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Joint Uplink and Downlink Coverage Analysis
of Cellular-based RF-powered IoT Network
Mustafa A. Kishk and Harpreet S. Dhillon
Abstract
Ambient radio frequency (RF) energy harvesting has emerged as a promising solution for powering
small devices and sensors in massive Internet of Things (IoT) ecosystem due to its ubiquity and cost
efficiency. In this paper, we study joint uplink and downlink coverage of cellular-based ambient RF
energy harvesting IoT where the cellular network is assumed to be the only source of RF energy. We
consider a time division-based approach for power and information transmission where each time-slot
is partitioned into three sub-slots: (i) charging sub-slot during which the cellular base stations (BSs) act
as RF chargers for the IoT devices, which then use the energy harvested in this sub-slot for information
transmission and/or reception during the remaining two sub-slots, (ii) downlink sub-slot during which
the IoT device receives information from the associated BS, and (iii) uplink sub-slot during which the
IoT device transmits information to the associated BS. For this setup, we characterize the joint coverage
probability, which is the joint probability of the events that the typical device harvests sufficient energy
in the given time slot and is under both uplink and downlink signal-to-interference-plus-noise ratio
(SINR) coverage with respect to its associated BS. This metric significantly generalizes the prior art on
energy harvesting communications, which usually focused on downlink or uplink coverage separately.
The key technical challenge is in handling the correlation between the amount of energy harvested
in the charging sub-slot and the information signal quality (SINR) in the downlink and uplink sub-
slots. Dominant BS-based approach is developed to derive tight approximation for this joint coverage
probability. Several system design insights including comparison with regularly powered IoT network
and throughput-optimal slot partitioning are also provided.
Index Terms
Stochastic geometry, Internet of Things, ambient RF energy harvesting, cellular network, Poisson
Point Process.
The authors are with Wireless@VT, Department of ECE, Virginia Tech, Blacksburg, VA (email: {mkishk, hdhillon}@vt.edu).
The support of the U.S. NSF (Grants CCF-1464293, CNS-1617896, and IIS-1633363) is gratefully acknowledged. This paper
was presented in part at the IEEE Globecom, Washington DC, 2016 [1]. Manuscript last updated: May 22, 2017.
arXiv:1705.06799v1 [cs.IT] 18 May 2017
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I. INTRODUCTION
Internet of Things (IoT) is a massive ecosystem of interconnected things (referred to as IoT
devices) with sensing, processing, and communication capabilities [2]. Due to its ubiquity, cellu-
lar network has emerged as an attractive option to provide reliable communication infrastructure
for supporting and managing these networks [3]–[6]. This new communication paradigm will
enable a new era of applications including medical applications, transportation, surveillance,
and smart homes to name a few. Unlike human-operated cellular devices, such as smart phones
and tablets, that can be charged at will, these IoT devices may be deployed at hard-to-reach
places, such as underground or in the tunnels, which makes it difficult to charge or replace
their batteries. This has led to an increasing interest in energy-efficient communication of IoT
devices, both from the system design [4]–[6], and hardware perspectives [7]. While these efforts
will increase the lifetime of these devices, they do not necessarily make them self-sustained in
terms of their energy requirements. One possible way to develop an almost self-perpetuating
IoT network is to complement or even circumvent the use of conventional batteries in the IoT
devices by energy harvesting. While one can use any energy harvesting method depending upon
the deployment scenario, such as solar energy, thermo-electronic, and mechanical energy [8],
we focus on the ambient RF energy harvesting [9], [10], where the IoT device harvests energy
through wireless RF signals. This is because of the ubiquity of RF signals even at hard-to-reach
places where the other popular sources, such as solar or wind, may not be available. Besides,
RF energy harvesting modules are usually cheaper to implement, which is another consideration
in the deployment of IoT devices [11]. Now if RF energy harvested from the communication
network (cellular network in this case) is the only source of energy, there will obviously be some
new design considerations due to the limitations in the energy availability and the correlation in
the communication and energy harvesting performance [12]. In this paper, we concretely expose
these design considerations using tools from stochastic geometry. In particular, we define and
analyze a new joint coverage probability metric, which significantly generalizes prior art in this
area. Before going into the details of our contributions, we discuss prior art next.
A. Prior Art
Owing to their remarkable tractability and realism, tools from stochastic geometry have
received significant attention over the past few years for the system-level analysis of cellular
networks. Interested readers are advised to refer to [13]–[16] and the references therein for a
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more pedagogical treatment of this topic. More relevant subset of these works for this paper
is the one that focuses on characterizing the performance of energy harvesting communication
networks; see [17]–[22] for a small subset. In this Subsection, we will discuss these works in
the broader context of uplink, downlink, and joint uplink/downlink coverage analyses.
Uplink analysis. Most of the stochastic geometry-based works in this area are focused on
the setups in which the device of interest first harvests ambient RF energy and then transmits
information to its designated node (which will be its serving BS in the uplink cellular network)
using this energy. Since the device that harvests energy is also the one that transmits information,
we discuss all these works under the category of uplink analysis to put things in the correct
context. The general theme of these works is to study the joint energy and uplink SINR coverage,
which is defined as the joint probability of harvesting sufficient ambient RF energy to enable
uplink transmission, and having uplink SINR above a predefined threshold. The energy and
uplink SINR coverage events are independent by construction if one assumes that the ambient RF
sources are placed independently of the communication network [17]–[19]. A few representative
works in this direction are discussed next. Authors in [17] studied a system of energy harvesting
wireless sensor network where a sensor node harvests ambient RF energy from the broadcast TV,
radio, and cellular signals. The sensor node uses this energy to transmit information to a data
sink located at a fixed distance. Authors in [18] studied a point-to-point (source-destination)
communication link consisting of an energy harvesting source that is powered by a power
beacon (PB). In particular, the source harvests power from the RF signals of PB using which
it transmits information to its destination. The assumption of the existence of dedicated PBs
was then generalized in [19] which studied the uplink performance of a cellular network in
which mobile users are powered by a network of PBs. The other general setup, in which the
prior art is significantly sparser, is the one where the same network of BSs is used for charging
and communication [20], [21]. This naturally correlates the energy and uplink SINR coverage
events. However, to maintain tractability, all prior works study energy and uplink coverage events
separately with [20] justifying it by assuming full channel inversion power control. While such
simplifications may work in specific system setups, it is desirable to handle correlation in the
two coverage events properly, which will be done as a special case of our analysis.
Downlink analysis. Another general theme in the literature is to explore setups in which the
device of interest first harvests ambient RF energy and then uses it to receive information. We will
discuss all these works under the general category of downlink analysis. In small devices with
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severely limited power budgets, which is the case for IoT devices, energy consumption during
information reception can be almost as important as the energy consumption during uplink
transmission. For instance, many recent works have shown that receiver energy consumption
scales noticeably with the data rates due to increase in the length of decoder interconnects [23]–
[25]. Motivated by this general fact, some aspects of system design have already been explored
with the consideration of receiver energy consumption, e.g., see [26]–[28] for a subset. For
instance, authors in [26] used tools from stochastic geometry to study the SINR outage probability
and average energy harvested under power splitting at the receiver in a system of randomly placed
transmitter-receiver pairs where each transmitter has a unique receiver at a fixed distance. The
main objective is to minimize the SINR outage probability subject to a constraint on the minimum
average harvested energy. Authors in [27] explored power splitting receiver architecture in a
point-to-point system to study the tradeoff between the average harvested energy and the average
data rate. For this setup, the achievable rate-energy regions are also derived for different types
of receiver architectures. Finally, [28] explored power control policies for outage minimization
in a point-to-point link assuming energy harvesting at both the transmitter and the receiver. The
outage is said to occur if the signal-to-noise ratio (SNR) is low or the energy harvested at the
transmitter or receiver is not high enough. Contrary to all these works, which are more applicable
to ad hoc or decentralized networks, the joint analysis of harvested energy and downlink SINR
in a cellular setup was recently performed in [29], [30]. In [29], since the exact analysis does
not provide insightful results, authors use Frechet’s inequality to derive an upper bound on the
joint downlink energy and SINR coverage probability. In this paper, we will derive joint energy
and downlink SINR coverage probability as the special case of our general result.
