Joint Load Balancing and Interference Mitigation in 5G Heterogeneous Networks
Abstract
We study the problem of joint load balancing (user association and user scheduling) and interference management (beamforming design and power allocation) in heterogeneous networks (HetNets) in which massive multipleinput multipleoutput (MIMO) macro cell base station (BS) equipped with a large number of antennas, overlaid with wireless selfbackhauled small cells (SCs) are assumed. Selfbackhauled SC BSs with fullduplex communication employing regular antenna arrays serve both macro users and SC users by using the wireless backhaul from macro BS in the same frequency band. We formulate the joint load balancing and interference mitigation problem as a network utility maximization subject to wireless backhaul constraints. Subsequently, leveraging the framework of stochastic optimization, the problem is decoupled into dynamic scheduling of macro cell users, backhaul provisioning of SCs, and offloading macro cell users to SCs as a function of interference and backhaul links. Via numerical results, we show the performance gains of our proposed framework under the impact of SCs density, number of BS antennas, and transmit power levels at low and high frequency bands. It is shown that our proposed approach achieves a gain in terms of celledge performance as compared to the closedaccess baseline in ultradense networks with SC BSs per .
I Introduction
To meet the massive data traffic demands in next generation G wireless networks a number of emerging technologies are currently investigated: ) higher frequency spectrum (mmWave); ) advanced spectralefficiency techniques (massive MIMO); and ) ultradense small cell deployments [2]. In this paper, we focus on the interplay between massive MIMO and a dense deployment of SCs in higher frequency bands. Massive MIMO plays an important role in wireless networks due to an improvement in energy and spectral efficiency [3]. In massive MIMO, a macro base station (MBS) equipped with a few hundreds antennas simultaneously serves tens of user equipments (UEs) and provides wireless backhaul to SCs, while the remaining degree of freedom of massive MIMO can be used to mitigate the crosstier interference. Ultra dense SC deployment provides an effective solution to increase network capacity by a factor of or more and offloads the wireless data from the MBS [4]. In order to reduce the deployment cost of SC, wireless backhaul has been considered as an attractive solution. In parallel to that, recent advances in fullduplex (FD) enables doubling spectral efficiency and lowering latency in which FDenabled SCs relay data from the massive MIMO MBS to the UEs in the same frequency band [5].
MmWave with short wavelength enables Massive MIMO to pack more antennas into highly directional footprint and to smartly do beamforming [6], making Massive MIMO practically feasible in real deployments. Recently, the efficiency of combining massive MIMO and inband wireless backhaulbased SC networks was studied in [5, 7], focusing on minimizing power consumption. The problem of user association for load balancing in heterogeneous networks (HetNets) has been studied in [8]. Although, users can be associated to more than one BS in order to reduce the load on the macro cell, deploying ultradense small cell networks makes user association more challenging. The work in [8] did not consider other important aspects in G such as Massive MIMO, FDenabled SCs, and mmWave communications. Recent work in [9] has addressed the usercell association for Massive MIMO HetNets, which did not consider the joint optimization of load balancing, precoder design, and power allocation. Also the wireless backhaul faces the problem of limitedbackhaul; hence the backhaul constraint needs to be considered. Thus far, the key challenge of how to dynamically optimize the overall network performance taking into account the backhaul dynamics and constraints, and load balancing utilizing the combination of Massive MIMO, FDenabled SCs, and mmWave communications has not been fully addressed [10].
User association taking into account dynamic backhaul in G HetNets faces a new challenge due to selfbackhauled SCs, i.e., guaranteeing wireless backhaul capacity between MBS and SCs in order to offload the traffic from MBSs to SCs. It raises the following important question: Should MBS serve all macro UEs (MUEs) even though it is highly loaded or offload some MUEs to SCs subject to the wireless backhaul capacity? Due to the random deployment of massive number of devices, UEs around hotspots (i.e. airport lounges, shopping malls, stations, and other crowded places) may receive poor services from afarMBS with multiple beams focused on the same location. On the contrary, these UEs will receive better services from nearby SCs with a reliable wireless backhaul composed of strong single beam from the MBS or multiple received antennas at SCs.
