# Coverage Analysis of Broadcast Networks with Users Having Heterogeneous
Content/Advertisement Preferences

Kanchan K. Chaurasia, Reena Sahu, Abhishek K. Gupta  The authors are with the
Department of Electrical Engineering, Indian Institute of Technology Kanpur,
Kanpur, India 208016. Email<EMAIL_ADDRESS>
###### Abstract

This work is focused on the system-level performance of a broadcast network.
Since all transmitters in a broadcast network transmit the identical signal,
received signals from multiple transmitters can be combined to improve system
performance. We develop a stochastic geometry based analytical framework to
derive the coverage of a typical receiver. We show that there may exist an
optimal connectivity radius that maximizes the rate coverage. Our analysis
includes the fact that users may have their individual content/advertisement
preferences. We assume that there are multiple classes of users with each user
class prefers a particular type of content/advertisements and the users will
pay the network only when then can see content aligned with their interest.
The operator may choose to transmit multiple contents simultaneously to cater
more users’ interests to increase its revenue. We present revenue models to
study the impact of the number of contents on the operator revenue. We
consider two scenarios for users’ distribution- one where users’ interest
depends on their geographical location and the one where it doesn’t. With the
help of numerical results and analysis, we show the impact of various
parameters including content granularity, connectivity radius, and rate
threshold and present important design insights.

###### Index Terms:

Stochastic geometry, Broadcast networks, Coverage.

##  I Introduction

Broadcasting networks provide society with various services including TV
communication, delivery of critical information and alerts, general
entertainment, and educational services and thus have been a key wireless
technology. With the recent advancement in wireless technologies and handheld
electronic devices including smartphone and tablets, the use of broadcasting
services has been extended to include many modern applications including
delivery of traffic information to vehicles in vehicle-to-infrastructure
networks, advertisement industry and mobile TV services. Many broadcasting
standards have been recently proposed including the digital video broadcast-
terrestrial standard (DVB-T2), the advanced television systems committee
standard (ATSC 3.0), and the DVB-next generation handheld standard (DVB-NGH)
to assist delivering of TV broadcasting services to mobile devices [1][2][3].
The advent of digital broadcasting has lead to a significant increase in the
demand for multimedia services for handheld devices including mobile TV, live
video streaming, and on-demand video in the last decade [1]. From these
applications’ perspective, broadcasting based multimedia services can provide
better data rate and performance compared to the uni-cast cellular network
based mobile TV. Note that in a cellular network where the desired data is
transmitted to each user via orthogonal resources, users may suffer from
spectral congestion in regions with high density due to limit bandwidth
resulting in the performance degradation. But, in a broadcasting network
providing multimedia services, all transmitters transmit identical data to all
users and hence, do not require orthogonal resource. In these networks, each
transmitter can use the complete spectrum to serve their users, and hence,
these are also called single frequency networks (SFN). Due to this, users may
experience a better quality of service.

###  I-A Related Work

Given increasing demand of broadcasting services, it is very interesting to
analyze the broadcast networks in terms of signal-to-noise-ratio (SINR) and
achievable data rate to understand their limitations and potential to meet
these demands. There have been some recent works in the system-level analysis
of broadcast networks. In [4], the authors evaluated the blocking probability
for users accessing a network delivering mobile TV services over a hybrid
broadcast unicast communication. In [5], the authors studied a cellular
network with uni-cast and multicast-broadcast deployments. However, these
works didn’t include the effect of transmitters’ locations in the evaluations
which is required for the system-level analysis of broadcast networks.
Stochastic geometry framework can be utilized to analyze wireless networks
from system level perspective [6, 7, 8]. Stochastic geometry based models have
been validated for various types of networks including cellular networks,and
ad hoc networks [9, 6, 10, 11]. In [12], the authors describe the analytical
approach to calculate the coverage probability of a hybrid broadcast and uni-
cast network, however, the authors have only considered a single broadcast
transmitter along with many uni-cast transmitters. In our past work [13], we
have considered a broadcast network with multiple broadcasting transmitters to
compute the coverage performance of users. However, the work assumed a static
connectivity region around the user where transmitters need to be located to
be able to serve the users. As shown in this paper, this connectivity region
is of variable size depending on the location of the first closest
transmitter. To the best of our knowledge, there exists no other past work
which analyzes the SINR and rate performance of a broadcast network with
multiple broadcasting transmitters which is one of the main focuses of this
paper.

Another important metric to evaluate broadcast networks is the revenue earned
by the network operator. In a broadcast network, the revenue is generated
either from subscribers as network access fees for viewing content of their
choice or from advertisers to show their advertisements to interested
subscribers. With digital broadcast, subscriptions and user-targeted
advertisements added a new dimension in the revenue. Due to advancements in
technologies over the past few decades, the advertising has become more user
targeted and location-adaptive which can be planned according to the user
demographics and their preferences to improve network revenue. In [14], the
authors studied location-based mobile marketing and advertising to show the
positive interest of mobile consumers in receiving relevant promotions. It is
intuitive that a targeted and localized content will have a better engagement
factor. It is interesting to analyze the network revenue earned from the users
with their preference dependent on their choices and geographical location. As
far we know, there does not exist any past work that analyses the network
revenue of a broadcast network with subscribers having preferences for content
and advertisement which is another focus of this paper.

###  I-B Contributions

In this paper, we derive an analytical framework to evaluate the performance
of a broadcast network with multiple broadcasting transmitters with users
having content preference. We also present a revenue model to quantify the
network revenue earned by the network operator. In particular, the
contributions of this paper are as follows:

  1. 1.

We consider a broadcast network with multiple transmitters. Since all
transmitters in a broadcast network are transmitting the same signal, received
signals from multiple transmitters from a certain connectivity region around
the user can be combined to improve the coverage at this user. Using tools
from stochastic geometry, we derive the expression for SINR coverage and rate
coverage of a typical receiver located at the origin. Due to the contribution
in the desired signal power from multiple transmitters, the analysis is
significantly different and difficult than their cellular counterpart. Our
main contribution lies in developing the framework and deriving techniques to
evaluate the analytical expressions of SINR and rate coverage. We show that
this connectivity region depends on network bandwidth.

  2. 2.

We present some numerical results to validate our analysis and present design
insights. We show the impact of connectivity region size, path-loss exponents,
and the network density on the SINR and rate coverage. We also find that there
exists an optimal size of connectivity region that maximizes the rate
coverage.

  3. 3.

In this paper, we also include the fact that users may have their individual
content or advertisement preferences. We assume that there are multiple
classes of users with each class of users prefers a particular type of
content/advertisements and the users will pay the network only when then can
see a particular content of their interest. We assume that one unit of revenue
comes to the network from a particular class of users if every user of this
class can see the content as per the preference of this class. We study the
revenue thus obtained by the network from users. The broadcast operator may
choose to transmit multiple contents simultaneously to cater more users’
interest to increase its revenue. However, given the limited resources, the
network can cater only to few classes and this capability depends on how these
user classes are distributed spatially. There are two scenarios considered for
users’ distribution. In one scenario, users’ interest depend on their
geographical position in the network and in the second scenario it does not.
We calculate the analytical expression for SINR coverage and rate coverage at
a typical user and evaluate the total revenue. We present many important
design insights via numerical results.

Notation: Let $\mathcal{B}({\mathbf{x}},r)$ denote the ball of radius $r$ with
center at ${\mathbf{x}}$. $\|{\mathbf{x}}\|$ denotes the norm of the vector
${\mathbf{x}}$ and $\|{\mathbf{x}}_{i}\|=r_{i}$ denotes the random distance of
BBS located at ${{\mathbf{x}}_{i}}$. Let $\mathbf{o}$ denote the origin.
$\mathsf{B}\left(x,y;z\right)$ is the incomplete Beta function which is
defined as

$\mathsf{B}\left(x,y;z\right)=\int_{0}^{z}u^{x-1}(1-u)^{y-1}\mathrm{d}u.$

Let $c$ denote the speed of EM waves in the media. $\mathsf{A}^{\complement}$
denotes the complement of set $\mathsf{A}$.

##  II System Model

In this paper, we consider a broadcast network with multiple broadcasting base
stations (BBSs), deployed in the 2D region $\mathsf{R}=\mathbb{R}^{2}$. The
considered system model is as follows:

Figure 1: Illustration of system model of a broadcast network. A typical user
is considered at the origin. ${X_{0}}$ is the distance of the nearest BS from
the typical user. The 2D region
$\mathcal{B}(\mathbf{o},{X_{0}}+R_{\mathrm{s}})$ denotes the connectivity
region of the user.

