# High-rate Reliable Communication using Multi-hop and Mesh THz/FSO Networks

Praveen Kumar Singya1, Behrooz Makki2, , Antonio D’Errico3, , and Mohamed-Slim
Alouini1 1P. K. Singya and M.-S. Alouini are with the Computer, Electrical,
and Mathematical Science and Engineering (CEMSE) Division, King Abdullah
University of Science and Technology (KAUST), Thuwal 23955-6900, Saudi Arabia
<EMAIL_ADDRESS>slim.alouini@kaust.edu.sa)2B. Makki is
with Ericsson Research, Ericsson, 41756 Göteborg, Sweden
(e-mail:behrooz.makki@ericsson.com)3D’Errico is with Ericsson Research,
Ericsson, 56124 Pisa, Italy<EMAIL_ADDRESS>
###### Abstract

In this work, we consider multi-hop and mesh hybrid teraHertz/free-space
optics (THz/FSO)-based backhaul networks for high data-rate communications.
The results are presented for the cases with both out-band integrated access
and backhaul (IAB) and non-IAB based communication setups. We consider
different deployments of the THz and FSO networks and consider both switching
and combining methods between the hybrid FSO/THz links. We study the impact of
atmospheric turbulence, atmospheric attenuation, and the pointing error on the
FSO communication. The THz communication suffers from small scale fading,
path-loss, and the misalignment error. Finally, we evaluate the effects of
atmospheric attenuation/path-loss, pointing/misalignment error, small-scale
fading, atmospheric turbulence, number of antennas, number of user equipments,
number of hops, and the threshold data-rates on the performance of considered
systems. As we show, with different network deployments and
switching/combining methods, the hybrid implementation of the THz/FSO links
improves the network reliability significantly.

###### Index Terms:

Free-space optics (FSO), teraHertz (THz) communication, integrated access and
backhaul (IAB), sub-THz, multi-hop, mesh, beyond 5G, 5G advanced, 6G.

##  I Introduction

A tremendous growth in wirelessly connected Internet devices and mobile
traffic is expected in beyond fifth generation (5G) communications. For this,
along with the large available bandwidth in millimeter wave (mmWave) and
beyond communication bands, various solutions are provided to achieve the high
data rates with large capacity and coverage. One such solution is network
densification [1] by deploying multiple access points in an area. This
increases the need for wireless backhauling, as fiber installation may not be
possible or economically viable. Wireless backhaul currently shares a large
portion of the backhaul market and is mainly based on well-planned point-to-
point communication in the range of up to 80 GHz. This may need improvements
from different aspects in 5G and beyond. For instance, alternative
technologies such as free space optics (FSO) can be combined with existing
networks to boost the reliability. Also, higher, e.g., sub-THz and THz bands
provide more bandwidth for backhauling. Finally, the so called integrated
access and backhaul (IAB) enables multi-hop backhauling. IAB provides flexible
wireless backhauling as well as the existing cellular services to the user
equipments (UEs) in the same node. Therefore, IAB complements the microwave
point-to-point backhaul deployments in dense urban and suburban regions. Both
in-band and out-band backhaul operations are supported by the IAB. In the in-
band IAB (resp. out-band IAB) operation, the backhaul and the access links
operate in the same (resp. different) frequency bands [2]. Note that, in 5G
NR, IAB has not been standardized for the sub-THz or THz bands. However, while
the scopes of 6G are still not clear, due to the demands on high rate
backhauling, in 6G the IAB implementation in high bands may be considered.

In the literature, various works study the IAB-enabled communication systems
[2, 3, 4]. Considering multiple hops, [5, 6, 7] study the resource allocation
and routing in IAB networks. Also, [8] discusses the node placement strategies
in multi-hop IAB network. In [9], the downlink sum-rate is maximized by joint
resource allocation and node placement strategies. Further, [10] investigates
the performance of mmWave IAB networks and validates their feasibility through
simulations. Then, [11] investigates the user throughput of a pico-cell based
multi-hop mmWave network which is helpful to deploy the IAB networks for
network densification. In [12, 13], the throughput of an IAB network is
analyzed by allocating the resources through dynamic time division duplexing
(TDD). To handle the interference in mobile IAB networks, [14] proposes a TDD
design. Then, [15] proposes various challenges and solutions towards the
mobile IAB design. Considering the infinite Poisson point processes, coverage
probability of IAB-based mmWave networks is characterized in [16, 17].

For the upcoming 6G communications, (sub-)THz communication is expected to
play a role in high data rate wireless backhauling. The THz bands (100 GHz to
10 THz including sub-THz bands) enables highly secured communication with
large directional gain, and provides large operational bandwidth [18, 19, 20].
However, THz communication may be limited to a short operational range (a few
hundred meters) due to large path-loss, misalignment error, and blockage in a
dense environment. This may limit the (sub-)THz operations in backhaul only,
and access communications may be difficult at THz bands.

In parallel, the demand for high capacity and data rates guide us towards FSO
communication which provides a low-cost alternative to optical fiber cables
for backhauling. The FSO enables a secured line-of-sight (LoS) communication
in a license-free optical band with colossal bandwidth and is easily deployed
[21]. However, the FSO communication is severely affected by the pointing
error, atmospheric attenuation, atmospheric turbulence, and weather conditions
like fog and smoke, which limits its operations mainly in backhaul
applications. Hence, a parallel deployment of the THz and FSO links can
provide a robust and highly reliable backhaul solution.

Considering mmWave bands for the RF link, various works study the hybrid
FSO/RF systems, where both the FSO and the RF links are used separately in
different hops or in parallel in one of the hops. For instance, [22, 23, 24,
25, 26, 27, 28, 29, 30] study various dual or multi-hop hybrid FSO/RF systems
by considering the FSO and the RF links in separate hops. On the other hand,
[31, 32, 33, 34, 35, 36, 37, 38] investigate the hybrid FSO/RF systems with
the parallel deployment of the FSO and the RF links in at-least one hop, and
either switching or simultaneous transmission is performed. Note that [22, 23,
24, 25, 26, 27, 28, 29, 30, 31, 39, 32, 33, 35, 34, 36, 37, 38] do not study
the THz channel characteristics and its performance in the considered
communication systems. Further, [40, 20, 41] investigate the performance of
dual-hop communication systems with at-lease one THz link, however, do not
consider the FSO link in any hop. Considering the FSO and the THz links in
separate hops, [42] analyzes the performance of a hybrid FSO/THz system. On
the other hand, [43] considers a dual-hop FSO/THz system, where both the FSO
and the THz links are deployed in parallel in backhaul and mmWave in access
link, and both the soft and hard switching schemes are studied.

In this work, we consider mesh and multi-hop hybrid THz/FSO networks for high-
rate reliable communications. We present the results for different models of
network deployment and for both cases with out-band IAB and non-IAB based
communication setups. We deploy both the FSO and the THz transceivers at each
node according to the network models shown in Figs. 1-2. We consider Gamma-
Gamma distribution for the FSO link’s atmospheric turbulence with Rayleigh
distributed pointing error. The $\alpha-\mu$ distribution is considered for
the THz link’s small-scale fading, along with the Rayleigh distributed
pointing error. The THz link is supported with a multi-antenna configuration
to reach the same order of rates as in the FSO links. Further, the
successfully decoded information signal at each node is forwarded to the next
node through DF relaying. At each node in the multi-hop or the mesh setup, if
both the THz and the FSO signals are available, we consider different decoding
methods based on either switching between the two signals or their
combination. Finally, multiple RF antennas are deployed at each micro base
station (MiBS)/child IAB to serve multiple UEs through the mmWave access
links, which are Nakagami-m distributed.

Figure 1: Multi-hop hybrid THz/FSO-based backhaul network with NLoS
communication between the macro BS/donor IAB node and the final micro BS/child
IAB node. Figure 2: Multi-hop hybrid THz/FSO-based backhaul network with LoS
communication between the macro BS/donor IAB node and the final micro BS/child
IAB node.

From this perspective, this study focuses on the following:

  * •

We first obtain closed-form and high signal-to-noise ratio (SNR) approximated
probability density function (PDF) and cumulative distribution function (CDF)
expressions of the FSO, the THz, and the mmWave access links’ SNR.

  * •

Based on the derived PDF and CDF expressions, we derive the outage probability
expressions for both system models of Figs. 1-2 as well as the mesh networks
with both switching and combining methods between the parallel THz and FSO
signals.

  * •

Based on the high SNR approximated PDF and CDF expressions, we derive the
asymptotic outage probability expressions for the considered system models
with switching and combining methods between the parallel THz and FSO signals.
Also, we find the diversity order for all considered setups.

  * •

For both system models, the multi-hop analysis is extended to the mesh network
by increasing the number of routes to improve the end-to-end (E2E) UE
performance.

  * •

Finally, we show the impact of atmospheric attenuation/path-loss,
pointing/misalignment error, small-scale fading, atmospheric turbulence,
number of RF antennas, number of UEs, number of hops, and the threshold data-
rates on the performance of considered systems.

From the results, we observe that the parallel deployment of the THz/FSO links
in the backhaul with proper system configuration improves the overall
performance of the communication system. Also, the diversity order for
combining and switching methods remains the same. However, depending on the
network deployment, the combining method may provide a considerable
performance gain compared to the switching method. Further, higher jitter
standard deviation drastically reduces the performance of the THz, the FSO,
and consequently the backhaul hybrid THz/FSO link. Then, the increase in
number of hops increases the coverage area, however, reduces the E2E
throughput. Also, mesh networks enable multiple routes and improve the
coverage.

Table I consists of the notations considered throughout this work. The rest of
the paper is organized as follows. Section II discusses the considered system
and channel models. Then, the performance analysis of System-1 and System-2,
shown in Figs. 1-2, are discussed in details in Section III and Section IV,
respectively. Section V extends the performance of the considered systems to
mesh networks. Then, the numerical and simulations results are shown and
discussed in Section VI, and finally, the conclusions are drawn in Section
VII.

TABLE I: Notations considered throughout the work. Notation | Definition | Notation | Definition
---|---|---|---
$\text{j}\in\\{\text{T},\text{F},\text{acc}\\}$ | The THz, FSO, or mmWave link’s selection parameter | $n$ | Number of hops
$N_{\text{t}}$ | Transmit antennas at the $n^{\text{th}}$ MiBS/child IAB | Q | Number of routes
$N_{\text{r}}$ | Antennas at the $n^{\text{th}}$ MiBS/child IAB THz receiver | $\kappa$ | FSO detector type
$P_{\text{j}_{n}}$ | Transmit power at source j of $n^{\text{th}}$ hop | $L_{\text{j}_{n}}$ | The j link length of $n^{\text{th}}$ hop
$G_{\text{t}}^{\text{j}_{n}}$ | Transmit antenna gain at node j of $n^{\text{th}}$ hop | $G_{\text{r}}^{\text{j}_{n}}$ | Receive antenna gain at node j of $n^{\text{th}}$ hop
$\alpha_{n}$,$\mu_{n}$ |  $n^{\text{th}}$ $\alpha-\mu$ distributed THz link’s fading coefficients | m | mmWave link’s fading parameter
$\alpha_{\text{F}_{n}}$,$\beta_{\text{F}_{n}}$ |  $n^{\text{th}}$ FSO link’s atmospheric turbulence parameters | $f_{\text{j}}$ | Operating frequency of the j link
$\xi_{\text{j}}$,$A_{0,\text{j}}$ | Pointing error parameters for the j link in $n^{\text{th}}$ hop | $\lambda_{\text{j}}$ | Wavelength of the j link
$\omega_{\text{j}_{n}}$,$a_{\text{j}_{n}}$,$\varepsilon_{\text{j}_{n}}$ | Beamwidth, receiver radius, and jitter standard deviation of j link | $\varrho_{\text{ox}}$ | Oxygen absorption in mmWave link
$h_{\text{l}_{n}}$,$I_{\text{l}_{n}}$,$p_{\text{l}_{n}}$ | Path-loss of the $n^{\text{th}}$ THz, FSO, and mmWave links | $\varrho_{\text{ra}}$ | Rain attenuation in mmWave link
$\gamma_{\text{j}_{n}}$ | Received SNR at node j in $n^{\text{th}}$ hop | c | Speed of light
$\gamma_{\text{th},n}$ | SNR threshold at the $n^{\text{th}}$ FSO and THz receiver | $\gamma^{\text{UE}}_{\text{th}}$ | SNR threshold at the mmWave UE

##  II System and Channel Models

We consider multi-hop hybrid THz/FSO networks to provide reliable high data-
rates to the mmWave UEs (for the extension of the results to the cases with
mesh networks, see Section V). We consider both non-IAB and out-band IAB-based
communication setups. With a non-IAB setup, the relay nodes in between do not
serve UEs and only forward backhaul data to the end relay node which is
responsible for access communication to the UEs. With out-band IAB, on the
other hand, while each relay node forwards backhaul data to following IAB
nodes, it also provides access links to its surrounding UEs. With an IAB
setup, the IAB donor is the node connecting to the core network via, for
example, fiber links. Then, the nodes following the IAB donor in a multi-hop
fashion are called IAB nodes (see Figs. 1-2). With a non-IAB setup, these
nodes are referred to as macro BS (MaBS) and micro BS (MiBS), respectively.

For both non-IAB and IAB models, we consider two different network
configurations:

  * •

System 1: The MaBS/donor IAB node and the final MiBS/child IAB node
communicate in NLoS manner. In this case, illustrated in Fig. 1, there are
hybrid THz/FSO links in each backhaul link. Such a setup is of interest when
there is no strong direct link between the MaBS/donor IAB and the final
MiBS/child IAB node.

  * •

System 2: The MaBS/donor IAB node and the final MiBS/child IAB node
communicate in LoS manner. In this case, there is a direct FSO link between
the MaBS and the final MiBS, as the FSO links can support long ranges.
However, because with large hop distances, the THz links may not support the
same order of rates as in the FSO links, we do not consider a direct THz link
between the MaBS and final MiBS. Instead, in the THz, the data is forwarded to
the final MiBS through multiple hops, as illustrated in Fig. 2.

Note that System-2 is of interest specially because, although we present the
results for the general case with different number of hops, as we explain in
the following, in practice multi-hop systems are of interest with a maximum of
two hops. In such cases, it is probable to have a strong LoS connection
between the first and last nodes, while due to the hop distance, the direct
THz connection between these two nodes is not desirable. Moreover, the
comparison between System 1 and 2 gives the chance to evaluate the benefits of
multi-hop FSO communication.

Both heterodyne detection and intensity modulation/direct detection (IM/DD)
methods are considered in the FSO links. Further, we consider DF relaying at
each node to forward the successfully decoded signal from the previous node to
the next node. For System-1 (resp. System-2) as shown in Fig. 1 (resp. Fig.
2), for each MiBS (resp. for the final MiBS), we consider different techniques
for signal reception:

  * •

Switching method: The receiver selects the best signal received through the
THz or the FSO links and decodes it.

  * •

Combination method: Based on the individual links SNR, the receiver may
combine the signals received through the THz and FSO links.

