 Research
 Open Access
 Published:
Distributed resource allocation with imperfect spectrum sensing information and channel uncertainty in cognitive femtocell networks
EURASIP Journal on Wireless Communications and Networking volume 2017, Article number: 201 (2017)
Abstract
To meet the everincreasing demands of mobile traffic, femtocells are considered as one of the promising solutions. In this paper, we study a sensingbased resource allocation scenario in cognitive femtocell networks and present an efficient distributed imperfectspectrumsensingbased resource allocation (DIRA) algorithm while considering the channel uncertainty to maximize the total data rate of cognitive femtocell networks by jointly optimizing both subchannel assignment and power allocation taking into account the influence of the sensing accuracy. However, the general optimization problem turns out to be a mixed integer programming problem. In order to make it tractable, the original optimization problem is divided into two suboptimization problems,namely,optimal subchannel allocation and optimal power allocation. Specifically, the proposed distributed fairnessbased subchannel allocation (DFSA) algorithm guarantees fairness by introducing channel condition difference and satisfaction degree as the indicators of subchannel allocation. Additionally, optimal power allocation with the consideration of imperfect spectrum sensing and interference uncertainty is performed using the proposed chanceconstrained power optimization (CPO) algorithm. Bernstein’s approximation is conducted to make the chance constraint tractable. Simulation results illustrate that the distributed imperfectspectrumsensingbased resource allocation (DIRA) algorithm can provide considerable fairness among femtocells and at the same time maximize the total data rate of the cognitive femtocell network.
Introduction
To accommodate with this everincreasing demand for mobile data transmission, the mobile network operators (MNOs) is facing with urgent requirement of seeking for new technologies to enhance the capacity by 1000 times [1]. In this context, small cell deployment has been viewed as one of the most effective and costefficient solutions [2]. Small cell is an umbrella term for lowpowered radio access nodes with a range of 10 m up to several hundred meters. It can be generally categorized into femtocells, picocells, or microcells according to their coverage range in ascending order. As several studies show more than 70% of data traffic occurs indoors [3], this has led to increasing interest in femtocells, which is known as home base station. The lowpower, shortrange, easy plugandplay, and selforganization features of femtocells benefit both the users and operators.
Embedding femtocells in traditional cellular system helps offloading the overloaded traffic in macrocells, expanding coverage, and boosting network capacity. However, the scarcity of the available wireless spectrum resource becomes a challenging issue in the development of wireless communication technologies [4], which urges MNOs to optimally utilize the bandwidth in order to obtain the maximization of network capacity. In this case, dedicatedchannel deployment of femtocells is no longer preferable from the operator’s perspective because of radio resource shortage and implementation difficulty. Compared with dedicatedchannel deployment, cochannel deployment is more attractive due to easy implementation and more efficient utilization of wireless spectrum. However, cochannel deployment might introduce severe crosstier interference between macrocells and femtocells where they share the same spectrum [5], especially when they are close to each other [6], and cotier interference between cochannel deployed femtocells in dense deployment scenario. Therefore, interference mitigation in such twotier networks needs to be tackled. Moreover, the random deployment, restricted/closed access of femtocells, and no coordination between two tiers turn this problem into a hard nut to crack [7]. Facing with the above challenges, cognitive radio has emerged as a potential technology which enables radio devices to monitor the radio environment and dynamically adjust transmission parameters according to the sensing result. A femtocell base station (FBS) equipped with cognitive function will be capable of identifying and choosing the suitable subchannel that provides the least harmful influence to others.
Related works
A considerable amount of literature is available for resource allocation and interference management in cognitive femtocell system. In [8], a hybrid overlay/underlay spectrum access mechanism was proposed to improve the overall system performance of cognitive femtocell networks. The subchannel allocation problem was formulated as a coalition formation game, and a modified recursive core algorithm was proposed to achieve stable and efficient allocation. In [9], a distributed joint power control method came up with a proper solution to manage the interference in twotier femtocell networks.
Usually, subchannel allocation and power control are jointly considered in literatures. In [10], a resource allocation scheme was proposed to maximize the total data rate of cognitive small cells without causing intolerable interference to macrocell users, in which femtocell user equipments (FUEs) could estimate the interference channel through the pilot signals broadcasted by macrocell base station (MBS). Moreover, the proposed algorithm could ensure the fairness with only a tiny reduction in throughput performance. In [11], a decentralized approach for dynamic subchannel and power allocation was considered. Reinforcement learningbased algorithm was applied to solve the uncoordinated spectrum sharing problems. The spectrum allocation was based on reinforcement learning algorithm, while the power allocation applied convex optimization algorithm which was decided by each femtocell independently. Besides, game theory is also a feasible solution which has been widely investigated in existing researches to jointly investigate the subchannel and power allocation in femtocell networks. In [12], a cooperative Nash bargaining game model was developed to study the subchannel allocation and power control problem jointly for cognitive small cell networks with the consideration of crosstier interference mitigation, outage limitation, imperfect CSI, and fairness.
