Joint Frame Design, Resource Allocation and User Association for Massive MIMO Heterogeneous Networks with Wireless Backhaul

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1 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL.XXX, NO.XXX, MONTH YEAR 1 Joint Frame Design, Resource Allocation and User Association for Massive MIMO Heterogeneous Networks with Wireless Backhaul Mingjie Feng, Student Member, IEEE, Shiwen Mao, Senior Member, IEEE, and, Tao Jiang Senior Member, IEEE Abstract In this aer, we investigate the roblem of frame design, resource allocation, and user association in a massive MIMO heterogeneous network HetNet) with wireless backhaul WB) and linear rocessing. The objective is to maximize the sum downlink rate of all users, subject to constraints on data rate of WB and fairness-aware constraints. Such a roblem is formulated as an integer rogramming roblem with both couled variables and couled constraints. We first develo a centralized scheme in which we decomose the original roblem into two subroblems and iteratively solve them until convergence to achieve a near-otimal solution. We then roose a distributed scheme by formulating a reeated game among all users and rove that the game converges to a Nash Equilibrium NE). Simulation studies show that the roosed schemes are adative to different network scenarios and traffic atterns, and achieve considerable gains over several benchmark schemes. Index Terms 5G Wireless; Massive MIMO; HetNet; Crosslayer Otimization; Wireless Backhaul. I. INTRODUCTION With the fast growing oularity of smart mobile devices and the exlosion of data-intensive services, the wireless system is exected to rovide a 1000x mobile data rate in the near future. To suort such high data rate with limited sectrum, aggressive sectrum reuse must be realized to achieve high sectral efficiency. To this end, massive MIMO Multile Inut Multile Outut) and small cell are recognized as two key technologies for emerging 5G wireless systems [2]. Massive MIMO refers to a cellular system with more than 100 antennas equied at the base station BS), which serves multile users with the same time-frequency resource [3]. A massive MIMO system can dramatically imrove the energy and sectral efficiency comared to traditional wireless systems due to highly efficient satial multilexing [4] [6]. Small cell Manuscrit received June 23, 2017; revised Oct. 31, 2017; acceted Dec. 17, This work was suorted in art by the NSF under Grants CNS and CNS , the Wireless Engineering Research and Engineering Center at Auburn University, and in art by the National Science Foundation for Distinguished Young Scholars of China NSFC) with Grant number , and the NSFC with Grant number M. Feng and S. Mao are with the Deartment of Electrical and Comuter Engineering, Auburn University, Auburn, AL USA. T. Jiang is with the School of Electronic Information and Communications, Huazhong University of Science and Technology, Wuhan, China. mzf0022@auburn.edu, smao@ieee.org, tao.jiang@ieee.org. This work was resented in art at IEEE SECON 2017, San Diego, CA, June Coyright c 2018 IEEE. Personal use of this material is ermitted. However, ermission to use this material for any other uroses must be obtained from the IEEE by sending a request to ubs-ermissions@ieee.org. Digital Object Identifier XXXX/YYYYYY deloyment, which forms a heterogeneous network HetNet), is another efficient aroach to enhance sectral efficiency. Due to the short transmission range, high signal to noise ratio SNR) and dense sectrum reuse can be achieved, resulting in significantly imroved network caacity. As an integration of these two techniques, massive MIMO HetNet has drawn considerable attention recently [7] [11], where the macrocell BS MBS) is equied with massive MIMO. In a massive MIMO HetNet, the MBS is equied with a large number of antennas. The MBS and small cell BS s SBS) collectively serve users in the cell. With such a network architecture, the MBS with massive MIMO can rovide a good quality of service QoS) to users located in the coverage holes of SBS s. Moreover, the SBS s can offload some traffic from the MBS so that the overhead and comlexity of rocessing at MBS can be reduced, resulting in erformance enhancement of users that are still served by the MBS. With the exected massive deloyment of small cells, connecting all SBS s to the core network directly with dedicated otical fiber may not be feasible due to significantly increased cost. Alternatively, the SBS s can be connected to the core network by transmitting data to the MBS through backhaul links. In this case, the design of backhaul system is an imortant issue of a HetNet. Although a massive MIMO HetNet can rovide high data rate links between users and BS s, the transmissions between MBS and SBS s may become the bottleneck of the network. Without a reliable backhaul, the aggregated data rates of small cell user equiments SUE) would be limited by the data rate of the backhaul link. For services with stringent delay requirements, the QoS of users may become unaccetable or even causing outages. Most existing works have considered wired backhaul between SBS s and MBS, since a wired connection can suort high data rate and it is more reliable in general. However, in a HetNet with large number of SBS s, wired connections to each SBS may not be cost-effective or even may be infeasible due to ractical constraints. Moreover, the wired backhaul deloyment may be highly inefficient when the wireless service rovider needs to ugrade or extend the network. Thus, the wireless backhaul WB) has the otential to lay an increasingly imortant role in 5G networks due to its easy and fast deloyment, flexibility, and low cost [12] [14]. In fact, WB in a massive MIMO HetNet can be quite reliable with roer configurations, esecially when massive MIMO are alied with linear rocessing techniques. From

2 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL.XXX, NO.XXX, MONTH YEAR 2 the ersective of an MBS, suorting a WB transmission is equivalent to serving a macrocell user equiment MUE). With linear rocessing, e.g., maximum ratio combination MRC) and maximum ratio transmission MRT), the recetion and recoding are based on linear functions of channel resonse matrices. When the number of antennas goes to infinity, the inner roducts of channel vectors of different links grow at a lower rate than that of the number of antennas, the interference between different WBs or MUEs can be averaged out [3]. Thus, the MBS can rovide high data rate links to multile WBs with simle linear rocessing techniques. The use of WB in massive MIMO HetNet has drawn some attentions recently [15] [18]. In [15], a joint user association and bandwidth allocation scheme was roosed to maximize the downlink sum logarithmic data rate in a massive MIMO HetNet with zero-forcing ZF) at MBS. A comarison of three WB deloyment strategies are resented in [16], namely comlete time division dulex, zero division dulex, and zero division dulex with interference rejection. An analytical framework based on stochastic geometry was resented in [17] to study the WB erformance in a massive MIMO HetNet with full-dulex small cells, and a closed-form exression of coverage robability was derived. In [18], the network architecture and feasibility issues of WB on the mmwave band were investigated in a dense HetNet with massive MIMO. Although these works resented several highly efficient aroaches, otimal frame design on ilots, i.e., the number of symbols used for ilots in each frame, has not been considered. Here, a frame is defined as a time-frequency resource block and the size of each frame is determined by the coherence time and coherence frequency of all UEs. In each frame, a certain fraction of time is used to transmit symbols that are used as ilots, and these ilots are sent by MUEs and WBs to estimate their channel gains to the MBS. While existing works assume a fixed fraction of time dedicated for ilot, the ilot length, i.e., the number of symbols used for ilots in each frame, can be adative to the traffic attern for erformance enhancement. There is clearly a trade-off on ilot length here. As discussed, the WBs and MUEs are equivalent from the MBS s oint of view. When the ilot length is large, more time is sent on channel estimation at MBS, and a large number of MUEs and WBs can be suorted. Moreover, the MUEs and WBs can be allocated with more channels since there is enough time to estimate all these channels. However, as a large fraction of time is dedicated to ilots, the remaining time for data transmission is small, resulting in a low data rate. When the ilot length is small, the fraction of time for data is increased, but the MUEs and WBs are allocated with less number of channels, which limits the data rates of MUEs and WBs. With a small data rate for WBs, the