network based on orthogonal frequency-division multiple access (OFDMA), as shown in Fig. 1.

Fig. 1
figure 1

Femtocell networks

Full size image

The interference from the macrocell base station (MBS) is regarded as an additive white Gaussian noise. All femtocell base stations (FBS) are connected to a femtocell gateway (FGW), which manages the operation and maintenance (OAM) information such as the femtocell location and identification through a backhaul link. FGW also manages the spectrum resource in the femtocell network based on the OAM information. Let ℕ = {0, 1, ⋯,N} denote the set of femtoells. Without loss of generality, we assume only one femtocell user (FUE) is connected to the FBS in a femtocell. The transmission power of FBS i (i ∈ ℕ) is assumed to be fixed at p i . The SINR of FUE i can be denoted as

$${r_i} = {{{p_i}{H_{ii}}} \over {\sum\limits_{j \ne i} {{p_j}{H_{ij}} + {\sigma ^2}}}}$$
(1)

where H ii denotes the channel gain from FBS i to FUE i, H ij denotes the channel gain from FBS j (j ∈ ℕ, j ≠ i) to FUE i, and σ2 is the single-sided spectral density of the independent white Gaussian noise.

Bandwidth allocation algorithm

In this section, we develop a bandwidth allocation algorithm for the OFDMA-based femtocell network. The cluster-based bandwidth allocation algorithm (CBBAA) can reduce the inter-femtocell interference and improve the spectrum efficiency. First, a weighted interference graph is constructed based on the topology of the femtocell network. Then, the femtocells are divided into several clusters. Finally, a bandwidth allocation algorithm is developed to meet the minimum rate constraint of each cluster.

Weighed interference graph and clustering

Based on the topology of the given femtocell network, we can construct a weighted interference graph G = (V, E, W), where V is the node set, E is the bi-directional edge set, and W is a weight matrix. We define V = {v1, v2, ⋯, v n }, E = {e ij , i, j ∈{0, 1, ⋯, N}, i ≠ j}, and W = {w ij , i, j ∈ {0, 1, ⋯, N}, i ≠ j}, where v i denotes the node corresponding to femtocell i, e ij is the edge between node v i and node v j , and w ij is the weight of the edge e ij . Then, the interference relationships among the femtocells are simplified to the connections among the nodes. Specifically, FUE i is interfered by FBS j when node vi and node v j are connected by a directed edge e ij , and w ij denotes the interference strength.

The weight is determined by the FUE measurement report based on the reciprocal of SINR. We consider two femtocells i and j in a femtocell network, where FUE i is interfered by FBS j. Assuming that the transmission power of FBS i and FBS j are pi and p j , respectively. Then, the weight w ij between femtocell i and femtocell j is denoted as

$$w_{ij} = \left\{ {\begin{array}{*{20}c} {\frac{{p_j H_{ij} + \sigma ^2 }} {{p_i H_{ii} }},} & {\frac{{p_j H_{ij} + \sigma ^2 }} {{p_i H_{ii} }} \geqslant \delta } \\ {0,} & {\frac{{p_j H_{ij} + \sigma ^2 }} {{p_i H_{ii} }} < \delta } \\ \end{array} } \right.$$
(2)

where δ is the interference threshold. The weight is zero when the interference among femtocells is lower than the threshold. Then, the femtocells are divided into clusters based on the weights. We assume the number of clusters is M and denote the set of clusters as \(\mathbb{M} = \{1,2, \cdots, M\}\). The interference strength of femtocell i is calculated as w i = ∑j≠ i w ij . The set of femtocells in cluster \(m(m \in \mathbb{M})\) is denoted as ℕ m = {1, 2, ⋯, N m }, where N m is the number of femtocells in cluster m. The interference strength between femtocell i and cluster m is calculated as \(w_i^m = \sum\nolimits_{j \in {\mathbb{N}_m}} {{w_{ij}}}\). We first calculate the interference strength w i for all femtocells and sort w i of a sequence in descending order. Then, the first M femtocells in the sequence are assigned into the M clusters, respectively. The rest of femtocells are assigned to the cluster with the minimal \(w_i^m\) one after another. When more than one cluster has the same minimal \(w_i^m\), the femtocell is assigned to one of them randomly. The clustering algorithm is given in Fig. 2.

Fig. 2
figure 2

The clustering algorithm

Full size image

Bandwidth allocation based on NBS

In this section, we will allocate bandwidth to each cluster based on NBS. Next, we first give the definitions of NBS and Pareto optimality[11, 12].

  • Definition 1. Let \(\mathbb{M} = \{1,2, \cdots, M\}\) denote the set of players and S denote the set of feasible payoff allocations, which should be a closed and convex set on RM. \(E^{\min} = (E_1^{\min}, \ \ldots, \ E_m^{\min}, \ \ldots, \ E_M^{\min})\), where \(E_m^{\min}\) denotes minimum payoff of player m. Then, (S, Emin)is a M-player bargaining problem.

Within the feasible set S, we define the notion of Pareto optimality.

