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Joint load balancing and interference coordination in multi-antenna heterogeneous networks
The Journal of China Universities of Posts and Telecommunications December 2016, 23(6): 34–40 www.sciencedirect.com/science/journal/10058885
http://jcupt.bupt.edu.cn
Joint load balancing and interference coordination in multi-antenna heterogeneous networks Zhu Wenfeng, Qiu Ling (), Chen Zheng, Liang Xiaowen 1. Key Laboratory of Wireless-Optical Communications, Chinese Academy of Sciences, University of Science and Technology of China, Hefei 230026, China 2. School of Information Science and Technology, University of Science and Technology of China, Hefei 230026, China
Abstract In heterogeneous networks (HetNets), it is desirable to offload users from macro cells to small cells to achieve load balancing. However, the offloaded users suffer a strong inter-tier interference. To guarantee the performance of the offloaded users, the interference from macro cells should be carefully managed. In this paper, we jointly optimize load balancing and interference coordination in multi-antenna HetNets. Different from previous works, instead of almost blank subframes (ABS) on which the macro cells waste time resource, the macro cells suppress the interference to the offloaded users by zero-forcing beamforming (ZFBF) on interference nulling subframes (INS). Considering user association cannot be conduct frequently, we derive the long-term throughput of users over Rayleigh fading channels while previous works focused on instantaneous rate. From the perspective of the spectrum efficiency and user fairness, we formulate a long-term network-wide utility maximization problem. By decomposing the problem into two subproblems, we propose an efficient joint load balancing and interference coordination strategy. Simulation results show that our proposal can achieve good system performance gains over counterparts in term of the network utility, cell edge throughput and average throughput. Keywords heterogeneous networks, multi-antenna, load balancing, interference coordination, zero-forcing beamforming
1
Introduction
Wireless data demand has witnessed an exponential growth in recent years due to the increasing of data hungry applications and the trend will continue in the foreseeable future [1]. HetNets, consisting of macro base stations (BSs) (MBSs) overlaid with small BSs (SBSs), along with maturity of multi-antenna transmission techniques are promising solutions to handle the current data deluge. Due to the different transmit power of MBSs and SBSs, when user association metric like reference signal received power (RSRP) is used, most of the users will associate to MBSs, which makes the SBSs extremely underutilized and limit the cell splitting gain. To tackle load balancing in HetNets, range expansion (RE) is introduced to expand the coverage of SBSs through adding a positive bias to their Received date: 07-06-2016 Corresponding author: Qiu Ling, E-mail: [email protected] DOI: 10.1016/S1005-8885(16)60067-5
measured signal strength during user association [2]. In Ref. [3], Ye et al. proposed a distribute user association scheme that achieves load balancing in HetNets through a network-wide utility maximization problem. The optimal user association for downlink multi-antenna HetNets was exploited in Ref. [4]. Although the load balancing can be tackled through the schemes in Refs. [2–4], they may cause the offloaded users suffer a stronger interference. To overcome this issue, ABS, on which the MBSs keep silent to reduce the interference to offloaded users, has been widely adopted [5]. User association and ABS have been jointly considered in Refs. [6–8]. In Ref. [6], the optimal cell RE bias and ABS ratio are obtained by considering the fairness effect among different UEs in a two-tier HetNet. The joint optimization of ABS and user association has been studied in Ref. [7] to strike a balance between retaining cell splitting gain and mitigating interference. In Ref. [8], Jin et al. revealed that the optimal ABS density is the ratio between the number of
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Zhu Wenfeng, et al. / Joint load balancing and interference coordination in multi-antenna heterogeneous networks
offloaded users and total users then proposed a user association scheme through a greedy approach. However, MBSs will waste time resource on ABS. On the other hand, deploying multi-antenna at MBSs provides a more effective alternative to interference coordination through beamforming. For example, in Ref. [9], Wu et al. investigated an interference nulling (IN) scheme by ZFBF in multi-antenna HetNets and obtained the optimal degrees of freedom (DoF) used for IN through stochastic geometry analysis. In this paper, we consider load balancing and interference coordination in a downlink two-tier multi-antenna HetNet. We propose INS, on which the MBSs use ZFBF to mitigate the interference to the offloaded users. Meanwhile, each MBS can use the remaining DoF to serve its own users on INS. Pervious researches mostly focus on instantaneous rate in user association. However, the user association operation is generally conducted much less frequently due to the handover cost in wireless networks. The average effect of time-varying channels should be carefully manipulated. In this paper, firstly, we derive the average rate of users over Rayleigh fading channels. Then, we formulate a long-term network-wide utility maximization problem, where user association and INS ratio are jointly optimized. Although the formulated problem is NP-hard, we decompose the problem into two subproblems and propose an efficient iterative algorithm to solve the original problem. Numerical results show that the proposed strategy can significantly improve the network utility, cell edge throughput and average throughput.
2
System model
Consider a downlink two-tier multi-antenna HetNet, where a macro cell tier and a small cell tier share the same radio spectrum. Each MBS and SBS are equipped with M m and M s antennas respectively and each user employs a single antenna. As shown in Fig. 1(a), consider a typical MBS B0 1,
35
= {1, 2,.., K } . In this paper, the SBSs users are classified into two complementary subsets, one for offloaded users and the other for unoffloaded users. Literally, offloaded users are users who are offloaded to SBSs and interfered seriously by MBSs. The offloaded users suffering dominate interference is a bottleneck towards improving the network performance. To protect offloaded users from strong inter-tier interference, we divide the frames into two subframes, i.e., normal subframes (NS) and INS as Fig. 1(b). Without loss of generality, we expand the meaning of offloaded users by denoting all users scheduled by SBSs on INS as offloaded users.