Due to the inherent hardware limitations and variable wireless environment, spectrum sensing errors are inevitable, causing interference to macrocell user equipments (MUEs) in cognitive femtocell networks [13]. Hence, owing to the imperfectness of the spectrum sensing, the traditional resource allocation algorithms might experience performance decrease. In [14], authors used the particle swarm optimization (PSO) algorithm to solve a joint uplink resource allocation problem for cognitive networks under the consideration of imperfect spectrum sensing. In [15], a multiobjective optimization problem that jointly considered the femtocell throughput maximization and transmit power minimization was formulated, subject to interference constraints on both femtocell and macrocell including the cochannel interference and adjacent channel interference constraints under spectrum sensing error probabilities. In [16], pricing and power allocation strategies were studied in a twotier femtocell network with the aim of maximizing energy efficiency, where both perfect and imperfect spectrum sensing cases were considered.
Besides, the information uncertainty is also significantly important for the variable wireless environment, including channel state information uncertainty [17], network access state uncertainty [18], and background noise uncertainty [19]. Particularly, those kind of uncertainty problems can be presented as a probabilistic problem by relaxing the constraint into an equivalent chance constraint [20–22]. Based on the aforementioned solutions, jointly considered subchannel allocation and power allocation are a proper way to solve resource allocation problems. The objectives of resource allocation mainly focus on interference management [9, 10, 12, 13, 19–22], capacity enhancement [8, 10, 13, 15, 20–22], power efficiency improvement [15, 16, 19], and fairness [10, 12]. However, to the best of our knowledge, resource allocation for cognitive femtocell network jointly considering interference management, fairness, imperfect spectrum sensing, and interference uncertainty has not been studied in previous works.
Contributions
Traditionally, the distribution of femtocells is modeled by a hexagonal grid. However, in twotier femtocell networks, topological randomness causes the gridbased model too idealized for both macrocells and femtocells, especially when most of femtocells are installed by their subscribed users. Recently, a new analytical method has gathered considerable attention, named stochastic geometry. It can not only capture the topological randomness but also provide tractable analytical results [23]. Hence, we model the cognitive femtocell network using stochastic geometric tools [24]. In this paper, resource allocation problem including subchannel scheduling and power allocation was jointly investigated for cognitive femtocell networks, where the overall objective is to maximize the total data rate, taking into account the influence of the imperfect spectrum sensing and channel uncertainly. The main contributions of the paper are the following:


Firstly, the distribution of femtocells is modeled by a hexagonal grid in the existing literature. However, the topological randomness causes the gridbased model too idealized for both macrocells and femtocells, especially when most of femtocells are installed by their subscribed users. In this paper, the cognitive femtocell network topology is based on stochastic geometry; macrocell and femtocell base stations are modeled as two independent Voronoi tessellations, where MBSs and FBSs follow Poisson point process (PPP) distribution in their own tessellation.


Secondly, the reliability of the spectrum sensing is taken into consideration which will affect the interference constraint. Moreover, the femtotomacro interference constraint under channel uncertainty can be cast as the chance constraint. Bernstein’s approximation is conducted to make the chance constraint tractable.


Thirdly, in order to maximize the total data rate of femtocells under the consideration of imperfect spectrum sensing and channel uncertainty while balancing the fairness of networks, a distributed fairnessbased subchannel allocation (DFSA) algorithm is implemented which uses channel condition difference as an indicator of subchannel allocation, taking the satisfaction degree of each FBS into consideration. To further increase the total capacity of femtocell network, transmit power of femtocells on each subchannel will be adapted subject to interference constraints and minimum quality of service (QoS) requirements using the chanceconstrained power optimization (CPO) algorithm.
The layout of this paper is outlined as follows. Section 2 describes the network model. Section 3 formulates our optimization task with imperfect spectrum sensing and interference uncertainty. In Section 4, the DIRA algorithm is proposed. Section 5 evaluates the performance of our proposed algorithm, and numerical results are presented with discussions. Section 6 concludes this paper.
System model
Network topology
In this paper, a cognitive femtocell network where macrocells are overlaid with multiple cognitive femtocells is considered. We model this twotier femtocell network by stochastic geometry tools. Figure 1 shows an example of a twotier network scenario; the coverage area of each cell depends on its location and other cells’ locations. Assuming that the MBSs and FBSs are distributed via an independent Poisson point process (PPP) of density d _{ m } and d _{ f }, respectively. The twotier network model can be abstracted as two Voronoi tessellations as shown in Fig. 2. By construction, each user located in the intersection of two cells will associate with either the MBS or the FBS of the Voronoi cell covering that user. We denote the sets of MBSs and FBSs as \(\mathcal {M}=[1,...,M]\) and \(\mathcal {F}=[1,...,F]\). In each femtocell, there are K FUEs. An orthogonal frequencydivision multiple access (OFDMA) downlink system is considered, where the total bandwidth of B _{ w } is divided into N subchannels.
