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1
Coverage Analysis of Spatially Clustered
RF-powered IoT Network
Mohamed A. Abd-Elmagid, Mustafa A. Kishk, and Harpreet S. Dhillon
Abstract—Owing to the ubiquitous availability of radio-
frequency (RF) signals, RF energy harvesting is a promising
candidate for powering IoT devices, some of which may be
deployed at difficult-to-reach places thus making it inconvenient
or even impossible to replace or recharge their batteries. In this
paper, we model and analyze an IoT network which harvests RF
energy and receives information from the same wireless network.
In order to enable this operation, each time slot is partitioned
into charging and information reception phases. For this setup,
we characterize two performance metrics: (i) energy coverage,
and (ii) joint signal-to-interference-plus-noise (SINR) and energy
coverage. This analysis is performed using a spatial model that
captures coupling between the locations of the IoT devices and
the nodes of the wireless network (referred henceforth as the IoT
gateways), which is usually ignored in the existing literature. In
particular, we model the locations of the IoT devices using a
general Poisson cluster process (PCP) and assume that the IoT
gateways (GWs) are located at the cluster centers. Our results
concretely demonstrate that both energy and joint coverage
probabilities decrease as the size of the clusters increases. As
expected, the performance converges to the case of modeling the
locations of the IoT devices and the GWs as two independent
PPPs when the cluster sizes go to infinity.
Index Terms—Stochastic geometry, wireless power transmis-
sion, Poisson cluster process, coverage probability.
I. INTRODUCTION
Due to the massive scale of Internet-of-things (IoT), it is
considered highly inefficient and even impractical to replace
or recharge batteries in IoT devices [1]. This has naturally
led to the consideration of energy harvesting to circumvent or
supplement conventional power sources, such as replaceable
batteries, in these devices. Due to its ubiquity and cost efficient
implementation, RF energy harvesting has quickly emerged as
an appealing solution for powering IoT devices [2].
The system-level performance analysis of an RF-powered
IoT network depends strongly on the choice of the spatial
models for the locations of both the RF sources and the IoT
devices. Thus far, the existing literature has been limited to
the spatial models in which the locations of the IoT devices
and the IoT GWs (as noted above already, IoT GWs simply
refer to the nodes of the wireless network that is powering the
IoT network) are modeled by two independent Poisson point
processes (PPPs). This model, however, lacks the ability to
capture natural spatial coupling between the IoT devices and
the GWs which results from deploying GWs in the areas with
higher density of IoT devices. Motivated by this, considering
the GWs as the only dedicated source of RF energy in the
system, we provide the first analysis for a spatially clustered
RF-powered IoT network with the GWs located at the cluster
The authors are with Wireless@VT, Department of ECE, Virginia Tech,
Blacksburg, VA. Email: {maelaziz, mkishk, hdhillon}@vt.edu. The support
of the U.S. NSF (Grant CCF-1464293) is gratefully acknowledged.
centers. Note that as the cluster sizes increase, this setup
converges to the independent PPP model used in the literature,
which renders the existing results in the literature as the special
cases of the results derived in this paper.
A. Related Work
Energy harvesting wireless networks are studied in literature
from different perspectives and with the focus on different
performance aspects [3]–[9]. Recalling the system setup con-
sidered in this paper where the locations of the RF-powered
IoT devices are modeled by PCP, the most relevant literature
can be categorized into two sets: (i) stochastic geometry
based analysis of energy harvesting wireless networks, and
(ii) analysis of wireless networks using PCP. Each of the two
categories is discussed next.
Stochastic geometry has been widely used for the analysis
of energy harvesting wireless networks due to its tractability
and realism [10]–[17]. However, most of the existing literature
focuses on using independent PPPs for modeling the locations
of both BSs and users [10]–[12], [14]. Although relatively
sparse, some works did consider setups in which either the
RF-powered users or the BS locations are modeled using a
different point process (other than PPP), such as modeling
the locations of RF sources using Ginibre α-determinantal
point process in [15] and using Poisson hole process (PHP)
in [16]. Authors in [17] used PCP for modeling the locations
of power beacons in a backscatter communication system.
However, due to the use of beamforming at the power beacons,
only the energy received from the cluster head is considered.