aggregated data rates of SUEs are limited, resulting in a oor erformance. In this aer, we investigate the roblem of joint frame design, resource allocation, and user association to maximize the downlink sum rate of all users under the WB and fairness constraints. We develo efficient centralized and distributed schemes to obtain the near-otimal solutions to the formulated roblem. The main contributions of this aer are as follows. We consider joint ilot length otimization, resource allocation, and user association in a massive MIMO HetNet with WB and linear rocessing, and rovide a rigorous roblem formulation. We roose a centralized iterative algorithm. The original roblem is decomosed into two subroblems and we iteratively solve them until convergence. The first roblem is joint ilot length otimization and resource allocation for MUEs and WBs, and we emloy a rimal decomosition aroach to obtain its otimal solution. The second roblem is user association, and we obtain its near-otimal solution with a cutting lane aroach. An iterative framework is designed to udate the arameters of the two subroblems in each iteration to minimize the erformance ga between the two roblems and guarantee that all constraints are satisfied. We roose a distributed scheme by formulating a reeated game among all users, and rove that the game converges to a Nash Equilibrium NE). The erformances of the roosed schemes are comared with several benchmark schemes. The simulation results show that erformance gains can be as much as more than 100% under certain circumstances. In the remainder of this aer, we resent the system model and roblem formulation in Section II. The centralized and distributed schemes are resented in Sections III and IV, resectively. We discuss our simulation study in Section V. Section VII concludes this aer. II. PROBLEM FORMULATION We consider a noncooerative multi-cell cellular system with focus on a tagged macrocell denoted as macrocell 0). Macrocell 0 is a two-tier HetNet consisting of an MBS with massive MIMO indexed by j = 0) and J single-antenna SBS s indexed by j = 1,2,...,J). The ayload data of SUEs is transmitted to the core network via WBs between the MBS and SBS s. Then, the reversed time division dulex RTDD) scheme is a natural choice for the MBS and SBS s [15]. With RTDD, the ulink and downlink transmissions of MBS and SBS s are erformed in a reversed attern, so that an SBS can transmit ulink data to receive downlink data from) the MBS, and transmit downlink data to receive ulink data from) SUEs simultaneously. The RTDD scheme is easy to imlement in a ractical system since it does not require interference cancellation at SBS s. There are K single-antenna mobile users indexed by k = 1,2,...,K). Each user can be served by either the MBS or an SBS. We define binary variables for user association as x k,j. = { 1, user k is associated with BS j 0, otherwise, k = 1,2,...,K, j = 0,1,...,J. 1) The sectrum band owned by the wireless service rovider WSP) is divided into N channels, and the bandwidth of each channel is defined to be the coherence bandwidth of massive MIMO terminals [21]. We assume the MBS adots linear rocessing schemes with MRC at receiver and MRT at transmitter [3], [19]. From the oint of view of MBS,

3 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL.XXX, NO.XXX, MONTH YEAR 3 a WB is equivalent to a user to be served. Thus, we can take advantage of the favorable roerties of massive MIMO by serving all MUEs and WBs on a same set of channels. This way, they can be ut into the beamforming grous on these channels. Due to the law of large numbers, the interference between any two links in a beamforming grou can be averaged out. From the ersective of an SBS, a WB is also equivalent to a user to be served. However, since the SBS s are assumed to be equied with single antenna, they cannot erform interference mitigation in the satial domain or self-interference cancelation. Hence, orthogonal resources must be assigned between WBs and SUEs to avoid mutual interference. Consequently, we assume that a roortion of α of the whole bandwidth is allocated to WBs and MUEs, and the rest 1 α) is allocated to SUEs. Note that α needs to be consistent across all macrocells to avoid cross-tier interference, and it is redetermined by the service rovider. We assume both the bandwidth of each frame and the bandwidth of a channel equal to the coherence bandwidth of all MUEs and WBs, given as W c. Then, each frame corresonds to a secific interval on a channel. The duration of a frame is T c seconds, which equals to the coherence time of all MUEs and WBs. Thus, the channel gains are constant in a frame and each frame can be viewed as a coherence block. The interval of a symbol is T s seconds, which consists of T u seconds for useful symbols and T g = T s T u seconds for guard interval. Let f be the sacing of subcarriers, then T u is given as T u = 1/ f. Within a coherence bandwidth, there are W c / f subcarriers. Hence, the channel resonse is constant over N sm = W c / f consecutive subcarriers in each symbol. Let τ be the ilot length, i.e., the number of OFDM symbols dedicated for ilots in each frame. Then, the number of terminals that can be suorted in each frame is τn sm. Therefore, the total number of MUEs and WBs that can be served by the MBS on each channel within the interval of a frame is uer bounded by τn sm. Given the available sectrum band for MUEs and WBs, we define the following resource allocation indicators a k,n. = { 1, channel n is allocated to MUE k 0, otherwise, k = 1,2,...,K, n = 1,...,αN. 2) b j,n. = { 1, channel n is allocated to SBS j s WB 0, otherwise, According to our analysis, we have K a k,n + j = 1,2,...,J, n = 1,...,αN. 3) J b j,n τn sm, n = 1,...,αN. 4) j=1 In a massive MIMO system, the effects of fast fading and noise vanish as the number of antennas goes to infinity; the only ossible interference comes from the UEs that share the same ilot sequence [3]. Thus, the only factor that limits the erformance of a massive MIMO system with linear rocessing is ilot contamination. For user k connecting to the MBS in macrocell 0, let macrocell l be the neighboring macrocells) that uses the same ilot sequence as user k. The downlink signal to interference ratio SIR) of user k when it connects to the MBS in the tagged macrocell, γ k,0, is γ k,0 = β 2 k,0/ l 0 β 2 k,l, 5) where β k,0 is the factor accounting for the roagation loss and shadowing effects between the MBS and user k, and β k,l accounts for the roagation loss and shadowing factor between user k and the MBS in macrocell l. When neighboring macrocells use different values of τ, an MBS receives not only the ilot signals of users from other cells, but also ulink data signals from other cells. As analyzed in [19], the nonorthogonal ulink data signals also contaminate the channel estimation of other cells, and the resulting interference is a random variable bounded by the interference caused by ilot signals. Hence, we use 5) as a worst-case aroximation in case the SIR cannot be measured by the MBS due to technical limits. When the values of τ are close to each other in different macrocells, such aroximation would be highly reliable. Due to the mobility of users, we assume that γ k,0 is udated with a eriod of T seconds. The data rate of user k is given by [3] R k,0 = a k,n 1 T T c τ ) Tu T s ) log1+γ k,0 ), 6) where T is the time sent to transmit ilot for one user and T = T s. 1 Due to channel recirocity of the TDD mode, the CSI is acquired by the MBS using ulink ilots. Then, γ k,0 and R k,0 can be obtained by the MBS. Similarly, let γ j be the downlink SIR of WB between the MBS and SBS j, it is given by γ j = β 2 j,0/ l 0 β 2 j,l, 7) where β j,0 is the factor accounts for the roagation loss and shadowing effects between the MBS and SBS j, and β j,l is the roagation loss and shadowing factor between SBS j and the MBS in macrocell l. The data rate of the WB for SBS j is then given as C j = b j,n 1 T T c τ ) Tu T s ) log1+γ j ). 8) We assume that the time interval for ulink ilots of MUEs and WBs are used to send control information from SBS s to SUEs, including CSI, ower and channel schedule of SUEs. We also assume that equal resource allocation is alied to SUEs served by the same SBS so that roortional fairness can be achieved [15]. Let γ k,j be the average signal to noise lus interference ratio SINR) of user k connecting to SBS j 1 The value of T can also be otimized based on hysical layer analysis. According to 6), a small value of T reduces the channel estimation overhead and increasesr k,0. However, the channel estimation quality may be degraded, resulting in decreased R k,0. Due to sace limit, we focus on frame level analysis and network scheduling roblems, the otential of otimizing T with hysical layer analysis can be investigated in future work.