Channel model
Cognitive FBSs opportunistically access the licensed subchannels belong to the macrocells. Through periodic spectrum sensing performed by the cognitive FBS, subchannels can be identified as busy or idle. In this paper, we apply overlay spectrum sharing mode between MBSs and FBSs, which means subchannels which are determined as idle can be utilized by the FBS.
We assume that the knowledge of interference is uncertain for each FBS due to the uncertainty of channel gain. Let g _{ f,k,n } be the channel gain of the kth FUE in femtocell \(f\in {\mathcal {F}}\) on subchannel \(n\in {\mathcal {N}}\). Hence, the received signaltointerferneceplusnoise ratio (SINR) of FUE k in femtocell f on subchannel n is given as
where \(p^{F}_{{f,k,n}}\) is the transmit power of FBS f to FUE k on subchannel n; I is the received interference from other FBSs, where \(I=\sum _{e=1,e\neq f}^{F}{p_{{e,k,n}}^{F} g_{{e,k,n}}}\); and N _{0} is the noise power. The data rate of FUE k in femtocell f on subchannel n is presented as
Assume that the estimated channel gain g _{ f,k,n } can be acquired accurately through traditional channel estimation techniques. However, the channel gain g _{ f,n } from FBS f to the MUE on subchannel n is difficult to estimate due to the lack of cooperation between MBSs and FBSs. Thus, we model g _{ f,n } as
where \(\overline {g}_{f,n}\) denotes the estimated channel gain on subchannel n obtained by averaging g _{ f,k,n } and \({\tilde {g}}_{f,n}\) represents the uncertain part of the channel gain.
Problem formulation with imperfect spectrum sensing
In this section, firstly, we discuss the crosstier interference caused by femtocells, which consists of cochannel interference caused by imperfect spectrum sensing and outofband interference introduced by sidelobe power leakage of orthogonal frequency division modulation (OFDM) signals. Secondly, an optimization framework which aims to maximizing the total data rate of the cognitive femtocell network is formulated with the consideration of imperfect spectrum sensing and interference uncertainty.
Interference from imperfect spectrum sensing and outofband emission
To ensure the QoS performance of MUEs, the interference caused by opportunistic access of FBSs in the licensed channel should be controlled. The interference introduced to the macrotier consists of two parts: (i) cochannel interference as a result of imperfect spectrum sensing and (ii) cognitive radio outofband (OOB) emission.
Due to the inherent hardware limitations and variable wireless environment, the spectrum usage can be falsely detected by cognitive femtocells. The result of spectrum sensing on subchannel n is a binary event denoted by \(S^{n}_{0}(\tilde {x}_{n}=0)\) and \(S^{n}_{1}(\tilde {x}_{n}=1)\), where \(\tilde {x}_{n}=0\) or 1 identifies that subchannel n is sensed to be idle or busy. Similarly, the actual state of the subchannel n can be denoted as \(H^{n}_{0}(x_{n}=0)\) and \(H^{n}_{1}(x_{n}=1)\), where x _{ n } represents the actual status of subchannel n, with x _{ n }=0 or 1 indicating that the subchannel n is vacant or occupied. Generally, there are four types of spectrum sensing probabilities: (i) idle channel detection probability: \(p^{n}_{nd}=Pr\{S^{n}_{0}H^{n}_{0}\}\); (ii) false alarm probability: \(p^{n}_{f}=Pr\{S^{n}_{1}H^{n}_{0}\}\); (iii) miss detection probability, \(p^{n}_{m}=Pr\{S^{n}_{0}H^{n}_{1}\}\); and (iv) detection probability, \(p^{n}_{d}=Pr\{S^{n}_{1}H^{n}_{1}\}\). Among the above sensing cases, false alarm and miss detection are considered as sensing errors. Since FBSs make access decisions based on the results of spectrum sensing, leading to four possible cases of spectrum sensing, which are given in Table 1, where the sensing results of femtocells are considered as priori probabilities.
Among the possible cases shown in Table 1, subchannel n is occupied by the MBSs only in cases 3 and 4. In case 3, the spectrum sensing result of subchannel n is vacant; however, the actual state of n is busy, resulting in great crosstier interference from FBS to MUE on subchannel n since FBS has no idea that this subchannel is not vacant as detected. In case 4, although the detection made by FBS is correct, MUE on subchannel n can still suffer from OOB emissions due to sidelobe power leakage.
We can calculate P _{3,n } and P _{4,n } using Bayes’ theorem. Thus, we have
where \(p^{n}_{s}\) represents the occupation probability of MBS on subchannel n.
