Joint Multiuser Downlink Beamforming and ... - Semantic Scholar

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arXiv:1610.04851v1 [cs.IT] 16 Oct 2016

Joint Multiuser Downlink Beamforming and Admission Control for Green Cloud-RANs with Limited Fronthaul Based on Mixed Integer Semi-definite Program Zhi Yu, Ke Wang, Lei Chen, and Hong Ji, Senior Member, IEEE, Key Laboratory of Universal Wireless Commun., Ministry of Education, Beijing University of Posts and Telecomm., Beijing, P. R. China

Abstract—Cloud radio access network (Cloud-RAN) is expected to be a promising architecture for next generation access network. In this architecture, the functionalities of conventional Baseband Units (BBUs) are shifted to a BBU pool by using cloud computing technology, which enables coordinated multipoint transmission through the low-cost low-power remote radio heads (RRHs) to improve spectrum and energy efficiency. However, with the dense deployment of RRHs, the huge network power consumption has become a great challenge for green CloudRAN, and multiuser downlink beamforming has been proposed as a promising solution. Moreover, the increasing number of mobile users (MUs) causes that admission control (AC) is essential for Cloud-RAN with limited fronthaul capacity and predefined power budget from system design perspective. In this paper, we consider the problem of joint multiuser downlink beamforming and admission control (JBAC) to enhance the admitted MUs in the network and reduce the network power consumption, while taking into account the Quality of Service (QoS) requirements of the MUs, the power budget constraints and fronthaul limitation. It is shown that the JBAC problem is a mixed integer nonlinear problem (MINLP), and still non-convex even though the continuous relaxation is adopted. Therefore, we first transform the JBAC problem into a Mixed-Integer Semidefinite Program (MISDP). Then, we propose a bound improving Branch and Bound (BnB) algorithm to yield the near-optimal solution. For practical application, a polynomial-time heuristic algorithm is proposed to derive the sub-optimal solution. Extensive simulations are conducted with different system configurations to show the effectiveness of the proposed two schemes. Furthermore, simulation results also show that the performance of the heuristic algorithm is very close to that of bound improving BnB algorithm, while the complexity of the former algorithm is much lower than the latter one. Index Terms—Cloud-RAN, admission control, downlink beamforming, green communication, limited fronthaul, Mixed-Integer Semidefinite Program.

I. I NTRODUCTION AND M OTIVATIONS To address high volume of wireless traffic and high data rate demand driven by applications of smart equipments, the traditional single-layer network architecture has shift to heterogeneous one with more densely deployed access points (APs) [1]. However, the increasing number of APs will present great challenges to the design of 5G wireless communication networks in terms of capital expenditure (CAPEX), operating expenditure (OPEX), and interference management [2]. Therefore, to meet the data demand of mobile users (MUs) in an

efficient way, a paradigm shift is required to develop 5G wireless networks. The cloud radio access network (Cloud-RAN) [3], leveraging cloud computing to accomplish large-scale coordinated multipoint (CoMP) communication, is regarded as a promising technology to address the key challenges in 5G wireless networks [4]. In a Cloud-RAN, baseband data and channel state information (CSI) are processed in a cloud data processor, called Baseband Unit (BBU) pool, and shared among the densely deployed remote radio heads (RRHs) via fronthaul links, which allows the RRHs to cooperatively transmit the data to the MUs. As a result, significant CAPEX and OPEX reduction can be achieved due to centralized signal processing, collaborative radio, real-time cloud computing [5]. However, as there are more and more RRHs deployed in Cloud-RAN, the overall network power consumption, including both transmission and circuit power consumption [6], will increase enormously. Therefore, green C-RAN has quickly attracted wide attention. The authors in [7] researched the problem of downlink beamforming to improve energy efficiency of C-RANs by using weighted mixed norm minimization. A three-stage algorithm based on the group-sparsity inducing norm was presented to minimize the power consumption for multicast Cloud-RANs [8]. Combining virtualized network resources and virtualized functional entities of baseband processing, the authors in [9] proposed an energy-saving scheme for C-RANs based on formation of Virtual Base Station. The authors in [10] studied the problem of joint MU-AP association and beamforming design to implement downlink and uplink energy minimization. Nevertheless, these excellent works do not take into account limitation of fronthaul links, which is becoming the a bottleneck for realizing the potential performance gain of Cloud-RANs [11]. Moreover, with the huge number of MUs involved in the network, enormous baseband signals and signaling overheads are required to be transmitted in fronthaul links, which incurs that the limited fronthaul problem becomes more severe. Hence, finding a solution of network power consumption minimization problem under limited fronthaul is critical to achieve green communication and commercial deployment of Cloud-RANs. Some works have been done on energy saving with constrained fronthaul/backhaul links in wireless networks. The authors in [12] investigated the tradeoff between total trans-

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mission power and sum backhaul capacity over all BSs in a network MIMO system. To minimize downlink transmission power, a low complexity algorithm with constraints on fronthaul capacity was presented in [13]. Nevertheless, these works assume that all the RRHs are involved to cooperatively transmit data to MUs, i.e., the RRH selection mechanism is not considered. Therefore, the schemes proposed by these works can not efficiently reduce the circuit power consumption, which accounts for a large part of network power consumption [14]. Furthermore, all the works mentioned above do not take into account admission control (AC), which is very important for downlink beamformer design in Cloud-RAN due to the fact that some MUs may be dropped when there are too many MUs in the networks with limited power budget and fronthaul links. In fact, joint AC and downlink beamforming for green communication in wireless systems has been a research topic of many works in literature. The authors in [15] proposed two computationally efficient convex approximation algorithms to maximize the number of users and minimize the transmission power in a single-cell MISO network. Three algorithmic schemes based on the levels of coordination between the macrocell base station and femtocell base stations were presented to perform AC and beamforming optimization for heterogeneous networks [16]. The authors in [17] proposed a holistic sparse optimization framework which considered the power minimization and user admission control for a multicast Cloud-RAN. However, since these excellent works assume that the capacity of fronthaul/backhaul is big enough to serve all the MUs, the proposed schemes can not apply to green Cloud-RAN with limited fronthaul. A joint resource allocation and AC framework for an orthogonal frequency division multiple access (OFDMA)-based two-tier Cloud-RAN with limited fronthaul is presented to reduce the transmission power consumption [18], however, the circuit power consumption, which is very important for green Cloud-RAN design, did not be taken into account. In this paper, we will study the network power consumption (including both transmission and circuit power consumption) minimization problem in Cloud-RAN, subject to the power budget constraints, the quality-of-service (QoS) requirements of MUs and limited fronthaul constraints. Since the problem may be infeasible when there are too many MUs accessing to the networks, we consider the problem of joint multiuser downlink beamforming and admission control (JBAC) for Cloud-RAN. In particular, we also take into account RRH selection and MU association in the JBAC problem formulation to reduce the circuit power consumption and meet the constraints of the fronthaul links, respectively, which is not considered in the previous works [7]- [17]. The major contributions of this paper can be summarized as follows: • We first research joint multiuser downlink beamforming and admission control problem in Cloud-RAN with limited fronthaul from energy saving perspective, which can achieve admitted MUs maximization and the associated network power consumption minimization. In addition, we also consider switching off the RRHs and associating the admitted MUs and active RRHs in the JBAC problem

formulation. Since the JBAC problem is mixed combinatorial and nonconvex, we transform the problem into a Mixed-Integer Semidefinite Program (MI-SDP) which is a Lagrangian bidual of the JBAC problem when all the binary variables are fixed. • We propose a bound improving Branch and Bound (BnB) algorithm with some complexity reduction mechanisms to derive the optimal solution of the MI-SDP problem. Based on these results, we extract the near-optimal beamforming vector through randomization method. Furthermore, we propose a polynomial-time heuristic algorithm to yield a suboptimal solution with low computational complexity, which is suitable for practical application. • Extensive simulations are conducted with different system configurations to verify the effectiveness of the proposed scheme. Specifically, it is shown that the performance of proposed heuristic algorithm is close to that of the near-optimal algorithm, while the runtime of the former algorithm is significantly lower than that of the latter one. The rest of this paper is organized as follows. In Section II, we presents the system, power model. The joint multiuser downlink beamforming and admission control problem is formulated in Section III. In Section IV, we transform the JBAC problem into a MI-SDP. We then introduce bound improving BnB algorithm and randomization method to derive the near-optimal solution in Section V. In Section VI, a low complexity heuristic algorithm is proposed. Simulation results are discussed in Section VII. Finally, we conclude the paper in Section VIII. Notation: Throughout this paper, boldface lower case and upper case letters represent vectors and matrices, respectively. R and C denote respectively the sets of real and complex numbers. The transpose and trace operators are denoted by (·)H and T r(·), respectively. Im{·} represents the imaginary part of a complex variable. We denote |S| as the cardinality of a set S. Finally, the notations In and enm denote, respectively, the n × n identity matrix and the n-th unit vector in Rm . •

