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Joint Precoding and Load Balancing Optimization
for Energy-Efficient Heterogeneous Networks
Jingya Li, Student Member, IEEE, Emil Björnson, Member, IEEE, Tommy Svensson, Senior Member, IEEE,
Thomas Eriksson, and Mérouane Debbah Fellow, IEEE
Abstract—This paper considers a downlink heterogeneous
network, where different types of multi-antenna base stations
(BSs) communicate with a number of single-antenna users.
Multiple BSs can serve the users by spatial multiflow transmission
techniques. Assuming imperfect channel state information at both
BSs and users, the precoding, load balancing, and BS operation
mode are jointly optimized for improving the network energy
efficiency. We minimize the weighted total power consumption
while satisfying quality of service constraints at the users. This
problem is non-convex, but we prove that for each BS mode
combination, the considered problem has a hidden convexity
structure. Thus, the optimal solution is obtained by an exhaustive
search over all possible BS mode combinations. Furthermore,
by iterative convex approximations of the non-convex objective
function, a heuristic algorithm is proposed to obtain a suboptimal
solution of low complexity. We show that although multi-cell joint
transmission is allowed, in most cases, it is optimal for each user
to be served by a single BS. The optimal BS association condition
is parameterized, which reveals how it is impacted by different
system parameters. Simulation results indicate that putting a BS
into sleep mode by proper load balancing is an important solution
for energy savings.
Index Terms—Energy efficiency, heterogeneous networks, load
balancing, precoding design.
I. INT ROD UC TI ON
The rapid growth of data traffic in wireless networks im-
pose great challenges on future wireless communication sys-
tems [2]–[4], in particular on improving the spectral efficiency
as well as the energy efficiency. At the same time, the users
are expecting that future networks will provide a uniform
quality of service (QoS) over the coverage area. In many
challenging scenarios, e.g., in shopping malls, dense urban
environments, or during the occurrence of traffic jams, the
users are non-uniformly distributed over the network [5]. One
widely acknowledged cost- and energy-efficient approach to
J. Li, T. Svensson and T. Eriksson are with the Department of Signals and
Systems, Chalmers University of Technology, 412 96 Gothenburg, Sweden
(e-mail: {jingya.li, tommy.svensson, thomase}@chalmers.se).
E. Björnson was with Supélec, Gif-sur-Yvette, France, and with the Depart-
ment of Signal Processing, KTH Royal Institute of Technology, Stockholm,
Sweden. He is currently with the Department of Electrical Engineering (ISY),
Linköping University, Linköping, Sweden (e-mail: emil.bjornson@liu.se).
M. Debbah is with CentralSupélec, Gif-sur-Yvette, France (e-mail: mer-
ouane.debbah@centralsupelec.fr).
This work has been performed in the framework of the FP7 project ICT-
317669 METIS, which is partly funded by the EU. This research has also
been supported by the project 621-2009-4555 Dynamic Multipoint Wireless
Transmission and the International Postdoc Grant 2012-228 from the Swedish
Research Council, and the ERC Starting Grant 305123 MORE (Advanced
Mathematical Tools for Complex Network Engineering).
Parts of this work have been published at the IEEE International Conference
on Communications (ICC), London, UK, June 2015 [1].
tackle these challenges is the concept of heterogeneous dense
networks, where the traditional macro base stations (BSs) are
complemented with a dense deployment of low-cost and low-
power BSs [6]–[8]. By adding such a large number of small
cells, the corresponding low-power BSs can offload traffic
from the macro BSs, reduce the average distance between
users and transmitters, and thereby improve the data rates
and/or reduce the average transmit power. Since the data
traffic load fluctuates greatly over the day [9], both macro
and small cells might be needed at peak hours while there
is an opportunity to turn off some BSs when there is little
traffic in the corresponding coverage areas. Load balancing
is the technique that maps the current traffic load to the
available transmission resources, i.e., associates users with
BSs. Mathematically speaking, the network would like to find
the BS association that maximizes some performance metric,
under the condition that the QoS requirements of all users are
fulfilled.
Different from the traditional cellular networks, the densely
deployed BSs will be heterogeneous in the number of anten-
nas, transmit power, backhaul capacity and reliability, coverage
area, etc. Moreover, the channel state information (CSI) at each
BS is likely to be different and imperfect. In this complex sce-
nario, a major research problem is to design low-complexity
and robust coordinated multi-BS transmission schemes that
minimize the total power consumption, while satisfying the
QoS expectations of the users.
The total power consumption of the network can be modeled
with a circuit part that depends on the transceiver hardware
and a dynamic part that is a function of the transmitted signal
power [10]–[13]. Adding more low-power BSs can reduce the
dynamic power consumption due to the shorter propagation
distances, but require more hardware; thus, it will increase the
circuit power part. Note that the circuit power consumption
also depends on the operational mode of each BS, i.e., whether
the BS is active or in sleep mode. It has been shown that,
putting a BS into sleep mode when there is nothing to transmit
or receive is an important solution for energy savings [11].
Therefore, to actually improve the overall power efficiency
of a heterogeneous network, the cooperation scheme, the BS
operational modes, and the load balancing must be properly
and jointly optimized.
Simulation-based studies for load balancing in heteroge-
neous networks have been performed within 3GPP, and several
biased-received-power based criteria were proposed to control
the number of users associated with the low-power BSs [6],
[7]. Moreover, load balancing was analyzed in [14]–[23] for
2
systems where the BSs are distributed according to stochas-
tic point processes. Using stochastic geometry tools, these
works have compared how different BS association rules (e.g.,
the nearest-BS based, the highest-received-power based, the
maximum signal-to-interference-and-noise ratio (SINR) based,
and the biased-SINR based cell selection) affect the downlink
SINR distribution [15]–[17] and the average achievable rate
[17]–[21]. We note that the results in [14]–[19], [21], [22]
are limited to BSs with single antennas, while contempo-
rary and future networks use multiple antennas for downlink
precoding. The papers [20] and [23] consider the practically
important case of multi-antenna BSs, but these results are
restricted to single-cell zero-forcing precoding with perfect
CSI; in contrast, imperfect CSI and inter-cell interference
coordination are essential properties of future heterogeneous
networks. Moreover, shadowing has a great impact on the
system performance of heterogeneous networks, but was not
considered in [14], [15], [17], [18], [20]–[23], probably due
to mathematical intractability.