4 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL.XXX, NO.XXX, MONTH YEAR 4 over a time eriod. The achievable data rate is given as R k,j = 1 T ) ) Tu 1 α)n τ T c T K s x log1+γ k,j ). 9) k,j We assume that the owers of SBS s and SUEs are adjusted to roer values so that the interference between different small cell users are controlled at an accetable level. Unlike the MBS with massive MIMO, the effect of fast fading exists on the channel between an SUE and an SBS, resulting in frequently varying CSI. Therefore, it is infeasible to use the instantaneous CSI for scheduling uroses. To this end, γ k,j is based on the time-averaged CSI measured by the SBS over T seconds in the revious eriod, and it is udated every T seconds. We aim to maximize the sum rate of a massive MIMO HetNet. Let x, a, and b denote the matrices of {x k,j }, {a k,n }, and {b j,n }, resectively. The roblem is formulated as P1 : max {x,a,b,τ} J j=0 x k,jr k,j 10) subject to: J j=0 x k,j 1, k = 1,2,...,K 11) x k,j S j, j = 0,1,...,J 12) a k,n + J j=1 b j,n τn sm, n = 1,...,αN 13) a k,n E k, k = 1,2,...,K 14) b j,n F j, j = 1,2,...,J 15) x k,jr k,j C j, j = 1,2,...,J 16) τ τ max, τ N + 17) a k,n {0,1}, b j,n {0,1}, x k,j {0,1}, n = 1,...,αN,k = 1,...,K,j = 0,...,J. 18) In roblem P1, constraint 11) is because each user can connect to at most one BS. We enforce an uer bound on the number of users that can be served by each BS in 12) to guarantee the QoS of users. Constraint 13) is directly from 4). By enforcing an uer bound on the number of channels that can be accessed by user k, constraint 14) is to guarantee fairness among the MUEs. Without such constraint, MUEs with high SIRs would be allocated with more channels than those with low SIRs, resulting in oor fairness erformance. 2 Thus, the value of E k for an MUE with high SIR is set to be lower than an MUE with low SIR. 3 Similarly, constraint 15) is to guarantee fairness among the WBs. Constraint 16) is due to the fact that the data rate of 2 Due to the channel hardening effect, the channel gains across different frequencies are close to each other [31]. Thus, the dominant factor that imacts the erformance of WBs and MUEs is the number of allocated channels. However, in other alication scenarios where the channel resonse varies significantly over different frequencies, e.g., in a mmwave network, frequency domain scheduling should be considered. Some existing aroaches can be alied include roortional fairness scheduling [22] and biartite matching based algorithm [40]. 3 The roer values of E k and F j deend on network toology, traffic attern, and QoS requirement of users. In a secific system, E k and F j can be dynamically adjusted based on the QoS of users. When the data rate of a MUE or WB at the edge of cell is lower than a threshold, the values of E k and F j for the MUEs and WBs with highest data rates will be lowered in the next eriod. The adjustment strategy of E k and F j can be done with an offline training rocess for each cell. WB for SBS j should be larger than or equal to the sum rate of all SUEs served by SBS j. Constraint 17) enforces an uer bound for the number of symbols for ilot transmissions. Since both γ k,0 and γ k,j are udated with the eriod of T, roblem P1 is also solved with the eriod of T. III. CENTRALIZED SOLUTION ALGORITHM In this section, we develo a centralized iterative scheme to obtain the near otimal solution of P1. Problem P1 is an integer rogramming roblem with both couling variables and couling constraints, and constraint 16) is a nonlinear couling constraint of two sets of variables. Thus, standard otimization techniques cannot be directly alied for the otimal solution. To make the roblem tractable, we decomose roblem P1 into i) WB and MUE resource allocation and ilot length otimization roblem and ii) user association roblem, and iteratively solve the two roblems until convergence. At each iteration, we udate the constraints of each roblem to satisfy all constraints of the original roblem. A. Resource Allocation and Pilot Otimization As can be seen in 6) and 9), R k,0 is determined by a; and R k,j, j = 1,...,J, is limited by b. Due to constraint 16), the sum rate of all MUEs and WBs naturally serves as an uer bound for the sum rate of all users. Thus, it is reasonable to try to maximize this uer bound and iteratively tighten the ga, so that the final solution is a close aroximation for the otimal solution of Problem P1. The roblem of maximizing the sum rate of all MUEs and WBs for a given x is as follows. P2 : max K {a,b,τ} 1 T ) τ 19) T c J a k,n log1+γ k,0 )+ b j,n log1+γ j ) subject to: 13) 18). j=1 Note that, constraint 16) can be written as b j,n x k,jr k,j log1+γ j). Since b j,n is always an integer, 16) is x k,jr k,j log1+γ j) equivalent to b j,n Suose constraint 16) has already been satisfied for the WB of SBS j, then allocating more resources to this WB can not imrove the actual sum rate of the users served by SBS j, while it otentially increases the value of τ, resulting in degraded system erformance. Thus, 16) is an active constraint in roblem P2. We have αn b K j,n = x k,jr k,j log1+γ j). Combining this constraint with 15), we have { K } b j,n = min x k,jr k,j,f j.j = 1,2,...,J. log1+γ j ). 20) To solve roblem P2, we first relax the integer constraints of a, b, and τ by allowing them to take any values in [0,1]. Lemma 1. The relaxed roblem of P2, P2-Relaxed, is a convex otimization roblem.

5 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL.XXX, NO.XXX, MONTH YEAR 5 Proof: The objective function of P2-Relaxed is a sum of quadratic terms and linear functions, with the quadratic terms given as τa k,n and τb j,n. It can be easily verified that the Hessian matrices of such quadratic terms are negative definite. Thus, the objective function is concave. Since all constraints are linear, P2-Relaxed is a convex otimization roblem. Since the decision variables are couled in the constraints, we use a rimal decomosition to transform roblem P2- Relaxed into two levels of roblems [26]. At the lower level, we find otimal solution of a and b for a given τ. Based on the solution of the lower level roblem, the otimal value of τ is then obtained with a subgradient aroach. 1) Otimal Solution of a and b for Given τ: With given τ, we have the following lower level roblem of P2-Relaxed. P3 :max K {a,b} + J j=1 a k,n log1+γ k,0 ) b j,n log1+γ j ) 21) subject to: 13),14),18), and 20). We can see that P3 is a linear rogramming LP), which can be solved with efficient methods such as simlex method. To analyze its roerty, we transform P3 into the standard form by concatenating the columns of a and b alternately, given as ỹ = [a 1,1,...,a K,1,b 1,1,...,b J,1,a 1,2,...,a K,2, 22) b 1,2,...,b J,2,...,a 1,αN,...,a K,αN,b 1,αN,...,b J,αN ] T. Let Z be the constraint matrix corresonding to ỹ, as Z = ) The right hand side RHS) of the LP is a αn +J +K) 1 vector, given by d = [τn sm,...,τn sm,e 1,...,E K,θ 1,...,θ J ] T, 24) { K where θ j = min x k,jr k,j /log1+γ j ),F j }. Lemma 2. The constraint matrix Z is totally unimodular. Proof: Omitted due to lack of sace, a similar case and roof can be found in [11]. Proerty 1. If the constraint matrix of an LP satisfies totally unimodularity, and the RHS is integral, then it has all integral vertex solutions [24]. Proerty 2. If an LP has feasible otimal solutions, then at least one of the feasible otimal solutions occurs at a vertex of the olyhedron defined by its constraints [25]. Lemma 3. All the decision variables in the otimal solution to the relaxed LP, roblem P3, are integers in {0,1}. Proof: This lemma directly follows Lemma 2, Proerty 1, and Proerty 2. 2) Otimal Value of τ: Denote gaτ),bτ),τ) and f aτ),bτ)) as the values of objective functions of P2- Relaxed and P3 for a given τ, which are given in 19) and 21), resectively. Letg τ) andf τ) be their otimal values for a given τ, resectively. At the higher level of roblem P2-Relaxed, we find the otimal value of τ by solving the following roblem. P4 : max {τ} g τ). 25) Consider the objective function of P2-Relaxed, given as gaτ),bτ),τ) = 1 T ) τ f aτ),bτ))). 26) T c Maximizing 26) is equivalent to maximizing the following log 1 T ) τ +log[f aτ),bτ))]. 27) T c Hence, roblem P4 is equivalent to the following roblem { max log 1 T ) } τ +log[f a τ),b τ))] 28) {τ} T c subject to: 17). ) Let h 1 τ) = log 1 T T c τ, h 2 τ) = log[f a τ),b τ))], and hτ) = h 1 τ) + h 2 τ). Since P2-Relaxed is a convex roblem according to Lemma 1, we can aly rimal decomosition to otimize h 1 τ) and h 2 τ) searately [26]. It can be easily verified that h 1 τ) is a differentiable concave function. For any τ and τ, we have log 1 T ) τ log 1 T ) τ T T c T c T c T τ τ τ ). Then, τ can be udated with the following gradient aroach to maximize h 1 τ). τ [t+1] = τ [t] T T c T τ [t]ρ[t], 29) where t is the index of iteration and ρ [t] is the ste size. To obtain the otimal solution of h 2 τ), we consider the following otimization roblem P5 : maxlog[f aτ),bτ))] {a,b} subject to: 13),14),18), and 20). Lemma 4. Strong duality holds for roblem P5. Proof: Since roblem P5 is a convex roblem, all the constraints are linear and the Slater condition reduces to feasibility [15], [23]. Thus strong duality holds. Let λ n be the otimal value of Lagrangian multilier corresonding to the constraint a k,n + J j=1 b j,n τn sm. We consider the otimal solutions to P5 for two different