II. S YSTEM M ODEL A. Network Model We consider a downlink Cloud-RAN with L remote radio heads (RRHs) and K mobile users (MUs), where each RRH is connected to a BBU Pool by a limited fronthaul link, as shown in Fig.1. Let L = {1, 2, . . . , L} and K = {1, 2, . . . , K} denote the set of RRH and MU indices, respectively. We assume that RRH l, l ∈ L, is equipped with nl ≥ 1 antennas, and all MUs are equipped with single antenna. In this network architecture, the centralized signal processing is performed at the BBU pool, which results in efficient cooperation of downlink transmission among all the RRHs. Let wl,k ∈ Cnl ×1 be the transmission beamforming vector at RRH l for transmitting data to MU k. Then the transmitted signal of RRH l can be generally expressed as K X (1) xl = wl,k sk , ∀l ∈ L k=1

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sleeping, i.e., there is no MUs served by this RRH. Moreover, since the state of RRH is determined by the association states of MUs, we define bl,k as the RRH-MU association indicator: bl,k = 1 means that the MU k is served by the RRH l, and bl,k = 0 otherwise, ∀k ∈ K, ∀l ∈ L. Then, according to the relationships among al , bl,k and wl,k , we have the following constraints: ( {bl,k = 0, ∀k ∈ K} ⇔ al = 0, ∀l ∈ L (4) bl,k = 0 ⇔ wl,k = 0, ∀l ∈ L, ∀k ∈ K

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The equivalent form of (4) can be written as: ( bl,k 6= 0, ∃k ∈ K ⇔ al = 1, ∀l ∈ L bl,k = 1 ⇔ wl,k 6= 0, ∀l ∈ L, ∀k ∈ K

Fig. 1: An example of downlink C-RAN with limited-capacity fronthaul links.

According to [21], the power consumption of RRH l in transmitting, denoted by PlT , comprises both circuit power consumption and transmission power consumption. With (4), PlT can be expressed as PlT = PlC +

where sk ∈ C denotes the data symbol for MU k with unit power, i.e. E[|sk |2 ] = 1. In this paper, we consider a quasi-static fading environment, and denote hl,k ∈ Cnl ×1 as the channel vector from RRH l to MU k. Then, the received baseband signal at MU k is given by yk =

L X

i=1,i6=k l=1

l=1

|

K L X X

hH l,k wl,k sk + {z

Desired Signal

}

|

hH l,k wl,i si +zk , ∀k ∈ K

{z

Interference

}

(2) where zk ∈ C is the additive Gaussian noise at the MU k, with mean zero and variance σk2 , i.e., zk ∼ CN (0, σk2 ), k ∈ K. We assume that the data symbols and noise for different MUs are mutually statistically independent and single user detection is adopted at the MUs, i.e., the interference in (2) is treated as noise. Then, the corresponding signal-tointerference-plus-noise ratio (SINR) at MU k is given by

SIN Rk =

| K P

L P

2 hH l,k wl,k |

l=1 L P

|

i=1,i6=k l=1

2 2 hH l,k wl,i | + σk

=

PlC

When we design coordinated beamforming vectors of RRHs for green Cloud-RAN, it is critical to consider not only the transmission power consumption but also the circuit power consumption which is dominated by the states of RRHs. According to [6], [14], [19], [20], there are two states for each RRH, which are sleeping and transmitting. Generally, we can know that the RRH is in sleeping if and only if there is no MUs associated with it. We denote the binary variable al as the state indicator of RRH l. Specifically, if some MUs are served by RRH l, we set al = 1, which means RRH l is active. Otherwise, al = 0 stands for that the RRH l is

1 + ηl

k=1 K X k=1

(6)

kwl,k k22

where and ηl denote the circuit power consumption in transmitting and the efficiency of the radio frequency power amplifier for RRH l, respectively. In addition, let the constant PlS denote the power consumption of RRH l when this RRH is in sleep. For Pico base station, the typical values are PlC = 6.8W , PlS = 4.3W , and ηl = 0.25 [21]. With (4) and (6), the power consumption of RRH l, denoted by Pl , can be expressed as Pl = al PlT + (1 − al )PlS = al PlC + = al PlC + PlS

+

K al X bl,k kwl,k k22 + (1 − al )PlS ηl

1 ηl

k=1 K X k=1

al PlCMS

(3)

B. Power Model

K 1 X bl,k kwl,k k22 ηl

PlC

= , ∀k ∈ K

(5)

kwl,k k22 + (1 − al )PlS

(7)

K 1 X + kwl,k k22 , ∀l ∈ L ηl k=1

where PlCMS = PlC − PlS . We denote Φ as the parameter tuple {al , bl,k , wl,k , ∀l ∈ L, ∀k ∈ K}. As a result, the network power consumption of the Cloud-RAN, denoted by F0 (Φ), is given by F0 (Φ) =

L X

Pl =

l=1

L X

PlS +

(al PlCMS +

l=1

l=1

Then, since the term

L X

L P

l=1

K 1 X kwl,k k22 ) ηl k=1 (8)

PlS is a constant if the network archi-

tecture is given, we redefine the network power consumption function F1 (Φ) by omitting the constant term PlS : F1 (Φ) =

L X l=1

(al PlCMS +

K 1 X kwl,k k22 ) ηl k=1

(9)

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III. P ROBLEM F ORMULATION Based on the power consumption model, we will formulate the network power consumption minimization problem for Cloud-RAN, and take into account admission control to ensure that the formulated problem is feasible.

power consumption minimization problem can be formulated as follows: P1 : min F1 (Φ) vΦ u K L u X X 2 2 t | hH s.t. l,k wl,i | + σk ≤ i=1,i6=k

A. Network Power Consumption Minimization for Cloud-RAN with Limited Fronthaul As previously mentioned, we jointly consider the following constraints in Cloud-RAN: the QoS requirement of each MU, power budget and the fronthaul limitation of each RRH. Similar to [10], [22]–[24], we employ the QoS constraints for MUs as follows SIN Rk ≥ γk , ∀k ∈ K,

(10)

where γk > 0 denotes the target SINR of MU k. Furthermore, since the phases of wl,k have no impact on the network power consumption and the QoS constraints, i.e. (9) and (10), the QoS constraints could be rewritten as the following second order cone (SOC) constraints [22]: v u K L u X X 2 t 2 | hH l,k wl,i | + σk i=1,i6=k

l=1

L

X 1 hH ≤ √ Re{ l,k wl,k }, ∀k ∈ K γk

(11a)

l=1

Im{

L X l=1

hH l,k wl,k } = 0, ∀k ∈ K

(11b)

k=1

that the transmission power of RRH l has to meet the following constraint: K X (12) ||wl,k ||22 ≤ PlM , ∀l ∈ L k=1

where PlM represents the power budget of RRH l. To quantify the fronthaul cost, one obvious metric is the average bits/sec. However, considering that the information exchanged in the fronthaul includes not only user data but also signaling overhead, this metric reveals too much detail, and the associated problem is highly combinatorial [25]. Hence, similar to [18], [26] and [27], we adopt the number of MUs served by each RRH which gives a first order measurement of fronthaul load as the metric of fronthaul cost instead. Then, we have the following constraints:

k=1

bl,k ≤ Sl , ∀l ∈ L

Im{

L X l=1

K X

k=1 K X k=1

l=1

L X

hH l,k wl,k }, ∀k ∈ K

(14b)

hH l,k wl,k } = 0, ∀k ∈ K

(14c)

l=1

||wl,k ||22 ≤ PlM , ∀l ∈ L

(14d)

bl,k ≤ Sl , ∀l ∈ L

(14e)

al = 0 ⇔ {bl,k = 0, ∀k ∈ K}, ∀l ∈ L

bl,k = 0 ⇔ wl,k = 0, ∀l ∈ L, ∀k ∈ K al ∈ {0, 1}, bl,k ∈ {0, 1}, ∀l ∈ L, ∀k ∈ K

(13)

where Sl denotes the maximum number of MUs that RRH l can serve. With the constraints of the QoS requirement, the maximum power budget and the limitted fronthaul and (4), the network