In addition to the effect of imperfect spectrum sensing, OOB emission can also cause interference to MUEs. The amount of OOB interference introduced to MUEs on subchannel j by the femtocell transmission on subchannel n, with unit transmit power, is given as
where \(\varphi (f)=T_{s}{\left [\frac {\sin \left (\pi f T_{s}\right)}{\pi f T_{s}}\right ]}^{2}\) represents the power spectral density (PSD) of an OFDM signal; T _{ s } is the duration of an OFDM signal.
Based on the above analysis, to jointly consider the effect of imperfect spectrum sensing and OOB emission in the interference constraint, we formulate the crosstier interference from FBS to MUE s with unit transmit power as
where \(\mathcal {N}_{v}\) represents the set of vacant subchannel in \(\mathcal {N}\) and \(\mathcal {N}_{o}\) is the set of subchannel occupied by MBSs. Considering the uncertainty of interference, we introduce ε∈(0,1) to guarantee the interference constraint in probability. Hence, the interference constraint can be written as a chance constraint as follows:
where I _{thre,s } is the interference limitation from FBSs to MUE s.
General optimization framework
In this paper, our major target is to maximize the total data rate of the femtocell network under the constraints of minimum QoS requirement and crosstier interference, taking into account the influence of sensing accuracy as well as channel uncertainty. Thus, the general optimization problem can be formulated as follows:
where τ _{ f,k,n }=1 or 0 indicates whether subchannel n is allocated to FUE k in femtocell f or not, \(p^{F}_{\text {max}}\) is the maximum transmit power of a FBS, and \(R^{0}_{f,k}\) is the minimum QoS requirement of FUE k in femtcoell f.
In our optimization problem, C1–C2 are power constraints indicating the transmit power of a FBS should not exceed the maximum transmit power \(p^{F}_{\text {max}}\). C3 is the minimum QoS requirement for FUEs. C4–C5 are constraints of subchannel allocation, representing that a subchannel n can not be assigned to two different FUEs in the same femtocell. C6 is the crosstier interference constraint represented by chance constraint, taking both imperfect spectrum sensing and interference uncertainty into consideration. Via C6, the crosstier interference from FBSs to MUE s will be limited below the threshold I _{thre,s } with a probability not less than 1−ε. In practice, it is difficult to acquire channel information accurately; thus, it is more reasonable to present the interference constraint as a chance constraint. According to the above analysis, C4 and C5 are integer constraints, which, as a result, lead the optimization problem to be a mixed chanceconstrained integer programming problem.
Distributed imperfectspectrumsensingbased resource allocation
The optimization problem in (9) is a mixed chanceconstrained integer programming problem which is computationally complex to address. To make it tractable, we divide the original problem into two suboptimization problems and solve them in two steps. Firstly, in order to remove the integer constraints of the optimization problem, we address the subchannel allocation problem using DFSA algorithm with the consideration of fairness among FBSs. In this way, the mixed chanceconstrained integer programming problem is simplified as a chanceconstrained programming problem without integer variables. Secondly, optimal power allocation to the subchannels is achieved by CPO algorithm. Bernstein’s approximation is applied to transform the chance constraint into a convex constraint, and the Lagrangian dual algorithm is used to solve the convex power optimization problem.
Subchannel allocation by DFSA algorithm
We propose a distributed fairnessbased subchannel allocation algorithm to assign the subchannels with unit power allocation. The aim of the DFSA scheme is to achieve a fair subchannel allocation, while maintaining considerable data rate of the whole network.
Through spectrum sensing, cognitive femtocells can identify the available subchannels and estimate their channel conditions. For FBS f, the channel condition c _{ f,n } of subchannel n is considered as the average estimated channel gain from FBS f to all FUEs associated with this FBS, given by
As a result, the channel condition table (CCT) is created based on this channel condition. FBS f sorts its sensing results, namely, the available subchannels in a descending order with respect to the c _{ f,n }, as shown in Table 2. By calculating the channel condition difference between certain subchannel and the next subchannel ranked in CCT, we can obtain the channel condition difference table (CCDT), as shown in Table 3.
The DFSA algorithm is based on CCDT instead of CCT. In the CCTbased subchannel allocation scheme, FBS f may suffer large quality degradation if FBS l accesses f’s first rank subchannel, although there are not too much differences between the second rank and first rank subchannel of FBS l, leaving FBS f with no choice but to access the second rank subchannel with relatively worse channel condition. However, our DFSA algorithm is based on CCDT; the FBS with larger first rank difference value may have the prior chance to access its first rank subchannel, which decrease the risk of suffering performance degradation due to the loss of the preferred subchannel. By taking channel condition difference as the indicator of subchannel allocation, the fairness between FBSs based on the average data rate can be guaranteed.