6 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL.XXX, NO.XXX, MONTH YEAR 6 values, τ and τ. Then, we have h 2 τ ) = log[f a τ ),b τ ))] a) = La τ ),b τ ),λ τ ),µ τ ),ν τ ),η τ )) b) La τ),b τ),λ τ ),µ τ ),ν τ ),η τ )) = log[f a τ),b τ))]+ +Φ+ λ nτ )τ N sm τn sm ) λ nτ )τn sm δ nτ)) c) αn h 2 τ)+n sm λ nτ )τ τ), 30) where δnτ) = a k,n τ) + J j=1 b j,n τ), µ, ν, and η are the Lagrangian multiliers corresonding to other constraints. Φ is given as ) K Φ = µ kτ ) E k a k,n τ) + + J νj τ ) F j j=1 b j,n τ) J ) ηj τ R k,j ) b j,n τ). 31) log1+γ j ) j=1 In 30), equalitya) is due to strong duality, inequalityb) is due to the otimality of a τ ) and b τ ), and inequality c) is due to the constraints of roblem P5 and the nonnegativity of all Lagrangian multiliers. It follows 30) that αn h 2 τ) h 2 τ )+N sm λ nτ )τ τ ). 32) By definition, N sm λ nτ) is a subgradient of h 2 τ). The maximum value of h 2 τ) can be obtained by αn τ [t+1] = τ [t] +N sm λ n[t] ρ [t] 33) Lemma 5. Problem P4 can be solved by the following subgradient method. ) αn τ [t+1] = τ [t] + N sm λ n[t] T ρ [t]. 34) T c T τ [t] N sm Proof: According the rinciles of rimal decomosition, λ n [t] T T c T τ is a subgradient of hτ), τ can be [t] udated by combining 29) and 33). The otimal value of τ can be achieved until iteration converges. There is a nice interretation for 34). In each udate, αn N sm λ n [t] indicates the erformance gain obtained by allocating more ilot symbols to WBs and MUEs, i.e., to increase τ. The second art, T T c T τ indicates the erformance [t] loss caused by the reduced number of data symbols. Denote η [t] as the subgradient of hτ), η [t] = ) N sm λ n [t] T T c T τ, the convergence of the τ is [t] shown in the following lemma. Lemma 6. With ste size set as ρ [t] = hτ ) hτ [t] ) η [t] ) 2, the sequence hτ [t] ) converges to its otimal value hτ ) with a seed faster than {1/ t} as t. Proof: Consider the otimality ga of τ, we have τ [t+1] τ ) 2 τ [t] + hτ ) hτ [t] ) η [t] ) 2 η [t] τ hτ ) hτ [t] ) ) 2 = τ [t] τ ) 2 + η [t] ) 2 +2τ [t] τ )η [t]hτ ) hτ [t] ) η [t] ) 2 hτ τ [t] τ ) 2 ) hτ [t] ) ) 2 η [t] ) 2 hτ τ [t] τ ) 2 ) hτ [t] ) ) 2 η 2, where η is an uer bound of η [t]. The first inequality is because τ [t+1] should roject to [0,τ max ], the second inequality is due to the roerty of subgradient, given as τ [t] τ )η [t] hτ [t] ) hτ ). Summing the above inequality from t = 1 to t, we have 2 hτ ) hτ )) [t] η 2 τ [1] τ ) 2. 35) t=1 Suose for contradiction, lim hτ ) hτ [t] ) ) t > 0. t Then, there must be a sufficiently large t and a ositive number ξ such that hτ ) hτ [t] ) ) t > ξ, t t. Taking the square sum from t to, we have ) 2 hτ ) hτ [t] ) ξ 2 1 =. 36) t t=t t=t It can seen that 36) contradicts 35). Thus, the hyothesis does not hold, we have ) 2 hτ ) hτ [t] ) lim t 1/ = 0, 37) t this indicates that hτ [t] ) converges with a seed faster than that of 1/ t. Note that, the otimal τ to P2-Relaxed may not be an integer. Since P2-Relaxed is a convex roblem, a simle way to find the otimalτ to P2 is to comare the objective values of roblem P2 under τ and τ, and select the larger one. As discussed in Lemma 3, the otimal solution to P2-Relaxed are integers for any given integer value of τ. Thus, such solution is also otimal to P2, we conclude that the otimal solution of P2 can be obtained. The rocedure of the roosed WB and MUE resource allocation and ilot length otimization scheme is summarized in Algorithm 1. Lemma 7. The comlexity of Algorithm 1 is uer bounded by 1/ε 2 1ε 2 2, where ε 1 is the threshold of convergence for τ, ε 2

7 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL.XXX, NO.XXX, MONTH YEAR 7 Algorithm 1: WB and MUE Resource Allocation and Pilot Length Otimization 1 Initialize τ ; 2 do 3 Solve roblem P5 to obtain λ nτ) ; 4 Udate τ with 34) ; 5 while τ does not converge and τ τ max); 6 if τ < τ max then 7 Solve P3 with τ and τ to obtain a τ ), b τ ), a τ ), and b τ ) ; 8 Use the results to comare the values of objective functions of P2. Then, τ = argmax { τ, τ }{g τ ),g τ )} ; 9 else 10 Set τ = τ max ; 11 end 12 Use τ to solve roblem P3, and obtain the otimal a and b ; is the threshold of convergence for λ. Proof: According to Lemma 6 and 37), for a sufficiently large t and a sufficiently small ε 1, we have hτ ) hτ [t] ) < 1/ t. Thus, when 1/ t > ε 1, hτ ) hτ [t] ) is guaranteed to be smaller than ε 1. Consequently, it takes less than 1/ε 2 1 stes for the sequence hτ [t] ) to achieve an otimality ga that is less than ε 1, t < 1/ε 2 1. In the same way, the number of iterations for the convergence of λ is uer bounded by 1/ε 2 2. In Algorithm 1, each udate of τ requires a set of otimal λ. Thus, the total number of variable udates is uer bounded by 1/ε 2 1ε 2 2, the comlexity of Algorithm 1 is uer bounded by 1/ε 2 1ε 2 2. B. User Association under WB Constraints For a given set of a, b, and τ, P1 is reduced to the following user association roblem. P6 :max {x} K j=0 J x k,j R k,j 38) subject to: 11),12), and 16) x k,j {0,1}, k = 1,2,...,K, j = 0,1,...,J. Constraint 16) can be rewritten as K ) αn x k,j log1+γ k,j ) b j,nlog1+γ j ) 0, 1 α)n j = 1,2,...,J, 39) which is a linear constraint on x. To solve P6, we first relax the integer constraint of x by allowing all x k,j to take any value between [0,1]. Denote the relaxed roblem as P6-Relaxed. The objective function of x k,j log1+γ k,j ) x k,j P6-Relaxed includes a weighted sum of, which is non-convex. Thus, only local otimal solution can be achieved with standard otimization techniques. However, if the values of Q j = x k,j are given, P6-Relaxed reduces to an LP. Since Q j S j, the otimal solution of P6-Relaxed can be obtained by searching all ossible combinations of Q = {Q 1,...,Q J } and solve the corresonding LPs. However, this results in a high comlexity as J j=1 S j LPs need to be solved. We thus use this aroach to obtain the initial otimal values of Q and udate it with a more efficient aroach. Recall that the system states are udated every T. Thus, in a low mobility environment, we can make use of Q in the revious eriod as an aroximation to the Q of the current eriod. Then, {R k,j } becomes indeendent of x, given as R k,j = 1 1 T ) ) Tu τ 1 α)n log1+γ k,j ). Q j T c P6-Relaxed is thus transformed to the following LP. P7 :max {x} K j=0 T s J x k,j R k,j 40) subject to: 11),12), and 39) x k,j [0,1], k = 1,2,...,K, j = 0,1,...,J. Since P7 is an LP, the cutting lane method [27] can be alied to obtain its otimal integer solution, and such solution is also otimal to P6 for a given Q. As users may dynamically join or leave the network, the traffic load of each BS varies over time, the aroximation of Q might be inaccurate. However, a key observation is that load balancing can be achieved by solving P7. When Q j is larger than its otimal value, R k,j would be small. Then fewer users would be connected to SBS j after the udate with the solution of P7, resulting in a decreased Q j. Thus, the value of Q j is exected to stay close to its otimal value, and the solution is exected to be near-otimal. In case the user distribution drastically changes and handover frequently haen e.g., during rush hours), which can be detected by each BS when measuring the CSI of nearby users, Q should be udated by solving P6-Relaxed with searching over all Q. Due to its high comlexity, such udate is carried out at a timescale much larger than T. C. Iterative Scheme with Near-Otimal Solution In this section, we roose an iterative aroach to obtain the near-otimal solution of the original roblem by solving the WB and MUE resource allocation and ilot length otimization roblem and the user association roblem iteratively until convergence. The iterative scheme is a three-stage rocess to guarantee that all constraints are satisfied as well as minimizing the ga of the two roblems. The roosed threestage rocess is based on the following facts. Lemma 8. Under otimal user association solutions, given fixed values of Q j of other BS s, the sum rate of all users served by SBS j decreases as Q j increases. Proof: According to 9), R k,j is roortional to x k,j log1+γ k,j ), which can be interreted as the average x k,j sectral efficiency of users served by SBS j. Consider an otimal user association with a given feasible set of Q. To maximize the sum rate, the users served by SBS j must be the firstq j users with the highest sectral efficiencies, i.e., the highest SINRs. Thus, when the values of Q j for other