Ia Main Contributions
The main contributions of this work are to study the problem of joint load balancing, interference mitigation, and inband wireless backhauling taking into account dynamic backhaul and traffic load, which are listed as follows:

The problem of joint load balancing (user association and user scheduling) and interference management (beamforming design and power allocation) for G HetNets is modeled in which a DL scheduler is designed at the MBS to schedule macro UEs and provide backhaul to inband FDenabled SCs, with FD capability SCs serve both MUEs and small cell UEs in the same frequency band. Moreover, an interference management scheme is proposed to mitigate both cotier and crosstier interference from the MBS and FDenabled SCs by designing a hierarchical precoding scheme and controlling the transmission of SCs. The problem is cast as a network utility maximization (NUM) problem subject to dynamic wireless backhaul constraints, traffic load, and imperfect channel state information (CSI). To make problem tractable, by invoking results from random matrix theory (RMT), we derive a closedform expression of the signaltointerferenceplusnoiseratio () and transmit power when the numbers of MBS antennas and users grow very large.

A Lyapunov framework is applied in order to solve the NUM problem in polynomial time. The NUM problem is decomposed into dynamic scheduling of MUEs, backhaul provisioning of FDenabled SCs, and offloading MUEs to FDenabled SCs. The joint load balancing and operation mode (FD or halfduplex) subproblem, which is a nonconvex program with binary variables, is converted into a convex program by using the successive convex approximation (SCA) method. The motivations of using SCA are due to (i) its low complexity and fast convergence, and (ii) the obtained solution which yields many relaxed variables is close to zero or one.

A performance evaluation is carried out to compare the proposed algorithm with other baselines under the impact of SCs density, number of BS antennas, and transmit power levels at low/high frequency bands. The effect of pilot training and channel aging is also studied to show the performance of Massive MIMO.

A comprehensive performance analysis of our proposed algorithm based on the Lyapunov framework is provided. There exists an utilityqueue backlog tradeoff, which leads to an utilitydelay balancing [11], where is the Lyapunov control parameter. Moreover, a convergence analysis of the approximation method based on the SCA method is studied.
IB Related Work
The authors in [12] addressed the problem of dynamic resource control for HetNets with flexible backhaul (wired and wireless). However, the problem of load balancing when the number of antennas and users grows large is not considered. The user association problem has been studied for HetNets in [8, 9], which does not take into account backhaul constraints. As pointed out in [13, 10] the current solutions for user association problem ignore the backhaul constraints, which is very crucial since the capacity of open access SCs with either wired or wireless backhaul always faces the limited backhaul constraint. Moreover, the load balancing problem should take into account imperfect CSI due to mobility, which is ignored in the previous work. Our previous work in [1] has considered the problem of joint inband scheduling and interference mitigation in G HetNets without considering the user association. In this work, we extend [1] by considering the load balancing problem taking into account the backhaul constraint and imperfect CSI, and further provides insights into the performance analysis of our proposed algorithm based on the Lyapunov framework and convergence of the SCA method.
The rest of this paper is organized as follows.^{1}^{1}1The lowercase letters, boldface lowercase letters, (boldface) uppercase letters and italic boldface uppercase letters are used to represent scalars, vectors, matrices, and sets, respectively. and denote the Hermitian transpose and the rank of matrix , respectively. denotes the block diagonal matric whose diagonal blocks are given by and the identity matrix of size is denoted by . The cardinality of a set , is denoted by . denotes the Gaussian random distribution of zero mean and variance of . Section II describes the system model and Section III provides the problem formulation for load balancing and interference mitigation. Section IV introduces the Lyapunov framework used to solve our problem. In Section V, we present the numerical results. We conclude the paper in Section VI.
Ii System Model
Iia System Model
The downlink (DL) transmission of a HetNet scenario is considered as shown in Fig. 1 in which a MBS is underlaid with a set of uniformly deployed S FDenabled SCs, . Let denote the set of all base stations, where . The MBS is equipped with N number of antennas and serves a set of singleantenna M MUEs . Let denote the set of users associated with MBS , where . The user indices represent the corresponding MUEs indices , while the user indices represent the corresponding SCs indices . We assume open access policy at FDenabled SCs and each FDenabled SC is equipped with antennas: one receiving antenna is used for the wireless backhaul and transmitting antenna to serve its singleantenna small cell UEs (SUEs) or other MUEs at the same frequency band. Let denote the set of SUEs, where . Moreover, SCs are assumed to be FD capable with perfect selfinterference cancelation (SIC) capabilities^{2}^{2}2The case of imperfect SIC is left for future work.. Cochannel timedivision duplexing (TDD) protocol is considered in which the MBS and FDenabled SCs share the entire bandwidth, and the DL transmission occurs at the same time. In this work, we consider a large number of antennas at both macro and SC BSs and a dense deployment of MUEs and SCs, such that .