(14f) (14g) (14h)

We note that constraints (14f) and constraints (14g) are not closed form, which make problem P1 hard to be solved. As a result, according to our previous work ( [28], Theorem 1), problem P1 can be reformulated as follows: L K ζ XX bl,k P2 : min F2 (Φ) = F1 (Φ) + Φ L·K

(15a)

l=1 k=1

s.t. (14b), (14c),

For power budget constraint, since the transmission power K P ||wl,k ||22 according to (1). We assume of RRH l is equal to

K X

1 √ Re{ γk

(14a)

K X

k=1 K X k=1

(15b)

||wl,k ||22 ≤ PlM , ∀l ∈ L

(15c)

bl,k ≤ al Sl , ∀l ∈ L

(15d)

||wl,k ||22 ≤ bl,k PlMAX , ∀l ∈ L, ∀k ∈ K

al ∈ {0, 1}, bl,k ∈ {0, 1}, ∀l ∈ L, ∀k ∈ K

(15e) (15f)

where ζ is a positive constant with ζ → 0. B. Considering Admission Control for Downlink Beamforming As shown in [29], the problem P2 is in fact a MixedInteger Second-Order Cone Program (MI-SOCP), which can be efficiently solved by the method proposed in [28]. However, according to the literatures [15]–[18] which focus on the downlink beamformer design for green communication, problem P2 can easily become infeasible, since not all QoS requirements of MUs can be satisfied when the number of MUs served by the network is much larger than the total number of the antennas of RRHs or the fronthaul capacity. Accordingly, some of the users should be temporarily dropped via user admission control to guarantee that the desired QoS at each admitted MU is attainable. In such scenarios, it makes sense to require the network power consumption minimization problem to maximize the number of MUs that can be served,

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while support these MUs with the minimum network power consumption. As a result, the problem of interest can be described in two stages. In the first stage, we will select subsets of the MUs with largest number of elements, which is given by max Ψ

K X

ck

(16a)

k=1

Clearly, when the MU k is refused to access the network, i.e., ck = −1, bl,k = 0, wl,k = 0, ∀l ∈ L, the objective function value will increase due to the penalty factor α, and the QoS requirement constraint of MU k will be satisfied if we set a proper feasibility guarantee factor β. Lemma1 : When 0 ≤ β ≤ 4 holds, problem P is min 3 L P P k∈K γ ·( k

l=1

s.t. (15c), (15d), (15e) (16b) ck + 1 SIN Rk ≥ γk · , ∀k ∈ K (16c) 2 ck + 1 bl,k ≤ , ∀l ∈ L, ∀k ∈ K (16d) 2 al ∈ {0, 1}, ck ∈ {1, −1}, bl,k ∈ {0, 1}, ∀l ∈ L, ∀k ∈ K (16e) where ck ∈ {1, −1} denotes MU access indicator: ck = 1 represents that the MU k is admitted to access the network, and ck = −1 otherwise, ∀k ∈ K. In addition, the Ψ is denoted as the parameter tuple {al , ck , bl,k , wl,k , ∀l ∈ L, ∀k ∈ K}. It is worth note that the constraints (16d) and (15e) guarantee that bl,k = 0, wl,k = 0, ∀l ∈ L hold if MU k is temporarily dropped, i.e. ck = −1, ∀k ∈ K. In the second stage, with the fixed MU access indicators, we solve the following problem to minimize the network power consumption

PlM ·

L l=1

2) ||hl,k ||22 +σk

always feasible. Proof : Please refer to the Appendix A. 4 Theorem1 : If we take 1 − P L PM (PlCM S +

l=1

1 and 0 ≤ β ≤ min

k∈K γ ·( k

L P l=1

PlM ·

4 PL

l=1

l ηl

)+ζ+4

≤α≤

, solution of 2) ||hl,k ||22 +σk

problem P3 can achieve the maximum number of MUs served by the network and the associated minimum network power consumption. Proof : Please refer to the Appendix B. Theorem1 shows that problem P3 is a single stage optimization reformulation of the two-stage problem. Furthermore, it is worth to note that the main advantage of problem P3 is that it allows a convenient convex relaxation which will be shown in the following section. IV. MI-SDP T RANSFORMATION

min F2 (Φ)

(17a)

Φ

s.t. (15c) − (15f ), (16c), (16d)

(17b)

Despite the second stage problem is also a MI-SOCP which can be efficiently solved like the problem P2 , we have to first obtain the optimal set of admitted MUs by solving the first stage problem. Unfortunately, the first stage problem is obviously mixed combinatorial and non-convex due to the constraints (16e) and (16c), respectively, which make the two-stage above problem intractable. As a result, we will reformulate the two-stage problem as a convenient form in the following subsection. C. Joint Multiuser Downlink Beamforming and Admission Control In this subsection, inspired by the literature [15], we reformulate the two-stage problem mentioned above as a joint multiuser downlink beanforming (JBAC) problem by introducing two suitably small positive constants α, β, which are used as penalty factor for MU dropped and feasibility guarantee factor. Specifically, the JBAC problem can be expressed as follows: P3 : min F3 (Ψ) = (1 − α)F2 (Φ) + α Ψ

s.t. (15c), (15d), (15e), (16d), (16e) L P 2 −1 (ck − 1)2 hH | l,k wl,k | + β l=1

P

|

L P

i∈K,i6=k l=1

2 2 hH l,k wl,i | + σk

K X

k=1

(ck − 1)2 (18a) (18b)

≥ γk , ∀k ∈ K (18c)

Although we reformulate the two-stage problem as JBAC problem with a simple form, we note that problem P3 is still a MINLP non-convex problem. As a result, we will transform problem P3 into a Mixed-Integer Semidefinite Program (MISDP) in this section. For ease of presentation, we assume that the all RRHs are equipped with the identical number of antennas and efficiency of power amplifier, denoted as n and η, respectively. Note that the algorithms proposed in this paper can be easily extended to the scenario with different number of antennas and power amplifier efficiency for each RRH. Then, we denote by hk = H H nL×1 [hH the aggregative channel vector for 1,k , . . . , hL,k ] ∈ C nLK×1 MU k. Let w ∈ C denote the aggregative beamforming vector, which can be written as follows: H H H H H w = [w1,1 , w2,1 , . . . , wL,1 , . . . , w1,K , . . . , wL,K ]H

(19)

We introduce a new variable W = wwH which is a rankone positive semidefinite (PSD) matrix. In addition, we define some constant matrices, such as E1k ∈ RnL×nLK , E2l ∈ RnK×nLK and E3l,k ∈ Rn×nLK , ∀l ∈ L, ∀k ∈ K, and set them as follows:  1 k  Ek , eK ⊗ InL 2 (20) El , (IK ⊗ elL ) ⊗ In   3 k l El,k , (eK ⊗ eL ) ⊗ In where ⊗ means Kronecker product. According to matrix theory, we know that xH Cx = T r(xH Cx) = T r(CxxH ) holds for any matrix C and vector x with proper dimensions. In addition, it can be shown that (ck − 1)2 in problem P3 is equivalent to (2 − 2ck ) under

6

the constraint ck ∈ {1, −1}. Thus, with W and the constant matrices defined in (20), we can rewrite problem P3 as