Finally, a subchannel requirement table (SRT) is formulated, which consists of the subchannel requirement of a FBS, as shown in Table 4, where \(N^{f}_{A}\) is the number of total available subchannels of FBS f under spectrum sensing result and \(N^{f}_{D}\) is the number of subchannels desired by FBS f. The actual subchannel access will take place once the whole subchannel selection procedure is finished; otherwise, there exists an available subchannel list to be accessed by FBS f. \(N^{f}_{to}\) is the number of subchannels for FBSs to access, which is initialized as 0 at the beginning of the algorithm. counter^{f} represents the number of subchannels still required by FBS f, given by
which is initialized as \(N^{f}_{D}\) and should be updated during the allocation procedure. \(N^{f}_{R}\) is the number of remaining available subchannels of FBS f, which is initialized as \(N^{f}_{A}\) and also changes during the allocation procedure. SD^{f} is the satisfaction degree of FBS f, given by
Furthermore, with the consideration of satisfaction degree, the fairness in the number of subchannels allocated to FBSs can be guaranteed.
Once the DFSA algorithm starts, FBSs share the CCDT and SRT with other FBSs in the same macrocell via broadcating. As a result, FBSs in the same macrocell have the knowledge of CCDTs and SRTs of each other. With this information, each FBS can perform the subchannel allocation process individually without causing collisions on the same subchannel.
In the proposed DFSA algorithm, only one subchannel can be selected by a certain FBS each time. The detail procedure of DFSA algorithm is shown in algorithm 1. The CCT, CCDT, and SRT of FBS f is denoted by CCT_{ f }, CCDT_{ f }, and SRT_{ f }, respectively. Besides, the table update procedure is shown in algorithm 2.
Power allocation for femtocells by chanceconstrained power optimization
Once the subchannel allocation is obtained based on DFSA algorithm, the integer variable τ _{ f,k,n } in the original optimization problem in (9) can be substituted by the subchannel allocation result. To further maximize the total data rate of cognitive femtocell network, we optimize the transmit power under the constraints of power, QoS requirement, and crosstire interference. Thus, the optimization problem can be written as
where \(\tau ^{*}_{{f,k,n}}\) is the subchannel allocation achieved by employing the DFSA algorithm in the previous subsection.
However, the optimization problem in (13) is still intractable due to the nonconvex chance constraint in C6. To achieve a convex feasible set of C6, a convex approximation of C6 should be performed, which can be achieved by using the Bernstein approximation method [25, 27].
Bernstein’s approximation of the chance constraint
Consider a chance constraint written as a form of
Assuming that

1)
p is a vector of decision parameters.

2)
{ζ _{ n }} is the set of random variables, with marginal distribution denoted by {π _{ n }}.

3)
{π _{ n }} belongs to a given family of probability distribution with bounded support of [−1,1], which means ζ _{ n } varies in the range of [−1,1].
Let us set
It is shown that the function
is convex in (t>0,p) [27]. The Bernstein approximation of (14) can be formulated as
which is a safe convex approximation of the chance constraint, that is, p satisfies the chance constraint if it satisifies Eq. (17). This approximation is tractable if {Λ _{ n }(y)} can be evaluated efficiently [25]. Consider a case of Λ _{ n }(y) when
where both \(1\leq \mu ^{}_{n}\leq \mu ^{+}_{n}\leq 1\) and σ _{ n }≥0 are constants. By choosing appropriate \(\mu ^{}_{n}\), \(\mu ^{+}_{n}\), and σ _{ n } and replacing Λ _{ n } in (17) with its upper bound given in (18), Eq. (17) can be bounded by
Invoking the arithmeticgeometric inequality for (19), the convex constraint can be written as
which is a safe conservative approximation of Eq. (14). BenTal and Nemirovski [25] give some examples of the value of \(\mu ^{}_{n}\), \(\mu ^{+}_{n}\), and σ _{ n }≥0 based on some prior knowledge (e.g., support, unimodality, and symmetry) of the distributions.
Suppose \(I^{s}_{{f,k,n}}\) follows a given distribution with bounded support of \(\left [a^{s}_{{f,k,n}},b^{s}_{{f,k,n}}\right ]\). Introduce constants \(\alpha ^{s}_{{f,k,n}}\triangleq \frac {1}{2}\big (b^{s}_{{f,k,n}}a^{s}_{{f,k,n}}\big)\) and \(\beta ^{s}_{{f,k,n}}\triangleq \frac {1}{2}\big (b^{s}_{{f,k,n}}+a^{s}_{{f,k,n}}\big)\) to make a normalization of support of [−1,1]; that is, \(\alpha ^{s}_{{f,k,n}}\zeta ^{s}_{n}+\beta ^{s}_{{f,k,n}}\in [\alpha ^{s}_{{f,k,n}},\beta ^{s}_{{f,k,n}}]\), where \(\zeta ^{s}_{n}\triangleq \frac {I^{s}_{{f,k,n}}\beta ^{s}_{{f,k,n}}}{\alpha ^{s}_{{f,k,n}}}\). Let
Then, C6 in (13) is equivalent to (14). Substitute f _{0}(p) and f _{ n }(p) into (20), noting that \(p^{F}_{{f,k,n}}\geq 0\), then we obtain
According to the mean inequality theorem, arithmetic mean cannot be greater than quadratic mean, which means \(\sqrt {\sum ^{N}_{n=1}X^{2}_{n}}\geq \frac {\sum ^{N}_{n=1}X_{n}}{\sqrt N}\). Thus, we can further approximate the above inequality as
where \(\gamma ^{s}_{{f,k,n}}\triangleq \mu ^{+}_{n}\alpha ^{s}_{{f,k,n}}+\beta ^{s}_{{f,k,n}}\). Therefore, the chance constraint C6 in Eq. (13) is transformed into a convex constraint.