As is evident from the above discussion, all the prior works on stochastic geometry-based
analyses of cellular networks with energy harvesting users/devices are either focused on uplink
or downlink. To the best of our knowledge, there is no work that deals with joint uplink/downlink
coverage probability defined by the joint energy, uplink SINR, and downlink SINR coverage
probability, which is the main focus of this paper. That being said, the joint downlink and uplink
coverage has received some attention recently in the regularly powered networks1[31]–[33]. For
instance, authors in [31] use a 3GPP simulation model to determine whether it is appropriate
to assume independence in the uplink and downlink coverage events. The simulation results
1Throughput this paper, we will refer to the IoT networks in which the IoT devices have uninterrupted access to a reliable
energy source, such as power grid or a battery, as the regularly powered networks.
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demonstrate that the two events cannot be treated as independent. This is due to the correlation
that results from associating with the same BS in both uplink and downlink. Sometimes this
correlation is ignored in the interest of tractability. For instance, in [32], authors derive the
joint uplink/downlink coverage probability as the product of two coverage probabilities. For
more accurate analysis, one should of course capture this correlation explicitly, as done in [33],
where the authors provided the accurate joint distribution of uplink and downlink path-loss
for generalized uplink/downlink cell association policies (associating with the same BS in both
channels is a special case). Assuming independent interference levels over uplink and downlink
channels, they use this joint distribution to derive the joint uplink/downlink coverage.
In this paper we study the performance of on-the-fly reception/transmission in a cellular-based
IoT network where the IoT devices first harvest energy and then use it to receive/transmit infor-
mation in the same time slot. Assuming cellular transmissions to be the only source of RF energy
for the IoT devices, we study the joint probability of a typical IoT device harvesting sufficient
energy and achieving both uplink and downlink SINR thresholds with respect to its associated
base station in a given time slot. As noted already, we will refer to this as uplink/downlink
coverage probability in this paper. Since the same infrastructure (cellular BSs) is used for
charging and communication, there is inherent correlation in the energy and uplink/downlink
coverage events, which is carefully incorporated in our analysis. Please refer to Section II for
more details on the system setup. We now summarize the contributions of this paper.
B. Contributions and Outcomes
Cellular-based IoT model. We develop a comprehensive model for cellular-based RF-powered
IoT network in which the locations of the BSs and the IoT devices are modeled using two
independent Poisson point processes (PPPs). Each time slot is assumed to be partitioned into
three sub-slots: (i) charging sub-slot, in which the received power from the cellular network
is used for charging devices to enable them to perform information transmission/reception in
the next two sub-slots, (ii) downlink sub-slot, in which the devices receive information from
their associated BSs, and (iii) uplink sub-slot, in which the devices transmit information to
their associated BSs using fractional channel inversion power control. Contrary to the prior
works discussed above that focused on the separate analysis of uplink and downlink coverage,
in this paper we focus on the analysis of joint uplink/downlink coverage (defined as the joint
probability of energy coverage, uplink SINR coverage, and downlink SINR coverage). Since
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cellular network is assumed to be the only source of RF energy for the IoT devices, the energy
and uplink/downlink coverage events are tightly coupled through the locations of the cellular
BSs. In particular, the amount of energy harvested by each device is highly correlated with
both the uplink and downlink SINR achieved by that device. Naturally, the uplink and downlink
coverage events are also coupled. As discussed next, we carefully handle this correlation in our
analysis, which is also one of the main technical contributions of this paper.
Joint uplink/downlink coverage analysis. As stated already, we define joint uplink/downlink
coverage as the joint probability that the typical device harvests sufficient energy in the first
sub-slot, achieves high enough downlink SINR in the second sub-slot, and achieves high enough
uplink SINR in the third sub-slot. These three events are correlated because of their dependence
on the point processes modeling the devices and the base stations. That being said, if we assume
independent fading across the three sub-slots and condition on the point processes, the three
events become conditionally independent. We therefore, derive the conditional probabilities of the
three events first. The complexity of this problem should be evident from the following two facts:
(i) the exact characterization of uplink SINR in a conventional single-tier cellular setup is not
known in the stochastic geometry literature [13], and (ii) the total energy harvested is essentially
a power-law shot noise field whose probability distribution function is not known in general. On
top of these challenges, we need to jointly decondition (average) over the point processes in order
to obtain the joint uplink/downlink coverage, which adds to the complexity of the problem. We
overcome all these challenges by developing a dominant BS-based approximation approach that
not only provides a tight approximation for the power-law shot noise field (energy harvested) but
also facilitates joint deconditioning over the point processes. The tightness of the approximate
joint coverage expression is verified by comparing it with the simulation results.
Useful system insights. Our analytical results provide several useful system insights. First, we
demonstrate the existence of optimal time-slot partitioning that maximizes system throughput.
The effect of other system parameters on this optimal partitioning is studied numerically. We
then compare the performance of the RF-powered IoT system with the one in which IoT devices
have access to a reliable power source (termed regularly powered network). Our analytical
results reveal several interesting thresholds beyond which the performance of this RF-powered
network is similar to that of the regularly powered network. For instance, we show that if the
distance of the typical device to the second closest BS is below a certain threshold, its downlink
coverage performance would be the same as the regularly powered network. We further study the
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Tagged
BS
Tagged
BS
Charging sub-slot Downlink sub-slot Uplink sub-slot
𝜏1𝑇 𝜏2𝑇 𝜏3𝑇
IoT device Cellular Network BS
Fig. 1. Illustration of the system setup and the three sub-slots (charging, downlink, and uplink).
effect of other system parameters including time-slot partitioning parameters, cellular network
density, RF-DC conversion efficiency, and cellular network transmission power on the system
performance. We show how these parameters can be tuned in order to get the performance of this
RF-powered network closer to that of a regularly powered network. This is done by defining a
tuning parameter that captures the effect of the aforementioned system parameters. Our analysis
shows that in order to get the performance of this RF-powered network closer to the regularly
powered network, it is only required to make sure that this tuning parameter is large enough.
II. SY ST EM MO DE L
We consider a cellular-based IoT network in which the IoT devices are solely powered by
the ambient RF energy. In this work, we assume that the cellular transmissions are the only
source of ambient RF energy for these devices. Quite reasonably, the IoT devices are assumed
to be batteryless (similar to [34], [35]). The more general case of finite-sized battery is left for
future work. In particular, we assume that all the energy required for uplink and/or downlink
communication by a device in a given time slot will need to be harvested by that device in the
same time slot. More details will be provided shortly. The locations of the cellular network BSs
and the IoT devices are modeled by two independent PPPs Φb≡ {xi} ⊂ R2and Φu≡ {ui} ⊂ R2
with densities λband λu, respectively [36]. As will be the case in reality, we assume λu> λb.
As implied in Fig. 1, we assume that each IoT device adopts the time-switching receiver
architecture (see [10]) in which the antenna is used for energy harvesting for a given fraction
of time and for communication for the rest of the time. The time slot duration is assumed to be
T(seconds). As shown in Fig. 1, each time-slot is further divided into charging, downlink, and
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uplink sub-slots with durations Tch =τ1T,TDL
tr =τ2T, and TUL
tr =τ3T, respectively. During
the charging sub-slot, all the BSs in the network act as RF chargers for the IoT devices. In the
downlink and uplink sub-slots, each IoT device receives and sends information to its associated
BS, respectively. This system setup will facilitate the analysis of joint uplink/downlink coverage
probability thus generalizing the prior work on energy harvesting networks that focused on the
analysis of downlink and uplink separately. Naturally, if we substitute τ2= 0, we can focus only
on the uplink analysis, which we refer to as the uplink mode. Similarly, if we substitute τ3= 0,
we can focus only on the downlink analysis, which we refer to as the downlink mode. The general
case in which τ2and τ3are both non-zero will be referred to as the joint uplink/downlink mode.