A. Bound improving BnB Algorithm We first take solving minimization problem as a example to overview BnB algorithm. A typical BnB algorithm consists of L X 1 two main parts namely branching and bounding, respectively. P4 : min F4 (Γ) = (1 − α)[ al PlC + T r(W) In branching part, we will divide a large problem into a few Γ η l=1 smaller ones according to some rules which will be introduced L K K X ζ XX + bl,k ] + α (2 − 2ck ) (21a) in the following. In bounding part, we will estimate how good LK a solution we can get for each smaller problem. To do this, we l=1 k=1 k=1 may have to divide the problem further, until we get a problem 1 −1 T r((E1k )H hk hH (2 − 2ck ) k Ek W) + β P s.t. ≥ γ , ∀k ∈ K that we can handle. Then, we will update the best-known k 2 1 T r((E1i )H hk hH k Ei W) + σk objective value (i.e., the smallest upper bound for minimization i6=k (21b) problem) of the original problem. In BnB algorithm, some of the branches can be efficiently removed from the search tree T r((E2l )H E2l W) ≤ PlM , ∀l ∈ L (21c) if one of the following pruning conditions is satisfied: K X (1) The smaller problem corresponding to the branch is bl,k ≤ al Sl , ∀l ∈ L (21d) infeasible. k=1 (2) The optimal objective value of the smaller problem of 3 H 3 T r((El,k ) El,k W) ≤ bl,k PlM , ∀l ∈ L, ∀k ∈ K (21e) the branch is larger than the best-known objective value. ck + 1 According to the above two pruning conditions, we will , ∀l ∈ L, ∀k ∈ K (21f) bl,k ≤ 2 present two approaches to potentially reduce the complexity of al ∈ {0, 1}, ck ∈ {1, −1}, bl,k ∈ {0, 1}, ∀l ∈ L, ∀k ∈ K BnB algorithm. For the first pruning condition, we add three (21g) linear redundant constraints which can be removed without W  0, rank(W) = 1 (21h) loss of generality, and then problem P5 can be expressed as follows: where Γ denotes a parameter tuple {al , ck , bl,k , W, ∀l ∈ P6 : min F4 (Γ) (23a) Γ L, ∀k ∈ K}, and we use W  0 to indicate that W is PSD. s.t. 21c − 21g, 22c, 22d (23b) Although problem P4 is also mixed integer and nonconvex bl,k ≤ al , ∀l ∈ L, ∀k ∈ K (23c) like problem P3 , we note that the objective function and all constrains of problem P4 are convex if {al , ck , bl,k , ∀l ∈ K X L, ∀k ∈ K} are continuous variables, except the rank conbl,k ≥ al , ∀l ∈ L (23d) straint rank(W) = 1 in constraints (21h). Therefore, we will k=1 L drop the rank constraint to obtain the following problem: X ck + 1 , ∀k ∈ K (23e) bl,k ≥ 2 l=1 P5 : min F4 (Γ) (22a) Γ For the second pruning condition, we will carefully select s.t. 21c − 21g (22b) branch of the tree and the search direction to obtain a good X 2 1 best-known objective value as soon as posible, which will be γk ( T r((E1i )H hk hH k Ei W) + σk ) presented in further detail in the following. i6=k 1 −1 For ease of presentation, we define B as a set which is W) + β (2 − 2c ), ∀k ∈ K ≤ T r((E1k )H hk hH E k k k composed of all the RRH-MU pairs in the CRAN, i.e. B = (22c) {(l, k)|∀l ∈ L, ∀k ∈ K}, and introduce another two RRH-MU W0 (22d) pair sets which can be expressed as: ( B1 = {(l, k)|bl,k = 1, ∀l ∈ L, ∀k ∈ K} We note that problem P5 is the semidefinite relaxation (24) (SDR) of problem P4 which can be interpreted as a LaB2 = {(l, k)|bl,k = 0, ∀l ∈ L, ∀k ∈ K} grangian bidual of problem P4 (or problem P3 ) [30] if all the We note that al = 1 and ck = 1 hold if (l, k) ∈ B1 . With binary variables are fixed. In the following, we will derive the B1 and B2 , the integer constraints of problem P6 (constraints near-optimal solution of problem P3 according to the optimal (21g)) are equivalent to: solution of problem P5 .  bl,k = 1, ∀(l, k) ∈ B1 (25a)     b = 0, ∀(l, k) ∈ B (25b) l,k 2 V. N EAR - OPTIMAL B EAMFORMER D ESIGN  al ∈ {0, 1}, ck ∈ {−1, 1}, ∀l ∈ L, ∀k ∈ K (25c)    bl,k ∈ {0, 1}, (l, k) ∈ B\(B1 ∪ B2 ) (25d) In this section, we first solve the optimal solution of problem ∗ Then, we can transform problem P as follows: 6 P5 , denoted by W , by using bound improving branch and bound (BnB) algorithm. Then, we adopt randomization P7 : min F4 (Γ) (26a) Γ method to obtain the the near-optimal solution of problem s.t. 21c − 21f , 22c, 22d, 23c − 23e, 25a − 25d (26b) P3 , denoted by w∗ , based on W∗ .

7

In addition, when B1 and B2 are fixed, the continuous relaxation of problem P7 can be expressed as the following SDP:

Algorithm 1 Bound improving BnB algorithm for solving MISDP 1:

P8 : min Γ

F4 (Γ)

(27a)

s.t. 21c − 21f , 22c, 22d, 23c − 23e, 25a, 25b al ∈ [0, 1], ck ∈ [−1, 1], ∀l ∈ L, ∀k ∈ K bl,k ∈ [0, 1], (l, k) ∈ B\(B1 ∪ B2 )

(27b)

l,k

l,i l,i

l,k

3:

(27c) (27d)

Clearly, in CRANs with multiples MUs served jointly by cooperating RRHs in CoMP transmission, the optimal objective value of problem P8 is the lower bound of problem P7 . We denote problem P7 and problem P8 as the parameter tuples (z, B1 , B2 ) and (z, B1 , B2 )CR , respectively, where z represents the optimal objective value of the problem P8 , i.e. the lower bound of problem P7 . The pseudo code of the proposed bound improving BnB algorithm is outlined in Algorithm 1, in which some notation have been introduced above. Here, we will explain the other notations: Q is the set of problems (or branches); z b is the bestknown objective value, and B1b and B2b are the corresponding RRH-MU pair sets, where B1 ∪ B2 = B holds; W∗ represents the solution of problem (z b , B1b , B2b ), which is also the optimal solution of problem P5 . What is more, we can easily deduce the optimal binary variables of problem P5 , denoted by {a∗l , c∗k , b∗l,k , ∀l ∈ L, ∀k ∈ K}, from B1b and B2b according to (4) and (24). In the following, we will present the main steps of Algorithm 1. 1)Branching: At each iteration, the problem that achieves the minimum lower bound, denoted as (ˆ z , Bˆ1 , Bˆ2 ), is selected to branch. This rule is known to be the bound improving [31]. Then, the RRH-MU pair with the maximal priotiy is selected, and the problem is divided into two smaller problems according to the value of the user association indicator corresponding to the chosen RRH-MU pair is zero or one. It is obvious that how to select the proper RRH-MU pair is critical to reduce the complexity of the bound improving BnB algorithm, since many problems may be pruned if we obtain a best-known objective value soon. Thus, we define θˆl,k as the priority of a RRH-MU pair for problem (ˆ z , Bˆ1 , Bˆ2 ) which can be expressed as: 3 ˆ T r((E3l,k )H hl,k hH ηl Sl PlM l,k El,k W) · θˆl,k = P (28) ˆ PC T r((E3 )H h hH E3 W) i6=k

2:

l

When the network is fixed, we note that the priority level ˆ which is the solution of problem θˆl,k is a function of W (ˆ z , Bˆ1 , Bˆ2 )CR . The first part of (28) can be interpreted as the contribution of this RRH-MU pair to the network performance. Specifically, the numerator represents the power of the desire signal, and the denominator stands for the power of interference caused by the RRH-MU pair. The second part of (28) contains the key system parameter of RRHs. Clearly, the RRH-MU pair belonging to a RRH with higher energy efficiency and fronthaul capacity has higher priority level. 2)Bounding and P runing: Based on the chosen branch and search direction, we update the user association sets, such as B1n,1 , B2n,1 , B1n,2 and B2n,2 . Then, we compute the lower

Initialization: z b = +∞; B10 = ∅, B20 = ∅; Q = {(z 0 , B10 , B20 )};n = 0. while Q = 6 ∅ do Select the problem (ˆ z , Bˆ1 , Bˆ2 ) from Q with zˆ = ˆ ∀l ∈ min z, and denote by {ˆ al , cˆk , ˆbl,k , W, (z,B1 ,B2 )∈Q

4: 5:

L, ∀k ∈ K} the solution of problem (ˆ z , Bˆ1 , Bˆ2 )CR . ˆ According to W, compute the priority level θˆl,k of each (l, k) ∈ B\(Bˆ1 ∪ Bˆ2 ) by (28). Select (l∗ , k ∗ ) = argmax θˆl,k . (l,k)∈B\(Bˆ1 ∪Bˆ2 )