(a) A two-tier HetNet
(b) Example of INS Fig. 1 System model
On NS, MBS serves its users and SBSs schedule its unoffloaded users. We assume the perfect channel state information (CSI) is available at the BSs. Both MBS and SBSs utilize maximum ratio transmission scheme, i.e., the beamforming vector at BS n is vn = hnk hnk , where hnk is the channel vector between BS n and its scheduled user k. If BS n is a SBS, hnk ~ (0 M s ×1 , I M s ) , otherwise
hnk ~ (0 M m ×1 , I M m ) .
overlaid with N SBSs and K users within its coverage. The BS set consisting of MBS B0 and SBSs is denoted by
On INS, the offloaded users are scheduled by SBSs while MBS uses ZFBF to perform IN by at most L ( L≤N ) DoF. Let S denotes the number of scheduled
denote SBSs. We denote the set of all users as
offloaded users on a certain INS. If S≤L , all scheduled offloaded users can avoid inter-tier interference, otherwise the MBS will randomly select L users from scheduled offloaded users to perform IN scheme. We call scheduled
= {0,1,..., N } , where index 0 denotes B0 , 1, 2,..., N
1 The results herein can be simply extended to the multi-macrocell scenarios. In fact, we consider the more general multi-macrocell scenario in simulations.
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The Journal of China Universities of Posts and Telecommunications
offloaded users that can avoid inter-tier interference as IN users and the rest are referred as none IN (NIN) users. Therefore, the MBS will use LIN = min ( L, S ) DoF to
M m − LIN DoF is used to serve its users. For MBS B0 , let
perform IN scheme. The remaining
Cnk = ∫ C
INS nk
∞
1 − (1 + zPn g nk ) z
0
=∫
−Ms
∞
1 − (1 + zPn g nk )
1 e − zN0 dz + zP g 1 i∈B / {n} i ik
∏
(4)
1 e − zN0 dz i∈B / {0, n} 1 + zPi g ik
(5)
−Ms
∏
z
0
2016
where C0k and C0kINS are the average rate of macro users
H = h0 k , h0O1 ,..., h0OL denote the channel matrix of B0 , where hnk ~ (0 M m × , I M m ) denotes the channel vector
on NS and INS respectively. Cnk denotes the average rate
between MBS and its scheduled user k, h0Oi ~ (0 M m ×1 , I M m ) denotes the channel vector
corresponds to the average rate of IN users.
IN
between MBS to IN users Or ( r = 1, 2,..., LIN ) . The ZFBF vector at B0 is designed to be v0 = w0 w0 , where w0 is the first column of
H † ( HH † ) . −1
SBSs utilize
maximum ratio transmission beamforming to serve its users. The signal to noise plus interference ratio (SINR) of user k ∈ from BS n ∈ can be expressed as
g nk =
Pn g nk vn † hnk
∑ Pg i
i∈I k
ik
vi † hik
2
2
+ N0
to channel gain between BS n and user k including large scale fading, antenna gain and shadowing. According to 2
vi † hik
vn † hnk
exp(1) ,
Lu
distribution with parameters
vn † hnk
2
unoffloaded
2
has a gamma
and 1, denoted as
Γ ( Lu ,1) , where Lu is the number of DoF of
BS n to serve user k. If BS n is a MBS, Lu = M m on NS and = Lu M m − LIN on INS. If BS n is a SBS, Lu = M s .
users
or
NIN
users
while
f ik = vi hik
∫
∞ 0 ∞ 0
,
(a) =
(b) 1 − E e − zP0 g0 k f0 k − z i∈∑/{0} Pik gik fik − zN E e E e 0 dz = z
1 − (1 + zP0 g 0 k ) z
−Mm
1 e − zN0 dz. 1 + zP g i∈B / {0} i ik
∏
0
(1/ z ) (1 − e − zx ) ⋅
[11], (b) follows by the fact
f ik ~ exp (1) ,
where (a) follows from ln (1 + x= / y)
e − zy dz
2
2
P0 g 0 k f 0 k C0 k = E ln (1 + g 0 k ) = E ln 1 + Pi gik f ik + N 0 ∑ i B / 0 ∈ { } ∞ 1 − e − zP0 g0 k f0 k − z i∈∑/{0} Pi gik fik − zN e e 0 dz = E ∫ 0 z
∫
CnkINS
Take C0k as an example, let f 0 k = v0† h0 k
Proof
(1)
where Pi is the transmission power of BS i, N 0 is the power of additive white Gaussian noise, g nk corresponds
Ref. [10],
of
∫
∞
f 0 k ~ Γ ( M m ,1) . The proof of Eq. (3) to Eq. (5) is similar and omitted for brevity.