Contrary to the existing works, where the coupling between
the locations of RF-powered wireless devices and the locations
of the RF-sources is usually ignored, our paper captures this
coupling by modeling the locations of the IoT devices using
a PCP.
Before going into more details about our contributions, it
should be noted that the ability of PCP to capture spatial
coupling among different wireless network components has re-
cently made it a preferred choice for modeling the user and/or
base station locations in a heterogeneous cellular network
(HetNet). This coupling was already captured in simulators
used by wireless industry, such as those used by the 3GPP [18],
[19]. In a series of recent works, PCP is used to: (i) model
the locations of mobile users while the BSs are assumed to
be located at the cluster centers [20], [21], (ii) model the
locations of small cell base stations (SBSs) where the macro
BSs are located at the cluster centers [22], and (iii) model the
clustering of both users and SBSs [23], [24]. Different from
these existing works, where the focus is on deriving the SINR
coverage probability, this paper provides the first analysis of
2
the joint energy and SINR coverage probability for spatially-
clustered RF-powered networks.
Contributions. This paper studies an RF-powered IoT net-
work, where the IoT GWs are the only source of RF energy.
Furthermore, each time-slot is assumed to be divided into
two phases: (i) charging phase and (ii) information reception
phase. For this setup, we provide an accurate characterization
of the energy coverage probability of RF-powered IoT with
the locations of the IoT devices modeled using PCP and the
GWs deployed in the cluster centers. Compared to the usual
approach of modeling the locations of IoT devices and GWs
using two independent PPPs, the spatial coupling captured in
the proposed model adds another layer of complexity to the
performance analysis. We propose an accurate approximation
to handle this complexity and derive an expression for each of
the energy coverage probability and the joint energy and SINR
coverage probability. This analysis provides several useful
system design insights. For instance, we show the existence
of optimal partitioning between charging and information re-
ception phases that maximizes the average system throughput.
In addition, we show that the performance of the IoT network
in terms of energy and joint coverage probabilities is lower
bounded by the setup in which the locations of the GWs and
the IoT devices are modeled using two independent PPPs.
II. SY ST EM MO DE L
A. Network Model
We study an RF-powered IoT network in which the IoT
devices are solely powered by RF energy harvesting circuitries.
As discussed in Section I, inspired by the fact that the GWs are
more likely to be deployed in areas where the density of the
IoT devices is relatively high, we primarily focus on the setup
in which the locations of GWs and IoT devices are coupled. In
particular, the locations of IoT devices are modeled by a PCP
Φu, where the locations of cluster centers are modeled by a
PPP Φwith density λ. The locations of IoT devices forming
each cluster are independent and identically distributed (i.i.d.)
given the location of their cluster center [25]. Union of all
locations of IoT devices around cluster centers forms the PCP
Φu. The GWs are assumed to be deployed at the cluster centers
and, hence, the locations of the GWs are modeled by Φ. We
consider a generic setup where the IoT device location Yu∈
R2with respect to its cluster center follows some arbitrary
distribution with probability density function fYu(·).
Time is assumed to be slotted with the duration of each
slot being Tseconds. Each time slot is partitioned into two
phases: i) charging phase: during the first portion of each
time slot, τT seconds, all GWs act as RF chargers for the
IoT devices so that each IoT device could harvest a certain
amount of energy required for its communication needs, and
ii) information reception phase: using the harvested energy in
the charging phase, each IoT device connects to a certain GW
and receives the transmitted data signal by its serving GW
during the remaining (1 −τ)Tseconds.