Optimal power allocation for femtocells
Based on the above analysis, the chanceconstrained programming problem in (13) can be transformed into a convex optimization problem. Thus, we can obtain optimal power allocation by solving the following convex optimization problem:
which can be solved by applying the Lagrangian dual decomposition method. By introducing dual variables λ, ν, and δ, the Lagrangian function is given by
where λ, ν, and δ are the Lagrange multiplier vectors for C2, C3, and C7, respectively. In OFDMAbased cognitive femtocell system, a subchannel can be assigned to only one FUE in the same FBS; thus, the subchannel set allocated to each FUE in the same FBS is independent of each other. Hence, the Lagrangian dual problem can be decomposed into a master problem and F×N subproblems, which can be solved iteratively. Accordingly, we have
where
The KarushKuhnTucker (KKT) conditions of (26) can be expressed as
where
The optimal power allocation for FBS f to FUE k on subchannel n can be expressed as
where \([x]^{+}\triangleq \max (0,x)\); thus, the boundary constraint C1 in (25) is contained in Eq. (35).
The subgradient search algorithm is used to calculate the nonnegative Lagrange multipliers λ _{ f,k }, ν _{ f,k }, and δ, given by
where \(\varepsilon ^{(t)}_{1}\), \(\varepsilon ^{(t)}_{2}\), and \(\varepsilon ^{(t)}_{3}\) are step sizes and t is the iteration number.
Simulation results and discussions
In this section, we evaluate the performance of the proposed DIRA scheme. In the simulation, we consider the downlink of a cognitive femtocell network in which femtocells are overlaid with macrocells. We assume that there are multiple macrocells and femtocells distributed in a manner of PPP with density d _{ m } and d _{ f } in a 4000 × 4000 m scenario. MUEs and FUEs are distributed randomly in the scenario. The number of FUEs associated with an FBS is 4. The channel gains are modeled as i.i.d. exponential random variables. Shadowing effect is modeled as a lognormal variable with standard deviation 6 dB. The false alarm probability \(p^{n}_{f}\) and miss detection probability \(p^{n}_{m}\) are uniformly distributed over (0.05, 0.1) and (0.01, 0.05). The occupation probability of MBSs on subchannel n is uniformly distributed over (0, 1). The Bernstein approximation parameters \(\mu ^{}_{n}\), \(\mu ^{+}_{n}\), and σ _{ n } are chosen from [25]. The simulation parameters are given in Table 5.
In the simulation, a comparison between the proposed CCDTbased DFSA algorithm and the CCTbased subchannel allocation algorithm has been adopted to evaluate the fairness performance of the algorithm. Figure 3 shows the satisfaction degree fairness for different density of FBS, where the satisfaction degree fairness is represented by the variance of satisfaction degree. As shown in the figure, compared with the CCTbased algorithm, the proposed CCDTbased DFSA algorithm achieves lower satisfaction degree variance, which illustrates that the proposed algorithm outperforms the CCTbased algorithm in respect of satisfaction degree fairness.
Figure 4 shows the data rate fairness comparison of the proposed algorithm and the CCTbased algorithm. As we can observe from the figure, the proposed algorithm achieves a lower variance of average FUE data rate of FBSs than the CCTbased algorithm. The outstanding fairness performance for average data rate of the proposed algorithm is achieved by applying the channel condition difference as an allocation indicator. According to Figs. 3 and 4, it becomes harder to guarantee fairness among different FBSs with increasing density of FBSs. This is because more and more FBSs compete to access the same subchannel, resulting in difficulty for maintaining fairness.
The total data rate of the FBSs for different density of FBSs is shown in Fig. 5, where the data rate achieved by two algorithms is almost the same. Through this figure, obviously, the proposed algorithm can improve the fairness performance without influencing the data rate.