###  II-A FSO Link Model

Considering the $n^{\text{th}}$ FSO link, the transmitted signal $s(t)$
received at the MiBS/child IAB111Note that IAB is an RF technology and the
existing IAB systems as defined by 3GPP do not support FSO-based
communication. node is expressed as

$\displaystyle y_{{\text{F}_{n}}}\left(t\right)=\left({P_{\text{F}_{n}}}\eta
I_{\text{l}_{n}}I_{n}\right)^{\frac{\kappa}{2}}s(t)+v_{\text{F}_{n}}\left(t\right),$
(1)

where $P_{\text{F}_{n}}$ represents the transmit power at the $n^{\text{th}}$
FSO node and $\eta$ is the optical-to-electrical (O/E) conversion coefficient
considered to be the same for all hops. Further,
$I_{n}=I_{\text{p}_{n}}I_{\text{a}_{n}}$ represents the fading coefficient of
the $n^{\text{th}}$ FSO link, where $I_{\text{p}_{n}}$ is the pointing error
during the transmission and $I_{\text{a}_{n}}$ represents the fading due to
the atmospheric turbulence. Also, $I_{\text{l}_{n}}$ is the atmospheric
attenuation of the $n^{\text{th}}$ FSO link. Moreover, $\kappa=1$ and
$\kappa=2$ represent the heterodyne detection and IM/DD, respectively, to
write the generalized expressions for both the FSO detectors. Furthermore,
$v_{\text{F}_{n}}(t)\sim\mathcal{N}(0,\sigma_{\text{o}}^{2})$ represents the
additive white Gaussian noise (AWGN) associated with the $n^{\text{th}}$ FSO
link with zero mean and $\sigma_{\text{o}}^{2}$ variance. For simplicity, we
consider identical variance for all optical links. Hence, at the
$n^{\text{th}}$ FSO receiver, the instantaneous received SNR will be
$\gamma_{\text{F}_{n}}={\left(P_{\text{F}_{n}}\eta
I_{\text{l}_{n}}I_{n}\right)^{\kappa}}/{\sigma^{2}_{\text{o}}}.$

####  II-A1 Atmospheric Attenuation

The atmospheric attenuation of the FSO link follows the Beer-Lambert Law [44]
and is characterized as

$\displaystyle
I_{\text{l}_{n}}=\text{exp}\left(-\mathcal{C}_{\text{A}_{n}}L_{\text{F}_{n}}\right),$
(2)

where, $L_{\text{F}_{n}}$ represents the $n^{\text{th}}$ FSO link length.
Further, attenuation coefficient $\mathcal{C}_{\text{A}_{n}}$ is modeled as

$\displaystyle\mathcal{C}_{\text{A}_{n}}=\frac{3.912}{\text{Vi}_{n}}\left(\frac{\lambda_{\text{F}_{n}}}{550}\right)^{\text{q}\left(\text{Vi}_{n}\right)}.$
(3)

Here, for the $n^{\text{th}}$ FSO link, $\text{Vi}_{n}$ is the visibility with
the visibility coefficient $\text{q}\left(\text{Vi}_{n}\right)$, where

$\displaystyle\text{q}(\text{Vi}_{n})=\begin{cases}0.585\text{Vi}_{n}^{1/3},&\text{for
}\text{Vi}_{n}<6\,\text{km}\\\ 1.3,&\text{for
}6<\text{Vi}_{n}<50\,\text{km}\\\ 1.6,&\text{for
}\text{Vi}_{n}>50\,\text{km}.\end{cases}$ (4)

####  II-A2 Pointing/Misalignment Error

For the $n^{\text{th}}$-hop, we consider an $a_{\text{j}_{n}}$ aperture radius
receiver situated $L_{\text{j}_{n}}$ distance apart from the transmitter.
Hence, at the receiver, the received power from the transmitted
$\omega_{\text{j}_{n}}$ beamwidth Gaussian beam is given as [45]

$\displaystyle
I_{\text{p,j}_{n}}\left(r_{\text{d},\text{j}_{n}};L_{\text{j}_{n}}\right)\approx
A_{\text{0,j}_{n}}~{}\text{exp}\left(-{2r_{\text{d,j}_{n}}^{2}}/{\omega^{2}_{\text{eq},\text{j}_{n}}}\right),$
(5)

where $r_{\text{d,j}_{n}}$ is the radial displacement between the detector and
beam center for the $n^{\text{th}}$-hop. The
$A_{\text{0,j}_{n}}=\left[\text{erf}(v_{0,\text{j}_{n}})\right]^{2}$
represents the fraction of the collected power at $r_{\text{d,j}_{n}}=0$ for
the $n^{\text{th}}$-hop, where
$v_{\text{0,j}_{n}}=\sqrt{{a_{\text{j}_{n}}^{2}\pi}/{(2\omega_{\text{j}_{n}}^{2})}}$
and
$\text{erf}\left(x\right)=\frac{2}{\sqrt{\pi}}\int_{0}^{x}\exp\left(-z^{2}\right)\text{d}z$
is the error function. Further, the equivalent beamwidth is given as
$\omega^{2}_{\text{eq},\text{j}_{n}}=\frac{\sqrt{A_{\text{0,j}_{n}}\pi}}{2v_{\text{0,j}_{n}}~{}\text{exp}(-v_{\text{0,j}_{n}}^{2})}\omega_{\text{j}_{n}}^{2}$.
Note that for both FSO and THz links, the pointing/misalignment error modeling
is considered to be similar and subscript $\text{j}\in[\text{F,T}]$
corresponds to the parameters for the FSO or THz links, respectively. However,
the selection parameters may differ for both the THz and FSO links. Figure 3
depicts the pointing/misalignment error for the FSO/THz systems. Here, for the
$n^{\text{th}}$-hop, $d_{\text{x,j}_{n}}$ is the beam displacement in the
horizontal plane and $d_{\text{y,j}_{n}}$ is the beam displacement in the
vertical plane. Consequently, the radial displacement vector is
$r_{\text{d,j}_{n}}=\left[d_{\text{x,j}_{n}},d_{\text{y,j}_{n}}\right]^{\text{Tr}}$,
where Tr is the transpose of a vector. If
$d_{\text{x,j}_{n}}\approx\mathcal{N}\left(\varpi_{\text{x,j}_{n}},\varepsilon_{\text{x,j}_{n}}^{2}\right)$
and
$d_{\text{y,j}_{n}}\approx\mathcal{N}\left(\varpi_{\text{y,j}_{n}},\varepsilon_{\text{y,j}_{n}}^{2}\right)$
are independently distributed Gaussian random variables (RVs), then
$|r_{\text{d,j}_{n}}|=\sqrt{d_{\text{x,j}_{n}}^{2}+d_{\text{y,j}_{n}}^{2}}$
will be Beckmann distributed. Hence, different pointing error modeling are
possible for different boresight and jitter values [46].

Figure 3: The misalignment/pointing error in the THz/FSO links.

Here, we consider that there is no boresight error, and jitters in horizontal
and vertical directions are identical. This corresponds to
$\varpi_{\text{x,j}_{n}}$ = $\varpi_{\text{y,j}_{n}}=0$ and
$\varepsilon_{\text{x,j}_{n}}^{2}=\varepsilon_{\text{y,j}_{n}}^{2}=\varepsilon_{\text{j}_{n}}^{2}$
for the $n^{\text{th}}$-hop. Thus, the radial displacement is Rayleigh
distributed and the PDF of pointing error is

$\displaystyle
f_{\text{I}_{\text{p,j}_{n}}}(x)=\frac{\xi_{\text{j}_{n}}^{2}}{A_{0,\text{j}_{n}}^{\xi_{\text{j}_{n}}^{2}}}x^{\xi_{\text{j}_{n}}^{2}-1},~{}~{}~{}~{}\text{for
}0\leq x\leq A_{0,\text{j}_{n}}$ (6)

where
$\xi_{\text{j}_{n}}={\omega_{\text{eq},\text{j}_{n}}}/{(2\varepsilon_{\text{j}_{n}})}$
is a parameter related to the severity of pointing error.

####  II-A3 Exact PDF and CDF Expressions of FSO Link

We consider Gamma-Gamma distribution to model the atmospheric turbulence of
the $n^{\text{th}}$ FSO link $I_{\text{a}_{n}}$, whose PDF is defined as

$\displaystyle f_{I_{\text{a}_{n}}}(x)$
$\displaystyle=\frac{2\left(\alpha_{\text{F}_{n}}\beta_{\text{F}_{n}}\right)^{\frac{\alpha_{\text{F}_{n}}+\beta_{\text{F}_{n}}}{2}}}{\Gamma(\alpha_{\text{F}_{n}})\Gamma(\beta_{\text{F}_{n}})}x^{\frac{\alpha_{\text{F}_{n}}+\beta_{\text{F}_{n}}}{2}-1}$
$\displaystyle\times\text{K}_{\alpha_{\text{F}_{n}}-\beta_{\text{F}_{n}}}\left(2\sqrt{\alpha_{\text{F}_{n}}\beta_{\text{F}_{n}}x}\right).$
(7)

Here, $\Gamma(x)=\int_{0}^{\infty}z^{x-1}\exp\left(-z\right)\text{d}z$
represents complete Gamma function [47, (6.1.1)] and
$\text{K}_{l}(x)=\frac{\Gamma\left(l+\frac{1}{2}\right)}{\sqrt{\pi}}\left(2x\right)^{l}\int_{0}^{\infty}\frac{\text{cos}(z)}{\left(x^{2}+z^{2}\right)^{l+\frac{1}{2}}}\text{d}z$
is the modified Bessel function of second kind with ${l}^{\text{th}}$ order
[47, (9.6.25)]. The atmospheric turbulence’s fading coefficients for plane
wave propagation are given as

$\displaystyle\alpha_{\text{F}_{n}}=\left[\text{exp}\left(\frac{0.49\sigma_{\text{RV}_{n}}^{2}}{\left(1+1.11\sigma_{\text{RV}_{n}}^{12/5}\right)^{7/6}}\right)-1\right]^{-1},$
$\displaystyle\beta_{\text{F}_{n}}=\left[\text{exp}\left(\frac{0.51\sigma_{\text{RV}_{n}}^{2}}{\left(1+0.69\sigma_{\text{RV}_{n}}^{12/5}\right)^{5/6}}\right)-1\right]^{-1},$
(8)

where the Rytov variance
$\sigma_{\text{RV}_{n}}^{2}=1.23C_{{n}}^{2}k_{{0}_{n}}L_{\text{F}_{n}}^{11/6}$
represents the turbulence strength metric of the $n^{\text{th}}$ FSO link.
Further, $k_{\text{0}_{n}}=2\pi/\lambda_{\text{F}_{n}}$ is the wave number and
refractive index structure parameter $C_{{n}}^{2}$ corresponds to the strength
of the atmospheric turbulence.

The fading coefficient $I_{n}=I_{\text{p}_{n}}I_{\text{a}_{n}}$ of the
$n^{\text{th}}$ FSO link has the combined impact of the pointing error and the
atmospheric turbulence. Hence, considering zero boresight error with identical
jitters for the pointing error and Gamma-Gamma PDF for the atmospheric
turbulence, for both detection techniques, the generalized PDF of the
$n^{\text{th}}$ FSO link’s SNR is given by

$\displaystyle f_{\gamma_{\text{F}_{n}}}(\gamma_{\text{F}_{n}})$
$\displaystyle=\frac{\xi^{2}_{\text{F}_{n}}}{\kappa\Gamma(\alpha_{\text{F}_{n}})\Gamma(\beta_{\text{F}_{n}})\gamma_{\text{F}_{n}}}$
$\displaystyle\times\text{G}^{3,0}_{1,3}\left[\frac{\alpha_{\text{F}_{n}}\beta_{\text{F}_{n}}}{A_{0{\text{F}_{n}}}}\left(\frac{\gamma_{\text{F}_{n}}}{\delta_{{\kappa}_{n}}}\right)^{\frac{1}{\kappa}}\Big{|}^{\xi^{2}_{\text{F}_{n}}+1}_{\xi^{2}_{\text{F}_{n}},\alpha_{\text{F}_{n}},\beta_{\text{F}_{n}}}\right],$
(9)

where
$\text{G}^{\hat{m},\hat{n}}_{\hat{p},\hat{q}}\left[z\big{|}^{a_{1},...,a_{\hat{n}},a_{\hat{n}+1},...,a_{\hat{p}}}_{b_{1},...,b_{\hat{m}},b_{\hat{m}+1},...,b_{\hat{q}}}\right]$
represents the Meijer-G function. Applying
$F_{\gamma_{\text{F}_{n}}}(x)=\int_{0}^{x}f_{\gamma_{\text{F}_{n}}}(z)\text{d}z$
with [48, (07.34.21.0084.01)], the CDF of the $n^{\text{th}}$ FSO link’s SNR
is given by

$\displaystyle F_{\gamma_{\text{F}_{n}}}(\gamma_{\text{F}_{n}})$
$\displaystyle=\mathbb{D}^{n}_{1}\text{G}^{3\kappa,1}_{\kappa+1,3\kappa+1}\left[\frac{\mathbb{D}^{n}_{2}\gamma_{\text{F}_{n}}}{\left(A_{0{\text{F}}_{n}}\right)^{{\kappa}}\delta_{\kappa_{n}}}\Big{|}^{1,\Psi^{n}_{1}}_{\Psi^{n}_{2},0}\right],$
(10)

where

$\displaystyle\mathbb{D}^{n}_{1}$
$\displaystyle=\frac{\kappa^{\left(\alpha_{\text{F}_{n}}+\beta_{\text{F}_{n}}-2\right)}\xi_{\text{F}_{n}}^{2}}{\left(2\pi\right)^{\kappa-1}\Gamma\left(\alpha_{\text{F}_{n}}\right)\Gamma\left(\beta_{\text{F}_{n}}\right)},\mathbb{D}^{n}_{2}=\frac{\left(\alpha_{\text{F}_{n}}\beta_{\text{F}_{n}}\right)^{\kappa}}{\kappa^{2\kappa}},$
$\displaystyle\Psi^{n}_{1}$
$\displaystyle=\left[\frac{\xi_{\text{F}_{n}}^{2}+1}{\kappa},...,\frac{\xi_{\text{F}_{n}}^{2}+\kappa}{\kappa}\right],\Psi^{n}_{2}=\Big{[}\frac{\xi_{\text{F}_{n}}^{2}}{\kappa},...,\frac{\xi_{\text{F}_{n}}^{2}+\kappa-1}{\kappa},$
$\displaystyle\frac{\alpha_{\text{F}_{n}}}{\kappa},...,\frac{\alpha_{\text{F}_{n}}+\kappa-1}{\kappa},\frac{\beta_{\text{F}_{n}}}{\kappa},...,\frac{\beta_{\text{F}_{n}}+\kappa-1}{\kappa}\Big{]},$
(11)

####  II-A4 High SNR PDF and CDF Expressions of the FSO Link

At high SNRs, the Meijer-G function can be approximated as [48,
(07.34.06.0001.01)]. Hence, the PDF and CDF of the $n^{\text{th}}$ FSO link
are approximated as

$\displaystyle
f^{A}_{\gamma_{\text{F}_{n}}}(\gamma_{\text{F}_{n}})\approx\frac{\xi^{2}_{\text{F}_{n}}}{\kappa\Gamma(\alpha_{\text{F}_{n}})\Gamma(\beta_{\text{F}_{n}})\gamma_{\text{F}_{n}}}$
$\displaystyle\times\sum_{p=1}^{3}\left(\frac{\alpha_{\text{F}_{n}}\beta_{\text{F}_{n}}}{A_{0{\text{F}_{n}}}}\left(\frac{\gamma_{\text{F}_{n}}}{\delta_{{\kappa}_{n}}}\right)^{\frac{1}{\kappa}}\right)^{\Psi^{n}_{4,p}}\frac{\overset{3}{\underset{\underset{q\neq
p}{q=1}}{\prod}}\Gamma(\Psi^{n}_{4,q}-\Psi^{n}_{4,p})}{\overset{1}{\underset{q=1}{\prod}}\Gamma(\Psi^{n}_{3,q}-\Psi^{n}_{4,p})},$
(12)