3
Problem formulation
For each SBS, the frames are comprised of NS and INS. We assume that the unoffloaded users and offloaded users I k = / {0} . If k is unoffloaded or NIN users, can only be scheduled at NS and INS separately. Therefore, SBS n can be considered as two virtual SBS (VSBS) I k = / {n} , otherwise I k = / {0, n} . numbered n and n + N , where VSBS n schedules Since user association cannot be conducted frequently, unoffloaded users on NS and mutes on INS, the other to track the users performance in the long run with respect VSBS i + N serves offloaded users on INS and mutes on to time-varying channels, we derive the average achievable NS. Now, the BS set expands to rate of different types of users. Let model the = 0,1,..., N , N + 1,..., 2 N . x { } e nk Theorem 1 The average rate of different types of
I k is interfering BS set of user k. If k is macro user,
C0 k = ∫
∞
1 − (1 + zP0 g 0 k ) z
0
C0INS k = ∫
−Mm
∞ 0
1 − (1 + zP0 g 0 k ) z
1 e − zN0 dz 1 + zP g i∈B / {0} i ik
∏
LIN − M m
1 e − zN0 dz i∈B / {0} 1 + zPi g ik
∏
xnk = 1 if user k is associated with BS n, xnk = 0 otherwise. Let yn denote association indicator where
users can be expressed as follows (2) (3)
the number of users associated with BS n, i.e., yn = ∑ xnk . We define Q as the probability that a k ∈
offloaded user is an IN user on a certain INS,
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Zhu Wenfeng, et al. / Joint load balancing and interference coordination in multi-antenna heterogeneous networks
Q = LIN
∑
2N
n= N +1
Fn , where Fn is the indicator whether
there are users associated with BS n, i.e., Fn = 1 when yn > 0 , Fn = 0 , otherwise. The achievable rate d nk of user k from BS n can be described as (1 − α ) C0 k + α C0INS 0 k ; n = (6) 1, 2,..., N d nk = (1 − α ) Cnk ; n = INS α QC( n − N ) k + α (1 − Q ) C( n − N ) k ; otherwise where α is the ratio of INS to total subframes (hereafter referred to as INS ratio). We assume that the BSs employ round-robin scheduling policy. The long-term throughput of user k can be written as 1 (7) Rk ∑ xnk d nk ; ∀k ∈ = y n∈e n
37
Under this approximation we can obtain Q = L N . Our simulation results demonstrate this approximation is reasonable for we find there is always at least one offloaded user in each SBS. However, it still remains NP-hard. We propose an alternative optimization method to solve this problem effectively. 3.1
INS ratio
Firstly, we optimize the INS ratio α for a given user association indicator xnk , then problem Eq. (8) is transformed as
d ln nk a yn n 0= k 1 = ( ((( U (a ) (9) s.t. 0≤a ≤1 Theorem 2 The optimal INS ratio is given by 0; Z E = 0 * (10) = α 1;= Z C 0, u ′(1) > 0 is the solution of U ′ α = 0; otherwise ( ) where Z C and Z E is the number of unoffloaded users max
2N
K
∑∑ x
nk
Maximizing throughput can result in the starvation of users with poor signal strength, which is not a satisfactory solution. To strike a balance of network spectrum efficiency and individual user fairness, we adopt proportional fairness utility Ref. [12] in our paper, i.e, the K K network utility function is U ∑ = = ln Rk ∑ ln ∑ (1 yn ) ⋅ x k 1 k 1 = = n∈e N K 2 N K and offloaded users respectively, i.e., Z C = ∑∑ xnk , xnk d nk = ∑∑ xnk ln ( d nk yn ). n 1= k 1 = =n 0=k 1 K 2N Then the problem of joint INS ratio and user association Z E = ∑ ∑ xnk . U ′ (α ) is the first order derivative of n =N +1 k =1 with the target to maximize the utility is formulated as 2N K total utility function which can be computed as follows d max ∑∑ xnk ln nk K −ZC ZE 1 x ,a yn n 0= k 1 = (11) = + + U ′ (α ) ∑ x0 k C −α α 1 0k k =1 s.t. +α C0INS k − C0 k C1 0≤a ≤1 Proof The second order derivative of utility function K (8) C2 ∑ x= yn ; ∀n ∈ e can be formulated as nk k =1 K −ZC −Z −1 2N = + + 2E (12) U ′′ (α ) ∑ x0 k 2 2 α C3 ∑ xnk= 1; ∀k ∈ k =1 (1 − α ) C0 k n=0 α + INS C0 k − C0 k C4= xnk {0,1} ; ∀n ∈ e , ∀k ∈ � It is intuitive that U ′′ (α ) is a negative so U (α ) is a Constaints C1 corresponds to the INS ratio. Constraints concave function with respect to α . We can obtain the C3 and C4 ensure that one user can only access with one globally optimal α * by setting U ′ (α ) = 0 . If Z E = 0 , BS. However, Eq. (8) is a complex problem with the combination of α and the binary variables xnk , which is then U ′ (α ) < 0 , which indicates that U (α ) is an NP-hard problem. We can solve it by exhaustive search monotonic decreasing, the optimal INS ratio α * = 0 . but it is irrational in reality. In order to tackle the problem When Z C = 0 , U ′ ( 0 ) = +∞ , if U ′ (1) > 0 which means we make the following approximation. U (α ) is monotonic increasing then α * = 1. In other Approximation We assume that there is at least one situations, U ′ ( 0 )≥0, U ′ (1)≤0, there is always offloaded user in each SBS, i.e., F= 1, n= N + 1,..., 2 N . n
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The Journal of China Universities of Posts and Telecommunications
α ∈ [0,1] which satisfies U ′ (α ) = 0. We can calculate
2016
The optimal yn* can be derived as
α * through bisection method due to the monotonicity of U ′ (α ) .
