#  Coverage Probability and Spectral Efficiency Analysis of Multi-Gateway
Downlink LoRa Networks

Lam-Thanh Tu, Abbas Bradai and Yannis Pousset Institute XLIM, University of
Poitiers, France e-mail: {lam.thanh.tu, abbas.bradai, yannis.pousset}@univ-
poitiers.fr

###### Abstract

The system-level performance of multi-gateway downlink long-range (LoRa)
networks is investigated in the present paper. Specifically, we first compute
the active probability of a channel and the selection probability of an active
end-device (ED) in the closed-form expressions. We then derive the coverage
probability (Pcov) and the area spectral efficiency (ASE) under the impact of
the capture effects and different spreading factor (SF) allocation schemes.
Our findings show that both the Pcov and the ASE of the considered networks
can be enhanced significantly by increasing both the duty cycle and the
transmit power. Finally, Monte-Carlo simulations are provided to verify the
accuracy of the proposed mathematical frameworks.

###### Index Terms:

LPWAN, LoRa Networks, Stochastic Geometry, System-Level Analysis, Spectral
Efficiency.

##  I Introduction

It is expected that the number of connected devices via wireless technology
will be over 20 billion by 2022 [1] and this number is likely to increase
significantly to realize the Internet of Things (IoT). Additionally, there are
many IoTs devices located in remote areas that are not able to connect to the
power grid to replenish its battery. Thus, short-range technologies such as
ZigBee, Bluetooth Low Energy (BLE), and other technologies that belong to the
family of the IEEE 802.15.4 standards are not suitable for connecting such
long-range transmission, low power consumption devices. Recently, low power
wide area networks (LPWAN) technology has emerged as a promising solution to
fulfill these requirements [2]. Among all available LPWAN technologies, long-
range (LoRa) has received many attractions from both academia and industry
thanks to its advantageous properties, i.e., long-range transmissions, energy-
efficient, and wide coverage areas. LoRa is primarily designed for uplink
traffic, from end-devices (ED) to gateway. Hence, it is no surprise that most
of the existing literature concentrates on best-effort applications, showing
the scenarios with solely unconfirmed messages and leaving the downlink
traffic although it is vital as well. For example, the acknowledgment (ACK)
messages sent to EDs to confirm the data reception, the join-accept message to
accept the join-request of EDs to join the networks, and the control messages
to adjust the transmission parameters such as LinkADRReq, LinkCheckAns. etc
[3]. Moreover, all mission-critical applications require a reliable data
transport system where the ACK messages are unavoidable [4].

The performance of uplink LoRa networks was studied extensively in [5, 6, 7,
8, Thanh:WCL:01]. In [5], the system-level coverage probability (Pcov) of ED
was examined where EDs were modelled by a homogeneous Poisson point process
(PPP). The performance of both Pcov and area spectral efficiency (ASE) were
investigated in [6] where EDs are distributed according to the Matern cluster
process. Compared with PPP, the Matern cluster process provides more
accurately the characteristics of the EDs. It, however, leads to mathematical
intractability thus, numerical computations are employed for computing these
metrics. The Pcov and the energy efficiency of the uplink single gateway LoRa
networks were addressed under different MAC protocols, i.e., pure ALOHA,
slotted ALOHA, and CSMA [7]. The findings show that the slotted ALOHA attains
the highest performance among all considered protocols. In [8], the authors
proposed a novel spreading factor (SF) allocation scheme that ensures a fair
collision probability among all spreading factors. This scheme is superior to
the popular distance-based allocation scheme in terms of the packet error
rate. Although many papers have been published on the uplink performance of
LoRa networks, the performance of another direction is fairly attractive.
Indeed, the performance of downlink LoRa networks were studied in [3, 9, 10].
More precisely, Valentina et. al. in [3] addressed the downlink scalability of
LoRa via computed-based simulations rather than rigorous mathematical
frameworks. Their findings show that multi-gateway is beneficial for downlink
LoRa networks and overcomes the bottleneck of downlink networks, the gateway,
due to the constraint of the duty cycle. The authors in [9] investigated the
impact of downlink feedback of the uplink traffic via computer-based
simulations. In [10], authors examined the average waiting time of the
downlink frame of class B devices.

Building upon the aforementioned works the present paper investigates the
system-level performance of multi-gateway downlink LoRa networks where both
gateways and EDs are distributed according to a homogeneous PPP. More
precisely, our main contributions are summarized as follows: i) the impact of
the duty cycle on the availability of the channel is taken into consideration;
ii) both co- and inter-SF interference (capture effects) are taken into
account at the receiver; iii) the active probability of an arbitrary channel
and the selection probability of an active ED are computed in the closed-form
expressions; iv) Pcov and ASE under two SF allocation schemes, namely the
fair-collision and random schemes, as well as co- and inter-SF interference
are derived; and v) Monte-Carlo simulations are supplied to verify the
accuracy of the proposed frameworks.