$\displaystyle
F^{\text{A}}_{\gamma_{\text{F}_{n}}}(\gamma_{\text{F}_{n}})\approx\mathbb{D}^{n}_{1}\sum_{p=1}^{3\kappa}\left(\frac{\mathbb{D}^{n}_{2}\gamma_{\text{F}_{n}}}{(A_{{0\text{F}}_{n}})^{\kappa}\delta_{\kappa_{n}}}\right)^{\Psi^{n}_{6,p}}$
$\displaystyle\times\frac{\overset{3\kappa}{\underset{\underset{q\neq
p}{q=1}}{\prod}}\Gamma(\Psi^{n}_{6,q}-\Psi^{n}_{6,p})\overset{1}{\underset{q=1}{\prod}}\Gamma(1-\Psi^{n}_{5,q}+\Psi^{n}_{6,p})}{\overset{\kappa+1}{\underset{q=2}{\prod}}\Gamma(\Psi^{n}_{5,q}-\Psi^{n}_{6,p})\overset{3\kappa+1}{\underset{q=3\kappa+1}{\prod}}\Gamma(1-\Psi^{n}_{6,q}+\Psi^{n}_{6,p})},$
(13)

where, $\Psi^{n}_{3}=[\xi_{\text{F}_{n}}^{2}+1]$,
$\Psi^{n}_{4}=[\xi_{\text{F}_{n}}^{2},\alpha_{\text{F}_{n}},\beta_{\text{F}_{n}}]$,
$\Psi^{n}_{5}=[1,\Psi^{n}_{1}]$, and $\Psi^{n}_{6}=[\Psi^{n}_{2},0]$ for the
$n^{\text{th}}$ FSO link. Further, $\Psi^{n}_{3,p}$/$\Psi^{n}_{3,q}$,
$\Psi^{n}_{4,p}$/$\Psi^{n}_{4,q}$, $\Psi^{n}_{5,p}$/$\Psi^{n}_{5,q}$, and
$\Psi^{n}_{6,p}$/$\Psi^{n}_{6,q}$ represent the
$p^{\text{th}}$/$q^{\text{th}}$ entry of $\Psi^{n}_{3}$, $\Psi^{n}_{4}$,
$\Psi^{n}_{5}$, and $\Psi^{n}_{6}$, respectively. From (13), the high SNR CDF
can be represented as
$\approx(G_{\text{c}}\times\delta_{\kappa_{n}})^{-G_{\text{d}}}$ with
$G_{\text{d}}$ and $G_{\text{c}}$ being the diversity and coding gains,
respectively. Here, $G_{\text{d}}$ can be decided by
$\Psi^{n}_{6}=[\Psi^{n}_{2},0]=\left[\frac{\xi_{\text{F}_{n}}^{2}}{\kappa},...,\frac{\xi_{\text{F}_{n}}^{2}+\kappa-1}{\kappa},\frac{\alpha_{\text{F}_{n}}}{\kappa},...,\frac{\alpha_{\text{F}_{n}}+\kappa-1}{\kappa},\frac{\beta_{\text{F}_{n}}}{\kappa},...,\frac{\beta_{\text{F}_{n}}+\kappa-1}{\kappa},0\right]$.
Further, $F^{\text{A}}_{\gamma_{\text{F}_{n}}}(\gamma_{\text{F}_{n}})$ has
$3\kappa$ summation terms, hence, the diversity order of the $n^{\text{th}}$
FSO link is
$\min\left(\frac{\xi_{\text{F}_{n}}^{2}}{\kappa},\frac{\alpha_{\text{F}_{n}}}{\kappa},\frac{\beta_{\text{F}_{n}}}{\kappa}\right).$

###  II-B THz Link

To compensate for the large path-loss at the sub-THz bands, we deploy
$N_{\text{r}}$ antennas at the THz receiver with maximum ratio combining
(MRC). Considering the $n^{\text{th}}$ THz link, the transmitted signal $s(t)$
received at the MiBS/child IAB node is given by

$\displaystyle
y_{\text{T}_{n}}(t)=\sqrt{P_{\text{T}_{n}}}h_{\text{l}_{n}}\sum_{i=1}^{N_{\text{r}}}h^{i}_{{\text{T}}_{{n}}}~{}s(t)+v_{\text{T}_{n}}(t),$
(14)

where $P_{\text{T}_{n}}$ is the transmitted power from the $n^{\text{th}}$ THz
source node with $h_{\text{l}_{n}}$ being the path-loss. Further, for the
$n^{\text{th}}$-hop,
$h^{i}_{\text{T}_{n}}=h^{i}_{\text{f}_{n}}h^{i}_{\text{p}_{n}}$ is the fading
coefficient of the THz link to the $i^{\text{th}}$ receiving antenna that has
the combined impact of channel fading $h^{i}_{\text{f}_{n}}$ and antenna
misalignment $h^{i}_{\text{p}_{n}}$. The THz transmitter and receiver are not
very far and the receiving antennas are deployed close to each other. Hence,
the same pointing error is considered at each receiving antenna. Therefore,
the combined channel gain can be written as
$||h_{\text{T}_{n}}||^{2}=h_{\text{p}_{n}}^{2}\sum_{i=1}^{N_{\text{r}}}|h^{i}_{\text{f}_{n}}|^{2}$.
Further, $v_{\text{T}_{n}}(t)$ is the AWGN associated with the $n^{\text{th}}$
THz link having 0 mean and $\sigma_{\text{T}}^{2}$ variance. Note that, for
simplicity, the identical AWGN variance $\sigma_{\text{T}}^{2}$ is considered
in each hop. Performing MRC at the $n^{\text{th}}$ THz receiver, the
instantaneous received SNR will be
$\gamma_{\text{T}_{n}}=\sum_{i=1}^{N_{\text{r}}}\gamma^{i}_{\text{T}_{n}}={P_{\text{T}_{n}}h_{\text{l}_{n}}^{2}||h_{\text{T}_{n}}||^{2}}/{\sigma_{\text{T}}^{2}}.$

####  II-B1 Path-loss

The path-loss $h_{\text{l}_{n}}$ of the $n^{\text{th}}$ THz link is expressed
as [41, 49]

$\displaystyle
h_{\text{l}_{n}}=\frac{c\sqrt{G^{\text{T}_{n}}_{\text{t}}G^{\text{T}_{n}}_{\text{r}}}}{4\pi
f_{\text{T}_{n}}d_{\text{T}_{n}}}$
$\displaystyle\times\exp\left(-\frac{1}{2}d_{\text{T}_{n}}\left(\sum_{\hat{i}}\mathbb{F}_{\hat{i}}(f_{\text{T}_{n}},\nu)+\mathbb{G}(f_{\text{T}_{n}},\nu)\right)\right),$
(15)

where $f_{\text{T}_{n}}$, $d_{\text{T}_{n}}$, $G^{\text{T}_{n}}_{\text{t}}$,
and $G^{\text{T}_{n}}_{\text{r}}$ are the operational frequency, link length,
transmit antenna gain, and receive antenna gain of the $n^{\text{th}}$ THz
link, respectively. Here, $\mathbb{G}_{\hat{i}}(f_{\text{T}},\nu)$ is a
fitting polynomial to match (II-B1) with the actual response. Further, at 119,
183, 325, 380, 439, and 448 GHz frequencies, the six major absorption lines
are shown by the set of polynomials
$\mathbb{F}_{\hat{i}}(f_{\text{T}_{n}},\nu)$. These parameters are described
in [50, Section II].

####  II-B2 Exact PDF and CDF expressions of the THz Link

We consider the experimentally validated $\alpha-\mu$ fading to characterize
each of the THz link’s gain [51]. Performing MRC at the THz receiver, the sum
of $N_{\text{r}}$ RVs with $\alpha-\mu$ distributions is also approximated
with a single $\alpha-\mu$ RV. Further, the same statistical modeling can be
considered for the misalignment error associated with the THz link as shown in
Fig. 3. Hence, we consider the Rayleigh distributed misalignment error for the
THz link whose PDF is given in (6).

Considering the combined impact of $\alpha-\mu$ distributed channel fading
with Rayleigh distributed misalignment error, the PDF of the SNR of the
$n^{\text{th}}$ THz link is obtained as

$\displaystyle f_{\gamma_{\text{T}_{n}}}(\gamma_{\text{T}_{n}})$
$\displaystyle=\frac{\mathbb{C}^{n}_{1}}{2(\hat{\gamma}_{\text{T}_{n}})^{\frac{\xi_{\text{T}_{n}}^{2}}{2}}}\gamma_{\text{T}_{n}}^{\frac{\xi_{\text{T}_{n}}^{2}}{2}-1}\Gamma\left(\mathbb{C}^{n}_{2},\mathbb{C}^{n}_{3}\left(\frac{\gamma_{\text{T}_{n}}}{\hat{\gamma}_{\text{T}_{n}}}\right)^{\frac{\alpha_{n}}{2}}\right),$
(16)

where
$\Gamma\left(a,x\right)=\int_{x}^{\infty}z^{a-1}\exp\left(-z\right)\text{d}z$
is the upper incomplete Gamma function. Here,
$\mathbb{C}^{n}_{1}=\frac{\xi^{2}_{\text{T}_{n}}}{\left(A_{0,\text{T}_{n}}\right)^{\xi^{2}_{\text{T}_{n}}}}\frac{(N_{\text{r}}\mu_{n})^{\xi^{2}_{\text{T}_{n}}/\alpha_{n}}}{\Omega_{n}^{\xi^{2}_{\text{T}_{n}}}\Gamma(N_{\text{r}}\mu_{n})}$,
$\mathbb{C}^{n}_{2}=\frac{\alpha_{n}N_{\text{r}}\mu_{n}-\xi^{2}_{\text{T}_{n}}}{\alpha_{n}}$,
$\mathbb{C}^{n}_{3}=\frac{N_{\text{r}}\mu_{n}}{\left(A_{{0,\text{T}}_{n}}\right)^{\alpha_{n}}\Omega_{n}^{\alpha_{n}}}$,
$\hat{\gamma}_{\text{T}_{n}}=N_{\text{r}}{\bar{\gamma}}_{\text{T}_{n}}$, and
$\Omega_{n}$ is the $\alpha_{n}$-root mean value of the fading channel of the
$n^{\text{th}}$-hop RVs with $\alpha-\mu$ distributions. Applying
$F_{\gamma_{\text{F}_{n}}}(x)=\int_{0}^{x}f_{\gamma_{\text{F}_{n}}}(z)\text{d}z$,
CDF of the $n^{\text{th}}$ THz link is derived as

$\displaystyle F_{\gamma_{\text{T}_{n}}}(\gamma_{\text{T}_{n}})$
$\displaystyle=\frac{\mathbb{C}^{n}_{1}}{\xi_{\text{T}_{n}}^{2}}\Bigg{[}\left(\frac{\gamma_{\text{T}_{n}}}{{\hat{\gamma}}_{\text{T}_{n}}}\right)^{\frac{\xi_{\text{T}_{n}}^{2}}{2}}\Gamma\left(\mathbb{C}^{n}_{2},\mathbb{C}^{n}_{3}\left(\frac{\gamma_{\text{T}_{n}}}{{\hat{\gamma}}_{\text{T}_{n}}}\right)^{\frac{\alpha_{n}}{2}}\right)$
$\displaystyle+{\left(\mathbb{C}^{n}_{3}\right)}^{-\frac{\xi_{\text{T}_{n}}^{2}}{\alpha_{n}}}\Upsilon\left(N_{\text{r}}\mu_{n},\mathbb{C}^{n}_{3}\left(\frac{\gamma_{\text{T}_{n}}}{{\hat{\gamma}}_{\text{T}_{n}}}\right)^{\frac{\alpha_{n}}{2}}\right)\Bigg{]},$
(17)

where
$\Upsilon\left(a,x\right)=\int_{0}^{x}z^{a-1}\exp\left(-z\right)\text{d}z$ is
the lower incomplete gamma function.

In Meijer-G form, the CDF of the $n^{\text{th}}$ THz link’s SNR can also be
represented as

$\displaystyle F_{\gamma_{\text{T}_{n}}}(\gamma_{\text{T}_{n}})$
$\displaystyle=\frac{\mathbb{C}^{n}_{1}}{\alpha_{n}({\hat{\gamma}}_{\text{T}_{n}})^{\frac{\xi_{\text{T}_{n}}^{2}}{2}}}\gamma_{\text{T}_{n}}^{\frac{\xi_{\text{T}_{n}}^{2}}{2}}$
$\displaystyle\times\text{G}^{2,1}_{2,3}\left[\mathbb{C}^{n}_{3}\left(\frac{\gamma_{\text{T}_{n}}}{{\hat{\gamma}}_{\text{T}_{n}}}\right)^{\frac{\alpha_{n}}{2}}\Big{|}^{1-\frac{\xi_{\text{T}_{n}}^{2}}{\alpha_{n}},1}_{0,\mathbb{C}^{n}_{2},-\frac{\xi_{\text{T}_{n}}^{2}}{\alpha_{n}}}\right].$
(18)

####  II-B3 High SNR PDF and CDF expressions of the THz Link

Using $\Gamma(m,x)\underset{\overset{x\rightarrow
0}{}}{\approx}\left(\Gamma(m)-\frac{x^{m}}{m}\right)$ at high SNR, the
$n^{\text{th}}$ THz link’s PDF (16) can be approximated as

$\displaystyle f^{\text{A}}_{\gamma_{\text{T}_{n}}}(\gamma_{\text{T}_{n}})$
$\displaystyle\approx\frac{\mathbb{C}^{n}_{1}}{2({\hat{\gamma}}_{\text{T}_{n}})^{\frac{\xi_{\text{T}_{n}}^{2}}{2}}}\gamma_{\text{T}_{n}}^{\frac{\xi_{\text{T}_{n}}^{2}}{2}-1}$
$\displaystyle\times\left(\Gamma(\mathbb{C}^{n}_{2})-\frac{{\left(\mathbb{C}^{n}_{3}\right)}^{\mathbb{C}^{n}_{2}}}{\mathbb{C}^{n}_{2}}\left(\frac{\gamma_{\text{T}}}{{\hat{\gamma}}_{\text{T}_{n}}}\right)^{\frac{\mathbb{C}^{n}_{2}\alpha_{n}}{2}}\right).$
(19)

Using
$F^{\text{A}}_{\gamma_{\text{T}_{n}}}(\gamma_{\text{T}_{n}})=\int_{0}^{\gamma_{\text{T}_{n}}}f^{\text{A}}_{\gamma_{\text{T}_{n}}}(z)\text{d}z$,
at high SNRs, the $n^{\text{th}}$ THz link’s CDF is approximated as

$\displaystyle F^{\text{A}}_{\gamma_{\text{T}_{n}}}(\gamma_{\text{T}_{n}})$
$\displaystyle\approx\frac{\mathbb{C}^{n}_{1}\Gamma(\mathbb{C}^{n}_{2})}{\xi_{\text{T}_{n}}^{2}}\left(\frac{\gamma_{\text{T}_{n}}}{{\hat{\gamma}}_{\text{T}_{n}}}\right)^{\frac{\xi_{\text{T}_{n}}^{2}}{2}}$
$\displaystyle-\frac{\mathbb{C}^{n}_{1}{\left(\mathbb{C}^{n}_{3}\right)}^{\mathbb{C}^{n}_{2}}}{\mathbb{C}^{n}_{2}\left(\alpha_{n}N_{\text{r}}\mu_{n}\right)}\left(\frac{\gamma_{\text{T}_{n}}}{{\hat{\gamma}}_{\text{T}_{n}}}\right)^{\frac{\alpha_{n}N_{\text{r}}\mu_{n}}{2}}.$
(20)

From (II-B3), the high SNR CDF can be represented as
$\approx(G_{\text{c}}\times\hat{\gamma}_{\text{T}_{n}})^{-G_{\text{d}}}$.
Hence, the diversity order of the $n^{\text{th}}$ THz link is
$\min\left(\frac{\xi_{\text{T}_{n}}^{2}}{2},\frac{\alpha_{n}N_{\text{r}}\mu_{n}}{2}\right)$.