yn* = e(
µn −1)
Eq. (8) can be described as 2N K d max ∑∑ xnk ln nk x yn n 0= k 1 =
µn ( t + 1= ) µn ( t ) − δ ( t ) yn ( t ) − ∑ xnk ( t )
(17)
With constraint in Eq. (13), the optimal user association scheme is From Theorem 2, it is observed that the optimal INS 1; n = n* ratio is an increasing function with respect to the number (18) xnk = of offloaded users Z E , which corresponds with reality that 0; otherwise the more offloaded users, the larger optimal INS ratio is. where ( 19) = n* arg max ( ln d nk − m n ) n 3.2 User association To obtain the optimal solution of the dual problem, we Next, we optimize the user association indicator xnk adopt the gradient descent method to update Lagrange multiplier µ [13], by assuming that the INS ratio α is fixed. The problem
s.t. K = C1 yn ∑ xnk ; ∀n ∈ e (13) k =1 2N C2 ∑ xnk= 1; ∀k ∈ � n=0 C3= xnk {0,1} ; ∀n ∈ e , ∀k ∈ � First, we relax the association indicator xnk as a
continuous variable in interval [0,1], then Eq. (13) becomes a convex optimization problem. By introducing Lagrange multiplier for the couple constraint C3 in Eq. (13), the Lagrangian function is
= ( x, y , µ )
2N
K
∑∑ x ( ln d
n 0= k 1 =
nk
2N
nk
− µn ) + ∑ ( µ n yn − yn ln yn ) n 0 =
(14) The Lagrange dual function is computed as ( x, y , ) D ( mm ) = max x, y s.t. (15) 2N xnk= 1; ∀k ∈ ∑ n=0 0≤xnk ≤1; ∀n ∈ e , ∀k ∈ To tackle the Lagrange dual function Eq. (15), differentiating with respect to xnk and yn , we can obtain the necessary condition for xnk and yn
∂ = µ n − 1 − ln yn = 0 ∂yn
1 > 0; xnk = ∂ = ln d nk − µn = 0; 0 < xnk < 1 ∂xnk < 0; x = 0 nk
where δ ( t ) 3.3
k is the step size of the tth iteration.
(20)
Joint iterative algorithm
In this subsection, we propose an effective iterative algorithm to jointly optimize the user association and the INS ratio. Steps of the algorithm are elaborated as follows Step 1 Initialization set ε = 10−4 , a = rand(1,1) . Iterations t = 1, 2,... Step 2 Update user association x. Set= µ ( 0 ) rand (1, 2 N + 1) , ξ = 10−4 , iteration k = 0,1, 2... Substep 1 Update user association. The user will be associated with the target BS according Eqs. (18) and (19). Substep 2 Update the Lagrange multiplier. Using the gradient method to update µ ( j + 1) as
Eq. (20). Substep 3 Check the error of Lagrange multiplier. If µ ( j + 1) − µ ( j ) < ξ go to Step 3 else go to Substep 1. Step 3 Update INS ratio. Calculate α ( t + 1) as Eq. (10) using x ( t + 1) . Step 4 Check the error of INS ratio. If α ( t + 1) − α ( t ) < ε , then stop iteration else go to Step 1.
(16)
4 Simulation results Considering a downlink two-tier HetNet, the SBSs are uniformly located in a circle with distance 2/9 inter site distance (ISD) from the MBS [8]. Assume that the SBSs
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Zhu Wenfeng, et al. / Joint load balancing and interference coordination in multi-antenna heterogeneous networks
are hotspots within a 35 m radius, wherein 3/4 of the users are uniformly distributed at random. The rest are uniformly distributed in MBS coverage. The parameters about antennas are set as M m = 8 , M s = 4 [14] and the optimal L is obtained by exhaustive search in simulations. Other physical layer parameters are shown in Table 1, based on the the 3rd generation partnership project (3GPP) recommendations in Refs. [15–16]. For validation purposes, we simulate several load balancing and interference coordination schemes. ‘RSRP’ assigns users based on max RSRP. ‘RE’ adopts a fixed bias 12 dB [17]. ‘RE with ABS’ combines RE and a fixed ABS ratio 0.4. Table 1
System simulation parameters
Simulation parameters Cell layout SBS number per sector User number per sector Bandwidth/MHz MBS ISD/m Transmit power of MBS/dBm Transmit power of SBS/dBm Minimum distance between MBS to users/m Minimum distance between SBS to users/m Thermal noise density/ (dBm ⋅ Hz −1 ) Path loss from MBS to users/dB Path loss from SBS to users/dB
39
To further verify the performance of various schemes, the performance gain vs. ‘RSRP’ is shown in Fig. 3. It is observed that ‘RE’ achieves a better user fairness than ‘RSRP’ because the network load is more balanced with ‘RE’. However, under ‘RE’ association scheme without careful interference management, the offloaded users suffer strong inter-tier interference from the MBSs. ‘RE with ABS’ has a better performance in edge users and utility than ‘RE’. As we expected, our proposal has a significant gain in average rate, edge users (the worst 5%) rate and network utility over other schemes, especially the edge user throughput gain is quite large (e.g., 2.25 × vs. ‘RSRP’ association scheme).
Value 57 cell hexagonal (19 MBSs, 3 sectors per MBS with wrap-around) 2 16 10 500 43 30 35 10 − 174
128.1 + 37.6lgR, R is the distance in kilometer 140.7 + 36.7lgR, R is the distance in kilometer
To illustrate the performances for different schemes, Fig. 2 shows the cumulative distribution function (CDF) of long-term throughput in HetNets of various schemes respectively. Compared with other schemes, our proposal improves significantly at the low rate. There are much less low rate (e.g., 0.2 bit/(s ⋅ Hz) ) users of our proposal than other schemes. For example, there is only 4.3% users whose rate below 0.2 bit/(s ⋅ Hz) , much less than 7.2%, 10.2% and 54.1% in ‘RE with ABS’, ‘RE’ and ‘RSRP’.
Fig. 3
5
Performance gain for different algorithms
Conclusions
In this paper, we investigate the joint optimization problem of load balancing and interference coordination in downlink multi-antenna HetNets. Different from previous works, we utilize ZFBF to avoid inter-tier interference to offloaded users. Considering user association cannot be conducted frequently, long-term throughput is considered. We derive the average achievable rate of users in a long run over Rayleigh fading channel. Then, we formulate the joint user association and INS ratio problem corresponding to proportional fairness. An alternative optimization algorithm is proposed to solve it effectively. Simulation results demonstrate the effectiveness and superiority of the proposed scheme. Acknowledgements This work was supported by the National Natural Science Foundation of China (61672484), and the National Hi-Tech Research
Fig. 2
Rate CDF of different association and IN schemes
and Development Program of China (2014AA01A702).