B. Propagation Model and Metrics of Interest
We perform our downlink analysis at a typical IoT device,
which is a randomly chosen IoT device from a randomly
chosen cluster of Φu(referred to as the representative cluster,
and its center is denoted by x0). Due to stationarity of this
setup, the typical IoT device is assumed to be located at the
origin without loss of generality. Assuming that the transmitted
power by all GWs is the same, denoted by Pt, the received
power at the location of the typical IoT device from a GW
located at x∈R2is Ptgxkxk−α, where gxdenotes the small-
scale fading gain between the typical IoT device and the GW
located at x, and kxk−αrepresents standard power law path-
loss with exponent α > 2. Under Rayleigh fading assumption,
gxis an exponential random variable with unit mean, i.e.,
gx∼exp(1). Hence, the total harvested energy by the typical
IoT device during charging phase can be expressed as
EH=ητ T X
x∈Φ
Ptgxkxk−α,(1)
where 0≤η≤1is the efficiency of the energy harvesting
circuitry. Owing to its longer lifetime compared to regular
rechargeable batteries, we assume that a supercapacitor is
used for storing the harvested energy at each IoT device. The
supercapacitor’s large charging and discharging rates make it
possible to use the energy soon after it is harvested. However,
due to its high leakage current, it is reasonable to assume that
any residual energy left in the current time slot will not be
available for use in a future time slot [26]. In other words,
the energy harvested by each IoT device in a certain time slot
is available to be consumed during the same time slot only.
Considering the scenario in which each IoT device is equipped
with a finite capacity battery is left for future work.
Using the harvested energy in the charging phase, the typical
IoT device receives information in the information reception
phase under maximum average received power based cell asso-
ciation strategy. In particular, the typical IoT device connects
to the GW which provides maximum received power averaged
over small-scale fading, i.e, the typical IoT device is served
by its closest GW. Hence, the signal-to-interference-plus-noise
ratio (SINR) at the typical IoT device in the information
reception phase can be expressed as
SINR = Pthx∗kx∗k−α
σ2+Px∈Φ\x∗Pthxkxk−α,(2)
where x∗is the location of the serving GW, hx∼exp(1)
models the small-scale fading gain in the information reception
phase and is assumed to be independent from gx, and σ2
denotes the thermal noise power. For this setup, we character-
ize the performance of the RF-powered IoT network in terms
of energy coverage probability, joint coverage probability and
average downlink achievable throughput, which are formally
defined next.
Definition 1. During the charging phase, the energy coverage
event occurs when the energy harvested by the typical IoT
device is at least Erec. The typical IoT device needs this
amount of energy Erec to power its receiving circuitry and,
hence, receive data successfully during the information recep-
tion phase. Practically, Erec is an increasing function of the
target downlink data rate and the duration of the information
reception phase [27]. The probability of the energy coverage
3
event can be mathematically expressed as
Ecov =E[ (EH≥Erec)] ,(3)
where (·)is the indicator function.
Definition 2. The typical IoT device is said to be in joint
coverage if two conditions are satisfied: i) EH≥Erec, and
ii) the SINR is above a specific threshold value βduring
the information reception phase. Therefore, the joint coverage
probability can be mathematically expressed as
Pcov =E[ (EH≥Erec) (SINR ≥β)] .(4)
Definition 3. The average received number of bits by the IoT
device per unit time per unit bandwidth can be expressed as
R= log2(1 + β)E[ (EH≥Erec) (SINR ≥β)] .(5)
C. Mathematical Preliminaries
After deriving the general results in terms of fYu(defined
in Section II-B), we will specialize them to a special case of
interest where Φuis modeled as a Mat´
ern cluster process. In
a Mat´
ern cluster process [25], the locations of devices are
sampled uniformly at random independently of each other
within a circular disc of radius Rcaround their cluster centers,
hence
fYu(y) = (1
πR2
c,if kyk ≤ Rc
0 otherwise.(6)
where yis a realization of the random vector Yu.
Recall that the locations of the typical IoT device and
GW deployed at its cluster center are coupled. In order to
explicitly capture this fact, we define two point processes: i)
Φ0which consists of only the representative cluster center,
i.e., Φ0={x0}, and ii) Φ1which includes the rest of points
of Φ, i.e., Φ1= Φ \x0. By this construction, the link between
the typical IoT device and the GW located at its cluster center
can be handled separately, as done in [20]. Note that Φ1can
be argued to have the same distribution as Φby applying
Slivnyak’s theorem [25]. Since Φ0includes only the GW
located at the representative cluster center, the typical IoT
device either connects to the closest GW from Φ1located at
x∗
1or the GW located at its cluster center x∗
0=x0. Therefore,
the location of the serving GW is given by
x∗= arg max
x∈{x∗
0,x∗
1}kxk−α.(7)
Let Ri=kx∗
ik,i∈ {0,1}, denote the distance from
the typical IoT device to its closest GW from Φi. Then, the
distribution of the distance R1is given by [25]:
PDF : fR1(r1)=2πλ exp −πλr2
1, r1≥0,(8)
CCDF : ¯
FR1(r1) = exp −πλr2
1, r1≥0.(9)
On the other hand, since the typical IoT device is located
at the origin, the relative location of the representative cluster
center with respect to the typical IoT device (x0) will have
the same distribution as that of the IoT device location Yu.