8 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL.XXX, NO.XXX, MONTH YEAR 8 BS s are fixed, the average sectral efficiency of users served by SBS j decreases as Q j increases. Proerty 3. In most cases, the users served by SBS j are the first Q j users with highest SINRs, and the sum rate of all users served by SBS j decreases as Q j increases. Comared to Lemma 8, we remove the assumtion that the values of Q j for other BS s are fixed. The only excetion of Proerty 3 haens when a user k originally served by a neighboring SBS j is handed over to SBS j due to an increase of Q j, while the SINR of this user is higher than at least one of the users currently served by SBS j. Suose user k has a lower SINR than user k when served by SBS j. Then both users are likely to be cell-edge users, and the coverage areas of SBS j and SBS j are likely to overla. Hence, the excetion case haens when both Q j and Q j increase and a cell-edge user is handed over to SBS j. As a result, when the SBS s are not densely deloyed, the excetion case would not haen. a) Stage I: In the first stage, we aim to guarantee that constraints 12), x k,j S j, are always satisfied for all SBS s. We begin with solving the initial MUE and WB resource allocation and ilot length otimization roblem without considering the constraint on WB data rate, b K j,n = x k,jr k,j log1+γ j). This corresonds to the case of setting the initial values of the RHS of 20) to be F j. Let P8 be the LP generated by removing constraints 12) from P7, P8 can be solved by the same aroach as P7. Then, we find the otimal user association under WB constraints by solving P8. With such initial solution, C j may be low for SBS j, R k,j is bounded by a low value. As in Proerty 3, a large number of users are exected to be assigned to SBS j to achieve a low value of R k,j, which may violate constraint 12) and be infeasible to P7. Thus, we first solve P8 to find the set of SBS s that violates the WB constraint, and then enforce additional constraints to P8 for feasibility. With the solution of P8, if constraint 12) of SBS j is not satisfied, P8 is udated by adding constraint x k,j = S j. Then, we udate R k,j by keeing the first S j highest SINR users to be served by SBS j. After that, we udate constraint 20) for SBS j with the udated x k,j and R k,j. This way, both constraints for SBS j are satisfied; the WB resource allocation and user association for SBS j become feasible. Based on Proerty 3, by keeing the first S j highest SINR users, the value of R k,j is exected to be the largest under a feasible and otimal solution of P7. This results in the smallest change on the RHS of constraint 20) for SBS j. Thus, the change of the olyhedron defined by Z is minimized, resulting in a smallest reduction of the objective function. Then, we solve the MUE and WB resource allocation and ilot length otimization roblem with the udated constraint 20) for SBS j. After that, we use the solution to solve P8 in the next iteration. Such rocess is reeated until all constraints 12) are satisfied for all SBS s. After the rocess is converged, we enter the second stage. b) Stage II: In the second stage, we aim to minimize the erformance ga between the two roblems, so that C j x k,jr k,j is minimized. The motivation of minimizing such ga is because allocating more channels to WBs leads to increased value of τ and decreased data rates of all users, it is desirable that the data rates rovided by WBs are sufficiently utilized by each SBS. To minimize the ga at each SBS, we find the SBS s with b j,n > these constraints as b j,n = x k,jr k,j log1+γ j) K x k,jr k,j log1+γ j ), and udate. 41) Then, we obtain the otimal {a, b, τ} with the udated constraints as in Section III-A. With {a, b, τ}, we solve P7 to obtain the otimal x. Such rocess is reeated until x k,jr k,j log1+γ j) does not hold for any SBS. b j,n > c) Stage III: In the third stage, we aim to guarantee that the WB constraints of all SBS s are satisfied after the udates in the second stage. With the udate in the second stage, the values of b j,n are reduced, which may cause an increased ratio of a k,n/ b j,n for some users. Hence, under the otimal solution of P8, these users may switch to the MBS. According to Proerty 3, the sum rate of SBS s that served these users in the revious iteration are exected to increase, resulting violation of the WB constraints. To deal with this situation, we can adjust and udate the values of b j,n with 20), and we reeat this rocess until the WB constraints of all SBS s are satisfied. The rocedure of the roosed iterative scheme is summarized in Algorithm 2. D. Remarks on Practical Concerns 1) Quasi-Static Channel Between MBS and SBS s: Due to the fixed locations of SBS s, the channels between SBS s and MBS are quasi-static [7]. As a result, the CSI of WBs can be udated less frequently comared to that of MUEs. This roerty can be emloyed to enhance the system erformance. In most time eriods, the SBS s can use some channels for WB transmission without sending ilots on these channels, resulting in reduced ilot length. Thus, we can assign the WBs to use all the αn channels to increase the data rate. In such scenario, the roblem formulation can be derived from Problem P1 with modifications on the constraints. Due to the different frequencies of CSI udate for MUE and WB, there are two cases at different time eriods. First case: Both WBs and MUEs need to send ilots. When the CSI of WB needs to be udated, all WBs are allocated with one ilot sequence on each channel so that the CSI of WBs on all channels can be obtained. This corresonds to set b j,n = 1 for j = 1,...,J, n = 1,...,αN. In addition, the constraint b j,n F j can be removed from Problem P1 since b is given. Second case: Only the MUEs need to send ilots. In these eriods, the MBS uses the CSI obtained in the first case until the next udate of CSI of WB. Then, the constraint a k,n + J j=1 b j,n τn sm in Problem P1 should be modified to a k,n τn sm. Since the WBs are allocated with all channels, we have b j,n = 1

9 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL.XXX, NO.XXX, MONTH YEAR 9 Algorithm 2: Iterative Scheme to Obtain a Near-Otimal Solution to Problem P1 1 Initialize Set the RHS of 20) as F j ; 2 do 3 Obtain {a,b,τ} with Algorithm 1; 4 Solve P8 to obtain x ; 5 for j = 1 : J do 6 if x k,j > S j) then 7 Set x k,j = S j ; 8 Udate 20) for SBS j ; 9 Add constraint x k,j = S j to P8 ; 10 end 11 end 12 Solve P8 to obtain udated x ; 13 Obtain {a,b,τ} with udated x using Algorithm 1 ; 14 while x k,j S j does not hold for all j ); 15 do 16 for j = 1 : J do 17 if K αn bj,n > x k,j R k,j log1+γ j) 18 Udate bj,n with 20) ; 19 end 20 end 21 Udate {a,b,τ} with Algorithm 1 ; 22 Udate x by solving P8 ; 23 while bj,n > K x k,j R k,j 24 do 25 for j = 1 : J do log1+γ j) 26 if K αn bj,n x k,j R k,j log1+γ j) 27 Udate bj,n with 20) ; 28 end 29 end 30 Udate {a,b,τ} with Algorithm 1 ; 31 Udate x by solving P8 ; 32 while bj,n K x k,j R k,j log1+γ j) ) then holds for any j ); ) then holds for any j ); for j = 1,...,J, n = 1,...,αN. Same as the first case, the constraint b j,n F j is also removed. With given τ, it can be easily verified that the constraint matrix of the linear rogramming for solving {a k,j } is unimodular for both cases. Thus, we can obtain the otimal{a,τ} using the same aroach as in Algorithm 1. Then, we aly Algorithm 2 to obtain the solutions for both cases. 