The precoding design is of paramount importance in multi-
antenna cellular networks, since it determines the achievable
array gains and interference suppression [24]. Joint precoding
and load balancing was studied in [25] for a homogeneous
network, where all BSs are turned on and there is no explicit
power constraints. In [26] and [27], the authors investigated
joint load balancing and power control in heterogeneous
networks with single-antenna BSs, where different algorithms
were proposed to maximize the minimum rate subject to per-
BS power constraints. Considering heterogeneous networks
with multi-antenna BSs, joint load balancing and precoding
algorithms were designed in [28]–[30] to maximize various
system utilities. In [31], downlink linear precoding problems
were studied jointly with BS selection. The objective was
to either minimize the total transmit power or maximize
the sum rate performance. The results in [31] show that by
imposing certain sparsity patterns in the precoding vectors,
the number of active BSs in the network can be effectively
reduced. With the objective of improving network energy
efficiency, radio resource optimization was studied in [32] for
the downlink of an orthogonal frequency-division multiplexing
(OFDM) system. In particular, the power allocation, subcarrier
allocation and the number of activated transmit antennas were
jointly optimized for maximization of the energy efficiency of
data transmission (bit/Joule delivered to the users). However,
the work in [32] did not optimize the precoding vectors and
the results were limited to a single-cell scenario. In [33], using
a stochastic geometry based model, the energy efficiency of
both multi-cell homogeneous and heterogeneous networks was
analyzed by considering active and sleep modes for macro
BSs with fixed power control. Since both BSs and users are
assumed to have a single antenna in [33], precoding design
was not considered.
Joint precoding and load balancing design problem is typ-
ically a mixed-integer nonlinear programming problem, for
which finding the global optimum is challenging [34]. Inspired
by the compressive sensing literature, the reweighted l1-norm
technique has been adopted in [35]–[39], where different
heuristic algorithms were proposed for solving joint precoding
and BS clustering design problems. In [40], [41], group sparse
optimization has been used to improve the energy efficiency of
cloud radio access networks, where the weighted mixed l1/lp-
norm minimization is used to induce group sparsity on the
beamforming. The BSs are switched off based on the obtained
group sparsity patterns. Note that in [37]–[41] the algorithms
are designed based on the assumption of perfect CSI at both
BSs and users. In this paper, we study joint precoding and
load balancing optimization for energy efficient heterogeneous
networks with imperfect CSI. The goal is to minimize the
weighted total power consumption while satisfying QoS con-
straints at the users and transmit power constraints at the BSs.
Although it is practically convenient and desirable to associate
each user with only one BS per time-frequency resource block,
our system model allows for serving users by multiple BSs.
The paper investigates the following important system design
questions: 1) Which and how many BSs should each user be
associated with? 2) How should the precoding matrices be
selected when having imperfect CSI? 3) How can we decide
on the operational mode (active or sleep) for each BS? The
contributions of this paper can be summarized as follows:
•We formulate the joint load balancing and precoding
as a non-convex optimization problem. We show that
for a given combination of BS modes, the considered
optimization problem can be reformulated as a convex
semi-definite problem. Thus, we obtain the global optimal
solution by an exhaustive search over all possible BS
mode combinations. The obtained global optimal solution
serves as an upper bound for any other suboptimal pre-
coding and load balancing solutions, e.g., the strategies
proposed in [6], [7], [14]–[23].
•We derive the structure of the optimal solution, by
investigating the structure of the dual problem. Our result
verifies the intuition that, in most cases, it is optimal
for each user to be served by a single BS. However,
there are also occasions when multi-BS association is
beneficial. Moreover, we show that the load balancing
rules previously considered in [6], [7], [14]–[23] are not
optimal when minimizing the total power consumption
under per-BS transmit power constraints and per-user
QoS constraints. The optimal BS association rule con-
sists of comparing weighted channel norms, where the
weighting matrix depends on channel uncertainty, power
constraints, and QoS constraints.
•We propose an efficient iterative algorithm that resolves
the non-convexity of the original optimization problem by
iterative convex approximations of the power consump-
tion functions. Each iteration solves a convex problem
with a modified objective function. This convex objective
function is updated in each iteration such that most of
the BSs with small transmit powers in the solution are
driven to sleep mode. We show that the idea behind the
proposed algorithm is very similar to the reweighted l1-
norm minimization based methods used in [37]–[39].
•Numerical results are provided to show how putting
BSs into sleep mode by proper load balancing is a key
to energy savings in heterogeneous networks. The BS
3
Macro BS
Micro BS
Pico BS
Fig. 1. Illustration of a downlink three-tier heterogeneous network consisting
of macro, micro and pico BSs.
activation probability is shown to depend on the target
QoS requirements, as well as the ratio between the circuit
power consumed in the active mode and that consumed
in the sleep mode.
The remainder of this paper is organized as follows: Sec-
tion II introduces the system and signal model. In Section III,
we analyze the optimal precoding and load balancing design.
In Section IV, an iterative heuristic algorithm is proposed to
obtain a suboptimal solution with low complexity. Section V
provides a set of numerical results to illustrate our analytical
results and the proposed algorithms. Finally, the main results
of the paper are summarized in Section VI.
Notation: we use upper-case bold face letters, such as E,
for matrices and lower-case bold face letters, such as h, for
vectors. W≽0represents that the matrix Wis positive
semidefinite. |C| denotes the cardinality of a set C. The
operator E{·} stands for expectation. The notation ∼denotes
“distributed as”, ,is used to mark definitions, ∥·∥ represents
the Euclidean norm, and Tr(·)is the matrix trace.
II. SY ST EM A ND SI GNA L MOD EL
We consider the downlink of a heterogeneous network
consisting of MBSs and Ksingle-antenna users, as illustrated
in Fig. 1. The heterogeneity lies in the assumption that the
MBSs are different in terms of the number of transmit
antennas, the power consumption characteristics, the channel
propagation model, and the CSI quality. BSs with the same
characteristics can be said to belong to the same tier or
category (e.g., macro or small BS), but we stress that our
system model supports anything from 1to Mtiers. The
users are not pre-associated with any particular cell and are
randomly distributed in the network coverage area.
BS vis assumed to have Nvantennas. The channel from
BS vto user kis assumed to be flat-fading, and denoted
by hk,v ∈CNv×1for v= 1, . . . , M and k= 1, . . . , K.
In practice, these channels are imperfectly known at the
BSs. This is modeled as hk,v =ˆ
hk,v +ek,v , where ˆ
hk,v
is the known estimate of hk,v at BS v. The error vector
ek,v vCN (0,Ek ,v)is assumed to have zero-mean and a
covariance matrix Ek,v ∈CNv×Nv. This is, for example, a
good model of time-division duplex (TDD) systems where
the channels are Rayleigh fading, hk,v ∼ CN (0, gk,v INv),
and the BS uses uplink pilot signals for channel estimation.