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The Journal of China Universities of Posts and Telecommunications
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(Editor: Wang Xuying)
http://jcupt.bupt.edu.cn
Joint load balancing and interference coordination in multi-antenna heterogeneous networks Zhu Wenfeng, Qiu Ling (), Chen Zheng, Liang Xiaowen 1. Key Laboratory of Wireless-Optical Communications, Chinese Academy of Sciences, University of Science and Technology of China, Hefei 230026, China 2. School of Information Science and Technology, University of Science and Technology of China, Hefei 230026, China
Abstract In heterogeneous networks (HetNets), it is desirable to offload users from macro cells to small cells to achieve load balancing. However, the offloaded users suffer a strong inter-tier interference. To guarantee the performance of the offloaded users, the interference from macro cells should be carefully managed. In this paper, we jointly optimize load balancing and interference coordination in multi-antenna HetNets. Different from previous works, instead of almost blank subframes (ABS) on which the macro cells waste time resource, the macro cells suppress the interference to the offloaded users by zero-forcing beamforming (ZFBF) on interference nulling subframes (INS). Considering user association cannot be conduct frequently, we derive the long-term throughput of users over Rayleigh fading channels while previous works focused on instantaneous rate. From the perspective of the spectrum efficiency and user fairness, we formulate a long-term network-wide utility maximization problem. By decomposing the problem into two subproblems, we propose an efficient joint load balancing and interference coordination strategy. Simulation results show that our proposal can achieve good system performance gains over counterparts in term of the network utility, cell edge throughput and average throughput. Keywords heterogeneous networks, multi-antenna, load balancing, interference coordination, zero-forcing beamforming
1
Introduction
Wireless data demand has witnessed an exponential growth in recent years due to the increasing of data hungry applications and the trend will continue in the foreseeable future [1]. HetNets, consisting of macro base stations (BSs) (MBSs) overlaid with small BSs (SBSs), along with maturity of multi-antenna transmission techniques are promising solutions to handle the current data deluge. Due to the different transmit power of MBSs and SBSs, when user association metric like reference signal received power (RSRP) is used, most of the users will associate to MBSs, which makes the SBSs extremely underutilized and limit the cell splitting gain. To tackle load balancing in HetNets, range expansion (RE) is introduced to expand the coverage of SBSs through adding a positive bias to their Received date: 07-06-2016 Corresponding author: Qiu Ling, E-mail: [email protected] DOI: 10.1016/S1005-8885(16)60067-5
measured signal strength during user association [2]. In Ref. [3], Ye et al. proposed a distribute user association scheme that achieves load balancing in HetNets through a network-wide utility maximization problem. The optimal user association for downlink multi-antenna HetNets was exploited in Ref. [4]. Although the load balancing can be tackled through the schemes in Refs. [2–4], they may cause the offloaded users suffer a stronger interference. To overcome this issue, ABS, on which the MBSs keep silent to reduce the interference to offloaded users, has been widely adopted [5]. User association and ABS have been jointly considered in Refs. [6–8]. In Ref. [6], the optimal cell RE bias and ABS ratio are obtained by considering the fairness effect among different UEs in a two-tier HetNet. The joint optimization of ABS and user association has been studied in Ref. [7] to strike a balance between retaining cell splitting gain and mitigating interference. In Ref. [8], Jin et al. revealed that the optimal ABS density is the ratio between the number of
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Zhu Wenfeng, et al. / Joint load balancing and interference coordination in multi-antenna heterogeneous networks
offloaded users and total users then proposed a user association scheme through a greedy approach. However, MBSs will waste time resource on ABS. On the other hand, deploying multi-antenna at MBSs provides a more effective alternative to interference coordination through beamforming. For example, in Ref. [9], Wu et al. investigated an interference nulling (IN) scheme by ZFBF in multi-antenna HetNets and obtained the optimal degrees of freedom (DoF) used for IN through stochastic geometry analysis. In this paper, we consider load balancing and interference coordination in a downlink two-tier multi-antenna HetNet. We propose INS, on which the MBSs use ZFBF to mitigate the interference to the offloaded users. Meanwhile, each MBS can use the remaining DoF to serve its own users on INS. Pervious researches mostly focus on instantaneous rate in user association. However, the user association operation is generally conducted much less frequently due to the handover cost in wireless networks. The average effect of time-varying channels should be carefully manipulated. In this paper, firstly, we derive the average rate of users over Rayleigh fading channels. Then, we formulate a long-term network-wide utility maximization problem, where user association and INS ratio are jointly optimized. Although the formulated problem is NP-hard, we decompose the problem into two subproblems and propose an efficient iterative algorithm to solve the original problem. Numerical results show that the proposed strategy can significantly improve the network utility, cell edge throughput and average throughput.
2
System model
Consider a downlink two-tier multi-antenna HetNet, where a macro cell tier and a small cell tier share the same radio spectrum. Each MBS and SBS are equipped with M m and M s antennas respectively and each user employs a single antenna. As shown in Fig. 1(a), consider a typical MBS B0 1,
35
= {1, 2,.., K } . In this paper, the SBSs users are classified into two complementary subsets, one for offloaded users and the other for unoffloaded users. Literally, offloaded users are users who are offloaded to SBSs and interfered seriously by MBSs. The offloaded users suffering dominate interference is a bottleneck towards improving the network performance. To protect offloaded users from strong inter-tier interference, we divide the frames into two subframes, i.e., normal subframes (NS) and INS as Fig. 1(b). Without loss of generality, we expand the meaning of offloaded users by denoting all users scheduled by SBSs on INS as offloaded users.