2) A Combination of Wired and Wireless Backhaul: In case of dense SBS deloyment with heavy traffic load, a long ilot length i.e., large τ) is required, resulting in degraded system erformance. To mitigate such bottleneck as well as reserving the benefits of wireless backhaul, a combination of wired and wireless backhaul is desirable. With roer configuration, a good tradeoff between erformance and cost can be achieved. With a combination of wired and wireless backhaul, the roblem formulation needs to be adjusted accordingly. We assume that the data rate of wired backhaul is sufficiently high so that the constraint x k,jr k,j C j can always be satisfied. LetΩbe the set of SBS s that uses wireless backhaul, then the constraints x k,jr k,j C j and b j,n F j only aly to j Ω. The constraints 20) and 39), which are derived from x k,jr k,j C j, are also alied to j Ω only. For the constraint a k,n + J j=1 b j,n τn sm, we relace the term J j=1 b j,n to j Ω b j,n. The solution under such new scenario can be obtained with the same Algorithm 3: Distributed User Association Strategy for BS j 1 while convergence not achieved) do 2 if BS j holds more than S j roosals) then 3 Put the to S j users with the highest SINRs in the waiting list and reject the other users ; 4 else 5 Put all users in the waiting list ; 6 end 7 end aroach in Algorithm 2 with the udated constraints 20) and 39). Secifically, since the SBS s with wired backhaul have no imact on the ilot otimization, we still maximize the sum rate of wireless backhaul and MUE by solving Problem P2 with Algorithm 1. For user association, the constraint x k,jr k,j C j does not aly to the SBS s with wired backhaul, and the roblem can be solved with the same aroach resented before. IV. DISTRIBUTED SOLUTION SCHEME In the centralized scheme, global information is required for centralized control, which usually leads to better erformance. But acquiring the global information may incur considerable overhead, which may be infeasible in a large scale network. In this section, we roose a distributed scheme by formulating a noncooerative reeated game among all users. In the reeated game, each user distributively makes its own decision. We demonstrate that the game will converge to an NE. A. Distributed User Association We formulate a reeated game among all users, the strategy of each user is to decide its serving BS. Due to the tradeoff in MUE and WB resource allocation, we set a rice for using one channel such that the number of channels used by MUEs and WBs can be controlled at roer values. The utility of user k is defined as { Uk,0 = ω k logr k,0 ) αn U k,j = ω k logr k,j ) a k,n αn bj,n K x k,j, j = 0,...,J. 42) where ω k is the evaluation of user k for data rate and is the rice of using one channel. When user k is served by an SBS, the cost of channels for the WB is shared by all users that are served by the SBS. In 42), a k,n is set by each user to be a fixed value that maximizes its utility, given as a k,n = argmax { a k,n} {U k,0} = ω k /. For b j,n, it is a variable given by 41), which is affected by other users decisions. The strategy of each user is x k,j = 1, j = argmax j {U k,j }. 43) To maximize the sum rate under constraint x k,j = S j, it is reasonable to assume that each BS serves the to S j users with highest SINRs. The user association strategy of BS s is summarized in Algorithm 3. Each user has a reference list for all BS s, the order of the list is determined by the order of U k,j, e.g., the BS with the largest U k,j is the first in the reference list of user k. Since

10 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL.XXX, NO.XXX, MONTH YEAR 10 Q j is unknown before the reeated game, the initial reference list of each user is determined by values of SINRs when connecting to different BS s. The roosed reeated game has the following two stages. In the first stage, each user rooses to the to BS in its reference list. Then, BS s resond to the roosals according to Algorithm 3. In the second stage, each BS j broadcasts the value of Q j to all users. Then each user k udates its reference list with R k,j. A user rooses to another BS under the following cases. Case 1: The roosal of the user is rejected. Case 2: A higher utility can be achieved by switching to another BS j and one of the two conditions is satisfied: i) Q j < S j, ii) Q j = S j, and there is a user k currently in the waiting list of BS j such that R k,j > R k,j. If user k is rejected by BS j, it marks BS j as unavailable in its reference list. Then, users in these two cases roose to the to BS among remaining available BS s. Once receiving the roosals, each BS comares the new roosals with those in its waiting list, and makes decisions according to Algorithm 3. If a user switches from BS j to BS j as described in Case 1, the users that once marked BS j as unavailable change the status of BS j to available. Given the BS decisions, each user then udates its reference list and makes another round of roosal if one of the two cases is satisfied. The reeated game is continued until convergence of user association is achieved. After convergence, the MBS relaces constraint 14) with a k,n = ω k / and udate constraint 15) with 20). It then determines {a,b,τ} as in Section III-A. B. Convergence Analysis The convergence erformance of the reeated game is given in Theorem 1, which shows that an NE can be achieved. Theorem 1. The reeated game converges to a Nash equilibrium that is otimal for each user. Proof: Suose the game does not converge. Then, there must be a user k that is currently served by BS j who wishes to roose to another BS j. Obviously, Case 1 does not hold since user k is served by BS j. Then, Case 2 holds, there is another BS j such that U k,j > U k,j and BS j is marked as available by user k. If condition i) is satisfied, Q j < S j, then user k would have already switched to BS j, which contradicts to the fact that it is served by BS j. If condition ii) is satisfied, Q j = S j, then there must another user k that is served by BS j such that R k,j > R k,j, i.e., BS j refers user k over user k. Since user k is in the waiting list of BS j while user k is not, it must be the case that user k has never roosed to BS j before. However, since U k,j > U k,j, user k must have roosed to BS j before BS j, which is also a contradiction. Thus, the reeated game converges. From the above analysis, we can see that the utility of each user cannot be further imroved given the strategies of other users. Thus, the strategy of each user is the best resonse to the strategies of other users when the reeated game converges. We conclude that the reeated game converges to an NE. The order of users that start the roosed rocess affects the system erformance, as different NEs would be achieved. Such randomness results in erformance loss of distributed scheme comared to the centralized one. V. SIMULATION STUDY We validate the roosed centralized and distributed schemes with MATLAB simulations. The scenario is based on a cellular system with hexagonal macrocells, and we consider the sum rate of all users in a tagged macrocell area. The MBS is located at the center, the SBS s and users are randomly distributed in the macrocell area. The radius of a macrocell is 500 m. The slow fading factor, β k,0, is based on the ITU ath loss model [28] and a lognormal shadowing with standard deviation of 10 db. The coherence bandwidth is 150 khz. We use the arameters of downlink LTE symbol for each OFDM symbol. The sacing between subcarriers is 15 khz, then N sm = 10; the useful symbol duration T u = 1/ f = 66.7 ms; and T s = T = 72 ms. The coherence time is T c = 720 ms, so each frame has 10 OFDM symbols, and we set τ max = 5. The total bandwidth is 4 MHz, so the total number of channels is 40. We assume α = 1 2 ; then 20 channels are allocated to MUEs and WBs and the other 20 channels are allocated to SUEs. The owers of SBS s are set according to the iterative water-filling scheme [29], with an uer bound of 30 dbm. The uer bounds of x k,j are set to be S j = 20 for SBS s and S 0 = 50 for MBS, resectively. We comare the roosed schemes with a heuristic scheme, termed Heuristic, for user association. Heuristic is based on Proerty 3 and is derived by making a modification on the centralized scheme. Secifically, instead of solving P8 at each iteration of the centralized scheme, the set of users served by each SBS is determined with a greedy aroach. In each round, we select the user with highest SINR to be served by SBS j and udate the value of R k,j. We continue such rocess until R k,j C j is satisfied. We also consider the case based on [15], in which ilot