Here, gk,v denotes the average channel gain between BS v
and user k, including pathloss and shadowing. If the minimum
mean-squared error (MMSE) channel estimator is used [42]–
[45], then estimation errors are zero-mean complex Gaussian
distributed and the error covariance becomes
Ek,v =gk,v
1 + γp
k,v
INv(1)
where γp
k,v =pgk,v
σ2
k
denotes the pilot SNR, pis the total
pilot power and σ2
kis the noise power. The users also need to
acquire CSI, but only for the precoded channels; this is further
discussed in Section II-B.
The received signal at user kis
yk=
M
v=1
hH
k,v xv+nk(2)
where xv∈CNv×1is the transmitted signal from BS vand
nkvCN 0, σ 2
kis the independent additive receiver noise at
user k.
A main goal of this paper is to determine the optimal
association between users and BSs. It makes practical sense to
only associate one BS with each user, but we will not make this
limiting assumption at this point since we simply do not know
if it is optimal. Instead, we assume that all BSs are able to
transmit to all users at the same time-frequency resource block,
and then our analysis will tell which and how many BSs that
each user should be associated with. Motivated by the fact that
tight phase synchronization between BSs is extremely difficult
to achieve in practice, only linear spatial multiflow transmis-
sion is allowed [46]. This is a scheme for multiple access that
allows each user to receive different parallel data streams from
multiple BSs. These streams are detected sequentially at the
user, based on conventional successive interference cancelation
techniques [47]. Define V,{1,2, . . . , M }as the set of all
BSs in the network, and let Vk⊆ V denote the set of BSs
that provide data transmission to user k. Then, the set of users
associated with BS vcan be represented by Uv={k|v∈ Vk}.
Let sk,v vCN (0,1) be the coded independent information
symbols for user k, transmitted from BS v.1Then, the desired
signals for user ktransmitted by BS vis wk,v sk,v , where
wk,v ∈CNv×1is the linear precoding vector for user kat BS
v. The aggregated transmitted signal from BS vis
xv=
k∈Uv
wk,v sk,v .(3)
A. Power Consumption Model
From (3), the expected transmit power from BS vcan be
calculated as
Ptrans,v=
k∈Uv
∥wk,v ∥2E|sk,v |2=
k∈Uv
∥wk,v ∥2.(4)
In this paper, we adopt the linear approximated power con-
sumption model proposed in [11, Eq. (4-3)] for 10 MHz
1Note that, the data symbols sk,v for user kare independent for different
BS v. This spatial multiflow transmission scheme is different from the
traditional network MIMO scheme, which assumes that the same user data
skis transmitted from all BSs. The network MIMO scheme allows for
coherent joint transmission from all BSs, however, it requires tight phase
synchronization between all BSs.
4
bandwidth, where the total consumed power of BS v, for
v∈ V, is
Pv=NvPactive,v + ∆vPtrans,v,0< Ptrans,v≤Pv,max
NvPsleep,v, Ptrans,v= 0,(5)
where Pactive,v is the hardware power consumption at BS v
at the minimum non-zero transmit power, Psleep,v denotes
the sleep mode power consumption of BS vwith Psleep,v ≤
Pactive,v. Note that Psleep,v >0in the sleep mode (due
to the DC-DC power supply, mains supply, active cooling,
maintaining backhaul connections, and enabling fast turn on
control signaling) [10], [11]. Here, Pv,max is the peak transmit
power constraint for BS v. The scaling factor, ∆v≥1, models
the inefficiency of the power amplifier; that is, how much extra
power that is consumed at BSs when the transmitted power is
Ptrans,v. Some example values of Pactive,v,Psleep,v ,Pv,max and
∆vfor different BS types can be found in [11, Table 8], and
some of these are also given in Table I.
B. Aggregated Received SINR
Each user might receive multiple information symbols, thus
we need an aggregated performance measure for each user.
The natural choice is the sum spectral efficiency of the user
when successive interference cancellation is applied. 2
Lemma 1: Assume that user kknows the effective precoded
channels wH
l,v ˆ
hk,v (for all land v). Then, a lower bound on
the achievable ergodic sum spectral efficiency of user kis
Rk=E{log2(1 + γk)}where the expectation is with respect
to the aggregated instantaneous SINR
γk=v∈Vk
ˆ
hH
k,v wk,v
2
Ik+Ek+σ2
k
(6)
with
Ik,
v∈V
l∈Uv
l=k
wH
l,v ˆ
hk,v ˆ
hH
k,v +Ek,v wl,v (7)
being the co-user interference and
Ek,
v∈Vk
wH
k,v Ek,v wk,v (8)
is the effective estimation errors on the channels related this
user.
Proof: The achievable sum spectral efficiency is obtained,
similar to [47], [48], by decoding the Gaussian information
sequences from the different BSs in a sequential manner, using
conventional successive interference cancellation. Since the
users only know the effective channels wH
l,v ˆ
hk,v and not the
true channels wH
l,vhk,v, the channel uncertainty is handled by
computing a lower bound on the mutual information, using
the approach from [49] where all signals that are uncorrelated
with wH
k,v ˆ
hk,v sk,v are treated as Gaussian noise (which is the
worst case in terms of mutual information). This applies for
both inter-user interference and the part of the desired signals
2The power consumption at the user side might depend on how many
symbols that the user receives, but this paper has an operator perspective where
only the power consumptions at BSs is considered—this is the dominating
factor in the downlink.
that are conveyed over the zero-mean channel estimation error
vectors.
This lemma provides a lower bound on the achievable
capacity, since the latter is unknown under imperfect CSI. We
note that Lemma 1 assumes that the users know the effective
precoded channels. In practice, the users can estimate these
effective channels using downlink pilots, and get estimates
of wH
l,vhk,v that are at least as accurate wH
l,v ˆ
hk,v . Hence, it
might be possible to achieve higher spectral efficiencies than
in Lemma 1. Nevertheless, the aggregated SINR in (6) is the
most convenient one for precoding design, since the BSs can
only utilize their own CSI in the optimization.
C. Problem Formulation
The focus of this paper is on the joint design of load
balancing (i.e., the UE association in Uv) and precoding
vectors (wk,v) for v= 1, . . . , M and k= 1, . . . , K , which is
an optimization that takes place at every channel realization.
To this end, the goal is to minimize the weighted total
power consumption (for any given channel realization) while
satisfying a set of SINR constraints (or, equivalently, spectral
efficiency constraints) for each user and a set of transmit power
constraints for each BS. These constraints are referred to as the
QoS constraints. With (4), (5) and (6) in hand, the optimization
problem can be formulated as
minimize
{Uv},{wk,v
}
M
v=1
avPv
subject to γk≥Γk,∀k
Ptrans,v ≤Pv,max,∀v
(9)
where Γk>0is the target SINR value for user k. By satisfying
this QoS target for every channel realization, the ergodic
spectral efficiency is Rk≥log2(1 + Γk). In this paper, we
assume that the weights av>0are given. These weights can
be used to balance the power consumptions of different BSs.