###  II-A Network Model

The location of BBSs can be modeled as a homogeneous Poisson point process
$\Phi=\\{\mathbf{X}_{i}\in\mathbb{R}^{2}\\}$ with density $\lambda$ in the
region $\mathsf{R}$ (See Fig. 1). Let $R_{e}=1/\sqrt{\lambda\pi}$ which
represents the cell radius of an average cell. The subscribers (users) of the
broadcasting service are assumed to form a stationary point process. We assume
a typical user located at the origin $\mathbf{o}$. Consider each BS is
operating in the same frequency band with the transmission bandwidth $W$. Let
$T_{\mathrm{s}}$ is the symbol time of the transmitted symbol which is
inversely proportional to the bandwidth $W$. Assume the transmit power of each
BBS be $p_{\mathsf{t~{}\\!}}$ and all devices are equipped with a single
isotropic transmit antenna. The analysis can be extended for finite networks
by taking $\mathsf{R}=\mathcal{B}(\mathbf{o},R)$ with a finite $R$.

###  II-B Channel Model

We assume the standard path-loss model. Hence, the received signal power from
the $i$th BBS at the typical user at origin is given as

$\displaystyle P_{i}$
$\displaystyle=p_{\mathsf{t~{}\\!}}a\beta_{i}{\|\mathbf{X}_{i}\|}^{-\alpha},$
(1)

where $X_{i}=\|\mathbf{X}_{i}\|$ denotes the random distance of this BBS from
the typical user. Here, $\alpha$ is the path-loss exponent and $a$ is near-
field gain which depends on the propagation environment. $\beta_{i}$ denotes
the fading between the $i^{th}$ BBS and the user. We assume Rayleigh fading,
i.e. , $\beta_{i}\sim\mathrm{Exp}(1)$ for tractability.

###  II-C Serving Signal and Interference Model

In a broadcast system, multiple BBSs may transmit the same data at the one
frequency band (as suggested by the name SFN). Therefore, at the receiver end,
it can be seen as a single transmission with multi-path propagation and
signals transmitted from multiple BBSs can be combined at the user. However,
since the signals from different BBSs are delayed according to time delays
dependent on their distance, some of these signals may be delayed
significantly and may overlap with the next transmission slots. Therefore,
only those signals that have delay within a certain limit can be combined to
successfully decode the received symbol [15]. The rest BBSs contribute to the
ISI (inter-symbol interference) which can be significant depending on the BBSs
density.

Let $\mathbf{X}_{0}$ denotes the nearest serving BBS. The probability density
function of the distance $X_{0}=\|\mathbf{X}_{0}\|$ to the nearest BS from the
user is given as[7]

$\displaystyle f_{X_{0}}(u)=2\pi\lambda ue^{-\pi\lambda
u^{2}}\mathbbm{1}\left({u\geq 0}\right).$ (2)

The time taken by the signal to reach from the $i^{th}$ BBS located at
$\mathbf{X}_{i}$ to the typical user at $\mathbf{o}$ be $T_{i}={X_{i}}/c$. In
particular, $T_{0}$ denotes the time taken by signal to reach from
$\mathbf{X}_{0}$ to a typical user at $\mathbf{o}$. Let the propagation delay
of transmitted signal from $i^{th}$ BBS compared to the nearest serving BBS is
$\Delta_{i}=T_{i}-T_{0}$.

We assume that the receiver design allows the maximum delay of $\delta
T_{\mathrm{s}}$ for the received signals to be combined at the user where
$\delta\in[0\ 1]$ is a design parameter. This means that the received signal
from the $i^{th}$ BBS may contribute in the serving signal power if
$\Delta_{i}\leq\delta T_{\mathrm{s}}$. This condition is equivalent to the
condition $\|\mathbf{X}_{i}\|-\|\mathbf{X}_{0}\|\leq
R_{\mathrm{s}}\stackrel{{\scriptstyle\Delta}}{{=}}T_{\mathrm{s}}\delta c$ on
the BBSs location $\mathbf{X}_{i}$. In other words, this means that all the
BBSs that are located in the 2D region
$\\{\mathbf{X}:\|\mathbf{X}\|\leq\|\mathbf{X}_{0}\|+R_{\mathrm{s}}\\}=\mathcal{B}(\mathbf{o},{X_{0}}+R_{\mathrm{s}})$
can contribute to the serving signal at the typical receiver at origin
$\mathbf{o}$. We term this region
$\mathcal{B}(\mathbf{o},{X_{0}}+R_{\mathrm{s}})$ as the connectivity region
for the user and ${X_{0}}+R_{\mathrm{s}}$ can be termed as the connectivity
radius. Let $\overline{m}^{2}={\lambda\pi R_{\mathrm{s}}^{2}}$ denote the mean
number of BBSs in this connectivity radius.

On the other hand, all the BBSs located outside
$\mathcal{B}(\mathbf{o},{X_{0}}+R_{\mathrm{s}})$ i.e. all the BBSs with
${X_{i}}\geq{X_{0}}+R_{\mathrm{s}}$ will contribute to the interference power
even when they are transmitting the same data as their signal will be delayed
beyond the specified limit.

###  II-D Modeling Content Preferences of Users

In this paper, we also include the fact that users may have their individual
content or advertisement preferences. We assume that there are
$N_{\mathrm{c}}$ classes of users. Here, $N_{\mathrm{c}}$ is termed
content/advertisement granularity. Each class of users prefers a particular
type of content/ advertisements. We assume that the users will pay the network
only when then can see a particular content of their interest. Each class
consists of some quanta of users. For simplicity, we assume that each class
has the same number of users, however, the presented framework can be
trivially extended to include user classes with unequal sizes. We assume that
one unit of revenue comes to the network from a particular class of users if
every user of this class can see the content as per the preference of this
class. Given the limited resources, the network can cater only to few classes
and this capability depends on how these user classes are distributed
spatially. We will consider two types of users class distributions over the
geographical space. We will also discuss a revenue model to characterize the
network’s revenue to help us understand optimal scheduling policies for the
two scenarios.

##  III Coverage Analysis for Common Content Transmission

We first start with the scenario that all users seek the same content, hence,
all BBSs are transmitting the same content to everyone. Examples include
systems transmitting emergency information, or traffic data which is common to
every user. In this section, we will derive the SINR and rate coverage
probability for a typical user at the origin $\mathbf{o}$ for such system.

###  III-A SINR

Since all BBSs are transmitting the same content, all BBSs located inside the
connectivity region $\mathcal{B}(0,X_{0}+\mathbb{R}_{\mathrm{s}})$ contribute
to the signal power. Therefore, the desired received signal power for the
typical user at origin is given as

$\displaystyle S^{\prime}$
$\displaystyle=p_{\mathsf{t~{}\\!}}a\beta_{0}{\|\mathbf{X}_{0}\|}^{-\alpha}+\sum_{\mathbf{X}_{i}\in\Phi\cap\mathcal{B}(0,{X_{0}}+R_{\mathrm{s}})\setminus\mathbf{X}_{0}}p_{\mathsf{t~{}\\!}}a\beta_{i}{\|\mathbf{X}_{i}\|}^{-\alpha}.$
(3)

Similarly, the total interference can be given as

$\displaystyle I^{\prime}$
$\displaystyle=\sum_{\mathbf{X}_{j}\in\Phi\cap\mathcal{B}(0,\,X_{0}+R_{\mathrm{s}})^{\complement}}p_{\mathsf{t~{}\\!}}a\beta_{j}{\|\mathbf{X}_{j}\|}^{-\alpha}.$
(4)

The signal-to-interference-plus-noise ratio (SINR) at the typical receiver is
given as

$\displaystyle\mathtt{SINR}$
$\displaystyle=\frac{S^{\prime}}{I^{\prime}+N}=\frac{\beta_{0}{{X_{0}}}^{-\alpha}+\sum_{\mathbf{X}_{i}\in\Phi\cap\mathcal{B}(0,{X_{0}}+R_{\mathrm{s}})\setminus\mathbf{X}_{0}}\beta_{i}{{X_{i}}}^{-\alpha}}{\sum_{\mathbf{X}_{j}\in\Phi\cap\mathcal{B}(0,{X_{0}}+R_{\mathrm{s}})^{\complement}}\beta_{j}{{X_{j}}}^{-\alpha}+\sigma^{2}}.$
(5)

Here, $\sigma^{2}$ is the normalized noise power given as
$\sigma^{2}=N/(p_{\mathsf{t~{}\\!}}a)$ where $N$ is the noise power. Similarly
normalized desired received signal power and interference are denoted by $S$
and $I$ which are given as $S=S^{\prime}/(p_{\mathsf{t~{}\\!}}a)$ and
$I=I^{\prime}/(p_{\mathsf{t~{}\\!}}a)$. Hence, the SINR is equal to

$\displaystyle\mathtt{SINR}$ $\displaystyle=\frac{S}{I+\sigma^{2}}.$ (6)

Let $K=\sigma^{2}/R_{e}^{-\alpha}$ which represents the SNR at cell edge of an
average cell.