Our analysis will be performed under the following assumptions: (i) each IoT device connects
to its nearest BS (referred to as tagged BS in the rest of the paper), (ii) fading gains across all
links are independent, (iii) fading gains across the same link in charging sub-slot (denoted by
gx), downlink sub-slot (denoted by hx), and uplink sub-slot (denoted by wx) are independent,
(iv) all channels suffer from Rayleigh fading. This means that gx,hx, and wxare all independent
exponential random variables with mean 1. Under these assumptions, we focus our analysis on
a typical device placed at the origin (without loss of generality due to the stationarity of PPP).
We now enrich our notation to express key metrics of interest for each sub-slot.
In the charging sub-slot, we are interested in measuring the amount of energy harvested by
the typical device. In order to do that, we first model the received power at the typical device
from a BS located at x∈Φbas Ptgxkxk−α, where gx∼exp(1) is the fading gain, Ptis the
transmission power (assumed to be the same for all the BSs), and kxk−αmodels standard power
law path-loss with exponent α > 2. The total energy harvested by the typical device is thus
EH=τ1T η X
x∈Φb
Ptgxkxk−αJoules,(1)
where η < 1represents the efficiency of the RF-to-DC conversion.
In the downlink and uplink sub-slots, we are interested in the expressions for the respective
SINRs. For the downlink sub-slot, the the SINR at the typical device is
SINRDL =Pthx1kx1k−α
Px∈Φb\x1Pthxkxk−α+σ2
DL
=Pthx1kx1k−α
I1+σ2
DL
,(2)
where hx∼exp(1) represents the fading gain between the typical device and the BS located
at x,x1is the location of the nearest (tagged) BS, I1denotes the interference power, and σ2
DL
models thermal noise power. For successful reception in the downlink sub-slot, the received
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Tagged BS
𝑢𝑖
𝑥1
(𝑖)
Typical IoT device
𝑢𝑜
𝑥1
Fig. 2. Key variables used in the uplink analysis.
SINR needs to be greater than a modulation-and-coding specific target SINR βDL. In addition,
the IoT device needs a minimum amount of energy Erec in order to be able to activate its
receiving chain circuitry and receive data successfully during the downlink sub-slot.
In the uplink sub-slot, each IoT device is assumed to perform uplink fractional channel
inversion power control. Hence, if the distance between the IoT device and its serving BS
is R, then the transmitted power is ρRα, where ρis the BS sensitivity, and ∈[0,1] is the
power control parameter. Therefore, the typical IoT device requires τ3T ρRα energy in order to
perform uplink transmission. We refer to IoT devices that have enough energy to transmit in
the uplink sub-slot as active devices. Focusing our analysis on the typical device located at the
origin, the uplink SINR for this device measured at its tagged BS is:
SINRUL =woρkx1k(−1)α
P
ui∈Φa\uo
δiwiρR(i)
1α
D−α
i+σ2
UL
,(3)
where Φais the point process representing all the devices (including the typical device) that are
scheduled on the same time-frequency resource as the typical device, σ2
UL models thermal noise
power, wi∼exp(1) is the channel fading gain between the device located at uiand the tagged
BS during uplink information sub-slot, x1is the location of the tagged BS, Di=kui−x1kis
the distance between the device located at uiand the tagged BS, R(i)
1is the distance between
the device located at uiand its serving BS (which is the closest BS to this device by definition),
and uois the location of the typical device. Also δiis an indicator function that equals to 1if
the IoT device located at uiis active, and 0otherwise. Please refer to Fig. 2 for the summary of
this uplink-specific notation. As was the case in downlink, this SINR needs to be greater than
a modulation-and-coding specific target SINR threshold βUL for successful transmission.
Joint uplink/downlink coverage. With this, we are now ready to formally introduce the key
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TABLE I
TABL E OF NOTATIO NS
Notation Description
Φb;λbPPP modeling the locations of BSs in the cellular network; density of the BSs
Φu;λuPPP modeling the locations of IoT devices; density of the IoT devices
τ1;τ2;τ3Time-slot division parameter for charging sub-slot; downlink sub-slot; uplink sub-slot
EHAmount of energy harvested from ambient RF signals (in Joules)
Davg Average data rate (throughput)
PDL,RP
cov ;PUL,RP
suc Downlink coverage probability; uplink coverage probability in a regularly powered network
PDL
cov ;PUL
suc ;PJ
suc Downlink coverage probability; uplink coverage probability; joint uplink/downlink coverage probability in the ambient RF powered network
gx;hx;wxFading gains during charging; downlink; uplink sub-slots (assumed to be i.i.d. across all links). Rayleigh fading is assumed
WD(WU)Bandwidth of the downlink channel (uplink channel)
Pt;ρBS transmission power; BS sensitivity
;αPower control parameter; path loss exponent (α > 2)
r1(r2) Distance between typical IoT device and its nearest BS (2nd nearest BS)
metric of interest for this paper: the joint uplink/downlink coverage probability. Recall that the
total energy harvested by a device in the charging sub-slot is used by that device to receive
information from the tagged BS during the downlink sub-slot and transmit information to the
tagged BS during the uplink sub-slot. Hence, the energy coverage condition for this case is:
EH≥Emin,(4)
where Emin =Erec +τ3T ρr1α,r1=kx1kis the distance between the typical IoT device and its
nearest BS. For completeness, three conditions need to be satisfied for uplink/downlink coverage:
(i) EH> Emin, (ii) SINRDL > βDL in the downlink sub-slot, and (iii) SINRUL > βUL in the
uplink sub-slot. Therefore, the joint uplink/downlink coverage probability is defined as:
PJ
suc =E[
1
(SINRDL ≥βDL)
1
(SINRUL ≥βUL)
1
(EH≥Emin)] .(5)
As noted already, when τ2and τ3are both non-zero, we call this a joint uplink/downlink mode.
As discussed next, if one of them is zero, we can specialize the above definition of joint coverage
to study downlink or uplink coverage probability separately.
Downlink coverage. If we substitute τ3= 0, each time slot is partitioned into charging and
downlink sub-slots. We referred to this as the downlink mode above. Since each device in this
mode only needs to perform downlink transmission, the energy coverage condition reduces to
EH> Erec. Consequently, the downlink coverage probability for this case can be defined as:
PDL
cov =E[
1
(SINRDL ≥βDL)
1
(EH≥Erec)] .(6)
Clearly (6) can be obtained from (5) by substituting τ3= 0,βUL = 0 and using EH> Erec as
the energy condition. If SINRDL ≥βDL and EH≥Erec (i.e., the IoT device is able to establish
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a communication link with the BS), the downlink data rate is R=WDlog(1 + βDL)bps in the
information sub-slot, where WDis the bandwidth of the downlink channel.
Uplink coverage. Similarly, if we substitute τ2= 0, there is no downlink sub-slot and each time
slot is partitioned into only charging and uplink sub-slots. This was referred to as the uplink mode
earlier in this Section. Since each IoT device now needs to perform only uplink communication,
the energy coverage condition for this case is EH> τ3T ρRα. This along with the uplink SINR
coverage condition gives the following definition for the uplink coverage probability:
PUL
suc =E[
1
(SINRUL ≥βUL)
1
(EH≥τ3T ρr1α)] .(7)
As was the case for downlink coverage above, (7) can be obtained from (5) by substituting
τ2= 0,βDL = 0 and using EH> τ3T ρrα
1as the energy condition. If the two coverage conditions
(SINRUL ≥βUL and EH≥τ3T ρr1α) are satisfied, the uplink data rate is R=WUlog(1 + βUL )
bps in the information sub-slot, where WUis the bandwidth of the uplink channel.