For the rest of the paper, we assume that the problem (9)
has at least one feasible solution, which is reasonable in
dense networks with an over-provisioning of access points.
In practice, if no feasible solution exists, the SINR constraints
have to be relaxed either by decreasing the target SINRs or
by removing users [25].
III. OPTIMAL PRE CO DI NG A ND LOA D BALANCING
In this section, we solve the optimization problem in (9). As
a first step, we show that the set variables Uvcan be eliminated
by optimizing over all precoding vectors.
Lemma 2: The original problem (9) is equivalent to 3
minimize
{wk,v }
M
v=1
avPv
subject to γk≥Γk,∀k
K
k=1
∥wk,v ∥2≤Pv,max,∀v
(10)
3Here, “equivalent” means that the minimal value of problems (9) and (10)
is the same and that the solution to (9) can be obtained from the solution to
(10).
5
where Pvcan be rewritten as a function of wk,v by substituting
Ptrans,v =K
k=1 ∥wk,v ∥2into (5), and γkis reformulated as
γk=M
v=1
ˆ
hH
k,v wk,v
2
Ik+Ek+σ2
k
(11)
with Ikrewritten as
Ik,
M
v=1
K
l=1
l=k
wH
l,v ˆ
hk,v ˆ
hH
k,v +Ek,v wl,v (12)
and Ekreplaced by
Ek,
M
v=1
wH
k,v Ek,v wk,v.(13)
Proof: Note that if BS jdoes not serve a particular user
k(i.e., k∈ Ujand j∈ Vk), then all terms that would have
contained wk,j in the SINR of (6) and the transmit power (4)
are missing. This is equivalent to setting wk,j =0and adding
said terms (which then are zero). Hence, the sets Uvand Vkare
fully determined by checking which of the precoding vectors
are zero:
Uv={k|wk,v = 0, k ∈ {1, . . . , K}} ,(14)
Vk={v|wk,v = 0, v ∈ V} .(15)
The sets Uvcan therefore be removed as optimization variables
from (9), if we add the missing terms in (4) and (6). The
corresponding equivalent problem is the one stated in this
lemma.
This lemma shows that we do not need to optimize the BS
association sets Uvsince these are implicitly determined by
checking which precoding vectors that are non-zero. Note that
although the expressions for Pv,γk,Ik, and Ekin Lemma 2
are different from the expressions in Section II, the values are
identical for every selection of precoding vectors {wk,v}. As
will be shown later, even if all BSs are allowed to transmit to
all users at the same time-frequency resource block, in most
cases, at the optimal point, each user kwill be connected to
only one BS.
The optimization problem (10) is not convex. In particular,
the power consumption function in (5) leads to a hard combi-
natorial problem [50]. Moreover, the SINR constraints of (10)
do not have a standard convex form. In the following, we
first show that, for each combination of BS modes (active or
sleep), problem (10) can be reformulated as a convex problem.
Then, the global optimum can be found by solving this convex
problem for all 2Mcombinations of modes.
Define wk,wT
k,1,wT
k,2, . . . , wT
k,M T∈C(∑M
v=1 Nv)×1
as the aggregated precoding vector for user kfrom all BSs.
We notice that the received SINR, γkin (11), can be expressed
as
γk=wH
kˆ
Rkwk
K
l=1
l=k
wH
lˆ
Rk+Ekwl+wH
kEkwk+σ2
k
.
using the block-diagonal matrices
Ek,diag (Ek,1,Ek,2, . . . , Ek,M )(16)
ˆ
Rk,
ˆ
Rk,10. . . 0
0ˆ
Rk,20.
.
.
.
.
.0...0
0.
.
.0ˆ
Rk,M
(17)
with the diagonal blocks ˆ
Rk,v ,ˆ
hk,v ˆ
hH
k,v ∈CNv×Nvfor
v= 1, . . . , M .
Similarly, the power constraints in (10) are written in terms
of wkas K
k=1 wH
kQvwk, where
Qv,diag (Q1,v,Q2,v , . . . , QM ,v)(18)
Qi,v ,INv,if i=v
0Nv×Nv,otherwise.(19)
With this notation, the optimization problem (10) looks like
a classical precoding optimization problem of the type in [51],
but with the important difference that ˆ
Rkhas rank Mand
not rank 1 as in the case with one BS per user. Hence, we
cannot use the second-order cone techniques from [51], but
the following semi-definite relaxation approach.4
Lemma 3: Let zvbe the BS mode indicator for v∈ V:
zv= 1 if BS vis active, and zv= 0 if BS vis in sleep
mode. Define Wk,wkwH
k≽0. Consider the following
semi-definite relaxation of (10) for fixed BS modes:
minimize
{Wk≽0}
K
k=1
Tr (AWk) + J(z)
subject to Tr ˆ
RkWk−Γk
K
l=1
l=k
Tr (ˆ
Rk+Ek)Wl
−ΓkTr (EkWk)≥Γkσ2
k,∀k
K
k=1
Tr (QvWk)≤zvPv,max,∀v
(20)
where zv∈ {0,1},∀vand
A,diag (a1∆1IN1, a2∆2IN2, . . . , aM∆MINM)(21)
J(z) =
v
avNv(Pactive,vzv+Psleep,v (1 −zv)) .(22)
The problem (20) is a convex semi-definite program and it
always has a rank one solution, if the problem is feasible.
Proof: For any fixed combination of BS modes z=
[z1, . . . , zM],J(z)in (22) is fixed. Then, the problem (20) is
on the form of (P2) in [52]. Based on [52, Theorem 1], this
type of optimization problems always has optimal solutions
with rank one if it is feasible.
Based on this lemma, we solve the original precoding and
load balancing problem as follows.
Theorem 1: The global optimum to (9) is obtained by
solving (20) for each of the 2Mmode combinations (zv= 0
or zv= 1 for each v) and selecting the solution that provides
4Semi-definite relaxation means that the optimization variables changed to
Wk,wkwH
k≽0instead of wk. This would require an additional rank
constraint, rank (Wk)=1,∀k, but this one is dropped as a relaxation.
6
the lowest weighted total power consumption.