Therefore, the distribution of R0=kx∗
0kcan be obtained
by applying the standard transformation from Cartesian to
polar coordinates, to the joint distribution of x0expressed in
Cartesian domain. We provide the distribution of the distance
R0for a Mat´
ern cluster process in the following remark.
Remark 1. When Φuis a Mat´
ern cluster process, the distri-
bution of R0is given by
PDF : fR0(r0) = 2r0
R2
c
,0≤r0≤Rc,(10)
CCDF : ¯
FR0(r0) = R2
c−r2
0
R2
c
,0≤r0≤Rc.(11)
Let us call AΦias the association event of the typical
IoT device with Φiin the information reception phase. Given
that the typical IoT device is associated with Φi, the serving
distance Wiis the distance between the typical IoT device
and its closest GW in Φi, i.e., Wi=Ri| AΦi. Then,
the distributions of the serving distance conditioned on the
association with Φ0and Φ1are given respectively by [20]:
PDF : fW0(w0) = ¯
FR1(w0)fR0(w0)
A0
,(12)
PDF : fW1(w1) = ¯
FR0(w1)fR1(w1)
A1
,(13)
where A0and A1denote the association probabilities of the
typical IoT device with Φ0and Φ1, respectively, i.e., Ai=
P(AΦi).
III. ENERGY COVE RAG E PROBAB IL IT Y ANALYSI S
This section is dedicated to studying the energy coverage
probability, as defined in Definition 1. Since the typical IoT
device is associated with either Φ0or Φ1, from the total
probability law, the energy coverage probability, given by (3),
can be expressed as
Ecov =E[ (EH≥Erec)] =
1
X
i=0
E[ (EH≥Erec)| AΦi]Ai
=
1
X
i=0
E(i)
covAi.(14)
Deriving an accurate closed-form expression for the energy
coverage probability is very challenging. This is attributed to
the fact that the CDF of the power-law shot noise process,
which represents the total amount of harvested energy by the
typical IoT device, is not known in closed form [25]. To
lend tractability, we consider that EHis approximated by the
harvested energy from the serving GW located at x∗plus the
conditional mean of the energy harvested from other GWs.
Thus, EHcan be expressed as
EH≈ητ T
gx∗kx∗k−α+E
X
x∈Φ\x∗
gxkxk−α
kx∗k
.
(15)
The energy coverage probability conditioned on the associ-
ation of the typical IoT device with Φi,i∈ {0,1}, is given
by the following two Lemmas.
Lemma 1. Given that the typical IoT device associates with
Φ1, the energy coverage probability conditioned on Φis given
by
E(1)
cov |Φ=P(EH≥Erec | AΦ1,Φ) = e−[wα
1(C(τ)−Ψ(w1))]+,
(16)
4
while the unconditional probability is given by
E(1)
cov =Z∞
0
e−[wα
1(C(τ)−Ψ(w1))]+fW1(w1)dw1,(17)
where C(τ) = Erec
ητ T Pt,[x]+= max{0, x}and Ψ(w1)is
defined as
Ψ(w1) = Z∞
r0>w1
r−α
0
fR0(r0)
¯
FR0(w1)dr0+2πλ
α−2w2−α
1.(18)
Proof: See Appendix A.
Lemma 2. Given that the typical IoT device associates with
Φ0, the energy coverage probability conditioned on Φis given
by
E(0)
cov |Φ=P(EH≥Erec | AΦ0,Φ) = e−[wα
0(C(τ)−θ(w0))]+,
(19)
while the unconditional probability is given by
E(0)
cov =FW0(A) + Z∞
A
e−C(τ)wα
0−2πλw2
0
α−2fW0(w0)dw0,
(20)
where A=2πλ
C(τ)(α−2) 1
α−2and θ(w0) = 2πλ
α−2w2−α
0.