As evident from the above discussion, joint uplink/downlink coverage probability encompasses
the other two as special cases. We will therefore start with the analysis of this general case. The
results for the downlink and uplink modes will be provided as special cases of this general setup
to provide useful system design insights.
III. JOINT UPL IN K AN D DOWNLINK MODE
This is the first technical section of the paper in which we will evaluate the joint up-
link/downlink coverage probability defined in Eq. 5. In particular, our goal is to evaluate the
joint probability of the following three events: (i) SINRDL ≥βDL, (ii) SINRUL ≥βUL , and
(iii) EH≥Emin. Keeping the joint treatment aside, the complexity of this analysis should be
evident from the following two facts: (i) the exact characterization of P(SINRUL ≥βUL)is
not known in the stochastic geometry literature [13], [37], and (ii) the total energy harvested
is essentially a power-law shot noise field whose probability density function is not known in
general. To make matters worse, all these events depend upon the point process Φbmodeling
the locations of the BSs, which necessitates their joint analysis. This dependence on Φbis quite
evident for both EHand SINRDL from their expressions given by Eqs. 1 and 2. While SINRUL
may not appear to depend on Φbon the first look (see Eq. 3), the point processes of the devices
and BSs are correlated though cell selection and resource scheduling (see [13] for the detailed
discussion), which couples the uplink coverage event with the other two events. Therefore, the
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main challenge in our analysis is the joint treatment of these three coverage events. That being
said, since the main source of this correlation, as evident from Eqs. 1, 2, 3, is the dependence
of the three events on Φb, they can be treated as independent when conditioned on Φbsince the
fading gains (hx,gx, and wx) in the three sub-slots are assumed independent. Consequently, the
joint uplink/downlink coverage probability defined in Eq. 5 can be expressed as
PJ
suc =EΦbhPSINRDL ≥βDLΦbPSINRUL ≥βULΦbPEH≥EminΦbi.(8)
In the following subsections, we carefully approximate the three conditional probability terms
using a dominant BS-based approach. The resulting expressions will then be used to derive our
main result for the the joint uplink/downlink coverage probability in Theorem 1.
A. Conditional Energy Coverage Probability
As discussed above, SINRDL and EHboth depend upon Φbexplicitly. However, due to pathloss,
the BSs located far away from the typical device do not contribute as much to both these terms as
the BSs located close to the typical point. Therefore, we reduce the dimensionality of this problem
by considering the effect of closest two BSs to the typical device exactly and approximating the
effect of the rest of the BSs. It will be clear shortly why we chose two and not any other number.
This dominant BS-based approach is useful when the exact analysis is either too difficult or leads
to unwieldy results. It has been used in the past to analyze the coverage of ad hoc networks [38],
coverage of downlink cellular networks [39], k-coverage of localization networks [40], [41], and
downlink coverage of wireless networks of unmanned aerial vehicles [42], [43]. Since all these
works focused on some form of (marginal) SINR-based coverage, they are not applicable to our
analysis because of the need to perform conditional analysis of each term separately and then
decondition jointly over all the terms. These works are listed here mainly for completeness.
We apply this approach to approximate the total energy harvested by the typical device in the
charging sub-slot (given by Eq. 1) by the energy harvested from the nearest two BSs (located
at distances r1=kx1kand r2=kx2kfrom the typical device) and the conditional mean
(conditioned on the location of the nearest 2 BSs) of the rest of the terms as follows:
EH=τ1T ηPtX
x∈Φb
gxkxk−α≈τ1T ηPtgx1kx1k−α+gx2kx2k−α+ Ψ(r2),(9)
where Ψ(r2) = EhPx∈Φb\x1,x2gxkxk−αx1, x2i. We will use this approximation to compute
the conditioned energy coverage probability P(EH≥Emin|Φb)which is necessary for the
computation of PJ
suc as explained above. In addition to enabling the joint coverage analysis,
13
this approximation will also lead to several crisp system design insights. For instance, as a result
of using this approximation, we will be able to define a threshold on r2(as well as λband time
switching parameters) below which the performance is approximately equivalent to that of a
regularly powered cellular-based IoT (further discussion will be provided in Remarks 5 and 8).
As discussed already, the typical IoT device needs to harvest a minimum amount of energy
Emin =Erec +τ3T ρkx1kα to be able to receive and transmit information. If it is able to harvest
this energy, it is said to be in energy coverage. In the following Lemma we derive an expression
for the conditional energy coverage probability using the approximation in Eq. 9.
Lemma 1 (Conditional Energy Coverage Probability).Probability that the harvested energy
during the charging sub-slot is greater than Emin conditioned on the point process Φbis
PEH≥EminΦb=rα
2exp −rα
1[F(r1, r2)]+−rα
1exp −rα
2[F(r1, r2)]+
rα
2−rα
1
,(10)
while the unconditioned probability is
P(EH≥Emin) =
∞
Z
0Z
r1∈Nr2
fR1,R2(r1, r2)dr1dr2(11)
+
∞
Z
0Z
r1∈Pr2
fR1,R2(r1, r2)rα
2exp (−rα
1F(r1, r2)) −rα
1exp (−rα
2F(r1, r2))
rα
2−rα
1
dr1dr2,
where F(r1, r2) = hC(τ1) + τ3ρrα
1
τ1ηPt−2πλb
α−2r2−α
2i,C(τ1) = Erec
τ1T ηPt,[x]+= max{0, x},fR1,R2(r1, r2) =
(2πλb)2r1r2e−λbπr22,Nr2={r1:F(r1, r2)≤0, r1< r2}, and Pr2={r1:F(r1, r2)≥0, r1<
r2}.
Proof: See Appendix A.
As explained before, the above expression can be used to compute the energy coverage
probability in the downlink mode by eliminating uplink conditions and vice versa for the uplink
mode. The complete results for these special cases will be presented in Lemmas 4 and 5.
B. Conditional SINR Coverage Probability
As a result of using the approximation introduced in Eq. 9, the conditional energy coverage
probability in Eq. 10 is only a function of the distances r1and r2(between the typical device
and its nearest two BSs). Keeping in mind that we will have to jointly decondition on all the
coverage events at the end (as evident from Eq. 8), it will be useful to derive conditional downlink
SINR coverage also in terms of r1and r2. In order to do that, we use the same dominant
14
BS-based approach that we used in the previous Subsection. In particular, we approximate
the interference in the denominator of SINR in Eq. 2 by the interference from the second
nearest BS (strongest interferer) and the expectation of the interference from the rest of the
BSs. Under this approximation, the conditional downlink SINR coverage probability becomes
PSINRDL ≥βDLr1, r2. A tractable expression for this conditional probability is derived next.
Lemma 2 (Conditional Downlink SINR Coverage Probability).Probability that the downlink
SINR at the typical device exceeds βDL, conditioned on r1and r2, is
PSINRDL ≥βDLr1, r2= exp (−G(r1, r2)) 1
1 + βDLrα
1
rα
2
,(12)
where G(r1, r2) = βDLσ2
DLrα
1
Pt+2πλβDL rα
1
(α−2)rα−2
2
.
Proof: See Appendix B.
With this, we are now left with deriving the conditional uplink probability, which we do next.