To summarize, Lemma 3 shows that semi-definite relaxation
is tight for the problem at hand. For each fixed mode z, we
can solve (20) using standard convex optimization software,
such as CVX [53] or YALMIP [54]. By doing this for all 2M
mode combinations, the global optimum to (9) is obtained. We
stress that (9) optimizes the precoding, load balancing (i.e., BS
association), and BS modes jointly. The global optimum to (9)
is a benchmark for any suboptimal heuristic load-balancing
and precoding algorithms; for example, the ones proposed
in [6], [7], [14]–[16], [18], [20], [21], [23].
A. Structure of the Optimal Load Balancing
Theorem 1 shows how to solve the joint precoding and load
balancing optimization problem (9) using convex optimization
techniques. Although it provides the truly optimal solution,
it brings little insight on the structure of the optimal load
balancing. In the following, we will analyze the dual problem
of (20) and thereby shed light on the optimal BS association.
Recall from Lemma 3 that (20) is a semi-definite optimiza-
tion problem. This problem is convex and satisfies Slater’s
condition, which implies strong duality [55, Sec. 5.2.3]. The
dual problem has the same optimal objective value as the
original problem. Define A,{v|zv= 1, v ∈ V} as the set
of active BSs, and S,{v|zv= 0, v ∈ V} as the set of BSs
in the sleep mode. The Lagrangian of (20) is
L({Wk, λk, µi, νj})
=J(z) +
K
k=1
Tr (AWk)−
K
k=1
λkTr ˆ
RkWk
+
K
k=1
λkΓk
K
l=1
l=k
Tr (ˆ
Rk+Ek)Wl
+
K
k=1
λkΓkTr(EkWk) + σ2
k
+
i∈A
µiK
k=1
Tr (QiWk)−Pi,max
+
j∈S
νj
K
k=1
Tr QjWk(23)
where λk, µi, νj≥0are the Lagrange multipliers associated
to the k-th user’s SINR constraint, the power constraint for
BS iin set A, and the power constraint for BS jin set S,
respectively. The dual problem to (20) is an unconstrained
maximization of the dual function, defined as
g({λk, µi, νj}) = minimize
{Wk}L({Wk, λk, µi, νj}).(24)
Define
Bk,A+λkΓkEk+
K
l=1
l=k
λlΓl(ˆ
Rl+El)+
i∈A
µiQi+
j∈S
νjQj,
(25)
which is a block-diagonal matrix whose v-th block is
Bk,v ,av∆vINv+λkΓkEk,v +
K
l=1
l=k
λlΓl(ˆ
Rl,v +El,v )
+
i∈A
µiQi,v +
j∈S
νjQj,v.(26)
From (23), it is easy to show that g({λk, µi, νj}) = J(z) +
K
k=1 λkΓkσ2
k−i∈A µiPi,max, if Bk−λkˆ
Rk≽0for all
k= 1, . . . , K; otherwise, g({λk, µi, νj}) = −∞. Hence, the
dual problem of (20) becomes
maximize
{λk,µi,νj≥0}J(z) +
K
k=1
λkΓkσ2
k−
i∈A
µiPi,max
subject to Bk−λkˆ
Rk≽0,∀k.
(27)
Lemma 4: Let {λ∗
k, µ∗
i, ν∗
j}denote the optimal Lagrange
multipliers to (27), and let B∗
k,v be the value of Bk,v in (26)
for these multipliers. The optimal precoding vectors are
w∗
k,v =
αk,v B∗
k,v −1ˆ
hk,v ,if λ∗
k=1
ˆ
hH
k,v (B∗
k,v )−1ˆ
hk,v
,
0,otherwise,
(28)
where αk,v ≥0is a scaling factor.
Proof: Since for any fixed z, strong duality holds for (20)
and the solution has rank one as W∗
k=w∗
k(w∗
k)H, the optimal
w∗
kcan be calculated by setting the first-order derivative of the
Lagrangian in (23) with respect to wkto zero; that is,
∂LWk, λ∗
k, µ∗
i, ν∗
j
∂wkw∗
k= 2 B∗
k−λ∗
kˆ
Rkw∗
k=0
(29)
from which we have the condition
B∗
k,v w∗
k,v =λ∗
kˆ
hk,v ˆ
hH
k,v w∗
k,v ,∀v. (30)
Hence,
w∗
k,v =αk,v B∗
k,v −1ˆ
hk,v (31)
for all kand v, where αk,v ,λ∗
kˆ
hH
k,v w∗
k,v is a scalar. Recall
that we assume that the problem (9) has at least one feasible
solution. Thus, λ∗
k>0for all k. If we now multiply (29) by
(w∗
k)Hfrom the left, we obtain the equivalent condition
2(w∗
k)HB∗
k−λ∗
kˆ
Rkw∗
k= 0 ⇔
2(w∗
k,v )HB∗
k,v −λ∗
kˆ
hk,v ˆ
hH
k,v w∗
k,v = 0.(32)
By plugging (31) into (32), we obtain the condition
α2
k,v ˆ
hH
k,v B∗
k,v −1ˆ
hk,v −λ∗
kˆ
hH
k,v B∗
k,v −1ˆ
hk,v 2= 0
(33)
which is satisfied when either λ∗
kˆ
hH
k,v B∗
k,v −1ˆ
hk,v = 1 or
αk,v = 0. These two cases correspond to the two cases in
(28).
Lemma 4 gives the structure of the optimal precoding
vectors. In particular, it helps us to understand the optimal
BS association (i.e., which precoding vectors w∗
k,v that are
non zero).
7
Theorem 2: The optimal BS association for user kfalls into
one of the following two cases:
1) It is only served by one BS v∗, with v∗=
arg max
vˆ
hH
k,v B∗
k,v −1ˆ
hk,v , that is, Vk={v∗};
2) It is served by a set of BSs Vk=
v∗v∗= arg max
vˆ
hH
k,v B∗
k,v −1ˆ
hk,v where
|Vk|>1.
Proof: We know from (28) in Lemma 4 that user kis
associated with BSs vonly if
λ∗
k=1
ˆ
hH
k,v B∗
k,v −1ˆ
hk,v
.(34)
Dual feasibility requires that Bk−λkˆ
Rk≽0for all k,
or equivalently that uH
k,v B∗
k,v −λ∗
kˆ
Rk,v uk,v ≥0for
all vectors uk,v. By selecting uk,v =B∗
k,v −1ˆ
hk,v , this
conditions becomes
ˆ
hH
k,v B∗
k,v −1ˆ
hk,v −λ∗
kˆ
hH
k,v B∗
k,v −1ˆ
hk,v 2≥0
⇒λ∗
k≤1
ˆ
hH
k,v B∗
k,v −1ˆ
hk,v
.