length is not considered for otimization and τ is set as a fixed value termed Static ilot). For Static ilot, the solution of {a,b,τ} is based on the solution rocedure in Section III-A. For Heuristic, we aly the same rocedure of the roosed centralized scheme excet the user association strategy. Since the erformance of the distributed scheme deends on the value of, we set to the value that achieves the maximal sum rate. We also consider the value of the objective function of roblem P2 under otimal solution as an uer bound for comarison. The sum rate erformances of different schemes are resented in Figs. 1 and 2. In Fig. 1, it can be seen that the erformances of all schemes first increase and then decrease as the number of SBS s grows. This is because a larger τ is required as the number of SBS s increases, and the interference between neighboring small cells degrades the average SINRs of SUEs. Both the centralized and distributed schemes outerform Static ilot, demonstrating that a erformance gain can be achieved with dynamically adjusted τ. The erformance of the centralized scheme is close to its uer bound, since we iteratively minimize the erformance ga of two roblems in the second stage of the iterative scheme given in Algorithm 2. It is also observed that the erformance of Heuristic is close to that of the centralized scheme when the number of SBS s

11 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL.XXX, NO.XXX, MONTH YEAR 11 Average sum rate bs) 11 x Number of SBS Uer bound Centralized Heuristic Distributed Static ilot with τ=1 Static ilot with τ=5 Average sum rate bs) 15 x Uer bound Centralized Heuristic Distributed Static ilot with τ=1 Static ilot with τ= Number of users Fig. 1. Average sum rates of different schemes versus the number of SBS 200 users). is small, due to the fact that Proerty 3 is more reliable when SBS s are not close to each other, and a user would not have close rates by connecting to different SBS s. The distributed scheme also achieves a satisfactory erformance since users are charged for using channels, resulting in efficient resource utilization. For Static ilot, the case of τ = 1 achieves better erformance than the case of τ = 5 when the number of SBS s is small, a small τ can accommodate the requirements of all WBs. However, when the number of SBS s is large, a larger τ rovides better erformance since the increased demand for WB data rates can be satisfied. Fig. 2 shows the erformances under different numbers of users, where similar trends can be observed. When the number of users increases, the sum rate of users with τ = 1 remains constant. This is because the resources for MUEs and WBs are quite limited. As a result, a considerable roortion of users cannot be served by any BS. In Fig. 3, the erformance of Static ilot with different τ values is evaluated. When τ is large, the erformance with 100 users is significantly worse than that with 400 users; however, when τ is small, the erformance with 400 users becomes worse than that with 100 users. This shows that a small value of τ significantly limit the system erformance in case of larger number of users, and τ needs to be dynamically adjusted to revent considerable erformance loss of Static ilot under different traffic atterns. The otimal values of τ under different numbers of SBS s and users are resented in Fig. 4. The otimal τ increases with both the number of SBS s and the number of users, since the more resources are required to satisfy the increasing demand. An examle of the reeated game is given in Fig. 5. We can see that the game converges after several rounds and a maximum sum utility is achieved uon convergence. We also resent an examle to evaluate the imact of rice on the system erformance in Fig. 6. By setting to a roer value, each user makes rational decision on channel usage, the value of τ can be set to a roer value. VI. RELATED WORK Fig. 2. Average sum rates of different schemes versus the number of users 20 SBS s). The Massive MIMO technology has attracted lots of interests in recent years. A comrehensive introduction on the fundamentals of signal rocessing issues can be found in [30]. An overview and analysis for uer layer techniques in massive MIMO systems is resented in [6]. The otentials, limits and ossible research roblems of massive MIMO were resented in [31]. As an imortant alication scenario, massive MIMO HetNet has also been widely studied. The key technical asects of massive MIMO HetNet include user association [10], [20] and interference management [7], [34]. The RTDD architecture for massive MIMO HetNet was first considered in [7]. With RTDD, the interference occurs between MBS and SBS s. Then, the MBS can emloy zero-forcing beamforming based on the estimated channel covariance beween MBS and SBS s. Comared to these works, we consider wireless connection between the MBS and each SBS, and roose a cross-layer otimization framework. The HetNet with WB has been studied in several rior works. Since another tye of transmission is added over transmissions between users and BS s, interference management becomes a key issue under certain system assumtions and has been investigated in [35], [36]. In [37], an load-aware design on satial multilexing was roosed to imrove the energy efficiency of a HetNet with WB. A recent overview on resource management of 5G HetNet with WB was resented in [14]. In this aer, we integrate massive MIMO with WB and deal with the challenges with an adative frame design. User association in HetNet is another closely related issue. A recent survey on user association of 5G network can be found in [38]. In [39], a near-otimal user association scheme was roosed, and the roosed scheme can be imlemented distributively with a dual decomosition. To deal with the integer constraint, several aroximation algorithms were roosed in [40] with the objective of minimizing the maximum load among all BS s. In [41], user association was considered from the ersective of maximizing the utilities of users, and different sectrum allocation strategies were jointly considered. In [42], a traffic-aware dynamic user association was considered through the cooeration of SBS s. Due to the secial architecture of a massive MIMO HetNet with WB, we otimize the network erformance with joint frame design, resource allocation, and user association in this aer.

12 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL.XXX, NO.XXX, MONTH YEAR Case of 100 users Case of 400 users 4 Average sum rate bs) Otimal value of τ Case of 60 SBS s Case of 20 SBS s τ Number of users Fig. 3. Average sum rates versus the value of τ under 2 different numbers of users 20 SBS s). 2.5 Fig. 4. Otimal value of τ under different numbers of SBS s 200 users). 5.5 Normalized utility Normalized sum rate Round of the reeated game Fig. 5. Convergence of the reeated bidding game 200 users and 20 SBS s). Fig. 6. Normalized sum rate versus the value of 200 users and 20 SBS s). VII. CONCLUSIONS In this aer, we considered the roblem of joint frame design, resource allocation, and user association to maximize the sum rate of a massive MIMO HetNet. We formulated a nonlinear integer rogramming roblem and roosed a centralized iterative scheme to obtain a near-otimal solution. We also roosed a distributed scheme by formulating a reeated game among all users and rove that the game converges to an NE. Simulation results show that the roosed schemes outerform several benchmark schemes. REFERENCES [1] M. Feng and S. Mao, Adative Pilot Design for Massive MIMO HetNets with Wireless Backhaul, in Proc. IEEE SECON 17, San Diego, CA, June 2017,.1 9. [2] J. Andrews, S. Buzzi, W. Choi, S. V. Hanly, A. Lozano, A. C. K. Soong, and J. C. Zhang, What will 5G be? IEEE J. Sel. Areas Commun., vol.32, no.6, , June [3] T. L. Marzetta, Noncooerative cellular wireless with unlimited numbers of base station antennas, IEEE Trans. Wireless Commun., vol.9, no.11, , Nov [4] H. Q. Ngo, E. G. Larsson, and T. L. Marzetta, Energy and sectral efficiency of very large multiuser MIMO systems, IEEE Trans. Commun., vol.61, no.4, , Ar [5] Y. Xu, G. Yue, and S. Mao, User grouing for massive MIMO in FDD systems: New design methods and analysis, IEEE Access J., vol.2, no.1, , Set [6] M. Feng and S. Mao, Harvest the otential of massive MIMO with multi-layer techniques, IEEE Network, vol.30, no.5,.40 45, Set./Oct [7] K. Hosseini, J. Hoydis, S. ten Brink, and M. Debbah, Massive MIMO and small cells: How to densify heterogeneous networks, in