It is noteworthy that uplink analysis is known to be a challenging problem even for regularly
powered networks. The locations of the devices scheduled in the same time frequency resource
block as the typical device (modeled as point process Φa\uoin Eq. 3) are correlated with the
locations of the BSs due to the structure of the Poisson Voronoi tessellation. This correlation
is further enhanced due to uplink power control, where the transmission power of each device
is a function of its distance to its serving BS. As discussed in [13], the exact analysis of this
setup in not known. It has, however, been shown that modeling the locations of the devices by
an independent PPP and handling dependence between the distances Diand R(i)
1(as defined
in Eq. 3) appropriately leads to a fairly tight approximation. For the latter, it is sufficient to
just account for the fact that R(i)
1< Di, i.e., the serving BS must be closer to the interfering
device than the tagged BS. Please refer to [13] for more details. Using this general idea, the
Laplace transform of the aggregate interference I2=P
ui∈Φa\uo
wiR(i)
1α
D−α
iat the tagged BS
in a regularly powered network was given in [13] as follows:
LI2(s) = Ee−I2s= exp −2πλbZ∞
0Zx2
0
1
1+(s)−1u−α/2xαπλbe−λbπuduxdx!,(13)
where Φais the point process modeling the locations of the selected devices in a given time-
frequency resource. This expression was used to derive the uplink coverage probability for
regularly powered networks in [13] as follows:
PUL,RP
suc =Z∞
0
fR1(r1)e−βULσ2
UL
ρr1(−1)αLI2βUL
r1(−1)αdr1,(14)
15
where fR1(r1) = 2πλbr1exp(−πλbr2
1). We will use this expression to compare the performance
of the proposed setup to that of the regularly powered networks.
Coming to the conditional uplink coverage in the proposed energy harvesting setup, note that
the dominant source of correlation between uplink SINR and the other two terms (downlink
SINR and the amount of energy harvested) is the serving distance r1. If we condition on r1and
treat Φaand Φbas independent point processes (as done above), the conditional uplink coverage
probability reduces to PSINRUL ≥βULr1, which is derived in the next Lemma.
Lemma 3 (Conditional Uplink SINR Coverage Probability).Probability that the uplink SINR
of the typical device at the tagged BS is greater than βUL, conditioned on r1, is
PSINRUL ≥βULr1=e−βUL σ2
UL
ρr1(−1)αL˜
I2βUL
r1(−1)α,(15)
where L˜
I2(s)is given by Eq. 13 by replacing λbwith ˜
λb=Phλb, where Ph=P(EH≥Emin).
Proof: See Appendix B.
C. Joint Uplink/Downlink Coverage Probability
Having derived the three conditional probability terms appearing in Eq. 8 in Lemmas 1, 2,
and 3, we are now ready to derive the joint uplink/downlink coverage probability. The only
remaining step is to uncondition their product with respect to the joint distribution of r1and r2,
which results in the following Theorem.
Theorem 1 (Joint uplink/downlink coverage probability).The joint uplink/ downlink coverage
probability PJ
suc of the typical IoT device with downlink and uplink SINR thresholds βDL and
βUL respectively is given by:
PJ
suc =
∞
Z
0Z
r1∈Nr2
fR1,R2(r1, r2)e−βULσ2
UL
ρr1(−1)αL˜
I2βUL
r1(−1)αexp (−G(r1, r2)) 1
1 + βDLrα
1
rα
2
dr1dr2
+
∞
Z
0Z
r1∈Pr2
fR1,R2(r1, r2)e−βULσ2
UL
ρr1(−1)αL˜
I2βUL
r1(−1)αexp (−G(r1, r2)−rα
1F(r1, r2))
rα
2−rα
1
rα
2
1 + βDLrα
1
rα
2
dr1dr2
−
∞
Z
0Z
r1∈Pr2
fR1,R2(r1, r2)e−βULσ2
UL
ρr1(−1)αL˜
I2βUL
r1(−1)αexp (−G(r1, r2)−rα
2F(r1, r2))
rα
2−rα
1
rα
1
1 + βDLrα
1
rα
2
dr1dr2,
(16)
16
where fR1,R2(r1, r2) = (2πλb)2r1r2e−πλbr2
2,F(r1, r2),Nr2,Pr2are as introduced in Lemma 1,
G(r1, r2)is as introduced in Lemma 2, and L˜
I2(s)is as introduced in Lemma 3.
Proof: This result follows directly by substituting Eq. 10, 12, 15 in Eq. 8 and integrating
over r1and r2using the joint distribution fR1,R2(r1, r2)as defined in [44, Eq. 28].
Remark 1. This general result can be used to derive both downlink coverage and uplink coverage
probabilities defined in Eqs. 6 and 7. For instance, if we remove all the uplink conditions by
putting βUL = 0 (note that LI2(0) = 1) and τ3= 0, then Eq. 16 will represent the downlink
coverage probability PDL
cov for the downlink mode. Similarly, if we remove all the downlink
conditions by putting βDL = 0 (note that G(r1, r2) = 0 in that case), Erec = 0, and τ2= 0, then
Eq. 16 will represent the uplink coverage probability PUL
suc for the uplink mode.
D. Average Throughput
We now derive expressions for both the uplink and the downlink average throughput in the
joint uplink/downlink mode. The average downlink throughput is
DDL
avg =τ2RDL
avg =τ2E[WDlog(1 + βDL)
1
(SINRDL ≥βDL)
1
(SINRUL ≥βUL)
1
(EH≥Emin)]
=τ2WDlog(1 + βDL)PJ
suc,(17)
where RDL
avg is the average data rate during downlink sub-slot in the joint mode. The multiplication
by τ2accounts for the fact that downlink sub-slot lasts for τ2fraction of the total time-slot
duration. Similarly, the average uplink throughput in the joint mode is:
DUL
avg =τ3RUL
avg =τ3E[WUlog(1 + βUL)
1
(SINRUL ≥βUL)
1
(SINRDL ≥βDL)
1
(EH≥Emin)]
=τ3WUlog(1 + βUL)PJ
suc,(18)
where RUL
avg is the average data rate during uplink sub-slot in the joint mode.
Remark 2. Note that for a given τ3, it is easier to satisfy the energy constraint for larger values
of τ1. This means both PJ
suc and RDL
avg are the increasing functions of τ1. However, increasing τ1
decreases τ2(for a given τ3), which reduces the downlink transmission time and may therefore
reduce average data rate DDL
avg. This indicates the existence of an optimal slot partitioning for
maximizing DDL
avg. Similar conclusions can be drawn about the relation between τ1and DUL
avg for
a given τ2. We will discuss more about this optimal slot partitioning in the sequel.
17
In the next two Sections, we will specialize the general results of this Section to the downlink
and uplink modes, which will provide several useful system design insights.
IV. DOWNLINK MOD E
The coverage probability in the downlink mode defined in Eq. 6 can be expressed as
PDL
cov =EΦbhPSINRDL ≥βDLΦbPEH≥ErecΦbi,(19)
which is the special case of Eq. 8. As discussed in Section II and later in Remark 1, we can
obtain this definition for PDL
cov by simply substituting τ3= 0,βUL = 0 and using EH> Erec
as the energy condition in the definition of joint uplink/downlink coverage probability given in
Eq. 5 (and hence Eq. 8). Therefore, PDL
cov can be derived directly by making these substitutions
in Eq. 16. Similarly, we can derive the energy coverage for downlink mode by using the same
substitutions in Eq. 11. We first state this energy coverage result next.
Lemma 4 (Energy coverage probability in the downlink mode).Probability that the harvested
energy during the charging sub-slot is greater than the value Erec is
P(EH≥Erec) = 1 −exp −πλbA2−πλbA2exp −πλbA2
+
∞
Z
A
r2
Z
0
fR1,R2(r1, r2)rα
2exp (−rα
1FDL(r1, r2)) −rα
1exp (−rα
2FDL(r1, r2))
rα
2−rα
1
dr1dr2,
(20)
where fR1,R2(r1, r2) = (2πλb)2r1r2e−πλbr2
2,FDL(r1, r2) = C(τ1)−2πλbr2−α
2
α−2,C(τ1) = Erec
τ1T ηPt, and
A=2πλb
C(τ1)(α−2) 1
α−2.