###  III-B SINR Coverage Probability

The SINR coverage probability $\mathrm{p_{c}}(\tau,\lambda)$ of a user is
defined as the probability that the SINR at the user is above the threshold
$\tau$ i.e.

$\displaystyle\mathrm{p_{c}}(\tau,\lambda)$
$\displaystyle=\mathbb{P}\left[\mathtt{SINR}>\tau\right]$ (7)

Using the conditioning on the nearest serving BBS’s location $\mathbf{X}_{0}$,
the SINR coverage for typical user at $\mathbf{o}$ is given as

$\displaystyle\mathrm{p_{c}}(\tau,\lambda)$
$\displaystyle=\mathbb{E}_{\mathbf{X}_{0}}\left[\mathbb{P}\left(\mathtt{SINR}>\tau\right)|\,\mathbf{X}_{0}\right]$
$\displaystyle=\mathbb{E}_{\mathbf{X}_{0}}\left[\mathbb{P}\left(\frac{S}{I+\sigma^{2}}>\tau\right)|\,\mathbf{X}_{0}\right]=\mathbb{E}_{\mathbf{X}_{0}}\left[\mathbb{P}\left(S>(I+\sigma^{2})\tau|\,\mathbf{X}_{0}\right)\right].$
(8)

Using the distribution of $\|\mathbf{X}_{0}\|=X_{0}$, the SINR coverage
probability can be further written as

$\displaystyle\mathrm{p_{c}}(\tau,\lambda)$
$\displaystyle=\mathbb{E}_{\mathbf{X}_{0}}\left[\mathbb{P}\left(S>\tau\left(I+N\right)\,|\,\mathbf{X}_{0}\right)\right]$
$\displaystyle=\int_{0}^{\infty}2\pi\lambda ue^{-\pi\lambda
u^{2}}\mathbb{P}\left(S>\tau\left(I+\sigma^{2}\right)\,|\,\|\mathbf{X}_{0}\|=u\right)\mathrm{d}u.$
(9)

To solve the inner term further, we will use Gil Pelaez’s Lemma [16] which
states that the CDF of a random variable $Y$ can be written in term of its
Laplace transform $\mathcal{L}_{Y}(t)$ as

$\displaystyle F_{Y}(s)=\mathbb{P}\left[Y\leq
s\right]=\frac{1}{2}-\frac{1}{\pi}\int_{0}^{\infty}\frac{1}{t}\mathsf{Im}\left[e^{-jts}\mathcal{L}_{Y}(-jt)\right]\mathrm{d}t.$
(10)

Using this Lemma, we get,

$\displaystyle\mathbb{P}\left(S>(I+\sigma^{2})\tau|\,\mathbf{X}_{0}\right)$
$\displaystyle=\mathbb{E}_{I|\,\mathbf{X}_{0}}\left[\frac{1}{2}+\frac{1}{\pi}\int_{0}^{\infty}\frac{1}{t}\mathsf{Im}\left[e^{-jt\tau(I+\sigma^{2})}\mathcal{L}_{S}(-jt)\right]\mathrm{d}t\right]$
$\displaystyle=\frac{1}{2}+\frac{1}{\pi}\int_{0}^{\infty}\frac{1}{t}\mathsf{Im}\left[\mathbb{E}_{I|\,\mathbf{X}_{0}}\left[e^{-jt\tau(I+\sigma^{2})}\right]\mathcal{L}_{S|\,\mathbf{X}_{0}}(-jt)\right]\mathrm{d}t$
$\displaystyle=\frac{1}{2}+\frac{1}{\pi}\int_{0}^{\infty}\frac{1}{t}\mathsf{Im}\left[\mathcal{L}_{I|\,\mathbf{X}_{0}}(jt\tau)e^{-jt\tau\sigma^{2}}\mathcal{L}_{S|\,\mathbf{X}_{0}}(-jt)\right]\mathrm{d}t.$
(11)

Now, using in (9), the SINR coverage probability is

$\displaystyle\mathrm{p_{c}}(\tau,\lambda)$
$\displaystyle=\frac{1}{2}+\frac{1}{\pi}\int_{0}^{\infty}\int_{0}^{\infty}2\pi\lambda
ue^{-\pi\lambda
u^{2}}\frac{1}{t}\mathsf{Im}\left[\mathcal{L}_{I|\,\mathbf{X}_{0}}(jt\tau)e^{-jt\tau\sigma^{2}}\mathcal{L}_{S|\,\mathbf{X}_{0}}(-jt)\right]\mathrm{d}t\
\mathrm{d}u.$ (12)

Here, $\mathcal{L}_{I|\,\mathbf{X}_{0}}(.)$ and
$\mathcal{L}_{S|\,\mathbf{X}_{0}}(.)$ are the Laplace transform of the sum
interference $I$ and of the desired received signal power $S$ respectively
which are given in the following Lemma.

######  Lemma 1.

The Laplace transforms of the desired signal power and the sum interference at
the receiver located at origin $\mathbf{o}$ are given as

$\displaystyle\mathcal{L}_{S\,|\,\mathbf{X}_{0}}(s)$
$\displaystyle=\frac{1}{1+sX_{0}^{-\alpha}}\exp\left(-2\pi\lambda\int_{X_{0}}^{X_{0}+R_{\mathrm{s}}}\frac{sr^{-\alpha}}{1+sr^{-\alpha}}r\,\mathrm{d}r\right)$
(13) $\displaystyle\mathcal{L}_{I}(s)$
$\displaystyle=\exp{\left(-2\pi\lambda\int_{X_{0}+R_{\mathrm{s}}}^{\infty}\frac{sr^{-\alpha}}{1+sr^{-\alpha}}r\,\mathrm{d}r\right)}$
(14)

###### Proof.

See Appendix A. ∎

Using Lemma 1 in (12), we can get the SINR coverage which is given in Theorem
1.

######  Theorem 1.

The probability of the SINR coverage for the user located at the origin in a
broadcast network with $\lambda$ density of BBSs, is given as

$\displaystyle\mathrm{p_{c}}(\tau,\lambda)=$
$\displaystyle\frac{1}{2}+\frac{1}{\pi}\int_{0}^{\infty}2\pi\lambda
ue^{-\pi\lambda
u^{2}}\int_{0}^{\infty}\frac{1}{t}\mathsf{Im}\left[{\color[rgb]{0,0,0}\frac{e^{-jt\tau\sigma^{2}}}{1-jtu^{-\alpha}}}\right.$
$\displaystyle\hskip
56.9055pt\times\left.\exp\left(-2\pi\lambda\left(\int_{u}^{u+R_{\mathrm{s}}}\frac{-jtr^{-\alpha}}{1-jtr^{-\alpha}}r\,\mathrm{d}r+\int_{u+R_{\mathrm{s}}}^{\infty}\frac{jt\tau
r^{-\alpha}}{1+jt\tau
r^{-\alpha}}r\,\mathrm{d}r\right)\right)\right]\mathrm{d}t\ \mathrm{d}u$
$\displaystyle=$
$\displaystyle\frac{1}{2}+\frac{1}{\pi}\int_{0}^{\infty}\int_{0}^{\infty}2v\frac{1}{s}\left[\frac{1}{1+s^{2}v^{-2\alpha}}\right]e^{-v^{2}}e^{-s^{\frac{2}{\alpha}}M_{d}(s,v)}$
$\displaystyle\times\left[sv^{-\alpha}\cos\left(s^{\frac{2}{\alpha}}N_{d}(s,v)+\tau
sK\right)-\sin\left(s^{\frac{2}{\alpha}}N_{d}(s,v)+\tau
sK\right)\right]\mathrm{d}v\;\mathrm{d}s$ (15)

where $M_{d}(t,\,u)$ and $N_{d}(t,\,u)$ are given as

$\displaystyle M_{d}(s,v)$
$\displaystyle=\frac{1}{\alpha}\left[Q\left(\frac{1}{\alpha},s^{2}(v+\overline{m})^{-2\alpha},s^{2}v.^{-2\alpha}\right)+\tau^{2/\alpha}Q\left(\frac{1}{\alpha},0,\tau^{2}s^{2}(v+\overline{m})^{-2\alpha}\right)\right]$
(16) $\displaystyle N_{d}(s,v)$
$\displaystyle=\frac{1}{\alpha}\left[-Q\left(\frac{1}{\alpha}+\frac{1}{2},s^{2}(v+\overline{m})^{-2\alpha},s^{2}v^{-2\alpha}\right)+\tau^{\frac{2}{\alpha}}Q\left(\frac{1}{\alpha}+\frac{1}{2},0,\tau^{2}s^{2}(v+\overline{m})^{-2\alpha}\right)\right]$
(17) with $\displaystyle
Q\left(z,a,b\right)=\mathsf{B}\left(z,-z+1;\frac{1}{1+a}\right)-\mathsf{B}\left(z,-z+1;\frac{1}{1+b}\right).$
(18)

###### Proof.

See Appendix B. ∎

Theorem 1 provides the SINR coverage in terms of two parameters: $K$ which
denotes the inverse of SNR at the cell edge and $\overline{m}^{2}$ which
denotes the mean number of BBSs in connectivity radius circle. Further we can
derive the following remarks.