(a) A two-tier HetNet
(b) Example of INS Fig. 1 System model
On NS, MBS serves its users and SBSs schedule its unoffloaded users. We assume the perfect channel state information (CSI) is available at the BSs. Both MBS and SBSs utilize maximum ratio transmission scheme, i.e., the beamforming vector at BS n is vn = hnk hnk , where hnk is the channel vector between BS n and its scheduled user k. If BS n is a SBS, hnk ~ (0 M s ×1 , I M s ) , otherwise
hnk ~ (0 M m ×1 , I M m ) .
overlaid with N SBSs and K users within its coverage. The BS set consisting of MBS B0 and SBSs is denoted by
On INS, the offloaded users are scheduled by SBSs while MBS uses ZFBF to perform IN by at most L ( L≤N ) DoF. Let S denotes the number of scheduled
denote SBSs. We denote the set of all users as
offloaded users on a certain INS. If S≤L , all scheduled offloaded users can avoid inter-tier interference, otherwise the MBS will randomly select L users from scheduled offloaded users to perform IN scheme. We call scheduled
= {0,1,..., N } , where index 0 denotes B0 , 1, 2,..., N
1 The results herein can be simply extended to the multi-macrocell scenarios. In fact, we consider the more general multi-macrocell scenario in simulations.
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The Journal of China Universities of Posts and Telecommunications
offloaded users that can avoid inter-tier interference as IN users and the rest are referred as none IN (NIN) users. Therefore, the MBS will use LIN = min ( L, S ) DoF to
M m − LIN DoF is used to serve its users. For MBS B0 , let
perform IN scheme. The remaining
Cnk = ∫ C
INS nk
∞
1 − (1 + zPn g nk ) z
0
=∫
−Ms
∞
1 − (1 + zPn g nk )
1 e − zN0 dz + zP g 1 i∈B / {n} i ik
∏
(4)
1 e − zN0 dz i∈B / {0, n} 1 + zPi g ik
(5)
−Ms
∏
z
0
2016
where C0k and C0kINS are the average rate of macro users
H = h0 k , h0O1 ,..., h0OL denote the channel matrix of B0 , where hnk ~ (0 M m × , I M m ) denotes the channel vector
on NS and INS respectively. Cnk denotes the average rate
between MBS and its scheduled user k, h0Oi ~ (0 M m ×1 , I M m ) denotes the channel vector
corresponds to the average rate of IN users.
IN
between MBS to IN users Or ( r = 1, 2,..., LIN ) . The ZFBF vector at B0 is designed to be v0 = w0 w0 , where w0 is the first column of
H † ( HH † ) . −1
SBSs utilize
maximum ratio transmission beamforming to serve its users. The signal to noise plus interference ratio (SINR) of user k ∈ from BS n ∈ can be expressed as
g nk =
Pn g nk vn † hnk
∑ Pg i
i∈I k
ik
vi † hik
2
2
+ N0
to channel gain between BS n and user k including large scale fading, antenna gain and shadowing. According to 2
vi † hik
vn † hnk
exp(1) ,
Lu
distribution with parameters
vn † hnk
2
unoffloaded
2
has a gamma
and 1, denoted as
Γ ( Lu ,1) , where Lu is the number of DoF of
BS n to serve user k. If BS n is a MBS, Lu = M m on NS and = Lu M m − LIN on INS. If BS n is a SBS, Lu = M s .
users
or
NIN
users
while
f ik = vi hik
∫
∞ 0 ∞ 0
,
(a) =
(b) 1 − E e − zP0 g0 k f0 k − z i∈∑/{0} Pik gik fik − zN E e E e 0 dz = z
1 − (1 + zP0 g 0 k ) z
−Mm
1 e − zN0 dz. 1 + zP g i∈B / {0} i ik
∏
0
(1/ z ) (1 − e − zx ) ⋅
[11], (b) follows by the fact
f ik ~ exp (1) ,
where (a) follows from ln (1 + x= / y)
e − zy dz
2
2
P0 g 0 k f 0 k C0 k = E ln (1 + g 0 k ) = E ln 1 + Pi gik f ik + N 0 ∑ i B / 0 ∈ { } ∞ 1 − e − zP0 g0 k f0 k − z i∈∑/{0} Pi gik fik − zN e e 0 dz = E ∫ 0 z
∫
CnkINS
Take C0k as an example, let f 0 k = v0† h0 k
Proof
(1)
where Pi is the transmission power of BS i, N 0 is the power of additive white Gaussian noise, g nk corresponds
Ref. [10],
of
∫
∞
f 0 k ~ Γ ( M m ,1) . The proof of Eq. (3) to Eq. (5) is similar and omitted for brevity.