Proof: See Appendix C.
Remark 3. The effect of the duration of the charging sub-slot Tch =τ1Tappears in the value
of C(τ1). Consistent with intuition, as this duration increases, the value of C(τ1)decreases and
the energy coverage probability approaches 1.
We now state the (downlink) coverage result for the downlink mode (defined in Eq. 6).
Theorem 2 (Downlink coverage probability in the downlink mode).The downlink coverage
18
probability with SINR threshold βDL and minimum required energy Erec is given by
PDL
cov =
A
Z
0
r2
Z
0
fR1,R2(r1, r2) exp(−G(r1, r2)) 1
1 + βDLrα
1
rα
2
dr1dr2(21)
+
∞
Z
A
r2
Z
0
fR1,R2(r1, r2) exp(−G(r1, r2)) rα
2exp (−rα
1FDL(r1, r2)) −rα
1exp (−rα
2FDL(r1, r2))
(rα
2−rα
1)1 + βDLrα
1
rα
2dr1dr2,
where G(r1, r2)is defined in Lemma 2, C(τ1),A, and FDL(r1, r2)are defined in Lemma 4.
Proof: See Appendix C.
Remark 4. The effect of the duration of the charging sub-slot Tch =τ1Tappears mainly in the
value of A(implicitly in the value of C(τ1)). It can be observed that as this duration increases,
the value of Aincreases and PDL
cov approaches the coverage probability of regularly powered
network PDL,RP
cov =P(SINR ≥βDL). This is because as Tch increases, it becomes easier to
satisfy the energy constraint and the energy coverage probability ultimately approaches unity.
Remark 5. Conditioned on Φb, the variable Arepresents an important system parameter. In
Eq. 21, it can be interpreted as a threshold on the value of r2. In particular, as long as the
distance to the second nearest BS (which is also the second dominant RF source on average)
is less than this threshold, the energy coverage condition is satisfied and the only condition
required for coverage is SINRDL ≥βDL, which is represented by the first term in Eq. 21. This
useful system insight is a result of using the approximation in Eq. 9 that defines the amount of
energy harvested in terms of distances r1and r2. This provides useful characterization of the
regime in which the performance of this RF-powered IoT network will be similar to the regularly
powered network. Similar observations will be provided for the uplink case in the next Section.
The general expression for average throughput given in Eq. 17 can be specialized for the
downlink mode as follows:
DDL
avg =τ2WDlog(1 + βDL)PDL
cov .(22)
Remark 6. Similar to our comments in Remark 2, RDL
avg is an increasing function of τ1. On
the other hand, the duration τ2Tof the downlink sub-slot decreases with increase in τ1. This
indicates the existence of an optimal value of τ1that maximizes DDL
avg.
19
V. UPLINK MODE
In this Section, we specialize the results of Section III to the uplink mode. Recall that in
the uplink mode, each time-slot is partitioning into two sub-slots: charging sub-slot and uplink
sub-slot. As discussed in Section II and Remark 1, the uplink coverage probability, defined in
Eq. 7, can be obtained from the definition of joint uplink/downlink coverage, given by Eq. 5, by
substituting τ2= 0,βDL = 0 and using EH> τ3T ρrα
1as the energy condition. Consequently,
the results for the uplink mode can be obtained from the general results of Section III by making
these substitutions. While these substitutions are quite similar to the ones that we made in the
previous Section for the downlink mode, there is a subtle difference in the energy conditions,
which is the reason why the final results are slightly different in the two cases. In particular,
while the minimum required energy in the downlink mode was fixed (Erec), it is a function of
the nearest BS location in the uplink mode (due to power control). As in the previous Section,
we first state the energy coverage result for the Uplink mode next.
Lemma 5 (Energy coverage probability in the uplink mode).Energy coverage probability is
P(EH≥τ3T ρkx1kα)=1−exp −πλb˜
A2−πλb˜
A2exp −πλb˜
A2+
∞
Z˜
A
H(r2)
Z
0
fR1,R2(r1, r2)dr1dr2
+
∞
Z˜
A
r2
Z
H(r2)
fR1,R2(r1, r2)rα
2exp (−rα
1FUL(r1, r2)) −rα
1exp (−rα
2FUL(r1, r2))
rα
2−rα
1
dr1dr2,(23)
where fR1,R2(r1, r2) = (2πλb)2r1r2e−πλbr2
2,H(r2) = 2πλb
(α−2) ˜
C(τ1)1
α r
2−α
α
2,FUL(r1, r2) = ˜
C(τ1)rα
1−
2πλbr2−α
2
α−2,˜
C(τ1) = τ3ρ
τ1ηPt,˜
A=2πλb
˜
C(τ1)(α−2) 1
(+1)α−2.
Proof: See Appendix D.
Remark 7. It is easy to see that increasing the density λbof the BS PPP Φbincreases energy
coverage probability due to two reasons. First, it reduces the distance r1between the typical
device and its serving BS, which reduces the transmission power r1α of this device, this making
it easier to satisfy the energy coverage condition. Second, increasing λbalso increases the
aggregate energy EHharvested by the typical device. This is also evident from Eq. 23 where all
the terms can be shown to be decreasing functions of λb.
We now present the uplink coverage probability (defined in Eq. 7) next. Using this, we will
20
discuss the differences between the regularly powered and energy harvesting networks.
Theorem 3 (Uplink coverage probability in the uplink mode).The uplink coverage probability
PUL
suc of the IoT device with SINR threshold βUL and uplink transmission power ρkx1kα is
PUL
suc =
˜
A
Z
0
r2
Z
0
fR1,R2(r1, r2)e−βULσ2
UL
ρr1(−1)αL˜
I2βUL
r1(−1)αdr1dr2
+
∞
Z˜
A
H(r2)
Z
0
fR1,R2(r1, r2)e−βULσ2
UL
ρr1(−1)αL˜
I2βUL
r1(−1)αdr1dr2
+
∞
Z˜
A
r2
Z
H(r2)
fR1,R2(r1, r2)
rα
2exp −rα
1FUL(r1, r2)−βUL σ2
UL
ρr1(−1)α
rα
2−rα
1
L˜
I2βUL
r1(−1)αdr1dr2
−
∞
Z˜
A
r2
Z
H(r2)
fR1,R2(r1, r2)
rα
1exp −rα
2FUL(r1, r2)−βUL σ2
UL
ρr1(−1)α
rα
2−rα
1
L˜
I2βUL
r1(−1)αdr1dr2,(24)
where H(r2),˜
C(τ1),FUL(r1, r2), and ˜
Aare as defined in Lemma 5, and L˜
I2(s)is defined in
Lemma 3.
Proof: See Appendix C.
By comparing the above result with the uplink coverage probability of the regularly powered
network given by Eq. 14, we note that the effect of energy harvesting mainly appears in the
term ˜
A. For instance, if we try to exclude the energy coverage condition (EH≥τ3T ρkx1kα)
by putting τ3= 0, we will get ˜
C(τ1) = 0, which will tend ˜
Ato ∞. This will eventually make
all the terms in Eq. 24 tend to zero except the first term which will be equivalent to Eq. 14.
Remark 8. Similar to Remark 5, the value of ˜
Ahere represents a threshold on the distance to
the second nearest BS r2. In particular, as long as r2≤˜
A, the uplink coverage probability of
the RF-powered network is exactly the same as that of the regularly powered network. This can
be deduced from the first term of Eq. 24.
Similar to Eq. 18, the average uplink throughput DUL
avg =τ3RUL
avg can be expressed as
DUL
avg =τ3WUlog(1 + βUL)PUL
suc .(25)
Note that, similar to Remark 6, the time-slot division parameter τ1has an optimum value that
maximizes the throughput DUL
avg. We conclude this Section with the following remark.