Proof: See Appendix B.
Remark 2. Intuitively, increasing the allocated portion of time
slot for charging phase, i.e., τ T , allows the IoT devices to
harvest more energy during the charging phase and, hence, the
energy coverage probability is improved. This can be clearly
seen from (17) and (20), where as τincreases, C(τ)decreases
and, hence, the energy coverage probability increases.
From the results given by Lemmas 1 and 2, the uncon-
ditional energy coverage probabilities for a Mat´
ern cluster
process are presented in the next corollary.
Corollary 1. When Φuis a Mat´
ern cluster process, the
unconditional energy coverage probabilities are given by
E(1)
cov =2πλ
A1ZRc
0
e−([wα
1(C(τ)−Ψ(w1))]++πλw2
1)
×w1
R2
c−w2
1
R2
c
dw1,(21)
E(0)
cov =1−e−πλA2
πλR2
cA0
+1
A0ZRc
A
e−(C(τ)wα
0+(πλ−2πλ
α−2)w2
0)2w0
R2
c
dw0,(22)
where A1=e−πλR2
c+πλR2
c−1
πλR2
c
,A0= 1 −A1and Ψ(w1)is
given by
Ψ(w1) = 2
α−2w2−α
1−R2−α
c
R2
c−w2
1
+πλw2−α
1.(23)
Proof: For a Mat´
ern Cluster process, Ψ(w1)can be
derived as follows
Ψ(w1) = Z∞
r0>w1
r−α
0
fR0(r0)
¯
FR0(w1)dr0+2πλ
α−2w2−α
1
(a)
=2w2−α
1−R2−α
c
(α−2) (R2
c−w2
1)+2πλ
α−2w2−α
1,(24)
where (a) follows from (10) and (11). The final results are
obtained by substituting (12) and (13) into (17) and (20),
respectively.
Remark 3. In the case of a Mat´
ern cluster process, as Rc→
∞,A1approaches 1 and Ψ(w1), given by (23), approaches
2πλw2−α
1
α−2.
Using Lemmas 1 and 2, the energy coverage probability is
formally stated in the following Theorem.
Theorem 1. The energy coverage probability can be obtained
as
Ecov =A0E(0)
cov +A1E(1)
cov,(25)
where E(1)
cov and E(0)
cov are given respectively by (17) and (20).
IV. JOI NT COVERAG E PROBABILITY ANALYS IS
In this section, using the conditional energy coverage prob-
ability results obtained in Section III, we derive the joint cov-
erage probability given by Definition 2. Afterwards, using the
joint coverage probability result, we characterize the average
downlink achievable throughput. From total probability law,
the joint coverage probability can be expressed as
Pcov =E[ (SINR ≥β) (EH≥Erec)] =
1
X
i=0
E[ (SINR ≥β) (EH≥Erec)| AΦi]Ai=
1
X
i=0
P(i)
covAi.
(26)
A. Joint Coverage Probability
In this subsection, our primary objective is to derive the joint
coverage probability experienced by the typical IoT device.
Towards this objective, we start by deriving the joint coverage
probability conditioned on the association of the typical IoT
device with Φi. The conditional joint coverage probabilities
are given by the following two Lemmas.
Lemma 3. Conditioned on the association of the typical IoT
device with Φ1, the joint coverage probability with SINR
threshold βand energy threshold Erec is given by
P(1)
cov =1
A1Z∞
w1=0
e−βσ2wα
1
Pt+[wα
1(C(τ)−Ψ(w1))]++2πλw2
1ρ(β,α)
×fR1(w1)Z∞
r0>w1
1
1 + βwα
1r−α
0
fR0(r0)dr0dw1,(27)
where Ψ(w1)is given by (18) and ρ(β, α)is defined as
ρ(β, α) = β2
α
2R∞
β
−2
α
1
1+uα
2du.
Proof: See Appendix C.