  • Definition 2. The payoff allocation E =(E1, ⋯, E M ) is Pareto optimal if and only if there is no other allocation E′ m such that E′ m > E m , ∀m, i.e., there exists no other allocation that leads to superior performance for some player without inferior performance for some other player.

The bargaining problem (S, Emin) has many Pareto optimal solutions denoted by f(S, Emin). A critical criterion is the fairness of resource allocations. In this study, we employ NBS with proportional fairness. Assuming that the minimum payoff is \(E_m^{\min}\). Then, the NBS can be defined as the solution of the following optimization problem.

$$E_m^{\ast} = \arg \mathop {\max}\limits_{{E_m} \in S} \prod\limits_{m = 1}^M {({E_m} - E_m^{\min}).} $$
(3)

We assume the allocated fraction of bandwidth for cluster m is α m , 0 ≤ α m ≤ 1. The sum of the fractions should satisfy \(\sum\nolimits_{m = 1}^M {{\alpha _m} \leq 1}\). Thus, the set of feasible payoff allocations can be denoted as

$$\matrix{{Q = \{\Gamma = ({\Gamma _1}, \cdots, {\Gamma _m}, \cdots, {\Gamma _M})\vert 0 \leq {\alpha _m} \leq 1} \hfill \cr {\quad \quad \sum\limits_{m = 1}^M {{\alpha _m}} \leq 1,m = 1,2, \cdots, M\}} \hfill \cr}$$
(4)

where Γ m is the transmission rate of cluster m.

$${\Gamma _m} = {\alpha _m}W\sum\limits_{i \in {\mathbb{N}_m}} {\ln (1 + {r_i})} $$
(5)

where \(W = {{\alpha {W_0}} \over {\ln 2}}\), α is the gap between the achievable rate and the actual transmission rate, and W0 is the total transmission bandwidth for the femtocell network. Next, we will prove Q is a convex subset of RM. According to the rate function (5) of cluster m, Q is closed subset on RM. Assuming that \({\Gamma ^a} = (\Gamma _1^a, \cdots, \Gamma _m^a, \cdots \Gamma _M^a) \in Q,\;{\Gamma ^b} = (\Gamma _1^b, \cdots, \Gamma _m^b, \cdots \Gamma _M^b) \in Q\), and 0 ≤ θ ≤ 1, we have

$$\theta \Gamma _m^a + (1 - \theta)\Gamma _m^b = [\theta \alpha _m^a + (1 - \theta)\alpha _m^b]W\sum\limits_{i \in {\mathbb{N}_m}} {\ln} (1 + {r_i}).$$

According to \(\Gamma _m^a \in Q\) and \(\Gamma _m^b \in Q\), we have \(0 \leq \alpha _m^a \leq 1,\;0 \leq \alpha _m^b \leq 1,\;\sum\nolimits_{m = 1}^M {\alpha _m^a \leq 1}\), and \(\sum\nolimits_{m = 1}^M {\alpha _m^b \leq 1}\). Then, we obtain

$$0 \leq \theta \alpha _m^a + (1 - \theta)\alpha _m^b \leq 1$$

and

$$\sum\limits_{m = 1}^M {(\theta \alpha _m^a + (1 - \theta)\alpha _m^b) = \theta} \sum\limits_{m = 1}^M {\alpha _m^a + (1 - \theta)\sum\limits_{m = 1}^M {\alpha _m^b \leq 1.}} $$

Then, we can conclude that Q is closed and convex on RM. The net rate function of cluster m can be denoted as

$${U_m} = {\Gamma _m} - \Gamma _m^{\min}$$
(6)

where \(\Gamma _m^{\min}\) is the minimum rate of cluster m. Then, the NBS-based bandwidth allocation problem can be denoted as

$$\matrix{{\max \quad \prod\limits_{m = 1}^M {{U_m}}} \hfill \cr {{\rm{s}}.{\rm{t}}.\quad \sum\limits_{m = 1}^M {{\alpha _m} \leq 1.}} \hfill \cr}$$
(7)

The optimization problem (7) is equivalent to solving the following optimization problem.

$$\matrix{{\max \quad \prod\limits_{m = 1}^M {\ln {U_m}}} \hfill \cr {{\rm{s}}.{\rm{t}}.\quad \sum\limits_{m = 1}^M {{\alpha _m} \leq 1.}} \hfill \cr}$$
(8)

The first and second derivative of lnU m with respect to α m can be denoted as

$${{{\rm{dln}}{U_m}} \over {{\rm{d}}{\alpha _m}}} = {{W\sum\limits_{i \in {\mathbb{N}_m}} {\ln} (1 + {r_i})} \over {{\Gamma _m} - \Gamma _m^{\min}}} > 0$$
(9)

and

$${{{{\rm{d}}^2}\ln {U_m}} \over {{\rm{d}}\alpha _m^2}} = - {{{{[W\sum\limits_{i \in {\mathbb{N}_m}} {\ln} (1 + {r_i})]}^2}} \over {{{({\Gamma _m} - \Gamma _m^{\min})}^2}}} < 0.$$
(10)