21
Charging sub-slot duration τ1
0 0.1 0.2 0.3 0.4 0.5
Downlink energy coverage probability P(EH>Erec)
0.2
0.4
0.6
0.8
1
Simulation
Theoretical
Fig. 3. Energy coverage probability in the downlink mode as a function of τ1.
Charging sub-slot duration τ1
0 0.05 0.1 0.15 0.2
Downlink coverage probability Pcov
DL
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.55
Simulation
Theoretical
Increasing βDL: [0, 1, 2] dB
Fig. 4. Downlink coverage probability PDL
cov in the downlink mode as a function of τ1.
Remark 9. By comparing the results for downlink and uplink modes (given in Theorems 2 and 3,
respectively), with those of the regularly powered network, we conclude that A=2πλb
C(τ1)(α−2) 1
α−2
and ˜
A=2πλb
˜
C(τ1)(α−2) 1
(+1)α−2can be used as tuning parameters for the energy harvesting
network. The closer we need the downlink or uplink performance to be to the regularly powered
network, the larger the values of Aand ˜
Aneed to be. These tuning parameters capture in their
definitions the effects of all system parameters including Pt,λb,τ1,τ2,τ3, and η.
VI. SIMULATION RESU LTS AND DISCUSSION
Unless specified otherwise, we will consider the following values for the simulation parameters
throughout this section: Erec = 10−5Joules, λb= 1,α= 4,η= 10−3,WD= 1 MHz, βDL = 1
22
Distance to nearest interferer r2
0.2 0.4 0.6 0.8 1 1.2 1.4
Downlink coverage probability Pcov
DL
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Energy harvesting network
Regularly powered network
r2 threshold=0.56
Fig. 5. Downlink coverage probability conditioned on the value of r2.
Charging sub-slot duration τ1
0 0.05 0.1 0.15 0.2
Davg
DL bits/sec
×105
2
2.5
3
3.5
4
4.5
5
5.5
Simulation
Theoretical
Increasing βDL: [0, 1, 2] dB
Fig. 6. Downlink average throughput DDL
avg in the downlink mode as a function of τ1.
dB, Pt= 0 dB, Pt
σ2
DL = 20 dB, ρ= 1 dBm, ρ
σ2
UL = 20 dB, λu= 30λb,βUL = 1 dB, = 0.8, and
T= 10−2sec.
A. Downlink Mode
In this subsection, we evaluate the performance of the downlink mode using the performance
metrics derived in Section IV. First, in Fig. 3 we plot the energy coverage probability result
derived in Lemma 4. As discussed in Remark 3, energy coverage probability is clearly an
increasing function of the time division parameter τ1. The theoretical results are also shown
to match perfectly with the simulation results obtained from Monte-Carlo trials, which verifies
the accuracy of the dominant BS-based approach used to approximate the energy EHin our
analysis. The downlink coverage result derived in Theorem 2 is plotted in Fig. 4. Comparisons
23
with simulation results again verify the accuracy of the dominant BS-based approximation. As
discussed in Remark 4, we notice that the coverage probability PDL
cov starts converging to the
coverage probability of regularly powered networks, given by P(SINRDL ≥βDL), at high values
of τ1. To glean sharper insights, we recall Remark 5, where we referred to Aas a threshold on
the value of distance to the second nearest BS r2, below which this RF-powered IoT network has
the same downlink coverage as the regularly powered network. In Fig. 5, we verify this insight
by plotting the coverage probabilities for both RF-powered and regularly powered networks
conditioned on r2(for τ1= 0.1). As predicted in Remark 5, the performance of both the
networks is the same when r2is below the threshold value, which in this case is r2=A= 0.56.
Even thought this insight was a byproduct of dominant BS-based approximation, we notice that
it is remarkably accurate. Right after the threshold value of r2=A= 0.56, the two curves
start diverging. Finally, we plot our results in Eq. 22 for the average throughput in Fig. 6.
Comparisons with the simulation results verify the accuracy of our analysis. The results also
illustrate the existence of an optimum value for τ1that maximizes the average throughput in the
downlink mode, as predicted in Remark 6.
B. Uplink Mode
In this section, we focus on the performance analysis of uplink mode. In particular, we will
study the effect of τ1and λbon the performance metrics derived in Section V. In Fig. 7, we
plot the energy coverage probability in the uplink mode as a function of λb. Consistent with
Remark 7, the energy coverage probability increases with λband saturates to unity when λbis
above a specific value, which we denote by λ∗
b. Beyond this value of density, the energy coverage
condition is satisfied with high probability. Consequently, the uplink coverage probability PUL
suc
is expected to converge to the SINR coverage probability, defined as P(SINRUL ≥βUL ), at
λ∗
b. This is verified in Fig. 8, where starting from λb=λ∗
b, the energy coverage condition is
satisfied most of the time and the uplink coverage probability reduces to SINR coverage, i.e.,
P(SINRUL ≥βUL, EH≥ρrα
1)'P(SINRUL ≥βUL). We also note that the SINR coverage
probability in Fig. 8 initially decreases with λbuntil it becomes constant starting from about
λb=λ∗
b. This is due to the increase in energy coverage probability which leads to increase
in the density of active devices, hence increasing the interference value. The value to which
they converge starting from λb=λ∗
bis the uplink coverage probability for the case of regularly
powered network (PUL,RP
suc in Eq. 14). Similar trends are observed in Fig. 9, where we note that
24
Cellular network density λb
100101102
Uplink energy coverage probability P(EH>τ3Tρ r1
ǫα)
0.2
0.4
0.6
0.8
1
Simulated
Theoretical
λb=λb
*
Fig. 7. Uplink energy coverage probability as a function of cellular network density λb.
Cellular network density λb
100101102
Uplink coverage probability PUL
suc
0.25
0.3
0.35
0.4
0.45
0.5
0.55
0.6 Uplink coverage probability
SINR coverage probability
Regularly powered network
λb=λb
*
Fig. 8. Uplink coverage probability PUL
suc in the uplink mode as a function of λb.
the uplink coverage and the SINR coverage probabilities converge at about τ1= 0.5, which can
be interpreted as the minimum value of τ1at which the energy coverage condition is satisfied
with a high probability. Also, similar to our discussion above on the effect of λb, the SINR
coverage probability in Fig. 9 initially decreases due to the increase in the energy coverage
probability which increases the density of active devices and, consequently, the interference.
C. Joint Uplink and Downlink Mode
In Fig. 10 we provide a 3D plot for DDL
avg as a function of τ1and τ3. Recall that τ2= 1−τ1−τ3.
We note that for any given value of τ1, the value of DDL
avg decreases as τ3increases (equivalently
τ2decreases). As discussed in Remark 2, for any given value of τ3, there exists optimal τ1(and
25
Charging sub-slot duration τ1
0 0.2 0.4 0.6 0.8
Uplink coverage probability PUL
suc
0.4
0.45
0.5
Uplink coverage probability
SINR coverage probability
Fig. 9. Uplink coverage probability as a function of uplink time-slot division parameter τ1.
1
0.8
0.6
τ1
0.4
0.2
0
1
0.5
×105
τ3
2
0
4
6
0
D avg
DL
Fig. 10. Downlink average throughput during joint uplink and downlink mode as a function of τ1and τ3.
hence optimal τ2) that maximizes DDL
avg. A similar behavior has already been seen in Fig. 6.
Similar observations can be made about the behavior of DUL
avg.
