# Energy Efficient Operation of Adaptive
Massive MIMO 5G HetNets

Siddarth Marwaha, Eduard A. Jorswieck, Mostafa Jassim, Thomas Kuerner, David
Lopez Perez, Xilnli Geng, Harvey Bao

###### Abstract

For energy efficient operation of the massive multiple-input multiple-output
(MIMO) networks, various aspects of energy efficiency maximization have been
addressed, where a careful selection of number of active antennas has shown
significant gains. Moreover, switching-off physical resource blocks (PRBs) and
carrier shutdown saves energy in low load scenarios. However, the joint
optimization of spectral PRB allocation and spatial layering in a
heterogeneous network has not been completely solved yet. Therefore, we study
a power consumption model for multi-cell multi-user massive MIMO 5G network,
capturing the joint effects of both dimensions. We characterize the optimal
resource allocation under practical constraints, i.e., limited number of
available antennas, PRBs, base stations (BSs), and frequency bands. We observe
a single spatial layer achieving lowest energy consumption in very low load
scenarios, whereas, spatial layering is required in high load scenarios.
Finally, we derive novel algorithms for energy efficient user (UE) to BS
assignment and propose an adaptive algorithm for PRB assignment and power
control. All results are illustrated by numerical system-level simulations,
describing a realistic metropolis scenario. The results show that a higher
frequency band should be used to support UEs with large rate requirements via
spatial multiplexing and assigning each UE maximum available PRBs.

###### Index Terms:

Wireless communications, resource allocation, optimization, energy efficiency,
multiple antenna networks, system-level assessment

##  I Introduction

00footnotetext: Parts of this work have been published in [1].

5G technology is engineered to deliver superior performance, offering gigabit
data rates, ultra-reliable low latency, and massive connectivity to link
everyone and everything [2]. The advanced hardware and performance of 5G
networks contribute to improved energy efficiency (EE) measured in bits per
Joule, surpassing previous technology generations. However, despite this
enhanced EE, the surge in data traffic, utilization of wider bandwidths,
increased antenna count, and higher levels of base station (BS) densification
in 5G currently lead to heightened energy consumption. This has resulted in an
augmented carbon footprint of networks in the 5G era [3]. To meet ambitious
targets for net-zero greenhouse gas (GHG) emissions, it is crucial to design
future wireless networks with a primary focus on enhanced energy efficiency.

###  I-A Motivation

It has been reported in [4] that $57\%$ of the total network energy is
consumed by the base stations (BSs), where power amplifiers (PAs),
transceivers, baseband units, and cables account for approximately $65\%$ of
the BS energy [2]. Therefore, significant attention has been directed towards
enhancing these modules to reduce BS energy consumption in recent years,
leading to the proposal and investigation of various energy-saving techniques.

Energy-saving techniques are generally aimed at adapting the time, space, and
frequency resources of the BS to traffic demands. Over the last 15 years,
solutions to improve spectral efficiency (SE) in all these dimensions have
been deployed. However, the wireless network does not always require maximum
SE. Instead, the SE should be carefully chosen to satisfy Quality of Service
(QoS) requirements while saving energy.

In the time domain, symbol shutdown turns off all PAs when it detects that a
downlink (DL) orthogonal frequency division multiplexing (OFDM) symbol has no
data to send, reducing BS power consumption without affecting QoS [5].

In the spatial domain, massive MIMO is the key technology for improving the
coverage, throughput, and reliability of 5G networks. The SE of massive MIMO-
aided 5G sites is reported to be three to five times greater than that of 4G
sites using traditional radio solutions. However, it is essential to select
the best massive MIMO configuration according to traffic needs to avoid
performance over-provisioning and maximize energy efficiency (EE). A good
scaling of massive MIMO energy consumption can be achieved by switching off
part of the massive MIMO PAs when traffic requirements are low [6], thus
adapting to the load and QoS requirements to avoid coverage and performance
degradation.

In the frequency domain, techniques including carrier aggregation and multi-
connectivity have been developed to increase available bandwidth for enhanced
data rates and/or reliability in 5G. As described in [7], in heterogeneous
networks (HetNets) operating multiple frequencies, carrier shutdown can
periodically assess the service load of the multiple carriers managed by the
BS. If the load in some capacity layers is lower than a specified threshold,
all the massive MIMO PAs and the baseband processing can be deactivated,
resulting in significant energy savings. However, QoS may be impacted if
active base stations cannot manage the increased traffic resulting from
carrier shutdown.

In this paper, we focus on the online configuration of hardware components in
the massive MIMO BS in spatial and frequency domains.

###  I-B State-Of-The-Art

For an overview of energy-efficient wireless communications and fundamental
green trade-offs, such as spectral efficiency (SE) versus energy efficiency
(EE), deployment efficiency versus EE, delay versus power, and bandwidth
versus power, as well as energy harvesting for sustainable green 5G networks,
see [8], [9], and [10].

While the overview in [11] delves into power-saving mechanisms for User
Equipment (UE) in great detail, our specific focus is directed towards
reducing network energy consumption under UE quality of service (QoS)
requirements. Detailed insights into different network power consumption
models, EE metrics, and main EE-enabling technologies provided by the third
generation partnership project (3GPP) new radio (NR), as well as power-saving
techniques in 5G NR, can be found in [2]. Additionally, [12] provides an
overview of means to monitor and evaluate EE.

Numerous recent papers address single-cell EE bounds and trade-offs, with some
providing analytical results and EE bounds and trade-offs with SEs.
Efficiently solving typical EE optimization problems often involves the use of
fractional programming (see [13] for an overview and references therein). In
multi-cell scenarios, the model is more challenging due to complex topology,
UE distributions, traffic models, wireless fading channels, protocols,
algorithms, and a large number of parameters. As a result, explicit and
general closed-form expressions describing the EE bounds and trade-offs of a
multi-cell massive MIMO network are lacking. In the following, we summarize
selected recent state-of-the-art works where the energy consumption of multi-
cell wireless networks is studied.

For massive MIMO systems, a simple linear or affine model as a function of the
transmit power has been identified as inadequate because it leads to unbounded
EE as the number of antennas grows large [14]. Therefore, in massive MIMO
systems, it is important to incorporate power consumed by different BS
components, such as PAs, transceivers, analog filters, and oscillators.
Accounting for circuit power consumption, in [15] and [16], EE has been shown
to be a quasi-concave function of the number of antennas, number of UEs, and
the transmit power, where EE is optimized by increasing the transmit power
with the number of antennas. In [17], it is advised to turn off a fraction of
antennas to reduce the total power consumption late at night when the traffic
demand is low. Whereas in [18], the downlink EE is maximized by adapting the
number of antennas to temporal load variations over a day. In [19], the
authors adjusted the number of antennas and transmit data rate to maximize EE
for uplink energy-efficient resource allocation in very large multi-user MIMO
systems. While in [20], the authors aimed to find the optimal densified
network configuration, concluding that reducing the cell size leads to higher
EE; however, the EE saturates when the circuit power dominates over transmit
power.

Instead of binary decisions to switch on and off complete BS, the activation
of sleep modes with finer granularity levels [21] is considered for EE
optimization too. In [22], EE with the introduction of several levels of sleep
depths is optimized. The BS density for enhancing EE through traffic-aware
sleeping strategies in both one- and two-tier cellular networks is optimized
in [23]. In [24], the most energy-efficient deployment solution for meeting
certain minimum service criteria is developed, and the corresponding power
savings through dynamic sleep modes are analyzed. In [25], the rate of
failures triggered by fatigue processes of BSs subject to sleep modes is
controlled by an algorithm called LIFE, applied to HetNets (LTE and legacy
UMTS).

In [26], the power control and EE in downlink multi-cell massive MIMO systems
are investigated and optimized. [27] optimizes the underlying multi-user
transport layer of the frequency division duplex orthogonal frequency division
multiple access (FDD-OFDMA) massive MIMO system. The goal of [28] is to
maximize the non-convex EE in a downlink massive MIMO system using a proposed
energy-efficient low-complexity algorithm. The optimization of the power
assignment to achieve the maximum EE for a downlink system of single-cell
massive MIMO based on a tight approximate expression for the achievable sum-
rate is studied in [29]. In [30], the authors jointly optimize the transmit
beamforming vectors of femto base stations (FBSs) and the phase-shift matrices
of reconfigurable intelligent surfaces (RISs) for an RIS-aided HetNet under
channel uncertainties and residual hardware impairments to maximize the
minimum EE of femtocells.

In addition, we refer the reader to [31] for a detailed overview of UE to BS
association techniques covering HetNets, massive MIMO, millimeter wave
(mmWave), and energy harvesting and various UE association metrics, including
EE. The authors in [32] investigate the reassignment of UEs between adjacent
small cells to enable spatial multiplexing gains via multi-user MIMO and
reduce energy consumption through small cell sleep state. To maximize the EE
of the system, the authors in [33] investigate the optimal on-off switching
and UE association in a HetNet with massive MIMO by formulating an integer
programming problem. In [34] and [35], the authors considered a comprehensive
power consumption model for massive MIMO HetNets covering UE to BS association
but did not consider PRB allocation.

While EE has been studied from various aspects, most works reported above do
not consider multi-carrier transmission with spatial and spectral resource and
component allocation, and the matching of UEs to BSs and frequency bands.

###  I-C Contribution and Problem Statement

In particular, our work, differently than previous ones, investigates the EE
of a HetNet massive MIMO network, presenting and working with a more
comprehensive power consumption model, which encompasses intricate factors
such as UE to BS and frequency band association, PRB allocation, distributed
power control, and the number of antennas/RF chains used. We propose a method
that deconstructs a mixed-integer non-convex programming problem into sub-
problems accounting for joint energy-efficient UE to BS assignment as well as
carrier, PRB, and transmit power allocation. Our proposed method uses long-
term channel statistics obtained from network level simulator SiMoNe
(Simulator for Mobile Networks) [36], which describes a realistic metropolis
(Berlin) scenario with realistic antenna deployments.

Our analysis and results show that:

  1. 1.

Under the assumption of homogeneous BS deployment and zero-forcing (ZF) with
the same energy consumption model and with the same resource allocation
strategy, there exists a simple sufficient condition to optimally decide on a
BS carrier shutdown, and assign the UEs of the shutting down BS to its
neighboring BSs.

  2. 2.

Under the assumption of fixed inter-cell interference and the use of ZF
massive MIMO precoding with equal power allocation, there exists a sufficient
condition to optimally allocate power to MIMO layers and PRBs. Whether a
single or multiple spatial layers are used per PRB, the optimal power
allocation requires using all available PRBs.

The rest of the paper is organized as follows. Section II introduces the
system model, elaborating on the deployment (II-A), signal (II-B), channel
(II-C), power consumption (II-D), time and download (II-E) models used, and
presents the problem statement (II-F). Section III discusses the main
theoretical results and the derived UE to BS assignment (III-A1 and III-A2)
and resource allocation (III-B1 and III-B2) algorithms. In Section IV, the
simulation setup (IV-A) is described, and the numerical results (IV-C and
IV-D) are presented. Finally, the conclusion and future work are summarized in
Section V.

##  II System Model

###  II-A System Architecture

Figure 1 shows an example realization of the system architecture, where a BS
manages a single cell, which can operate different frequency bands, e.g.,
$700$ MHz band with $20$ MHz bandwidth and $2.6$ GHz band with $100$ MHz
bandwidth. We highlight the same by showing a single BS in a cell, i.e., $2.6$
GHz band in blue or two BSs in a cell, i.e., $700$ MHz band in green and $2.6$
GHz band in blue. Each frequency band possesses a certain number of PRBs which
can be assigned to the associated UEs.

(a) Macroscopic view: Multiple BSs, each one managing multiple cells and
serving multiple UEs.

(b) Microscopic view: Single BS managing two cells operating at different
frequency bands, where each frequency band has a given set of PRBs to assign
to the UEs associated to the BS.

Figure 1: An example system scenario: 6 BSs consisting of one or two cells
operating different frequency bands (e.g. $700$ MHz band in green and $2.6$
GHz band in blue.), where the UEs are uniformly distributed and the lower
frequency band is sparsely deployed.

In the system model, $\mathcal{M}=\\{1,\dots,M\\}$ and
$\mathcal{B}=\\{1,\dots,B\\}$ represent the set of BSs and the set of
available frequency bands per BS, respectively. $\mathcal{N}_{m}^{b}$ denotes
the set of available PRBs in the $m^{th}$ BS and $b^{th}$ frequency band, with
cardinality $N_{m}^{b}$.

The number of active PRBs at the $m^{th}$ BS in the $b^{th}$ frequency band is
denoted by $\alpha_{m}^{b}$, i.e., $0\leq\alpha_{m}^{b}\leq N_{m}^{b}$. Note
that the triplet $\alpha=(m,b,n)$ represents the PRB $n$ operated by the
$m^{th}$ BS in the $b^{th}$ frequency band, and the functions $m(\alpha)$,
$b(\alpha)$ and $n(\alpha)$ return the BS, frequency band and PRB index of PRB
$\alpha$, respectively. The number of active transmit antennas at the $m^{th}$
BS in the $b^{th}$ frequency band is denoted by $a_{m}^{b}$, with a maximum of
${A}_{m}^{b}$ transmit antennas, i.e., $1\leq a_{m}^{b}\leq A_{m}^{b}$.

Furthermore, $\mathcal{K}=\\{1,\dots,K\\}$ denotes the set of UEs. The
assignment of the $k^{th}$ UE to the available PRBs in the $m^{th}$ BS and
$b^{th}$ frequency band is realized by a matching function
$\mu(k)=[\alpha_{1},...,\alpha_{N_{k}}]$, which returns the set of PRBs
assigned to the $k^{th}$ UE, with cardinality $N_{k}$. The matching function
$\mu$ is overloaded, and also returns the set of UEs assigned to a PRB
$\alpha$, i.e.,

$\mu(\alpha)=\mu(m,b,n)=\mu(m(\alpha),b(\alpha),n(\alpha))=\begin{cases}\mathcal{K}_{m,b,n}&\textrm{if
UEs assigned}\\\ \emptyset&\textrm{if unassigned or not used}.\end{cases}$ (1)

###  II-B Signal Model

We can distinguish between original massive MIMO beamforming models from [37],
with recent application in [38], or the recent (new) massive MIMO beamforming
models from [39].

Typical forms of single-cell SINR expressions using ZF, often used in current
networks, are111 It should be noted that the SINR expression in [37], [38] and
[39] considers pilot contamination and intra-cell interference, which,
however, are not accounted for in this work and hence the expressions have
been adapted accordingly. However, pilot contamination can be included via
lower channel gains or additional noise terms.

$\displaystyle\textrm{SINR}_{k}^{\textrm{ZF}}$ $\displaystyle=$
$\displaystyle{(a-K)\beta_{k}p_{k}},$

where $a$ is the number of active antennas, $K$ is the number of UEs,
$\beta_{k}$ is the large-scale fading, and $p_{k}$ is the allocated transmit
power. The multi-cell scenario studied in [39, Section VI] derived the
effective signal to interference and noise ratio (SINR) for ZF as

$\displaystyle\textrm{SINR}_{m,k}^{\textrm{ZF}}=\frac{(a_{m}-K)\beta_{m,k}p_{m,k}}{1+\sum_{m^{\prime}\in\mathcal{P}_{m}\setminus\\{m\\}}(a_{m^{\prime}}-K)\beta_{m^{\prime},k}p_{m^{\prime},k^{\prime}}},$
(2)

where $a_{m}$ is the number of active antennas at the $m^{th}$ BS,
$\beta_{m,k}$ is the large scale fading of the $k^{th}$ UE at the $m^{th}$ BS,
$p_{m,k}$ is the assigned transmit power to the $k^{th}$ UE by the $m^{th}$
BS, $\mathcal{P}_{m}$ is a set consisting of all the BSs, and
$m^{\prime}\in\mathcal{P}_{m}\setminus\\{m\\}$ represents the set of BSs
creating interference.

It is important to note that these SINR expressions are derived for
independent and identically distributed (iid) Rayleigh small scale fading. In
particular, the channel matrices of different UEs are assumed to be
independent, and the number of antennas at each BS large enough. In [40], the
case with a large number of antennas but line-of-sight (LOS) conditions is
studied. In the LOS scenario, channel correlation among UEs can happen. By
proper UE scheduling, and dropping highly correlated UEs, these cases can be
avoided.

In the following, we will consider the ZF achievable SINR expression from
above along with the power and PRB allocation. Moreover, we will assume that a
UE can only be associated to one BS and allocated to one frequency band, and
thus, for the sake of simplicity, we partially remove the BS and frequency
band indexes $m$ and $b$ in the following expressions.

In the downlink, the transmitted signal is generated at the BS by precoding
and scaling the data symbols [41]. Let $p_{k,\alpha}$ be the normalized
transmit power applied on PRB $\alpha=(m,b,n)$ when serving the $k^{th}$ UE.
Then, the data rate achieved by the $k^{th}$ UE over the set of assigned PRBs
$\mu(k)$ is calculated using their average SINR222The SINR expression has been
normalized by the noise power, and thus the noise variance has been included
as part of the large scale fading. Additionally, the SINR expressions are
computed under small scale Rayleigh fading. However, the large scale fading is
later simulated via ray tracing. $\gamma_{k}$, as all these PRBs are coded
together, i.e.

$R_{k}=(\bar{b}\cdot
N_{k})\log_{2}\left(1+\frac{1}{N_{k}}\left(\sum_{\begin{subarray}{c}\alpha\in\mu(k)\end{subarray}}\frac{(a_{m(\alpha)}^{b(\alpha)}-|\mu(\alpha)|)\cdot
p_{k,\alpha}\cdot\beta_{k,\alpha}}{1+\sum\limits_{\begin{subarray}{c}\alpha^{\prime}:m(\alpha^{\prime})\neq
m(\alpha)\\\ n(\alpha^{\prime})=n(\alpha)\\\
b(\alpha^{\prime})=b(\alpha)\end{subarray}}\sum\limits_{k^{\prime}}p_{k^{\prime},\alpha^{\prime}}\cdot\beta_{k,\alpha^{\prime}}}\right)\right),$
(3)

where $\bar{b}$ is the PRB bandwidth333 The PRB bandwidth is $180$KHz in LTE
and NR when using a sub-carrier spacing of $15$KHz.,
$a_{m(\alpha)}^{b(\alpha)}$ is the number of active transmit antennas at the
$m^{th}$ BS in the $b^{th}$ frequency band, $|\mu(\alpha)|$ is the number of
UEs spatially multiplexed on PRB $\alpha$, and $\beta_{k,\alpha}$ is the large
scale fading experienced by the $k^{th}$ UE on PRB $\alpha=(m,b,n)$.

###  II-C Ray Tracing based Channel Model

We assume that the system operates in time division duplex (TDD) mode, where
the receiver can directly estimate the channel information from the received
signal [42], whereas, the transmitter relies on the reciprocity principle to
obtain the channel information on the same flat-fading subcarrier [14]. We
consider ZF precoding for massive MIMO based on perfect channel state
information at the transmitter (CSIT). Afterwards, we use the achievable rate
expression obtained with channel hardening which only depends on the long-term
channel gains for our algorithm design.

Highly accurate channel estimates can be determined using ray tracing [43]. In
this work, we use a ray tracing based large-scale fading predictor, called
Femto Predictor (FemtoPred), which was developed at the Institute for
Communications Technology at TU Braunschweig. As input data, the ray tracer
uses the following data sets, among others: building data, antenna diagrams,
frequency of operation, subscriber locations. The ray search is then conducted
using state-of-the-art ray tracing techniques and optimizations, as described
in [44]. Based on the results of the ray search, the influence of the free
space path loss as well as that of other propagation effects such as
diffraction, scattering, reflection and transmission are evaluated in order to
determine each ray’s contribution to the overall received signal strength. The
ray with the highest amplitude and phase shift at the receiver side was
selected as the dominant ray. This ray, together with other selected paths,
whose strength is larger than a threshold, contribute to the final received
signal strength at the receiver side444The details for the large-scale fading
computations can be found in Appendix D..

###  II-D Power Consumption Model

The total power consumption across all the BSs and frequency bands considering
load dependent and independent power consumption can be defined as [1] [15]
[45],

$\displaystyle P_{tot}=P_{LD}+P_{LI},$ (4)

where the load dependent power consumption $P_{LD}$ is calculated as

$\displaystyle P_{LD}$ $\displaystyle=$
$\displaystyle\sum_{m\in\mathcal{M}}\frac{1}{\eta_{PA_{m}}}\sum_{b\in\mathcal{B}}\sum_{k\in\mu(\alpha)}\sum_{\alpha\in\alpha_{m}^{b}}p_{k,\alpha},$
(5)

and the load-independent power consumption $P_{LI}$ is computed as

$\displaystyle P_{LI}$ $\displaystyle=$
$\displaystyle\sum_{m\in\mathcal{M}}\sum_{b\in\mathcal{B}}I_{m}^{b}\left(\frac{1}{\lambda_{m,0}^{b}}P_{m,FIX}^{b}+\frac{1}{\lambda_{m,1}^{b}}P_{m,SYNC}^{b}\right)+\sum_{m\in\mathcal{M}}\sum_{b\in\mathcal{B}}a_{m}^{b}D_{m,0}^{b}$
(6)
$\displaystyle+\sum_{m\in\mathcal{M}}\sum_{b\in\mathcal{B}}\left(a_{m}^{b}D_{m,1}^{b}\left(\sum_{k}N_{k}\right)\right)+C,$

where $\eta_{PA_{m}}$ is the combined power amplifier and antenna efficiency
of the $m^{th}$ BS, $I_{m}^{b}$ is an indicator function, indicating if any
PRB of the $m^{th}$ BS and $b^{th}$ frequency band is being used555 Note that,
with this indicator function, we realise an ideal carrier shutdown in which
the BS does not consume anything when no PRB is used., $P_{m,FIX}^{b}$ is the
load-independent power consumption in the $m^{th}$ BS and $b^{th}$ frequency
band required for site-cooling, control signaling, backhaul infrastructure and
base-band processing, $P_{m,SYNC}^{b}$ is the load-independent power consumed
by the local oscillator in the $m^{th}$ BS and $b^{th}$ frequency band,
$D_{m,0}^{b}$ is the power consumed by the RF chain attached to an antenna in
the $m^{th}$ BS and $b^{th}$ frequency band, including converters, mixers,
filters, etc., $D_{m,1}^{b}$ is the power consumed by the signal processing of
a MIMO layer across a PRB in the $m^{th}$ BS and $b^{th}$ frequency band,
through $\lambda_{m,0}^{b}$ and $\lambda_{m,1}^{b}$ the power consumption
model can capture a linear, sub-linear or an independent relationship between
$P_{m,FIX}^{b}$, $P_{m,SYNC}^{b}$ and $D_{m,0}^{b}$, indicating the level of
hardware sharing across the $B$ frequency bands in the $m^{th}$ BS, and $C$ is
the fixed power consumed by the coding at a massive MIMO BS and backhaul
support.

###  II-E Time and Download Model

We assume a simple transmission model where ${X}_{k}$ bits of data are
transmitted for each UE $k$, and the time ${T}_{k}$ to transfer this data is
computed based on the achievable rate ${R}_{k}$ as

$T_{k}=\frac{{X}_{k}}{{R}_{k}}.$ (7)

Based on this, the EE of the system can be defined as

$\displaystyle EE=\frac{\sum_{k\in\mathcal{K}}\frac{R_{k}}{T_{k}}}{P_{tot}},$
(8)

where $R_{k}$ and $P_{tot}$ are defined in (3) and (4), respectively.
Furthermore, it should be noted that the UEs whose rate requirements cannot be
satisfied, i.e. the outages, are not considered while computing the sum rate
in the numerator of (8). Their contribution to the power consumption in the
denominator of (8) is accounted for. The outage probability is reported in
Sections IV-C and IV-D separately.

###  II-F Problem Statements and Preliminaries

In this subsection, we state the problem, under consideration and we collect
some of the preliminary results from [1] on the optimization of simple special
cases of the scenario outlined above.

If rate requirements of all UEs should be fulfilled with minimum transmit
power, the following optimization problem is considered for all
$m\in\mathcal{M}$ and $b\in\mathcal{B}$

$\displaystyle\min_{p,\alpha,\mu}$ $\displaystyle P_{tot}$ (9a) subject to
$\displaystyle{R_{k}\geq\underline{R}_{k}}$ (9b) $\displaystyle
p_{k,\alpha}\geq 0$ (9c)
$\displaystyle\sum_{k}\sum_{\alpha:m(\alpha)=m}p_{k,\alpha}\leq P^{\max}$ (9d)
$\displaystyle 0\leq\alpha_{m}^{b}\leq N_{m}^{b}$ (9e)
$\displaystyle\max|\mu(\alpha)|<a_{m}^{b}$ (9f)
$\displaystyle\left\\{\bigcup_{n}\mu(m,b,n)\right\\}\cap\left\\{\bigcup_{n}\mu(m^{\prime},b^{\prime},n)\right\\}=\emptyset,\forall(m^{\prime},b^{\prime})\neq(m,b)$
(9g)

where (9b) corresponds to the minimum rate constraints, (9c) to the non-
negativeness of the power constraints, (9d) to the sum power constraints per
BS and band, the maximum number of available PRBs constraints in (9e), and the
minimum number of required antennas constraints in (9f), and the last
constraint (9g) means that the intersection between the the set containing all
the users assigned to BS $m$ and the set containing all the users assigned to
BS $m^{\prime}$ must be an empty set. This excludes coordinated multi-point
(CoMP) or multi-connectivity.

Single User Scenario: As the first step, a single macro cell with one active
frequency band serving a single UE is studied [1]. The minimum rate constraint
for a single UE is defined as,

$\underline{R}\leq\bar{b}\cdot|\alpha|\log_{2}\left(1+\frac{1}{|\alpha|}\sum_{n=1}^{|\alpha|}(a-1)\beta
p_{1}\right),$ (10)

where, $\underline{R}$ is the QoS requirement for the user, $\bar{b}$ is the
PRB bandwidth, $\alpha$ is the number of PRBs allocated to the user, $a$ is
the number of antennas, $\beta$ is the large scale fading, and $p_{1}$ is the
transmit power allocated to the user. Assuming uniform power allocation for
each PRB assigned to the user, i.e., $p_{1}=p$, the total power consumption of
the macro cell can be expressed as,

$P_{tot}=\frac{\alpha}{\eta_{PA}}p+D_{0}a+D_{1}\alpha a$ (11)

where, the constant $C$ is neglected for simplification and $p$ is obtained
from (10) as,

$p\geq\frac{2^{\frac{\underline{R}}{\bar{b}\alpha}}-1}{(a-1)\beta}.$ (12)

To understand the behavior of the objective function, $P_{tot}$ is further
analyzed and minimized with respect to number of antennas $a$ and number of
PRBs $\alpha$ separately. It was observed that the optimum number of PRBs
$\alpha^{*}$ and the optimum number of antennas $a^{*}$ can be computed by
equating the derivative of the corresponding objective function with respect
to $\alpha$ and $a$, respectively, to zero, which yields,

$\alpha^{*}=\frac{\underline{R}\log(2)}{\bar{b}}\frac{1}{W\left({\frac{(D_{1}\eta_{PA}a(a-1)\beta)-1}{e}}\right)+1}$
(13)

$a^{*}=\sqrt{\frac{\alpha(2^{\frac{\underline{R}}{\bar{b}\alpha}}-1)}{\eta_{PA}\beta(D_{0}+D_{1}\alpha)}}+1$
(14)

where, $W(\cdot)$ in (13) is the Lambert W function and $e$ is the Euler’s
number.

######  Remark II.1.

Depending on the parameters $\eta_{PA}$, $D_{0}$, $D_{1}$, $\beta$, and
$\underline{R}$ there is an optimal number of active antennas. However,
numerical evidence suggests that minimum number of antennas should be used as
long as sufficient number of PRBs is available.

######  Remark II.2.

Looking at the properties of the optimum $a^{*}$ and $\alpha^{*}$, we observe
that there must be an optimal number of PRBs to achieve a given rate while
consuming lowest total power.

Simple Multiuser Scenario: The total power consumption for a multi-user
scenario can be computed as

$P_{tot}=\frac{1}{\eta_{PA}}\sum_{k=1}^{K}\alpha_{k}p_{k}+D_{0}a+D_{1}aK\sum_{k=1}^{K}\alpha_{k}.$
(15)

The following analysis considers a symmetric scenario, such that
$\beta_{1}=\beta_{2}=\beta$, and
$\underline{R}_{1}=\underline{R}_{2}=\underline{R}$ to compute the total power
consumed with and without spatial multiplexing. For convenience, we start with
this assumption and later relax it.

######  Lemma II.3 (Lemma 1 [1]).

Consider a symmetric scenario with $K$ users, where
$\beta_{1}=\ldots=\beta_{K}=\beta$ and
$\underline{R}_{1}=\dots=\underline{R}_{K}=R$. Then, the minimum power
consumption is achieved for $\alpha_{1}=\ldots=\alpha_{K}=\frac{\pi}{K}$, if
$\pi$ is the total number of PRBs used.

######  Theorem II.4 (Proposition 1 in [1]).

For a symmetric scenario, if the rate requirement $\underline{R}$ approaches
small values, i.e. $\underline{R}\to 0$, the total power consumed when the
users are not spatially multiplexed is lower than when the users are spatially
multiplexed. Then, the optimal number of PRBs is one and minimum number of
antennas should be used (for example $3$ antennas for $2$ users if spatially
multiplexed or $2$ antennas for $2$ users if not spatially multiplexed)

######  Theorem II.5 (Proposition 2 in [1]).

There exists a specific number of PRBs $\alpha^{*}$ for which
$P_{tot}^{NSM}=P_{tot}^{SM}$. For all $\alpha<\alpha^{*}$,
$P_{tot}^{NSM}<P_{tot}^{SM}$, whereas, for all $\alpha>\alpha^{*}$,
$P_{tot}^{NSM}>P_{tot}^{SM}$.

######  Remark II.6.

The preliminary results above indicate that there exists a dichotomy between
SM and NSM depending on the load of the cell. Furthermore, it seems more
efficient to first fill up the spectral domain (allocating PRBs) than
activating the next antennas and switch on spatial layers. The results in
Section III also confirm this underlying intuition for multi-cell systems.

##  III UE Matching and Resource Allocation

The programming problem in (9) is a mixed-integer non-convex programming
problem which is difficult to solve jointly and globally. Therefore, we
consider a divide-and-conquer approach. As illustrated in Figure 2, the
workflow starts with the generated (or measured) long-term channel parameters
and rate requirements obtained from network level simulation. The first
computational step is to perform the UE assignment to the BS. After the UE
assignment is fixed, the component allocation (PRBs and spatial layers)
together with power control is performed, assuming equal power is allocated
across all PRBs assigned to a UE.

Figure 2: Approach to deconstruct the problem complexity into UE assignment
and resource and component allocation.

###  III-A UE to BS Assignment

The baseline algorithm for UE assignment consists of choosing the band and BS
combination which results in the minimum fading or maximum (channel) gain,
i.e.,

$\mu(k)=\max_{\alpha}\beta_{k,\alpha}.$ (16)

In the following, the matching found by the basline algorithm is improved.

####  III-A1 Analytical Results

The load-independent power $P_{LI}$ leads to the sub-optimality of (16) when
trying to solve (9). Furthermore, we conjecture that a lower number of UEs
served by a BS corresponds to lower EE, because the constant power consumption
dominates. In particular, BSs serving a single UE in one band can achieve a
very low EE. Note that the EE of a BS can be defined as the ratio of the sum
of rates delivered by the BS divided by the total consumed energy for
delivering this rate, i.e., $E_{m}=\frac{\sum_{k\in\mu(m)}R_{k}}{P_{m}}$,
where, the power consumed $P_{m}$ corresponds to the part of $P_{tot}$ in (4)
for the one term related to BS $m$ in band $b$, i.e.,

$\displaystyle P_{m}$ $\displaystyle=$
$\displaystyle\frac{1}{\eta_{PA_{m}}}\sum_{k}N_{k}\,p_{k,\alpha}+I_{m}\left(\frac{1}{\lambda_{m,0}}P_{m,FIX}+\frac{1}{\lambda_{m,1}}P_{m,SYNC}\right)$
(17)
$\displaystyle+D_{m,0}a_{m}+\left(D_{m,1}a_{m}\left(\sum_{k}N_{k}\right)\right)+C.$

In the best case, the individual rates achieved should exactly correspond to
the rate requirements, therefore, we could replace $R_{k}$ in (8) by the
requirement $\underline{R}_{k}$. Assuming that the UE to BS assignment has
been fixed and equal power is allocated across all the PRBs assigned to a UE
associated with the BS $m$, it can be observed that only the first term in
(17) is dependent on the transmit power allocated to the UEs. Therefore, for
simplicity, the remaining terms can be dropped and a power minimization
problem can be formulated subjected to minimum rate requirement, sum power
budget and non-negative power allocation constraints. Analyzing the KKT
conditions for the power minimization problem, it can be shown that the
transmit power is assigned to each UE to maintain its minimum rate
demand666The proof of the same has been provided in Appendix A..

The idea for improving the UE to BS assignment consists in identifying the BSs
that have the lowest EE, switch them off and distribute the formerly assigned
UEs to the next best BSs and bands.

######  Theorem III.1.

Under the assumption of homogeneous BS with the same load-independent energy
consumption model, and with the same PRB allocation and power control
strategy, it is energy optimal to switch off BS $m$ with $\ell$ UEs
$\mu(m)=\\{1,...,\ell\\}$ assigned and assign them to their next best BSs
$n_{1},...,n_{\ell}$ if the following conditions are satisfied simultaneously

$\displaystyle\beta_{k,\alpha}^{-1}+\Delta_{l}\geq\gamma_{l}\beta_{k,\alpha^{\prime}}^{-1}$
(18)

for all $1\leq l\leq\ell$, with $m(\alpha^{\prime})=n_{\ell}$ and where
$\Delta_{l}$ corresponds to the EE gain777$\Delta_{l}$ and
$\gamma_{l}=\frac{\gamma_{2l}}{\gamma_{1l}}$ are computed explicitly for
spatial multiplexing (SM) and no spatial multiplexing (NSM). However, due to
space constraints, the proof is provided in Appendix B..

###### Proof.

The proof is based on an inequality chain to bound the loss of EE of the new
assigned BS $n$ compared to the EE gain of the switched-off BS $m$ and is
provided in Appendix B. ∎

The result in Theorem 7 motivates the algorithm to switch off BSs that have
low EE, as described in Section III-A2. Furthermore, comparing the assignment
of UEs for different bands, we observe that the SINR distribution over the UEs
for different bands shows significant different support. This stems from the
higher path losses for higher frequencies. The exact numbers are reported in
Section IV-A. The assignment rule in (16) leads to sparsely filled higher
frequency bands and crowded lower frequency bands. Therefore, the second idea
is to improve the assignment in (16) by adding a large-scale fading bias [46]
to the higher frequency channels, i.e.,

$\tilde{\beta}_{k,\alpha}=\beta_{k,\alpha}+\Theta_{b(\alpha)},$

where $\Theta_{b(\alpha)}\geq 0$ is the bias for the band $b$. The UE
assignment is then based on the modified large-scale fading gains
$\tilde{\beta}_{k,\alpha}$. However, the PRB assignment and power control is
performed afterwards on the true large-scale fading channels
$\beta_{k,\alpha}$.

####  III-A2 Algorithms

The derivations above lead to four different algorithms which are implemented
for the UE assignment. The first simple baseline algorithm is called the
greedy UE assignment algorithm, as shown in Algorithm 1. It is the baseline
scheme in (16).

Algorithm 1 Greedy UE assignment

for $i=1..K$ do

$\mu(k)\leftarrow\max\limits_{\alpha}\beta_{k,\alpha}$.

end for

The second algorithm performs the re-matching of the UEs, which is implemented
by removing the BSs with the lowest EE from the list of available BSs.
Thereby, infinite loops, where a UE is re-matched to the second best (which
might serve a single UE or multiple UEs) and then re-matching to the first one
is avoided. The pseudo code can be found in Algorithm 2.

Algorithm 2 Re-Matching UE assignment

1:Run Algorithm 1

2:while $\exists m:|\mu(m)|\leq j$ do

3: $k\leftarrow\mu(m)$ ; $\mathcal{M}\leftarrow\mathcal{M}\setminus m$

4:
$\mu(k)\leftarrow\max\limits_{m(\alpha^{\prime})\in\mathcal{M}}\beta_{k,\alpha^{\prime}}$.

5:end while

Algorithm 3 Threshold UE assignment

1:Run Algorithm 1

2:while $\exists m:|\mu(m)|\leq j$ do

3: if
$\frac{\beta_{k,\alpha}-\beta_{k,\alpha^{\prime}}}{\beta_{k,\alpha^{\prime}}}<\delta$
then

4: $k\leftarrow\mu(m)$; $\mathcal{M}\leftarrow\mathcal{M}\setminus m$

5:
$\mu(k)\leftarrow\max\limits_{m(\alpha^{\prime})\in\mathcal{M}}\beta_{k,\alpha^{\prime}}$.

6: end if

7:end while

Algorithm 2 is further extended, where we propose a threshold based UE to BS
assignment. In Algorithm 3, the UEs are reassigned to a BS only if the ratio
$\frac{\beta_{k,\alpha}-\beta_{k,\alpha^{\prime}}}{\beta_{k,\alpha^{\prime}}}$,
is less than a threshold $\delta$ and the BS with no UE assigned are switched-
off. Through the ratio
($\frac{\beta_{k,\alpha}-\beta_{k,\alpha^{\prime}}}{\beta_{k,\alpha^{\prime}}}$)
and the threshold ($\delta$), we avoid assignment of the UEs to those BSs that
provide weak channel gain resulting in low EE.

The fourth algorithm, as shown in Algorithm 4, is the bias algorithm, where
the UE to BS assignment is improved by adding a bias888Note that, as shown in
[46], it is possible to formulate a large-scale fading bias model to chose a
suitable bias value per BS, however, such a formulation further increases the
complexity of the problem under consideration. Therefore, in this work, the
bias is heuristically chosen by comparing the channel values of the $700$ MHz
and $2.6$ GHz frequency bands. The exact value of the bias is provided in
Section IV-D. to the higher frequency channels. This not only enables the
lower frequency band to be sparsely filled and the higher frequency band to be
crowded, but it can also enable the selection of a better serving BS.

Algorithm 4 Bias UE assignment

for $i=1..K$ do

$\tilde{\beta}_{k,\alpha}=\beta_{k,\alpha}+\Theta_{b(\alpha)}$

$\mu(k)\leftarrow\max\limits_{\alpha}\tilde{\beta}_{k,\alpha}$.

end for

Finally, the fifth algorithm is a combination of the bias algorithm with the
re-matching, as shown in Algorithm 5, where first an artificial bias is
injected and then the BSs are switched-off. Similarly, as shown in Algorithm
6, UE to BS assignment is also performed after injecting the bias and re-
matching based on the threshold.

Algorithm 5 Bias and Re-Matching UE assignment

1:Run Algorithm 4

2:while $\exists m:|\mu(m)|=j$ do

3: $k\leftarrow\mu(m)$ ; $\mathcal{M}\leftarrow\mathcal{M}\setminus m$

4:
$\mu(k)\leftarrow\max\limits_{m(\alpha^{\prime})\in\mathcal{M}}\tilde{\beta}_{k,\alpha^{\prime}}$.

5:end while

Algorithm 6 Bias and Threshold UE assignment

1:Run Algorithm 4

2:while $\exists m:|\mu(m)|=j$ do

3: if
$\frac{\tilde{\beta}_{k,\alpha}-\tilde{\beta}_{k,\alpha^{\prime}}}{\tilde{\beta}_{k,\alpha^{\prime}}}<\delta$
then

4: $k\leftarrow\mu(m)$ ; $\mathcal{M}\leftarrow\mathcal{M}\setminus m$

5:
$\mu(k)\leftarrow\max\limits_{m(\alpha^{\prime})\in\mathcal{M}}\tilde{\beta}_{k,\alpha^{\prime}}$.

6: end if

7:end while

###  III-B PRB, Spatial Layers and Power Optimization

Only part of the optimization problem in (9) is obtained after UE to BS
assignment. Allocation of PRBs of BS $m$ to all assigned UEs $k\in\mu(m)$ and
power allocation has to be performed, too. It should be noted that the PRB and
power allocation is performed for all six UE to BS assignment algorithms
proposed in Section III-A2.

####  III-B1 Analytical Results

We make the following assumptions partly based on the current state of the art
as well as based on the goal to perform the allocation and control more
efficiently.

At first, all UEs assigned to one BS $m$ obtain the same power allocation,
i.e., $p_{k,\alpha}=p_{m}\;\forall k,\alpha(k)$, because the granularity to
adapt at one BS is obtained from the assignment of the number of PRBs. From
the results in the preliminary results section (Section II-F), we know that
there exists an optimum number of PRBs $\alpha^{*}$ assigned to fulfill the
rate requirement of the UE $k$. By computing the closed form expression for
this number from (13), we adapt to the large-scale fading and rate
requirements of a particular UE. Furthermore, we consider only two PRB
assignment strategies, namely no spatial multiplexing (NSM) and spatial
multiplexing (SM), as illustrated in Figure 3. In NSM, the total number of
PRBs available for BS $m$ in band $b$ are optimally distributed among the UEs
according to their rate requirement and channel gains. In contrast, via SM
each UE obtains a separate spatial layer and can use up to the maximum number
of available PRBs, which reduces the spatial degrees of freedom to
$(a_{m}-|\mu(\alpha)|)$.

Figure 3: PRB assignment strategies: SM where each UE is assigned a spatial
layer and NSM with a single spatial layer.

To characterize the proposed PRB allocation and power control strategy, we
have the following result, where the sufficient condition for optimal power
allocation for NSM and SM is given by

$\displaystyle\alpha_{1}(\tilde{p}_{1})\tilde{p}_{1}-\alpha_{1}(p_{1}^{*})p_{1}^{*}$
$\displaystyle\geq$ $\displaystyle\delta
C\eta_{PA},\;\text{where},\;\delta=\alpha_{1}(p_{1}^{*})-\alpha_{1}(\tilde{p}_{1}),$
(19)

which states that the difference between load-dependent part with $\tilde{p}$
and $p^{*}$, i.e.,
$\alpha_{1}(\tilde{p}_{1})\tilde{p}_{1}-\alpha_{1}(p_{1}^{*})p_{1}^{*}$ must
be greater than or equal to the load-independent part $\delta C\eta_{PA}$
(Appendix C), otherwise the load-independent power consumption dominates.

######  Theorem III.2.

Within a cell applying ZF massive MIMO precoding, for fixed inter-cell
interference, if the condition in (19) is fulfilled for all $\tilde{p}>p^{*}$,
then, in the case of NSM, the optimal power allocation is obtained by finding
the power $p_{m}^{*}$ such that the sum of all PRBs assigned to the UEs equals
the maximum number of available PRBs $N_{m}$, i.e.,

$\displaystyle\sum_{k\in\mu(m)}\alpha_{k}^{*}(p_{m}^{*})=N_{m}$ (20)

and for SM, the optimal individual power allocation, is obtained by finding
the power $\boldsymbol{p}^{*}=[p_{1}^{*},\dots,p_{k}^{*}]$ such that each UE
is assigned the maximum number of available PRBs $N_{m}$, i.e.,

$\displaystyle(\alpha_{1}^{*}(p_{1}^{*}),\dots,\alpha_{k}^{*}(p_{k}^{*}))_{k\in\mu(m)}=N_{m}.$
(21)

###### Proof.

The proof demonstrates by contradiction that deviating from the strategies in
(20) and (21) cannot save energy and is provided in Appendix C. The proof
shows that any other power allocation $\tilde{p}_{m}$ does not satisfy (20)
and (21) for NSM and SM, respectively, either leading to outages, if
$\tilde{p}_{m}<p_{m}^{*}$ or leading to a lower EE (due to an increase in the
total power consumption of the BS), if $\tilde{p}_{m}>p_{m}^{*}$.

∎

######  Remark III.3.

The sufficient condition in (19) depends on the energy consumption parameters.
Usually, it is fulfilled, because $D_{m,1}$ is very small and therefore, the
costs for using additional PRBs are small compared to the gain in terms of
power consumption.

####  III-B2 Algorithms

This result directly leads to the Algorithm 7, which is implemented for the
assignment of PRBs and power control. It should be noted that the PRB
assignment and power control can be executed independently at each cell in a
multi-cell network in a decentralized manner [47] because each BS only
requires two pieces of local information: the channel gains between itself and
its associated UEs and the interference plus noise power at the associated
UEs. Therefore, each BS can independently perform the PRB allocation and power
control decisions for the UEs it serves, taking into account its own local
conditions and optimizing its performance based on the available resources and
UE requirements, without relying on control from a central entity. By
implementing the proposed algorithms in a distributed manner, the
computational complexity and the need for extensive inter-cell communication
can be avoided. This approach enables scalability and improves the overall
efficiency of the multi-cell network.

Algorithm 7 PRB allocation and power control for NSM and SM

$p_{m}^{0}=p_{m}$ for all $m\in\mathcal{M}$

while $\sum_{m}|p_{m}^{t+1}-p_{m}^{t}|\neq 0$ do

for $m=1..M$ do

if NSM then

Interference $I_{m}^{t}$ at BS $m$ is computed based on the power allocation
of the other cells

$p_{m}^{*}\leftarrow
p_{m}^{t}:\sum_{k\in\mu(m)}\alpha_{k}^{*}(p_{m}^{t},I_{m}^{t})=N_{m}$

else

Interference $I_{m}^{t}$ at BS $m$ is computed based on the power allocation
of the other cells

$p_{m}^{*}\leftarrow
p_{m}^{t}:(\alpha_{k}^{*}(p_{m}^{t},I_{m}^{t}))_{k\in\mu(m)}=N_{m}$

end if

end for

end while

The convergence of the Algorithm 7 depends on the feasibility of the rate
requirements under the power constraints [41]. Before the algorithm is
started, we check the feasibility of the requested rates using Perron-
Frobenius theory [48]. If feasible, the iterative algorithm converges when the
allocated power per BS or UE does not change between the current and previous
iterations, i.e., an optimum power allocation is achieved.

###  III-C Computational Complexity of the Proposed Algorithms

We make a distinction between the computational complexity of the UE to BS
assignment (Algorithms 1-6) and the PRB allocation and power control
(Algorithm 7).

A summary of the computational complexity can be found in Table I, where $K$
and $M$ represent the number of UEs and the number of BSs, respectively.

Algorithm Complexity Benchmark $O(K)$ Algorithm 1 & 4 $O(K)$ Algorithm 2 & 5
$O(K(M+1))$ Algorithm 3 & 6 $O(K(M+1))$ Algorithm 7
$TMT_{B};T_{B}=\begin{cases}\max(n_{1},n_{2},n_{3})*K&\quad\text{NSM}\\\
K&\quad\text{SM }\end{cases}$

TABLE I: Computational Complexity of Algorithms 1-7.

  * •

UE to BS Assignment: Firstly, the Greedy UE assignment algorithm scales
linearly with the number of UEs, and therefore, has the computational
complexity of order $K$, i.e. O($K$). Secondly, the RE-Matching UE assignment
consists of two steps, where the first step has the the same complexity as
Greedy UE assignment, i.e. $O(K)$, and the second step has the complexity
$O(MK)$, where $M$ is the number BSs. Therefore, the overall complexity of RE-
Matching UE assignment is $O(K(M+1))$. Similarly, the Threshold UE assignment
has the complexity of $O(K(M+1))$. Thirdly, the addition of the bias in the
Bias UE assignment has negligible computational complexity, and therefore, the
Bias UE assignment has the same complexity as the Greedy UE assignment, i.e.,
$O(K)$. Finally, following the above, the computational complexity of Bias and
RE-Matching and Bias and Threshold UE assignment algorithms have similar
complexity as Re-Matching and Threshold UE assignment, respectively, i.e.
$O(K(M+1))$.

  * •

PRB Allocation and Power Control: Let $T$ be the number of iterations required
for the algorithm to converge, i.e. the stopping criteria in the while loop.
Then, the computational complexity of Algorithm $7$ is $O(TMT_{B})$, where $M$
is the number of BSs. The computational complexity for $T_{B}$ is dependent on
the multiplexing technique. In the case of NSM, we use bisection search to
allocate power to the UEs, such that with optimal power
$\sum_{k\in\mu(m)}\alpha_{k}^{*}(p_{m}^{*})=N_{m}$ is satisfied. Whereas, for
SM with fixed interference and the condition
$(\alpha_{1}^{*}(p_{1}^{*}),\dots,\alpha_{k}^{*}(p_{k}^{*}))_{k\in\mu(m)}=N_{m}$
at optimal power, we use the achievable rate expression to compute the
allocated power per UE. Therefore, the computational complexity of
$T_{B}=\max(n_{1},n_{2},n_{3})\cdot K$ for NSM, where $n_{1}$, $n_{2}$ and
$n_{3}$ represent the number of iterations needed for the three while loops
used for bisection search and $T_{B}=K$ for SM.

From the analysis above, it can be seen that our proposed UE to BS assignment,
PRB allocation and power control have low complexity, and can be efficiently
implemented in a distributed manner. In addition, to compare our proposed
distributed PRB allocation and power control algorithm, we propose a benchmark
algorithm, where the maximum available power per BS is equally distributed
among the UEs assigned and the UEs are spatially multiplexed. We then compute
the number of PRBs required to meet the minimum rate requirement of each UE.
Any UE requiring more than the maximum available PRBs is considered as an
outage.

The complexity of the proposed benchmark scheme scales linearly with the
number of UEs, i.e. $O(K)$, where the computation of the number of PRBs is
negligible. However, the performance of such an algorithm is poor in
comparison to our proposed algorithm, as shown in Figure 13. Furthermore, to
solve the optimization problem in (9) (see manuscript), any direct search
method would require an exhaustive search of all possible PRB assignment,
followed by finding the optimal power allocation for each such assignment. For
such an approach, it is apparent that the complexity is exponential in the
number of BSs, UEs and PRBs[49].

##  IV Numerical Results

All network simulations were conducted using SiMoNe [36], which is designed to
simulate complex wireless communication scenarios as realistically as
possible.

###  IV-A Simulation Setup

Varying network structures were implemented, and multiple simulation
parameters were considered. The metrics measured for coverage optimization
include reference signal received power (RSRP) and SINR. In LTE networks, RSRP
is used to measure the UE’s received signal strength over the common reference
signal, with a fixed threshold of (RSRP $>-115$ dBm) to measure the network’s
coverage quality. SINR measures the the UE signal quality, and a threshold of
(SINR $>-6.5$dB) was set to measure the outage probability.

In addition, the metrics describing the macroscopic scenario simulation
include the location, antennas and subscribers/UEs. An outdoor scenario of the
size ($7*13$) km was created in the city center of Berlin, Germany. 3D
building data was taken from the actual city data, and realistic antenna
deployment was implemented across the city based on an actual network antenna
density distribution.

Macroscopic antennas were deployed within the scenario, where the number of
antennas is dependent on their frequencies; the higher the frequency, the more
antennas were inserted. Multiple antenna deployment were tested to maintain a
threshold of $RSRP_{ue}<-115$ dBm and ensure an efficient antenna deployment
from an energy perspective. The antenna layout was fixed to a $3$-sector
antenna per site with azimuth angles of (0, 120, 240)°. Additionally,
different mechanical tilts of (2, 4, 6)° for the antenna main lobe were tested
following the antenna deployment procedure. The metrics were measured, and the
deployment was calibrated based on the results. Furthermore, UEs were deployed
homogeneously across the map to simulate outdoor subscribers.

(a) 700 MHz

(b) 2.6 GHz

Figure 4: Antennas distributed based on a realistic antenna distribution in
downtown Berlin.

To obtain sufficient data from the simulation, multiple scenarios with two
different frequencies, i.e., $700$ MHz and $2.6$ GHz, were tested. Table II
shows the evaluation parameters for the Berlin scenario. For $700$ MHz band
(see Figure 4(a)), $13$ outdoor BS were deployed since lower frequencies
travel greater distances and provide higher coverage, which is reflected in
Figure 5. Furthermore, it was observed that a mechanical tilt of $2^{\circ}$
provides the best results. Therefore, the RSRP values at this mechanical tilt
were selected.

Parameters |  Values
---|---
Carrier Frequency |  700 MHz, 2.6 GHz
Antenna Type |  Directional Antenna
Antenna Gain (dBi) |  8 for 700 MHz, 18.6 for 2.6 GHz
Cell Layout |  700 MHz - 13 sites, 2.6 GHz - 42 sites; 3 sectors per site
Bandwidth (MHz) |  10 for 700 MHz, 100 for 2.6 GHz
Antenna Diagram |  3GPP specified for 700 MHz, Commscope-HWXX-6516DS-VTM2600 for 2.6 GHz
Antenna Transmit Power (dBm) |  46
Tilt Type |  Mechanical Tilt (2, 4, 6)°
Effective Isotropic Radiated Power (dBm) |  63.5
BS Antenna Height (m) |  Different Heights depending on building data with $\max_{h}=59,356$ and $\min_{h}=5$
Number of UEs |  195
UE Transmit Power (dBm) |  23
UE Distribution |  Homogeneous
UE Height (m) |  1.5
UE Type & Mobility |  Pedestrian - Stationary
TABLE II: 700 MHz and 2.6 GHz simulation parameters.

(a) Reference Signal Received Power

(b) Signal to Interference and Noise Ratio

Figure 5: Prediction of RSRP and SINR with 13 BS distribution scenario in the
700 MHz frequency.

Similarly, for the 2.6 GHz band, the number of outdoor sector antennas was
increased to $42$ (see Figure 4(b)) to ensure a higher coverage, which is
reflected in Figure 6. In addition, Figure 7 shows the CDF of the RSRP values
at different mechanical tilts, where, mechanical tilt = $2^{\circ}$ provided
the best results and, therefore, the RSRP values were selected based on this
tilt.

(a) Reference Signal Received Power

(b) Signal to Interference & Noise Ratio

Figure 6: Prediction of RSRP and SINR with 42 BS distribution scenario in the
2.6 GHz frequency. Figure 7: CDF of the 2.6 GHz band at three different
mechanical tilts.

###  IV-B Convergence of the Proposed Distributed Algorithm

In Section III-B2, we mention that the convergence of the proposed distributed
Algorithm 7 is related to the power constraints. Figure 8 verifies this
statement, showing the number of iterations required for convergence. The
y-axis represents the power consumed by the BS. The two curves can be
distinguished based on the initial power assignment, where maximum power
initialization represents that the maximum available transmit power is equally
split among each assigned UE and, with zero power initialization each UE
assigned to the BS is initialized with zero transmit power.

We observe that the algorithm converges within $6$ or $7$ iterations in this
case, depending on the targeted accuracy level. However, with a slightly
relaxed accuracy level, the algorithm can converge in $3$ or $4$ iterations,
as inferred from the figure.

Figure 8: Convergence of the proposed distributed Algorithm 7.

###  IV-C Simulation Results - Individual Bands

In this section, we present the simulation results for the setup explained in
Section IV-A, considering one active band and $195$ UEs. Additionally, we
assume that a $100$ Megabit file is to be transferred, and the maximum number
of available antennas are used per BS. Furthermore, each UE has the same
minimum rate requirement.

$D_{0}$ $D_{1}-700$ MHz $D_{1}-2.6$ GHz $P_{FIX}$ $P_{SYNC}$ $\eta_{PA}$ 4.49
W 0.00312 W 0.01560 W 300.0 W 34.0 W 0.48

TABLE III: Parameters for the computation of energy consumption.

Tables III and IV display the value of the parameters used throughout the
simulations. In Table IV, $P^{max}$ represents the maximum transmit power per
BS, $M$ denotes the number of BSs, and $RL$ and $SL$ are the abbreviations
used to represent the location of the BSs. Here $RL$ denotes the refreshed
(different) locations, while $SL$ means same locations999Please note that the
locations of BSs in $2.6$ GHz band were fixed, while the locations of BSs in
$700$ MHz band were changed. Therefore, the locations of BSs in $700$ MHz are
relative to the locations of BSs in $2.6$ GHz band. In this context,
’refreshed location’ indicates that BS locations in the 700 MHz band differ
from those in the 2.6 GHz band..

${A}_{m}^{b}$ $N_{m}^{b}$ $P^{max}$ $\bar{b}$ M - RL M - SL 700 MHz 4 100 200
W 180 KHz 39 126 2.6 GHz 64 273 120 W 360 KHz 126 126

TABLE IV: Parameters highlighting the maximum number of available resources.

####  IV-C1 Simulation Results for $700$ MHz Band

Figure 9 illustrates the distribution of UEs among the BSs using the greedy
approach described in Algorithm 1. About $10\%$ of the BSs have no UEs
assigned when the BSs in $700$ MHz band are located differently (RL), while
more than $60\%$ of the BSs have no UEs assigned when BSs in $700$ MHz band
are located at the same positions (SL). This is due to the variation in the
number of BSs in the $700$ MHz band in the two scenarios. Figure 10(a) further
illustrates the impact of having fewer BSs. A lower total number of BSs in the
network results in more outages due to highly congested BSs, leading to a
lower EE.

Figure 9: CDF of number of active UEs in 700 MHz: RL - different BS location
and SL - same BS location relative to BSs in 2.6 GHz band.

(a) Overall outages.

(b) Outages for low data rates.

Figure 10:  Outages over different rate requirements in 700 MHz band: RL -
different BS location and SL - same BS location relative to BSs in 2.6 GHz
band.

(a) Overall EE of the network.

(b) EE of the network for low data rates.

Figure 11:  EE in $700$ MHz band: RL - different BS location and SL - same BS
location relative to BSs in 2.6 GHz band.

Figure 11(a) shows the EE of the network. Firstly, it is evident that
spatially multiplexing UEs achieve higher EE for large data rates, while a
single spatial layer results in higher EE for small data rates. Such a
behavior results from the difference in the number of outages between SM and
NSM (see Figure 10(a)) and also aligns with the conclusions mentioned in [1].
Secondly, low EE is achieved for lower rates, which increases as the rates
increase. This results from the longer time required to transmit $100$
Megabits of data for lower data rates, leading to higher power consumption
and, consequently, reduced EE. Finally, as depicted in Figure 10(a), an
increase in the number of outages is observed for higher rates. Since the sum
rate computation does not consider the impact of outages, the total EE should
theoretically be zero in the case of a $100\%$ outage. Such a behavior can be
observed for NSM (with BSs in RL) when the data rate is greater than $35$
Mbps. Furthermore, for NSM, when the number of UEs to be assigned to BSs is
much larger than the number of available BSs and each UE requires a large
rate, a higher percentage of outages are observed because the limited
available PRBs- to be distributed among the UEs- are insufficient to satisfy
the minimum rate requirement. Thus, indicating that higher rates cannot be
supported in $700$ MHz frequency band.

####  IV-C2 Simulation Results for $2.6$ GHz Band

Figure 12(a) shows the distribution of UEs over BSs and Table V explains the
meaning of the labels in the legend. It can be seen that more than $60\%$ of
the BSs are assigned no UEs in all assignment techniques. Since BSs with only
one UE are turned-off in the Rematched 1 assignment, BSs are always assigned
two or more UEs. Similarly, three or more UEs are assigned to a BS via
Rematched 2 assignment.

Baseline Greedy approach Rematched 1 BSs with 1 UE switched-off Rematched 2
BSs with 1 and 2 UEs switched-off Threshold 1 and 2 BSs switched-off based on
a threshold value

TABLE V: Naming of UE to BS assignment techniques.

(a) CDF of number of active UEs in 2.6 GHz.

(b) Outages over different rates in 2.6 GHz band.

Figure 12: UE distribution over BSs and outages in $2.6$ GHz band.

Figure 13 shows the changes EE over different rates. Firstly, note that higher
rates can be supported in $2.6$ GHz band. Secondly, the EE improves as the
rates increase, resulting from less amount of time required to transmit $100$
Mbits of data and lower number of outages (see Figure 12(b)). It is apparent
that SM significantly outperforms NSM, achieving approximately $3-5$ times
larger EE for larger data rates, where, even the benchmark scheme outperforms
NSM. Thirdly, among the threshold based UE to BS assignment techniques, it is
observed that a higher threshold value, i.e. $20$, results in a higher EE for
SM compared to a lower threshold. For NSM, a lower threshold value, i.e.
$0.5$, yields a higher EE, thus, indicating that the choice of the threshold
depends on the multiplexing scenario. Finally, via the proposed threshold
based re-matching algorithm, it is clearly evident that simply switching-off
BSs with one and two UEs is not sufficient to yield large EE. A
straightforward switch-off of BSs with an arbitrary number of UEs (in these
simulations BSs with one and two UEs) can lead to the assignment of UEs to BSs
with weak channel gain, thus, resulting in more outages and a lower EE.
Therefore, to achieve the highest EE, UEs should be spatially multiplexed and
re-matched to the BSs using a carefully selected threshold value.

(a) Overall EE of the network.

(b) EE of the network for smaller rates.

Figure 13: Total EE of the Network in $2.6$ GHz band.

Figures 14(a) and 14(b) show the impact of the number of active antennas/RF
chains on the EE and the percentage of outages of the network, respectively,
when the UEs are spatially multiplexed. It can be observed that the number of
active antennas/RF chains affects both the EE and the percentage of outages. A
lower number of active antennas/RF chains lead to a lower EE because the
number of outages are large. As the number of active antennas/RF chains
increases, the number of outages decrease and the EE improves, where
proportionality between number of active antennas/RF chains and number of
outages is observed. However, if the number of active antennas/RF chains grows
really large (exemplified by the case with $64$ antennas) the load-independent
power consumption begins to dominate, resulting in a lower EE. Furthermore, it
should be noted that for each active antenna/RF chain value in Figutre 14(a),
the UEs are assigned all available PRBs. The results from Figure 14(a) suggest
that from an EE perspective it is better to assign all PRBs before turning on
additional antennas/RF chains.

(a) Overall EE of the network.

(b) Overall outages of the network.

Figure 14: Total EE and outages of the network in $2.6$ GHz band.

###  IV-D HetNet Simulation Results - Multiple Bands

Shown below are the simulation results for the setup explained in Section IV-A
for two active bands with a total of $195$ UEs. We assume that a $100$
Megabits of data is to be transferred, maximum number of available antennas
are used per BS and each UE has the same rate requirement. Furthermore, Tables
III and IV show the system parameters used for the simulations.

(a) CDF of number of active UEs in $700$ MHz band.

(b) CDF of number of active UEs in $2.6$ GHz band.

Figure 15: Distribution of UEs in a HetNet scenario for different UE to BS
matching techniques.

(a) Outages in $700$ MHz band.

(b) Outages in $2.6$ GHz band.

Figure 16: Outages in a HetNet scenario for different data rates.

Figures 15(a) and 15(b) show the distribution of the UEs over the BSs based on
the UE to BS assignment algorithms described in Section III-A. It can be
observed that the (greedy) UE to BS assignment rule in (16) leads to a
sparsely filled $2.6$ GHz frequency band and a crowded $700$ MHz frequency
band. However, as discussed in Section IV-C and depicted in Figures 16(a) and
16(b), a crowded $700$ MHz band leads to higher outages resulting in lower EE.
In addition, it is infeasible to assign more than three UEs per PRB when the
UEs are spatially multiplexed101010For SM, zero-forcing cannot be applied if
the BSs are assigned more than $3$ UEs per PRB due to limited number of
antennas. and high rates cannot be supported. Therefore, a bias of $35$ dB is
injected to improve the UE to BS assignment111111However, note that the PRB
assignment and power control is performed on the true large-scale fading
values. such that we obtain a sparsely filled $700$ MHz band and a crowded
$2.6$ GHz band (Figures 15(a) and 15(b)) via the distribution of UEs over BSs
and through Table VI via the number of UEs assigned in each band.

UE Assignment Algorithm No. of UEs in 700 MHz No. of UEs in 2.6 GHz Basline
(Greedy) 195 0 Biased 12 183 Biased Rematched 1 5 190 Biased Rematched 2 3 192
Biased Threshold (val = 1) 16 179 Biased Threshold (val = 15) 18 177

TABLE VI: Total number of UEs assigned to BSs in $700$ MHz and $2.6$ GHz
bands.

(a) Overall EE of the network.

(b) EE of the network for smaller rates.

Figure 17: Total EE of the HetNet for different UE to BS assignment
algorithms.

Figure 17 shows the EE of the HetNet network121212Note that the rates shown on
the x-axis have the following format: (Rate in $700$ MHz band + Rate in $2.6$
GHz band). For example, all UEs in $700$ MHz band having a rate of $1$ Mbps
and all UEs in $2.6$ GHz band having a rate of $10$ Mbps is represented by
$(1+10)$ Mbps.. It can be observed that SM significantly outperforms NSM. In
addition, off-loading the UEs to a higher frequency band by adding a bias
yields much larger EE gains relative to the greedy baseline matching.
Furthermore, for SM, switching-off BSs help lower the power consumption,
thereby, increasing the EE of the network. Among the re-matching techniques,
for lower rates, a simple switch-off of BSs with low EE (in these simulation
BSs with one and two UEs were switched-off) provides a higher EE, whereas, for
larger data rates, a threshold-based reassignment of the UEs yields a higher
EE. In contrast, a straightforward switch-off of BSs for NSM leads to much
larger outages (Figure 16) and lower EE because the available PRBs-
distributed among the UEs- are insufficient to support the rate requirements.
Overall, to achieve high EE the UEs should be spatially multiplexed and each
UE should be assigned maximum available PRBs. Depending on the required data
rate, either BSs with one and two UEs assigned should be switched-off or a
threshold-based reassignment of the UEs to BSs should be adopted. Furthermore,
lower frequency band should be sparsely filled, while higher frequency bands
should be assigned more UEs.

##  V Conclusion and Future Work

In this work, we derive novel algorithms for energy efficient UE to BS
assignment and propose a re-matching algorithm to switch-off BSs with low EE
for massive MIMO HetNets. In addition, we propose a PRB assignment and power
control algorithm, which can be implemented in a distributed way in a multi-
cell network. Our analysis and simulations suggest the following: _i)_ $700$
MHz band alone should only be used for very low data rates (for example
control signaling, RRM). It cannot support higher rates even for a small
number of UEs. The low frequency band, without having a higher frequency band,
is inadequate to support large data rates and number of UEs, _ii)_ $2.6$ GHz
band alone should be used to support UEs with large rates, where the UEs
should be spatially multiplexed, each UE should be assigned the maximum
available PRBs and a threshold-based reassignment of the UEs to BSs should be
adopted to achieve higher EE. In addition, a dense deployment of BSs is a
necessary condition to enable the advantages of the $2.6$ GHz described above,
and _iii)_ the combined $700$ MHz and $2.6$ GHz band performance is worse than
$2.6$ GHz band alone, because UEs assigned to the $700$ MHz band are not
flexible and achieve lower rates. Overall, it is recommended that if a BS has
multiple antennas, spatial multiplexing should be always used and each UE
should be assigned the maximum available PRBs (also for low load and a low
number of UEs).

For future work, numerical assessments with larger number of UEs and higher
loads should be performed. We anticipate that NSM might be insufficient to
support a large number of UEs given its limited resources. Furthermore, the
proposed re-matching and bias UE assignments might need to be reevaluated to
accommodate a larger variety of data rates. While considering the bias UE
assignment, it’s possible to formulate a large-scale fading bias model to
optimize the large-scale fading bias value per BS, as demonstrated in [46].
Additionally, the number of active antennas per BS can also be optimized.

Another intriguing avenue for future research is understanding the impact of
the latest developments in BS hardware on performance and investigating the
optimality of the proposed algorithms under these new settings. This
exploration can be conducted both numerically and analytically based on the
derived rate and energy consumption expressions.

##  Appendix A Proof Justifying the replacement of $\underline{R}_{k}$ with
$R_{k}$ in (8)

In the following, we justify the replacement of $\underline{R}_{k}$ with
$R_{k}$ by analysing the KKT conditions for a single cell and extending the
results to the multi-cell scenario.

We begin by analysing the total power consumption (P) for a single cell, which
with equal power allocated across all the PRBs assigned to a UE associated
with this cell, can be written as

$\displaystyle
P=\frac{1}{\eta_{PA}}\sum_{k}N_{k}\,p_{k}+I\left(\frac{1}{\lambda_{0}}P_{FIX}+\frac{1}{\lambda_{1}}P_{SYNC}\right)+D_{0}a+\left(D_{1}a\left(\sum_{k}N_{k}\right)\right)+C.$
(22)

Note that only the first term in (22) depends on the transmit power allocated
to the UEs in this cell. Therefore, for simplicity, we drop the remaining
terms. Under the assumption that the UE to BS assignment has been fixed, the
total power minimization can be defined as

$\displaystyle\min_{\boldsymbol{p}\geq 0}$
$\displaystyle\sum_{k=1}^{K}N_{k}p_{k}$ (23a) s.t.
$\displaystyle\sum_{k=1}^{K}p_{k}\leq P^{\max}$ (23b)
$\displaystyle(\bar{b}\cdot
N_{k})\log_{2}(1+(M-K)p_{k}\tilde{\beta_{k}})\geq\underline{R}_{k}\;,\forall
k=1,\dots,K,$ (23c)

where $N_{K}$ is the number of PRBs assigned to a UE, $\tilde{\beta_{k}}$
represents the large scale fading normalized by the noise power, $M$ is the
number of antennas, and $K$ is the number of multiplexed UEs. The minimum rate
constraint (23c) can be rewritten as
$p_{k}\geq\left(\frac{2^{\frac{\underline{R}_{k}}{(\bar{b}\cdot
N_{k})}}-1}{\tilde{\beta_{k}}(M-K)}\right),\forall k=1,\dots,K$. Let us denote
$P^{\min}_{k}=\left(\frac{2^{\frac{\underline{R}_{k}}{(\bar{b}\cdot
N_{k})}}-1}{\tilde{\beta_{k}}(M-K)}\right),~{}\forall k=1,\dots,K$. Then,
Problem (23) can be reformulated as

$\displaystyle\min_{\boldsymbol{p}\geq 0}$
$\displaystyle\sum_{k=1}^{K}N_{k}p_{k}$ (24a) s.t.
$\displaystyle\sum_{k=1}^{K}p_{k}\leq P^{\max}$ (24b) $\displaystyle p_{k}\geq
P^{\min}_{k},\forall k=1,\dots,K$ (24c)

Problem (24) is convex in $\boldsymbol{p}$ with an affine feasible set. It can
be shown by analyzing the KKT conditions that in the power minimization
problem, the allocated power to each UE is assigned to maintain its minimum
rate demand. The Slater’s condition holds in (24) since it is convex and there
exists $\boldsymbol{p}\geq 0$ satisfying (24b) and (24c) with strict
inequalities. Therefore, the strong duality in (24) holds. Hence, the KKT
conditions are satisfied and the optimal solution $\boldsymbol{p}^{*}$ can be
obtained using the Lagrange dual method [50]. The Lagrange function (lower-
bound) of (24) is given by

$L(\boldsymbol{p},\boldsymbol{\mu},\boldsymbol{\delta},\nu)=\sum\limits_{k=1}^{K}N_{k}p_{k}+\sum\limits_{k=1}^{K}\mu_{k}\left(P^{\min}_{k}-p_{k}\right)+\sum\limits_{k=1}^{K}\delta_{k}(-p_{k})+\nu\left(\sum\limits_{i=k}^{K}p_{k}-P^{\text{max}}\right),$
(25)

where $\boldsymbol{\mu}=[\mu_{1},\dots,\mu_{K}]$,
$\boldsymbol{\delta}=[\delta_{1},\dots,\delta_{K}]$, and $\nu$ are the
Lagrangian multipliers corresponding to the constraints (24c), (24b), and
$p_{k}\geq 0,~{}k=1,\dots,K$, respectively. The Lagrange dual problem is given
by

$\displaystyle\max_{\boldsymbol{\mu},\boldsymbol{\delta},\nu}~{}~{}$
$\displaystyle\inf\limits_{\boldsymbol{p}}\left\\{L(\boldsymbol{p},\boldsymbol{\mu},\boldsymbol{\delta},\nu)\right\\}$
s.t. $\displaystyle\mu_{k}\geq 0,~{}\forall k=1,\dots,K,$
$\displaystyle\delta_{k}\geq 0,~{}\forall i=k,\dots,K.$

The KKT conditions are listed below.

  1. 1.

Feasibility of the primal problem (24):

$\textbf{C-1.1:}~{}p^{*}_{k}\geq P^{\min}_{k}\;,\forall
k,~{}~{}~{}~{}~{}\textbf{C-1.2:}~{}p^{*}_{k}\geq 0,~{}\forall
k,~{}~{}~{}~{}~{}\textbf{C-1.3:}~{}\sum\limits_{k=1}^{K}p^{*}_{k}\leq
P^{\text{max}}.$

  2. 2.

Feasibility of the dual problem:

$\textbf{C-2.1:}~{}\mu^{*}_{k}\geq 0,~{}\forall
k=1,\dots,K,~{}~{}~{}~{}~{}~{}\textbf{C-2.2:}~{}\delta^{*}_{k}\geq
0,~{}\forall k=1,\dots,K,~{}~{}~{}~{}~{}~{}\textbf{C-2.3:}~{}\nu^{*}\geq 0.$

  3. 3.

The complementary slackness conditions:

$\textbf{C-3.1:}~{}\mu^{*}_{k}\left(P^{\min}_{k}-p^{*}_{k}\right)=0,\forall
i=1,\dots,K,$ $\textbf{C-3.2:}~{}\delta^{*}_{k}p^{*}_{k}=0,~{}\forall
i=1,\dots,K,~{}~{}~{}~{}~{}~{}~{}~{}~{}\textbf{C-3.3:}~{}\nu^{*}\left(\sum\limits_{i=k}^{K}p^{*}_{k}-P^{\text{max}}\right)=0.$

  4. 4.

The condition
$\nabla_{\boldsymbol{p}^{*}}L(\boldsymbol{p}^{*},\boldsymbol{\mu}^{*},\boldsymbol{\delta}^{*},\nu^{*})=0$,
which implies that

$\textbf{C-4:}~{}\frac{\partial L}{\partial
p^{*}_{k}}=N_{k}-\mu_{k}^{*}+\delta^{*}_{k}+\nu^{*}=0,~{}\forall k=1,\dots,K.$

The primal dual $\delta^{*}_{k},~{}\forall k=1,\dots,K,$ acts as a slack
variable in C-4 (due to the KKT condition C-2.2), so it can be eliminated by
reformulating the KKT conditions (C-4,C-2.2) and C-3.2 respectively as

$\nu^{*}\geq\mu_{k}^{*}-N_{k},~{}\forall k=1,\dots,K,$ (27)

and

$p^{*}_{k}\left(\nu^{*}-\left(\mu_{k}^{*}-N_{k}\right)\right)=0,~{}\forall
k=1,\dots,K.$ (28)

If problem (24) is feasible with $\sum_{k=1}^{K}p_{k}<P^{\max}$, then the
optimal solution satisfies $\sum_{k=1}^{K}p^{*}_{k}<P^{\max}$, meaning that
$\nu^{*}=0$, due to condition C-3.3. In this case,
$\mu_{k}^{*}-N_{k}=0,~{}\forall k$, meaning that $\mu_{k}^{*}=N_{k},~{}\forall
k$, so $\mu_{k}^{*}>0,~{}\forall k$ as $N_{k}>0$. According to condition
C-3.1, we conclude that $P^{\min}_{k}-p^{*}_{k}=0,~{}\forall k$, thus
$p^{*}_{k}=P^{\min}_{k}~{}\forall k$, which is equivalent to $(\bar{b}\cdot
N_{k})\log_{2}(1+(M-K)p_{k}\tilde{\beta_{k}})=\underline{R}_{k},~{}\forall k$.
Therefore, replacing $\underline{R}_{k}$ with $R_{k}$ is justifiable. In the
case $\underline{R}_{k}=0$, no transmit power is allocated to the UE, i.e.
$p_{k}^{*}=0$.

The analysis above can be easily extended to the multi-cell scenario
considering fixed inter-cell interference in each iteration of the Algorithm
7, where
$P^{\min}_{k}=\left(\frac{(2^{\frac{\underline{R}_{k}}{(\bar{b}N_{k})}}-1)(I_{k}+\sigma_{k})}{{\beta_{k}}(M-K)}\right),\forall
k$.

##  Appendix B Proof of Theorem 7

We illustrate the proof via the considered setup in Figure 18, where BS
$m_{1}$ serves few users $k_{1},\dots,k_{3}$, whereas, BS $m_{2}$ serves users
$k_{4},\dots,k_{6}$. We aim at showing that switching-off BS $m_{1}$ and
assigning users $k_{1},\dots,k_{3}$ to BS $m_{2}$, given that
$k_{1},\dots,k_{3}$ has the next best channel with BS $m_{2}$, is energy
optimal under certain conditions (see (43) and (64)) and the assumption of
homogeneous BS with same load-independent energy consumption model131313Note
that the proof can be easily extended to heterogeneous BSs with different
load-independent energy consumption model., and with same PRB allocation and
power control strategy.

Figure 18: Example scenario to illustrate the proof of Theorem 7.

The proof is shown separately for SM and NSM as follows:

###### Proof.

_i)_ Spatial Multiplexing: As depicted in Figure 18, consider BS $m_{1}$ that
has only a few users $\\{k_{1},...,k_{\ell}\\}$ assigned, facing interferences
$I_{k_{1}},...,I_{k_{\ell}}$ from BS $m_{2}$. For the given minimum rate
requirements $\underline{R}_{k_{1}},...,\underline{R}_{k_{\ell}}$, the
required transmit power for user $k_{l}$ with PRBs $\alpha_{l}$ in BS $m_{1}$
can be computed as

$p_{k_{l},\alpha_{l}}=\frac{(2^{\underline{R^{\prime}_{k_{l}}}}-1)}{a-\ell}\frac{(I_{k_{l}}+N)}{\beta_{k_{l},\alpha_{l}}},$
(29)

for $1\leq l\leq\ell$ with
$\underline{R^{\prime}_{k_{l}}}=\frac{\underline{R_{k_{l}}}}{\bar{b}\cdot
N_{m}}$ and maximum available PRBs $N_{m}$. The total power consumption of BS
$m_{1}$, including the load independent power ($P_{LI}$), is calculated as

$P_{m_{1}}=\frac{N_{m}}{\eta_{PA}}\sum_{l=1}^{\ell}p_{k_{l},\alpha_{l}}+P_{LI}.$
(30)

Denote $m_{2}(l)$ as the second best channel
$\beta_{k_{l},\alpha^{\prime}_{l}}$ for user $k_{l}$. If all users
$k_{1},...,k_{\ell}$ are assigned to their second best BSs
$m_{2}(1),...,m_{2}(\ell)$ then $P_{m_{1}}$ amount of power can be saved.
However, to support user $k_{1},...,k_{\ell}$ by BS
$m_{2}(1),...,m_{2}(\ell)$, additional power is needed because _a)_ users
$k_{1},...,k_{\ell}$ needs to be served and _b)_ users served by
$m_{2}(1),...,m_{2}(\ell)$ need more power.

Let us focus on the re-matching of user $l$. Assuming $K_{2}(l)>0$ users,
$\mu(m_{2}(l))=\\{k_{l1},\dots,k_{lK_{2}}\\}$, are already assigned to
$m_{2}(l)$, when user $k_{l}$ joins BS $m_{2}(l)$ with PRBs
$\alpha^{\prime}_{l}$ assigned, the required transmit power can be computed
as,

$p_{k_{l},\alpha^{\prime}_{l}}=\frac{(2^{\underline{R^{\prime}_{l}}}-1)}{a-K_{2}(l)-1}\frac{(\tilde{I}_{k_{l}}+N)}{\beta_{k_{l},\alpha^{\prime}_{l}}}$
(31)

where, it is observed that $\tilde{I}_{k_{l}}<I_{k_{l}}$ and
$\beta_{k_{l},\alpha^{\prime}_{l}}\leq\beta_{k_{l},\alpha_{l}}$, and the
difference between the transmit powers before and after user $k_{l}$ joins BS
$m_{2}(l)$ can be computed as

$\Delta_{k_{2l}}=p^{\prime}_{k_{2l},\alpha^{\prime}_{2l}}-p_{k_{2l},\alpha^{\prime}_{2l}},$
(32)

where $p_{k_{2l},\alpha^{\prime}_{2l}}$ is the required transmit power before
user $k_{l}$ joins BS $m_{2}(l)$ and is computed as

$p_{k_{2l},\alpha^{\prime}_{2l}}=\frac{(2^{\underline{R^{\prime}_{k_{2l}}}}-1)}{a-K_{2}(l)}\frac{(I_{k_{2l}}+N)}{\beta_{k_{2l}},\alpha^{\prime}_{2l}},$
(33)

and $p^{\prime}_{k_{2l},\alpha^{\prime}_{2l}}$ is the required transmit power
after user $k_{l}$ joins BS $m_{2}(l)$ and is computed as

$p^{\prime}_{k_{2l},\alpha^{\prime}_{2l}}=\frac{(2^{\underline{R^{\prime}_{k_{2l}}}}-1)}{a-K_{2}(l)-1}\frac{(\tilde{I}_{k_{2l}}+N)}{\beta_{k_{2l},\alpha^{\prime}_{2l}}}.$
(34)

Then,

$\Delta_{k_{2l}}=\frac{(2^{\underline{R^{\prime}_{k_{2l}}}}-1)}{\beta_{k_{2l},\alpha^{\prime}_{2l}}}\left(\frac{\tilde{I}_{k_{2l}}+N}{a-K_{2}(l)-1}-\frac{I_{k_{2l}}+N}{a-K_{2}(l)}\right),$
(35)

and the total difference in power accounting for all the users assigned to BS
$m_{2}(l)$ can be calculated as

$\sum_{k\in\mu(m_{2}(l))}\Delta_{k}+\Delta_{k_{l}}$ (36)

with

$\Delta_{k_{l}}=(2^{\underline{R^{\prime}_{k_{l}}}}-1)\left(\frac{\tilde{I}_{k_{l}}+N}{\beta_{k_{l},\alpha^{\prime}_{l}}}\frac{1}{a-K_{2}(l)-1}-\frac{I_{k_{l}}+N}{\beta_{k_{l},\alpha_{l}}}\frac{1}{a-\ell}\right).$
(37)

Assuming the load independent power consumption for all involved BSs is the
same, the difference of power consumption before and after re-matching can be
obtained and compared to zero as

$\displaystyle\sum_{l=1}^{\ell}\sum_{k\in\mu(m_{2}(l))}\Delta_{k}+\Delta_{k_{l}}-P_{LI}$
$\displaystyle\leq 0,$ (38) $\displaystyle\implies$
$\displaystyle\sum_{l=1}^{\ell}\Delta_{k_{2}(l)}+(2^{\underline{R^{\prime}_{k_{l}}}}-1)\left(\frac{\tilde{I}_{k_{l}}+N}{\beta_{k_{l},\alpha^{\prime}_{l}}}\frac{1}{a-K_{2}(l)-1}-\frac{I_{k_{l}}+N}{\beta_{k_{l},\alpha_{l}}}\frac{1}{a-\ell}\right)-P_{LI}$
$\displaystyle\leq 0,$ (39) $\displaystyle\implies$
$\displaystyle\sum_{l=1}^{\ell}\frac{\tilde{I}_{k_{l}}+N}{a-K_{2}(l)-1}\beta_{k_{l},\alpha^{\prime}_{l}}^{-1}-\frac{I_{k_{l}}+N}{a-\ell}\beta_{k_{l},\alpha_{l}}^{-1}$
$\displaystyle\leq\sum_{l=1}^{\ell}\frac{-\Delta_{k_{2}(l)}}{2^{\underline{R^{\prime}_{k_{l}}}}-1}+P_{LI},$
(40) $\displaystyle\implies$
$\displaystyle\sum_{l=1}^{\ell}\frac{(a-\ell)(\tilde{I}_{k_{l}}+N)\beta_{k_{l},\alpha^{\prime}_{l}}^{-1}-(a-K_{2}(l)-1)(I_{k_{l}}+N)\beta_{k_{l},\alpha_{l}}^{-1}}{(a-K_{2}(l)-1)(a-\ell)}$
$\displaystyle\leq\sum_{l=1}^{\ell}\frac{-\Delta_{k_{2}(l)}}{2^{\underline{R^{\prime}_{k_{l}}}}-1}+P_{LI},$
(41)

where $\Delta_{k_{2}(l)}=\sum_{k\in\mu(m_{2}(l))}\Delta_{k}$. Let
$\gamma_{1l}=(a-K_{2}(l)-1)(I_{k_{l}}+N)$,
$\gamma_{2l}=(a-\ell)(\tilde{I}_{k_{l}}+N)$ and

$\Delta_{l}=\sum_{l=1}^{\ell}\frac{(-\Delta_{k_{2}(l)})}{(2^{\underline{R^{\prime}_{k_{l}}}}-1)\gamma_{1l}}(a-K_{2}(l)-1)(a-\ell)+P_{LI}\sum_{l=1}^{\ell}\frac{(a-K_{2}(l)-1)(a-\ell)}{\gamma_{1l}}.$
(42)

Then the following inequalities for $1\leq l\leq\ell$ are sufficient for (41)

$\frac{\gamma_{2l}}{\gamma_{1l}}\beta_{k_{l},\alpha^{\prime}_{l}}^{-1}\leq\beta_{k_{l},\alpha_{l}}^{-1}+\Delta_{l}.$
(43)

∎

The following remarks hold for all $\frac{\gamma_{2l}}{\gamma_{1l}}$ with
$1\leq l\leq\ell$, however, we only state it for a single user.

######  Remark B.1.

From (44), it can be inferred that at least $a>K_{2}+1$ antennas are required.

$\frac{\gamma_{21}}{\gamma_{11}}=\frac{a-1}{a-K_{2}-1}\frac{\tilde{I}_{k_{1}}+N}{I_{k_{1}}+N}$
(44)

Let $a=K_{2}+N$, where $N>1$, then, $\frac{\gamma_{21}}{\gamma_{11}}$ can be
written as,

$\frac{\gamma_{21}}{\gamma_{11}}=\left(\frac{K_{2}}{N-1}+1\right)\left(\frac{\tilde{I}_{k_{1}}+N}{I_{k_{1}}+N}\right).$
(45)

Since, $I_{k_{1}}>\tilde{I}_{k_{1}}$, the ratio of the interference is
bounded, i.e, $0<\left(\frac{\tilde{I}_{k_{1}}+N}{I_{k_{1}}+N}\right)\leq 1$,
and therefore, $\frac{\gamma_{21}}{\gamma_{11}}$ is bounded, i.e.,

$0\leq\frac{\gamma_{21}}{\gamma_{11}}\leq\left(\frac{K_{2}}{N-1}+1\right).$
(46)

######  Remark B.2.

Under the assumption that
$\beta_{k_{1},\alpha^{\prime}}\leq\beta_{k_{1},\alpha}$, it can be observed in
(43) that $\beta_{k_{1},\alpha^{\prime}}^{-1}\geq\beta_{k_{1},\alpha}^{-1}$.
Thus, for (43) to hold, $\Delta$ in (43) must be positive, i.e,

$\displaystyle\Delta$ $\displaystyle>$ $\displaystyle 0$ (47)
$\displaystyle\frac{(-\Delta_{k_{2}})}{(2^{\underline{R^{\prime}_{1}}}-1)\gamma_{11}}(a-K_{2}-1)(a-1)+P_{LI}\frac{(a-K_{2}-1)(a-1)}{\gamma_{11}}$
$\displaystyle>$ $\displaystyle 0.$ (48)

From (48), it can be inferred that

$\Delta_{k_{2}}<P_{LI}(2^{\underline{R^{\prime}_{1}}}-1).$ (49)

###### Proof.

_ii)_ No Spatial Multiplexing: Similarly, for the given minimum rate
requirements $\underline{R}_{k_{l}}$, the required transmit power for user
$k_{l}$ with PRBs $\alpha_{l}$ in BS $m_{1}$ can be computed as

$p_{k_{l},\alpha_{l}}=\frac{(2^{\underline{R^{\prime}_{l}}}-1)}{a-1}\frac{(I_{k_{l}}+N)}{\beta_{k_{l},\alpha_{l}}},$
(50)

with $\underline{R^{\prime}_{k_{l}}}=\frac{\underline{R_{k_{l}}}}{\bar{b}\cdot
N_{m}}$ and maximum available PRBs $N_{m}$. The total power consumption of BS
$m_{1}$, including the load independent power ($P_{LI}$), is calculated as

$P_{m_{1}}=\frac{N_{m}}{\eta_{PA}}\sum_{l=1}^{\ell}p_{k_{l},\alpha_{l}}+P_{LI}.$
(51)

Under the assumption that BS $m_{2}(l)$ has the second best channel
$\beta_{k_{l},\alpha^{\prime}_{l}}$ for user $k_{l}$, if all users
$\\{k_{1},...,k_{\ell}\\}$ are assigned to their second best BS
$m_{2}(1),...,m_{2}(\ell)$, $P_{m_{1}}$ amount of power/energy can be saved.
However, to support user $k_{1},...,k_{\ell}$ by BS
$m_{2}(1),...,m_{2}(\ell)$, additional power is needed because _a)_ users
$k_{1},...,k_{\ell}$ need to be served and _b)_ users served by
$m_{2}(1),...,m_{2}(\ell)$ need more power. Assuming $K_{2}(l)>0$ users,
$\mu(m_{2}(l))=\\{k_{l1},\dots,k_{lK_{2}}\\}$, are already assigned to
$m_{2}(l)$, when user $k_{l}$ joins BS $m_{2}(l)$ with PRBs
$\alpha^{\prime}_{l}$ assigned, the required transmit power can be computed as

$p_{k_{l},\alpha^{\prime}_{l}}=\frac{(2^{\underline{R^{\prime}_{k_{l}}}}-1)(\tilde{I}_{k_{l}}+N)}{\beta_{k_{l},\alpha^{\prime}_{l}}(a-1)},$
(52)

with
$\underline{R^{\prime}_{k_{l}}}=\frac{\underline{R_{k_{l}}}}{\bar{b}\cdot\alpha^{\prime}_{l}}$,
where $\alpha^{\prime}_{l}$ represents the number of PRBs assigned to user
$k_{l}$. It is observed that $\tilde{I}_{k_{l}}<I_{k_{l}}$ and
$\beta_{k_{l},\alpha^{\prime}_{l}}\leq\beta_{k_{l},\alpha_{l}}$, and the
difference between the transmit powers after and before user $k_{l}$ joins BS
$m_{2}(l)$ can be computed as

$\Delta_{k_{2l}}=p^{\prime}_{k_{2l},\alpha^{\prime}_{2l}}-p_{k_{2l},\alpha^{\prime}_{2l}},$
(53)

where $p_{m_{2}(l),k_{2l}}=p_{k_{2l},\alpha^{\prime}_{2l}}$ is the required
transmit power before user $k_{l}$ joins BS $m_{2}(l)$ and is computed as

$p_{k_{2l},\alpha^{\prime}_{2l}}=\frac{(2^{\underline{R^{\prime}_{k_{2l}}}}-1)(I_{k_{2l}}+N)}{\beta_{k_{2l},\alpha^{\prime}_{2l}}(a-1)},$
(54)

with
$\underline{R^{\prime}_{k_{2l}}}=\frac{\underline{R_{k_{2l}}}}{\bar{b}\cdot\alpha^{\prime}_{2l}}$
and $p^{\prime}_{m_{2}(l),k_{2l}}=p^{\prime}_{k_{2l},\alpha^{\prime}_{2l}}$ is
the required transmit power after user $k_{l}$ joins BS $m_{2}(l)$ and is
computed as

$p^{\prime}_{k_{2l},\alpha^{\prime}_{2l}}=\frac{(2^{\underline{R^{\prime\prime}_{k_{2l}}}}-1)(\tilde{I}_{k_{2l}}+N)}{\beta_{k_{2l},\alpha^{\prime}_{2l}}(a-1)},$
(55)

with
$\underline{R^{\prime\prime}_{k_{2l}}}=\frac{\underline{R_{k_{2l}}}}{\bar{b}\cdot\tilde{\alpha^{\prime}}_{2l}}$
Then,

$\displaystyle\Delta_{k_{2l}}=\frac{1}{\beta_{k_{2l},\alpha^{\prime}_{2l}}(a-1)}\left((2^{\underline{R^{\prime\prime}_{k_{2l}}}}-1)(\tilde{I}_{k_{2l}}+N)-(2^{\underline{R^{\prime}_{k_{2l}}}}-1)(I_{k_{2l}}+N)\right),$
(56)
$\displaystyle=\frac{1}{\beta_{k_{2l},\alpha^{\prime}_{2l}}(a-1)}\left(2^{\underline{R^{\prime\prime}_{k_{2l}}}}(\tilde{I}_{k_{2l}}+N)-2^{\underline{R^{\prime}_{k_{2l}}}}(I_{k_{2l}}+N)+I_{k_{2l}}-\tilde{I}_{k_{2l}}\right),$
(57)

and the total difference in power accounting for all the users assigned to BS
$m_{2}$ can be calculated as

$\sum_{k\in\mu(m_{2}(l))}\Delta_{k}+\Delta_{k_{l}}$ (58)

with

$\Delta_{k_{l}}=\frac{1}{(a-1)}\left(\frac{(2^{\underline{R^{\prime\prime}_{k_{l}}}}-1)(\tilde{I}_{k_{l}}+N)}{\beta_{k_{l},\alpha^{\prime}_{l}}}-\frac{(2^{\underline{R^{\prime}_{k_{l}}}}-1)(I_{k_{l}}+N)}{\beta_{k_{l},\alpha_{l}}}\right).$
(59)

Assuming the load independent power consumption for both the BSs is the same,
the difference of power consumption before and after re-matching can be
obtained and com[pared to zero as

$\displaystyle\sum_{l=1}^{\ell}\sum_{k\in\mu(m_{2}(l))}\Delta_{k}+\Delta_{k_{l}}-P_{LI}$
$\displaystyle\leq 0,$ (60) $\displaystyle\implies$
$\displaystyle\sum_{l=1}^{\ell}\Delta_{k_{2}(l)}+\frac{1}{(a-1)}\left(\frac{(2^{\underline{R^{\prime\prime}_{k_{l}}}}-1)(\tilde{I}_{k_{l}}+N)}{\beta_{k_{l},\alpha^{\prime}_{l}}}-\frac{(2^{\underline{R^{\prime}_{k_{l}}}}-1)(I_{k_{l}}+N)}{\beta_{k_{l},\alpha_{l}}}\right)-P_{LI}$
$\displaystyle\leq 0,$ (61) $\displaystyle\implies$
$\displaystyle\sum_{l=1}^{\ell}(2^{\underline{R^{\prime\prime}_{k_{l}}}}-1)(\tilde{I}_{k_{l}}+N)\beta_{k_{l},\alpha^{\prime}_{l}}^{-1}-(2^{\underline{R^{\prime}_{k_{l}}}}-1)(I_{k_{l}}+N)\beta_{k_{l},\alpha_{l}}^{-1}$
$\displaystyle\leq$ (62)
$\displaystyle\left(\sum_{l=1}^{\ell}(-\Delta_{k_{2}(l)})+P_{LI}\right)(a-1),$

where $\Delta_{k_{2}(l)}=\sum_{k\in\mu(m2(l))}\Delta_{k}$. Let
$\gamma_{1l}=(2^{\underline{R^{\prime}_{k_{l}}}}-1)(I_{k_{l}}+N)$,
$\gamma_{2l}=(2^{\underline{R^{\prime\prime}_{k_{l}}}}-1)(\tilde{I}_{k_{l}}+N)$
and

$\Delta_{l}=\sum_{l=1}^{\ell}\frac{(-\Delta_{k_{2}(l)})}{\gamma_{1l}}(a-1)+\frac{P_{LI}(a-1)}{\sum_{l=1}^{\ell}\gamma_{1l}}.$
(63)

Then the following inequalities for $1\leq l\leq\ell$ are sufficient for (62)

$\frac{\gamma_{2l}}{\gamma_{11}}\beta_{k_{l},\alpha^{\prime}_{l}}^{-1}\leq\beta_{k_{l},\alpha_{l}}^{-1}+\Delta_{l}.$
(64)

∎

The following remarks hold for all $\frac{\gamma_{2l}}{\gamma_{1l}}$ with
$1\leq l\leq\ell$, however, we only state it for a single user.

######  Remark B.3.

From (65), it can be inferred that at least $\underline{R^{\prime}_{1}}>0$.

$\frac{\gamma_{21}}{\gamma_{11}}=\frac{(2^{\underline{R^{\prime\prime}_{1}}}-1)}{(2^{\underline{R^{\prime}_{1}}}-1)}\frac{(\tilde{I}_{k_{1}}+N)}{(I_{k_{1}}+N)}.$
(65)

######  Remark B.4.

Under the assumption that
$\beta_{k_{1},\alpha^{\prime}}\leq\beta_{k_{1},\alpha}$, it can be observed in
(64) that $\beta_{k_{1},\alpha^{\prime}}^{-1}\geq\beta_{k_{1},\alpha}^{-1}$.
Thus, for (64) to hold, $\Delta$ in (64) must be positive, i.e,

$\displaystyle\Delta$ $\displaystyle>$ $\displaystyle 0$ (66)
$\displaystyle\frac{(-\Delta_{k_{2}}+P_{LI})}{\gamma_{11}}(a-1)$
$\displaystyle>$ $\displaystyle 0.$ (67)

From (67), it can be inferred that

$\Delta_{k_{2}}<P_{LI}.$ (68)

##  Appendix C Proof of Theorem III.2

###### Proof.

The proof for both NSM and SM is done by contra-diction as follows:

_i)_ No Spatial Multiplexing: Let $\tilde{p}_{m}$ be the transmit power
allocated to the users in a cell with fixed interference. For any
$\tilde{p}_{m}>0$, $\sum_{k\in\mu(m)}\alpha_{k}(\tilde{p}_{m})$ can be
expressed as

$\sum_{k\in\mu(m)}\alpha_{k}(\tilde{p}_{m})=\sum_{k\in\mu(m)}\frac{\underline{R_{k}}}{\bar{b}\log_{2}\left(1+\frac{(a_{m}-|\mu(\alpha)|)\beta_{k}\tilde{p}_{m}}{I_{k}+N}\right)},$
(69)

assuming equal power allocation per PRB. If $\tilde{p}_{m}<p_{m}^{*}$, it can
be observed from (69), that

$\displaystyle\sum_{k\in\mu(m)}\alpha_{k}(\tilde{p}_{m})>N_{m},$ (70)

where $N_{m}$ is the maximum available PRBs. However, since the system is
constrained by the maximum number of available PRBs, any
$\tilde{p}_{m}<p_{m}^{*}$ leads to outages because the minimum rate
requirement of the users is not satisfied.

For any $\tilde{p}_{m}>p_{m}^{*}$, it can be observed that

$\displaystyle\sum_{k\in\mu(m)}\alpha_{k}(\tilde{p}_{m})<N_{m},$ (71)

leading to a lower EE due to an increase in the total power consumption of the
BS, i.e.,

$\displaystyle P_{m}(\tilde{p}_{m})>P_{m}(p_{m}^{*}).$ (72)

To show that (72) holds, it suffices to show that updating the allocated
transmit power for a single user results in a lower EE. Thus, between the
vectors $\boldsymbol{\tilde{p}_{m}}$ and $\boldsymbol{p_{m}^{*}}$, the
transmit power for only one user is updated, i.e.,
$\boldsymbol{\tilde{p}_{m}}\rightarrow\boldsymbol{p_{m}^{*}}=[p_{m1}^{*},\tilde{p}_{m2},\dots,\tilde{p}_{mk}]$
and (71) and (20) can be expressed as141414The subscript for the BS $m$ has
been dropped for simplicity.

$\displaystyle\alpha_{1}(\tilde{p}_{1})+\sum_{k=2}^{K}\alpha_{k}(\tilde{p}_{k})$
$\displaystyle=$ $\displaystyle N_{m}-\delta$ (73)
$\displaystyle\alpha_{1}(p_{1}^{*})+\sum_{k=2}^{K}\alpha_{k}(\tilde{p}_{k})$
$\displaystyle=$ $\displaystyle N_{m},$ (74)

respectively, where $\delta\geq 0$ represents the change in the number of PRBs
due to an increase in the allocated transmit power for a single user.
Subtracting (73) from (74), $\delta$ can be obtained as

$\delta=\alpha_{1}(p_{1}^{*})-\alpha_{1}(\tilde{p}_{1})$ (75)

and the total power consumption of the BS151515It should be noted that the
terms independent of PRBs $\alpha_{k}$ and transmit power $p_{k}$ can be
treated as constants and therefore, have been omitted here. with the optimal
power $p_{1}^{*}$ can be computed as

$\displaystyle P_{m}(\boldsymbol{p_{m}}^{*})$ $\displaystyle=$
$\displaystyle\frac{1}{\eta_{PA}}\alpha_{1}(p_{1}^{*})p_{1}^{*}+C\alpha_{1}(p_{1}^{*})+\frac{1}{\eta_{PA}}\sum_{k=2}^{K}\alpha_{k}(\tilde{p}_{k})\tilde{p}_{k}+C\sum_{k=2}^{K}\alpha_{k}(\tilde{p}_{k})$
(76) $\displaystyle=$
$\displaystyle\alpha_{1}(p_{1}^{*})\left(\frac{1}{\eta_{PA}}p_{1}^{*}+C\right)+\sum_{k=2}^{K}\alpha_{k}(\tilde{p}_{k})\left(\frac{1}{\eta_{PA}}\tilde{p}_{k}+C\right),$
(77)

whereas, the total power consumption of the BS with
$\boldsymbol{\tilde{p}_{m}}$ can be computed as

$\displaystyle P_{m}(\boldsymbol{\tilde{p}_{m}})$ $\displaystyle=$
$\displaystyle\frac{1}{\eta_{PA}}\sum_{k=1}^{K}\alpha_{k}(\tilde{p}_{k})\tilde{p}_{k}+C\sum_{k=1}^{K}\alpha_{k}(\tilde{p}_{k})$
(78) $\displaystyle=$
$\displaystyle\alpha_{1}(\tilde{p}_{1})\left(\frac{1}{\eta_{PA}}\tilde{p}_{1}+C\right)+\sum_{k=2}^{K}\alpha_{k}(\tilde{p}_{k})\left(\frac{1}{\eta_{PA}}\tilde{p}_{k}+C\right)$
(79) $\displaystyle=$
$\displaystyle(\alpha_{1}(p_{1}^{*})-\delta)\left(\frac{1}{\eta_{PA}}\tilde{p}_{1}+C\right)+\sum_{k=2}^{K}\alpha_{k}(\tilde{p}_{k})\left(\frac{1}{\eta_{PA}}\tilde{p}_{k}+C\right),$
(80)

where, in (77) and (80), $C=D_{m,1}a_{m}\left(\sum_{k}N_{k}\right)$ as shown
in (17). From (72), after simplifying and using (75), it can be seen that

$\displaystyle\alpha_{1}(p_{1}^{*})\left(\frac{1}{\eta_{PA}}\tilde{p}_{1}+C\right)-\alpha_{1}(p_{1}^{*})\left(\frac{1}{\eta_{PA}}p_{1}^{*}+C\right)-\delta\left(\frac{1}{\eta_{PA}}\tilde{p}_{1}+C\right)$
$\displaystyle\geq 0$ (81) $\displaystyle\implies$
$\displaystyle\alpha_{1}(p_{1}^{*})(\tilde{p}_{1}-p_{1}^{*})-\delta(\tilde{p}_{1}+C\eta_{PA})$
$\displaystyle\geq 0$ (82) $\displaystyle\implies$
$\displaystyle\tilde{p}_{1}(\alpha_{1}(\tilde{p}_{1})+\delta)-\alpha_{1}(p_{1}^{*})p_{1}^{*}-\delta\tilde{p}_{1}+\delta
C\eta_{PA}$ $\displaystyle\geq 0$ (83) $\displaystyle\implies$
$\displaystyle\alpha_{1}(\tilde{p}_{1})\tilde{p}_{1}-\alpha_{1}(p_{1}^{*})p_{1}^{*}$
$\displaystyle\geq\delta C\eta_{PA},$ (84)

which states that the difference between load-dependent part with $\tilde{p}$
and $p^{*}$, i.e.,
$\alpha_{1}(\tilde{p}_{1})\tilde{p}_{1}-\alpha_{1}(p_{1}^{*})p_{1}^{*}$ must
be greater than or equal to the load-independent part $\delta C\eta_{PA}$.

_ii)_ Spatial Multiplexing: Similarly, it can be shown for SM that the optimal
power allocation is obtained by finding the power $\boldsymbol{p^{*}}$ such
that each user is assigned maximum available PRBs.

From (85), it can be observed that any $\tilde{p}_{m}<p_{m}^{*}$ leads to an
increase in the number of PRBs exceeding maximum available PRBs $N_{m}$

$\alpha_{k}(\tilde{p}_{m})=\frac{\underline{R_{k}}}{\bar{b}\log_{2}\left(1+\frac{(a_{m}-|\mu(\alpha)|)\beta_{k}\tilde{p}_{m}}{I_{k}+N}\right)}$
(85)

However, since no more than maximum available PRBs can be assigned to a user,
$\tilde{p}_{m}<p_{m}^{*}$ leads to outages because the minimum rate
requirement of the users with $\tilde{p}_{m}$ cannot be satisfied.

For any $\tilde{p}_{m}>p_{m}^{*}$, it can be observed that

$\displaystyle(\alpha_{1}(\tilde{p}_{1}),\dots,\alpha_{k}(\tilde{p}_{k}))_{k\in\mu(m)}<N_{m},$
(86)

leading to a lower EE due to an increase in the total power consumption of the
BS, i.e.,

$\displaystyle P_{m}(\tilde{p}_{m})>P_{m}(p_{m}^{*}).$ (87)

To show that (87) holds, it suffices to show that updating the allocated
transmit power for a single user results in a lower EE. Thus, between the
vectors $\boldsymbol{\tilde{p}_{m}}$ and $\boldsymbol{p_{m}^{*}}$, the
transmit power for a only one user is updated, i.e.,
$\boldsymbol{\tilde{p}_{m}}\rightarrow\boldsymbol{p_{m}^{*}}=[p_{m1}^{*},\tilde{p}_{m2},\dots,\tilde{p}_{mk}]$
and (86) and (21) for this user can be expressed as161616The subscript for the
BS $m$ has been dropped for convenience.

$\displaystyle\alpha_{1}(\tilde{p}_{1})$ $\displaystyle=$ $\displaystyle
N_{m}-\delta$ (88) $\displaystyle\alpha_{1}({p}_{1}^{*})$ $\displaystyle=$
$\displaystyle N_{m},$ (89)

respectively, where $\delta\geq 0$ represents the change in the number of PRBs
due to an increase in the allocated transmit power for a single user.
Subtracting (88) from (89), $\delta$ can be obtained as

$\delta=\alpha_{1}(p_{1}^{*})-\alpha_{1}(\tilde{p}_{1})$ (90)

and the total power consumption of the BS171717It should be noted that the
terms independent of PRBs $\alpha_{k}$ and transmit power $p_{k}$ can be
treated as constants and therefore, have been omitted here. with the optimal
power $p_{1}^{*}$ can be computed as

$\displaystyle P_{m}(\boldsymbol{p_{m}}^{*})$ $\displaystyle=$
$\displaystyle\frac{1}{\eta_{PA}}\alpha_{1}(p_{1}^{*})p_{1}^{*}+C\alpha_{1}(p_{1}^{*})+\frac{1}{\eta_{PA}}\sum_{k=2}^{K}\alpha_{k}(\tilde{p}_{k})\tilde{p}_{k}+C\sum_{k=2}^{K}\alpha_{k}(\tilde{p}_{k})$
(91) $\displaystyle=$
$\displaystyle\alpha_{1}(p_{1}^{*})\left(\frac{1}{\eta_{PA}}p_{1}^{*}+C\right)+N_{m}(K-1)\left(\frac{1}{\eta_{PA}}\tilde{p}_{k}+C\right),$
(92)

whereas, the total power consumption of the BS with
$\boldsymbol{\tilde{p}_{m}}$ can be computed as

$\displaystyle P_{m}(\boldsymbol{\tilde{p}_{m}})$ $\displaystyle=$
$\displaystyle\frac{1}{\eta_{PA}}\sum_{k=1}^{K}\alpha_{k}(\tilde{p}_{k})\tilde{p}_{k}+C\sum_{k=1}^{K}\alpha_{k}(\tilde{p}_{k})$
(93) $\displaystyle=$
$\displaystyle\alpha_{1}(\tilde{p}_{1})\left(\frac{1}{\eta_{PA}}\tilde{p}_{1}+C\right)+\sum_{k=2}^{K}\alpha_{k}(\tilde{p}_{k})\left(\frac{1}{\eta_{PA}}\tilde{p}_{k}+C\right)$
(94) $\displaystyle=$
$\displaystyle(\alpha_{1}(p_{1}^{*})-\delta)\left(\frac{1}{\eta_{PA}}\tilde{p}_{1}+C\right)+N_{m}(K-1)\left(\frac{1}{\eta_{PA}}\tilde{p}_{k}+C\right),$
(95)

where, in (92) and (95), $C=D_{m,1}a_{m}\left(\sum_{k}N_{k}\right)$ as shown
in (17). From (87), after simplifying and using (90), it can be seen that

$\displaystyle\alpha_{1}(p_{1}^{*})\left(\frac{1}{\eta_{PA}}\tilde{p}_{1}+C\right)-\alpha_{1}(p_{1}^{*})\left(\frac{1}{\eta_{PA}}p_{1}^{*}+C\right)-\delta\left(\frac{1}{\eta_{PA}}\tilde{p}_{1}+C\right)$
$\displaystyle\geq 0$ (96) $\displaystyle\implies$
$\displaystyle\alpha_{1}(p_{1}^{*})(\tilde{p}_{1}-p_{1}^{*})-\delta(\tilde{p}_{1}+C\eta_{PA})$
$\displaystyle\geq 0$ (97) $\displaystyle\implies$
$\displaystyle\tilde{p}_{1}(\alpha_{1}(\tilde{p}_{1})+\delta)-\alpha_{1}(p_{1}^{*})p_{1}^{*}-\delta\tilde{p}_{1}+\delta
C\eta_{PA}$ $\displaystyle\geq 0$ (98) $\displaystyle\implies$
$\displaystyle\alpha_{1}(\tilde{p}_{1})\tilde{p}_{1}-\alpha_{1}(p_{1}^{*})p_{1}^{*}$
$\displaystyle\geq\delta C\eta_{PA},$ (99)

which states that the difference between load-dependent part with $\tilde{p}$
and $p^{*}$, i.e.,
$\alpha_{1}(\tilde{p}_{1})\tilde{p}_{1}-\alpha_{1}(p_{1}^{*})p_{1}^{*}$ must
be greater than or equal to the load-independent part $\delta C\eta_{PA}$. ∎

##  Appendix D Large-scale Fading Computation

Highly accurate channel estimates can be determined using ray tracing [43]. In
this work, we use the in-house developed ray tracing based large-scale fading
predictor, called Femto Predictor (FemtoPred), which was developed at the
Institute for Communications Technology at TU Braunschweig. To describe an
outdoor model with an arbitrary number of UEs, the FemtoPred processes 3D
building data accounting for reflection, diffraction and transmission, along
with different antenna diagrams. The FemtoPred starts by creating pairs
between cell sites and map pixels, and deriving the reflected, diffracted and
transmitted paths, even if a LoS path is detected. Then, the channels are
calculated considering path losses, shadow and multi-path fading, while
accounting for LoS and non line-of-sight (NLoS).

####  D-1 Large-scale Fading

Within our scenario, a macroscopic approach was adopted, where, every pixel of
the map represents a large-scale fading value. Since we are simulating
multiple frequencies ($700$ MHz and $2.6$ GHz), ray tracing is well-suited for
simulating different wave lengths of different frequencies accounting for the
distance between the $T_{x}$ and $R_{x}$ in case of a LOS.

The large-scale fading prediction inside the FemtoPred can be divided into
three steps [51]: firstly, via geometrical approach, the image method is used
to find the only qualified rays to speed up the calculations. Secondly, for
each ray electromagnetic calculations are performed to determine the path
loss. In addition, the angel-of-arrival (AoA) at the receiver and the angle-of
departure (AoD) at the transmitter are derived. Finally, to calculate the
antenna pattern, the 3D antenna gain is multiplied by the complex received
power.

For NLOS, the large-scale fading calculations consider multiple components
including diffraction, reflection and transmission, as described below.

##### Diffraction

The knife edge model by Deygout [52] was used to calculate the buildings’ roof
diffraction. For a direct ray path, all the intersected surfaces are
determined on the assumption that a diffracted edge always belongs to the
surface of the intersected ray path. Therefore, in case of a single obstacle
intersecting the path of the ray, the diffraction $C$ is computed as [53]:

$\displaystyle C_{K,E}(dB)=6.9+20\log(\sqrt{(v-0.1)^{2}+1}+v-0.1)$ (100)

where, $v$ is the Fresnel diffraction. As shown in Figure181818Source: LTE-
Advance and Next Generation Wireless Networks 19, the Fresnel diffraction is
dependant on the height $h$ of the obstacle that intercepts the ray, the
distance between the transmitter $T_{x}$ and the obstacle $r1$ (along the LoS)
and the distance between the obstacle $r2$ and the receiver $R_{x}$ (along the
LoS).

Figure 19: Calculating Fresnel diffraction

Thus, the Fresnel diffraction is computed as,

$\displaystyle v=h\sqrt{\frac{2}{\lambda}\frac{s1+s2}{r1r2}}.$ (101)

where, and $\lambda$ is the wavelength and $s1=\sqrt{h^{2}+r1^{2}}$ and
$s2=\sqrt{h^{2}+r2^{2}}$ are the Pythagorean distance between the $T_{x}$ and
$R_{x}$.

For multiple obstacles that intersect the path of the ray, the Deygout method
describes the following algorithm:

  * •

Step 1 : The main obstacle between the $T_{x}$ and the $R_{x}$ is calculated
based on the highest $v$ values from (101), where only a single knife edge is
considered in between and the diffraction loss191919In this implementation,
the diffraction loss was also multiplied by $k$ which is the _empirical
correction factor_ with values ranging between $0.2$ to $0.5$. is
$C_{2}(v_{2})$.

  * •

Step 2 : The filtering procedure of the qualified points to the left
$\vec{C}(v_{1,left})$ and the right of the ray $\vec{C}(v_{3,right})$.

  * •

Step 3 : If more obstacles are still available, then Step $2$ is repeated.

In case of three obstacles between the BS and the mobile station, total
diffraction loss, using the Deygout method, can be calculated as,

$\displaystyle
C_{d}(dB)=C_{2}(v_{2})+C_{1}{(v_{1,left})}+C_{3}{(v_{3,right})}$ (102)

##### Reflection and Transmission

In our simulator, the reflection and transmission of the signals is dependant
on the signal frequency. For this to be as accurate as possible, a dielectric
layer needs to be implemented based on the material of the buildings inside
the simulated environment (in this work, concrete buildings).

The direction of the incidence can be defined as the angle between the wave
vector and the normal to the interface [54]. The angle of incident
($\Theta_{e}$), transmitted ($\Theta_{t}$) and reflected ($\Theta_{r}$) wave
can be defined as,

$\displaystyle\frac{\sin{\Theta_{t}}}{\sin{\Theta_{e}}}=\frac{\sqrt{\epsilon_{1}}}{\sqrt{\epsilon_{2}}}$
(103)

where, $\Theta_{r}=\Theta_{e}$, and $\epsilon_{1}$, $\epsilon_{2}$ are the
dielectric constants of the each side of the material.

To compute the magnitude of the wave, a distinction between whether the
electric field vector is parallel to the interface of the dielectrics (TE
wave) and whether the magnetic field is parallel to the interface transversal
magnetic field (TM wave) is necessary, where $\rho_{TE}$, $T_{TE}$,
$\rho_{TM}$, and $T_{TM}$ can be computed as,

$\displaystyle\rho_{TE}=\frac{\sqrt{\epsilon_{1}}\cos{\Theta_{e}}-\sqrt{\epsilon_{2}}\cos{\Theta_{t}}}{\sqrt{\epsilon_{1}}\cos{\Theta_{e}}+\sqrt{\epsilon_{2}}\cos{\Theta_{t}}}$
(104)

$\displaystyle
T_{TE}=\frac{2\sqrt{\epsilon_{1}}\cos{\Theta_{e}}}{\sqrt{\epsilon_{1}}\cos{\Theta_{e}}+\sqrt{\epsilon_{2}}\cos{\Theta_{t}}}$
(105)

$\displaystyle\rho_{TM}=\frac{\sqrt{\epsilon_{2}}\cos{\Theta_{e}}-\sqrt{\epsilon_{1}}\cos{\Theta_{t}}}{\sqrt{\epsilon_{2}}\cos{\Theta_{e}}+\sqrt{\epsilon_{1}}\cos{\Theta_{t}}}$
(106)

$\displaystyle
T_{TM}=\frac{2\sqrt{\epsilon_{1}}\cos{\Theta_{e}}}{\sqrt{\epsilon_{2}}\cos{\Theta_{e}}+\sqrt{\epsilon_{1}}\cos{\Theta_{t}}}$
(107)

Conclusively, the overall reflection and transmission values can be calculated
for a dielectric layer of thickness $d_{layer}$ as,

$\displaystyle T=\frac{T_{1}T_{2}e^{-ja}}{1+\rho_{1}\rho_{2}e^{-2ja}},$ (108)

$\displaystyle\rho=\frac{\rho_{1}+\rho_{2}e^{-ja}}{1+\rho_{1}\rho_{2}e^{-2ja}},$
(109)

$\displaystyle\alpha=\frac{2\pi}{\lambda}\sqrt{\epsilon_{2}}d_{layer}\cos({\Theta_{t}})$
(110)

where $\rho_{1},\rho_{2}$, $T_{1}$, and $T_{2}$ are the reflection and
transmission coefficients respectively and $\alpha$ is the electrical length
of the layer.

####  D-2 Antenna Modeling

In our simulation, directional antenna patterns have been implemented for both
frequencies. For the $700$ MHz band, a $3$GPP antenna model [55] was
generated, where, the antenna element $A_{E}$ consists of two main components,
i.e., vertical and horizontal propagation patterns.

The antenna diagram can be modeled based on (111) and (112),

$\displaystyle
A_{E,V}(\theta)\text{(dB)}=-\min\Bigg{[}12\Bigg{(}\frac{\theta-90^{\circ}}{\theta_{3dB}}\Bigg{)}^{2},SLA_{v}\Bigg{]}$
(111)

$\displaystyle
A_{E,H}(\phi)\text{(dB)}=-\min\Bigg{[}12\Bigg{(}\frac{\phi^{\prime\prime}}{\phi_{3dB}}\Bigg{)}^{2},A_{m}\Bigg{]}$
(112)

where, $A_{E,V}$ and $A_{E,H}$ represent the vertical and horizontal
propagation pattern, respectively. A $3$D antenna element pattern can be
acquired by combining the vertical and horizontal patterns and adding the
maximum antenna gain as,

$\displaystyle
A(\theta,\phi)\text{(dB)}=G_{E,max}-\min\bigg{\\{}-\bigg{[}A_{E,V}(\theta)+A_{E,H}(\phi)\bigg{]},A_{m}\Bigg{\\}}$
(113)

where, $\theta$ is the zenith angle ([0 , 180] degrees), $\phi$ is the azimuth
angle ([-180 , 180] degrees), $\phi_{3dB}=65^{\circ}$ is the $3$dB beamwidth
of the horizontal antenna pattern, $\theta_{3dB}$ = $15^{\circ}$ is the $3$dB
beamwidth of the vertical antenna pattern, $SLA_{v}$ = 30 is the sidelobe
attenuation, $A_{m}$ is the maximum attenuation of the main lobe, and
$G_{E,max}=8$ dBi is the maximum antenna element gain202020These parameter
values have been specified for the $700$ MHz antenna diagram..

For the 2.6 GHz band a single band sector ”Commscope (HWXX-6516DS-VTM2600)”
antenna diagram pattern was chosen.

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