##  II System Model

###  II-A LoRa Networks Modeling

Let us consider a downlink multi-gateway LoRa networks where gateways and EDs
follow by two independent homogeneous Poisson distribution processes denoted
by ${\Psi_{{\rm{GW}}}},{\Psi_{{\rm{ED}}}}$ with densities
${\lambda_{\rm{GW}}}\ll{\lambda_{\rm{ED}}}$, respectively. The PPP modelling
is adopted due to its mathematical tractability and is a lower bound of the
practical measurement [11]. Thus, it can be applied to any practical scenario.
We employ a slotted ALOHA protocol and all messages are taken place within one
time-slot [7]. The slotted ALOHA is chosen since it achieved the best
performance among all available protocols as proven in [7]. Additionally, each
gateway has $N_{\text{Ch}}=8$ channels for downlink transmission and all
channels are regulated by the duty cycle $\rho$ ranging from 0.1% to 10%
depending on the regulator [12]. Additionally, we adopt an equal power
allocation among all channels, the transmit power of each channel is then
computed as $P_{{\rm{tx}}}^{{\rm{Ch}}}={{P_{{\rm{tot}}}}}/{{N_{{\rm{Ch}}}}}$
where $P_{\text{tot}}$ is the total transmit power. The developed mathematical
frameworks are for the typical ED denoted by $\text{ED}_{0}$ which is located
at the origin (Slivnyak theorem [14, Th. 1.4.5]) and the gateway which serves
the $\text{ED}_{0}$ is referred to $\text{GW}_{0}$. The cell association
criterion is provided in Section II-C. Interference created from other
technologies that operate at the same industrial, scientific, and medical
(ISM) band does not take into account as it is a typical case in the
literature [5, 13].

###  II-B Channel Modelling

Considering a generic link from an arbitrary gateway to a generic end-device,
it is subjected to both small-scale fading and large-scale path-loss. The
impact of the shadowing is not taken into account since it can implicitly be
examined by appropriately scaling the density of GWs [15].

####  II-B1 Small-scale fading

Let us denote $h_{n}$ as the small-scale fading of an arbitrary link from a
gateway to ED and is modelled by a Rayleigh distribution. The channel gain
$|h_{n}|^{2}$, as a result, follows an exponential distribution with mean
$\Omega$. For simplicity, we assume that $\Omega_{n}=1,\forall n$.

####  II-B2 Large-scale path-loss

Let us consider a generic link from a gateway to an ED the large-scale path-
loss is defined as

$\displaystyle L_{n}=l\left(r_{n}\right)=K_{0}r_{n}^{\beta}.$ (1)

Here $r_{n}$ is the distance from the gateway to the ED;
$K_{0}=\left(4\pi{f_{c}}/c\right)^{2}$ and $\beta>2$ are the path-loss
constant and the path-loss exponent, respectively. $f_{c}$ is the carrier
frequency (in Hz) and $c=3\times 10^{8}$ (in meters per second) is the speed
of light.

###  II-C Cell Association Criterion

ED is served by the gateway which has the smallest path-loss. Let us denote
$\Psi_{{\rm{GW}}}^{\left({\rm{A}}\right)}$ as the set of available gateways. A
gateway is considered as available if it has at least one available channel at
each time slot. The serving GW, $\text{GW}_{0}$, is then formulated as
follows:

$\displaystyle{\rm{G}}{{\rm{W}}_{0}}=\mathop{\text{argmin}}\limits_{n\in\Psi_{{\rm{GW}}}^{\left({\rm{A}}\right)}}\left\\{{{L_{n}}}\right\\}.$
(2)

It is noted that $\Psi_{{\rm{GW}}}^{\left({\rm{A}}\right)}$ is also a
homogeneous PPP with density
$\lambda_{{\rm{GW}}}^{\left({\rm{A}}\right)}=\mu{\lambda_{{\rm{GW}}}}$
according to the thinning property of PPP [14].
$\mu=1-{\left({1-\rho}\right)^{{N_{{\rm{Ch}}}}}}$ is the probability having
one available channel of a gateway.

###  II-D Spreading Factor Allocation

####  II-D1 Fair-collision scheme

We consider the fair-collision SF allocation scheme that was proposed in [8].
The considered SF allocation method guarantees a fair-collision probability
among all available SFs. Additionally, it is proven that the selected scheme
is far better than the popular distance-based allocation scheme [8]. The
probability that an ED is assigned to SF$k$ from its serving gateway is then
computed as

$\displaystyle{p_{k}^{\text{fa}}}=\left(k/2^{k}\right)/\sum\nolimits_{i=7}^{12}\left(i/2^{i}\right),\;\;\;k\in\left\\{7,\ldots,12\right\\}.$
(3)

As a consequence, the density of EDs utilized SF$k$ under this scheme is
calculated as
$\lambda_{\text{ED,fa}}^{k}=p_{k}^{\text{fa}}\lambda_{\text{ED}}$.

####  II-D2 Random scheme

Differently under a random scheme, each ED is randomly assigned to an
arbitrary SF from its serving gateway. Thus, the probability of EDs utilized
SF$k$ is $p_{k}^{\text{ra}}=1/6,\forall k$ and the density of EDs utilized
SF$k$ is given as
$\lambda_{\text{ED,ra}}^{k}=p_{k}^{\text{ra}}\lambda_{\text{ED}}$.

###  II-E Interference Modeling

In the present work, we consider the aggregate interference from all active
gateways apart from the serving GW, $\text{GW}_{0}$. A gateway is defined as
active if it does not belong to one of the two following cases: i) there is no
ED in its coverage area; ii) EDs are presented in its coverage area,
nevertheless, no downlink transmission happens. The second condition is
originated from the main purpose of LoRa networks. Particularly, LoRa is
designed for uplink transmission, thus downlink traffic is very limited
compared to its counterpart and occurs solely when EDs request acknowledges or
control messages from the gateway. An ED demanding information is called
active ED and the set of active EDs is denoted by
$\Psi_{{\rm{ED}}}^{\left({\rm{A}}\right)}$ that follows by a homogeneous PPP
with density
$\lambda_{{\rm{ED}}}^{\left({\rm{A}}\right)}=\theta\lambda_{{\rm{ED}}}$ due to
the thinning property of PPP, $\theta$ is the active probability of EDs. The
adopted interference modelling is certainly more accurate than one considering
the strongest interference [5, 13], especially when a number of interferers go
up without bound. Moreover, we also consider a practical scenario where the
orthogonality between SFs is imperfect thus, the receiver suffers from both
co- and inter-SF interference instead of only co-SF interference like in [5,
13].