###  II-C mmWave Access Links

We deploy multiple RF antennas ($N_{\text{t}}$) at each MiBS/child IAB node
for mmWave transmission to the UEs, which has single antenna due to the small
size. Then, with $\hat{s}(t)$ being the successfully decoded signal at the
$n^{\text{th}}$ MiBS/child IAB node, the $l^{\text{th}}$ UE receives the
signal after maximal ratio transmission (MRT) as

$\displaystyle
y_{\text{I}_{n}\text{U}_{l}}\left(t\right)=\sqrt{P_{\text{acc}}p_{\text{l}}}\sum_{k=1}^{N_{\text{t}}}h_{{\text{k}_{n}\text{U}_{l}}}\hat{s}(t)+v_{\text{U}_{l}}(t),$
(21)

where $p_{\text{l}}$ represents the gain of the path-loss in mmWave access
link and $P_{\text{acc}}$ is the transmit power at the MiBS/child IAB node.
Further, $h_{{\text{k}_{n}\text{U}_{l}}}$ is the fading coefficient of the
$k^{\text{th}}$ antenna in $n^{\text{th}}$ hop to the $l^{\text{th}}$ UE and
$v_{\text{U}_{l}}(t)$ is the zero mean and $\sigma_{\text{acc}}^{2}$ variance
AWGN of the $l^{\text{th}}$ UE’s receiver. We consider Gamma distributed
mmWave access link’s gain, i.e., $\mathcal{G}(m,\Omega_{\text{acc}})$, where
$m$ is the fading severity and $\Omega_{\text{acc}}$ is the average power. We
consider that all UEs are independent and identically distributed, hence, the
same fading severity $m$ and average power is considered for each UE. At the
$l^{\text{th}}$ UE, a single Gamma RV
$\approx\mathcal{G}(mN_{\text{t}},\Omega_{\text{acc}})$ can approximate the
sum of $N_{\text{t}}$ Gamma RVs. Hence, the received SNR at $l^{\text{th}}$ UE
is
$\gamma_{\text{acc}}={P_{\text{acc}}p_{\text{l}}||h_{\text{acc}}||^{2}}/{\sigma_{\text{acc}}^{2}}$
with
$||h_{\text{acc}}||^{2}=\sum_{k=1}^{N_{\text{t}}}|h_{{\text{k}_{n}\text{U}_{l}}}|^{2}$
with the following PDF and CDF

$\displaystyle f_{\gamma_{\text{I}_{n}\text{U}_{l}}}(\gamma_{\text{acc}})$
$\displaystyle=\frac{1}{\Gamma\left(mN_{\text{t}}\right)}\left(\frac{m}{\bar{\gamma}_{\text{acc}}}\right)^{mN_{\text{t}}}\gamma_{\text{acc}}^{mN_{\text{t}}-1}\exp{\left(-\frac{m}{\bar{\gamma}_{\text{acc}}}\gamma_{\text{acc}}\right)},$
(22)
$\displaystyle~{}~{}~{}F_{\gamma_{\text{I}_{n}\text{U}_{l}}}(\gamma_{\text{acc}})$
$\displaystyle=1-\frac{1}{\Gamma\left(mN_{\text{t}}\right)}\Gamma\left(mN_{\text{t}},\frac{m}{\bar{\gamma}_{\text{acc}}}\gamma_{\text{acc}}\right),$
(23)

respectively, where
$\bar{\gamma}_{\text{acc}}={\Omega_{\text{acc}}P_{\text{acc}}p_{\text{l}}}/{\sigma_{\text{acc}}^{2}}$
is the average received SNR.

####  II-C1 Path-loss

The gain of the path-loss $p_{\text{l}}$ for the mmWave access link is given
as [52]

$\displaystyle
p_{\text{l}}[\text{dB}]=G^{\text{acc}}_{\text{t}}+G^{\text{acc}}_{\text{r}}-20\log_{10}\left(\frac{4\pi
L_{\text{acc}}}{\lambda_{\text{acc}}}\right)-\left(\varrho_{\text{ox}}+\varrho_{\text{ra}}\right)L_{\text{acc}},$
(24)

where $G^{\text{acc}}_{\text{t}}$, $G^{\text{acc}}_{\text{r}}$,
$\lambda_{\text{acc}}$, and $L_{\text{acc}}$ are the transmit antenna gain,
receiver antenna gain, wavelength, and the link distance of the mmWave access
link, respectively. Also, $\varrho_{\text{ra}}$ and $\varrho_{\text{ox}}$
represent respectively the rain attenuation and the oxygen absorption.

##  III Performance Analysis of System-1

For System-1 (as shown in Fig. 1), we consider the MaBS/donor IAB and the
final MiBS/child IAB are in NLoS. Hence, we deploy both the THz and the FSO
links in parallel in each hop and the overall backhaul communication is
completed through the serial multi-hop hybrid THz/FSO links with DF relaying.
Further, we consider both switching and combining methods at the receiver, as
explained in the following. We start the analysis for the multi-hop networks.
Then, as explained in Section V, the results are extended to the cases with
mesh networks.

###  III-A Combining Method

####  III-A1 Outage Probability

For the considered setup, the THz link is given higher priority and the FSO
link is deployed to provide the backup to the THz link, if required. Hence,
considering the $n^{\text{th}}$ hybrid THz/FSO backhaul link with combining
method, the THz link works alone if its SNR is greater than or equal to an SNR
threshold, i.e., $\gamma_{\text{T}_{n}}\geq\gamma_{\text{th},n}$, where
$\gamma_{\text{th},n}$ is the threshold. However, with
$\gamma_{\text{T}_{n}}<\gamma_{\text{th},n}$, receiver generates a feedback to
activate the FSO link for simultaneous transmission along with the THz link if
$\left(\gamma_{\text{T}_{n}}+\gamma_{\text{F}_{n}}\right)\geq\gamma_{\text{th},n}$.
Therefore, for the $n^{\text{th}}$ backhaul hop, the instantaneous received
SNR $\left(\gamma^{\text{CO}}_{\text{I}_{n}}\right)$ of the hybrid THz/FSO
link is given as

$\displaystyle\gamma^{\text{CO}}_{\text{I}_{n}}=\begin{cases}\gamma_{\text{T}_{n}},&\text{for
}\gamma_{\text{T}_{n}}\geq\gamma_{\text{th},n}\\\
\left(\gamma_{\text{T}_{n}}+\gamma_{\text{F}_{n}}\right),&\text{for
}\gamma_{\text{T}_{n}}<\gamma_{\text{th},n}~{}\&~{}\left(\gamma_{\text{T}_{n}}+\gamma_{\text{F}_{n}}\right)\geq\gamma_{\text{th},n}\\\
0,&\text{for
}\left(\gamma_{\text{T}_{n}}+\gamma_{\text{F}_{n}}\right)<\gamma_{\text{th},n}.\end{cases}$
(25)

With combining, outage probability of the $n^{\text{th}}$ hybrid THz/FSO
backhaul link is calculated as

$\displaystyle\mathcal{P}^{\text{CO}}_{\gamma_{\text{I}_{n}}}(\gamma_{\text{th},n})$
$\displaystyle=F^{\text{CO}}_{\gamma_{\text{I}_{n}}}(\gamma_{\text{th},n})=\mathcal{P}\left[\left(\gamma_{\text{T}_{n}}+\gamma_{\text{F}_{n}}\right)<\gamma_{\text{th},n}\right].$
(26)

Then, (26) can be derived as

$\displaystyle\mathcal{P}^{\text{CO}}_{\gamma_{\text{I}_{n}}}(\gamma_{\text{th},n})$
$\displaystyle=\mathbb{I}_{1}(\gamma_{\text{th},n})-\mathbb{I}_{2}(\gamma_{\text{th},n})+\mathbb{I}_{3}(\gamma_{\text{th},n}).$
(27)

Here,

$\displaystyle\mathbb{I}_{1}(\gamma_{\text{th},n})$
$\displaystyle=\mathbb{D}^{n}_{1}\frac{\mathbb{C}^{n}_{1}\Gamma\left(\mathbb{C}^{n}_{2}\right)}{\xi_{\text{T}_{n}}^{2}{{\bar{\gamma}}_{\text{T}_{n}}}^{\frac{\xi^{2}_{\text{T}_{n}}}{2}}}\Gamma(\mathbb{C}^{n}_{4})\gamma_{\text{th},n}^{\mathbb{C}^{n}_{4}-1}$
$\displaystyle\times\text{G}^{3\kappa,1}_{\kappa+1,3\kappa+1}\left[\frac{\mathbb{D}^{n}_{2}\gamma_{\text{F}_{n}}}{\left(A_{0{\text{F}_{n}}}\right)^{{\kappa}}\delta_{\kappa_{n}}}\Big{|}^{1,\Psi^{n}_{1}}_{\Psi^{n}_{2},1-\mathbb{C}^{n}_{4}}\right],$
$\displaystyle\mathbb{I}_{2}(\gamma_{\text{th},n})$
$\displaystyle=\mathbb{D}^{n}_{1}\frac{\mathbb{C}^{n}_{1}}{\xi_{\text{T}_{n}}^{2}{{\bar{\gamma}}_{\text{T}_{n}}}^{\frac{\xi^{2}_{\text{T}_{n}}}{2}}}\sum_{z_{1}=0}^{\infty}\frac{(-1)^{z_{1}}}{z_{1}!\left(\mathbb{C}^{n}_{2}+z_{1}\right)}\left(\frac{\mathbb{C}^{n}_{3}}{{{\bar{\gamma}}_{\text{T}_{n}}}^{\frac{\alpha_{n}}{2}}}\right)^{\mathbb{C}^{n}_{2}+z_{1}}$
$\displaystyle\times\Gamma(\mathbb{C}^{n}_{5})\gamma_{\text{th},n}^{\mathbb{C}^{n}_{5}-1}\text{G}^{3\kappa,1}_{\kappa+1,3\kappa+1}\left[\frac{\mathbb{D}^{n}_{2}\gamma_{\text{F}_{n}}}{\left(A_{0{\text{F}_{n}}}\right)^{{\kappa}}\delta_{\kappa_{n}}}\Big{|}^{1,\Psi^{n}_{1}}_{\Psi^{n}_{2},1-\mathbb{C}^{n}_{5}}\right],$
$\displaystyle\mathbb{I}_{3}(\gamma_{\text{th},n})$
$\displaystyle=\mathbb{D}^{n}_{1}\frac{\mathbb{C}^{n}_{1}{\left(\mathbb{C}^{n}_{3}\right)}^{-\frac{\xi_{\text{T}_{n}}^{2}}{\alpha_{n}}}}{\xi_{\text{T}_{n}}^{2}}\sum_{z_{2}=0}^{\infty}\frac{(-1)^{z_{2}}}{z_{2}!\left(N_{\text{r}}\mu_{n}+z_{2}\right)}$
$\displaystyle\times\left(\frac{\mathbb{C}^{n}_{3}}{{{\bar{\gamma}}_{\text{T}_{n}}}^{\frac{\alpha_{n}}{2}}}\right)^{N_{\text{r}}\mu_{n}+z_{2}}\Gamma(\mathbb{C}^{n}_{6})\gamma_{\text{th},n}^{\mathbb{C}^{n}_{6}-1}$
$\displaystyle\times\text{G}^{3\kappa,1}_{\kappa+1,3\kappa+1}\left[\frac{\mathbb{D}^{n}_{2}\gamma_{\text{F}_{n}}}{\left(A_{0{\text{F}_{n}}}\right)^{{\kappa}}\delta_{\kappa_{n}}}\Big{|}^{1,\Psi^{n}_{1}}_{\Psi^{n}_{2},1-\mathbb{C}^{n}_{6}}\right],$
(28)

where $\mathbb{C}^{n}_{4}=\frac{\xi^{2}_{\text{T}_{n}}}{2}+1$,
$\mathbb{C}^{n}_{5}=\frac{\xi^{2}_{\text{T}_{n}}}{2}+\frac{\alpha_{n}}{2}\left(\mathbb{C}^{n}_{2}+z_{1}\right)+1$,
and
$\mathbb{C}^{n}_{6}=\frac{\alpha_{n}}{2}\left(N_{\text{r}}\mu_{n}+z_{1}\right)+1$.

Substituting the identities $\mathbb{I}_{1}(\gamma_{\text{th},n})$,
$\mathbb{I}_{2}(\gamma_{\text{th},n})$, and
$\mathbb{I}_{3}(\gamma_{\text{th},n})$ in (27), outage probability of the
$n^{\text{th}}$ hybrid THz/FSO backhaul link for the combining method is
obtained.

Proof: See Appendix A.

With DF relaying, outage probability of the $N$-hop hybrid THz/FSO-based
backhaul system in a non-IAB setup is obtained as

$\displaystyle\mathcal{P}^{\text{CO}}_{\text{BH,S-1}}$
$\displaystyle=1-\prod_{\underset{}{n=1}}^{\overset{}{N}}\left(1-F^{\text{CO}}_{\gamma_{\text{I}_{n}}}\left(\gamma_{\text{th},n}\right)\right).$
(29)

Also, the E2E outage probability of the considered network at $l^{\text{th}}$
UE is derived as

$\displaystyle\mathcal{P}^{\text{CO}}_{\text{E2E,S-1}}\left(\gamma^{\text{UE}}_{\text{th}}\right)$
$\displaystyle=1-\left(1-\mathcal{P}^{\text{CO}}_{\text{BH,S-1}}\right)\times\left(1-F_{\gamma_{\text{acc}}}\left(\gamma^{\text{UE}}_{\text{th}}\right)\right),$
$\displaystyle=\mathcal{P}^{\text{CO}}_{\text{BH,S-1}}+F_{\gamma_{\text{acc}}}\left(\gamma^{\text{UE}}_{\text{th}}\right)-\mathcal{P}^{\text{CO}}_{\text{BH,S-1}}\times
F_{\gamma_{\text{acc}}}\left(\gamma^{\text{UE}}_{\text{th}}\right).$ (30)

Note. The outage probability in (III-A1) is for a non-IAB setup. However, with
an IAB setup, we need to average the performance over the outage probabilities
in each hop to find the overall network outage probability, i.e., with an
$N$-hop network, the total outage probability, assuming the same number of UEs
per node, is given by

$\displaystyle\mathcal{P}_{\text{out}}^{\text{total}}=\frac{1}{N}\sum_{n=1}^{N}\mathcal{P}_{\text{out}}^{n},$
(31)

with $\mathcal{P}_{\text{out}}^{n}$ given by (34), except for the case where
the donor IAB serves the UEs. In that case, a direct access link exists
between the donor and UEs, with outage probability given in (23).

Note. In an alternative method, one can always use both links simultaneously.
In that case, outage probability is given by
$\mathcal{P}\left[\left(\gamma_{\text{T}_{n}}+\gamma_{\text{F}_{n}}\right)<\gamma_{\text{th},n}\right]$
which ends up the same as in (27).