IiB Channel Model
We denote the propagation channel between the MUE and the antennas of the MBS in which is the channel between the MUE and the MBS antenna. Let denote the channel matrix between all MUEs and the MBS antennas. Moreover, we assume imperfect CSI for MUEs due to mobility and we denote as the estimate of in which the imperfect CSI can be modeled as [14]:
(1) 
where is the estimate of the smallscale fading channel matrix and is the spatial channel correlation matrix that accounts for path loss and shadow fading. Here, and are the real channel and the channel noise, respectively, modeled as Gaussian random matrix with zero mean and variance . The channel estimate error of MUE is denoted by ; in case of perfect CSI, . Similarly, let and denote the channel matrices from the MBS antennas to SCs and SUEs, respectively. Let denote the channel propagation from SC to any receiver . Let denote the SUE served by the SC .
Iii Load Balancing and Interference Mitigation
In this section, we formulate the joint optimization of user association, user scheduling, beamforming design, and power allocation. To that end, we first derive the received signal, data rate, and power transmit for each receiver (SCs are also treated as macro BS’s UEs). We then formulate the problem as a network utility maximization subject to wireless backhaul constraints. However, the formulated problem does not have closedform expressions for the objective and constraints. Hence, we apply RMT [15] to get these closedform expressions. We finally utilize the tool of stochastic optimization to decouple our problem into several solvable subproblems.
The problem of user scheduling and user association for load balancing in the DL is addressed in which the MBS simultaneously provides data transmission to MUEs and wireless backhaul to the FDenabled SCs, while the SCs with FD capability serve both SUEs and MUEs. For each MUE , let binary variable indicate the transmission association from BS to MUE , i.e., when MUE is associated with BS , otherwise . Similarly, let binary variables and denote the transmission association indicators from MBS to SC and from SC to SUE , respectively. We assume that each MUE connects to one BS (either MBS or SC ) at time slot . Each SC is equipped with transmitting antennas, and we assume that each SC serves up to active users (either SUE or MUE) at each time slot, such that , where the superscript stands for “active users”. Hence, we have the following constraints for load balancing:
(2) 
We define vector containing all transmission indicators between BSs and UEs. Let be the total number of transmissions at SC, where superscript stands for “transmissions”, and thus the latter of (2) becomes .
Iiia Downlink Transmission Signal
The MBS serves two types of users: MUEs with imperfect CSI and FDenabled SCs with perfect CSI. Let , , and denote the DL MBS transmit power assigned to MUE , the DL MBS transmit power assigned to SC , and the maximum transmit power at the MBS, respectively. We focus on the multipleinput singleoutput (MISO) channel, where the MBS with N antennas can serve K UEs. Here, we take into account user scheduling and association, and our proposal can apply to any special case when number of UEs is larger than number of antennas, i.e., . SC exploits FD capability to double capacity, FDenabled SC causes unwanted FD interference: crosstier interference to adjacent MUEs (or other SCs), and cotier interference to other UEs. Hence, in order to convert the interference channel to the MISO channel, we design a precoder at the MBS and propose an operation mode policy to control FD interference in order to treat the total FD interference as additional noise.
Definition 1
[Operation Mode Policy] We define as the operation mode to control the FDenabled SC transmission to reduce FD interference. The operation mode is expressed as . Here, indicates SC operates in FD mode and for halfduplex (HD) mode.