######  Remark 1.

For interference limited scenario, $K=0$, which means the coverage probability
is a function of $\overline{m}$ only. In case $\overline{m}$ is fixed,
individual variation of $\lambda$ and $R_{\mathrm{s}}$ will not change the
coverage.

######  Remark 2.

For a broadcast network, an increase in the BBS density $\lambda$ improves
both the desired signal power and the interference power. However, due to
increase in number of serving BBSs due to increase in $\lambda$ which improve
the overall SINR coverage (which can also be seen in the numerical results).
This behavior is different than conventional cellular case. Recall that with
single serving BS density, the SINR in an interference-limited cellular
network does not get affected by any increase in the BS density which is known
as SINR invariance [6]. This can be shown from (15) by performing a
comparative study between $\lambda$ and $\lambda(1+\epsilon)$ with
$\epsilon<1$ for some $\lambda$.

######  Remark 3.

It can been observed that the SINR coverage probability increases with an
increases connectivity radius $R_{\mathrm{s}}$ as it increases the serving
power and decreases the interference.

###  III-C Rate Coverage Probability

The rate coverage probability of a user is defined as the probability that the
maximum achievable rate for the considered user is above some threshold $\rho$
i.e.

$\displaystyle\mathrm{r_{c}}(\rho)$
$\displaystyle=\mathbb{P}\left[\mathtt{Rate}>\rho\right].$

Note that the maximum achievable rate for the typical user is given as

$\displaystyle\mathtt{Rate}$ $\displaystyle=\xi W\log_{2}(1+\mathtt{SINR})$
(19)

where $\xi$ is some coefficient that denotes the spectrum utilization. $W$
denotes the system bandwidth available to each BBS. Hence, the rate coverage
for the typical user is

$\displaystyle\mathrm{r_{c}}(\rho)$
$\displaystyle=\mathbb{P}\left[\mathtt{Rate}>\rho\right]$
$\displaystyle=\mathbb{P}\left[\xi W\log_{2}(1+\mathtt{SINR})>\rho\right]$
$\displaystyle=\mathbb{P}\left[\mathtt{SINR}>2^{\rho/(\xi
W)}-1\right]=\mathrm{p_{c}}(2^{\rho/(\xi W)}-1)$ (20)

where $\mathrm{p_{c}}$ is the SINR coverage probability given in (15). Note
that the available bandwidth $W$ affects $T_{\mathrm{s}}$ and hence,
$R_{\mathrm{s}}$. If the BBSs use orthogonal frequency division multiplexing
(OFDM) for transmission with FFT size $N_{\mathrm{s}}$, then, $W$ is related
to $T_{\mathrm{s}}$ as

$W=\frac{N_{\mathrm{s}}}{T_{\mathrm{s}}}.$

Hence, the connectivity radius is

$\displaystyle R_{\mathrm{s}}=T_{\mathrm{s}}\delta
c=\frac{N_{\mathrm{s}}\delta c}{W}.$ (21)

Hence, an increase in the system bandwidth increases the pre-log factor in
(19), however, it also decreases the connectivity radius resulting in the
lower SINR coverage probability. Therefore, we can observe a trade-off on the
rate coverage with increasing bandwidth.

###  III-D Numerical Results

We now validate our results for SINR and rate coverage probabilities through
numerical simulation. We will also explore the impact of different parameters
on the coverage probabilities via numerical evaluations of derived expressions
to develop design insights. The default parameters are given in Table I which
are according to [17, 18].

TABLE I: Default parameters for numerical evaluations Parameters |  Numerical value |  Parameters |  Numerical value
---|---|---|---
$R_{\mathrm{s}}$ |  19.18 km |  $\lambda$ |  0.0014 BBSs / $\text{km}^{2}$
$N$ |  0 |  Path-loss $a,\ \alpha$ |  $1.6\times 10^{-3},\ 4$
$W$ |  8 MHz |  $p_{\mathsf{t~{}\\!}}$ |  20 dB
$N_{\mathrm{s}}\delta$ |  512 |  Coefficients |  $a=10^{-3},\ \xi=1$
$N_{\mathrm{c}}$ |  15 |  Simulation radius |  $800$ km
Figure 2: SINR coverage vs. SINR threshold ($\tau$) for various BBS density
$\lambda$ in a broadcast system with multiple BBSs. Here, the solid lines
represent the analytical expression and markers represent simulation values.
The parameters are according to Table I. It can be seen that the analysis
matches with simulation results.  Figure 3: Rate coverage vs. rate threshold
($\rho$) for various BBS density $\lambda$ in a broadcast system with multiple
BBSs. Here, the solid lines represent the analytical expression and markers
represent simulation values. The parameters are according to Table I. It can
be seen that the analysis matches with simulation results.

Validation of results: Fig. 2 shows the SINR coverage probability vs SINR
threshold ($\tau$) for different values of BBSs density ($\lambda$). Here, the
solid lines represent the analytical expression and markers represent
simulation values. It can be seen that the analysis matches with simulation
results which establishes the validity of the presented analytical results.
From Fig. 2, it can be seen that SINR coverage increases with an increase in
the BBS density consistent with Remark 2. Similarly, Fig. 3 shows the rate
coverage probability vs rate threshold ($\rho$) for different values of
$\lambda$. It can be seen that the rate coverage increases with an increase in
$\lambda$ which is expected due to the SINR coverage behavior with $\lambda$.

Figure 4: SINR coverage vs. $R_{\mathrm{s}}$ for different values of SINR
threshold ($\tau$) in a broadcast network. Here, bandwidth varies with
$R_{\mathrm{s}}$ according to (21) with maximum value at 80 MHz. The rest of
the parameters are according to Table I. It is observed that the SINR improves
with $R_{\mathrm{s}}$.  Figure 5:  Rate coverage vs. $R_{\mathrm{s}}$ for
different values of rate threshold ($\rho$ in Mbps) in a broadcast network.
Here, bandwidth varies with $R_{\mathrm{s}}$ according to (21) with maximum
value at 50 MHz. The rest of the parameters are according to Table I. A trade-
off is seen in rate the with varying $R_{\mathrm{s}}$.

Impact of connectivity radius on SINR and rate coverage: Fig. 4 shows the
variation of SINR coverage with the connectivity radius ($R_{\mathrm{s}}$) for
different values of target SINR threshold. It is observed that the SINR
coverage increases with $R_{\mathrm{s}}$. It can be justified as
$R_{\mathrm{s}}$ increases the number of contributing BBSs increases and the
number of interfering BBSs decreases.

Fig. 5 shows the variation of the rate coverage with $R_{\mathrm{s}}$. We can
observe that with $R_{\mathrm{s}}$, the rate coverage first increases up to a
certain value of $R_{\mathrm{s}}$ and afterwards starts decreasing again. From
(21), increase in $R_{\mathrm{s}}$ requires a decrease in the bandwidth $W$ in
order to allow larger symbol time. This causes a trade-off in the system
performance. As bandwidth is a pre-log factor in the rate expression, it has a
negative and larger impact on the rate coverage. Hence, beyond a certain value
of $R_{\mathrm{s}}$, the impact of decrease in $W$ dominates the increase in
SINR caused by increased $R_{\mathrm{s}}$ which results in the decrease in the
rate. Due to the same reasons, there may exist an optimal $R_{\mathrm{s}}$
that maximizes the rate coverage. The knowledge of optimal $R_{\mathrm{s}}$
can be helpful in designing the broadcast network.

Impact of network density on SINR coverage: Fig. 6(a) and (b) show the
variation of SINR and rate coverage with the network density $\lambda$ for
different value of target SINR threshold and connectivity radius (which is
achieved by changing bandwidth while keeping other parameters the same as
Table I). The coverage while ignoring the noise is also shown. From Fig. 5(a),
it can be seen that densification of the network helps in SINR coverage. When
the BBS density is very small, network is noise limited. As $\lambda$
increases, BBSs comes closer to the user improving SINR coverage while SIR
coverage remains constant. After a certain $\lambda$, the increase in
$\lambda$ reduces the interference also. Hence, both SIR and SINR coverage
improves. At high value of $\lambda$, coverage becomes 1 as all dominant BBSs
provide serving power. The behavior of SIR coverage is similar to as seen in
networks with dual-slope pathloss [19].