###  II-F Load Modeling

Let us denote ${N_{{\rm{ED}}}}\in\mathbb{N}$ and
$\tau\in\left\\{1,\ldots,N_{\text{Ch}}=8\right\\}$ are the number of active
EDs and the number of available channels in a generic cell at one time-slot.
If ${N_{{\rm{ED}}}}\geq\tau$ then the serving gateway will randomly select
$\tau$ active EDs to serve. It is noted that due to the hardware constraint,
the gateway can serve merely one ED per channel in one time-slot. Thus, there
is no intra-cell interference and the inter-cell interference is present on a
channel basis. On the contrary, if ${N_{{\rm{ED}}}}<\tau$ all
${N_{{\rm{ED}}}}$ active EDs will be served by its serving BS. The consider
random scheduling at each transmission instance ensures that all active EDs
associated with a gateway are scheduled for transmission in the long term
manner.

##  III Preliminary results

In this section, we summarize some intermediate results which are important
for computing the two considered metrics, namely the coverage probability and
the area spectral efficiency. More precisely, the active probability of a
generic channel of an available gateway is employed to identify the set of
active interferer when computing the SIR condition of the Pcov, i.e., the
probability that the intended signal is greater than the sum of active
gateways transmitting at the same channel. On the other hand, the selection
probability of an active ED is used to compute the ASE which are formulated in
(29).

###  III-A Active probability of a channel of an available gateway

The active probability of a channel of an available gateway refers to the
probability that a channel is randomly chosen among all available channels to
serve an active ED and is provided by Lemma 1 as follows:

###### Lemma 1

Considering
$\mathcal{A}=\lambda_{{\rm{ED}}}^{\left({\rm{A}}\right)}/\lambda_{{\rm{GW}}}^{\left({\rm{A}}\right)}$
the active probability of a channel denoted by $P_{\text{Act}}$ is calculated
as

$\displaystyle{P_{{\rm{Act}}}}=$
$\displaystyle\sum\limits_{i=1}^{{N_{\rm{Ch}}}}{}\left({1-\left({\frac{{{N_{{\rm{Ch}}}}-i}}{{{N_{{\rm{Ch}}}}}}+\frac{i}{{{N_{{\rm{Ch}}}}}}\sum\limits_{k=0}^{i-1}{}\left({1-\frac{k}{i}}\right){\mathcal{T}_{1}}\left(k\\!\right)}\\!\right)}\\!\right)$
$\displaystyle\;\;\;\;\;\;\;\;\;\times\mathcal{V}\left({{N_{{\rm{Ch}}}},i,\rho}\right)/\mu$
$\displaystyle\mathcal{T}_{1}\left(k\right)=$
$\displaystyle\frac{{{{\left({3.5}\right)}^{3.5}}\Gamma\left({k+3.5}\right){\mathcal{A}^{k}}}}{{\Gamma\left({3.5}\right)k!{{\left({\mathcal{A}+3.5}\right)}^{k+3.5}}}},$
(4)

where
$\mathcal{V}\left({{N_{{\rm{Ch}}}},i,\rho}\right)=\left({\begin{array}[]{*{20}{c}}{{N_{{\rm{Ch}}}}}\\\
i\end{array}}\right){\rho^{i}}{\left({1-\rho}\right)^{{N_{{\rm{Ch}}}}-i}}$ is
the probability having $i$ available channel out of $N_{\rm{Ch}}$ channels,
$\mathcal{T}_{1}\left(k\right)$ is the probability having $k$ active EDs in a
generic cell [16], and $\Gamma\left(.\right)$ is the Gamma function.

###### Proof:

The proof can be derived in two main steps. In the first step, we apply the
total probability theorem for the number of available channels conditioned on
at least one available channel. In the second step, given the number of
available channels, we compute the active probability of a channel via the
inactive probability and employ the total probability theorem for the number
of active EDs, $N_{\text{ED}}$, associated with the gateway. ∎

###  III-B Selection probability of an active ED

The selection probability of an active ED refers to the probability that an
active ED is scheduled for transmission in an available channel. This
probability accounts for the fact that the number of active EDs is greater
than the number of available channels. Thus, some active EDs can be served and
the others are blocked. The selection probability denoted by ${P_{\rm{Sel}}}$
is computed in Lemma 2 as follows:

###### Lemma 2

Considering
$\mathcal{A}=\lambda_{{\rm{ED}}}^{\left({\rm{A}}\right)}/\lambda_{{\rm{GW}}}^{\left({\rm{A}}\right)}$
the selection probability of an active ED can be computed as follows:

$\displaystyle{P_{{\rm{Sel}}}}=$
$\displaystyle\sum\limits_{i=1}^{{N_{\rm{Ch}}}}{}\frac{\mathcal{V}\left({{N_{{\rm{Ch}}}},i,\rho}\right)}{\mu}\left({1-{\mathcal{B}_{1}\left(i\right)}\left({{\mathcal{B}_{2}\left(i\right)}-{\mathcal{B}_{3}\left(i\right)}}\right)}\right),$
(5)

where

$\displaystyle{\mathcal{B}_{1}}\left(i\right)=$
$\displaystyle\frac{{{{\left({3.5}\right)}^{3.5}}\Gamma\left({i+4.5}\right){\mathcal{A}^{i}}}}{{\Gamma\left({3.5}\right){{\left({\mathcal{A}+3.5}\right)}^{i+4.5}}}}$
$\displaystyle{\mathcal{B}_{2}}\left(i\right)=$
$\displaystyle\frac{{{\;{}_{2}}{F_{1}}\left({1,i+4.5,i+1,\mathcal{A}/\left({\mathcal{A}+3.5}\right)}\right)}}{{\Gamma\left({i+1}\right)}}$
$\displaystyle{\mathcal{B}_{3}}\left(i\right)=$
$\displaystyle\frac{i{{\;{}_{2}}{F_{1}}\left({1,i+4.5,i+2,\mathcal{A}/\left({\mathcal{A}+3.5}\right)}\right)}}{{\Gamma\left({i+1}\right)}},$
(6)

where ${\;{}_{2}}{F_{1}}\left(.\right)$ is the Gaussian hypergeometric
function.