The second derivative of lnU m with respect to α m is always negative. Therefore, lnU m is concave in α m , and the problem (8) is a convex optimization problem. For a convex optimization problem, the optimal solution can be obtained by the Karush-Kuhn-Tucker (KKT) conditions. We give the Lagrange function.

$$L({\alpha _m},\mu) = \sum\limits_{m = 1}^M {\ln {U_m} - \mu (\sum\limits_{m = 1}^M {{\alpha _m} - 1})}$$
(11)

where μ is the Lagrange multiplier. We take the derivative of the Lagrange function with respect to α m and μ, respectively, and let them equal to zero.

$$\left\{{\matrix{{\mu = {{{A_m}} \over {{\alpha _m}{A_m} - \Gamma _m^{\min}}}} \hfill \cr {\sum\limits_{m = 1}^M {{\alpha _m} = 1}} \hfill \cr}} \right.$$
(12)

from which, the optimal bandwidth allocation solution is obtained.

$$\left\{{\matrix{{\mu \ast = {M \over {1 - \sum\limits_{m = 1}^M {{{\Gamma _m^{\min}} \over {{A_m}}}}}}} \hfill \cr {\alpha _m^{\ast} = {{\Gamma _m^{\min}} \over {{A_m}}} + {{1 - \sum\limits_{m = 1}^M {{{\Gamma _m^{\min}} \over {{A_m}}}}} \over M}} \hfill \cr}} \right.$$
(13)

where A m is denoted as

$${A_m} = W\sum\limits_{i \in {\mathbb{N}_m}} {\ln} (1 + {r_i}).$$
(14)

Thus, the corresponding net rate can be denoted as

$${U_m} = {{{A_m}\left({1 - \sum\limits_{m = 1}^M {{{\Gamma _m^{\min}} \over {{A_m}}}}} \right)} \over M}.$$
(15)

The net rate (15) is related to the number of clusters and the minimum rates. Specifically, the net rate is gradually decreased with the minimum rates when the number of clusters is fixed.

Numerical results

We assume the femtocells are densely deployed in coverage of macrocell, and the radius of macrocell is 700 m. We use P a to represent the percentage of active FBSs and investigate the performance with P a = 40%. Each active FBS provides service to an FUE in one slot. The minimum separation distance between the FBS and the FUE is 3 m. The penetration loss is set to 10 dB. The parameters settings are given in Table 1.

Table 1 Simulation parameters
Full size table

Before clustering based on the weights, we need to know the value of m. The target SINR of FUEs is set to 20 dB. In Fig. 3, we obtain the total net rates versus the number of clusters based on the Monte Carlo method. We find that the total net rates are maximal when M is five.

Fig. 3
figure 3

The total net rates versus the number of clusters

Full size image

The SINR of FUEs with and without clustering are shown in Fig. 4 when the number of clusters is set to 5. We assume the target SINR is 20 dB, which is the minimum SINR corresponding to the minimum rates of FUEs. We see that only three FUEs meet the target SINR without clustering and all of the FUEs meet the target SINR after clustering.

Fig. 4
figure 4

The SINR of FUEs with and without clustering

Full size image

The optimal number of clusters versus the target SINR of FUEs is shown in Fig. 5. We see that the optimal number of clusters increases with the target SINR. Clustering is not needed when the target SINR is set to 5 dB. The femtocells are divided into 10 clusters when the target SINR is higher than 40 dB. The FUEs have achieved their maximal SINR when the target SINR is higher than 40 dB.

Fig. 5
figure 5

The optimal number of clusters versus the target SINR of FUEs

Full size image

The total net rates of FUEs versus the number of clusters are given in Fig. 6. The total net rates of FUEs increase at first and decrease with the number of clusters. We also compare the CBBAA with the average bandwidth allocation algorithm (ABAA). The ABAA outperforms the CBBAA when the number of clusters is less than 5. However, the CBBAA is much better when the number of clusters is more than 5. The FUEs cannot meet the total minimum rates when the number of clusters is 2 in both two bandwidths allocation schemes, and the total net rates of FUEs are maximal when the number of clusters is 5.

Fig. 6
figure 6

The total net rates of FUEs versus the number of clusters

Full size image

The clustering results with different number of femtocells (NOF) are shown in Table 2. We give the number of clusters (NOC), the number of FUEs (NOE) in each cluster, and the fraction of bandwidth allocated (FBA) to each cluster. We see that the number of clusters is increased with the number of femtocells. Furthermore, an optimal fraction of bandwidth is allocated to each cluster. For example, the femtocells are divided into 11 clusters when the number of femtocells are 20. Although the allocated bandwidth of each cluster is small, the minimum rate constraint is satisfied and the fairness is achieved.

Table 2 The clustering results
Full size table

Conclusion

In this paper, we study the inter-femtocell interference management problem when femtocells are densely deployed. We propose a cluster-based bandwidth allocation algorithm. Specifically, we construct the weighted interference graph and divide the femtocells into clusters. Then, the bandwidth is allocated to the clusters based on NBS. Simulation results show that the cluster-based bandwidth allocation can reduce the inter-femtocell interference and meet the minimum rate constraint of each cluster.