####  III-A2 Diversity Order for the Combining Method

For combining method, the asymptotic outage probability of System-1 is derived
as

$\displaystyle\mathcal{P}^{\text{CO,A}}_{\gamma_{\text{I}_{n}}}(\gamma_{\text{th},n})$
$\displaystyle\approx\int_{0}^{\gamma_{\text{th},n}}f^{\text{A}}_{\gamma_{\text{F}}}(\gamma)\times
F^{\text{A}}_{\gamma_{\text{T}}}(\gamma_{\text{th},n}-\gamma)\text{d}\gamma.$
(32)

This can be simplified as

$\displaystyle\mathcal{P}^{\text{CO,A}}_{\gamma_{\text{I}_{n}}}(\gamma_{\text{th},n})\approx\sum_{p=1}^{3}\Theta_{1}(p)$
$\displaystyle\times\left[\Theta_{2}\left(\frac{\gamma_{\text{th},n}}{\bar{\gamma}}\right)^{\frac{\xi_{\text{T}_{n}}^{2}}{2}+\frac{\Psi_{4,p}}{\kappa}}-\Theta_{3}\left(\frac{\gamma_{\text{th},n}}{\bar{\gamma}}\right)^{\frac{\alpha_{n}N_{\text{r}}\mu_{n}}{2}+\frac{\Psi_{4,p}}{\kappa}}\right],$
(33)

where
$\Theta_{1}(p)=\frac{\xi^{2}_{\text{F}_{n}}}{\kappa\Gamma(\alpha_{\text{F}_{n}})\Gamma(\beta_{\text{F}_{n}})}\frac{\overset{3}{\underset{\underset{q\neq
p}{q=1}}{\prod}}\Gamma(\Psi_{4,q}-\Psi_{4,p})}{\overset{1}{\underset{q=1}{\prod}}\Gamma(\Psi_{3,q}-\Psi_{4,p})}\left(\frac{\alpha_{\text{F}_{n}}\beta_{\text{F}_{n}}}{A_{0{\text{F}_{n}}}\Delta_{1}^{\frac{1}{\kappa}}}\right)^{\Psi_{4,p}}$,
$\Theta_{2}=\frac{\mathbb{C}^{n}_{1}\Gamma(\mathbb{C}^{n}_{2})}{\xi_{\text{T}_{n}}^{2}{\left(\Delta_{2}\right)}^{\frac{\xi_{\text{T}_{n}}^{2}}{2}}}\text{B}\left(\frac{\xi_{\text{T}_{n}}^{2}}{2}+1,\frac{\Psi_{4,p}}{\kappa}\right)$,
and
$\Theta_{3}=\frac{\mathbb{C}^{n}_{1}{\left(\mathbb{C}^{n}_{3}\right)}^{\mathbb{C}^{n}_{2}}}{\mathbb{C}^{n}_{2}\left(\alpha_{n}N_{\text{r}}\mu_{n}\right){\left(\Delta_{2}\right)}^{\frac{\alpha_{n}N_{\text{r}}\mu_{n}}{2}}}\text{B}\left(\frac{\alpha_{n}N_{\text{r}}\mu_{n}}{2}+1,\frac{\Psi_{4,p}}{\kappa}\right).$

Proof: See Appendix B

From (III-A2), the diversity order of the $n^{\text{th}}$ backhaul link with
combining is obtained as

$\displaystyle G^{\text{CO},n}_{\text{d}}=\underset{\overset{\text{diversity
order of the nth FSO
link}}{}}{\underbrace{\min\left(\frac{\xi_{\text{F}_{n}}^{2}}{\kappa},\frac{\alpha_{\text{F}_{n}}}{\kappa},\frac{\beta_{\text{F}_{n}}}{\kappa}\right)}}+\underset{\overset{\text{diversity
order of the nth THz
link}}{}}{\underbrace{\min\left(\frac{\xi_{\text{T}_{n}}^{2}}{2},\frac{\alpha_{n}N_{\text{r}}\mu_{n}}{2}\right)}}.$
(34)

Here, the first and the second terms represent the diversity order of the
$n^{\text{th}}$ FSO and THz links, respectively. Further, the sum operation
represents that both FSO and THz links work in parallel. Also, the diversity
order of the $N$-hop hybrid THz/FSO-based backhaul system is given as

$\displaystyle G^{\text{CO}}_{\text{d,S-1}}=$
$\displaystyle{\underset{\forall_{n=1}^{N}}{\min}}\left[\underset{\overset{\text{diversity
order of the nth hybrid THz/FSO
link}}{}}{\underbrace{\min\left(\frac{\xi_{\text{F}_{n}}^{2}}{\kappa},\frac{\alpha_{\text{F}_{n}}}{\kappa},\frac{\beta_{\text{F}_{n}}}{\kappa}\right)+\min\left(\frac{\xi_{\text{T}_{n}}^{2}}{2},\frac{\alpha_{n}N_{\text{r}}\mu_{n}}{2}\right)}}\right].$
(35)

Here, the final minimum operation outside the bracket denotes that the
diversity order of a multi-hop network is limited by the worst hop. Note that
the diversity order is the same for the IAB and non-IAB setup because, at
high-SNR, the performance is determined by the worst-case scenario which is
the outage probability at the end of the multi-hop chain.

###  III-B Switching Method

####  III-B1 Outage Probability

Considering the $n^{\text{th}}$ hybrid THz/FSO backhaul link with switching,
the THz link is active, if $\gamma_{\text{T}_{n}}\geq\gamma_{\text{th},n}$.
However, with $\gamma_{\text{T}_{n}}<\gamma_{\text{th},n}$, receiver generates
a feedback to switch off the THz link and activate the FSO link if
$\gamma_{\text{F}_{n}}\geq\gamma_{\text{th},n}$. Hence, for the
$n^{\text{th}}$ hybrid THz/FSO backhaul link, the instantaneous received SNR
at the MiBS/IAB $\left(\gamma^{\text{SW}}_{\text{I}_{n}}\right)$ is

$\displaystyle\gamma^{\text{SW}}_{\text{I}_{n}}=\begin{cases}\gamma_{\text{T}_{n}},&\text{for
}\gamma_{\text{T}_{n}}\geq\gamma_{\text{th},n}\\\
\gamma_{\text{F}_{n}},&\text{for
}\gamma_{\text{T}_{n}}<\gamma_{\text{th},n}~{}\&~{}\gamma_{\text{F}_{n}}\geq\gamma_{\text{th},n}\\\
0,&\text{for
}\gamma_{\text{T}_{n}}<\gamma_{\text{th},n}~{}\&~{}\gamma_{\text{F}_{n}}<\gamma_{\text{th},n}.\end{cases}$
(36)

The hybrid THz/FSO link is in outage when the instantaneous received SNRs of
both the THz and the FSO links (i.e., $\gamma_{\text{T}_{n}}$ and
$\gamma_{\text{F}_{n}}$) are below thresholds. Therefore, the outage
probability of the $n^{\text{th}}$ hybrid THz/FSO backhaul link is calculated
as

$\displaystyle\mathcal{P}^{\text{SW}}_{\gamma_{\text{I}_{n}}}(\gamma_{\text{th},n})$
$\displaystyle=F^{\text{SW}}_{\gamma_{\text{I}_{n}}}(\gamma_{\text{th},n})=\mathcal{P}\left[\gamma_{\text{T}_{n}}<\gamma_{\text{th},n}~{}\&~{}\gamma_{\text{F}_{n}}<\gamma_{\text{th},n}\right].$
(37)

The THz and FSO links are statistically independent, hence, outage probability
is obtained as

$\displaystyle\mathcal{P}^{\text{SW}}_{\gamma_{\text{I}_{n}}}\left(\gamma_{\text{th},n}\right)$
$\displaystyle=F_{\gamma_{\text{T}_{n}}}\left(\gamma_{\text{th},n}\right)\times
F_{\gamma_{\text{F}_{n}}}\left(\gamma_{\text{th},n}\right).$ (38)

Substituting (10) and (II-B2) in (38), outage probability of the
$n^{\text{th}}$ hop is derived as

$\displaystyle\mathcal{P}^{\text{SW}}_{\gamma_{\text{I}_{n}}}\left(\gamma_{\text{th},n}\right)$
$\displaystyle=\frac{\mathbb{C}^{n}_{1}\mathbb{D}^{n}_{1}}{\alpha_{\text{T}_{n}}({\bar{\gamma}}_{\text{T}_{n}})^{\frac{\xi_{\text{T}_{n}}^{2}}{2}}}\gamma_{\text{th},n}^{\frac{\xi_{\text{T}_{n}}^{2}}{2}}$
$\displaystyle\times\text{G}^{2,1}_{2,3}\left[\mathbb{C}^{n}_{3}\left(\frac{\gamma_{\text{th},n}}{{\bar{\gamma}}_{\text{T}_{n}}}\right)^{\frac{\alpha_{\text{T}_{n}}}{2}}\Big{|}^{1-\frac{\xi^{2}_{\text{T}_{n}}}{\alpha_{n}},1}_{0,\mathbb{C}^{n}_{2},-\frac{\xi^{2}_{\text{T}_{n}}}{\alpha_{n}}}\right]$
$\displaystyle\times\text{G}^{3\kappa,1}_{\kappa+1,3\kappa+1}\left[\frac{\mathbb{D}^{n}_{2}\gamma_{\text{th},n}}{\left(A_{{0\text{F}_{n}}}\right)^{\kappa}\delta_{\kappa_{n}}}\Big{|}^{1,\Psi^{n}_{1}}_{\Psi^{n}_{2},0}\right].$
(39)

Also, outage probability of the $N$-hop hybrid THz/FSO-based backhaul system
is obtained as

$\displaystyle\mathcal{P}^{\text{SW}}_{\text{BH,S-1}}$
$\displaystyle=1-\overset{N}{\underset{n=1}{\prod}}\left(1-F^{\text{SW}}_{\gamma_{\text{I}_{n}}}\left(\gamma_{\text{th},n}\right)\right).$
(40)

Finally, the E2E outage probability of System-1 at $l^{\text{th}}$ UE is
obtained as

$\displaystyle\mathcal{P}^{\text{SW}}_{\text{E2E,S-1}}\left(\gamma^{\text{UE}}_{\text{th}}\right)$
$\displaystyle=1-\left(1-\mathcal{P}^{\text{SW}}_{\text{BH,S-1}}\right)\times\left(1-F_{\gamma_{\text{acc}}}\left(\gamma^{\text{UE}}_{\text{th}}\right)\right).$
(41)

Note. Equation (41) gives the outage probability for the non-IAB setup where
the UEs are served only by the end access point. The same as in (31), the
outage probability in the out-band IAB setup is given by averaging the outage
probability over all IAB donor and child IAB nodes.

####  III-B2 Diversity Order for the Switching Method

For the $n^{\text{th}}$ hybrid THz/FSO backhaul link, the asymptotic outage
probability can be derived as

$\displaystyle\mathcal{P}^{\text{SW,A}}_{\gamma_{\text{I}_{n}}}(\gamma_{\text{th},n})$
$\displaystyle\approx
F^{\text{A}}_{\gamma_{\text{F}}}(\gamma_{\text{th},n})\times
F^{\text{A}}_{\gamma_{\text{T}}}(\gamma_{\text{th},n}).$ (42)

Substituting $F^{\text{A}}_{\gamma_{\text{F}}}(\gamma_{\text{th},n})$ and
$F^{\text{A}}_{\gamma_{\text{T}}}(\gamma_{\text{th},n})$ from (13) and
(II-B3), respectively, in (42), we obtain

$\displaystyle\mathcal{P}^{\text{SW,A}}_{\gamma_{\text{I}_{n}}}(\gamma_{\text{th},n})\approx\mathbb{D}^{n}_{1}\sum_{p=1}^{3\kappa}\left(\frac{\mathbb{D}^{n}_{2}\gamma_{\text{th},n}}{(A_{0,\text{F}})^{\kappa}\delta_{{\kappa}}}\right)^{\Psi^{n}_{6,p}}$
$\displaystyle\times\frac{\overset{3\kappa}{\underset{\underset{q\neq
p}{q=1}}{\prod}}\Gamma(\Psi^{n}_{6,q}-\Psi^{n}_{6,p})\overset{1}{\underset{q=1}{\prod}}\Gamma(1-\Psi^{n}_{5,q}+\Psi^{n}_{6,p})}{\overset{\kappa+1}{\underset{q=2}{\prod}}\Gamma(\Psi^{n}_{5,q}-\Psi^{n}_{6,p})\overset{3\kappa+1}{\underset{q=5\kappa+1}{\prod}}\Gamma(1-\Psi^{n}_{6,q}+\Psi^{n}_{6,p})}$
$\displaystyle\times\left[\frac{\mathbb{C}^{n}_{1}\Gamma(\mathbb{C}^{n}_{2})}{\xi_{\text{T}_{n}}^{2}}\left(\frac{\gamma_{\text{th},n}}{{\bar{\gamma}}_{\text{T}_{n}}}\right)^{\frac{\xi_{\text{T}_{n}}^{2}}{2}}-\frac{\mathbb{C}^{n}_{1}{\left(\mathbb{C}^{n}_{3}\right)}^{\mathbb{C}^{n}_{2}}}{\mathbb{C}^{n}_{2}\left(\alpha_{n}N_{\text{r}}\mu_{n}\right)}\left(\frac{\gamma_{\text{th},n}}{{\bar{\gamma}}_{\text{T}_{n}}}\right)^{\frac{\alpha_{n}N_{\text{r}}\mu_{n}}{2}}\right].$
(43)

At high SNRs, we have $\bar{\gamma}\rightarrow\infty$. Consequently, the
average received SNR of the THz and the FSO links tend to infinity, i.e,
$\bar{\gamma}_{\text{T}_{n}},\delta_{\kappa_{n}}\rightarrow\infty$. Hence,
considering $\bar{\gamma}_{\text{T}_{n}}=\Delta_{1}\bar{\gamma}$ and
$\delta_{\kappa_{n}}=\Delta_{2}\bar{\gamma}$, where $\Delta_{1}$ and
$\Delta_{2}$ are constants, (III-B2) can be modified as

$\displaystyle\mathcal{P}^{\text{SW,A}}_{\gamma_{\text{I}_{n}}}(\gamma_{\text{th},n})\approx\sum_{p=1}^{3\kappa}\Theta_{4}(p)\times\bigg{[}\Theta_{5}\left(\frac{\gamma_{\text{th},n}}{\bar{\gamma}}\right)^{\Psi^{n}_{6,p}+\frac{\xi_{\text{T}_{n}}^{2}}{2}}$
$\displaystyle-\Theta_{6}\left(\frac{\gamma_{\text{th},n}}{\bar{\gamma}}\right)^{\Psi^{n}_{6,p}+\frac{\alpha_{n}N_{\text{r}}\mu_{n}}{2}}\bigg{]},$
(44)

where
$\Theta_{4}(p)=\mathbb{D}^{n}_{1}\frac{\overset{3\kappa}{\underset{\underset{q\neq
p}{q=1}}{\prod}}\Gamma(\Psi^{n}_{6,q}-\Psi^{n}_{6,p})\overset{1}{\underset{q=1}{\prod}}\Gamma(1-\Psi^{n}_{5,q}+\Psi^{n}_{6,p})}{\overset{\kappa+1}{\underset{q=2}{\prod}}\Gamma(\Psi^{n}_{5,q}-\Psi^{n}_{6,p})\overset{3\kappa+1}{\underset{q=5\kappa+1}{\prod}}\Gamma(1-\Psi^{n}_{6,q}+\Psi^{n}_{6,p})}\\\
\left(\frac{\mathbb{D}^{n}_{2}}{(A_{0,\text{F}})^{\kappa}\Delta_{1}}\right)^{\Psi^{n}_{6,p}}$,
$\Theta_{5}=\frac{\mathbb{C}^{n}_{1}\Gamma(\mathbb{C}^{n}_{2})}{\xi_{\text{T}_{n}}^{2}\Delta_{2}^{\frac{\xi_{\text{T}_{n}}^{2}}{2}}}$,
and
$\Theta_{6}=\frac{\mathbb{C}^{n}_{1}{\left(\mathbb{C}^{n}_{3}\right)}^{\mathbb{C}^{n}_{2}}}{\mathbb{C}^{n}_{2}\left(\alpha_{n}N_{\text{r}}\mu_{n}\right)\Delta_{2}^{\frac{\alpha_{n}N_{\text{r}}\mu_{n}}{2}}}$.

From (III-B2), the diversity order of the $n^{\text{th}}$ hybrid THz/FSO
backhaul link is given as

$\displaystyle
G^{\text{SW},n}_{\text{d}}=\min\left(\frac{\xi_{\text{F}_{n}}^{2}}{\kappa},\frac{\alpha_{\text{F}_{n}}}{\kappa},\frac{\beta_{\text{F}_{n}}}{\kappa}\right)+\min\left(\frac{\xi_{\text{T}_{n}}^{2}}{2},\frac{\alpha_{n}N_{\text{r}}\mu_{n}}{2}\right)$
(45)

From (34) and (45), it is clear that the diversity order of the
$n^{\text{th}}$ hybrid THz/FSO backhaul link for both the combining and
switching methods is the same. However, combining provides additional SNR gain
over switching, which we will see in Section VI. Finally, the diversity order
of the $N$-hop hybrid THz/FSO-based backhaul system is given as

$\displaystyle G^{\text{SW}}_{\text{d,S-1}}=$
$\displaystyle\underset{\overset{\forall_{n=1}^{N}}{}}{\min}\left[\underset{\overset{\text{diversity
order of the nth hybrid THz/FSO
link}}{}}{\underbrace{\min\left(\frac{\xi_{\text{F}_{n}}^{2}}{\kappa},\frac{\alpha_{\text{F}_{n}}}{\kappa},\frac{\beta_{\text{F}_{n}}}{\kappa}\right)+\min\left(\frac{\xi_{\text{T}_{n}}^{2}}{2},\frac{\alpha_{n}N_{\text{r}}\mu_{n}}{2}\right)}}\right].$
(46)

##  IV Performance Analysis of System-2

For System-2 (as shown in Fig. 2), we consider the MaBS/donor IAB and the
final MiBS/child IAB are in LoS. Hence, the backhaul communication takes place
through $N$ cascaded THz links and a single FSO link, with proper combining at
the final MiBS/IAB node. We analyze the performance of the proposed System-2
by combining both signals coming from the $N$ cascaded THz links and the
single FSO link, or by switching between $N$ cascaded THz links or to the
single FSO link for signal reception. Both cases are analyzed in the
following.