###### Proof:

The proof is also derived in two main steps and the first step is similar to
Lemma 1. In the second step, the selection probability of an active ED given
$i$ available channels denoted by ${P_{{\rm{Sel}}}}\left(i\right)$ is then
formulated as

$\displaystyle{P_{{\rm{Sel}}}}\left(i\right)=\sum\limits_{k=0}^{i-1}{}\Pr\left\\{{N_{{\rm{ED}}}^{{}^{\prime}}=k}\right\\}+\sum\limits_{k=i}^{\infty}\frac{i\Pr\left\\{{N_{{\rm{ED}}}^{{}^{\prime}}=k}\right\\}}{{k+1}},$
(7)

where $N_{{\rm{ED}}}^{{}^{\prime}}$ is the number of other active EDs in the
coverage area of a gateway conditioned on an ED has already associated to this
gateway; $\Pr\left\\{.\right\\}$ is the probability operator;
$\Pr\left\\{{N_{{\rm{ED}}}^{{}^{\prime}}=k}\right\\}$ is held with the help of
[16, Lemma 3] and by formulating the summations in terms of the Gaussian
hypergeometric function we finish the proof. ∎

##  IV Performance Analysis

The present analysis concentrates on two vital metrics of LoRa networks, i.e.,
the Pcov and the ASE. The Pcov particularly focuses on the performance of LoRa
networks from the viewpoint of an ED while the ASE is from the network’s point
of view. Thus, the examination of two metrics permits a holistic assessment of
the network performance.

###  IV-A Coverage Probability

An active ED of SF$k$ is considered as successfully received information from
its serving gateway, provided that both signal-to-noise ratio (SNR) and
signal-to-interference ratio (SIR) conditions are satisfied.

####  IV-A1 SIR condition analysis

The SIR condition is the probability that the ratio of the intended signal
utilized SF$k$ to the aggregate interference from all active GWs transmitting
on the same channel utilized SF$\widetilde{k}$ is greater than a pre-defined
threshold. More precisely, we investigate two aggregate interference cases: i)
only co-SF interference; ii) both co- and inter-SF interference. The former
corresponds to the perfect orthogonality between signals with different SFs,
while the latter considers imperfect orthogonality between these signals.

##### Co-SF interference analysis

The SIR condition under co-SF interference of an SF$k$ ED under SF allocation
scheme $s$ is formulated as follows:

$\displaystyle P_{{\mathop{\rm
cov}}}^{{\rm{SIR,co}}}\left({k,s}\right)=\Pr\left\\{{{\rm{SI}}{{\rm{R}}_{k,k}^{s}}\geq{\Delta_{k,k}}}\right\\}={P_{{\rm{SIR}}}}\left({k,k,s}\right),$
(8)

where ${\Delta_{k,\widetilde{k}}}$ (in dB),
$k,\widetilde{k}\in\left\\{{7,\ldots,12}\right\\}$ is the $k$-row,
$\widetilde{k}$-column of matrix $\Delta$ in (10) and accounts for the
interference rejection threshold that depends on the SF$k$ of the intended
signal and SF$\widetilde{k}$ of the interferer [17];
${\rm{SI}}{{\rm{R}}_{k,\widetilde{k}}^{s}}=\left({\left|{{h_{0}}}\right|}^{2}/L_{0}\right)/{I_{\widetilde{k},s}}$;
${I_{\widetilde{k},s}}=\sum\nolimits_{z\in\psi_{{}_{\widetilde{k},s}}^{\left(\rm{I}\right)\backslash
0}}{}\left({\left|{{h_{z}}}\right|}^{2}/L_{z}\right)\textbf{1}\left({{L_{z}}>{L_{0}}}\right)$
is the aggregate interference from all active gateways apart from the GW0
transmitting on the same channel as the typical link and utilizing
SF$\widetilde{k}\in\left\\{7,\ldots,12\right\\}$ under $s$ SF allocation
method; $\textbf{1}\left(.\right)$ is the indicator function; the condition
$\textbf{1}\left({{L_{z}}>{L_{0}}}\right)$ ensures that all interferer lie
outside the circle whose radius is the distance from GW0 to ED0 and center at
ED0; ${{{\left|{{h_{0}}}\right|}^{2}}}$, ${{{\left|{{h_{z}}}\right|}^{2}}}$
and ${{L_{0}}}$, ${{L_{z}}}$ are the small-scale fading and large-scale path-
loss of the serving GW0 and interferer GWz to ED0;
$\psi_{{}_{\widetilde{k},s}}^{\left(\rm{I}\right)\backslash 0}$ is the set of
interference gateways that serves ED of SF$\widetilde{k}$ under SF allocation
scheme $s$ and is approximated as a homogeneous PPP with density
$\lambda_{{\rm{GW}}}^{\left({\rm{I}}\right),\widetilde{k},s}=p_{\widetilde{k}}^{s}{P_{{\rm{Act}}}}\lambda_{{\rm{GW}}}^{\left({\rm{A}}\right)}$.