Proc. ICC 13, Budaest, Hungary, June 2013, [8] E. Björnson, M. Kountouris, and M. Debbah, Massive MIMO and small cells: Imroving energy efficiency by otimal soft-cell coordination, in Proc. ICT 13, Casablanca, Morocco, May 2013,.1 5. [9] D. Bethanabhotla, O. Y. Bursalioglu, H. C. Paadooulos, and G. Caire, Otimal user-cell association for massive MIMO wireless networks, in IEEE Trans. Wireless Commun., vol.15, no.3, , Mar [10] Y. Xu and S. Mao, User association in massive MIMO HetNets, IEEE Systems J., vol.11, no.1,.7 19, Mar [11] M. Feng, S. Mao, and T. Jiang, BOOST: Base station on-off switching strategy for energy efficient massive MIMO HetNets, in Proc. INFO- COM 16, San Francisco, CA, Ar. 2016, [12] X. Ge, H. Cheng, M. Guizani, and T. Han, 5G wireless backhaul networks: Challenges and research advances, IEEE Network, vol.28, no.6,.6 11, Nov [13] U. Siddique, H. Tabassum, E. Hossain, and D. I. Kim, Wireless backhauling of 5G small cells: Challenges and solution aroaches, IEEE Wireless Commun., vol.22, no.5,.22 31, Oct [14] N. Wang, E. Hossain, and V. K. Bhargava, Backhauling 5G small cells: A radio resource management ersective, IEEE Wireless Commun., vol.22, no.5,.41 49, Oct [15] N. Wang, E. Hossain, and V. K. Bhargava, Joint downlink cell association and bandwidth allocation for wireless backhauling in twotier HetNets with large-scale antenna arrays, IEEE Trans. Wireless Commun., vol.15, no.5, , May [16] B. Li, D. Zhu, and P. Liang, Small cell in-band wireless backhaul in massive MIMO systems: A cooeration of next-generation techniques,

13 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL.XXX, NO.XXX, MONTH YEAR 13 IEEE Trans. Wireless Commun., vol.14, no.12, , Dec [17] H. Tabassum, A. H. Sakr, E. Hossain, Analysis of massive MIMOenabled downlink wireless backhauling for full-dulex small cells, IEEE Trans. Commun., vol.64, no.6, , June [18] Z. Gao, L. Dai, D. Mi, Z. Wang, M.A. Imran, and M.Z. Shakir, MmWave massive-mimo-based wireless backhaul for the 5G ultradense network, IEEE Wireless Commun., vol.22, no.5,.13 21, Oct [19] F. Fernandes, A. Ashikhmin, and T. L. Marzetta, Inter-cell interference in noncooerative TDD large scale antenna systems, IEEE J. Sel. Areas Commun., vol.31, no.2, , Feb [20] Q. Ye, O. Y. Bursalioglu, H. C. Paadooulos, C. Caramanis, and J. G. Andrews, User association and interference management in massive MIMO HetNets, IEEE Trans. Commun., vol.64, no.5, , May [21] E. Björnson, E. G. Larsson, and M. Debbah, Massive MIMO for maximal sectral effciency: How many users and ilots should be allocated?, IEEE Trans. Wireless Commun., vol.15, no.2, , Feb [22] H. Kim and Y. Han, A roortional fair scheduling for multicarrier transmission systems, IEEE Commun. Lett., vol.9, no.3, , Mar [23] S. Boyd and L. Vandenberghe, Convex Otimization,, Cambridge, UK: Cambridge University Press, [24] A. Schrijver, Theory of Linear and Integer Programming, Hoboken, NJ: John Wiley & Sons, June [25] C. Berenstein and R. Gay, Comlex Variables: An Introduction, New York City, NY: Sringer, [26] D. P. Palomar and M. Chiang, A tutorial on decomosition methods for network utility maximization, IEEE J. Sel. Areas Commun., vol.24, no.8, , Aug [27] R. Gomory, Outline of an algorithm for integer solutions to linear rograms, Bull. Amer. Math. Soc., vol.64, no.5, , Set [28] ITU, Guidelines for Evaluation of Radio Transmission Technologies for IMT-2000, Recommendation ITU-R M. 1225, [29] W. Yu, Multiuser water-flling in the resence of crosstalk, in Proc. IEEE ITA Worksho, San Diego, CA, Jan. 2007, [30] F. Rusek, D. Persson, B. Lau, E. Larsson, T. Marzetta, O. Edfors, and F. Tufvesson, Scaling u MIMO: Oortunities and challenges with very large arrays, IEEE Sig. Proc., vol.30, Jan. 2013, [31] E. G. Larsson, F. Tufvesson, O. Edfors, and T. L. Marzetta, Massive MIMO for next generation wireless systems, IEEE Commun., vol.52, no.2, , Feb [32] L. Lu, G. Y. Li, A. L. Swindlehurst, A. Ashikhmin, and R. Zhang, An overview of massive MIMO: benefits and challenges, IEEE J. Sel. Toics Sig. Process., vol.8, no.5, , Oct [33] K. Zheng, L. Zhao, J. Mei, B. Shao, W. Xiang, and L. Hanzo, Survey of large-scale MIMO systems, IEEE Commun. Sur. Tut., vol.17, , third quarter [34] A. Adhikary, H. S. Dhillon, and G. Caire, Massive-MIMO meets HetNet: Interference coordination through satial blanking, IEEE J. Sel. Areas Commun., vol.33, no.6, , June [35] S. Samarakoon, M. Bennis, W. Saad, and M. Ltva-aho, Backhaul-aware interference management in the ulink of wireless small cell networks, IEEE Trans. Wireless Commun., vol.12, no.11, , Nov [36] L. Sanguinetti, A. L. Moustakas, and M. Debbah, Interference management in 5G reverse TDD hetnets with wireless backhaul: A large system analysis, IEEE J. Select. Areas Commun., vol.33, no.6, , June [37] H. H. Yang, G. Geraci, T. Q. S. Quek, Energy-efficient design of MIMO heterogeneous networks with wireless backhaul, IEEE Trans. Wireless Commun., vol. 15, no. 7, , July [38] D. Liu, L. Wang, Y. Chen, M. Elkashlan, K.-K. Wong, R. Schober and L. Hanzo, User association in 5G networks: A survey and an outlook, IEEE Commun. Surveys Tuts., vol.18, no.2, , second quarter [39] Q. Ye, B. Rong, Y. Chen, M. A.-Shalash, C. Caramanis, and J. G. Andrews, User association for load balancing in heterogeneous cellular networks, IEEE Trans. Wireless Commun., vol.12, no.6, , June [40] H. Zhou, S. Mao, and P. Agrawal, Aroximation algorithms for cell association and scheduling in femtocell networks, IEEE Transactions on Emerging Toics in Comuting, vol.3, no.3, , Set [41] Y. Lin, W. Bao, W, Yu, B. Liang, Otimizing user association and sectrum allocation in HetNets: A utility ersective, IEEE J. Select. Areas Commun., vol.33, no.6, , June [42] M. Feng, T. Jiang, D. Chen, and S. Mao, Cooerative small cell networks: High caacity for hotsots with interference mitigation, IEEE Wireless Commun., vol.21, no.6, , Dec Mingjie Feng [S 15] received his B.E. and M.E. degrees from Huazhong University of Science and Technology in 2010 and 2013, resectively, both in electrical engineering. He was a visiting student in the Deartment of Comuter Science, Hong Kong University of Science and Technology, in He is currently a Ph.D. student in the Deartment of Electrical and Comuter Engineering, Auburn University, AL. His research interests include cognitive radio networks, heterogeneous networks, massive MIMO, mmwave network, and full-dulex communication. He is a reciient of a Woltosz Fellowshi at Auburn University. Shiwen Mao [S 99-M 04-SM 09] received his Ph.D. in electrical and comuter engineering from Polytechnic University, Brooklyn, NY in He is the Samuel Ginn Distinguished Professor and Director of the Wireless Engineering Research and Education Center WEREC) at Auburn University, Auburn, AL. His research interests include wireless networks and multimedia communications. He is a Distinguished Lecturer of the IEEE Vehicular Technology Society. He is on the Editorial Board of IEEE Transactions on Multimedia, IEEE Internet of Things Journal, IEEE Multimedia, ACM GetMobile, among others. He received the 2015 IEEE ComSoc TC-CSR Distinguished Service Award, the 2013 IEEE ComSoc MMTC Outstanding Leadershi Award, and the NSF CAREER Award in He is a co-reciient of the Best Demo Award from IEEE SECON 2017, the Best Paer Awards from IEEE GLOBECOM 2016 & 2015, IEEE WCNC 2015, and IEEE ICC 2013, and the 2004 IEEE Communications Society Leonard G. Abraham Prize in the Field of Communications Systems. Tao Jiang [M 06-SM 10] is currently a Chair Professor in the School of Electronics Information and Communications, Huazhong University of Science and Technology, Wuhan, P. R. China. He received Ph.D. degree in information and communication engineering from Huazhong University of Science and Technology, Wuhan, P.R. China, in Ar He has authored or co-authored over 200 technical aers in major journals and conferences and 9 books/chaters in the areas of communications and networks. He has served or is serving as associate editor of some technical journals in communications, including in IEEE Transactions on Signal Processing, IEEE Communications Surveys and Tutorials, IEEE Transactions on Vehicular Technology, IEEE Internet of Things Journal, and he is the associate editor-in-chief of China Communications. He is a reciient of the NSFC Distinguished Young Scholars Award in He was awarded as the Most Cited Chinese Researchers announced by Elsevier in 2014, 2015 and He is a senior member of IEEE.