###  IV-A Combining Method

####  IV-A1 Outage Probability

Since the FSO link is working as a backup for the cascaded THz links, first we
check the availability of THz links in the multi-hop hybrid THz/FSO-based
network. If all THz links experience good channel quality, the equivalent
cascaded THz link serves irrespective of the FSO link. If the final THz link
is below threshold given that all intermediate THz links experience good
channel quality, we consider simultaneous transmission through the cascaded
THz links and the FSO link. Also, if any of the intermediate THz link fails,
the FSO link becomes active if $\gamma_{\text{F}}\geq\gamma_{\text{th},N}$.
Therefore, for the considered System-2 with combining method, the
instantaneous received SNR at the final MiBS/child IAB is given as (47), at
the top of the next page.

$\displaystyle\gamma^{\text{CO}}_{\text{CI}}=\begin{cases}\gamma_{\text{T}_{\text{eq}}},&\text{for
}\forall_{n=1}^{N}~{}\gamma_{\text{T}_{n}}\geq\gamma_{\text{th},n}\\\
\gamma_{\text{T}_{n}}+\gamma_{\text{F}},&\text{for
}\forall_{n=1}^{N}\left({\gamma_{\text{T}_{n}}}<\gamma_{\text{th},n}~{}\&~{}\left(\gamma_{\text{T}_{n}}+\gamma_{\text{F}}\right)\geq\gamma_{\text{th},N}~{}\&~{}\gamma_{\text{T}_{n-1}}\geq\gamma_{\text{th},(n-1)}\right)\\\
\gamma_{\text{F}},&\text{for
}\forall_{n=1}^{N-1}\big{(}\gamma_{\text{T}_{n}}<\gamma_{\text{th},n}~{}\&~{}\gamma_{\text{F}}\geq\gamma_{\text{th},N}~{}\&~{}\gamma_{\text{T}_{n-1}}\geq\gamma_{\text{th},(n-1)}\big{)}\\\
0,&\text{for
}\forall_{n=1}^{N-1}\big{(}\gamma_{\text{T}_{n}}<\gamma_{\text{th},n}~{}\&~{}\gamma_{\text{F}}<\gamma_{\text{th},N}~{}\&~{}\gamma_{\text{T}_{n-1}}\geq\gamma_{\text{th},(n-1)}\big{)}\\\
&~{}{\text{or
}}\forall_{n=1}^{N}\left(\left(\gamma_{\text{T}_{n}}+\gamma_{\text{F}}\right)<\gamma_{\text{th},N}~{}\&~{}\gamma_{\text{T}_{n-1}}\geq\gamma_{\text{th},(n-1)}\right),\end{cases}$
(47)

Hence, the outage probability of the multi-hop THz/FSO backhaul network is
found as

$\displaystyle\mathcal{P}^{\text{CO}}_{\gamma_{\text{CI}}}=$
$\displaystyle\mathcal{P}\bigg{[}\forall_{n=1}^{N-1}\big{(}\gamma_{\text{T}_{n}}<\gamma_{\text{th},n}~{}\&~{}\gamma_{\text{F}}<\gamma_{\text{th},N}~{}\&~{}\gamma_{\text{T}_{n-1}}\geq\gamma_{\text{th},(n-1)}\big{)}$
$\displaystyle~{}{\text{or
}}\forall_{n=1}^{N}\left(\left(\gamma_{\text{T}_{n}}+\gamma_{\text{F}}\right)<\gamma_{\text{th},N}~{}\&~{}\gamma_{\text{T}_{n-1}}\geq\gamma_{\text{th},(n-1)}\right)\bigg{]}.$
(48)

This can be solved as

$\displaystyle\mathcal{P}^{\text{CO}}_{\gamma_{\text{CI}}}$
$\displaystyle=F_{\gamma_{\text{F}}}\left(\gamma_{\text{th},N}\right)\times\bigg{[}\sum_{n=1}^{N-1}F_{\gamma_{\text{T}_{n}}}\left(\gamma_{\text{th},n}\right)$
$\displaystyle\times\prod_{n_{1}=1}^{n-1}\left(1-F_{\gamma_{\text{T}_{(n-n_{1})}}}\left(\gamma_{\text{th},(n-n_{1})}\right)\right)\bigg{]}$
$\displaystyle+\prod_{n=1}^{N-1}\left(1-F_{\gamma_{\text{T}_{n}}}\left(\gamma_{\text{th},n}\right)\right)\times
F^{\text{CO}}_{\gamma_{\text{I}_{N}}}.$ (49)

Here, $F^{\text{CO}}_{\gamma_{\text{I}_{N}}}$ is the CDF by performing the
combining between the $N^{\text{th}}$ THz link and the single FSO link at the
final node which is calculated the same as in (26). Further,
$F_{\gamma_{\text{T}_{n}}}\left(\gamma_{\text{th},n}\right)$ is the CDF of the
$n^{\text{th}}$ THz link (II-B2) and
$F_{\gamma_{\text{F}}}\left(\gamma_{\text{th},N}\right)$ is the CDF of the FSO
link (10). Substituting (10), (II-B2), and (26) in (IV-A1), outage probability
of backhaul links in System-2 is obtained.

Finally, the E2E outage probability of the considered network at
$l^{\text{th}}$ UE is derived as

$\displaystyle\mathcal{P}^{\text{CO}}_{\text{E2E,S-2}}\left(\gamma^{\text{UE}}_{\text{th}}\right)$
$\displaystyle=1-\left(1-\mathcal{P}^{\text{CO}}_{\gamma_{\text{CI}}}\right)\times\left(1-F_{\gamma_{\text{acc}}}\left(\gamma^{\text{UE}}_{\text{th}}\right)\right).$
(50)

Note that (50) gives the E2E outage probability for the cases with non-IAB
based communication setup. Following (31), one can extend the results to the
cases with IAB setup.

Finally, in an alternative method, one can always use both links
simultaneously. In that case, the outage probability is given by
$\mathcal{P}\Big{[}\forall_{n=1}^{N-1}\big{(}\gamma_{\text{T}_{n}}<\gamma_{\text{th},n}~{}\&~{}\gamma_{\text{F}}<\gamma_{\text{th},N}~{}\&~{}\gamma_{\text{T}_{n-1}}\geq\gamma_{\text{th},(n-1)}\big{)}~{}{\text{or
}}\forall_{n=1}^{N}\left(\left(\gamma_{\text{T}_{n}}+\gamma_{\text{F}}\right)<\gamma_{\text{th},N}~{}\&~{}\gamma_{\text{T}_{n-1}}\geq\gamma_{\text{th},(n-1)}\right)\Big{]}$
which ends up in the same result as in (IV-A1).

####  IV-A2 Diversity Order for the Combining Method

The diversity order of the $N$ cascaded THz links is
$\underset{\forall_{n=1}^{N}}{\min}\left(\frac{\xi_{\text{T}_{n}}^{2}}{2},\frac{\alpha_{n}N_{\text{r}}\mu_{n}}{2}\right)$.
Further, the diversity order of the single FSO link is
$\min\left(\frac{\xi_{\text{F}}^{2}}{\kappa},\frac{\alpha_{\text{F}}}{\kappa},\frac{\beta_{\text{F}}}{\kappa}\right)$.
Thus, the overall diversity order of System-2 for the combining method is
obtained as

$\displaystyle G^{\text{ST}}_{\text{d,S-2}}=\underset{\overset{\text{diversity
order of the single E2E FSO
link}}{}}{\underbrace{\min\left(\frac{\xi_{\text{F}}^{2}}{\kappa},\frac{\alpha_{\text{F}}}{\kappa},\frac{\beta_{\text{F}}}{\kappa}\right)}}+\underset{\overset{\text{diversity
order of the N cascaded THz
links}}{}}{\underbrace{\underset{\forall_{n=1}^{N}}{\min}\left(\frac{\xi_{\text{T}_{n}}^{2}}{2},\frac{\alpha_{n}N_{\text{r}}\mu_{n}}{2}\right)}}.$
(51)

###  IV-B Switching Method

####  IV-B1 Outage Probability

$\displaystyle\gamma^{\text{SW}}_{\text{CI}}=\begin{cases}\gamma_{\text{T}_{\text{eq}}},&\text{for }{\forall_{n=1}^{N}~{}\gamma_{\text{T}_{n}}}\geq\gamma_{\text{th},n}\\\ \gamma_{\text{F}},&\text{for }\forall_{n=1}^{N}\big{(}\gamma_{\text{T}_{n}}<\gamma_{\text{th},n}~{}\&~{}\gamma_{\text{F}}\geq\gamma_{\text{th}}~{}\&~{}\gamma_{\text{T}_{n-1}}\geq\gamma_{\text{th},(n-1)}\big{)}\\\ 0,&\text{for }\forall_{n=1}^{N}\big{(}\gamma_{\text{T}_{n}}<\gamma_{\text{th},n}~{}\&~{}\gamma_{\text{F}}<\gamma_{\text{th}}~{}\&~{}\gamma_{\text{T}_{n-1}}\geq\gamma_{\text{th},(n-1)}\big{)}.\end{cases}$ (52) TABLE II: Considered parameters for the simulations. THz link | FSO link | mmWave access link
---|---|---
Parameters | Value | Parameters | Value | Parameters | Value
$f_{\text{T}_{n}}$ | 119 GHz | $\lambda_{\text{F}_{n}}$ | 1550 nm | $f_{\text{acc}}$ | 28 GHz
$G_{\text{t}}^{\text{T}_{n}}$ | 55 dBi | Strong: $C_{n}^{2}$ | 1$\times 10^{-12}$m-2/3 | $G_{\text{t}}^{\text{acc}}$ | 44 dBi
$G_{\text{r}}^{\text{T}_{n}}$ | 55 dBi | Moderate: $C_{n}^{2}$ | 5$\times 10^{-13}$m-2/3 | $G_{\text{r}}^{\text{acc}}$ | 44 dBi
$p$ | 101325 Pa | Strong: $\alpha_{\text{F}_{n}}$,$\beta_{\text{F}_{n}}$ | 4.343, 2.492 | $\varrho_{\text{ra}}$ | 0 dB/km
$T$ | 298 K | Moderate: $\alpha_{\text{F}_{n}}$,$\beta_{\text{F}_{n}}$ | 5.838, 4.249 | $\varrho_{\text{ox}}$ | 15.1 dB/km
$\chi$ | 50$\%$ | $\eta$ | 1 |  |
${a_{\text{T}_{n}}}$ | ${\lambda_{\text{T}_{n}}/{\left(2\pi\right)}\sqrt{G^{\text{T}_{n}}_{\text{t}}}}$ | ${a_{\text{F}_{n}}}$ | 20 cm |  |
${\omega_{\text{T}_{n}}}$ | 50 cm | ${\omega_{\text{F}_{n}}}$ | 40 cm |  |
$\varepsilon_{\text{T}_{n}}$ | 6 cm | $\varepsilon_{\text{F}_{n}}$ | 5 cm |  |

Since the FSO link is working as a backup for the THz link, first we check the
availability of the THz link in the multi-hop hybrid THz/FSO-based backhaul
network. If all THz nodes can decode the data correctly, the equivalent multi-
hop THz link will serve irrespective of the FSO link. However, if any of the
THz links fails, the FSO link becomes active if
$\gamma_{\text{F}}\geq\gamma_{\text{th}}$. Therefore, for the considered
System-2 with switching, the instantaneous received SNR at the final
MiBS/child IAB is given as (52), at the top of this page. From this, the
outage probability is calculated as

$\displaystyle\mathcal{P}^{\text{SW}}_{\gamma_{\text{CI}}}\left(\gamma_{\text{th}}\right)=$
$\displaystyle\mathcal{P}\left[\forall_{n=1}^{N}\big{(}\gamma_{\text{T}_{n}}<\gamma_{\text{th},n}~{}\&~{}\gamma_{\text{F}}<\gamma_{\text{th}}~{}\&~{}\gamma_{\text{T}_{n-1}}\geq\gamma_{\text{th},(n-1)}\big{)}\right].$

This can be solved as

$\displaystyle\mathcal{P}^{\text{SW}}_{\gamma_{\text{CI}}}\left(\gamma_{\text{th}}\right)$
$\displaystyle=F_{\gamma_{\text{F}}}\left(\gamma_{\text{th}}\right)\times\bigg{[}\sum_{n=1}^{N}F_{\gamma_{\text{T}_{n}}}\left(\gamma_{\text{th},n}\right)$
$\displaystyle\times\prod_{n_{1}=1}^{n-1}\left(1-F_{\gamma_{\text{T}_{(n-n_{1})}}}\left(\gamma_{\text{th},(n-n_{1})}\right)\right)\bigg{]}.$
(54)

With non-IAB setup, the E2E outage probability of the considered network is
derived as

$\displaystyle\mathcal{P}^{\text{SW}}_{\text{E2E,S-2}}\left(\gamma^{\text{UE}}_{\text{th}}\right)$
$\displaystyle=1-\left(1-\mathcal{P}^{\text{SW}}_{\gamma_{\text{CI}}}\right)\times\left(1-F_{\gamma_{\text{acc}}}\left(\gamma^{\text{UE}}_{\text{th}}\right)\right).$
(55)

Finally, the same as in (31), the outage probability in the out-band IAB setup
is given by averaging the outage probability over all IAB donor and child IAB
nodes.

####  IV-B2 Diversity Order for the Switching Method

With DF relaying, the diversity order of the considered System-2 with
switching is given as

$\displaystyle G^{\text{SW}}_{\text{d,S-2}}=\underset{\overset{\text{diversity
order of the single E2E FSO
link}}{}}{\underbrace{\min\left(\frac{\xi_{\text{F}}^{2}}{\kappa},\frac{\alpha_{\text{F}}}{\kappa},\frac{\beta_{\text{F}}}{\kappa}\right)}}+\underset{\overset{\text{diversity
order of the N cascaded THz
links}}{}}{\underbrace{\underset{\forall_{n=1}^{N}}{\min}\left(\frac{\xi_{\text{T}_{n}}^{2}}{2},\frac{\alpha_{n}N_{\text{r}}\mu_{n}}{2}\right)}}.$
(56)

Note that the diversity order of System-2 for both combining and switching
methods is the same, which is already proved in Section III for System-1.

##  V Mesh Network

We consider a mesh network with $Q$ independent and non-overlapping multi-hop
routes from the MaBS to the UE. The $q^{\text{th}}$ route ($q=1,2,...,Q$)
consists of the multi-hop hybrid THz/FSO links according to System-1 and
System-2. Thus, for the mesh network, outage probability is

$\displaystyle\mathcal{P}^{\text{mesh}}_{\text{out}}=\prod_{q=1}^{Q}\mathcal{P}_{\text{out},q},$
(57)

where $\mathcal{P}_{\text{out},q}$ is the outage probability of the
$q^{\text{th}}$ E2E path from the MaBS to the destination UE. For System-1,
this is derived in (III-A1) and (41) for the combining and the switching
methods, respectively. For System-2, this is derived in (50) and (55) for the
combining and the switching methods, respectively. Note that in (57) we have
used the fact that in a mesh network an outage occurs if the data is correctly
transferred to the destination through none of the routes and there is no
overlap between the routes.