6: 7: 8: 9: 10: 11: 12: 13: 14: 15: 16: 17: 18: 19: 20: 21: 22: 23: 24: 25:

Set n = n + 1. Set B1n,1 = Bˆ1 ∪ {(l∗ , k ∗ )}, B2n,1 = Bˆ2 and B1n,2 = Bˆ1 , B2n,2 = Bˆ2 ∪ {(l∗ , k ∗ )}. Solve problem (z n,i , B1n,i , B2n,i )CR , i = 1, 2. Set z n,i = +∞ if problem (z n,i , B1n,i , B2n,i )CR is infeasible, i = 1, 2. if z n,1 < z b , then if B == B1n,1 ∪ B2n,1 , then Set z b = z n,1 , B1b = B1n,1 , B2b = B2n,1 else Set Q = Q ∪ {(z n,1 , B1n,1 , B2n,1 )} end if end if if z n,2 < z b , then if B == B1n,2 ∪ B2n,2 , then Set z b = z n,2 , B1b = B1n,2 , B2b = B2n,2 else Set Q = Q ∪ {(z n,2 , B1n,2 , B2n,2 )} end if end if end while return z b , B1b , B2b , W∗

bounds of the two problems, i.e., (z n,1 , B1n,1 , B2n,1 )CR and (z n,1 , B1n,1 , B2n,1 )CR , by solving the corresponding problem P8 , respectively. According to [31], global convergence will be guaranteed with the simple bound. After finding the lower bounds (z n,1 , z n,2 ) and updating the current best-known objective value z b , the problem whose lower bound is smaller than z b can be added to the problem set Q, otherwise, the problem will be dropped. This step is refereed to as pruning. B. Randomization Method for Near-optimal Solution Extracting When the Algorithm 1 has been done, we can obtain an optimal solution of problem P5 which contains a PSD matrix W∗ . However, W∗ may not be a feasible solution of problem P4 , since the rank of W∗ may not be equal to one. As a result, we now discuss how to extract the near-optimal aggregative beamforming vector w∗ by randomization method [32]. If the W∗ is rank-one, the optimal solution of problem P3 can be achieved. Specifically, we apply the eigenvalue√decomposition to W∗ as W∗ = λe we weH , and take w∗ = λe we . Otherwise, we first generate a set of candidate beamforming

8

vectors v1 , v2 , . . . , vM , where vm ∼ CN (0, W∗ ), m = 1, 2, . . . , M . We then need to do scaling to yield the beamforming vector which can meet the constraints of problem P3 . For the vector vm , the scaling problem can be formulated as follows: min

s.t.

x

x H 1 vm ((E1k )H hk hH k Ek

Algorithm 2 Polynomial-time Heuristic Algorithm 1: 2: 3:

i6=k

(29a) −

X

x>0

P

(29b) (29c) (29d)

Clearly, the problem above is a linear program, and we can easily solve the optimal solution, denoted by xm . Then, we calculate the aggregative beamforming vector by wm = √ xm vm , m = 1, 2, . . . , M , and have the following theorem. Theorem2 : {a∗l , c∗k , b∗l,k , wm , ∀l ∈ L, ∀k ∈ K} is a feasible solution of problem P3 , m = 1, 2, . . . , M . Proof : Please refer to the Appendix C. With Theorem 2, we find w∗ = min{w1 , w2 . . . , wM }, and set w∗ as the aggregative beamforming vector. According to [32], the aggregative beamforming vector w∗ can achieve near optimality if M ≥ 15.

i

k

k

i

k

each k ∈ K. If so, go to step 4; else, remove the MU k˜ with largest gap to its target SINR, i.e. k˜ = argmax(γk −

1 (E1i )H hk hH k Ei )vm · x

i6=k + β −1 (2 − 2ck ) ≥ γk σk2 , ∀k ∈ K H vm (E2l )H E2l vm · x ≤ PlM , ∀l ∈ L

Initialization: Set K = {1, 2, . . . , K}. Solve problem P8 with B1 = ∅, B2 = ∅ and K, and ˜ the solution. denote by W ˜ T r((E1 )H h hH E1 W) Check whether P T r((E1k)H h khHk E1kW)+σ 2 ≥ γk holds, for ˜

i6=k

4: 5:

6:

7:

1 ˜ T r((E1k )H hk hH k Ek W) 1 ˜ 2 ), T r((E1i )H hk hH k Ei W)+σk

k∈K

from the set K, and go to step

2. ˜ for ∀l ∈ L, ∀k ∈ K. Compute priority level θl,k = Θ(W) For each l ∈ L, sort θl,k , ∀l ∈ L, ∀k ∈ K in the descending order, and set the RRH-MU association indicators (bl,k ) corresponding to the Sl largest priorities as 1 and the others as 0; compute al , ∀l ∈ L according to bl,k , ∀l ∈ L, ∀k ∈ K and (4). Check whether problem P2 with fixed al , bl,k , ∀l ∈ L, ∀k ∈ K is feasible or not; if so, go to step 7; else, remove the MU with largest gap to its target SINR k˜ from the set K and go to step 2. With K, adopt the inflation algorithm proposed in [28] to compute the suboptimal solution of problem P2 .

VI. S UBOPTIMAL B EAMFORMER D ESIGN Despite the bound improving BnB algorithm presented in section V can obtain the near-optimal solution of problem P3 , the computational complexity required may be prohibitive for large network. For practical application, we propose a polynomial-time heuristic algorithm which is numerically shown to yield a suboptimal solution of problem P3 . The pseudo code of the proposed low complexity heuristic algorithm is outlined in Algorithm 2. The main idea of Algorithm 2 is to divide the suboptimal beamformer design problem into two phases. In the first phase, we assume that all the RRHs are active, and the admission control procedure is implemented, which corresponds to the top six steps of Algorithm 2. Specifically, Step 3, Step 6 and constraints (21c) guarantee that there is at least one point satisfying the QoS constraints, the fronthaul limitation constraints and the power budget constraints of problem P2 , respectively. Thus, when the six step have been done, problem P2 with admitted MUs set K is feasible. In the second phase, since problem P2 is feasible, we can use the inflation algorithm presented in our previous work [28] to solve the suboptimal solution, which may further reduce the overall network power consumption by sleeping the idle RRH. Because of the space constraint, the details of the inflation algorithm are omitted here. Furthermore, since the result of Algorithm 2 is a suboptimal solution of problem P3 , we can set it as the initialization of z b to reduce the complexity of Algorithm 1. In the following, we will analysis the computational complexity of Algorithm 2. For each of SDP problem P8 , the 3.5 computational complexity is O(DSDP log(1/ǫ)) by using a custom-built interior-point algorithm [33] with a solution accuracy ǫ > 0, where DSDP represents the dimension of

the positive semidefinite cone of problem P8 , i.e. DSDP = nLK + LK + K + L. Since there are K iterations at most, the worst-case computational complexity of the top six steps is polynomial. In addition, according to [28], the inflation algorithm is a polynomial-time algorithm also. Therefore, Algorithm 2 is a polynomial-time algorithm, meanwhile, it converges in finite iterations. VII. N UMERICAL R ESULTS In this section, we present numerical results to verify the two proposed algorithms from three perspectives: ensuring feasibility by adopting admission control; serving more MUs for downlink transmissions in Cloud-RAN; and achieving network power saving with proper user association and downlink beamforming. We consider a network with L = 4 multiantenna RRHs and some single-antenna MUs uniformly and independently distributed in the square region [−100 100] × [−100 100] meters. Similar to the existing works [34] [7], we adopt the following channel model: (i) the path-loss (PL) at distant d(km): P L = 148.1 + 37.6log10d (dB), (ii) lognorm shadowing with zero mean, 8dB variance, (iii) smallscale fading distribution: CN (0, 1), (iv) transmission antenna power gain 9dB and noise power σk = −104dBm [35]. In addition, for the RRH l, we use homogeneous setting: the power budget PlM = 0.13W , the power amplifier efficiency ηl = 0.25, circuit power consumption PlC = 6.8W , and power consumption in sleep PlS = 4.3W [21]. The small constant ζ in the objective function of problem P2 is set to 10−3 . The algorithms proposed in this paper is also suitable for the heterogeneous setting scenarios. Each point of the

9

10

Average number of served MUs

Average number of served MUs

9.5

9

8.5

8

PP−Algorithm, n=2 PP−Algorithm, n=4 PB−Algorithm, n=2 PB−Algorithm, n=4

7.5

7

6.5

2

3

4

5

6

7

8

9

9.5 9 8.5 8

7 6.5

10

PP−Algorithm, n=2 PP−Algorithm, n=4 PB−Algorithm, n=2 PB−Algorithm, n=4

7.5

4

4.5

5

Target SINR (dB)

Fig. 2: Number of served MUs versus Target SINR. Average network power consumption (W)

5.5

6

6.5

7

7.5

8

Fronthaul Capacity

Fig. 4: Number of served MUs versus Fronthaul Capacity.