Figure 6:  Coverage vs. BBS density $\lambda$ for different values of
$R_{\mathrm{s}}$ in a broadcast network. Here, rest of the parameters are
according to Table I. (a) SINR coverage. Dashed lines indicate SIR coverage
while ignoring noise. (b) Rate coverage. Dashed lines indicate rate coverage
ignoring the noise.

##  IV Scenario I: Coverage Analysis for Networks with Users having a High
Level of Spatial Heterogeneity in Content Preference

We now extend the system model to include networks with users having their
individual content or advertisement preferences. In this section, we consider
the first scenario where there is high level of spatial heterogeneity in
users. This means that all classes of users are present in any region. Given
the limitation of resources, network selects $n$ classes of users and shows
$n$ contents (one for each class) at any point of time. Here, $n$ is a design
parameter decided by the broadcast network. Since the user classes are
spatially inseparable, each BBSs should transmit to the same $n$ contents to
improve SINR coverage. We have assumed OFDM based transmission where a BBS
transmits the $n$ number of contents on orthogonal resources.

###  IV-A SINR and SINR Coverage

To improve coverage, the network can use the same bands for a particular
content across all BBSs. Since all the BBSs are transmitting the same data in
a band, the SINR for a typical user is the same as given in (5). Similarly, in
this case, the SINR coverage probability of a typical user is the same as
given in Theorem 1.

###  IV-B Rate Coverage

Now, the available resources are divided among $n$ contents. If the total
available bandwidth is $W$, the bandwidth available for each content is $W/n$.
The instantaneous achievable rate for a typical user located at origin, for
each content is

$\displaystyle\mathtt{Rate}$
$\displaystyle=\xi\frac{W}{n}\log_{2}(1+\mathtt{SINR}).$ (22)

From (20), the rate coverage probability is given as:

$\displaystyle\mathrm{r_{c}}(\rho)$
$\displaystyle=\mathbb{P}\left[\mathtt{SINR}>2^{n\rho/(\xi
W)}-1\right]=\mathrm{p_{c}}(2^{n\rho/(\xi W)}-1)$ (23)

where $\mathrm{p_{c}}$ is given in (15).

###  IV-C Network Revenue

Let $\rho$ denote the minimum rate required for a user to be able to view the
content. Then, the rate coverage $\mathrm{r_{c}}$ at $\rho$ denotes the
fraction of users that are able to view this content. Therefore,
$\mathrm{r}_{c}$ unit of revenue will be earned by the network from a
particular class, since only $\mathrm{r_{c}}$ fraction of users can watch it.
Therefore, the network’s total revenue $\mathrm{R}_{\mathrm{n}}$ can be given
as:

$\displaystyle\mathrm{R}_{\mathrm{n}}$ $\displaystyle=n\mathrm{r}_{c}(\rho).$
(24)

Figure 7: Variation of total revenue with respect to allowed number of user
classes for different rate threshold $\rho$ (Mbps) (which is a proxy for
content quality requested). Content granularity is $N_{\mathrm{c}}=15$. Other
parameters are according to Table I. It is observed that an optimal value of
$n$ can provide the maximum revenue to the network which depends on the
content quality requested.

###  IV-D Numerical Results: Impact of $n$ on Total Revenue

Fig. 7 shows the variation of the network revenue $\mathrm{R}_{\mathrm{n}}$
with $n$ for a system with $N_{\mathrm{c}}=15$. We can observe that initially,
the revenue increases with an increase in $n$ up to a certain value, and
thereafter, starts decreasing. This can be justified in the following way. If
$n$ increases, the following two effects take place– (1) more user classes are
served, causing a linear increase in $\mathrm{R}_{\mathrm{n}}$, and (2)
available bandwidth $W/n$ for each content/advertisement decreases causing the
rate coverage to drop. As a combined effect dictated by (24), the revenue is
optimal at a particular value of $n$. However, this behavior also depends on
the target rate threshold. At a higher rate threshold,
$\mathrm{R}_{\mathrm{n}}$ decreases with $n$, showing $n=1$ as the optimal
choice. This implies that the optimal number of user classes that can be
served, depends on the quality of the content. If the quality requested is
high, then it may be better to serve less user classes, while more user
classes can be served when the quality requested is lower.

##  V Scenario II: Coverage Analysis for Networks with Users having Spatially
Separated Classes for Content Preference

The second scenario we consider corresponds to the case where user classes are
spatially separated. For tractability, we assume that coverage area of each
BBS comprises of users from a single class. Thus, users of different classes
are spatially separated. Let $S_{i}$ denote the class of the $i$th BBS located
at $\mathbf{X}_{i}$ which is a uniform discrete random variable with PMF given
as

$\displaystyle p_{S_{i}}(k)$
$\displaystyle=\frac{1}{N_{\mathrm{c}}}\mathbbm{1}\left({1\leq k\leq
N_{\mathrm{c}}}\right).$ (25)

We assume that $S_{i}$’s are independent of each other.

The network can cater to all user classes by letting each BBS to transmit
content according to the class of users lying in its coverage area. Note that
for a typical user, only those BBSs that transmit the same content, can
contribute to the serving signal power at this user. Therefore, this strategy
will reduce the number of serving BSs and hence reduce the SINR. On the other
hand, network can decide to cater only one user class by forcing all BBSs to
transmit only one content, will reduce the revenue as users of only one class
will receive their preferred content. It will be interesting to find the
optimal number of user-classes that can be catered by the network. As a
general problem, we consider that the network decides to cater $n$ user
classes out of total $N_{\mathrm{c}}$ classes. Let us denote the set of all
selected index by $\mathcal{M}$.

Let us denote the content transmitted by $i$th BS by $M_{i}$. If the BBS’s
user class is one of the $n$ selected classes (i.e. $S_{i}\in\mathcal{M}$), it
will transmit the content corresponding to its user class i.e. $M_{i}=S_{i}$.
If the BBS’s user class is not one of the selected classes, it will transmit
the content corresponding to a randomly selected user class to help boost its
signal strength.

Let us consider a typical user at $\mathbf{o}$. Without loss of generality,
assume that its user class is 1. The probability that its class is one of the
$n$ selected classes is $n/N_{\mathrm{c}}$. Let us condition on the fact that
it is one of the selected classes.

Let the tagged BS of this typical user transmits the content $M_{0}=1$. Then,
for the $i$th BBS, $M_{i}=M_{0}$ if

  1. 1.

$S_{i}=M_{0}$ which occurs with probability $\frac{1}{N_{\mathrm{c}}}$, or

  2. 2.

$S_{i}\notin\mathcal{M}$ and $S_{i}=M_{i}$ which occurs with probability
$\frac{N_{\mathrm{c}}-n}{N_{\mathrm{c}}}\cdot\frac{1}{n}$.

Therefore, the probability that the $i$th BBS is transmitting the content as
per the preference of this typical user is

$\displaystyle p$
$\displaystyle=\mathbb{P}\left[M_{i}=M_{0}\right]=\frac{1}{N_{\mathrm{c}}}+\frac{N_{\mathrm{c}}-n}{N_{\mathrm{c}}}\cdot\frac{1}{n}=\frac{1}{n}.$
(26)

###  V-A SINR

Now, note that the BBSs that are transmitting the same content as $M_{0}$ and
are located inside $\mathcal{B}(\mathbf{o},{X_{0}}+R_{\mathrm{s}})$ will
contribute to the desired signal power to the typical user at origin
$\mathbf{o}$. Therefore, the desired signal power for the typical user at
origin is given as

$\displaystyle S=$ $\displaystyle
p_{\mathsf{t~{}\\!}}a\beta_{0}{{X_{0}}}^{-\alpha}+\sum_{\mathbf{X}_{i}\in\Phi\cap\mathcal{B}(0,X_{0}+R_{\mathrm{s}})\setminus\mathbf{X}_{0}}p_{\mathsf{t~{}\\!}}a\beta_{i}{{X_{i}}}^{-\alpha}\mathbbm{1}\left({M_{i}=M_{0}}\right).$
(27)

Similarly, the interference for the typical user is caused by the BBSs that
are either located outside $\mathcal{B}(\mathbf{o},{u}+R_{\mathrm{s}})$ or
located inside the $\mathcal{B}(\mathbf{o},{X_{0}}+R_{\mathrm{s}})$ but
transmitting a different content than $M_{0}$. Hence, the total interference
is given as