VII. CONCLUSIONS
In this paper, we developed an analytical framework to study joint uplink and downlink
coverage performance of a cellular-based ambient RF energy harvesting network in which IoT
devices are solely powered by the downlink cellular transmissions. Each time-slot is assumed
to be partitioned into charging, downlink, and uplink sub-slots. Within each time-slot, the IoT
devices (assumed batteryless) first harvest RF energy from cellular transmissions and then use
this energy to perform downlink and uplink communication in the subsequent sub-slots. For this
setup, we derived the joint probability that the typical device harvests sufficient energy in the
26
charging sub-slot and achieves sufficiently high downlink and uplink SINRs in the following two
sub-slots. The main technical contribution is in handling the correlation between these energy
and SINR coverage events. Using this result, we also studied system throughput as a function
of the time-slot division parameters. Optimal slot partitioning that maximizes this throughput is
also discussed. Using these results, we also compared the performance of this RF-powered IoT
network with a regularly powered network in which the IoT devices have uninterrupted access
to reliable power source, such as a battery. We derived thresholds on several system parameters
beyond which the performance of this RF-powered IoT network converges to that of the regularly
powered network.
Finally, we defined a tuning parameter, which incorporates the effect of all system parameters,
and needs to be sufficiently high for the coverage performance of this RF-powered network to
converge to that of the regularly powered network.
This work can be extended in multiple directions. From the energy harvesting perspective,
the system model can be extended to include rechargeable batteries (with finite capacities) at
the devices. This will require explicit consideration of the temporal dimension, as done in [22],
where the BSs were assumed to be self-powered with access to batteries with finite capacities.
From the modeling perspective, it is important to consider other BS-device configurations, such
as the ones in which devices are clustered around the BSs [45].
APPENDIX A
The value of Ψ(r2)can be derived as follows:
Ψ(r2) = E
X
x∈Φb\x1,x2
gxkxk−αx1, x2
(a)
=E
X
xi∈Φb\x1,x2
kxik−α
(b)
= 2πλbZ∞
r2
1
rαrdr=2πλb
α−2(r22−α),
(26)
where (a) follows from the assumption that all {gx}are independent and exponentially distributed
random variables with mean one, and (b) follows from Campbell’s theorem [46] with conversion
from Cartesian to polar coordinates and using r2=kx2k. Using the approximation introduced
in Eq. 9, the conditional energy coverage probability can be expressed as:
PEH≥EminΦb=Pτ1T η Ptgx1r−α
1+gx2r−α
2+2πλb
α−2r2−α
2≥Erec +τ3T ρrα
1
=Pgx1r−α
1+gx2r−α
2≥C(τ1) + τ3ρrα
1
τ1ηPt
−2πλb
α−2r2−α
1=Pgx1r−α
1+gx2r−α
2≥ F(r1, r2)
27
(c)
=rα
2exp(−rα
1[F(r1, r2)]+)−rα
1exp(−rα
2[F(r1, r2)]+)
rα
2−rα
1
,(27)
where step (c) is due to hypo-exponential distribution of gx1r−α
1+gx2r−α
2(sum of two exponential
random variables with rates rα
1and rα
2), C(τ1) = Erec
τ1T ηPt, and [x]+= max{0, x}. This concludes
the proof of Eq. 10. Noting that PEH≥EminΦb= 1 when F(r1, r2)≤0and integrating
over r1and r2with fR1,R2(r1, r2) = (2πλb)2r1r2e−λbπr22[44], the result in Eq. 11 follows.
APPENDIX B
Using the definition of SINRDL in Eq. 2 and approximating the interference I1by the sum
of interference from the nearest interferer and the expectation of the interference from the rest
of the interference field, we get
P(SINRDL ≥βDL|r1, r2) = PPthx1kx1k−α
I1+σ2
DL
≥βDLr1, r2
=PPthx1r−α
1
Pthx2r−α
2+PtΨ(r2) + σ2
DL
≥βDLr1, r2(d)
=P
Pthx1r−α
1
Pthx2r−α
2+Pt
2πλbr2−α
2
α−2+σ2
DL
≥βDLr1, r2
=Phx1r−α
1≥βDLσ2
DL
Pt
+2πλbβDLr2−α
2
α−2+βDLhx2r−α
2(28)
(e)
=Ehx2exp −rα
1βDLσ2
DL
Pt
+2πλbβDLr2−α
2
α−2+βDLhx2r−α
2(f)
= exp(−G(r1, r2)) 1
1 + βDL
rα
1
rα
2
,
where (d) follows from substituting for Ψ(r2)as derived in Eq. 26, and steps (e) and (f) follow
from the assumption that hx∼exp(1), and defining G(r1, r2) = βDLσ2
DLrα
1
Pt+2πλbβDL r2−α
2rα
1
α−2.
In the uplink sub-slot, the locations of active IoT devices (IoT devices in energy coverage)
in a given time-frequency resource can be approximately modeled by the PPP ˜
Φuwith density
˜
λu=Ph×λbwhere Ph=P(EH≥Emin). This will lead to the following expression for SINRUL:
SINRUL =wokx1k(−1)α
P
ui∈˜
Φu\uo
wiR(i)
1α
D−α
i+σ2
UL
ρ
.(29)
Defining ˜
I2=P
ui∈˜
Φu\uo
wiR(i)
1α
D−α
i, we have:
PSINRUL ≥βULr1=P
w0r1(−1)α
˜
I2+σ2
UL
ρ
≥βULr1
=E˜
I2
P
w0≥(˜
I2+σ2
UL
ρ)βUL
r1(−1)αr1,˜
I2
(g)
=E˜
I2
exp
−(˜
I2+σ2
UL
ρ)βUL
r1(−1)α
(h)
=e−βULσ2
UL
ρr1(−1)αL˜
I2βUL
r1(−1)α,(30)
28
where step (g) is due to the assumption that w0is exponentially distributed with mean one, and
step (h) results from using the Laplace transform of ˜
I2, which can be found by replacing λb
with ˜
λb=Phλbin Eq. 13, where Ph=P(EH≥Emin).
APPENDIX C
We apply the substitutions in Remark 1 for the downlink case to both Lemma 1 and Theorem 1
to get both energy coverage probability and PDL
cov . Applying these substitutions reduces the value
of F(r1, r2)to FDL(r1, r2) = C(τ1)−2πλbr2−α
2
α−2, where C(τ1)is as defined in Lemma 1. Letting
A=2πλb
(α−2)C(τ1)1
α−2, we note that the set Nr2will be empty set for r2≥ A while for r2≤ A
the set will be simply Nr2={r1:r1≤r2}. Similarly, the set Pr2will be empty set for r2≤ A
while for r2≥ A the set will reduce to Pr2={r1:r1≤r2}. Applying these integration
limits on our result in Lemma 1 leads to the final result in Lemma 4. Similarly, applying these
new integration limits to the result in Theorem 1 and noting that the substitutions explained in
Remark 1 include βUL = 0 (which leads to L˜
I2(0) = 1 in Eq. 16), the final result in Theorem 2
follows.
APPENDIX D
Similar to the approach in the downlink case, we apply the substitutions in Remark 1 for
the uplink case to both Lemma 1 and Theorem 1. Applying these substitutions reduces the
value of F(r1, r2)to FUL(r1, r2) = ˜
C(τ1)rα
1−2πλbr2−α
2
α−2, where ˜
C(τ1) = τ3ρ
τ1ηPt. Letting ˜
A=
2πλb
˜
C(τ1)(α−2) 1
(+1)α−2, we note that the set Nr2={r1:r1<2πλb
˜
C(τ1)(α−2) 1
α r
2−α
α
2}for r2≥˜
Awhile
for r2≤˜
Athe set will be simply Nr2={r1:r1≤r2}. Similarly, the set Pr2will be empty set
for r2≤˜
Awhile for r2≥˜
Athe set will reduce to Pr2={r1:2πλb
˜
C(τ1)(α−2) 1
α r
2−α
α
2≤r1≤r2}.
Applying these integration limits on our result in Lemma 1 leads to the final result in Lemma 5.
Similarly, applying these new integration limits to the result in Theorem 1 and noting that the
substitutions explained in Remark 1 include βDL = 0 (which makes G(r1, r2) = 0 in Eq. 16),
the final result in Theorem 3 follows.
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