27.6

27.4

27.2

PP−Algorithm, n=2 PP−Algorithm, n=4 PB−Algorithm, n=2 PB−Algorithm, n=4

27

26.8

26.6

2

3

4

5

6

7

8

9

10

Target SINR (dB)

Fig. 3: Network power consumption versus Target SINR.

Average network power consumption (W)

27.8 27.8 27.7 27.6

PP−Algorithm, n=2 PP−Algorithm, n=4 PB−Algorithm, n=2 PB−Algorithm, n=4

27.5 27.4 27.3 27.2 27.1

4

4.5

5

5.5

6

6.5

7

7.5

8

Fronthaul Capacity

Fig. 5: Network power consumption versus Fronthaul Capacity. simulation results is averaged over 100 randomly generated network realization. A. Network Performance versus Target SINR In this subsection, we evaluate the performance of the proposed bound improving BnB algorithm (PB-Algorithm) and polynomial-time heuristic algorithm (PP-Algorithm), where the number of antennas is two or four, i.e., n = 2 or n = 4. In addition, we set the total number of MUs and the fronthaul capacity as 10 and 6, respectively. In Fig.2 and Fig.3, we compare the various performance of the two proposed algorithms, such as the number of served MUs and network power consumption. As shown in these figures and table, PB-Algorithm always outperforms PPAlgorithm. This is because the later algorithm can derive suboptimal solution, while the former one is able to yield nearoptimal solution. Even so, the performance of PP-Algorithm are very close to that of PB-Algorithm, which confirms the effectiveness of the proposed PP-Algorithm with only polynomial-time complexity. Furthermore, we also observe that the number of antennas has great impact on the various

network performance, since more antennas can obtain greater diversity gain. Specifically, as shown in Fig.2, the number of served MUs decreases sharply with the growth of target SINR. Because excessive target SINR will lead to few users’ requirements can be satisfied under the limitations of maximum transmission power and fronthaul capacity. In addition, Fig.3 depicts the variation in network power consumption versus target SINR. As we have seen, the network power consumption increases with the fronthaul capacity. This is due to that more active RRHs and higher transmission power are needed to meet the growing QoS requirements of MUs. Moreover, the curves ascends slowly when the target SINR is hight, since the power budget is limited for each RRH. This figure demonstrates that the performance gap between PP-Algorithm and PB-algorithm becomes larger when the target SINR is low. Because more RRHs can be in sleep mode in PP-Algorithm, and more circuit power consumption can be saved when the target SINR is low.

Average network power consumption (W)

10

Average number of admitted MUs

11 10 9 8 7

PP−Algorithm, n=2 PP−Algorithm, n=4 PB−Algorithm, n=2 PB−Algorithm, n=4 EL−Algorithm, n=2 EL−Algorithm, n=4

6 5 4 3

4

5

6

7

8

9

10

11

28 27.5 27 26.5 26 25.5

PP−Algorithm, n=2 PP−Algorithm, n=4 PB−Algorithm, n=2 PB−Algorithm, n=4 EL−Algorithm, n=2 EL−Algorithm, n=4

25 24.5 24 23.5 23

4

5

6

7

8

9

10

11

12

Total Number of MUs

12

Total Number of MUs

Fig. 6: Number of admitted MUs versus Total Number of MUs.

Fig. 7: Network power consumption versus Total Number of MUs.

250

TABLE I: Number of active RRHs versus Total Number of MUs.

EL-Algorithm, EL-Algorithm, PP-Algorithm, PB-Algorithm, PP-Algorithm, PB-Algorithm,

n=2 n=4 n=2 n=2 n=4 n=4

4

6

8

10

12

4 4 3.6897 3.6724 2 1.9724

4 4 3.9943 3.6868 3.7449 3.7246

4 4 4 4 3.9293 3.9141

4 4 4 4 4 4

4 4 4 4 4 4

200

CPU time (s)

Total Number of MUs

PP−Algorithm, n=4 PB−Algorithm, n=4 GB−Algorithm, n=4

150

100

50

B. Network Performance versus Fronthaul Capacity From Fig.4 and Fig.5, we evaluate the effect of fronthaul capacity on the number of served MUs and the network power consumption, respectively. We consider that there are 10 MUs deployed in the network, and the target SINR is 6dB. Based on the reasons mentioned above, we can easily explain that PB-Algorithm outperform PP-Algorithm, and the advantages of more antennas are also obvious. Specifically, in Fig.4, we have observed that the slopes of the curves are decreasing with the fronthaul capacity. This is due to that the number of served MUs is limited by the power budget of each RRH. Moreover, Fig.5 depicts the network power consumption is increasing with fronthaul capacity. Because the RRHs have to enhance their transmission power to serve more MUs when the fronthaul capacity is high. In particular, different from the trend of the curves in Fig.4, the network power consumption with 4 antennas is decreasing when the fronthaul capacity is high. This is because that not all the RRHs need to be switched on in that scenario. As a result, we can decrease the circuit power consumption. C. Network Performance versus Total Number of MUs In this subsection, we first evaluate the network performance of the two proposed schemes by comparing with the scheme presented in [13] (EL-Algorithm). In EL-Algorithm, the authors focus on implement transmission power consumption

0

4

5

6

7

8

9

10

11

12

Total Number of MUs

Fig. 8: Algorithm Run-time versus Total Number of MUs.

minimization for Cloud-RAN with fronthaul capacity limitation, and they assume that the total MUs in the network can be cooperatively served by all the RRHs. Then, we compare the run-time of general BnB algorithm (GB-Algorithm), PBAlgorithm and PP-Algorithm. In particular, the target SINR and fronthaul capacity are configured as 6dB and 6, respectively. From Fig.6, Fig.7 and Table I, we can also observe that the three kinds of performance of PB-Algorithm is better than those of PP-Algorithm, since the former algorithm can derive better solution. Fig.6 shows that the number of served MUs is proportional to total number of MUs in the network. With more and more MUs deployed in the network, the growth of the number of admitted MUs becomes slow, since some MUs have to be temporarily dropped. In particular, from this figure, we can easily know that the maximal number of admitted MUs of the proposed algorithms with 2 and 4 antennas is almost eight and ten, respectively. Moreover, this figure also demonstrates that EL-Algorithm can not be applied in the scenario with too many MUs (e.g., 8 MUs with n = 2 and 10 MUs with n = 4), even though this algorithm can serve all the

11

MUs when total number of MUs is small. This is due to that EL-Algorithm do not consider admission control in downlink beamforming problem. Fig.7 depicts the variation in network power consumption versus the number of MUs. As we have seen, the performance of the proposed algorithms is better than that of EL-Algorithm, especially when total number of MUs is small. Because our proposed schemes take into account both transmission power consumption and circuit power consumption. According, we can switch off some RRHs to save circuit power, while all the RRHs are always active in EL-Algorithm. Furthermore, we also notice that the network power consumption of the proposed algorithm with 4 antennas is significantly smaller than other cases. From Table I, we can know that even few active RRHs are able to serve the total MUs in the network with the limitations of maximum transmission power and fronthaul capacity. Fig.8 shows the average algorithm run-time versus total number of MUs in the network. As shown in this figure, the slope of PP-Algorithm is significantly smaller than that of the other two algorithms, since the computation complexity of the former algorithm is polynomial, which is demonstrated in Section V. In addition, the run-time of PB-Algorithm is much shorter than that of GB-Algorithm. This is because that we introduce some complexity reduction mechanisms into PB-Algorithm, such as adding linear redundant constraints, selecting proper branch and improving initial point. VIII. C ONCLUSIONS AND F UTURE W ORK In this paper, we investigated joint multiuser downlink beamforming and admission control problem for the CloudRANs with limited fronthaul capacity, which was aiming to minimize the network power consumption and ensure the feasibility of the problem. Since JBAC problem is mixed combinatorial and non-convex, we transformed the original problem into a MI-SDP which is a SDP when the integer variables are relaxed to continuous ones. Then, we proposed a bound improving BnB algorithm with some complexity reduction methods to get the near-optimal solution. To further lower the computational complexity, a polynomial-time heuristic algorithm was proposed to derive the sub-optimal solution. Simulation results depicted that our proposed algorithms can significantly enhance the number of served MUs and reduce the network power consumption. It was also shown that the performance of the proposed heuristic algorithm is very close to that of the near-optimal algorithm, while the complexity of the former algorithm is much smaller. In addition, simulation results demonstrated that the circuit power consumption dominated by the RRH mode (active or sleep) can not be ignored when designing green Cloud-RAN. Future work is in progress to consider the impact of imperfect CSI on our proposed schemes.