$\displaystyle I=$
$\displaystyle\sum_{\mathbf{X}_{i}\in\Phi\cap\mathcal{B}(0,X_{0}+R_{\mathrm{s}})\setminus\mathbf{X}_{0}}p_{\mathsf{t~{}\\!}}a\beta_{i}{{X_{i}}}^{-\alpha}\mathbbm{1}\left({M_{i}\neq
M_{0}}\right)+\sum_{\mathbf{X}_{j}\in\Phi\cap\mathcal{B}(0,{X_{0}}+R_{\mathrm{s}})^{\complement}}p_{\mathsf{t~{}\\!}}a\beta_{j}{{X_{j}}}^{-\alpha}.$
(28)

Now, the SINR for this user is given as

$\displaystyle\mathtt{SINR}=\frac{S}{I+N}$
$\displaystyle=\frac{\beta_{0}{\|\mathbf{X}_{0}\|}^{-\alpha}+\sum_{\mathbf{X}_{i}\in\Phi\cap\mathcal{B}(0,u+R_{\mathrm{s}})\setminus\mathbf{X}_{0}}\beta_{i}{\|\mathbf{X}_{i}\|}^{-\alpha}\mathbbm{1}\left({M_{i}=M_{0}}\right)}{\sum_{\mathbf{X}_{i}\in\Phi\cap\mathcal{B}(0,u+R_{\mathrm{s}})\setminus\mathbf{X}_{0}}\beta_{i}{\|\mathbf{X}_{i}\|}^{-\alpha}\mathbbm{1}\left({M_{i}\neq
M_{0}}\right)+\sum_{\mathbf{X}_{j}\in\Phi\cap\mathcal{B}(0,u+R_{\mathrm{s}})^{\complement}}\beta_{j}{\|\mathbf{X}_{j}\|}^{-\alpha}+\sigma^{2}}.$
(29)

Here, $\sigma^{2}$ is the normalized noise power which is given as
$\sigma^{2}=N/(p_{\mathsf{t~{}\\!}}a)$ where $N$ is the noise power.

###  V-B SINR Coverage Probability

We now calculate the SINR coverage for a typical user. Similar to Section
III-B, the SINR coverage probability is given as

$\displaystyle\mathrm{p_{c}}(\tau,\lambda)$
$\displaystyle=\frac{1}{2}+\frac{1}{\pi}\int_{0}^{\infty}\int_{0}^{\infty}2\pi\lambda
ue^{-\pi\lambda
u^{2}}\frac{1}{t}\mathsf{Im}\left[\mathcal{L}_{I|\,\mathbf{X}_{0}}(jt\tau)e^{-jt\tau\sigma^{2}}\mathcal{L}_{S|\,\mathbf{X}_{0}}(-jt)\right]\mathrm{d}t\
\mathrm{d}u$ (30)

where $S$ and $I$ are given in (27) and (28).

######  Lemma 2.

Conditioned on the location of the closest serving BBS, the Laplace transforms
of the desired signal power and the sum interference at the receiver are given
as

$\displaystyle\mathcal{L}_{S|\,\mathbf{X}_{0}}(s)=\frac{1}{1+su^{-\alpha}}\exp\left(-2\pi\lambda
p\int_{X_{0}}^{X_{0}+R_{\mathrm{s}}}\frac{sr^{-\alpha}}{1+sr^{-\alpha}}r\mathrm{d}r\right)$
(31)
$\displaystyle\mathcal{L}_{I|\mathbf{X}_{0}}(s)=\exp\left(-2\pi\lambda(1-p)\int_{X_{0}}^{X_{0}+R_{\mathrm{s}}}\frac{sr^{-\alpha}}{1+sr^{-\alpha}}r\mathrm{d}r-2\pi\lambda\int_{X_{0}+R_{\mathrm{s}}}^{\infty}\frac{sr^{-\alpha}}{1+sr^{-\alpha}}r\mathrm{d}r\right).$
(32)

###### Proof.

See Appendix C. ∎

Using Lemma 2 and (30) we can calculate the SINR coverage which is given in
Theorem 2.

######  Theorem 2.

The probability of SINR coverage for the user located at the origin in a
broadcast network with $\lambda$ density of BBSs, is given as:

$\displaystyle\mathrm{p_{c}}(\tau,\lambda)=$
$\displaystyle\frac{1}{2}+\frac{1}{\pi}\int_{0}^{\infty}2\pi\lambda
ue^{-\pi\lambda
u^{2}}\int_{0}^{\infty}\frac{1}{t}\mathsf{Im}\left[{\color[rgb]{0,0,0}\frac{e^{-jt\tau\sigma^{2}}}{1-jtu^{-\alpha}}}\right.$
$\displaystyle\indent\times\left.\exp\left(-2\pi\lambda\left(p\int_{u}^{u+R_{\mathrm{s}}}\frac{-jtr^{-\alpha}}{1-jtr^{-\alpha}}r\,\mathrm{d}r+(1-p)\int_{u}^{u+R_{\mathrm{s}}}\frac{jt\tau
r^{-\alpha}}{1+jt\tau r^{-\alpha}}r\,\mathrm{d}r\right.\right.\right.$
$\displaystyle\hskip
42.67912pt\left.\left.\left.+\int_{u+R_{\mathrm{s}}}^{\infty}\frac{jt\tau
r^{-\alpha}}{1+jt\tau
r^{-\alpha}}r\,\mathrm{d}r\right)\right)\right]\mathrm{d}t\mathrm{d}u$
$\displaystyle=$
$\displaystyle\frac{1}{2}+\frac{1}{\pi}\int_{0}^{\infty}\int_{0}^{\infty}2v\frac{1}{s}\left[\frac{1}{1+s^{2}v^{-2\alpha}}\right]e^{-v^{2}}e^{-s^{\frac{2}{\alpha}}M^{\prime}_{d}(s,v)}$
$\displaystyle\times\left[sv^{-\alpha}\cos\left(s^{\frac{2}{\alpha}}N^{\prime}_{d}(s,v)+\tau
sK\right)-\sin\left(s^{\frac{2}{\alpha}}N^{\prime}_{d}(s,v)+\tau
sK\right)\right]\mathrm{d}v\;\mathrm{d}s$ (33)

where $M^{\prime}_{d}(t,\,u)$ and $N^{\prime}_{d}(t,\,u)$ is given as

$\displaystyle M_{d}(s,v)=$
$\displaystyle\frac{1}{\alpha}\left[pQ\left(\frac{1}{\alpha},s^{2}(v+\overline{m})^{-2\alpha},s^{2}v^{-2\alpha}\right)+(1-p)\tau^{2/\alpha}Q\left(\frac{1}{\alpha},\tau^{2}s^{2}(v+\overline{m})^{-2\alpha},\tau^{2}s^{2}v^{-2\alpha}\right)\right.$
$\displaystyle\left.+\tau^{2/\alpha}Q\left(\frac{1}{\alpha},0,\tau^{2}s^{2}(v+\overline{m})^{-2\alpha}\right)\right]$
(34) $\displaystyle N_{d}(s,v)=$
$\displaystyle\frac{1}{\alpha}\left[-pQ\left(\frac{1}{\alpha}+\frac{1}{2},s^{2}(v+\overline{m})^{-2\alpha},s^{2}v^{-2\alpha}\right)\right.$
$\displaystyle\left.+(1-p)\tau^{2/\alpha}Q\left(\frac{1}{\alpha}+\frac{1}{2},\tau^{2}s^{2}(v+\overline{m})^{-2\alpha},\tau^{2}s^{2}v^{-2\alpha}\right)\right.$
$\displaystyle\left.+\tau^{2/\alpha}Q\left(\frac{1}{\alpha}+\frac{1}{2},0,\tau^{2}s^{2}(v+\overline{m})^{-2\alpha}\right)\right]$
(35)

###### Proof.

See Appendix D. ∎

###  V-C Rate Coverage

Since each BBS shows only one content, it can use the total available
bandwidth $W$. The instantaneous achievable rate for a typical user located at
origin, while receiving the content, is

$\displaystyle\mathtt{Rate}$ $\displaystyle=\xi{W}\log_{2}(1+\mathtt{SINR}).$
(36)

From (20), the rate coverage probability is given as:

$\displaystyle\mathrm{r_{c}}(\rho)$
$\displaystyle=\mathbb{P}\left[\mathtt{SINR}>2^{\rho/(\xi
W)}-1\right]=\mathrm{p_{c}}(2^{\rho/(\xi W)}-1)$ (37)

where $\mathrm{p_{c}}$ is given in Theorem 2.