Lemma 4. Conditioned on the association of the typical IoT
device with Φ0, the joint coverage probability with SINR
threshold βand energy threshold Erec is given by
P(0)
cov =ZA
0
e−βσ2wα
0
Pt+2πλw2
0ρ(β,α)fW0(w0)dw0
+Z∞
A
e−hβσ2
Pt+C(τ)iwα
0+[ρ(β,α)−1
α−2]2πλw2
0fW0(w0)dw0,
(28)
Proof: See Appendix D.
5
Remark 4. Similar to Remark 2 for the energy coverage
probability, from (27) and (28), it is clear that increasing τde-
creases C(τ), and consequently the joint coverage probability
increases.
Using Lemmas 3 and 4, the joint coverage probability is
formally stated in the following Theorem.
Theorem 2. The joint coverage probability can be obtained
as
Pcov =A0P(0)
cov +A1P(1)
cov,(29)
where P(1)
cov and P(0)
cov are given respectively by (27) and (28).
B. Average Downlink Throughput
Using the joint coverage probability obtained in the previous
subsection, the average downlink achievable throughput is
characterized in this subsection. We now specialize the generic
definition of the average achievable throughput, given by
Definition 3, to our setup in the following proposition.
Proposition 1. The average downlink achievable throughput
per IoT device, expressed in bits/sec/Hz, is given by R=
(1 −τ) log2(1 + β)Pcov, where Pcov is given by Theorem 2.
Here, the fraction 1−τis due to the fact that the typical
IoT device only receives data from its serving GW during the
information reception phase, which occupies 1−τfraction of
the total time slot duration.
Remark 5. As shown in Remark 4, Pcov is an increasing
function of τ. However, the portion of time slot dedicated for
information reception phase, i.e., (1 −τ)T, decreases with τ.
This suggests the existence of an optimal τthat maximizes the
average downlink achievable throughput.
V. NUMERICAL RE SU LTS
Now, we verify the accuracy of the expressions derived
in Sections III and IV by comparing them with simulation
results. Unless otherwise specified, the following simulation
setup is considered: α= 4,λ= 1,β= 1 dB, Erec =
(1 −τ)T(aR0+b)joules, R0= log2(1 + β),T= 10−2
seconds, a= 10−3,b= 5 ×10−4,η= 10−3,σ2= 0, and
Pt= 1.
In Fig. 1a, we plot the energy coverage probability. The
results support our comments in Remarks 2 and 4 that the
energy coverage probability increases as the duration of the
charging phase τT increases. We also notice that the energy
coverage probability increases as Rcdecreases. Recalling that
in our setup the GWs are deployed at the cluster centers,
where the locations of the cluster centers are modeled by
a PPP, we compare the performance of our setup with the
one in which the locations of the IoT devices and the GWs
are modeled using two independent PPPs. The latter setup,
which was studied in [11], is referred to in Fig. 1a as PPP.
As expected, we notice that the gap between the performance
of the considered setup and the PPP setup from [11] increases
as the cluster size decreases. Furthermore, we notice that
as the cluster size increases, the energy coverage probability
converges to that of the PPP setup of [11].
In Fig. 1b, we plot the joint coverage probability derived in
Theorem 2 against different values of τ. We notice that the
joint coverage probability converges to a fixed value as τin-
creases. This is expected due to the convergence of the energy
coverage probability to unity as τincreases, which reduces
the joint coverage probability to the probability of having the
SINR value above βat large values of τ. Similar to the energy
coverage, we note that the joint coverage probability is lower
bounded by that of the PPP setup considered in [11].
In Fig. 1c we plot the average throughput provided in
Proposition 1. We observe the existence of an optimal value of
τthat maximizes the throughput, as we discussed in Remark 5.
VI. CONCLUSION
In this paper, we studied a system setup consisting of two
networks: (i) RF-powered IoT, where the locations of the
IoT devices are modeled by PCP and (ii) the GWs, which
are deployed at the cluster centers. Assuming that the GWs
are the only sources of RF energy for the IoT devices, we
derived the energy coverage and the joint SINR and energy
coverage probability of the IoT device in the downlink. We
proposed an accurate approximation in order to handle the
derivation challenges that result from modeling the locations
of the IoT devices as a PCP. Our numerical results revealed
that the performance of the IoT network in terms of the
energy and joint coverage probabilities is better when the
GWs are deployed at the cluster centers compared to when
the locations of the IoT devices and GWs are modeled using
two independent PPPs.