We min

k∈K γ ·( k

APPENDIX A P ROOF OF L EMMA 1 will first demonstrate that β 4 guarantees L P P l=1

PlM ·

L l=1

2) ||hl,k ||22 +σk

≤ that

constraints (18c) are always satisfied even when ck = −1, wl,k = 0, ∀l ∈ L for every k ∈ K, which is equivalent to β≤

4 , ∀k ∈ K. L P 2 + σ2 ) | hH w | l,k l,i k

K P

γk · (

(30)

i=1,i6=k l=1

According to Cauchy-Schwartz inequality, we have L L L P P P 2 hH ≤ ( ||hl,k ||22 ) · ( ||wl,i ||22 ). Then, the | l,k wl,i | l=1

l=1

l=1

following inequations hold: K X

i=1,i6=k L X

=(

l=1

|

L X l=1

2 hH l,k wl,i | ≤ K X L X

||hl,k ||22 ) · (

L L K X X X ( ||hl,k ||22 ) · ( ||wl,i ||22 ) i=1 l=1

||wl,i ||22 ) ≤

i=1 l=1

l=1

L X l=1

L X PlM · ( ||hl,k ||22 ) l=1

(31) The last inequation is based on constraints (15c). 4 Then, if β ≤ min holds, we L P P k∈K γ ·( k

l=1

L l=1

PlM ·

2) ||hl,k ||22 +σk

have the following inequations: β ≤ min k∈K

γk · (

L P

l=1

= γk · max(

L P

k∈K l=1

≤

γk · max(

PlM ·

PlM ·

K P

|

4 PL

4 PL

4 L P

k∈K i=1,i6=k l=1

≤

γk · (

K P

l=1

l=1

||hl,k ||22 + σk2 )

||hl,k ||22 + σk2 )

(32)

2 2 hH l,k wl,i | + σk )

4 , ∀k ∈ K, L P H 2 2 hl,k wl,i | + σk ) |

i=1,i6=k l=1

which means that constraints (18c) are met even though ck = −1, wl,k = 0, ∀l ∈ L, ∀k ∈ K. Therefore, we can come to the conclusion that there is always at least one solution of 4 problem P3 with β ≤ min , i.e., L P P k∈K γ ·( k

l=1

PlM ·

L l=1

2) ||hl,k ||22 +σk

{al = 0, ck = 0, bl,k = 0, wl,k = 0, ∀l ∈ L, ∀k ∈ K}. APPENDIX B P ROOF OF T HEOREM 1 According to Lemma1, we know that problem P3 is 4 always feasible if β ≤ min holds. L P P k∈K γ ·( k

l=1

o

PlM ·

L l=1

2) ||hl,k ||22 +σk

o Hence, we denote Ψ = {aol , cok , bol,k , wl,k , ∀l f f f f and Ψf = {al , ck , bl,k , wl,k , ∀l ∈ L, ∀k

∈ L, ∀k ∈ K} ∈ K} as the optimal solution and a arbitrary feasible solution of problem P3 , respectively. In addition, we assume that the number of served MUs by using solution Ψf is more than that by K K P P using solution Ψf , i.e., 1(cfk = 1) > 1(cok = 1), k=1

k=1

where 1(·) is the indicator function. As a result, we can

12

deduce that

K P

k=1

(cfk − 1)2 ≤

K P

k=1

(cok − 1)2 − 4. Moreover,

according to constraints (15c) and constraints (15f), we have L P PM (1 − α)F2 (Ψf ) ≤ (1 − α){ (PlCMS + ηll ) + ζ}. Then, we l=1

can derive that

F3 (Ψf ) = (1 − α)F2 (Ψf ) + α(cfk − 1)2 L X

PM ≤ (1 − α){ (PlCMS + l ) + ζ} + α{(cok − 1)2 − 4} ηl l=1 ( ) L X PlM CMS = (1 − α){ (Pl + ) + ζ} + 4α + α(cok − 1)2 . ηl l=1 (33) 4 , we can drive that Recalling that α ≥ 1− P L PM (PlCM S +

l η

)+ζ+4

l l=1   L P PlM CMS + ηl ) + ζ} + 4α ≤ 0, therefore, we (1 − α){ (Pl

have

El,k vm = 0n×1 according to probability theory. Thus, we have ||E3l,k wm ||22 = xm ||E3l,k vm ||22 = 0 ≤ b∗l,k PlM .

l=1

F3 (Ψf ) ≤ α(cok − 1)2

≤ (1 − α)F2 (Ψo ) + α(cok − 1)2 = F3 (Ψo ),

(34)

which contradicts the assumption that Ψo = o {aol , cok , bol,k , wl,k , ∀l ∈ L, ∀k ∈ K} is the optimal solution of problem P3 . Therefore, we can deduce that there is no other solution that serves more MUs for problem P3 . ˇ as the set of MUs served by the network, i.e., We denote K o ˇ ˇ o as the K = {k|ck = 1, k ∈ K}. In addition, we define Ψ ˇ parameter tuple {al , ck , bl,k , wl,k , ∀l ∈ L, ∀k ∈ K}. Recalling ˇ o is a the constraints of problem P3 , it is easy to know that Ψ o feasible solution of problem P2 , as Ψ is feasible in problem o P3 . Furthermore, considering bol,k = 0, wl,k = 0, ∀l ∈ L with o o o ˇ ˇ o) + ck = −1, we have F3 (Ψ ) = F3 (Ψ ) = (1 − α)F2 (Ψ ˇ 4α(|K| − |K|), where the second part of the right hand side of the equation is a constant. Since Ψo is the optimal solution of ˇ o is also the optimal solution of problem P2 , problem P3 , Ψ o ˇ i.e., Ψ can achieve minimum network power consumption ˇ with the served MUs set K. APPENDIX C P ROOF OF T HEOREM 2 {a∗l , c∗k , b∗l,k , ∀l

∈ L, ∀k ∈ K} is feasible in problem Since P7 , we can easily know that these binary variables satisfy the constraints (15d),(16d) and (16e) of problem P3 . From constraints (29b) and constraints (29c), we can deduce that the vector wm meets the constraints (15c) and (18c) of problem P3 . Therefore, we just need to prove that wm is feasible in constraint (15e). On the one hand, if b∗l,k = 1, we need to prove 3 ||El,k wm ||22 ≤ PlM . Considering that the vector wm meets the constraint (15c), we have ||El2 wm ||22 ≤ PlM . And be3 cause of ||El,k wm ||22 ≤ ||El2 wm ||22 , it can be deduced that 3 2 ||El,k wm ||2 ≤ b∗l,k PlM . On the other hand, if b∗l,k = 0, we 3 H 3 have T r((El,k ) El,k W∗ ) ≤ 0. Since W∗ is PSD matrix, i.e., the diagonal elements of W∗ are all equal or greater than zero, we have E3l,k · diag(W∗ ) = 0n×1 . Combining with vm ∼ CN (0, W∗ ), we can come to the conclusion that