We assume that the MBS uses a precoding scheme, . To exploit the degrees of freedom of massive MIMO, the hierarchical interference mitigation scheme in [16, 17] is applied to design the precoder, i.e., , where is used to control cotier interference and capture the spatial multiplexing gain, and is used to suppress crosstier interference. Here, , where the subscript stands for “interference”. The precoder is chosen such that
(3) 
where is the sum of the correlation matrices between MBS antennas and users belong to SC . Here, is in the null space of . Note that determines that the transmission of FDenabled SC is enabled or not. The precoder is designed to adapt to the real time CSI based on , where . In this paper, we consider the regularized zeroforcing (RZF) precoding^{3}^{3}3Other precoders are left for future work. that is given by , where the regularization parameter is scaled by N to ensure that the matrix is well conditioned as . The precoder is chosen to satisfy the power constraint , where . We also assume that each SC uses ZF precoding to server its users, which reads such that is chosen to satisfy the equality power constraint ^{4}^{4}4We choose the equality constraints for transmit power at SCs to reach the optimal rate at maximum power rather than using , since the power at SCs is relatively small.. Here, . The channel propagation from the SC to the MUE (referred to as user ) is , where is the channel correlation matrix. Here, and are the real channel and the channel noise from SC to MUE , respectively, modeled as a Gaussian random matrix with zero mean and variance .
By utilizing a massive number of antennas at MBS, a large spatial degree of freedom is utilized to serve MUEs and FDenabled SCs, while the remaining degrees of freedom are used to mitigate crosstier interference. In massive MIMO system, the total number of antennas is considered as the degree of freedom [16]. Hence, we have the antenna constraint for user association and operation mode such that . For notational simplicity, we remove the time dependency from the symbols throughout the discussion. The received signal at each MUE at time instant is given by
(4) 
where is the signal symbol from the MBS to the MUE , is the precoding vectors of MUE , and is the thermal noise at MUE . While is the transmit signal symbol from SC to its user .
At time instant , the received signal at each SC suffers from selfinterference, crosstier and cotier interference, which is given by
(5) 
where is the signal symbol from the MBS to the SC , is the precoding vectors of SC , and is the thermal noise of the SC .
The received signal from the SC at receiver , , if the SC operates in HD mode, . For FD mode, , the received signal is given by
(6) 
where is the transmit data symbol from the SC to receiver and is the thermal noise at receiver . We imply that the receiver can be either a SUE or an MUE.
The precoder is designed at the MBS to null the cotier interference and to remove completely the crosstier interference to SCs’s users (3) and the selfinterference is well treated, while . Thus, according to (4)(6), the of an MUE served by MBS, a SC served by MBS, a receiver served by SC are given in (7)(9), respectively.
(7) 
(8) 
(9) 
IiiB Joint Load Balancing and Interference Mitigation Algorithm
Let us consider a joint optimization of load balancing , operation mode , interference mitigation , and transmit power allocation that satisfies the transmit power budget of MBS i.e. , . We define and as the FD interference to noise ratio (INR) from FDenabled SC to any scheduled receiver , and the allowed FD INR threshold, respectively. The FD interference threshold is defined such that , such that the total FD interference is considered as noise. Under the operation mode policy, we schedule the receiver and enable the transmission of SC as long as . Let be a composite control variable of user association and operation mode. We define as a composite control variable, which adapts to the spatial channel correlation matrix .
For a given that satisfies (3) and operation mode policy, the respective Ergodic data rates of SC and SUE are and . While from the constraint (2) the Ergodic data rate of MUE will depend on which BS the MUE is associated with, i.e., . In other words, the first term is the data rate from from the MBS to MUE when MUE is associated with the MBS, while the second term is when the FDenabled SCs allow MUE to connect (If MUE is connected to the FDenabled SC, then the rate of MUE should be the minimum between and data stream from the MBS via FDenabled SC to MUE, excepts other SC’s users).
Definition 2
For any vector , let denote the time average expectation of , where . Similarly, denotes the time average expectation of the Ergodic data rate.
For a given composite control variable that adapts to the spatial channel correlation matrix , the average data rate region is defined as the convex hull of the average data rate of users, which is expressed as:
where . Following the results from [18], the boundary points of the rate regime with total power constraint and no selfinterference are Paretooptimal^{5}^{5}5The Pareto optimal is the set of user rates at which it is impossible to improve any of the rates without simultaneously decreasing at least one of the others.. Moreover, according to [19, Proposition 1], if the INR covariance matrices approach the identity matrix, the Pareto rate regime of the MIMO interference system is convex. Hence, our rate regime is a Paretooptimal, and thus is convex with above constraints.