3
Problem formulation
For each SBS, the frames are comprised of NS and INS. We assume that the unoffloaded users and offloaded users I k = / {0} . If k is unoffloaded or NIN users, can only be scheduled at NS and INS separately. Therefore, SBS n can be considered as two virtual SBS (VSBS) I k = / {n} , otherwise I k = / {0, n} . numbered n and n + N , where VSBS n schedules Since user association cannot be conducted frequently, unoffloaded users on NS and mutes on INS, the other to track the users performance in the long run with respect VSBS i + N serves offloaded users on INS and mutes on to time-varying channels, we derive the average achievable NS. Now, the BS set expands to rate of different types of users. Let model the = 0,1,..., N , N + 1,..., 2 N . x { } e nk Theorem 1 The average rate of different types of
I k is interfering BS set of user k. If k is macro user,
C0 k = ∫
∞
1 − (1 + zP0 g 0 k ) z
0
C0INS k = ∫
−Mm
∞ 0
1 − (1 + zP0 g 0 k ) z
1 e − zN0 dz 1 + zP g i∈B / {0} i ik
∏
LIN − M m
1 e − zN0 dz i∈B / {0} 1 + zPi g ik
∏
xnk = 1 if user k is associated with BS n, xnk = 0 otherwise. Let yn denote association indicator where
users can be expressed as follows (2) (3)
the number of users associated with BS n, i.e., yn = ∑ xnk . We define Q as the probability that a k ∈
offloaded user is an IN user on a certain INS,
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Zhu Wenfeng, et al. / Joint load balancing and interference coordination in multi-antenna heterogeneous networks
Q = LIN
∑
2N
n= N +1
Fn , where Fn is the indicator whether
there are users associated with BS n, i.e., Fn = 1 when yn > 0 , Fn = 0 , otherwise. The achievable rate d nk of user k from BS n can be described as (1 − α ) C0 k + α C0INS 0 k ; n = (6) 1, 2,..., N d nk = (1 − α ) Cnk ; n = INS α QC( n − N ) k + α (1 − Q ) C( n − N ) k ; otherwise where α is the ratio of INS to total subframes (hereafter referred to as INS ratio). We assume that the BSs employ round-robin scheduling policy. The long-term throughput of user k can be written as 1 (7) Rk ∑ xnk d nk ; ∀k ∈ = y n∈e n
37
Under this approximation we can obtain Q = L N . Our simulation results demonstrate this approximation is reasonable for we find there is always at least one offloaded user in each SBS. However, it still remains NP-hard. We propose an alternative optimization method to solve this problem effectively. 3.1
INS ratio
Firstly, we optimize the INS ratio α for a given user association indicator xnk , then problem Eq. (8) is transformed as
d ln nk a yn n 0= k 1 = ( ((( U (a ) (9) s.t. 0≤a ≤1 Theorem 2 The optimal INS ratio is given by 0; Z E = 0 * (10) = α 1;= Z C 0, u ′(1) > 0 is the solution of U ′ α = 0; otherwise ( ) where Z C and Z E is the number of unoffloaded users max
2N
K
∑∑ x
nk
Maximizing throughput can result in the starvation of users with poor signal strength, which is not a satisfactory solution. To strike a balance of network spectrum efficiency and individual user fairness, we adopt proportional fairness utility Ref. [12] in our paper, i.e, the K K network utility function is U ∑ = = ln Rk ∑ ln ∑ (1 yn ) ⋅ x k 1 k 1 = = n∈e N K 2 N K and offloaded users respectively, i.e., Z C = ∑∑ xnk , xnk d nk = ∑∑ xnk ln ( d nk yn ). n 1= k 1 = =n 0=k 1 K 2N Then the problem of joint INS ratio and user association Z E = ∑ ∑ xnk . U ′ (α ) is the first order derivative of n =N +1 k =1 with the target to maximize the utility is formulated as 2N K total utility function which can be computed as follows d max ∑∑ xnk ln nk K −ZC ZE 1 x ,a yn n 0= k 1 = (11) = + + U ′ (α ) ∑ x0 k C −α α 1 0k k =1 s.t. +α C0INS k − C0 k C1 0≤a ≤1 Proof The second order derivative of utility function K (8) C2 ∑ x= yn ; ∀n ∈ e can be formulated as nk k =1 K −ZC −Z −1 2N = + + 2E (12) U ′′ (α ) ∑ x0 k 2 2 α C3 ∑ xnk= 1; ∀k ∈ k =1 (1 − α ) C0 k n=0 α + INS C0 k − C0 k C4= xnk {0,1} ; ∀n ∈ e , ∀k ∈ � It is intuitive that U ′′ (α ) is a negative so U (α ) is a Constaints C1 corresponds to the INS ratio. Constraints concave function with respect to α . We can obtain the C3 and C4 ensure that one user can only access with one globally optimal α * by setting U ′ (α ) = 0 . If Z E = 0 , BS. However, Eq. (8) is a complex problem with the combination of α and the binary variables xnk , which is then U ′ (α ) < 0 , which indicates that U (α ) is an NP-hard problem. We can solve it by exhaustive search monotonic decreasing, the optimal INS ratio α * = 0 . but it is irrational in reality. In order to tackle the problem When Z C = 0 , U ′ ( 0 ) = +∞ , if U ′ (1) > 0 which means we make the following approximation. U (α ) is monotonic increasing then α * = 1. In other Approximation We assume that there is at least one situations, U ′ ( 0 )≥0, U ′ (1)≤0, there is always offloaded user in each SBS, i.e., F= 1, n= N + 1,..., 2 N . n
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The Journal of China Universities of Posts and Telecommunications
α ∈ [0,1] which satisfies U ′ (α ) = 0. We can calculate
2016
The optimal yn* can be derived as
α * through bisection method due to the monotonicity of U ′ (α ) .
yn* = e(
µn −1)
Eq. (8) can be described as 2N K d max ∑∑ xnk ln nk x yn n 0= k 1 =
µn ( t + 1= ) µn ( t ) − δ ( t ) yn ( t ) − ∑ xnk ( t )
(17)
With constraint in Eq. (13), the optimal user association scheme is From Theorem 2, it is observed that the optimal INS 1; n = n* ratio is an increasing function with respect to the number (18) xnk = of offloaded users Z E , which corresponds with reality that 0; otherwise the more offloaded users, the larger optimal INS ratio is. where ( 19) = n* arg max ( ln d nk − m n ) n 3.2 User association To obtain the optimal solution of the dual problem, we Next, we optimize the user association indicator xnk adopt the gradient descent method to update Lagrange multiplier µ [13], by assuming that the INS ratio α is fixed. The problem
s.t. K = C1 yn ∑ xnk ; ∀n ∈ e (13) k =1 2N C2 ∑ xnk= 1; ∀k ∈ � n=0 C3= xnk {0,1} ; ∀n ∈ e , ∀k ∈ � First, we relax the association indicator xnk as a
continuous variable in interval [0,1], then Eq. (13) becomes a convex optimization problem. By introducing Lagrange multiplier for the couple constraint C3 in Eq. (13), the Lagrangian function is
= ( x, y , µ )
2N
K
∑∑ x ( ln d
n 0= k 1 =
nk
2N
nk
− µn ) + ∑ ( µ n yn − yn ln yn ) n 0 =
(14) The Lagrange dual function is computed as ( x, y , ) D ( mm ) = max x, y s.t. (15) 2N xnk= 1; ∀k ∈ ∑ n=0 0≤xnk ≤1; ∀n ∈ e , ∀k ∈ To tackle the Lagrange dual function Eq. (15), differentiating with respect to xnk and yn , we can obtain the necessary condition for xnk and yn
∂ = µ n − 1 − ln yn = 0 ∂yn
1 > 0; xnk = ∂ = ln d nk − µn = 0; 0 < xnk < 1 ∂xnk < 0; x = 0 nk
where δ ( t ) 3.3
k is the step size of the tth iteration.