R EFERENCES [1] N. Bhushan, J. Li, D. Malladi, R. Gilmore, D. Brenner, A. Damnjanovic, R. T. Sukhavasi, C. Patel, and S. Geirhofer, “Network densification: The dominant theme for wireless evolution into 5G,” IEEE Communications Magazine, vol. 52, no. 2, pp. 82–89, February 2014. [2] Y. Shi, J. Zhang, K. B. Letaief, B. Bai, and W. Chen, “Large-scale convex optimization for ultra-dense Cloud-RAN,” IEEE Wireless Communications, vol. 22, no. 3, pp. 84–91, June 2015. [3] “C-RAN: The road towards green RAN,” White Paper by China Mobile, ver. 2.5, Oct. 2011. [4] A. Checko, H. L. Christiansen, Y. Yan, L. Scolari, G. Kardaras, M. S. Berger, and L. Dittmann, “Cloud-RAN for mobile networks - technology overview,” IEEE Communications Surveys Tutorials, vol. 17, no. 1, pp. 405–426, Firstquarter 2015. [5] J. Wu, Z. Zhang, Y. Hong, and Y. Wen, “Cloud radio access network (C-RAN): A primer,” IEEE Network, vol. 29, no. 1, pp. 35–41, Jan 2015. [6] J. Wu, Y. Bao, G. Miao, S. Zhou, and Z. Niu, “Base station sleeping control and power matching for energy-delay tradeoffs with bursty traffic,” IEEE Transactions on Vehicular Technology, vol. 65, no. 5, pp. 3657–3675, May 2016. [7] Y. Shi, J. Zhang, and K. B. Letaief, “Group sparse beamforming for green cloud-ran,” IEEE Transactions on Wireless Communications, vol. 13, no. 5, pp. 2809–2823, May 2014. [8] J. Cheng, Y. Shi, B. Bai, W. Chen, J. Zhang, and K. B. Letaief, “Group sparse beamforming for multicast green Cloud-RAN via parallel semidefinite programming,” in Communications (ICC), 2015 IEEE International Conference on, June 2015, pp. 1886–1891. [9] X. Wang, S. Thota, M. Tornatore, H. S. Chung, H. H. Lee, S. Park, and B. Mukherjee, “Energy-efficient virtual base station formation in optical-access-enabled Cloud-RAN,” IEEE Journal on Selected Areas in Communications, vol. 34, no. 5, pp. 1130–1139, May 2016. [10] S. Luo, R. Zhang, and T. J. Lim, “Downlink and uplink energy minimization through user association and beamforming in C-RAN,” IEEE Transactions on Wireless Communications, vol. 14, no. 1, pp. 494– 508, Jan 2015. [11] M. Peng, C. Wang, V. Lau, and H. V. Poor, “Fronthaul-constrained cloud radio access networks: Insights and challenges,” IEEE Wireless Communications, vol. 22, no. 2, pp. 152–160, April 2015. [12] B. Dai and W. Yu, “Sparse beamforming for limited-backhaul network mimo system via reweighted power minimization,” in 2013 IEEE Global Communications Conference (GLOBECOM), Dec 2013, pp. 1962–1967. [13] V. N. Ha, L. B. Le, and N. D. Do, “Cooperative transmission in Cloud RAN considering fronthaul capacity and cloud processing constraints,” in 2014 IEEE Wireless Communications and Networking Conference (WCNC), April 2014, pp. 1862–1867. [14] Z. Niu, “TANGO: Traffic-aware network planning and green operation,” IEEE Wireless Communications, vol. 18, no. 5, pp. 25–29, October 2011. [15] E. Matskani, N. D. Sidiropoulos, Z. q. Luo, and L. Tassiulas, “Convex approximation techniques for joint multiuser downlink beamforming and admission control,” IEEE Transactions on Wireless Communications, vol. 7, no. 7, pp. 2682–2693, July 2008. [16] D. H. N. Nguyen, L. B. Le, and T. Le-Ngoc, “Multiuser admission control and beamforming optimization algorithms for miso heterogeneous networks,” IEEE Access, vol. 3, pp. 759–773, 2015. [17] Y. Shi, J. Cheng, J. Zhang, B. Bai, W. Chen, and K. B. Letaief, “Smoothed lp-minimization for green Cloud-RAN with user admission control,” IEEE Journal on Selected Areas in Communications, vol. 34, no. 4, pp. 1022–1036, April 2016. [18] A. Abdelnasser and E. Hossain, “Resource allocation for an OFDMA Cloud-RAN of small cells overlaying a macrocell,” IEEE Transactions on Mobile Computing, vol. PP, no. 99, pp. 1–1, 2016. [19] J. Wu, Y. Zhang, M. Zukerman, and E. K. N. Yung, “Energy-efficient base-stations sleep-mode techniques in green cellular networks: A survey,” IEEE Communications Surveys Tutorials, vol. 17, no. 2, pp. 803– 826, Secondquarter 2015. [20] S. Zhang, J. Gong, S. Zhou, and Z. Niu, “How many small cells can be turned off via vertical offloading under a separation architecture?” IEEE Transactions on Wireless Communications, vol. 14, no. 10, pp. 5440–5453, Oct 2015.

13

[21] V. Giannini, C. Desset, I. Godor, P. Skillermark, M. Olsson, M. Imran, D. Sabella, M. Gonzalez, O. Blume, and A. Fehske, “How much energy is needed to run a wireless network?” IEEE Wireless Communications, vol. 18, no. 5, pp. 40–49, Oct. 2011. [22] M. Bengtsson and B. Ottersten, Optimal and Suboptimal Transmit Beamforming. USA: CRC Press, 2001. [23] J. Zhao, T. Q. Quek, and Z. Lei, “Coordinated multipoint transmission with limited backhaul data transfer,” IEEE Transactions on Wireless Communications, vol. 12, no. 6, pp. 2762–2775, June 2013. [24] H. Dahrouj and W. Yu, “Coordinated beamforming for the multicell multi-antenna wireless system,” IEEE Transactions on Wireless Communications, vol. 9, no. 5, pp. 1748–1759, May 2010. [25] A. Sanderovich, O. Somekh, H. V. Poor, and S. Shamai, “Uplink macro diversity of limited backhaul cellular network,” IEEE Transactions on Information Theory, vol. 55, no. 8, pp. 3457–3478, Aug 2009. [26] V. N. Ha, L. B. Le, and N. D. Do, “Energy-efficient coordinated transmission for cloud-rans: Algorithm design and trade-off,” in Information Sciences and Systems (CISS), 2014 48th Annual Conference on, March 2014, pp. 1–6. [27] F. Zhuang and V. K. N. Lau, “Backhaul limited asymmetric cooperation for mimo cellular networks via semidefinite relaxation,” IEEE Transactions on Signal Processing, vol. 62, no. 3, pp. 684–693, Feb 2014. [28] Z. Yu, K. Wang, H. Ji, X. Li, and H. Zhang, “Joint user association and downlink beamforming for green Cloud-RANs with limited fronthaul,” in arXiv preprint arXiv:1606.07241, June 2016. [29] S. Drewes, “Mixed integer second order cone programming,” Ph.D. dissertation, Darmstadt Univ. of Technology, Darmstadt, Germany, 2009. [30] L. Vandenberghe and S. Boyed, “Semidefinite programming,” SIAM review, vol. 38, no. 1, pp. 49–95, 1996. [31] H. Tuy, F. Al-Khayyal, and P. Thach, “Monotonic optimization: Branch and cut methods,” in Essays and Surveys in Global Optimization. Springer US, 2005, pp. 39–78. [32] Z. q. Luo, W. k. Ma, A. M. c. So, Y. Ye, and S. Zhang, “Semidefinite relaxation of quadratic optimization problems,” IEEE Signal Processing Magazine, vol. 27, no. 3, pp. 20–34, May 2010. [33] C. Helmberg, F. Rendl, R. Vanderbei, and H. Wolkowicz, “An interiorpoint method for semidefinite programming,” SIAM Journal on Optimization, vol. 6, no. 2, pp. 342–361, 1996. [34] Y. Cheng, M. Pesavento, and A. Philipp, “Joint network optimization and downlink beamforming for CoMP transmissions using mixed integer conic programming,” IEEE Transactions on Signal Processing, vol. 61, no. 16, pp. 3972–3987, Aug 2013. [35] I. Hwang, B. Song, and S. S. Soliman, “A holistic view on hyperdense heterogeneous and small cell networks,” IEEE Communications Magazine, vol. 51, no. 6, pp. 20–27, June 2013.