(35)
Hence, the equality in (34) can only be achieved for the BSs
that have the largest value on ˆ
hH
k,v B∗
k,v −1ˆ
hk,v . This can
be one or multiple BSs, as reflected by the theorem.
Theorem 2 proves that single-BS association is optimal
in most cases, although our system model supports spatial
multiflow transmission from multiple BSs (a similar result
was obtained in [19] in for single-antenna BSs). The optimal
BS association for user kis the one with the largest value
of ˆ
hH
k,v B∗
k,v −1ˆ
hk,v . We notice that B∗
k,v in (26) is the
weighted sum of several terms; the spatial directions of in-
terfering channels, the noise variance, the channel uncertainty,
and the matrices from the power constraints. These terms are
weighted by the different Lagrange multipliers, which means
that the QoS and power constraints that are hard to satisfy will
have a large impact on B∗
k,v and vice versa. The BS association
rule is based on the norm of the channel ||ˆ
hH
k,v ||2from
BS v, which is then weighted through B∗
k,v . The weighing
will punish BSs with smaller power budget, lower estimation
quality, and/or many users with high QoS targets.
As seen from Case 2 in Theorem 2, it may happen that
multiple BSs are associated with a certain user. This occurs
when the most appropriate BS does not have the power
resources to satisfy the QoS target, thus another BS needs to
help out. This result stands in contrast to [25] where single-BS
association always occurs since there are no power constraints.
The probability of multi-BS association is evaluated in Section
V.
The optimal BS association rule is clearly a complicated
function of the channel quality, estimation quality, power
constraints, and QoS constraints. This stands in contrast
to heuristic association rules (e.g., the nearest-BS based,
the highest-received-power based, the max-SINR based, the
biased-received-power based and the biased-SINR based load
balancing criteria), which are generally not optimal in terms of
maximizing the energy efficiency under per-BS transmit power
constraints and per-user QoS constraints. These heuristic as-
sociation rules have been studied under various conditions
(different from our system model); see for example [6], [7],
[14]–[16], [20]–[23], [33]. Hopefully, these heuristics can
evolve in future works, based on insights on the optimal BS
association from Theorem 2.
IV. ITE RATIVE HEURISTIC ALGO RI TH M DES IG N
In this section, we tackle the non-convex problem (9) by
iterative convex approximations of the power consumption
functions. In particular, each iteration solves a problem with
a modified objective function, which is convex. This convex
objective function is updated in each iteration such that most of
the BSs with small transmit powers in the solution are driven
to sleep mode. The proposed algorithm will find a suboptimum
to the original problem in (9).
Note that 0≤Ptrans,v ≤Pv,max for each BS v,v∈ V. Thus,
the total consumed power of BS v,Pvin (5), can be relaxed
with its convex envelope, Pc.e.
vover the interval [0, Pv,max],
where
Pc.e.
v(Pt,v),NvPsleep,v + ∆′
vPtrans,v (36)
with
∆′
v,Nv(Pactive,v −Psleep,v)
Pv,max
+ ∆v(37)
which is the largest convex function smaller than or equal to
Pvover the interval. Replacing Pvwith Pc.e.
v, problem (9) and
(10) are relaxed to
minimize
{wk,v }
M
v=1
avPc.e.
v
subject to γk≥Γk,∀k
K
k=1
∥wk,v ∥2≤Pv,max,∀v.
(38)
The idea, which is based on replacing an indicator function of
a bounded variable with its convex envelope, is often referred
to as the l1-norm relaxation, where sparse solutions can be
obtained. The relaxed problem (38) can be reformulated as a
convex optimization problem
minimize
{Wk≽0}
K
k=1
Tr (A′Wk) +
M
v=1
avNvPsleep,v
subject to Tr ˆ
RkWk−Γk
K
l=1
l=k
Tr (ˆ
Rk+Ek)Wl
−ΓkTr (EkWk)≥Γkσ2
k,∀k
K
k=1
Tr (QvWk)≤Pv,max,∀v
(39)
where A′is a modified block diagonal matrix of A, with
∆vreplaced by ∆′
vfor each block v. Note that based on
Lemma 3, the rank-one constraints are dropped without loss
8
of optimality. Compared to the original problem (10), the
relaxed problem (39) has the same feasible set, but a modified
objective function. The optimal value of (39) is a lower bound
on the optimal value of the original problem (10).
The proposed iterative heuristic algorithm is as follows:
1) i:= 0; Initialize W(0)
kfor k= 1, . . . , K by solving the
relaxed convex problem (39).
2) i:= i+ 1; Obtain the transmit power of each BS vas
P(i−1)
trans,v =K
k=1 Tr QvW(i−1)
k.
Define ˆ
P(i)
v(Ptrans,v),NvPsleep,v + ∆(i)
vPtrans,v, where
∆(i)
v,Nv(Pactive,v −Psleep,v)
P(i−1)
trans,v +δ+ ∆v.(40)
Solve the modified optimization problem
minimize
{Wk≽0}
K
k=1
Tr A(i)Wk+
M
v=1
avNvPsleep,v
subject to Tr ˆ
RkWk−Γk
K
l=1
l=k
Tr (ˆ
Rk+Ek)Wl
−ΓkTr (EkWk)≥Γkσ2
k,∀k
K
k=1
Tr (QvWk)≤Pv,max,∀v
(41)
where A(i)is the modified block diagonal matrix of A,
with ∆vreplaced by ∆(i)
vfor each block v.
3) Let W(i)
kbe the solution to this problem.
4) If P(i−1)
trans,v and P(i)
trans,v are approximately5equal for each
v, return W∗
k:= W(i)
k.
Otherwise, go back to step 2).
Note that δin (40) is a non-negative small value, which can
be interpreted as a soft threshold for deciding when a BS
is set to sleep mode. Define P∗
trans,v ,K
k=1 Tr (QvW∗
k).
Thus, for P∗
trans,v ≫δ, we have ˆ
PvP∗
trans,v,NvPsleep,v +
Nv(Pactive,v−Psleep,v )
P∗
trans,v+δ+ ∆vP∗
trans,v ≈NvPactive,v+∆vP∗
trans,v =
PvP∗
trans,v, and BS vis in the active mode; while for
P∗
trans,v = 0,ˆ
PvP∗
trans,v,NvPsleep,v and BS vis under
the sleep mode.
For each iteration as shown in step 2), when P(i−1)
trans,v is small,
the modified ∆(i)
vin (40) becomes large, i.e., the derivative
of the power consumption function ˆ
P(i)
v(Ptrans,v)increases.
Therefore, the modified optimization problem (41) will push
the small P(i−1)
trans,v to zero; that is, the BSs with small transmit
powers in the solution to the previous problem are driven to
sleep mode. This leads to sparse solutions of W∗
k.