$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\begin{array}[]{*{20}{c}}\textrm{SF}_{7}&\textrm{SF}_{8}&\textrm{SF}_{9}&\textrm{SF}_{10}&\textrm{SF}_{11}&\textrm{SF}_{12}\end{array}$
(10) $\displaystyle\\!\\!\Delta\\!=$
$\displaystyle\\!\\!\begin{array}[]{*{20}{c}}\textrm{SF}_{7}\\\
\textrm{SF}_{8}\\\ \textrm{SF}_{9}\\\ \textrm{SF}_{10}\\\ \textrm{SF}_{11}\\\
\textrm{SF}_{12}\end{array}\\!\\!\\!\left[{\begin{array}[]{*{20}{c}}1&{-8}&{-9}&{-9}&{-9}&{-9}\\\
{-11}&1&{-11}&{-12}&{-13}&{-13}\\\ {-15}&{-13}&1&{-13}&{-14}&{-15}\\\
{-19}&{-18}&{-17}&1&{-17}&{-18}\\\ {-22}&{-22}&{-21}&{-20}&1&{-20}\\\
{-25}&{-25}&{-25}&{-24}&{-23}&1\end{array}}\\!\\!\\!\right].$ (23)

${P_{{\rm{SIR}}}}\left({k,k,s}\right)$ in (8) is computed in the closed-form
expression with the help of Lemma 3 and is given as

###### Lemma 3

Let us define a shorthand
$\Theta\left(x\right)={\;{}_{2}}{F_{1}}\left({1,-\delta,1-\delta,-x}\right)-1$,
$\delta=2/\beta$, the $P_{S{\rm{IR}}}\left({k,\widetilde{k}},s\right)$ is then
computed as follows

$\displaystyle
P_{S{\rm{IR}}}\left({k,\widetilde{k}},s\right)={\left({1+p_{\widetilde{k}}^{s}{P_{{\rm{Act}}}}\Theta\left({{\Delta_{k,\widetilde{k}}}}\right)}\right)^{-1}}.$
(24)

###### Proof:

See Appendix -A. ∎

By employing Lemma 3, $P_{{\mathop{\rm
cov}}}^{{\rm{SIR,co}}}\left({k,s}\right)$ in (8) is then evaluated as

$\displaystyle P_{{\mathop{\rm
cov}}}^{{\rm{SIR,co}}}\left({k,s}\right)={\left({1+p_{k}^{s}{P_{{\rm{Act}}}}\Theta\left({{\Delta_{k,k}}}\right)}\right)^{-1}}.$
(25)

##### Both co- and inter-SF interference analysis

The SIR condition under both co- and inter-SF interference of ED utilized
SF$k$ with $s$ SF allocation method is given as

$\displaystyle P_{{\mathop{\rm cov}}}^{{\rm{SIR,bo}}}\left({k,s}\right)=$
$\displaystyle\prod\nolimits_{\widetilde{k}=7}^{12}{\Pr\left\\{{{\rm{SIR}}_{k,\widetilde{k}}^{s}\geq{\Delta_{k,\widetilde{k}}}}\right\\}}$
$\displaystyle=$
$\displaystyle\prod\nolimits_{\widetilde{k}=7}^{12}{\left({1+p_{\widetilde{k}}^{s}{P_{{\rm{Act}}}}\Theta\left({{\Delta_{k,\widetilde{k}}}}\right)}\right)^{-1}}.$
(26)

####  IV-A2 SNR condition analysis

The SNR condition refers to the reception condition is the probability that
received signals at the ED of SF$k$ is greater than the quality-of-service
(QoS) requirement and is given as

$\displaystyle
P_{{\rm{cov}}}^{{\rm{SNR}}}\left(k\right)=\Pr\left\\{{\frac{{P_{{\rm{tx}}}^{{\rm{Ch}}}{{\left|{{h_{0}}}\right|}^{2}}}}{{{\sigma^{2}}{L_{0}}}}\geq{\gamma_{{\rm{D}},k}}}\right\\}=\delta\pi\lambda_{{\rm{GW}}}^{\left({\rm{A}}\right)}{\left({{K_{0}}}\right)^{-\delta}}$
$\displaystyle\times\int\limits_{x=0}^{\infty}{}{x^{\delta-1}}\exp\left({-\frac{{{\gamma_{{\rm{D}},k}}{\sigma^{2}}}}{{P_{{\rm{tx}}}^{{\rm{Ch}}}}}x-\pi\lambda_{{\rm{GW}}}^{\left({\rm{A}}\right)}{{\left({\frac{x}{{{K_{0}}}}}\right)}^{\delta}}}\right)dx.$
(27)

The integration in (IV-A2), unfortunately, is not able to compute in the
closed-form expression due to the arbitrary value of the path-loss exponent,
i.e., $\beta>2$. It, however, can be straightforwardly computed by employing
numerical methods via some popular software like Matlab or Mathematica.
${\gamma_{\rm{D}}}$ is the QoS threshold and is a function of the spreading
factor $k$, particularly, we have
${\gamma_{\rm{D}}}=\left\\{-6,-9,-12,-15,-17.5,-20\right\\}$ dBm for SF7 to
SF12, respectively.
${{\sigma^{2}}}=-174+\text{NF}+10\log_{10}\left(\text{Bw}\right)$ (in dBm) is
the noise variance of AWGN noise; NF is the noise figure (in dBm) and Bw is
the transmission bandwidth (in Hz).