(20)
Joint iterative algorithm
In this subsection, we propose an effective iterative algorithm to jointly optimize the user association and the INS ratio. Steps of the algorithm are elaborated as follows Step 1 Initialization set ε = 10−4 , a = rand(1,1) . Iterations t = 1, 2,... Step 2 Update user association x. Set= µ ( 0 ) rand (1, 2 N + 1) , ξ = 10−4 , iteration k = 0,1, 2... Substep 1 Update user association. The user will be associated with the target BS according Eqs. (18) and (19). Substep 2 Update the Lagrange multiplier. Using the gradient method to update µ ( j + 1) as
Eq. (20). Substep 3 Check the error of Lagrange multiplier. If µ ( j + 1) − µ ( j ) < ξ go to Step 3 else go to Substep 1. Step 3 Update INS ratio. Calculate α ( t + 1) as Eq. (10) using x ( t + 1) . Step 4 Check the error of INS ratio. If α ( t + 1) − α ( t ) < ε , then stop iteration else go to Step 1.
(16)
4 Simulation results Considering a downlink two-tier HetNet, the SBSs are uniformly located in a circle with distance 2/9 inter site distance (ISD) from the MBS [8]. Assume that the SBSs
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Zhu Wenfeng, et al. / Joint load balancing and interference coordination in multi-antenna heterogeneous networks
are hotspots within a 35 m radius, wherein 3/4 of the users are uniformly distributed at random. The rest are uniformly distributed in MBS coverage. The parameters about antennas are set as M m = 8 , M s = 4 [14] and the optimal L is obtained by exhaustive search in simulations. Other physical layer parameters are shown in Table 1, based on the the 3rd generation partnership project (3GPP) recommendations in Refs. [15–16]. For validation purposes, we simulate several load balancing and interference coordination schemes. ‘RSRP’ assigns users based on max RSRP. ‘RE’ adopts a fixed bias 12 dB [17]. ‘RE with ABS’ combines RE and a fixed ABS ratio 0.4. Table 1
System simulation parameters
Simulation parameters Cell layout SBS number per sector User number per sector Bandwidth/MHz MBS ISD/m Transmit power of MBS/dBm Transmit power of SBS/dBm Minimum distance between MBS to users/m Minimum distance between SBS to users/m Thermal noise density/ (dBm ⋅ Hz −1 ) Path loss from MBS to users/dB Path loss from SBS to users/dB
39
To further verify the performance of various schemes, the performance gain vs. ‘RSRP’ is shown in Fig. 3. It is observed that ‘RE’ achieves a better user fairness than ‘RSRP’ because the network load is more balanced with ‘RE’. However, under ‘RE’ association scheme without careful interference management, the offloaded users suffer strong inter-tier interference from the MBSs. ‘RE with ABS’ has a better performance in edge users and utility than ‘RE’. As we expected, our proposal has a significant gain in average rate, edge users (the worst 5%) rate and network utility over other schemes, especially the edge user throughput gain is quite large (e.g., 2.25 × vs. ‘RSRP’ association scheme).
Value 57 cell hexagonal (19 MBSs, 3 sectors per MBS with wrap-around) 2 16 10 500 43 30 35 10 − 174
128.1 + 37.6lgR, R is the distance in kilometer 140.7 + 36.7lgR, R is the distance in kilometer
To illustrate the performances for different schemes, Fig. 2 shows the cumulative distribution function (CDF) of long-term throughput in HetNets of various schemes respectively. Compared with other schemes, our proposal improves significantly at the low rate. There are much less low rate (e.g., 0.2 bit/(s ⋅ Hz) ) users of our proposal than other schemes. For example, there is only 4.3% users whose rate below 0.2 bit/(s ⋅ Hz) , much less than 7.2%, 10.2% and 54.1% in ‘RE with ABS’, ‘RE’ and ‘RSRP’.
Fig. 3
5
Performance gain for different algorithms
Conclusions
In this paper, we investigate the joint optimization problem of load balancing and interference coordination in downlink multi-antenna HetNets. Different from previous works, we utilize ZFBF to avoid inter-tier interference to offloaded users. Considering user association cannot be conducted frequently, long-term throughput is considered. We derive the average achievable rate of users in a long run over Rayleigh fading channel. Then, we formulate the joint user association and INS ratio problem corresponding to proportional fairness. An alternative optimization algorithm is proposed to solve it effectively. Simulation results demonstrate the effectiveness and superiority of the proposed scheme. Acknowledgements This work was supported by the National Natural Science Foundation of China (61672484), and the National Hi-Tech Research
Fig. 2
Rate CDF of different association and IN schemes
and Development Program of China (2014AA01A702).
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The Journal of China Universities of Posts and Telecommunications
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(Editor: Wang Xuying)