##  VI Simulation Results

We present our results with respect to the transmit power which is defined as
$10\log_{10}\left({P_{\text{j}}}\right)$ in the log-domain. Here,
$P_{\text{j}}$ with $\text{j}\in\\{\text{T,F,acc}\\}$ is the transmit power at
the $j$ node. For simplicity, we consider
$P_{\text{F}}=P_{\text{T}}=P_{\text{acc}}$ in each hop for the analysis.
Further,
$\sigma^{2}_{\text{o}}=\sigma^{2}_{\text{T}}=\sigma^{2}_{\text{acc}}=1$ is
assumed in each hop. Rest of the simulation parameters are given in Table II.
We present the results for the general cases with multi-hop and mesh networks.
However, as demonstrated in [2, 3], with multi-hop and IAB networks, the
system performance is significantly affected by increasing the number of hops,
as the traffic congestion and E2E latency increases. For this reason, except
for Fig. 4, where multi-hop results are shown for both the systems, only dual
backhaul hops are considered for rest of the results. In Figs. 7-10, we
present the results for the cases with IAB setup where each intermediate IAB
node in the multi-hop setup serves a number of surrounding UEs. In Figs. 4-6,
on the other hand, we present the results for the non-IAB setup where the UEs
are served only by the end MiBS. Note that if the SNR threshold is fixed in
all hops, we are in non-IAB case (even if the figure only presents the
backhaul performance).

Figure 4: Impact of number of hops on the outage probability of the backhaul
hybrid THz/FSO systems with switching. Here, we consider 200 m links in each
hop, 25 dB transmit power, $\gamma_{\text{th}}=1$ in each hop, and present the
results for both System 1 and 2. Also, $C_{n}^{2}=1\times
10^{-12}\text{m}^{-2/3}$ and $\alpha_{n}=2,\mu_{n}=3,N_{r}=2$.

In Fig. 4, we compare the outage probability of the multi-hop backhaul hybrid
THz/FSO systems for different turbulence/fading conditions. We study both
System-1 (S-1 in Figs.) and System-2 (S-2 in Figs.), as defined in Figs. 1-2,
respectively. From Fig. 4, we observe that the FSO link’s performance
decreases significantly for System-2 (shown with dotted red) as compared to
System-1 (shown with solid red), where $N$ FSO links are deployed in DF mode.
However, for both systems, $N$ THz links are deployed in series for the E2E
backhaul communication, hence, the THz link’s performance remains the same
(shown in blue for both systems). Consequently, the E2E performance of
System-2 decreases significantly as compared to System-1. As shown in Fig. 4,
for both systems, increasing the number of hops from 1 to 2 results in
significant outage probability increment. However, for more than 2 hops, the
outage probability is only slightly affected by increasing the number of hops.
On the other hand, as discussed before and shown in [2, 3], the E2E
delay/throughout are significantly deteriorated by increasing the number of
hops.

Figure 5: E2E outage probability of mesh network for both the systems with
different routes by fixing 3 hops per route (2 backhaul and one access link)
and by considering switching method. Here, $L_{\text{F}}=200$ m,
$L_{\text{T}}=200$ m, and $L_{\text{acc}}=100$ m in each hop.

In Fig. 5, we show the E2E outage probability of mesh networks for both system
models. Here, in each route, we fix the number of hops to three with two
hybrid THz/FSO-based backhaul hops and one mmWave access hop. We consider
strong turbulence/fading case by setting $C_{n}^{2}=1\times 10^{-12}$ m-2/3
which gives $\alpha_{\text{F}}=4.34$, $\beta_{\text{F}}=2.49$ for the FSO
links. Further, we consider $\alpha=2,\mu=2,N_{\text{r}}=2$ and
$m=2,N_{\text{t}}=3$ for the THz and mmWave access links, respectively.

Figure 6: Comparison of outage probability of both the multi-hop THz/FSO-based
systems against SNR threshold.

(a) Different cases with and without middle IAB node.

(b) Outage probability versus UE distance with respect to the middle IAB and
child IAB nodes.

(c) Outage probability versus jitter standard deviation.

Figure 7: Comparison of both the hybrid THz/FSO-based dual-hop IAB networks
with switching.

As compared to the single route multi-hop network (as shown in Fig. 4), whose
outage performance degrades with number of hops, a mesh network improves the
overall outage performance by enabling more routes. From Fig. 5, we observe
that for the considered set of parameters and an outage probability $10^{-6}$,
System-1 provides 3 dB gain over System-2 for a single route multi-hop
network. Also, by increasing the E2E routes (Q=5) in the mesh network and
outage probability $10^{-6}$, we obtain approximately 10 dB gain over the
single route multi-hop network for both systems. Adding more routes will
further improve the E2E performance, however, with a decreased rate than
before. The results are shown for the switching case. With combining, we will
achieve improved performance over switching, however, the qualitative insights
remain the same.

In Fig. 6, we show the impact of SNR threshold on the outage performance of
the dual-hop backhaul systems for both System-1 (solid lines) and System-2
(dotted lines) with both switching and combining methods. Here, we consider
the same SNR threshold at each receiving MiBS node, hence, it is a non-IAB
system. The FSO link of System-1 outperforms the FSO link of System-2, since,
for the dual-backhaul hops, the FSO link in System-2 is approximately twice
longer than the FSO link in System-1. Further, the cascaded THz link of
System-2 (dual-hop) has slightly lower performance than the single THz link of
System-1. Consequently, the dual-hop backhaul hybrid THz/FSO-based link of
System-1 has better performance than System-2 for both the switching and
combining methods. Further, we observe a significant performance improvement
in System-1 with combining over the switching method at the receiving MiBS as
compared to System-2. This is due to the fact, that in System-2, the
switching/combining is performed at the end of the second hop, and the single
THz link of the first hop dominates in the backhaul dual-hop communication.
Hence, in this case, we observe only slight improvement in combining over
switching in System-2.

(a) Different cases when UE lies in between the donor and middle IAB node.

(b) Outage probability versus MU distance with respect to the middle IAB and
child IAB nodes.

(c) Outage probability versus jitter standard deviation.

Figure 8: Comparison of both dual-hop IAB networks with switching.

In Figs. 4-6, we have shown the results of non-IAB multi-hop and mesh
networks, where identical SNR threshold is considered on each of the receiving
MiABs. On the other hand, Figs. 7-10 illustrate the results for IAB networks.
With IAB, each IAB node carries the information of all UEs to be served by the
IAB node itself and its following child IAB nodes. Different SNR thresholds
consideration at each IAB node is motivated with the fact that the SNR
threshold is directly related to the number of UEs associated with each IAB
node. Let us assume a desired rate $\mathcal{R}$ bps/Hz at each UE, and $L$
such UEs are being served by the $n^{\text{th}}$ IAB node or its following
child IAB nodes in the multi-hop chain. Then, the SNR threshold at the
$n^{\text{th}}$ IAB node will be
$\gamma_{\text{th},n}=2^{L\times\mathcal{R}}-1$. Thus, the SNR threshold of
each IAB node is scaled by the total number of UEs which are served by the IAB
node and its following child nodes.

Considering 10 UEs associated to each IAB node with $\mathcal{R}=0.1$ bps/Hz,
we set the threshold at each IAB node. In Fig. 7(a), we show the movement of
the UE in between the middle and last IAB nodes, and show the need of the
middle IAB node. We consider that the height of the donor and the child IAB
node is $H=30$ m, and the $l^{\text{th}}$ UE is at distance $d$ apart from the
middle IAB node and moving towards the final IAB node. We consider the cases
when the middle IAB node is serving the UE via 200 m hybrid THz/FSO link
(System-1 (first case of Fig. 7(a))), the middle IAB node is serving the UE
via 200 m THz link (System-2 (second case of Fig. 7(a))), and the final IAB
node is serving the UE via 400 m hybrid THz/FSO link (in the absence of middle
IAB (third case of Fig. 7(a))).

In Fig. 7(c), we show the results of the UE’s outage probability versus
distance $d$. Here, we observe that System-1 outperforms System-2 up-to some
distance when the middle IAB is serving the UE. This is due to the fact that
the hybrid THz/FSO link is serving the UE in System-1, as opposed to a single
THz link in System-2. Further, we observe after 100 m distance, the outage
performance at the UE through the middle IAB node is quite poor for both
systems. This is due to the fact that the UE is far from the middle IAB’s
coverage region. At the same time, it reaches in the final IAB’s coverage
region. Hence, we observe that the outage performance at the UE through the
final IAB node (via 400 m hybrid THz/FSO link) is improved (shown with green)
after 100 m distance, which was quite poor earlier. Hence, there is a handover
as the UE moves towards the final node. The UE which was initially served by
the middle IAB node is now served by the final IAB node and will not reach
below an acceptable performance limit. Also, we show that without the middle
IAB node, the UE’s outage performance is quite poor if it is directly being
served by the donor IAB node. This motivated for the IAB setups.

In Fig. 7(c), we show the impact of pointing error by varying the jitter
standard deviation
($\varepsilon_{\text{F}}=\varepsilon_{\text{T}}=\varepsilon$) on the
performance of the THz, the FSO, and the hybrid THz/FSO links. For this, we
fix the receiver radius $a_{\text{F}}=a_{\text{T}}=0.2$ m and beamwidth
$\omega_{\text{F}}=\omega_{\text{T}}=0.4$ m in each hop. From Fig. 7(c), we
observe that the 200 m hybrid THz/FSO link between the donor and middle IAB
node (System-1) has acceptable performance till $\varepsilon=0.25$ m (for an
outage probability of $10^{-2})$, and reaches into outage with further
increase in $\varepsilon$. The 400 m hybrid THz/FSO link between the donor and
final IAB node (when middle IAB is not used) has poorer performance than the
200 m hybrid THz/FSO link at lower $\varepsilon$, however, has nearly the same
performance at higher values of $\varepsilon$. Also, the 400 m hybrid THz/FSO
link has acceptable performance till $\varepsilon=0.25$ m (for an outage
probability of $10^{-2}$), and reaches into outage with further increase in
$\varepsilon$. On the other hand, 200 m THz link between the donor and middle
IAB node (System-2) has the worst performance, and reaches below $10^{-2}$
outage probability even at approximately $\varepsilon=0.15$ m. Further,
increase in $\varepsilon$ forces the THz link into complete outage. Hence,
from Fig. 7c, at low values of $\varepsilon$, System-1 is more sensitive to
pointing error, which is intuitively because, in System-1, in each hop both
the FSO and the THz links are affected by the pointing error.

(a)

(b)

Figure 9: Comparison of outage probability for both system models versus
transmit power.

In Fig. 8(a), we show the movement of the UE in between the donor and the
middle IAB nodes and its impact on the E2E outage performance. We consider
that the UE is at distance $d$ apart from the donor IAB node and is moving
towards the middle IAB node. We consider the cases when the donor IAB is
serving the UE directly, when the middle IAB node is serving the UE via 200m
hybrid THz/FSO link (System-1 (second case of Fig. 8(a))), or the middle IAB
node is serving the UE via 200 m THz link (System-2 (third case of Fig.
8(a))).

From Fig. 8(c), we observe that the outage performance at the UE is low till
100 m when it is served by the donor IAB directly. However, with further
increase in $d$, the UE reaches in the middle IAB’s coverage region. Hence,
the performance decreases rapidly when donor IAB serves the UE. At the same
time, it receives strong signal for the middle IAB node, and has improved
performance, compared to the case when the UE is served by the donor IAB node.
Hence, there exist a handover after 100 m distance, and the UE is served by
the middle IAB node, while the UE was earlier served by the donor IAB node.
Further, we have better performance with System-1 (with hybrid THz/FSO link)
as compared to System-2 (with single THz link) when the middle IAB serves the
UE. Finally, Fig. 8(c) shows the same impact of the jitter standard deviation
$\varepsilon$ on the 200 m hybrid THz/FSO link (System-1) and the 200 m THz
link (System-2) between the donor IAB and the middle IAB nodes, as also
discussed in Fig. 7(c).

Considering 10 UEs associated to each IAB node with $\mathcal{R}=0.1$ bps/Hz,
we set the threshold at each IAB and obtain the outage results for both
systems with switching/combining methods in Fig. 9. Here, Fig. 9(a) and Fig.
9(b) show the results for the strong and moderate turbulence/fading,
respectively. From Fig. 9(a), we observe that the FSO link of System-1 has
significant performance improvement (around 11 dB for an outage probability
$10^{-4}$) than the FSO link of System-2. This allows superior outage
performance for System-1, as compared to System-2 for both switching/combining
methods (around 2.5 dB gain with switching and 4.5 dB gain with combining
method for an outage probability $10^{-4}$). Also, the combining method
provides around 2 dB gain over the switching method for System-1. However, the
outage performance remains the same for both switching/combining methods for
System-2. This is because the switching/combining is performed at the second
hop and the single THz link of the first hop dominates in overall performance.
The similar insights can be obtained from Fig. 9(b) for the moderate
turbulence/fading case.

Figure 10: Theoretical and asymptotic outage probability of both the systems
for combining/switching methods.

In Fig. 10, the asymptotic outage probability results are shown for both
systems. The asymptotic outage results follow the exact outage probability at
medium and high SNRs which validates their accuracy. Further, at high SNRs,
the results of combining and switching have the same slope for both systems
which justifies the same diversity order for the combining and switching
methods. The diversity order of the FSO link is
$\text{min}\left(\frac{\xi_{\text{F}_{n}}^{2}}{\kappa},\frac{\alpha_{\text{F}_{n}}}{\kappa},\frac{\beta_{\text{F}_{n}}}{\kappa}\right)$
and the diversity order of the THz link is
$\min\left(\frac{\xi_{\text{T}_{n}}^{2}}{2},\frac{\alpha_{n}N_{\text{r}}\mu_{n}}{2}\right)$.
Thus, with multi-hop parallel THz/FSO deployment, the diversity order of
System-1 for both combining and switching is given by (III-B2). Similarly, the
diversity order of System-2 for both the combining/switching methods is
obtained by (51).

##  VII Conclusions

In this work, we consider multi-hop and mesh hybrid THz/FSO-based backhaul
networks with different deployments of the THz and the FSO links, and analyze
their performance for the cases with both IAB and non-IAB based communication
setups. For the analysis, we consider both the combining and switching methods
between the THz/FSO links, and the impact of various practical considerations
like atmospheric attenuation/path-loss, pointing/misalignment error,
atmospheric turbulence/fading, number of antennas, number of UEs, number of
hops, and the threshold data-rates are shown on the performance of considered
systems. From the results, we observe that the parallel deployment of the
THz/FSO links in the backhaul with proper system configuration improves the
overall performance of the communication system. Also, the diversity order for
combining and switching methods remains the same. However, combining provides
a considerable performance gain, compared to switching. Further, the increased
jitter standard deviation drastically reduces the performance of the THz, the
FSO, and consequently the backhaul THz/FSO link. This significantly
deteriorates the E2E performance. Then, an increase in number of hops
increases the coverage area, at the cost of traffic congestion and E2E
latency. Finally, mesh networks enable multiple routes and improve the E2E
performance significantly.