We investigate the effect of ε in the total data rate of FBSs, where d _{ f }=12BS/ km^{2}, I _{thre,s }=−110 dBm, \(R^{0}_{f,k}=5\) kbps. As shown in Fig. 6, the total data rate of FBSs increases when the ε gets larger. Obviously, a larger ε relaxed the interference constraints to FBSs, which results in higher tolerance for crosstier interference from FBSs to MUEs. It illustrates that the higher data rate of FBSs can be achieved at the cost of increasing interference to MUEs.
Furthermore, we compare the total crosstier interference from FBSs to MUEs for perfect and imperfect spectrum sensing cases, where d _{ f }=12BS/ km^{2}, I _{thre,s }=−110 dBm, and \(R^{0}_{f,k}=5\) kbps. As shown in Fig. 7, the crosstier interference is an increasing function of P _{max}. The crosstier interference from FBSs to MUEs in imperfect spectrum sensing case is lower than that in perfect spectrum sensing case. This is because the imperfect spectrum sensing case overestimates the crosstier interference with the consideration of sensing errors, such as false alarm and miss detection.
Figure 8 shows the total data rate of FBSs is an increasing function of maximum transmit power of FBS. As a comparison, the random subchannel allocation (RSA) and the equal power allocation (EPA) schemes are also evaluated in the simulation. In the RSA scheme, FBSs opportunistically access vacant subchannels in CSMA/CA manner. The equal power allocation allocates equal power for each subchannel, while the proposed algorithm performs an optimal power allocation with the consideration of imperfect spectrum sensing, crosstier interference, and QoS requirements. Figure 8 compares the following kinds of combination of subchannel allocation scheme and power allocation scheme: (i) RSA+EPA; (ii) RSA+CPO; (iii) DFSA+EPA; and (iv) DFSA+CPO(DIRA). In the simulation, d _{ f }=12BS/ km^{2}, I _{thre,s }=−110 dBm, and \(R^{0}_{f,k}=5\) kbps. It can be seen that the total data rate grows with the increase of P _{max}, and the proposed DIRA scheme significantly increases the total data rate of FBSs.
Conclusions
In this paper, we investigated the resource allocation problem in cognitive femtocell networks. A joint subchannel and power allocation algorithm was proposed to maximize the total data rate of femtocells with the consideration of fairness and imperfect spectrum sensing. Particularly, a CCDTbased subchannel allocation algorithm DFSA was developed to allocate subchannels to FBSs while guaranteeing the fairness among femtocells. We introduced spectrum sensing error probabilities to capture the imperfect spectrum sensing influence and combined them with OOB emission to formulate the crosstier interference constraint. Furthermore, due to the interference uncertainty, we formed the interference constraint as the chance constraint and implied Bernstein’s approximation to make it tractable. Finally, the optimal power allocation problem was solved by the Lagrangian dual method. Simulation results verified that our proposed algorithm can achieve fair subchannel allocation and significant data rate improvement.
References
 1
DT Ngo, T LeNgoc, Architectures of smallcell networks and interference management (Springer Publishing Company, Incorporated, Cham, 2014).
 2
X Huang, S Liu, Y Li, F Zhu, Q Chen, in 2016 IEEE 27th Annual International Symposium on Personal, Indoor, and Mobile Radio Communications (PIMRC). Dynamic cell selection and resource allocation in cognitive small cell networks (IEEE, Valencia, 2016), pp. 1–6.
 3
D LopezPerez, A Valcarce, RG lDe, J Zhang, OFDMA femtocells: a roadmap on interference avoidance. Commun. Mag. IEEE. 47(9), 41–48 (2009).
 4
X Huang, L Chen, Q Chen, B Shen, Coalition based optimization of resource allocation with malicious user detection in cognitive radio networks, KSII Trans. Internet Inf. Syst. 10(10), 4661–4680 (2016). doi:10.3837/tiis.2016.10.003.
 5
V Chandrasekhar, J Andrews, A Gatherer, Femtocell networks: a survey. Commun. Mag. IEEE. 46(9), 59–67 (2008).
 6
M Yavuz, F Meshkati, S Nanda, A Pokhariyal, Interference management and performance analysis of UMTS/HSPA+ femtocells. IEEE Commun Mag. 47(9), 102–109 (2009).
 7
SM Cheng, SY Lien, FS Chu, KC Chen, On exploiting cognitive radio to mitigate interference in macro/femto heterogeneous networks. IEEE Wirel. Commun. 18(3), 40–47 (2011).
 8
B Ma, HC Man, VWS Wong, J Huang, Hybrid overlay/underlay cognitive femtocell networks: a game theoretic approach. IEEE Trans. Wirel. Commun. 14(6), 3259–3270 (2015).
 9
DT Ngo, BL Long, T Lengoc, E Hossain, IK Dong, Distributed interference management in twotier CDMA femtocell networks. IEEE Trans. Wirel. Commun. 11(3), 979–989 (2012).
 10
H Peng, T Fujii, Joint resource allocation and interference avoidance with fairness consideration for multicell cognitive radio networks. Wirel. Commun. Netw. Conf, 1348–1353 (2012).