Having obtained the mathematical framework of two conditions, the coverage
probability of a SF$k$ ED under SF allocation scheme $s$ and
$o\in\left\\{\text{co, bo}\right\\}$ interference denoted by ${P_{{\mathop{\rm
cov}}}^{o}}\left(k,s\right)$ is computed as

$\displaystyle{P_{{\mathop{\rm cov}}}^{\rm{co}}}\left(k,s\right)=$
$\displaystyle P_{{\mathop{\rm
cov}}}^{{\rm{SNR}}}\left(k\right){\left({1+p_{k}^{s}{P_{{\rm{Act}}}}\Theta\left({{\Delta_{k,k}}}\right)}\right)^{-1}}$
(28) $\displaystyle{P_{{\mathop{\rm cov}}}^{\rm{bo}}}\left(k,s\right)=$
$\displaystyle P_{{\mathop{\rm
cov}}}^{{\rm{SNR}}}\left(k\right)\prod\nolimits_{\widetilde{k}=7}^{12}{\left({1+p_{\widetilde{k}}^{s}{P_{{\rm{Act}}}}\Theta\left({{\Delta_{k,\widetilde{k}}}}\right)}\right)^{\\!-1}}\\!.$

###  IV-B Area Spectral Efficiency

The area spectral efficiency (in bit/s/$\text{m}^{2}$) measures the network
information rate per unit area which satisfies the minimum QoS objectives, as
imposed by the reliability thresholds $\gamma_{\text{D},k}$ and
$\Delta_{k,\widetilde{k}}$. Mathematical speaking, the ASE of the considered
networks under $s\in\left\\{\text{fa, ra}\right\\}$ SF allocation scheme and
$o\in\left\\{\text{co, bo}\right\\}$ interference is given by

$\displaystyle{\rm{AS}}{{\rm{E}}_{s}^{\rm{o}}}=\sum\limits_{k=7}^{12}{p_{k}^{s}\lambda_{{\rm{ED}}}^{\left({\rm{A}}\right)}}{{\cal
R}_{k}}{P_{{\rm{Sel}}}}{P_{{\rm{cov}}}^{o}}\left({k,s}\right)=\sum\limits_{k=7}^{12}{{\rm{AS}}{{\rm{E}}_{k,s}^{o}}}$
(29)

where
$\mathcal{R}_{k}=k\left(\text{Bw}/2^{k}\right)\left(4/(4+\text{Cr})\right)$ is
the bit rate of SF$k$; $\text{Cr}\in\left\\{1,\ldots,4\right\\}$ is the coding
rate.

##  V Numerical Results

This section provides numerical results to verify the accuracy of the proposed
mathematical frameworks in Section IV. Unless otherwise stated, the following
setup is considered: $\beta=2.9$, Bw = 125 kHz, NF = 6 dBm, $f_{c}$ = 868 MHz,
$P_{\text{tot}}$ = 25 dBm, Cr = 1, $N_{\text{Ch}}=8$,
${\lambda_{{\rm{GW}}}}=2/\text{km}^{2}$, $\rho=1\%$,
${\lambda_{{\rm{ED}}}}=1000/\text{km}^{2}$, $\theta=0.01$.

Figure 1:  Coverage probability versus the total transmit power. Solid lines
are plotted by using (28). Markers are Monte-Carlo simulations.

Fig. 1 illustrates the coverage probability versus the total transmit power
$P_{\text{tot}}$. Firstly, it is evident that the proposed mathematical
frameworks are extremely tight compared with the Monte-Carlo simulations.
Secondly, turning up the transmit power is beneficial to the Pcov as it is a
monotonic increasing function. Thirdly, Fig. 1(a) reveals that under the fair-
collision allocation scheme the larger the SF, the better the Pcov owing to
the smaller $p_{k}^{\rm{fa}}$ of the larger $k$. Additionally, observing Fig.
1(b) we experience that $P_{{\mathop{\rm
cov}}}^{{\rm{bo}}}\left({11,{\rm{fa}}}\right)$ outperforms $P_{{\mathop{\rm
cov}}}^{{\rm{bo}}}\left({11,{\rm{ra}}}\right)$, although the gap is negligible
when $P_{\text{tot}}$ is fair and moderate.

Figure 2:  Area spectral efficiency versus the total transmit power. Solid
lines are plotted by using (29). Markers are Monte-Carlo simulations.

Fig. 2 sketches the ASE with respect to the $P_{\text{tot}}$. We again
experience that the developed framework is close to the exact computer-based
simulations. It is apparent that ASE goes up with the transmit power and has a
similar trend as the Pcov. The main reason is that ASE in (29) is equal to
Pcov multiplying with some factors that are independent of the
$P_{\text{tot}}$. Besides, the smaller the SF, the higher the bit rate
$\mathcal{R}$ is the main reason that increasing SF will improve the ASE.
Relying on the relation of the $p_{k}^{\rm{fa}}$ to the $p_{k}^{\rm{ra}}$, the
ASE of the fair-collision scheme maybe outperform or underperform compared
with its counterpart. Particularly, we have
$\text{ASE}_{8,\text{fa}}^{\rm{bo}}>\text{ASE}_{8,\text{ra}}^{\rm{bo}}$ due to
$p_{8}^{\rm{fa}}>p_{8}^{\rm{ra}}$ and
$\text{ASE}_{9,\text{fa}}^{\rm{bo}}<\text{ASE}_{9,\text{ra}}^{\rm{bo}}$
because of $p_{9}^{\rm{fa}}<p_{9}^{\rm{ra}}$ as shown in Fig. 2(b).

Figure 3:  $P_{\rm{Sel}}$ (a) and $P_{\rm{Act}}$ (b) versus the ratio of
$\lambda_{{\rm{ED}}}^{\left({\rm{A}}\right)}/{\lambda_{{\rm{GW}}}}$ with
various gateway channels. Solid lines are plotted by using (1) and (5).
Markers are Monte-Carlo simulations.