Lemma 5: The proposed iterative heuristic algorithm al-
ways converges.
Proof: The objective function of problem (41) is on the
form of the objective function in [50, Eq. (21)], which always
gives convergence; that is, with 0≤Ptrans,v ≤Pv,max a
5There are many different ways to define “approximately equal”, such as
max
v
P(i−1)
trans,v −P(i)
trans,v
≤εand ∑M
v=1
P(i−1)
trans,v −P(i)
trans,v
≤ε. The latter
is used as a stopping criterion in our simulation with ε= 10−6.
convex, compact set, and δ > 0, we can show that P(i)
trans,v −
P(i−1)
trans,v →0for v= 1, . . . , M . A proof of convergence for
this type of heuristic algorithms is given in [50, Appendix B].
Note that, upon convergence, the partial derivative with respect
to Ptrans,v of the function minimized in the last iteration is
given by Nv(Pactive,v −Psleep,v)
P∗
trans,v +δ+ ∆v,(42)
which is equal to the derivative of the function
f(Ptrans,v) =
M
v=1
αvlog (Ptrans,v +δ) +
M
v=1
∆vPtrans,v (43)
at Ptrans,v =P∗
trans,v, where αv,Nv(Pactive,v −Psleep,v).
From the equality of the first-order conditions for optimality,
we see that the iterative procedure finds a local minimum of
f(Ptrans,v). The log-sum function M
v=1 αvlog (Ptrans,v +δ)
is used as a smooth surrogate for the circuit power consump-
tion part of the objective function. Therefore, our proposed
heuristic algorithm is very similar to the weighted l1-norm
minimization methods, where the weighting factors are chosen
based on the log-sum surrogate function of the l0-norm [56].
V. NUMERICAL RE SU LTS
Numerical results are presented in this section to illustrate
our analytical results and the proposed algorithms. The pur-
pose of this section is not to provide a large-system analysis,
but to compare the heuristic algorithm from Section IV with
the optimal solution from Theorem 1, for which the complexity
of mode selection grows quickly with the number of BSs.
The propagation environment is a simplified version of the
dense urban information society model (TC2) used in the
METIS project [57], as illustrated in Fig. 2. The model consists
of four square-shaped buildings of dimensions 120 ×120 m,
each with 6 floors. A macro BS (MBS) is complemented with
4 small cell BSs (SBSs). The MBS has 4 transmit antennas,
and the SBSs have 2 transmit antennas each. Load-balancing
is particularly important in the lightly loaded cases that occur
during the majority of the day [9], because then there is an
opportunity to turn off BSs and associate users with other BSs
than the closest one. Hence, in most of the simulations, we
consider five users that are randomly and uniformly dropped
in the network, whereof 4 users are indoors and 1 user is
outdoors in every user drop. The system bandwidth is 10 MHz.
Here, we adopt the indoor and outdoor propagation models,
PS#1 - PS#4, identified in METIS. More details regarding
network deployment and propagation modes can be found
in [57, Table 3.7 and Section 8.1]. We assume independent
Rayleigh small-scale fading. The MMSE channel estimation
errors are calculated based on (1) with the total pilot power
p=Pv,max/2. Table I shows the power model parameters and
is based on [11, Table 6 and Table 8].
Three different joint precoding and load balancing schemes
are compared in the scenario depicted in Fig. 2. We name these
three schemes as “Optimal”, “Heuristic” and “All Active”
respectively. The “Optimal” scheme obtains the global optimal
solution as described in Theorem 1, by an exhaustive search
9
50 100 150 200 250
50
100
150
200
250
21
34
MBS
SBS 3
SBS 1
SBS 2
SBS 4
Fig. 2. The MBS (cross) and SBSs (circles) deployment considered in
Section V.
Table I
POWE R MO DEL PA RAM ETE RS F OR DI FFE REN T BS T YPE S.
BS type NvPv,max Psleep,v Pactive,v ∆v
MBS 4 39.8 W 75.0 W 130.0 W 4.7
SBS 2 6.3 W 39.0 W 56.0 W 2.6
over all 25possible BS mode combinations. The “Heuristic”
scheme follows the algorithm proposed in Section IV, and the
value of the soft threshold δis set to 10−4. The “All Active”
scheme is used as our performance baseline, which solves the
optimization problem (9) by assuming that all BSs are active,
i.e., the BS mode indicator zv= 1 for all BSs v∈ V. For each
scheme, the performance is averaged over 1000 independent
user drops that provide feasible solutions for our optimization
problem (9). For each user drop, the algorithms are evaluated
over 50 independent channel realizations. The weights avare
set to 1 for all BSs.
Define the dynamic part of total power consumption as
the total RF power (M
v=1 av∆vPtrans,v), and the remaining
part of the total power consumption as the circuit power
(vavNvPactive,vzv+vavNvPsleep,v (1 −zv)). Figs. 3 and
4 demonstrate the total RF power and the total power con-
sumption as a function of target spectral efficiency per user,
respectively. As expected, the total power consumption and the
RF power increase as the target spectral efficiency increases.
Fig. 3 shows that the RF power for the “All Active” scheme is
less than that of the “Heuristic” and “Optimal” schemes. This
is expected since all BSs are active in the “All Active” scheme,
whileas for the “Heuristic” and “Optimal” schemes, some BSs
are put into sleep mode. With more BSs being active, the
“All Active” scheme provides better energy-focusing and less
propagation losses between the users and the transmitters, and
will therefore reduce the total RF power. However, as can be
seen from Fig. 4, compared to the “All Active” scheme, the
“Heuristic” and “Optimal” schemes can substantially reduce
the total power consumption, especially when the target QoS
is small. This is because the circuit power consumption under
the sleep mode is much lower compared to the one under the
active mode, i.e., Psleep,v ≪Pactive,v. For the “All Active”
scheme, the increase in the circuit part from the extra power
consumed by activating BSs clearly outweighs the decrease in
1 1.5 2 2.5 3 3.5 4 4.5 5
0
5
10
15
20
25
30
Target spectral efficiency per user [bit/s/Hz]
Total RF power [W]
All Active
Heuristic
Optimal
Fig. 3. Total RF power (the dynamic part ∑M
v=1 av∆vPtrans,v) vs. target
spectral efficiency per user (Rk).
1 1.5 2 2.5 3 3.5 4 4.5 5
650
700
750
800
850
900
950
1000
Target spectral efficiency per user [bit/s/Hz]
Total power consumption [W]
All Active
Heuristic
Optimal
Fig. 4. Total power consumption (∑M
v=1 avPv) vs. target spectral efficiency
per user (Rk).
the dynamic part. This implies that putting a BS into sleep
mode by proper load balancing is an important solution for
energy savings in heterogeneous networks.