This work has many possible extensions. For instance, we
focused in this paper on only the downlink performance of the
clustered RF-powered IoT. One possible extension would be
to consider the joint uplink/downlink coverage probability of
this system setup. Another possible extension is to consider
battery-equipped IoT devices with finite battery sizes. In that
case, the dynamics and steady state distribution of the battery
levels would explicitly appear in the analysis.
APPENDIX
A. Proof of Lemma 1
The energy coverage probability, conditioned on Φand AΦ1,
can be expressed as follows
E(1)
cov |Φ=P(EH≥Erec | AΦ1,Φ)
=P ητ T X
x∈Φ
Ptgxkxk−α≥Erec | AΦ1,Φ!
(a)
=P gx∗kx∗k−α+E"X
x∈Φ\x∗
gxkxk−α| AΦ1,kx∗k#
≥C(τ)| AΦ1,Φ!
(b)
=P gx∗
1w−α
1+E"X
x1∈Φ1\x∗
1
gx1kx1k−α+gx0kx0k−α
| AΦ1, w1#≥C(τ)|w1!
6
Normalized duration of charging phase τ
0 0.1 0.2 0.3 0.4 0.5
Energy coverage probability
0
0.2
0.4
0.6
0.8
1
Theorem 1
Simulation
PPP
Rc increases:
[0.5, 0.75, 1.5]
(a)
Normalized duration of charging phase τ
0 0.1 0.2 0.3 0.4 0.5
Joint coverage probability
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Theorem 2
Simulation
PPP
Rc increases:
[0.5, 0.75, 1.5]
(b)
Normalized duration of charging phase τ
0 0.1 0.2 0.3 0.4 0.5
Throughput (bits/sec/Hz)
0
0.2
0.4
0.6
0.8
1Proposition 1
Simulation
PPP
Rc increases:
[0.5, 0.75, 1.5]
(c)
Fig. 1. Performance characterization of the typical device in terms of: (a) energy coverage probability, (b) joint coverage probability, and (c) average downlink
achievable throughput.
=Pgx∗
1w−α
1+ Ψ(w1)≥C(τ)|w1(c)
= e−[wα
1(C(τ)−Ψ(w1))]+,
(30)
where in step (a), the energy harvested at the typical device is
approximated by the sum of energy harvested from its serving
GW x∗and the conditional mean of the energy harvested from
the other GWs, and C(τ) = Erec
ητ T Pt. Step (b) follows from the
conditioning on AΦ1and (c) follows from the Rayleigh fading
assumption, i.e., gx∗
1∼exp(1).
The conditional mean of the energy harvested from all GWs
except the serving one, Ψ(w1), is derived as follows
E
X
x1∈Φ1\x∗
1
gx1kx1k−α+gx0kx0k−α| AΦ1, w1
(a)
=ER0kx0k−α|R0> w1+EΦ1
X
x1∈Φ1\x∗
1
kx1k−α|w1
(b)
=Z∞
r0>w1
r−α
0
fR0(r0)
¯
FR0(w1)dr0+ 2πλ Z∞
w1
r−αrdr
=Z∞
r0>w1
r−α
0
fR0(r0)
¯
FR0(w1)dr0+2πλ
α−2w2−α
1,(31)
where (a) follows from the fact that the channel gains are
assumed to be Rayleigh distributed along with conditioning
on AΦ1which implies that R0> w1, and (b) follows
from Campbell Mecke Theorem [25] with conversion from
Cartesian to polar coordinates. Finally, E(1)
cov is obtained by
taking the expectation of (30) with respect to the serving
distance W1. This completes the proof.
B. Proof of Lemma 2
Given that the typical device is associated with Φ0, i.e.,
the GW located at its cluster center, the energy coverage
probability conditioned on Φis obtained as follows
E(0)
cov |Φ=P(EH≥Erec | AΦ0,Φ) (a)
= e−[wα
0(C(τ)−θ(w0))]+,
(32)
where (a) follows from applying similar arguments as in (30)
and θ(w0)is given by
θ(w0) = E"X
x1∈Φ1
gx1kx1k−α|w0#=2πλ
α−2w2−α
0.(33)
Substituting θ(w0)from (33) into (32), we obtain
E(0)
cov |Φ=(e−(C(τ)wα
0−2πλ
α−2w2
0),if w0≥A
1,if w0< A (34)
where A=2πλ
C(τ)(α−2) 1
α−2. The final expression in (20)
follows from taking the expectation over the serving distance
W0along with applying the condition in (34). This completes
the proof.