Fig. 3 reveals the behavior of $P_{\text{Sel}}$ and $P_{\text{Act}}$ regarding
the ratio $\lambda_{{\rm{ED}}}^{\left({\rm{A}}\right)}/{\lambda_{{\rm{GW}}}}$.
We obverse that augmenting the number of channels $N_{\rm{Ch}}$ will
ameliorate the $P_{\text{Act}}$ but decrease the $P_{\text{Sel}}$. Concerning
the $P_{\text{Sel}}$, increasing
$\lambda_{{\rm{ED}}}^{\left({\rm{A}}\right)}/{\lambda_{{\rm{GW}}}}$ means that
there are more active EDs tagged to each cell. Thus, the probability that an
ED is selected by the GW will decline. On the contrary, the more active EDs,
the higher the active probability of each channel hence, $P_{\text{Act}}$
keeps increasing with
$\lambda_{{\rm{ED}}}^{\left({\rm{A}}\right)}/{\lambda_{{\rm{GW}}}}$.
Nevertheless, due to the constraint of the duty cycle $\rho$, the upper bound
of the $P_{\text{Act}}$ will never approach one, provided that the number of
active ED goes without bound.

Figure 4:  Area spectral efficiency versus the ratio of
$\lambda_{{\rm{ED}}}^{\left({\rm{A}}\right)}/{\lambda_{{\rm{GW}}}}$. Solid
lines are plotted by using (29). Markers are Monte-Carlo simulations.

Fig. 4 illustrates the ASE of specific SFs (a) and the summed ASE of the whole
networks (b) versus the ratio
$\lambda_{{\rm{ED}}}^{\left({\rm{A}}\right)}/{\lambda_{{\rm{GW}}}}$. The
figure unveils that the ASE is monotonically increasing with respect to
$\lambda_{{\rm{ED}}}^{\left({\rm{A}}\right)}/{\lambda_{{\rm{GW}}}}$. From
Figs. 2(b) and 4(a), we observe that ASE under random assignment scheme can be
either better or worse than another scheme. Nonetheless, observing Fig. 4(b),
the summed ASE of the whole networks under fair-collision scheme is superior
to its counterpart. Additionally, the ASE suffers only from co-SF interference
is always higher than one is subjected to both co- and inter-SF interference.
Nevertheless, the gap between the two schemes is negligible.

Figure 5:  Coverage probability (a) and summed ASE (b) versus the duty cycle
$\rho$. Solid lines are plotted by using (28) and (29).

Fig. 5 investigates the behavior of Pcov (a) and the summed ASE (b) as a
function of the duty cycle $\rho$. This figure highlights the benefits of
increasing the duty cycle as both metrics ameliorate significantly with
$\rho$. Particularly, the summed ASE under fair-collision scheme improves over
10-fold when $\rho$ increases from 0.1% to 1% for $P_{\text{tot}}$ = 25 dBm
while the improvement under random assignment scheme is also higher than
6-times. Again, Fig. 5(b) confirms the superior of fair-collision scheme
compared with random assignment one. Additionally, Fig. 5(a) highlights that
the gap of the Pcov under co- and all-SF interference is either negligible or
significantly relying on the utilized SF. Specifically,
$P_{\text{cov}}^{\rm{co}}\left(7,s\right)$ is higher than
$P_{\text{cov}}^{\rm{bo}}\left(7,s\right)$ nearly 0.1 while
$P_{\text{cov}}^{\rm{co}}\left(10,s\right)$ and
$P_{\text{cov}}^{\rm{bo}}\left(10,s\right)$ is nearly indistinguishable.

##  VI Conclusion

The mathematical frameworks of the downlink multi-gateway LoRa networks were
examined in the present paper. More precisely, two important metrics namely,
the coverage probability and the area spectral efficiency were computed under
both SF fair-collision and random assignment schemes and both co- and inter-SF
interference. Our results revealed that improving the duty cycle and the
transmit power was important to ameliorate both metrics. Numerical results
were also provided to clarify the correctness of the proposed mathematical
frameworks. The paper can be extended in several directions and one of the
most important ones is to do the practical experiments for comparing with the
proposed mathematical framework.

###  -A Proof of Eq. (24)

In this section, we derive (24), let us start with the definition of the SIR
condition as follows:

$\displaystyle
P_{S{\rm{IR}}}\left({k,\widetilde{k}},s\right)=\Pr\left\\{{{\left|{{h_{0}}}\right|}^{2}}/\left(L_{0}I_{\widetilde{k},s}\right)\geq{\Delta_{k,\widetilde{k}}}\right\\}$
$\displaystyle\mathop{=}\limits^{\left(a\right)}\int\nolimits_{l=0}^{\infty}{}\int\nolimits_{i=0}^{\infty}\exp\left({-il{\Delta_{k,\widetilde{k}}}}\right){f_{{I_{\widetilde{k},s}}}}\left(i\right){f_{{L_{0}}}}\left(l\right)didl$
$\displaystyle\mathop{=}\limits^{\left(b\right)}\int\nolimits_{l=0}^{\infty}{M_{{I_{\widetilde{k},s}}}}\left({\left.{l{\Delta_{k,\widetilde{k}}}}\right|l}\right){f_{{L_{0}}}}\left(l\right)dl,$
(30)

where $\left(a\right)$ is held by utilizing the cumulative distribution
function (CDF) of the exponential function; $\left(b\right)$ is obtained via
employing the definition of the moment generating function (MGF) of
${I_{\widetilde{k},s}}$ conditioned on $L_{0}$. Looking at (-A), in order to
solve this integration, we first identify the MGF of ${I_{\widetilde{k},s}}$
conditioned on $L_{0}$ as follows:

$\displaystyle{M_{{I_{\widetilde{k},s}}}}\left({\left.s\right|l}\right)=\mathbb{E}\left\\{{\exp\left({-s{I_{\widetilde{k},s}}}\right)}\right\\}\mathop{=}\limits^{\left(a\right)}\exp\left({\delta\pi\lambda_{{\rm{GW}}}^{\left({\rm{I}}\right),\widetilde{k},s}}/{\left({{K_{0}}}\right)}^{\delta}\right.$
$\displaystyle\left.\times{\mathbb{E}_{h}}\left\\{{\int\nolimits_{x=l}^{\infty}\left({1-\exp\left(-sh/x\right)}\right){x^{\delta-1}}dx}\right\\}\right)$
$\displaystyle\mathop{=}\limits^{\left(b\right)}\exp\\!\left(\\!-\pi\lambda_{{\rm{GW}}}^{\left({\rm{I}}\right),\widetilde{k},s}{{\left(l/K_{0}\right)}^{\delta}}{\mathbb{E}_{h}}\left\\{{\left({{}_{1}{F_{1}}\left({-\delta,1-\delta,-sh/l}\right)-1}\right)}\\!\right\\}\\!\right)$
$\displaystyle\mathop{=}\limits^{\left(c\right)}\exp\left(-\pi\lambda_{{\rm{GW}}}^{\left({\rm{I}}\right),\widetilde{k},s}{{\left(l/K_{0}\right)}^{\delta}}\Theta\left(s/l\right)\right),$
(31)

where $\mathbb{E}\left\\{.\right\\}$ is the expectation operator;
$\left(a\right)$ employs the probability generating functional (PGFL) of PPP
with density ${\lambda_{{\rm{GW}}}^{\left({\rm{I}}\right),\widetilde{k},s}}$;
$\left(b\right)$ utilities following result: given $v\in(0,1)$, $a>0$ and
$b\in\mathbb{R}$, we have
$\int\nolimits_{a}^{\infty}\left({\exp\left({b/x}\right)-1}\right){x^{v-1}}dx=\left({1/v}\right){a^{v}}\left({1-{\;{}_{1}}{F_{1}}\left({-v,1-v,b/a}\right)}\right)$;
and $\left(c\right)$ is held by following: given $a,b,c\in\mathbb{R}$,
$\alpha\in\mathbb{N}$, we have
$\int_{0}^{\infty}{}{x^{\alpha-1}}\exp\left({-cx}\right){\;{}_{1}}{F_{1}}\left({a,b,-x}\right)dx={c^{-\alpha}}\Gamma\left(\alpha\right){\;{}_{2}}{F_{1}}\left({\alpha,a,b,-1/c}\right)$;
${\;{}_{1}}{F_{1}}\left(.\right)$ is the confluent hypergeometric function. By
substituting (-A) and the probability density function (PDF) of $L_{0}$, i.e.,
${f_{{L_{0}}}}\left(l\right)=\delta\pi\lambda_{{\rm{GW}}}^{\left({\rm{A}}\right)}{\left({{K_{0}}}\right)^{-\delta}}{l^{\delta-1}}\exp\left({-\pi\lambda_{{\rm{GW}}}^{\left({\rm{A}}\right)}{{\left({l/{K_{0}}}\right)}^{\delta}}}\right)$
[15], (-A) is then evaluated as

$\displaystyle
P_{S{\rm{IR}}}\left({k,\widetilde{k}},s\right)=\delta\pi\lambda_{{\rm{GW}}}^{\left({\rm{A}}\right)}{\left(K_{0}\right)^{-\delta}}\int\nolimits_{l=0}^{\infty}\exp\\!\left(\\!{-\pi\lambda_{{\rm{GW}}}^{\left({\rm{A}}\right)}{{\left(l/K_{0}\right)}^{\delta}}}\right)$
$\displaystyle\times{l^{\delta-1}}\exp\left({-\pi\lambda_{{\rm{GW}}}^{\left({\rm{I}}\right),\widetilde{k},s}{{\left(l/K_{0}\right)}^{\delta}}\Theta\left({{\Delta_{k,\widetilde{k}}}}\right)}\right)dl$
$\displaystyle\mathop{=}\limits^{\left(a\right)}\pi\lambda_{{\rm{GW}}}^{\left({\rm{A}}\right)}{\left(1/K_{0}\right)^{\delta}}\int\nolimits_{t=0}^{\infty}\exp\bigg{(}-\pi\lambda_{{\rm{GW}}}^{\left({\rm{A}}\right)}t{{\left(1/K_{0}\right)}^{\delta}}\bigg{(}1+\bigg{.}\bigg{.}$
$\displaystyle\times\left.\left.\\!p_{\widetilde{k}}^{s}{P_{{\rm{Act}}}}\Theta\left({{\Delta_{k,\widetilde{k}}}}\right)\right)\right)dt\\!=\\!{\left({1\\!+p_{\widetilde{k}}^{s}{P_{{\rm{Act}}}}\Theta\left({{\Delta_{k,\widetilde{k}}}}\\!\right)}\\!\right)^{-1}}\\!,$
(32)

where $\left(a\right)$ follows from the change of variable $t=l^{\delta}$ and
employing
$\lambda_{{\rm{GW}}}^{\left({\rm{I}}\right),\widetilde{k},s}=p_{\widetilde{k}}^{s}{P_{{\rm{Act}}}}\lambda_{{\rm{GW}}}^{\left({\rm{A}}\right)}$
QED.

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