Fig. 5 plots the cumulative distribution function (CDF) of
the total power consumption for the considered three schemes.
The target spectral efficiency per user Rkis 4 bit/s/Hz. We
observe that compared to the “All Active” scheme, 20% of
the total power consumption can be saved by the “Optimal”
scheme with 70% probability and by the “Heuristic” scheme
with 55% probability. For some user drops, the energy con-
sumption can be reduced by 30% for both the “Optimal” and
“Heuristic” schemes.
Fig. 6 demonstrates the BS activation probability versus
the target spectral efficiency per user. Here, the activation
probability of the SBS is averaged over the probabilities of
the four SBSs depicted in Fig. 2. We see that for the “All
Active” scheme, the activation probabilities of the MBS and
SBS are always one, since all BSs are always active in this
scheme. Moreover, as anticipated, for both the “Heuristic”
and “Optimal” schemes, the BS activation probabilities of
10
650 700 750 800 850 900 950 1000 1050
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Total power consumption [W]
CDF
All Active
Heuristic
Optimal
Fig. 5. The CDF of total power consumption.
1 1.5 2 2.5 3 3.5 4 4.5 5
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Target spectral efficiency per user [bit/s/Hz]
BS activation probability
All Active, MBS
All Active, SBS
Heuristic, MBS
Heuristic, SBS
Optimal, MBS
Optimal, SBS
Fig. 6. BS activation probability vs. target spectral efficiency per user (Rk).
the MBS and SBS increase as the target spectral efficiency
per user increases. This is because in order to satisfy the
raised QoS expectations of all users, the probability that a BS
becomes active should increase so as to provide better energy-
focusing and less propagation losses. Over the considered
range of target spectral efficiency per user, the “Optimal”
scheme has lower activation probability for the MBS and
higher activation probability for the SBS as compared to the
“Heuristic” scheme. Note that the circuit power consumed
under the active mode Pactive,v for the MBS is much higher
than that of the SBSs. Thus, as shown in Fig. 4, the “Optimal”
scheme results in better energy saving as compared to the
“Heuristic” scheme.
Figs. 7-9 investigate the impact of the ratio η,
Psleep,v/Pactive,v on the overall energy efficiency for different
schemes. The values of Pactive,v are fixed to 130W and 56W for
the MBS and SBSs respectively. The target spectral efficiency
Rkis fixed to 3 bit/s/Hz. In Figs. 7 and 8, the total RF power
and the total power consumption are plotted as a function of
the ratio η, respectively. It can be seen from Fig. 7 that the
RF power of the “Optimal” and “Heuristic” schemes decreases
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
14
15
16
17
18
19
20
21
Psleep,v /Pactive,v
Total RF power [W]
All Active
Heuristic
Optimal
Fig. 7. Total RF power vs. η=Psleep,v /Pactive,v.
as the ratio η(or equivalently Psleep,v) increases, especially
when the ratio ηis large (close to 1). This is because it is
better to turn on more BSs, to reduce the RF power, when the
difference between the active and sleep modes decreases. The
BS activation probability increases more for the “Optimal”
scheme, compared to the “Heuristic” scheme. Hence, we
observe that the total RF power reduces more significantly
for the “Optimal” scheme. From Fig. 8, we see that the total
power consumption increases almost linearly as ηincreases.
This is mainly due to the increase of Psleep,v.
Although the system allows all BSs to transmit to all
users simultaneously at the same time-frequency resource
block, Fig. 9 shows that the probability that a user is served
by multiple BSs is less than 4.2% for all the considered
schemes over the entire range of ηwhen the target spectral
efficiency Rkis fixed to 3 bit/s/Hz. Not shown here, the joint
transmission probability has also been evaluated over different
targets of spectral efficiency, i.e., for Rk={1,2,3,4,5}
bits/s/Hz, while the ratio ηis fixed according to Table I. For
these cases, simulation shows that the probability of multi-BS
joint transmission is less than 4% over the considered range
of Rk. Fig. 10 shows the joint transmission probability as
function of the number of users, for a target spectral efficiency
of 1 bit/s/Hz. The probability increases with the number of
users, since it is harder to satisfy the QoS targets, but it is
still in the range of a few percentages. These observations
are in line with Theorem 2. From Fig. 9, we also observe
that, for the “Optimal” and “Heuristic” schemes, the joint
transmission probability increases as the ratio ηincreases.
This is expected since by increasing η, the BS activation
probability increases. Thus, the joint transmission probability
also increases. Compared to the “Heuristic” algorithm, the
“Optimal” scheme has a lower BS activation probabilities, and
therefore it also has a lower joint transmission probability.
VI. CO NC LU SI ON S
This paper analyzed the energy efficiency in heterogeneous
networks. More specifically, the downlink precoding vectors,
11
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
400
500
600
700
800
900
1000
Psleep,v /Pactive,v
Total power consumption [W]
All Active
Heuristic
Optimal
Fig. 8. Total power consumption vs. η=Psleep,v/Pactive,v . The target
spectral efficiency per user Rkis 3 bit/s/Hz.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0.03
0.032
0.034
0.036
0.038
0.04
0.042
Psleep,v /Pactive,v
Joint transmission probability
All Active
Heuristic
Optimal
Fig. 9. Joint transmission probability vs. η=Psleep,v /Pactive,v . The target
spectral efficiency per user Rkis 3 bit/s/Hz.
5 6 7 8 9 10
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
0.05
Number of users
Joint transmission probability
All Active
Heuristic
Optimal
Fig. 10. Joint transmission probability vs. the number of users. The target
spectral efficiency per user Rkis 1 bit/s/Hz.
load balancing (i.e., user-BS association), and BS operational
modes were jointly optimized to minimize the weighted total
power consumption. In order to verify how many BSs that
should serve a user at the optimal load balancing solution, each
user can be served by multiple BSs using spatial multiflow
transmission. We proved that the optimal BS association rule
consists of comparing weighted channel norms, where the
weighting matrices depend on channel uncertainty, power
constraints and QoS constraints. Moreover we proved that, in
most cases, it is optimal for each user to be served by a single
BS. Multiple BSs only serve a user when the primary BS
does not have the power resources to deliver the full QoS, in
which case neighboring BSs can cooperate in order to provide
the full QoS. An iterative heuristic algorithm was proposed to
find a suboptimal solution of relatively low complexity and it
achieves good performance in relation to the optimal scheme.
Our numerical results showed that the total power consumption
can be greatly reduced by putting a BS into sleep mode using
proper load balancing.
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