##  Appendix A Proof of Outage Probability for Combining Method

The (26) can be solved as

$\displaystyle\mathcal{P}^{\text{ST}}_{\gamma_{\text{I}_{n}}}(\gamma_{\text{th},n})=\int_{0}^{\gamma_{\text{th},n}}f_{\gamma_{\text{F}_{n}}}(\gamma)F_{\gamma_{\text{T}_{n}}}(\gamma_{\text{th},n}-\gamma)\text{d}\gamma,$
$\displaystyle=\frac{\xi^{2}_{\text{F}_{n}}}{\kappa\Gamma(\alpha_{\text{F}_{n}})\Gamma(\beta_{\text{F}_{n}})}\frac{\mathbb{C}^{n}_{1}}{\xi_{\text{T}_{n}}^{2}}$
$\displaystyle\times\int_{0}^{\gamma_{\text{th},n}}\gamma^{-1}\text{G}^{3,0}_{1,3}\left[\frac{\alpha_{\text{F}_{n}}\beta_{\text{F}_{n}}}{A_{0{\text{F}_{n}}}\delta_{{\kappa_{n}}}^{\frac{1}{\kappa}}}\gamma^{\frac{1}{\kappa}}\Big{|}^{\xi^{2}_{\text{F}_{n}}+1}_{\xi^{2}_{\text{F}_{n}},\alpha_{\text{F}_{n}},\beta_{\text{F}_{n}}}\right]$
$\displaystyle\times\Bigg{[}\left(\frac{\gamma_{\text{th},n}-\gamma}{{\bar{\gamma}}_{\text{T}_{n}}}\right)^{\frac{\xi_{\text{T}_{n}}^{2}}{2}}\Gamma\left(\mathbb{C}^{n}_{2},\frac{\mathbb{C}^{n}_{3}}{{{\bar{\gamma}}_{\text{T}_{n}}}^{\frac{\alpha_{n}}{2}}}\left(\gamma_{\text{th},n}-\gamma\right)^{\frac{\alpha_{n}}{2}}\right)$
$\displaystyle+{\left(\mathbb{C}^{n}_{3}\right)}^{-\frac{\xi_{\text{T}_{n}}^{2}}{\alpha_{n}}}\Upsilon\left(N_{\text{r}}\mu_{n},\frac{\mathbb{C}^{n}_{3}}{{{\bar{\gamma}}_{\text{T}_{n}}}^{\frac{\alpha_{n}}{2}}}\left(\gamma_{\text{th},n}-\gamma\right)^{\frac{\alpha_{n}}{2}}\right)\Bigg{]}\text{d}\gamma.$
(A.1)

Series expansion of the lower incomplete gamma function is given as [53,
(8.354.2)]

$\displaystyle\Gamma\left(a,bx\right)=\Gamma\left(a\right)-$
$\displaystyle\sum_{z_{1}=0}^{\infty}\frac{(-1)^{z_{1}}}{z_{1}!\left(a+z_{1}\right)}\left(bx\right)^{a+z_{1}},$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\text{for }\left[a\neq
0,-1,-2,...\right].$ (A.2)

Substituting (A) in (A), we obtain

$\displaystyle\mathcal{P}^{\text{ST}}_{\gamma_{\text{I}_{n}}}(\gamma_{\text{th},n})=\frac{\xi^{2}_{\text{F}_{n}}}{\kappa\Gamma(\alpha_{\text{F}_{n}})\Gamma(\beta_{\text{F}_{n}})}\frac{\mathbb{C}^{n}_{1}}{\xi_{\text{T}_{n}}^{2}}$
$\displaystyle\times\int_{0}^{\gamma_{\text{th},n}}\gamma^{-1}\text{G}^{3,0}_{1,3}\left[\frac{\alpha_{\text{F}_{n}}\beta_{\text{F}_{n}}}{A_{0{\text{F}_{n}}}\delta_{{\kappa_{n}}}^{\frac{1}{\kappa}}}\gamma^{\frac{1}{\kappa}}\Big{|}^{\xi^{2}_{\text{F}_{n}}+1}_{\xi^{2}_{\text{F}_{n}},\alpha_{\text{F}_{n}},\beta_{\text{F}_{n}}}\right]$
$\displaystyle\times\Bigg{[}\left(\frac{\gamma_{\text{th},n}-\gamma}{{\bar{\gamma}}_{\text{T}_{n}}}\right)^{\frac{\xi_{\text{T}_{n}}^{2}}{2}}\bigg{(}\Gamma\left(\mathbb{C}^{n}_{2}\right)-\sum_{z_{1}=0}^{\infty}\frac{(-1)^{z_{1}}}{z_{1}!\left(\mathbb{C}^{n}_{2}+z_{1}\right)}$
$\displaystyle\times\left(\frac{\mathbb{C}^{n}_{3}}{{{\bar{\gamma}}_{\text{T}_{n}}}^{\frac{\alpha_{n}}{2}}}\right)^{\mathbb{C}^{n}_{2}+z_{1}}\left(\gamma_{\text{th},n}-\gamma\right)^{\frac{\alpha_{n}}{2}\left(\mathbb{C}^{n}_{2}+z_{1}\right)}\bigg{)}$
$\displaystyle+{\left(\mathbb{C}^{n}_{3}\right)}^{-\frac{\xi_{\text{T}_{n}}^{2}}{\alpha_{n}}}\sum_{z_{1}=0}^{\infty}\frac{(-1)^{z_{1}}}{z_{1}!\left(N_{\text{r}}\mu_{n}+z_{1}\right)}\left(\frac{\mathbb{C}^{n}_{3}}{{{\bar{\gamma}}_{\text{T}_{n}}}^{\frac{\alpha_{n}}{2}}}\right)^{N_{\text{r}}\mu_{n}+z_{1}}$
$\displaystyle\times\left(\gamma_{\text{th},n}-\gamma\right)^{\frac{\alpha_{n}}{2}\left(N_{\text{r}}\mu_{n}+z_{1}\right)}\Bigg{]}\text{d}\gamma.$
(A.3)

This can be written as

$\displaystyle\mathbb{I}_{1}(\gamma_{\text{th},n})-\mathbb{I}_{2}(\gamma_{\text{th},n})+\mathbb{I}_{3}(\gamma_{\text{th},n}),$
(A.4)

where identities $\mathbb{I}_{1}(\gamma_{\text{th},n})$,
$\mathbb{I}_{2}(\gamma_{\text{th},n})$, and
$\mathbb{I}_{3}(\gamma_{\text{th},n})$ are shown in (A), at the top of the
next page.

$\displaystyle\mathbb{I}_{1}(\gamma_{\text{th},n})$
$\displaystyle=\frac{\xi^{2}_{\text{F}_{n}}}{\kappa\Gamma(\alpha_{\text{F}_{n}})\Gamma(\beta_{\text{F}_{n}})}\frac{\mathbb{C}^{n}_{1}\Gamma\left(\mathbb{C}^{n}_{2}\right)}{\xi_{\text{T}_{n}}^{2}{{\bar{\gamma}}_{\text{T}_{n}}}^{\frac{\xi^{2}_{\text{T}}}{2}}}\int_{0}^{\gamma_{\text{th},n}}\gamma^{-1}\left(\gamma_{\text{th},n}-\gamma\right)^{\frac{\xi^{2}_{\text{T}_{n}}}{2}}\text{G}^{3,0}_{1,3}\left[\frac{\alpha_{\text{F}_{n}}\beta_{\text{F}_{n}}}{A_{0\text{F}_{n}}\delta_{\kappa_{n}}^{1/\kappa}}\gamma^{\frac{1}{\kappa}}\Big{|}^{\xi^{2}_{\text{F}_{n}}+1}_{\xi^{2}_{\text{F}_{n}},\alpha_{\text{F}_{n}},\beta_{\text{F}_{n}}}\right]\text{d}\gamma,$
$\displaystyle\mathbb{I}_{2}(\gamma_{\text{th},n})$
$\displaystyle=\frac{\xi^{2}_{\text{F}_{n}}}{\kappa\Gamma(\alpha_{\text{F}_{n}})\Gamma(\beta_{\text{F}_{n}})}\frac{\mathbb{C}^{n}_{1}}{\xi_{\text{T}_{n}}^{2}{{\bar{\gamma}}_{\text{T}_{n}}}^{\frac{\xi^{2}_{\text{T}}}{2}}}\sum_{z_{1}=0}^{\infty}\frac{(-1)^{z_{1}}}{z_{1}!\left(\mathbb{C}^{n}_{2}+z_{1}\right)}\left(\frac{\mathbb{C}^{n}_{3}}{{{\bar{\gamma}}_{\text{T}_{n}}}^{\frac{\alpha_{n}}{2}}}\right)^{\mathbb{C}^{n}_{2}+z_{1}}$
$\displaystyle\times\int_{0}^{\gamma_{\text{th},n}}\gamma^{-1}\left(\gamma_{\text{th},n}-\gamma\right)^{\frac{\xi^{2}_{\text{T}}}{2}+\frac{\alpha_{n}}{2}\left(\mathbb{C}^{n}_{2}+z_{1}\right)}\text{G}^{3,0}_{1,3}\left[\frac{\alpha_{\text{F}_{n}}\beta_{\text{F}_{n}}}{A_{0\text{F}_{n}}\delta_{\kappa_{n}}^{1/\kappa}}\gamma^{\frac{1}{\kappa}}\Big{|}^{\xi^{2}_{\text{F}_{n}}+1}_{\xi^{2}_{\text{F}_{n}},\alpha_{\text{F}_{n}},\beta_{\text{F}_{n}}}\right]\text{d}\gamma,$
$\displaystyle\mathbb{I}_{3}(\gamma_{\text{th},n})$
$\displaystyle=\frac{\xi^{2}_{\text{F}_{n}}}{\kappa\Gamma(\alpha_{\text{F}_{n}})\Gamma(\beta_{\text{F}_{n}})}\frac{\mathbb{C}^{n}_{1}{\left(\mathbb{C}^{n}_{3}\right)}^{-\frac{\xi_{\text{T}_{n}}^{2}}{\alpha}}}{\xi_{\text{T}_{n}}^{2}}\sum_{z_{2}=0}^{\infty}\frac{(-1)^{z_{2}}}{z_{2}!\left(N_{\text{r}}\mu+z_{2}\right)}\left(\frac{\mathbb{C}^{n}_{3}}{{{\bar{\gamma}}_{\text{T}_{n}}}^{\frac{\alpha_{n}}{2}}}\right)^{N_{\text{r}}\mu+z_{2}}$
$\displaystyle\times\int_{0}^{\gamma_{\text{th},n}}\gamma^{-1}\left(\gamma_{\text{th},n}-\gamma\right)^{\frac{\xi^{2}_{\text{T}}}{2}+\frac{\alpha_{n}}{2}\left(N_{\text{r}}\mu+z_{2}\right)}\text{G}^{3,0}_{1,3}\left[\frac{\alpha_{\text{F}_{n}}\beta_{\text{F}_{n}}}{A_{0\text{F}_{n}}\delta_{\kappa_{n}}^{1/\kappa}}\gamma^{\frac{1}{\kappa}}\Big{|}^{\xi^{2}_{\text{F}_{n}}+1}_{\xi^{2}_{\text{F}_{n}},\alpha_{\text{F}_{n}},\beta_{\text{F}_{n}}}\right]\text{d}\gamma.$
(A.5)

Applying [48, (07.34.21.0085.01)] with mathematical computations, closed-form
solutions of the identities $\mathbb{I}_{1}(\gamma_{\text{th},n})$,
$\mathbb{I}_{2}(\gamma_{\text{th},n})$, and
$\mathbb{I}_{3}(\gamma_{\text{th},n})$ are obtained as given in (III-A1).

##  Appendix B Proof of Diversity Analysis for Combining Method

Substituting the high SNR PDF expression of the FSO link (12) and the high SNR
CDF expression of the THz link (II-B3) in (32), we obtain

$\displaystyle\mathcal{P}^{\text{ST,A}}_{\gamma_{\text{I}_{n}}}(\gamma_{\text{th},n})\approx\frac{\xi^{2}_{\text{F}_{n}}}{\kappa\Gamma(\alpha_{\text{F}_{n}})\Gamma(\beta_{\text{F}_{n}})}$
$\displaystyle\times\sum_{p=1}^{3}\frac{\overset{3}{\underset{\underset{q\neq
p}{q=1}}{\prod}}\Gamma(\Psi_{4,q}-\Psi_{4,p})}{\overset{1}{\underset{q=1}{\prod}}\Gamma(\Psi_{3,q}-\Psi_{4,p})}\left(\frac{\alpha_{\text{F}_{n}}\beta_{\text{F}_{n}}}{A_{0{\text{F}_{n}}}}\right)^{\Psi_{4,p}}\left(\frac{1}{\delta_{\kappa_{n}}}\right)^{\frac{\Psi_{4,p}}{\kappa}}$
$\displaystyle\times\int_{0}^{\gamma_{\text{th},n}}\gamma^{\frac{\Psi_{4,p}}{\kappa}-1}\bigg{[}\frac{\mathbb{C}^{n}_{1}\Gamma(\mathbb{C}^{n}_{2})}{\xi_{\text{T}_{n}}^{2}{{\bar{\gamma}}_{\text{T}_{n}}}^{\frac{\xi_{\text{T}_{n}}^{2}}{2}}}\left(\gamma_{\text{th},n}-\gamma\right)^{\frac{\xi_{\text{T}_{n}}^{2}}{2}}$
$\displaystyle-\frac{\mathbb{C}^{n}_{1}{\left(\mathbb{C}^{n}_{3}\right)}^{\mathbb{C}^{n}_{2}}}{\mathbb{C}^{n}_{2}\left(\alpha_{n}N_{\text{r}}\mu_{n}\right){\bar{\gamma}}_{\text{T}_{n}}^{\frac{\alpha_{n}N_{\text{r}}\mu_{n}}{2}}}\left(\gamma_{\text{th},n}-\gamma\right)^{\frac{\alpha_{n}N_{\text{r}}\mu_{n}}{2}}\bigg{]}\text{d}\gamma,$
(B.1)

Applying the identity [53, (3.191.1)] in (B) with mathematical computations,
we obtain

$\displaystyle\mathcal{P}^{\text{ST,A}}_{\gamma_{\text{I}_{n}}}(\gamma_{\text{th},n})\approx\frac{\xi^{2}_{\text{F}_{n}}}{\kappa\Gamma(\alpha_{\text{F}_{n}})\Gamma(\beta_{\text{F}_{n}})}$
$\displaystyle\times\sum_{p=1}^{3}\frac{\overset{3}{\underset{\underset{q\neq
p}{q=1}}{\prod}}\Gamma(\Psi_{4,q}-\Psi_{4,p})}{\overset{1}{\underset{q=1}{\prod}}\Gamma(\Psi_{3,q}-\Psi_{4,p})}\left(\frac{\alpha_{\text{F}_{n}}\beta_{\text{F}_{n}}}{A_{0{\text{F}_{n}}}}\right)^{\Psi_{4,p}}\left(\frac{1}{\delta_{\kappa_{n}}}\right)^{\frac{\Psi_{4,p}}{\kappa}}$
$\displaystyle\times\bigg{[}\frac{\mathbb{C}^{n}_{1}\Gamma(\mathbb{C}^{n}_{2})}{\xi_{\text{T}_{n}}^{2}{{\bar{\gamma}}_{\text{T}_{n}}}^{\frac{\xi_{\text{T}_{n}}^{2}}{2}}}\left(\gamma_{\text{th},n}\right)^{\frac{\xi_{\text{T}_{n}}^{2}}{2}+\frac{\Psi_{4,p}}{\kappa}}\text{B}\left(\frac{\xi_{\text{T}_{n}}^{2}}{2}+1,\frac{\Psi_{4,p}}{\kappa}\right)$
$\displaystyle-\frac{\mathbb{C}^{n}_{1}{\left(\mathbb{C}^{n}_{3}\right)}^{\mathbb{C}^{n}_{2}}}{\mathbb{C}^{n}_{2}\left(\alpha_{n}N_{\text{r}}\mu_{n}\right){\bar{\gamma}}_{\text{T}_{n}}^{\frac{\alpha_{n}N_{\text{r}}\mu_{n}}{2}}}\left(\gamma_{\text{th},n}\right)^{\frac{\alpha_{n}N_{\text{r}}\mu_{n}}{2}+\frac{\Psi_{4,p}}{\kappa}}$
$\displaystyle\times\text{B}\left(\frac{\alpha_{n}N_{\text{r}}\mu_{n}}{2}+1,\frac{\Psi_{4,p}}{\kappa}\right)\bigg{]},$
(B.2)

where $\text{B}\left(a,b\right)=\int_{0}^{1}z^{a-1}(1-z)^{b-1}\text{d}z$
represents the Beta function.

At high SNRs, the transmit SNR $\bar{\gamma}\rightarrow\infty$. Consequently,
the average received SNR of the THz and the FSO links also tend to infinity,
i.e, $\bar{\gamma}_{\text{T}_{n}},\delta_{\kappa_{n}}\rightarrow\infty$.
Hence, considering $\bar{\gamma}_{\text{T}_{n}}=\Delta_{1}\bar{\gamma}$ and
$\delta_{\kappa_{n}}=\Delta_{2}\bar{\gamma}$, where $\Delta_{1}$ and
$\Delta_{2}$ are constants, (B) can be finally simplified as (III-A2).

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