C. Proof of Lemma 3
Given AΦ1, the joint coverage probability is obtained as
follows
P(1)
cov =E[ (SINR ≥β) (EH≥Erec)| AΦ1]
(a)
=EΦ"P(SINR ≥β| AΦ1,Φ) P(EH≥Erec | AΦ1,Φ) #,
(35)
where (a) follows from the fact that the energy and SINR
coverage events are (conditionally) independent conditioned
on Φ. The SINR coverage probability conditioned on Φand
AΦ1,S(1)
cov |Φ, is derived as follows
P(SINR ≥β| AΦ1,Φ) = P Pthx∗
1w−α
1
I1+σ2≥β| AΦ1,Φ!
(a)
=Ehx0,hx1"exp −βwα
1I1+σ2
Pt!#(b)
= e−βσ2wα
1
Pt×
Ehx0he−βwα
1kx0k−αhx0i×Y
x1∈Φ1\x∗
1
Ehx1he−βwα
1kx1k−αhx1i
(c)
= e−βσ2wα
1
Pt1
1 + βwα
1kx0k−αY
x1∈Φ1\x∗
1
1
1 + βwα
1kx1k−α,
(36)
where Ii=Px∈Φ\x∗
iPthxkxk−α, (a) follows from the fact
that hx∗
1∼exp(1), (b) follows from the independence of the
channel power gains hx0and {hx1}, and (c) follows from the
assumption that the channel gains are Rayleigh distributed.
Therefore, from (16) and (36), P(1)
cov can be expressed as
P(1)
cov =EΦhE(1)
cov |ΦS(1)
cov |Φ| AΦ1i
7
=EΦ"e−βσ2wα
1
Pt1
1 + βwα
1kx0k−α
×Y
x1∈Φ1\x∗
1
1
1 + βwα
1kx1k−αe−[wα
1(C(τ)−Ψ(w1))]+| AΦ1#
(a)
=Ew1"e−βσ2wα
1
Pt+[wα
1(C(τ)−Ψ(w1))]+ER0h1
1 + βwα
1r−α
0
|R0> w1i×EΦ1\x∗
1hY
x1∈Φ1\x∗
1
1
1 + βwα
1kx1k−αi#
(b)
=Ew1"e−βσ2wα
1
Pt+[wα
1(C(τ)−Ψ(w1))]++2πλw2
1ρ(β,α)
×Z∞
r0>w1
1
1 + βwα
1r−α
0
fR0(r0)
¯
FR0(w1)dr0#,(37)
where (a) follows by distributing the expectation over the point
process Φ1\x∗
1and the rest of random quantities and (b)
follows from the PGFL of the PPP [25] where ρ(β, α) =
β2
α
2R∞
β
−2
α
1
1+uα
2du. The final expression in (27) follows from
(13). This completes the proof.
D. Proof of Lemma 4
The SINR coverage probability conditioned on Φand AΦ0
can be obtained as follows
S(0)
cov |Φ=P(SINR ≥β| AΦ0,Φ)
=PPthx0w−α
0
I0+σ2≥β| AΦ0,Φ
(a)
= e−βσ2wα
0
PtY
x1∈Φ1
1
1 + βwα
0kx1k−α,(38)
where (a) follows from applying similar arguments as in (36).
Therefore, from (19) and (38), P(0)
cov can be expressed as
P(0)
cov =EΦhE(0)
cov |ΦS(0)
cov |Φ| AΦ0i
(a)
=Ew0"e−βσ2wα
0
Pt+[wα
0(C(τ)−θ(w0))]++2πλw2
0ρ(β,α)#,
(39)
where (a) follows from applying the same approach used in
(37). The final expression in (28) follows from applying the
condition in (34). This completes the proof.
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