Let us assume that each FDenabled SC acts as a relay to forward data to its users. If the MBS transmits data to FDenabled SC , but the transmission of SC is disabled, it cannot serve its SUE. Hence, we define as a data queue at SCs, where at each time slot , the wireless backhaul queue at FDenabled SC is
(10) 
The SC offloads some MUEs from the MBS if the wireless backhaul capacity between the SCs and the MBS is guaranteed, and hence, for each SC we have the following wireless backhaul condition for all : “If the access link between the MUE and the MBS is better than the link between the MUE and the SCs, then the MUE connects with the MBS rather than with other SCs”, i.e.,^{6}^{6}6The queues of MUEs are handled at the MBS and SCs strictly handle data for SUEs, hence when SCs open connection for MUEs, they should have immediate capacity in terms of data rate. We do not include the constraint (11) for the closed access case in [1].
(11) 
Definition 3
[Queue stability] For any discrete queue over time slots and , is stable if . A queue network is stable if each queue is stable.
We define the network utility function to be nondecreasing, concave over the convex region for a given . The objective is to maximize the network utility under wireless backhaul constraints and imperfect CSI. Thus, the NUM problem is given by,
(12a)  
subject to  (12b) 
where with is the weight of user , is assumed to be twice differentiable, concave, and increasing Lipschitz function for all . Solving (12) is nontrivial since the average rate region does not have a tractable form. To overcome this challenge, we need to find closedform expressions of the data rate and the average transmit power. Inspired by [15], we invoke RMT to get the closedform expressions for the user data rate and transmit power as .
IiiC ClosedForm Expression via Deterministic Equivalent
We invoke recent results from RMT in order to get the deterministic equivalent of user rate and transmit power via Theorem 1.
Theorem 1
Recall that is the RZF parameter. As ; , by applying the technique in [15, Theorem 2], the deterministic equivalent of the asymptotic of MUE is
where denotes the almost sure convergence and . Here, forms the unique positive solution of which is the Stieltjes transform of nonnegative finite measure [15, Theorem 1], where . In addition, , and . , and are given by , , where . , are given by , , . Similarly, the SINR of SC is
The power constraint at the MBS can be calculated as . Moreover, following the analysis in the proof of [15, Theorem 3], [16, Lemma 6] for a small fixed^{7}^{7}7The deterministic equivalent holds for a small fixed as studied in [16], while the problem of finding the optimal value has been studied in [15, 17]. , and yield the deterministic equivalent of the asymptotic SINRs of UEs (7)(9) as
(13) 
(14) 
(15) 
Moreover, we obtain the closedform expression for the transmit power constraint, i.e.,
Although the closedform expressions of average data rate and transmit power are obtained, our problem considers a timeaverage optimization with a large number of control variables, and dynamic traffic load over the convex region for a given composite control variable and . Our aim is to maximize the aggregate network utility subject to queue stability in which the wellknown Lyapunov optimization yields an utility throughput optimality and stability [20]. Hence, we apply the driftpluspenalty technique [20] to solve load balancing, operation mode selection, and power allocation problems.
Iv Lyapunov Optimization Framework
The network operation is modeled as a queueing network that operates in discrete time . Let denote the bursty data arrival destined for each user , i.i.d over time slot . Let denote the vector of transmission queue blacklogs at MBS at slot . The queue evolution is given by
(16) 
Here, we consider the bound of the traffic arrival of user is bounded such that , for some constant . Futuremore, let be the upper bound of data rate for user at time slot , such that . The set at constraint (12b) is replaced by an another equivalent set by introducing auxiliary variables , that satisfies , where . The evolution of wireless backhaul queue is rewritten as
(17) 
For a given and , the optimization problem (12) subject to the network stability and dynamic backhaul can be posed as
(18a)  
subject to  (18b)  
(18c) 
In order to ensure the inequality constraint (18b), we introduce a virtual queue vector which evolves as follows
(19) 
We define the queue backlog vector as (whereas the stability of yields all constraints of problem (18) are hold). The Lyapunov function can be written as
For each time slot , denotes the Lyapunov drift, which is given by
Noting that and for any real positive number , and thus, by neglecting the index we have: