# Green Operations of SWIPT Networks:
The Role of End-User Devices

Gianluca Rizzo,
HES-SO Valais, Switzerland & Universita’ di Foggia, Italy -
<EMAIL_ADDRESS>Marco Ajmone Marsan,
Institute IMDEA Networks, Spain -<EMAIL_ADDRESS>Christian Esposito,
Biagio Boi  Universita’ di Salerno, Italy - [esposito<EMAIL_ADDRESS>
###### Abstract

Internet of Things (IoT) devices often come with batteries of limited capacity
that are not easily replaceable or rechargeable, and that constrain
significantly the sensing, computing, and communication tasks that they can
perform. The Simultaneous Wireless Information and Power Transfer (SWIPT)
paradigm addresses this issue by delivering power wirelessly to energy-
harvesting IoT devices with the same signal used for information transfer. For
their peculiarity, these networks require specific energy-efficient planning
and management approaches. However, to date, it is not clear what are the most
effective strategies for managing a SWIPT network for energy efficiency. In
this paper, we address this issue by developing an analytical model based on
stochastic geometry, accounting for the statistics of user-perceived
performance and base station scheduling. We formulate an optimization problem
for deriving the energy optimal configuration as a function of the main system
parameters, and we propose a genetic algorithm approach to solve it. Our
results enable a first-order evaluation of the most effective strategies for
energy-efficient provisioning of power and communications in a SWIPT network.
We show that the service capacity brought about by users brings energy-
efficient dynamic network provisioning strategies that radically differ from
those of networks with no wireless power transfer.

##  I Introduction

00footnotetext: This paper was presented in part at the 20th International
Symposium on Modeling and Optimization in Mobile, Ad Hoc, and Wireless
Networks (WiOpt) [1].

The advent of the Internet of Things (IoT) paradigm has significantly impacted
the energy consumption of Radio Access Networks (RAN) [2]. On one side, the
proliferation of IoT devices is leading to an exponential increase in the
number of connected devices and in the amount of data generated and exchanged
[3]. On the other, the wide heterogeneity of IoT communication requirements
and traffic patterns, together with the irregularity in the profiles of device
activity and sleep cycles, induce patterns of energy consumption within the
RAN that differ substantially from those of traditional, broadband-centric
cellular networks [4, 5], with a strong impact on the way the network is
planned and operated.
This impact is further amplified when the cellular network does not only
deliver connectivity to all its users but also power to IoT devices. In this
work, we consider scenarios in which IoT devices harvest RF energy from the
environment (passive energy harvesting \- EH) as well as from the signal
transmitted by their serving base station (active EH) [6]. Such configurations
are interesting as they allow for overcoming the power budget constraints that
limit the potential of battery-operated IoT in many application domains.
Indeed, these limitations often imply tight constraints on the amount of
sensing, computing, and actuation that IoT devices can perform, and negatively
affect their availability.
Among the available technologies for wireless power delivery to EH devices,
simultaneous wireless information and power transfer (SWIPT) [7, 8, 9, 10, 11]
is of particular interest, as it exploits the same cellular infrastructure
used to deliver data. This feature makes it easier and more cost-effective to
deploy than alternative approaches. However, different from traditional
single-service cellular networks (i.e., networks with no wireless power
transfer), energy efficiency in SWIPT networks encompasses both the energy
consumed by BSs and that consumed by IoT users, which, for their ubiquity and
rising numbers, are poised to play a key role in determining the overall
energy footprint of the network. Thus, SWIPT networks require the development
of specific approaches for energy-efficient network design and management,
which may substantially differ from traditional, single-service, RANs.
Several works on energy efficiency in SWIPT [12, 13, 14] consider simple
scenarios composed of a single base station (BS). They focus on accounting for
the impact of the interplay between power and information delivery on the
energy footprint of the network. Other works ([15, 16]) consider specific BS
deployment scenarios serving a population of EH IoT devices, and they propose
some heuristics for saving energy in those configurations. However, these
algorithm and scenario-specific approaches do not allow drawing general
conclusions on the impact of the main system parameters on energy efficiency
in a generic SWIPT network, and thus getting an idea of the actual potential
for energy saving of dynamic network management in SWIPT. Indeed, these
aspects are key for supporting any decision on how to effectively plan and
manage such networks.

Another set of results focuses on a system-level analysis of SWIPT networks
based on the tools of stochastic geometry [17, 18, 19]. Crucially, however,
none of these works account for the effects of resource scheduling among
users, nor do they address the key issue of determining the energy-optimal
strategies for SWIPT network provisioning and QoS-aware dynamic management.
In this paper, we address these issues and propose an analytical framework
based on stochastic geometry (SG) for system-level performance evaluation of
SWIPT cellular networks, which accounts for resource scheduling among
broadband and IoT EH users. The use of SG allows focusing on the average
behavior of the system over many realizations of the process of UE (user
equipment) and BS spatial distributions. This allows characterizing the
potential for QoS-aware energy optimization of SWIPT-enabled networks and the
main trade-offs between user-perceived performance and resource utilization in
a scenario-independent manner. Specifically, the results of this paper are as
follows:

  * •

We derive analytical expressions for the key statistics of the main
performance indicators of a SWIPT network.

  * •

We formulate the problem of deriving a configuration (in terms of density of
active BSs, split ratio, and transmit power) that minimizes the mean total
power consumption of the network while achieving a target QoS level for both
information transfer (in downlink and uplink) and wireless power delivery.

  * •

We propose a genetic algorithm (GA) approach for deriving energy-efficient
SWIPT network configurations, achieving the target user-perceived performance
levels.

  * •

We characterize the key performance and resource efficiency trade-offs of the
configurations resulting from our heuristic. This allows observing some rather
surprising effects, such as, in some cases, a decrease in the optimal density
of BSs required to serve an increasingly dense population of UEs and the
irrelevance of active EH for very dense populations of IoT devices.

The paper is organized as follows. In Section II, we introduce the related
works on system modeling for SWIPT networks. In Section III, we present the
system model, and in Section IV, we derive a model for user-perceived
performance for information and power transfer. In Section V, we formulate an
optimization problem for the maximization of the energy efficiency of the
network. In Section VI we assess numerically our results, and in Section VII,
we conclude the paper with some final remarks.

##  II Related works

The growing interest of the wireless communication community towards the SWIPT
paradigm is due to its potential for enabling ubiquitous and self-sustained
cellular-based IoT [7, 8, 9, 10, 11]. Moreover, differently from IoT energy
harvesting techniques based on solar or wind, those based on active wireless
power transfer have the advantage of being stable and available at any time
[6]. Several technical challenges however still need to be addressed for the
practical viability of SWIPT networks, related to energy-harvesting
transceiver and algorithm design, system integration, protocol design, and
energy-efficient network planning and operations [20, 21].

Initial works on energy efficiency in SWIPT networks considered scenarios
composed of a BS or a broadband unit with several remote radio heads, and a
population of EH devices. These works aim to optimize the system configuration
in terms of energy efficiency [12, 13, 22]. Among these, [12] characterizes
the data rate vs. energy trade-off in a single SWIPT BS scenario. [13]
proposes an approach for optimizing transmit power allocation to each user.
[22] proposes a Markov chain model to derive the joint power and connectivity
outage probability. [23] investigates strategies for transmit-power-efficient
resource allocation. However, the simple scenarios considered in these works
do not allow for extending the results or extrapolating them to different and
more realistic configurations, nor do they account for the impact of the
statistics of BS and user distribution.

Algorithms for deriving energy-optimal configurations of sample SWIPT systems
have been proposed for networks with CoMP [24], in NOMA networks [25], in full
duplex settings [26], as well as in scenarios with two-way relays [27]. Among
these, [15] proposes an algorithm to minimize the total energy consumption of
a set of BSs and a population of IoT devices, over sub-carrier and transmit
power allocation. However, the strategies proposed in these works focus on
deriving the optimal configuration for a specific set of deployed BS, with
their specific layout and user spatial distribution. As such, they do not
allow evaluating the overall potential for energy-saving enabled by their
schemes in a scenario-independent manner. Thus, they leave open the more
general issue of what are the most effective management strategies to achieve
system-level energy proportionality in a SWIPT network, as a function of the
main system parameters.

Similarly to ours, another set of works ([17, 18, 28]) performs system-level
analysis, modeling the average behavior of a SWIPT network, such as the data
rate vs. energy trade-off (e.g., see [18]), or optimizing the D2D successful
transmission rate [28]. These works however do not account for the effects of
resource allocation and scheduling among users, such as the statistics of the
sharing of BS time across all associated users, which are key for an accurate
modeling of user-perceived performance. Thus, they do not enable an accurate
and realistic characterization of the main performance trade-offs, and in
particular of the strategies for QoS-aware energy-efficient operation of SWIPT
networks.

##  III System Model

We assume that BSs are distributed in space according to a homogeneous planar
Poisson Point Process (PPP) with a density equal to $\lambda_{b}$ BSs per
$km^{2}$. UEs are distributed in space according to a homogeneous PPP with an
intensity equal to $\lambda_{u}$ UEs per $km^{2}$. UEs are either broadband
(BB) terminals, or IoT (Internet of Things) devices. We assume the latter are
a fraction $\gamma$ of the total number of UEs.

We consider the case in which IoT devices harvest the energy necessary for
their own operation, while BB users don’t. To this end, each IoT device is
equipped with two separate receivers, one for information and another for
energy, and it is capable of exploiting downlink signals for decoding its
intended information, as well as downlink and uplink signals from both BSs and
UEs for charging its battery. The receiver operating modes we consider are
[29]:

  * •

time switching (TS), by which a fraction $\mathbf{\eta}$ (the time switch
ratio, with $0\leq\eta\leq 1$) of the time dedicated by a BS to serve an
associated IoT device in downlink is devoted to active power transfer, i.e.,
it is used by the UE for harvesting energy from the signal received from the
BS. The rest of that time is used for receiving information.

  * •

static power splitting (SPS), by which a fraction $\mathbf{\nu}$ (the power
split ratio, with $0\leq\nu\leq 1$) of the power received at any time instant
by an IoT device is harvested.

  * •

dynamic power splitting (DPS), by which a fraction $\mathbf{\nu}$ of the power
transmitted by any source during the time the serving base station devotes to
a given IoT device is harvested. During the rest of the time, all of the
received power is harvested at the node.

We assume $\eta$ and $\nu$ are the same for all devices. Thus, IoT devices
harvest energy not only from their serving BS (active charging), but also from
the signal received from all of the other BSs, as well as from uplink
transmissions from both IoT and BB UEs (passive charging).

Figure 1: Downlink time scheduling scheme for a SWIPT BS, for a network with
time switching EH receiver architecture, with GPS weights.

We assume that BSs use a generalized processor sharing (GPS) mechanism to
divide BS time among all the connected UEs. In downlink, the GPS weights are
$1$ for IoT UEs, and $w_{d}$ for BB UEs. The time spent by the BS without
transmitting is modeled as a user with a GPS weight $\beta_{d}$. As for
uplink, in all configurations, the GPS weights are $1$ for IoT UEs,
$\delta_{u}$ for BB UEs, and $\beta_{u}$ for the uplink BS time not assigned
to any UE. IoT UEs periodically cycle between two operational states. During
the active state, they send and receive data, plus possibly they perform some
other task, such as sensing, while harvesting energy from active and passive
sources. In the low power state, IoT devices only harvest energy. We assume
that each IoT device is disconnected from the power grid, and possesses an
ideal battery (i.e., with efficiency equal to one) with enough capacity to
compensate for the fluctuations in the energy consumed and harvested. Note
however that our approach can be easily extended to consider battery
nonidealities and limitations due to finite battery capacity. With $\phi$ we
denote the fraction of time spent by an IoT device in the active state. BB
users are instead assumed to always be in the active state.
We assume the system is in saturation, i.e., BSs have always data to send to
users, and users in the active state have always data to transmit. A key
aspect of the EH process is the function $h=\Theta(h_{in})$ that maps the
received power $h_{in}$ to the harvested power $h$ for EH IoT devices.
$\Theta$ models the effect of several technological factors affecting the
efficiency of the energy harvesting process, ranging from receiver antenna
gain to exposure of the specific IoT device, to receiver architecture, among
others. In this work, we consider two different models:

  * •

a linear model, by which $h=\xi h_{in}$, with $\xi\leq 1$; and

  * •

a nonlinear model, where the mapping function is a normalized sigmoid [30],
given by

$\Theta(h_{in})=\max\left\\{\frac{h_{max}}{e^{-\chi
h_{s}+\iota}}\left(\frac{1+e^{-\chi h_{s}+\iota}}{1+e^{-\chi
h_{in}+\iota}}-1\right),0\right\\}$ (1)

$h_{max}$ is the maximum harvested power when the energy harvesting circuit is
saturated, while $h_{s}$ is the harvester’s sensitivity threshold, i.e. the
minimum amount of power able to activate the harvesting process. $\chi$ and
$\iota$ are parameters that control the steepness of the sigmoid.

Note however that our approach extends easily to other EH models.

TABLE I: Main notation used in the paper Name |  Description
---|---
$\bar{\tau}_{j}$, $j=d,u$ |  Mean ideal per-bit delay in downlink/uplink ($s^{-1}$)
$\tau_{j}$, $j=d,u$ |  Per-bit delay in downlink/uplink ($s^{-1}$)
$\tau_{j}^{0}$, $j=d,u$ |  Target per-bit delay in downlink/uplink ($s^{-1}$)
$\xi$ |  EH conversion efficiency ratio (linear model)
$h_{0}$ |  Minimum harvested power required by each IoT device ($W$)
$\mu$ |  Maximum acceptable ratio of IoT users which harvest less than $h_{0}$ W
$\phi$ |  Fraction of time in which IoT devices are active
$\lambda_{b}$ |  Mean BS density ($m^{-2}$)
$\lambda_{u}$ |  Mean density of user terminals ($m^{-2}$)
$\gamma$ |  Fraction of user terminals which are IoT devices
$\eta$ |  Fraction of the time dedicated by a BS to serve an IoT device in downlink used for harvesting energy from the signal received from the base station
$\nu$ |  Fraction of the power received at any time instant by an IoT device which is used for energy harvesting
$\beta_{j}$, $j=d,u$ |  GPS weight of the amount of time spent by a BS in idle mode in downlink/uplink
$w_{d}$ |  Ratio between the amount of time dedicated to serving a BB UE and the one dedicated to serving an IoT user in downlink
$\delta_{j}$, $j=d,u$ |  Ratio between the BS time dedicated to a BB user, and the base station time dedicated to an IoT user for data transfer in downlink/uplink
$k$ |  Frequency reuse factor
$\alpha$ |  Path loss exponent
$P$ |  BS transmit power ($W$)
$G$ |  BS antenna gain
$L$ |  BS antenna loss (out of beam)

Given that the cellular RAN delivers two classes of services to two different
types of UE, the performance metrics are defined as follows:

  * •

Information transfer: For all UEs, the end-user performance metric is the per-
bit delay $\tau$ of data transfers, defined as the inverse of the short-term
user throughput, i.e., the actual rate at which the user is served.

  * •

Power transfer: The performance parameter is the amount of power harvested by
an IoT device, denoted with $h$.

As a result of the scheduling strategy (both in downlink and uplink), the
amount of harvested energy may vary substantially across the various devices.
These metrics must be considered relative to the corresponding application
demands. As an example, the energy harvested by each device during a complete
on-off cycle (and therefore the value of the parameter $h$) must be sufficient
to compensate for the energy consumed during the cycle by such tasks as
sensing, processing, storing, and distributing data, while accounting for
battery efficiency and for fluctuations in consumption patterns due to changes
in the operational state of IoT devices. Accordingly, the target performance
for information transfer is related to application-specific requirements.

###  III-A Channel and Service Model

Our channel model only takes into account distance-dependent path loss. The
incorporation of the effects of fading and shadowing would not alter our
approach, and it is left for future consideration. We assume that random
frequency reuse is in place, with reuse factor $k$. That is, every BS is
assigned one out of $k$ frequency bands with equal probability.
We assume that UEs are associated with the BS that provides the largest SINR
at the user location. We consider urban scenarios, where the high capacity
demand justifies the use of strategies for energy efficient network planning
and management, and where the assumption of high attenuation (with exponent
$\alpha\geq 3$) typically holds. In these settings, as no fading is considered
and all BSs use the same transmit power, assuming that users associate to the
closest BS is a reasonable approximation [19].
We assume BS antennas use beamforming, and we denote with $G$ the beamforming
gain and with $L$ the side lobes attenuation. If with $a$ we denote the
aperture of the main lobe (in degrees), the relationship between $G$ and $L$
is given by

$L=1-(G-1)\frac{a}{360-a}$

As users are assumed to be distributed uniformly in space, and as base station
time is supposed to be shared in equal parts across users of the same type,
the mean power received by a user while it is not being served from the base
station to which it is associated (located at a distance $r$) is
$PL_{g}r^{-\alpha}$, with $L_{g}=\frac{Ga+L(360-a)}{360}$. Thus
$L_{g}r^{-\alpha}$ is the mean attenuation with which the power transmitted by
the serving base station is received by a user when it is not being served.

Denote by $S(x)$ the location of the BS that is closest to a UE located at
$x$. We denote the capacity of a user located at a distance $r$ from the BS by
$C(r,P,G,I)$ bit/s per Hertz, where $P$ is the BS transmit power, and $I$ the
total received interfering power. We model $C(r,P,G,I)$ using Shannon’s
capacity law:

$C(r,P,G,I)=(B/k)\log_{2}\left(1+\frac{PGr^{-\alpha}}{N_{0}+I(r,k)}\right)$
(2)

where $\alpha$ is the attenuation coefficient, $N_{0}$ the power spectral
density of the additive white Gaussian noise, and $k$ the reuse factor.

###  III-B Base Station and UE Energy Consumption Model

The power consumed by a BS depends on several factors, which vary according to
the BS type (e.g. macro, micro, femto) and the implementation technology (e.g.
standalone vs cloud-RAN), among others. In what follows, we adopt a very
flexible BS energy model, first proposed in [31, 32]:

$q_{1}+U_{d}[q_{2}+q_{3}(P-P_{min})]$ (3)

where $U_{d}$ is the downlink BS utilization and $P$ is its transmit power,
which we assume varies within the interval $[P_{min},P_{max}]$. $q_{1}$ models
the total power consumed when the BS is idle, which does not depend on load or
transmit power. In what follows we assume that, when transmitting, a BS is
using the whole bandwidth at its disposal. $q_{2}U_{d}$ models that fraction
of consumed power which depends on utilization, but not on transmit power
(such as part of baseband and RF processing). The third component models the
fraction of consumed power due to the power amplifier which depends, at the
same time, on the transmit power $P$ and on the amount of active resources at
the BS.

###  III-C Base station service model

We define the utilization $U(S(x))$ of the BS serving a UE at location $x$ as
the average fraction of time in which the BS is busy transmitting (in
downlink) or receiving (in uplink), respectively. Thus, the expression of the
utilization of the BS is given by the fraction of BS time dedicated to all
active users:

$U_{d}(S(x))=\frac{N_{iot}(S(x))+w_{d}N_{bb}(S(x))}{N_{iot}(S(x))+w_{d}N_{bb}(S(x))+\beta_{d}}$
(4)
$U_{u}(S(x))=\frac{N_{iot}(S(x))+\delta_{u}N_{bb}(S(x))}{N_{iot}(S(x))+\delta_{u}N_{bb}(S(x))+\beta_{u}}$
(5)

where $N_{iot}(S(x))$ and $N_{bb}(S(x))$ denote respectively the number of
broadband and IoT users in the active state, and associated with the same BS
as the UE at location $x$. Note that, as IoT users cycle between active and
inactive states, on average only a fraction $\phi$ of them is active at any
point in time. Given $w_{d}$ and $\delta_{u}$, by tuning $\beta_{j}$ it is
possible to vary the mean amount of service received by UEs for both
communication and energy transfer, and the overall BS utilization, both in
downlink and in uplink.

######  Definition.

The ideal per-bit delay perceived by a UE is the per bit delay which a UE
would perceive if the BS with which the UE is associated had utilization equal
to 1.

Note that the above definition does not assume that all of the BSs have
utilization equal to one, but only the BS serving the considered UE. From
expressions (4), (5) and the above definition, we have that for a UE at $x$,
the relationship between the ideal per bit delay $\tau_{j}^{id}$, $j\in\;d,u$
and the actual per bit delay, both in uplink and in downlink, is
$U_{j}(S(x))\tau_{j}(S(x))=\tau_{j}^{id}(S(x))$, with $j\in\;d,u$. Indeed,
from the expression of the utilization, the ratio between the ideal and the
actual per-bit delay is equal to the fraction of time the BS is active.

For each UE type, and for both uplink and downlink, a notion of target minimum
quality of service (QoS) is defined. Namely, UEs are said to perceive
satisfactory performance if the average per-bit delay experienced by a typical
BB user (resp. IoT device)111The definition of the typical user in the system
is provided by classical Palm theory [33]. are less than their respective
predefined target values. In what follows, for a given $\eta$, we assume each
BS adopts the values of $\beta_{j}$, $j\in\;d,u$ (and thus of utilization in
downlink and uplink) which achieve the target QoS for communications. This
makes the average of the actual per bit delays (over all the users associated
to that BS) coincide with the target values.

##  IV Modeling User Perceived Performance

In this section, we characterize the main performance parameters, e.g. the
per-bit delays and the harvested power perceived by a typical user who is just
beginning service222This assumption is standard in works exploiting stochastic
geometry. In fact there is no difference with respect of the case where users
reside in the system since a while, it is just a way of choosing the point of
view from which to carry out the calculations. , as well as the mean harvested
power, as a function of the main system parameters. The expression of the
power harvested by a user at $x$ is given by the following result.

######  Lemma 1.

Let $K(x)=[N_{iot}(S(x))+w_{d}N_{bb}(S(x))]^{-1}$. Then the power harvested by
a user at $x$ is $h(x)=\Theta(h_{in}(x))$, where the received power
$h_{in}(x)$ is given by:

  * •

TS:

$h_{in}(x)=PD(x)^{-\alpha}U_{d}(S(x))\left[G\eta
K(x)+L_{g}(1-K(x))\right]+\left[1-K(x)U_{d}(S(x))(1-\eta)\right](I(x)+O(x))$
(6)

  * •

SPS:

$h_{in}(x)=\nu
PD(x)^{-\alpha}U_{d}(S(x))\left[GK(x)+L_{g}(1-K(x))\right]+\nu(I(x)+O(x))$ (7)

  * •

DPS:

$h_{in}(x)=PD(x)^{-\alpha}U_{d}(S(x))\left[G\nu
K(x)+L_{g}(1-K(x))\right]+I(x)+O(x)$ (8)

$I(x)$ is the total power harvested by the user at $x$ from BSs other than the
one with which it is associated. $O(x)$ is the power harvested from UE
transmissions, averaged over time, and $U_{d}(S(x))$ is the downlink
utilization of the BS with which the user in $x$ is associated.

For a proof, please refer to Appendix -A. From its definition and from the
service model description, it derives that in downlink, the ideal per-bit
delay perceived by a BB UE at $x$ is given by

$\tau_{d}^{id}(x)=\frac{N_{iot}(S(x))+w_{d}N_{bb}(S(x))}{w_{d}C(x,P,G,I)}$ (9)

For IoT users, the per-bit delay is given by:

  * •

TS:

$\tau_{d,I,TS}^{id}(x)=\frac{N_{iot}(S(x))+w_{d}N_{bb}(S(x))}{(1-\eta)C(x,P,G,I)}$
(10)

  * •

SPS, DPS:

$\tau_{d,I,PS}^{id}(x)=\frac{N_{iot}(S(x))+w_{d}N_{bb}(S(x))}{C(x,(1-\nu)P,G,(1-\nu)I)}$
(11)

For the uplink instead, for broadband users, we have

$\tau_{u}^{id}(x)=\frac{N_{iot}(S(x))+\delta_{u}N_{bb}(S(x))}{\delta_{u}C(x,P_{I},1,0)}$
(12)

For IoT users, the ideal per-bit delay perceived in uplink is
$\tau_{u,I}^{id}(x)=\delta_{u}\tau_{u}^{id}(x)$. With $\bar{\tau}_{j}$
($\bar{\tau}_{j,I}$), $j=d,u$ we denote the average per-bit delay perceived in
downlink (resp. in uplink) by broadband and IoT users, respectively. The
following result derives an expression for the average per-bit delay perceived
by the typical user (BB or IoT) which is just beginning service.

######  Theorem 1.

The mean ideal per-bit delays in downlink and uplink, and the mean ideal per-
Joule delay perceived by a typical best-effort user joining the system are
given by:

$\displaystyle\bar{\tau}_{d}=$ $\displaystyle H(w_{d},w_{d},C(r,P,G,\bar{I}))$
(13) $\displaystyle\bar{\tau}_{d,I,TS}=$
$\displaystyle\bar{\tau}_{d}\frac{w_{d}}{(1-\eta)}$ (14)
$\displaystyle\bar{\tau}_{d,I,PS}=$ $\displaystyle
H(w_{d},1,C(r,(1-\nu)P,G,(1-\nu)\bar{I}))$ (15) $\displaystyle\bar{\tau}_{u}=$
$\displaystyle H\left(\delta_{u},\delta_{u},C(r,P_{I},1,0)\right)$ (16)
$\displaystyle\bar{\tau}_{u,I}=$ $\displaystyle\delta_{u}\bar{\tau}_{u}$ (17)

Where:

$H(y,z,g(r))=\int_{0}^{\infty}\frac{f(r,y)e^{-\lambda_{b}\pi
r^{2}}\lambda_{b}2\pi r}{zg(r)}\mathop{}\\!\mathrm{d}r.$ (18)

with

$f(r,y)=\lambda_{u}\left[y+\gamma\left(\phi-y\right)\right]\int_{0}^{\infty}\int_{0}^{2\pi}e^{-\lambda_{b}A(r,x,\theta)}xd\theta
dx$

$A(r,x,\theta)$ is the area of the circle centered at $(x,\theta)$ with radius
$x$ that is not overlapped by the circle centered at $(0,-r)$ with radius $r$.
$C(r,P,G,\bar{I})$ is given by (2), with the interference term $\bar{I}$ given
by

$\bar{I}(r,k)=\frac{PL_{g}\lambda_{b}2\pi
r^{2-\alpha}}{k(\alpha-2)}\frac{\bar{\tau}_{d}}{\tau_{d}^{0}}$ (19)

For the proof, please refer to appendix -B. We say that the considered system
is in the dense IoT regime when the density of IoT users is such that the
probability of having a cell without IoT users is negligible. This assumption
is coherent with many current projections on expected IoT deployments in 6G,
in which the expected device densities are of the order of 10 million
devices/km2 and more [34, 35, 36, 37].

######  Theorem 2.

In the dense IoT regime, the cumulative distribution function $CDF_{h}$ of the
power harvested by an IoT user who is just beginning service is
$CDF_{h}(h_{0})=CDF_{r}(g^{-1}(h_{0}))$ for all $h_{0}\geq 0$, where $CDF_{r}$
is the cumulative distribution function of the distance of the user to its
serving BS:

$CDF_{r}(r)=\int_{0}^{r}e^{-\lambda_{b}\pi y^{2}}\lambda_{b}2\pi ydy$

where $g(r)=\Theta\left(F_{j}(r)+\frac{Z_{j}(r)}{f(r,w_{d})}\right)$ with
$j\in\\{TS,SPS,DPS\\}$, and $f(r,w_{d})$ given by Theorem 1.

  * •

TS:

$F_{TS}(r)=Pr^{-\alpha}L_{g}\frac{\bar{\tau}_{d}}{\tau_{d}^{0}}+k\bar{I}(r,k)+\bar{O}$
$Z_{TS}(r)=\frac{\bar{\tau}_{d}}{\tau_{d}^{0}}\bigg{[}Pr^{-\alpha}(G\eta-
L_{g})-(1-\eta)(k\bar{I}(r,k)+\bar{O})\bigg{]}$
$\bar{O}=\frac{(1-\gamma)\delta_{u}P_{bb}+\phi\gamma
P_{I}}{(1-\gamma)\delta_{u}+\phi\gamma}\frac{\lambda_{b}\pi\alpha}{\alpha-2}\frac{\bar{\tau}_{u}}{\tau_{u}^{0}}$

  * •

SPS: $F_{SPS}(r)=\nu F_{TS}(r)$, and $Z_{SPS}(r)=\nu
Pr^{-\alpha}\frac{\bar{\tau}_{d}}{\tau_{d}^{0}}(G-L_{g})$.

  * •

DPS: $F_{DPS}(r)=F_{TS}(r)$, and
$Z_{SPS}(r)=Pr^{-\alpha}\frac{\bar{\tau}_{d}}{\tau_{d}^{0}}(\nu G-L_{g})$.

For the proof, please refer to Appendix -C. $P_{bb}$ and $P_{I}$ denote the
transmit power of BB and IoT UEs, respectively.

######  Theorem 3.

(Maximally fair GPS weights). The values of the GPS weight $w_{d}$ which
maximize fairness among users are given by $w_{d}=\delta_{d}(1-\eta)$ for the
TS mode, and

$w_{d}=\delta_{d}\frac{\log_{2}\left(1+\frac{(1-\nu)PG\bar{r}^{-\alpha}}{N_{0}+(1-\nu)\bar{I}(\bar{r},k)}\right)}{\log_{2}\left(1+\frac{PG\bar{r}^{-\alpha}}{N_{0}+\bar{I}(\bar{r},k)}\right)}$

for the PS modes. $\bar{r}$ is the mean distance of UEs from the serving base
station, and $\bar{I}(\bar{r},k)$ is given by Theorem 1.

###### Proof.

In our GPS scheduler, $w_{d}$ is the ratio between the amount of time
dedicated to serving a BB UE and the one dedicated to serving an IoT user (for
both information transfer and power transfer). We need to ensure fairness in
resource scheduling for information transfer between BB and IoT users (as for
power transfer, only IoT users are involved). This is implemented by having
the ratio between the (mean) amount of bits received by a BB user and an IoT
user be equal to $\delta_{d}$, i.e. to the ratio between the target values of
throughput of the two user classes. Thus we have, for the TS case, in any
location at a distance $r$ from the serving base station,

$\delta_{d}=\frac{w_{d}R(r)}{(1-\eta)R(r)}$

where with $R(r)$ we denote the amount of bits received by the user at $r$
during the time that its serving bases station dedicates to it. We recall that
$(1-\eta)$ is the GPS weight dedicated to information transfer to an IoT UE.
This brings to $w_{d}=\delta_{d}(1-\eta)$. For the PS cases, we proceed in the
same manner. We write down the ratio between the amount of bits received from
the serving base station, using the Shannon capacity formula:

$\frac{w_{d}\log_{2}\left(1+\frac{PGr^{-\alpha}}{N_{0}+I(r,k)}\right)}{\log_{2}\left(1+\frac{(1-\nu)PGr^{-\alpha}}{N_{0}+(1-\nu)I(r,k)}\right)}$

However such a ratio is a function of $r$, and of the interference experienced
by the specific user considered. Thus, we consider the mean user, and set
$w_{d}$ in such a way as to satisfy the equality

$\delta_{d}=\frac{w_{d}\log_{2}\left(1+\frac{PG\bar{r}^{-\alpha}}{N_{0}+\bar{I}(\bar{r},k)}\right)}{\log_{2}\left(1+\frac{(1-\nu)PG\bar{r}^{-\alpha}}{N_{0}+(1-\nu)\bar{I}(\bar{r},k)}\right)}$

where we approximate the mean interference perceived by the user with the
expression in Theorem 1. ∎

##  V Energy-Optimal Network Configuration

One of the main open issues in SWIPT networks is to determine, as a function
of the main system parameters as well as of the energy and traffic demands,
how the main network parameters should be tuned in order to optimize the
energy consumed by the system. To this end, we present below the formulation
of the optimization problem which provides, for a given BS energy model, as
well as for a given user mean density, the energy optimal BS transmit power,
the optimal density of active BSs, as well as the optimal amount of BS time
dedicated to power transfer, which satisfy the performance constraints.

######  Problem 1.

(TS version)

$min_{P,\lambda_{b},\eta}\;\;\lambda_{b}\left[q_{1}+\frac{\bar{\tau}_{d}(\eta,P,\lambda_{b})}{\tau^{0}_{d}}\left(q_{2}+q_{3}(P-P_{min})\right)\right]$

Subject to:

$\displaystyle\frac{\bar{\tau}_{d}(\eta,P,\lambda_{b})}{\tau^{0}_{d}}\leq
1,\quad$
$\displaystyle\frac{\bar{\tau}_{u}(\eta,P,\lambda_{b})}{\tau^{0}_{u}}\leq 1$
(20) $\displaystyle P_{min}\leq P\leq$ $\displaystyle P_{max}$ (21)
$\displaystyle 0\leq\eta\leq$ $\displaystyle 1$ (22) $\displaystyle
CDF_{h}(h_{0},\eta,P)\leq$ $\displaystyle\mu$ (23) $\displaystyle
0\leq\lambda_{b}\leq$ $\displaystyle\lambda_{b,max}$ (24)

where $h_{0}$ in constraint (23) is the minimum harvested power required by
each IoT device to operate, computed based on the amount of energy required by
the device during a whole on-off cycle. $\mu$ is the maximum acceptable ratio
of IoT users which harvest less than $h_{0}$ Joules per second.
$\bar{\tau}_{d}$, $\bar{\tau}_{u}$ are given by Theorem 1. $\lambda_{b,max}$
derives from practical constraints to BS deployments in urban settings. In the
PS configurations, Problem 1 takes a very similar structure, where the
optimization variable $\nu$ takes the place of variable $\eta$, with
$0\leq\nu\leq 1$. In both formulations, Problem 1, is clearly non-convex and
nonlinear, and it cannot be solved efficiently.
A simple heuristic approach to solve Problem 1 is based on defining, for each
of the three decision variables, a discrete set of values (e.g., based on
practical considerations on the maximal accuracy with which they can be varied
in practical scenarios), and on an exhaustive grid search on those sets.
However, such a brute-force approach gets quickly unfeasible when the decision
variables are varied with fine granularity. Thus, for such large problems, a
metaheuristic is required [38]. To this end, we have adopted a Genetic
Algorithm (GA) approach [39], where a population of chromosomes, each
constituting the binary representation of a triplet of values for the three
decision variables, is evaluated by assigning a fitness value. Such a value is
a score indicating the optimality of the solution, and it is used to feed the
archive of the best solutions found so far, by picking up the dominant one
(_i.e._ , the chromosome with the highest fitness value, with respect to our
problem). Then a new population is generated using proper genetic operators
(Gaussian Mutation and Crossover, which for each couple of chromosomes returns
a child that lies halfway on the segment defined by the two parents) for the
mating and mutations of the randomly selected chromosomes. Such a sequence of
steps is repeated until a termination condition (i.e. on the maximum number of
iterations) is met. In this way, guided by the fitness function, the GA
realizes a random exploration of the solution space, based on the evolutionary
operators’ output.
To facilitate the convergence of the GA, the fitness function has been defined
through the penalty method [40], by which, in addition to the cost function of
1, a penalty is introduced, proportional to the degree by which constraints
(20) (downlink and uplink) and (22) are violated. This ensures that the
metaheuristic is able to explore values of the decision variables that are
very close to the border of the feasibility region of Problem 1. The resulting
fitness function has the following expression:

$v(P,\lambda_{b},\eta)=\lambda_{b}\left(q_{1}+\frac{\bar{\tau}_{d}(\eta,P,\lambda_{b})}{\tau^{0}_{d}}\right)[q_{2}+q_{3}(P-P_{min})]+k_{1}\left(\frac{\bar{\tau}_{u}(\eta,P,\lambda_{b})}{\tau^{0}_{u}}-1\right)^{+}-k_{2}[CDF_{h}(h_{0},\eta,P)-\mu]^{-}+$
$+k_{3}\left(\frac{\bar{\tau}_{d}(\eta,P,\lambda_{b})}{\tau^{0}_{d}}-1\right)^{+}$

where $(x)^{+}=max(x,0)$, and $(x)^{-}=min(x,0)$. The (positive) values of
$k_{1}$, $k_{2}$, and $k_{3}$ may be tuned empirically to ensure the desired
behavior of the GA metaheuristic in proximity to the border of the feasibility
region.

##  VI Numerical Results

In this section, we validate numerically our analytical results, and we
investigate the properties of different strategies for energy efficiency in
SWIPT networks.

###  VI-A Setup

We assumed base stations work at a frequency of $1.5$ GHz, and use a bandwidth
of $50$ MHz, compatible with 4G+ standards. Unless otherwise indicated, we
assume a percentage of IoT devices equal to $80\%$ of the total number of UEs
(representative of many present-day scenarios), a transmit power equal to
$0.2$ W for both IoT and BB UEs, and a frequency reuse factor of $3$.
Moreover, by default, we assumed a beamforming gain equal to $10$, and
constant over the whole main lobe aperture (which we assume to be equal to
$45$ degrees), and a path loss exponent equal to $3$, typical of urban areas.
For linear energy harvesting, we assumed a conversion efficiency of $0.9$,
with no lower/upper threshold. We assume the deployed base stations to be of
the macro type, with a transmit power that varies between $1$ and $11$ W.
Unless otherwise stated, we set a target mean per-bit delay in downlink for BB
(resp. IoT) UEs equal to $10^{-5}$ s (resp. $10^{-3}$ s), and in uplink equal
to $10^{-4}$ s for all UEs, (e.g., typical of IoT systems for environmental
monitoring [15]). We consider the user density to vary from $10^{-4}$ users
per $m^{2}$ (modeling settings with a high share of BB users) to $10^{-1}$
users per $m^{2}$ (modeling scenarios with crowds of BB UEs and high-density
IoT deployments). Unless otherwise specified, we assume IoT UEs to be active
all the time ($\phi=1$). We set to $5\%$ the maximum acceptable share of IoT
users who are not able to harvest the target minimum energy.

The parameters of the BS energy model are chosen to fit two different types of
BSs. The first type (labeled LLP – low load proportionality) reflects the
behavior of the majority of current stand-alone BSs, and it is characterized
by a $27\%$ load proportionality (with $q_{1}=1100$, $q_{2}=100$, and
$q_{3}=30$). Conversely, the high load proportionality (HLP) BS type (with
$q_{1}=482.3$, $q_{2}=48.23$, and $q_{3}=144.69$) corresponds to a $75\%$ load
proportionality, achievable, e.g., through time-domain duty-cycling at the
sub-system level, i.e., through micro-sleep techniques involving modules of
the BS or of the BBU in cloud-RAN designs [41]. For the sake of comparison,
these parameters were chosen to fit a per-BS maximum consumed power of $1500$
W, typical of stand-alone macro BSs [42]. The value of BS density has been
varied with a granularity of $10^{-4}$ $m^{-2}$, to guarantee good accuracy of
the GA search process. We assumed the mean number of users per base station to
be lower bound to $5$. This models the simple energy-saving strategy common
among MNOs, which switch off those BSs that serve very few users to no user at
all, as they represent a very high energy cost per user and a small benefit
for performance. The size of the initial population in the GA algorithm has
been set to $100$ chromosomes. This choice has proven to be a good compromise
between computational load and convergence speed. In the fitness function,
parameters $k_{1}$, $k_{2}$ and $k_{3}$ have been set to $1,000$, $5,000$, and
$1,000$, respectively, as these values have proven appropriate to enable our
GA approach to explore effectively the borders of the feasibility region. The
termination condition of the GA has been set based on the convergence of the
fitness value. Specifically, we assumed that convergence is reached when the
geometric average of the relative change in the value of the spread over $50$
generations is less than $10^{-6}$, and the final spread is less than the mean
spread over the past $50$ generations.

###  VI-B Linear Energy Harvesting

To assess the accuracy of our results, we simulated the system for several
values of user and BS densities, in the linear EH case. For each of the three
power harvesting configurations, and over different values of user density,
Fig. 2 shows the values of power per $km^{2}$ consumed by the network at the
optima derived by our GA, as well as those derived by simulating the system in
those optima.
As the figure shows, the power consumed by EH IoT users has a strong impact on
the total power consumed by the network, irrespective of the degree of energy
proportionality of BSs. Indeed, at low user densities, consumed power more
than doubles when the target minimum harvested power passes from $1$ to $6$
mW. At high user densities instead, the difference is smaller because the
power consumed to provide information transfer dominates the overall energy
footprint of the network. This is due also to the growing inefficiency of
information transfer at high user densities, due to rising interference
levels.
Fig. 2 suggests also that our modeling approach is very accurate across very
diverse system configurations. In particular, very good accuracy is achieved
even for low values of user density, for which the dense IoT regime assumption
of Theorem 2 does not hold.

(a) Time Switching

(b) Dynamic Power Splitting

(c) Static Power Splitting

Figure 2:  Power per $km^{2}$ consumed by the network at the optimum vs user
density, for different target minimum harvested power and IoT UE receiver
configurations. Markers denote values from simulations, derived with a $95\%$
confidence interval of at most $8\%$.

These results suggest that, in sharp contrast to RANs delivering only
connectivity, as the density of IoT users increases, their contribution to the
service capacity of the whole SWIPT network (in the form of power delivered to
other IoT users) increases too. The effect is visible in all configurations
when comparing the setting in which the energy harvested originates only from
BS transmissions (the ”no UPC” curves in Fig. 2), with those in which users
can harvest energy also from transmissions of other users. In all
configurations, we find that as user density increases, its effects manifest
as a decrease in the power consumed by the network. Such a decrease continues
until a minimum is reached, after which the power consumed increases again
with increasing user density. Such behavior is quite surprising, but it is the
result of the interplay between, on one side, the detrimental effect of rising
interference on communications (which decreases the efficiency with which
network resources are used), and the beneficial effect of the increase in user
transmissions on the amount of power harvested by IoT users. Indeed, when
increasing user density from very low values, these two contrasting effects
cause the overall power consumed by the network to decrease, before increasing
again due to the effects of high levels of interference on user-perceived QoS
for data. The impact of user contribution to network service capacity is less
marked when the target minimum service capacity is low (e.g., in the $1$ mW
plots), due to the low impact of power delivery on the overall consumed power
of the network, and it increases for higher target minimum harvested power.
Such impact is limited also in the SPS mode, due to the lower efficiency of
passive EH in that configuration.

Figure 3:  Power consumed by the network for each user (broadband or IoT) vs.
user density, for different target minimum harvested power and IoT UE receiver
configurations.

Figure 4:  Increase in consumed power with respect to linear EH, as a function
of user density, for HLP $1$ mW target minimum harvested power, and different
IoT UE receiver configurations.

These features of optimal power consumption suggest that (possibly dynamic)
configuration tuning holds the potential to play a key role in minimizing the
energy footprint of a SWIPT network. Indeed, these features suggest that by
taking advantage of user-provided service capacity, it is possible to make the
network increasingly more energy efficient when the demand (of both energy and
communication) it must satisfy grows. This is suggested also by Fig. 4, which
shows indeed that for all configurations and target minimum harvested power,
the network energy efficiency increases when user density increases. Here too,
when the user-provided service capacity dominates the overall energy footprint
of the network, the decrease of per-user energy consumed is faster, while it
tends to slow down for those user densities for which the effects of
interference dominate. Quite surprisingly, for the same target minimum
harvested power, when user-provided service capacity dominates, the power per
user consumed by the network does not depend on the amount of load
proportionality of the BS energy model (red and blue curves are very close).
It is only when interference dominates that the higher BS densities required
to satisfy the target QoS for information transfer bring an increase in the
impact of the fixed energy costs of BSs on the overall energy footprint of the
network.
The importance of passive energy harvesting in SWIPT networks is also visible
from Fig. 4, which shows the increase in the network consumption when only
active charging from the serving BS is available. Indeed, when passive EH is
not available (e.g., due to limitations in the HW architecture of devices) the
energy consumed by the network can be several times larger than in the case in
which passive EH is available.
Another noteworthy feature of a SWIPT network, emerging from Fig. 2 is that
for each configuration of EH receiver architecture and target minimum
harvested power, there is a value of user density below which no feasible
solution of Problem 1 exists. This is due to the lower bound to the mean
number of users per base stations (equal to $5$ in our evaluations). Indeed,
as with increasing distance from the serving base station the power received
by users decays much faster than channel capacity, guaranteeing a target QoS
for harvested power at low user densities requires ever increasing BS
densities and thus very low mean number of users per BS, up to the point of
economic unfeasibility. Note however that this is relative to a scenario in
which users are distributed uniformly at random in space. In realistic
settings, this implies that deployments of small amounts of EH IoT devices are
of course feasible, but they should be, at least locally, dense enough to be
able to deliver power with the target QoS while satisfying the system
constraints.

(a) Time Switching

(b) Dynamic Power Splitting

(c) Static Power Splitting

Figure 5: Optimal BS density for different target minimum harvested power and
IoT UE receiver configurations.

(a) Time Switching

(b) Dynamic Power Splitting

(c) Static Power Splitting

Figure 6: Optimal transmit power for different target minimum harvested power
and IoT UE receiver configurations.

(a) Time Switching

(b) Dynamic Power Splitting

(c) Static Power Splitting

Figure 7:  Optimal time splitting factor (for TS) and power splitting factor
(for DPS and SPS) for different target minimum harvested power and IoT UE
receiver configurations.

In Fig. 7, 7, and 7 we plotted the optimal values of BS density, of transmit
power, and of time (respectively, power) split ratio, for the three EH
receiver configurations. As Fig. 7 shows, in the TS and DPS cases the
evolution of the optimal BS density as a function of user density is
characterized by three regimes. At first, for very low user densities, the
optimal BS density increases with user density as expected. For higher user
densities, the optimal BS density reaches a plateau in the case without user-
supplied service capacity, and it decreases in the case in which the full
potential of passive EH is available. This suggests (quite surprisingly!)
patterns of BS sleep modes which, in contrast to those for non-SWIPT RANs,
turn off BSs when demand increases. Finally, for even larger user densities,
in the interference-dominated regime, the optimal BS density increases again.
These three regimes are not present however for all values of target minimum
harvested power, due to the minimum user-per-BS limit adopted. These three
regimes are not present in the SPS case, in which the lower efficiency of
passive EH brings optimal BS densities which increase monotonically with user
density.
From Fig. 7 we see that the optimal transmit power varies with user density in
a specular manner to optimal BS density, decreasing when it increases, and
vice versa. That is, the energy optimal strategies emerging from our GA
algorithm tend to compensate for the thinning (resp. densification) of BS
density with cell zooming (resp. shrinking), via transmit power tuning.
Fig. 7 offers some key insights on how tuning split factor contributes to
achieving energy-optimal operation of a SWIPT network. For low-load
proportional BSs, as well as for configurations with high target minimum
harvested power, the bulk of the energy consumed must be supplied by the
serving BS, which thus has to allocate the majority of its time to
transferring energy to IoT users. As expected, such a share of BS time is
larger in scenarios with a larger target minimum harvested power. However, in
all configurations, for increasing user densities, the share of BS time
dedicated to active charging decreases, thanks to the combined effect of user-
supplied service capacity, and higher BS density. Beyond a given user density
(which depends on the BS energy model and receiver EH architecture) the SWIPT
network effectively stops actively delivering power to IoT users (the split
factor gets values close to zero), as passive power supply from ambient
sources is sufficient to achieve the target QoS for energy harvesting. Again,
note that for the SPS configurations, as harvesting from user transmissions is
less efficient, the decrease in the splitting factor with increasing user
density is substantially slower than in TS or DPS.

###  VI-C Impact of harvesting nonlinearities

A crucial aspect of the energy harvesting process is the presence of
nonlinearities in the relation between the output power of the energy
harvester circuit at the IoT devices and its input power. Indeed, such aspect
may affect heavily the amount of harvested power as a function of the network
settings, and thus the energy optimal configuration of the whole SWIPT
network. To investigate its impact on the energy optimal strategies derived by
our GA, in another set of experiments we have considered the energy harvesting
curve given by Equation 1. Following indications from the literature [30], we
have set the maximum output power $h_{max}$ to $10$ mW, $\chi$ to $274$, and
the sensitivity threshold $h_{s}$ to $0.064$ mW. Figure 9 shows the nonlinear
EH curve for the chosen parameters and for two values of $\iota$, which tunes
the maximum efficiency of the nonlinear harvesting process. Its values have
been chosen not to have a maximum efficiency larger than one, and to
approximate the linear harvesting curve which we have considered in the
previous experiments. In the figure indeed, we have also plotted the linear
harvesting functions whose slope approximates well that of the two sigmoids.
As the plot shows, both sigmoids are characterized by a range of input power
within which they are well approximated by a linear function (and which we
denote henceforth as a ”quasilinear regime”). Note that, with the given
settings, choosing $\iota=0.9$ (respectively, $1.30$) produces a sigmoid which
in the quasilinear regime approximates well a linear model with slope $0.9$
(respectively, $0.77$).

Figure 8: Harvested power as a function of power received by IoT EH user, for
a linear (with conversion efficiency of $0.9$ and $0.77$ respectively) as well
as for the nonlinear model (Equation 1) with $h_{s}=0.064$ mW, and $\chi=274$.

Figure 9: Increase in consumed power with respect to linear EH, as a function
of user density, for different values of the parameter $\iota$ of the
nonlinear EH curve, and for different network setups, in the time splitting
configuration.

Figure 9 illustrates the increase in power consumed at the optimum induced by
the nonlinear EH function versus that of its linear approximation, for the
given values of $\iota$, and for time splitting EH mode. As the plot suggests,
nonlinearity brings to an increase of the consumed power for target minimum
harvested power values which are much lower than $h_{max}$, and for low values
of user density. Indeed, for a target harvested power of $1$ mW the sigmoid is
in a regime in which the conversion efficiency is $20\%$ to $30\%$ less than
its maximum value, pushing the system to compensate with a more conservative
transmit power allocation, thus bringing to a higher overall consumed power.
This is more evident for lower user densities, as in those regimes the lower
mean amount of users per base station brings (by the law of small numbers) to
larger variance and thus to greatly amplify the effect of the $20\%$ to $30\%$
decrease in conversion efficiency on those users whose harvested power falls
on the tail of the distribution.

##  VII Conclusions

In this paper, we proposed an analytical framework for the characterization of
the performance of a SWIPT network serving a combination of broadband users
and EH IoT devices, and for the identification of energy-optimal network
configurations satisfying constraints on user-perceived QoS. A novel model
that includes the usage of non-linear harvesting has been introduced taking
into consideration aspects related to sensitivity and maximum capacity.
Numerical results suggest that substantial energy savings are possible with
schemes that adapt the main network parameters to fluctuations in user
density. We also showed that such schemes remain essential, even in more
energy-proportional cloud-RAN settings. The most interesting insight provided
by our results concerns the interplay between energy harvesting and
interference in cases of extreme densities of IoT devices. In such cases, we
observed that very high device densities facilitate energy harvesting before
interference becomes a problem. As a result, the optimal base station density
can be very low even when the network load is high. This implies that base
station sleep modes in SWIPT networks should follow different dynamics than
those of networks that do not provide energy harvesting to IoT devices.

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###  -A Proof of Lemma 1

We present here only the derivation of the TS case, as the proof in the SPS
and DPS cases is a minor variation of the former. Let us consider that BS
utilization is equal to 1. The power of the signal received by the user at $x$
by the base station, to get charged wirelessly, is $P_{BS}D(x)^{-\alpha}$.
$N_{iot}(S(x))+\delta_{d}(1-\eta)N_{bb}(S(x))$ is the sum of the GPS weights
of all users served by the base station serving the user at $x$. $\eta$ is the
product of the GPS weight of the IoT user by the factor $\eta$, which denotes
that fraction of BS time dedicated to the IoT user which is devoted to active
power transfer (to distinguish it from passive power transfer, i.e. from power
harvested by ambient radiation). Thus
$K(x)=\eta[N_{iot}(S(x))+\delta_{d}(1-\eta)N_{bb}(S(x))]^{-1}$ is the fraction
of total base station time dedicated to actively charging the user at $x$,
when downlink BS utilization is 1. In the more general case in which base
station utilization is $U_{d}(S(x))\leq 1$, the expression for $K(x)$ is
$K(x)=\eta U_{d}(S(x))[N_{iot}(S(x))+\delta_{d}(1-\eta)N_{bb}(S(x))]^{-1}$.
The amount of energy received by the user at $x$ during such time, and due to
active charging, is therefore $\xi P_{BS}D(x)^{-\alpha}GK(x)$.
$\frac{K(x)(1-\eta)}{\eta}$ is the fraction of total base station time
dedicated to transmitting information to the IoT user at $x$, and it accounts
for downlink base station utilization. Thus,
$\left(1-\frac{K(x)(1-\eta)}{\eta}\right)$ is the fraction of time during
which the user at $x$ can harvest energy, because it is not receiving
information. The fraction of time during which the BS to which the user at $x$
is associated is active and not serving it is
$U_{d}(S(x))\left(1-\frac{K(x)}{U_{d}(S(x))\eta}\right)$. The sources of
energy from which IoT users can charge passively are 1) the serving BS, due to
downlink transmission to other users belonging to the same cell; 2) other base
stations; 3) users in the uplink, from all cells. The amount of power received
due to (1) is thus obtained by the product of the power received by $x$ when
its BS is serving other users, $P_{BS}D(x)^{-\alpha}L$, multiplied by the
fraction of time during which the BS is active and serving other users:

$\xi
P_{BS}D(x)^{-\alpha}LU_{d}(S(x))\left(1-\frac{K(x)}{U_{d}(S(x))\eta}\right)$

The contribution (2) is given by $\xi I(D(x))$ multiplied by
$\left(1-\frac{K(x)(1-\eta)}{\eta}\right)$, i.e. by the fraction of base
station time dedicated to other than transmitting information to the user at
$x$. Note that $I(D(x))$ account also for the downlink utilization of the
interfering base stations. As for the contribution (3), we denote it with $\xi
O(x)\left(1-\frac{K(x)(1-\eta)}{\eta}\right)$.

###  -B Proof of Theorem 1

We consider the user at zero, but we drop this indication in what follows for
ease of notation (i.e. $S(0)$ becomes $S$, and $D(0)$ becomes $D$). We start
by computing the Palm expectation of the expression in (9):

$\bar{\tau}_{d}=E^{0}\bigg{[}\frac{N_{iot}(S)+w_{d}N_{bb}(S)}{w_{d}C(D,I)}\bigg{]}=\int_{0}^{\infty}E^{0}[\frac{N_{iot}(S)+w_{d}N_{bb}(S)}{w_{d}C(D,I)}|r\leq
D\leq r+dr]P(r\leq D\leq r+dr)$
$\approx\int_{0}^{\infty}\frac{E^{0}[N_{bb}(S)+w_{d}N_{iot}(S)|r\leq D\leq
r+dr]}{w_{d}C(r,\bar{I}(r,k))}P(B(0,r)=\phi)\lambda_{b}2\pi rdr$

where $\bar{I}(r,k)$ is the average interfering power for the typical user at
$r$, given by $\bar{I}(r,k)=\frac{PL\lambda_{b}2\pi
r^{2-\alpha}}{k(\alpha-2)}\frac{\bar{\tau}_{d}}{\tau_{d}^{0}}$ [43]. Thus
$\bar{\tau}_{d}\approx\int_{0}^{\infty}\frac{E^{0}[N_{bb}(S)+w_{d}N_{iot}(S)|r\leq
D\leq r+dr]}{w_{d}C(r,\bar{I}(r,k))}e^{-\lambda_{b}\pi r^{2}}\lambda_{b}2\pi
rdr$ , where $P(B(0,r)=\phi)$ is the probability that a ball of radius $r$
centered at the origin is empty. Let $N_{tot}(S)=N_{bb}(S)+N_{iot}(S)$. The
palm expectation at the numerator becomes $QE^{0}[N_{tot}(D)|r\leq D\leq
r+dr]$, with $Q=w_{d}+\gamma(\phi-w_{d})$. The random variable $N_{tot}(D)$ is
the total number of users present in the same cell as the user at the origin,
when his distance from its serving base station is $D$. As users are
distributed according to a PPP with intensity $\lambda_{u}$, $N_{tot}(D)$ is
Poisson, with an intensity given by the conditional Palm expectation inside
the integral. Using Campbell’s formula [33], this expectation becomes

$E^{0}\bigg{[}\int_{0}^{\infty}\int_{0}^{2\pi}\mathbf{1}_{(S(x,\theta)=S|r\leq
D\leq r+dr)}\lambda_{u}d\theta
xdxQ\bigg{]}=\int_{0}^{\infty}\int_{0}^{2\pi}\lambda_{u}P(S(x,\theta)=S|r\leq
D\leq r+dr)d\theta xdxQ$

The conditional probability within the integral is given by
$\int_{0}^{\infty}\int_{0}^{2\pi}\lambda_{u}e^{-\lambda_{b}A(r,x,\theta)}d\theta
xdx$, where $A(r,x,\theta)$ is given by:

$A(r,x,\theta)=\pi
x^{2}-\bigg{[}r^{2}\arccos\left(\frac{r+x\sin(\theta)}{d(r,x,\theta)}\right)+x^{2}\arccos\left(\frac{x+r\sin(\theta)}{d(r,x,\theta)}\right)+$
$-\frac{1}{2}(-(d(r,x,\theta)-x)^{2}+r^{2})^{\frac{1}{2}}((d(r,x,\theta)+x)^{2}-r^{2})^{\frac{1}{2}}\bigg{]}$

where d is the distance [44]. By substituting, we get the expression for
$\bar{\tau}_{d}$.

The derivation of $\bar{\tau}_{u}$ follows along the same lines.The
expressions of $\bar{\tau}_{d}$ and $\bar{\tau}_{u}$ obtained are implicit
(i.e. they are a function of these same parameters). Thus they constitute a
fixed point problem. It is easy to see however that the mapping operator
associated with this fixed point is contractive, and thus that the problem
admits a unique solution.

###  -C Proof of Theorem 2

######  Lemma 2.

The average power received from users for the typical user arriving in the
system at a distance $r$ from the serving BS is approximated by

$\bar{O}=\frac{(1-\gamma)\delta_{u}P_{bb}+\phi\gamma
P_{I}}{(1-\gamma)\delta_{u}+\phi\gamma}\frac{\lambda_{b}\pi\alpha}{\alpha-2}\frac{\bar{\tau}_{u}}{\tau_{u}^{0}}$
(25)

###### Proof.

The mean transmit power from a user is
$\frac{(1-\gamma)\delta_{u}P_{bb}+\gamma\phi
P_{I}}{(1-\gamma)\delta_{u}+\gamma\phi}$. Let $x^{\prime}_{i}(t),i\in\chi(j)$
be the position of the $i$-th user served by BS $j$ at time t,
$d(x,x^{\prime}_{i}(t))$ denote the distance between the two users considered,
and $u(x^{\prime}_{i}(t))$ the probability that the given user is transmitting
at time $t$. Then we can write the expression of $O(x,t)$ as

$\sum_{j}\sum_{i\in\chi(j)}\frac{(1-\gamma)\delta_{u}P_{bb}+\gamma\phi
P_{I}}{(1-\gamma)\delta_{u}+\gamma\phi}d(x,x^{\prime}_{i}(t))^{-\alpha}u(x^{\prime}_{i}(t))$
(26)

where $\chi(j)$ is the set of users served by BS j. As we assume the base
station has utilization $\frac{\bar{\tau}_{u}}{\tau_{u}^{0}}$ in the uplink,
$u(x^{\prime}_{i}(t))=\frac{1}{\lambda_{u}A_{j}}\frac{\bar{\tau}_{u}}{\tau_{u}^{0}}$,
where $A_{j}$ is the area of the Voronoi cell of BS j. Given that users are
uniformly distributed in the plane, the palm expectation of (26) is well
approximated by

$\frac{(1-\gamma)\delta_{u}P_{bb}+\gamma\phi
P_{I}}{(1-\gamma)\delta_{u}+\gamma\phi}E\left[\sum_{j}\frac{\int_{x^{\prime}\in
A_{j}}d(x,x^{\prime})^{-\alpha}dx^{\prime}}{\lambda_{u}A_{j}}\right]\frac{\bar{\tau}_{u}}{\tau_{u}^{0}}=$

As $A_{j}$ is statistically independent on $d(x,x^{\prime})$, the approximated
formula becomes

$=\frac{(1-\gamma)\delta_{u}P_{bb}+\gamma\phi
P_{I}}{(1-\gamma)\delta_{u}+\gamma\phi}\frac{\int_{0}^{+\infty}\lambda_{u}min(1,s^{-\alpha})2\pi
sds}{\lambda_{u}\bar{A}}\frac{\bar{\tau}_{u}}{\tau_{u}^{0}}$

where $\bar{A}$ is the mean area of a Voronoi cell for a BS density of
$\lambda_{b}$, which is equal to $\lambda_{b}^{-1}$. Assuming $\alpha>2$, we
get

$=\frac{(1-\gamma)\delta_{u}P_{bb}+\gamma\phi
P_{I}}{(1-\gamma)\delta_{u}+\gamma\phi}\frac{\lambda_{b}\pi\alpha}{\alpha-2}\frac{\bar{\tau}_{u}}{\tau_{u}^{0}}$

∎

###### Proof.

(Theorem 2) Let us rewrite expressions (5) to (7) in the form
$h(x)=F(x)+K(x)Z(x)$. We have:

  * •

In the TS mode, $F_{TS}(x)=PD(x)^{-\alpha}LU_{d}(x)+I(x)+O(x)$ and
$Z_{TS}(x)=PD(x)^{-\alpha}U_{d}(x)(G\eta-L_{g})-U_{d}(x)(1-\eta)(I(x)+O(x))$.

  * •

In the SPS mode, $F_{SPS}(x)=\nu F_{TS}(x)$ and $Z_{SPS}(x)=\nu
PD(x)^{-\alpha}U_{d}(x)(G-L_{g})$.

  * •

In the DPS mode, $F_{DPS}(x)=F_{TS}(x)$ and
$Z_{SPS}(x)=PD(x)^{-\alpha}U_{d}(x)(\nu G-L_{g})$.

Let us consider first the TS case. Let’s compute $E^{0}[F(D)|r\leq D\leq
r+dr]$. In the TS case, $\forall x$, we approximate
$U_{d}(x)=\frac{\bar{\tau}_{d}}{\tau_{d}^{0}}$. Thus
$PE^{0}[D(x)^{-\alpha}LU_{d}(x)|r\leq D\leq
r+dr]=Pr^{-\alpha}L_{g}\frac{\bar{\tau}_{d}}{\tau_{d}^{0}}$. As for the
remaining terms, $O(x)$ does not depend on the user distance from its serving
BS. The expression of its expected value is thus given by Lemma 2. As for the
conditional expectation of $I(x)$, it is given by (19) in Theorem 1,
multiplied by reuse factor $k$. We thus have $E^{0}[I(x)+O(x)|r\leq D\leq
r+dr]=k\bar{I}(r,k)+\bar{O}$. The SPS and DPS cases are derived similarly. As
for $K(x)Z_{TS}(x)$, $Z(x)$ is a function of $D$, except for $O(x)$, for which
we approximate $E^{0}\left[K(D)O(D)|r\leq D\leq
r+dr\right]=E^{0}\left[K(D)|r\leq D\leq r+dr\right]O(r)$. We thus have

$E^{0}\left[K(D)|r\leq D\leq
r+dr\right]Z(r)=E^{0}\left[\frac{1}{QN_{tot}(D)}|r\leq D\leq r+dr\right]Z(r)$

For Jensen’s inequality,  $E^{0}\left[\frac{1}{N_{tot}(D)}|r\leq D\leq
r+dr\right]\leq\frac{1}{E^{0}[N_{tot}(D)|r\leq D\leq r+dr]}$ . For a given
cell, $N_{tot}(D)$ is Poisson distributed. Thus the ratio of the standard
deviation over the mean of this variable decreases with increasing user
density. Therefore in the dense IoT regime, such an inequality is tight.
therefore, the denominator is then computed as a function of $r$ as in the
proof of Theorem 1. The derivation of the expressions for the DPS and SPS
cases follows along the same line. ∎