 11
T Ahmed, Y Le Moullec, in 2015 23rd Telecommunications Forum Telfor (TELFOR). Frequency and power allocation schemes for heterogeneous networks including femto cells (IEEE, Belgrade, 2015), pp. 277–280. doi:10.1109/TELFOR.2015.7377464.
 12
H Zhang, C Jiang, NC Beaulieu, S He, X Chu, in 2014 IEEE Global Communications Conference. Cooperative bargaining resource allocation for cognitive small cell networks (Austin, 2014), pp. 3080–3085. doi:10.1109/GLOCOM.2014.7037278.
 13
X Huang, B BeferullLozano, C Botella, Quasinash equilibria for nonconvex distributed power allocation games in cognitive radios. IEEE Trans. Wirel. Commun. 12(7), 3326–3337 (2013).
 14
HB Fan, EKL Chan, in IEEE Wireless Communications and NETWORKING Conference Vol.20. Joint uplink power and subchannel allocation in cognitive radio network (IEEE, Shanghai, 2012), pp. 306–311.
 15
E Bedeer, OA Dobre, MH Ahmed, KE Baddour, A multiobjective optimization approach for optimal link adaptation of OFDMbased cognitive radio systems with imperfect spectrum sensing. IEEE Trans. Wirel. Commun. 13(4), 2339–2351 (2014).
 16
Q Zhou, Z Chen, F Gao, JCF Li, M Lei, in ICC 2014  2014 IEEE International Conference on Communications Vol.99. Pricing and power allocation in sensingbased cognitive femtocell networks (IEEE, Sydney, 2014), pp. 5342–5347.
 17
Z Liu, J Wang, Y Xia, H Yang, Robust optimisation of power control for femtocell networks. Iet Signal Process. 7(5), 360–367 (2013).
 18
O Simeone, E Erkip, S Shamai Shitz, Robust transmission and interference management for femtocells with unreliable network access. IEEE J. Sel. Areas Commun. 28(9), 1469–1478 (2010).
 19
P Zhang, L Zhao, Z Liu, H Yang, in Control Conference. Robust power control for femtocell networks based on chanceconstrained linear programs (IEEE, Hangzhou, 2015), pp. 6536–6540.
 20
Y Zhang, S Wang, Resource allocation for cognitive radioenabled femtocell networks with imperfect spectrum sensing and channel uncertainty. IEEE Trans. Veh. Technol. 65(9), 1–1 (2016).
 21
NY Soltani, SJ Kim, GB Giannakis, Chanceconstrained optimization of OFDMA cognitive radio uplinks. IEEE Trans. Wirel. Commun. 12(3), 1098–1107 (2013).
 22
I Kim, HW Lee, in IEEE International Conference on Communications. Robust power allocation in cognitive radio networks with uncertain knowledge of interference (IEEE, Sydney, 2014), pp. 1609–1613.
 23
H Elsawy, E Hossain, M Haenggi, Stochastic geometry for modeling, analysis, and design of multitier and cognitive cellular wireless networks: a survey. IEEE Commun. Surv. Tutor. 15(3), 996–1019 (2013).
 24
S Lin, SJ Yoo, Distributed fair resource allocation for cognitive femtocell networks. Wirel. Pers. Commun, 1–20 (2017).
 25
A BenTal, A Nemirovski, Selected topics in robust convex optimization. Math. Program. 112(1), 125–158 (2008).
 26
3GPP TSG RAN WG4, (Radio) Meeting #51, R4092042, Simulation assumptions and parameters for FDD HeNB RF requirements (3GPP Press, San Francisco, 2009).
 27
A Nemirovski, A Shapiro, Convex approximations of chance constrained programs. SIAM J. Optim. 17(4), 969–996 (2006).
Acknowledgements
This work is supported by the National Natural Science Foundation of China (NSFC) (61401053) and Changjiang Scholars and Innovative Research Team in University (IRT1299).
Author information
Affiliations
Contributions
XH designed the study, conception, and main algorithms. LS drafted the article and designed the simulations and theoretical certifications. CZ worked with the data analysis and prepared the manuscript. QC reviewed and edited the manuscript. DZ worked with the data analysis and encoding. All authors have made substantive intellectual contributions to this study and approved the manuscript.
Corresponding author
Ethics declarations
Competing interests
The authors declare that they have no competing interests.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
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.
About this article
Cite this article
Huang, X., Shi, L., Zhang, C. et al. Distributed resource allocation with imperfect spectrum sensing information and channel uncertainty in cognitive femtocell networks. J Wireless Com Network 2017, 201 (2017). https://doi.org/10.1186/s1363801709851
Received:
Accepted:
Published:
Keywords
 Cognitive femtocell
 Resource allocation
 Imperfect spectrum sensing
 Interference uncertainty