###  V-D Network Revenue

Let $\rho$ denote the minimum rate required for a user to be able to view the
content. Then, the rate coverage $\mathrm{r_{c}}$ at $\rho$ denotes the
fraction of users that are able to view this content. Therefore,
$\mathrm{r}_{c}$ unit of revenue will be earned by the network from a
particular class, since only $\mathrm{r_{c}}$ fraction of users can watch it.
The probability that the typical user receives the content as per its
preference is $n/N_{\mathrm{c}}$ which is also the probability that the
network will receive revenue from this typical user. Similar to previous
sections, the total revenue can be computed as

$\displaystyle\mathrm{R}_{\mathrm{n}}$
$\displaystyle=\frac{n}{N_{\mathrm{c}}}\cdot\mathrm{r_{c}}(\rho),$ (38)

where the rate coverage $\mathrm{r_{c}}$ for the typical user is given by
(37).

###  V-E Numerical Results

We now present numerical results for the considered scenario II. The
parameters are stated in Table I.

Figure 8:  Variation of SINR and rate coverage probability with respect to
allowed number of user classes for different values of connectivity radius
($R_{\mathrm{s}}$ in km) at a typical user in a broadcast system with
geographically separated user classes. Content granularity is
$N_{\mathrm{c}}=15$, SINR threshold $\tau=10$ dB and rate threshold $\rho=15$
Mbps. Network density is $\lambda=.014$/km2. The bandwidth varies with
$R_{\mathrm{s}}$ according to (21) with maximum value at 80 MHz. Other
parameters are according to Table I. The coverage decreases with $n$ due to
increased interference at the typical user.

Impact of $n$ on SINR and rate coverage probability: Fig. 8(a) shows the
variation of SINR coverage with $n$ for different values of connectivity
radius $R_{\mathrm{s}}$. Here, SINR threshold $\tau=10$ dB and there are
$N_{\mathrm{c}}=15$ user classes. From Fig. 8, we observe that for a fix value
of $R_{\mathrm{s}}$, SINR coverage decreases with increase in $n$. This is due
to the fact that more BBSs interfere as $n$ increases. However, after a
certain $n$, the coverage doesn’t changes much with $n$. This is because
additional fraction of BBSs that interfere when $n$ increases by 1, is equal
to $\frac{n}{n+1}-\frac{n-1}{n}=\frac{1}{n(n+1)}$ which decreases very fast
with $n$. Therefore, after a certain $n$, there will not be a significant
increase in the interference, which makes the SINR constant with $n$.

It is also observed that the decrease in the SINR coverage probability is
faster when $R_{\mathrm{s}}$ is large. This can be justified as follows. First
note that $n$ only affects the BBSs that can either be a interferer or a
serving BBS, depending on the content they are showing. These BBSs lie in the
ring of $R_{\mathrm{s}}$ width denoting the region and their number
approximately scales as $\lambda\pi R_{\mathrm{s}}^{2}$. Note that this number
is large when $R_{\mathrm{s}}$ is large. When we allow BBSs to show more
advertisements/contents, a large number of these BBSs means that there is a
larger number of potential interferes. When $R_{\mathrm{s}}$ is small, there
are less number of these potential BBSs (or even 0), hence allowing more
advertisements doesn’t affect the coverage significantly. Fig. 8(b) shows the
variation of rate coverage with respect to $n$ for different size of
connectivity region. The rate coverage also follows similar behavior as SINR
coverage as described in (37).

Figure 9: Variation of the total network revenue with respect to allowed
number of user classes $n$ for different values of rate threshold $\rho$ (in
Mbps) with $R_{\mathrm{s}}=150\,\text{km}$ for a broadcast system with
geographically separated user classes. Content granularity is
$N_{\mathrm{c}}=15$. Here, bandwidth varies with $R_{\mathrm{s}}$ according to
(21). Other parameters are according to Table I.

Impact of $n$ on the total network revenue: Fig. 9 shows the impact of $n$ on
the total network revenue for different values of rate threshold. Increase in
$n$ means catering to more number of user classes. From Fig. 9, we can observe
that the revenue initially decreases and then, increases with $n$. The initial
decrease in the revenue seen from $n=1$ to $n=2$ for some configurations is
due to the decrease in rate coverage from $n=1$ to $n=2$, as observed in Fig.
8(b) which dominated the increase in the revenue generated due to catering to
an additional user class. However, this behavior may depend on the target rate
threshold and the value of $R_{\mathrm{s}}$.

Figure 10:  Variation of the total network revenue with respect to the
connectivity radius $R_{\mathrm{s}}$ (in km) for different values of $n$ with
$\rho=5\,\text{Mbps}$ for a broadcast system with geographically separated
user classes. Content granularity is $N_{\mathrm{c}}=15$. Here, bandwidth
varies with $R_{\mathrm{s}}$ according to (21) with maximum value at 50 MHz.
Other parameters are according to Table I.

Impact of $R_{\mathrm{s}}$ on the total network revenue Fig. 10 shows the
total revenue with respect to size of connectivity region $R_{\mathrm{s}}$ for
different $n$ with $\rho=10\,\text{Mbps}$ and $N_{\mathrm{c}}=15$. We can
observe that, the revenue decreases with increase in $R_{\mathrm{s}}$. Also
the revenue increases with $n$ for lower values of $R_{\mathrm{s}}$. For a
higher values of $R_{\mathrm{s}}$, the behavior with $n$ is not monotonic. The
revenue for $n\geq 1$ may fall below the revenue for $n=1$. As discussed
previously, the rate coverage decreases drastically from $n=1$ to $n=2$ for
large $R_{\mathrm{s}}$ which can dominate the increase in the revenue
generated due to catering to an additional user class.

##  VI Conclusions

In this paper, we presented an analytical framework for the system performance
of a broadcast network using stochastic geometry. Since all BBSs in the
broadcast network are transmitting the same signal, signals from multiple BSs
can be used to improve the coverage. We show that there exists a region such
that all BBSs lying in this region may contribute to the desired signal power.
We computed the SINR and rate coverage probability for a typical user located
at the origin. We validated our results using numerical analysis. Using these
results, we found that there exists an optimal region size which maximized the
rate coverage. When users consists of many user classes having heterogenous
content preference, network can schedule content to maximize its revenue. We
presented an analytical model of revenue thus obtained from users. The results
are validated through numerical analysis. We also present the variation of
total revenue with respect to various parameters including number of user
classes to be catered, size of connectivity region, and rate threshold. We
show how content quality also affects the network decision on variety of
content shown by the operator.

##  Appendix A Proof for Lemma 1

Using (4), the Laplace transform of the sum interference $I|\mathbf{X}_{0}$ is
given as

$\displaystyle\mathcal{L}_{I\,|\,\mathbf{X}_{0}}(s)=\mathbb{E}\left[e^{-sI}|\mathbf{X}_{0}\right]=\mathbb{E}\left[\exp{\left(-s\sum_{\mathbf{X}_{j}\in\Phi\cap\mathcal{B}(0,{X_{0}}+R_{\mathrm{s}})^{\complement}}\beta_{j}{\|\mathbf{X}_{j}\|}^{-\alpha}\right)}\right]$
$\displaystyle\overset{(a)}{=}\exp{\left(-\lambda\int_{\Phi\cap\mathcal{B}(0,{X_{0}}+R_{\mathrm{s}})^{\complement}}\left(1-\mathbb{E}_{\beta}\left[e^{-s\beta\|\mathbf{X}\|^{-\alpha}}\right]\right)\mathrm{d}{\mathbf{x}}\right)}$
$\displaystyle\stackrel{{\scriptstyle(b)}}{{=}}\exp{\left(-2\pi\lambda\int_{X_{0}+R_{\mathrm{s}}}^{\infty}\left(1-\mathbb{E}_{\beta}\left[e^{-s\beta
r^{-\alpha}}\right]\right)r\,\mathrm{d}r\right)}$ (39)

where (a) is due to the probability generating functional (PGFL) of
homogeneous PPP [7] and (b) is due to conversion to polar coordinates. Now,
since $\beta$’s are exponetially distributed, using their MGF, we get

$\displaystyle\mathcal{L}_{I}(s)=\exp{\left(-2\pi\lambda\int_{u+R_{\mathrm{s}}}^{\infty}\frac{sr^{-\alpha}}{1+sr^{-\alpha}}r\,\mathrm{d}r\right)}.$
(40)

Similarly, the Laplace transform of desired signal power conditioned on the
nearest is BS located at $\mathbf{X}_{0}$, is given as:

$\displaystyle\mathcal{L}_{S\,|\,\mathbf{X}_{0}}(s)=\mathbb{E}_{S|\,\mathbf{X}_{0}}\left[e^{-sS}|\mathbf{X}_{0}\right]$
$\displaystyle=$
$\displaystyle\mathbb{E}_{\beta_{i},\mathbf{X}_{i}}\left[\exp{\left(-s\left(\beta_{0}\|\mathbf{X}_{0}\|^{-\alpha}+\sum_{\mathbf{X}_{i}\in\Phi\cap\mathcal{B}{(\mathbf{o},X_{0}+R_{\mathrm{s}})}\setminus\mathbf{X}_{0}}\beta_{i}\|\mathbf{X}_{i}\|^{-\alpha}\right)\right)}\right]$
$\displaystyle=$
$\displaystyle\mathbb{E}_{\Phi\,|\,\mathbf{X}_{0}}\left[\mathbb{E}_{\beta_{0}|\,\mathbf{X}_{0}}\left[e^{-s\beta_{0}\|\mathbf{X}_{0}\|^{-\alpha}}\right]\right.$
$\displaystyle\times\left.\prod_{\mathbf{X}_{i}\in\Phi\cap\mathcal{B}{(\mathbf{o},X_{0}+R_{\mathrm{s}})}\setminus\mathbf{X}_{0}}\mathbb{E}_{\beta_{i}|X_{i},\mathbf{X}_{0}}\left[e^{-s\beta_{i}\|\mathbf{X}_{i}\|^{-\alpha})}\right]\right]$
(41)

Now, from Slivnyak theorem [7], we know that conditioned on $\mathbf{X}_{0}$,
$\\{\mathbf{X}_{i}:\mathbf{X}_{i}\in\Phi\\}\setminus\\{\mathbf{X}_{0}\\}$ is a
PPP with the same density. Therefore, using the PGFL of a PPP and noting that
$\beta_{i}$’s are exponential RVs, we get

$\displaystyle\mathcal{L}_{S\,|\,\mathbf{X}_{0}}(s)=$
$\displaystyle\frac{1}{1+s\|\mathbf{X}_{0}\|^{-\alpha}}\exp\left(-2\pi\lambda\int_{X_{0}}^{X_{0}+R_{\mathrm{s}}}\left(1-\frac{1}{1+sr^{-\alpha}}\right)r\mathrm{d}r\right)$
$\displaystyle=$
$\displaystyle\frac{1}{1+sX_{0}^{-\alpha}}\exp\left(-2\pi\lambda\int_{X_{0}}^{X_{0}+R_{\mathrm{s}}}\frac{sr^{-\alpha}}{1+sr^{-\alpha}}r\mathrm{d}r\right).$
(42)

##  Appendix B Proof of Theorem 1

Using Lemma 1, we get

$\displaystyle\mathcal{L}_{S|\,X_{0}=u}(-jt)$
$\displaystyle=\frac{1+jtu^{-\alpha}}{1+t^{2}u^{-2\alpha}}\exp\left(-2\pi\lambda\int_{u}^{u+R_{\mathrm{s}}}\frac{-jtr^{-\alpha}}{1-jtr^{-\alpha}}r\mathrm{d}r\right)$
(43) $\displaystyle\mathcal{L}_{I|\,X_{0}=u}(jt\tau)$
$\displaystyle=\exp{\left(-2\pi\lambda\int_{u+R_{\mathrm{s}}}^{\infty}\frac{jt\tau
r^{-\alpha}}{1+jt\tau r^{-\alpha}}r\,\mathrm{d}r\right)}.$ (44)

Substituting the above values in (11), we get

$\displaystyle\mathbb{P}\left(S>(I+\sigma^{2})\tau|\,X_{0}=u\right)$
$\displaystyle=\frac{1}{2}+\frac{1}{\pi}\int_{0}^{\infty}\frac{1}{t}\mathsf{Im}\left[\left(\frac{1+jtu^{-\alpha}}{1+t^{2}u^{-2\alpha}}\right)e^{-jt\tau\sigma^{2}}\times\right.$
$\displaystyle\hskip
14.22636pt\left.\exp\left(-2\pi\lambda\left(\int_{u+R_{\mathrm{s}}}^{\infty}\frac{jt\tau
r^{-\alpha}}{1+jt\tau
r^{-\alpha}}r\,\mathrm{d}r+\int_{u}^{u+R_{\mathrm{s}}}\frac{-jtr^{-\alpha}}{1-jtr^{-\alpha}}r\,\mathrm{d}r\right)\right)\right]\mathrm{d}t$
$\displaystyle=\frac{1}{2}+\frac{1}{\pi}\int_{0}^{\infty}\frac{1}{t}\mathsf{Im}\left[\left(\frac{1+jtu^{-\alpha}}{1+t^{2}u^{-2\alpha}}\right)e^{-jt\tau\sigma^{2}}\times\right.$
$\displaystyle\left.\exp\left(-2\pi\lambda\left(\int_{u+R_{\mathrm{s}}}^{\infty}\frac{t^{2}\tau^{2}r^{-2\alpha+1}}{1+t^{2}\tau^{2}r^{-2\alpha}}\mathrm{d}r+\int_{u}^{u+R_{\mathrm{s}}}\frac{t^{2}r^{-2\alpha+1}}{1+t^{2}r^{-2\alpha}}\mathrm{d}r\right.\right.\right.$
$\displaystyle\left.\left.\left.+j\int_{u+R_{\mathrm{s}}}^{\infty}\frac{t\tau
r^{-\alpha+1}}{1+t^{2}\tau^{2}r^{-2\alpha}}\mathrm{d}r\,-\,j\int_{u}^{u+R_{\mathrm{s}}}\frac{tr^{-\alpha+1}}{1+t^{2}r^{-2\alpha}}\mathrm{d}r\right)\right)\right]\mathrm{d}t$
(45)

where the last step is obtained using multiplication of conjugate terms. Now,
if we define

$\displaystyle
M(t,u)=2\alpha{t^{-2/\alpha}}\left[\int_{u}^{u+R_{\mathrm{s}}}\frac{t^{2}r^{-2\alpha+1}}{1+t^{2}r^{-2\alpha}}\mathrm{d}r+\,\int_{u+R_{\mathrm{s}}}^{\infty}\frac{t^{2}\tau^{2}r^{-2\alpha+1}}{1+t^{2}\tau^{2}r^{-2\alpha}}\mathrm{d}r\right]$
(46) $\displaystyle
N(t,u)=2\alpha{t^{-2/\alpha}}\left[\int_{u+R_{\mathrm{s}}}^{\infty}\frac{t\tau
r^{-\alpha+1}}{1+t^{2}\tau^{2}r^{-2\alpha}}\mathrm{d}r\,-\,\int_{u}^{u+R_{\mathrm{s}}}\frac{tr^{-\alpha+1}}{1+t^{2}r^{-2\alpha}}\mathrm{d}r\right]$
(47)

(45) can be written as

$\displaystyle\mathbb{P}\left(S>(I+\sigma^{2})\tau|\,X_{0}=u\right)$
$\displaystyle=\frac{1}{2}+\frac{1}{\pi}\int_{0}^{\infty}\frac{1}{t}\mathsf{Im}\left[\left(\frac{1+jtu^{-\alpha}}{1+t^{2}u^{-2\alpha}}\right)\right.$
$\displaystyle\times\left.\exp\left(-2\pi\lambda{\color[rgb]{0,0,0}t^{2/\alpha}}M(t,u)-j2\pi\lambda{\color[rgb]{0,0,0}t^{2/\alpha}}N(t,u)-jt\tau\sigma^{2}\right)\right]\mathrm{d}t.$
(48)

Now, with some trivial manipulations and substituting (48) in (9), we get

$\displaystyle\mathrm{p_{c}}(\tau,\lambda)=$
$\displaystyle\frac{1}{2}+\frac{1}{\pi}\int_{0}^{\infty}\int_{0}^{\infty}2\pi\lambda
ue^{-\pi\lambda
u^{2}}\cdot\frac{1}{t}\cdot\left[\frac{1}{1+t^{2}u^{-2\alpha}}\right]e^{-2\pi\lambda
t^{2/\alpha}M(t,\,u)/2\alpha}$
$\displaystyle\times\left[tu^{-\alpha}\cos{\left(\frac{\pi}{\alpha}\lambda{t^{2/\alpha}}N(t,\,u)+\,t\tau\sigma^{2}\right)}-\sin{\left(\frac{\pi}{\alpha}\lambda{t^{2/\alpha}}N(t,\,u)+\,t\tau\sigma^{2}\right)}\right]\mathrm{d}t\mathrm{d}u.$
(49)

Further, the forms of $M$ and $N$ can be simplified by trivial manipulations
and definition of incomplete Beta function in (46) and (47) to get

$\displaystyle
M(t,u)=Q\left(\frac{1}{\alpha},t^{2}(u+R_{\mathrm{s}})^{-2\alpha},t^{2}u^{-2\alpha}\right)+\tau^{2/\alpha}Q\left(\frac{1}{\alpha},0,t^{2}\tau^{2}(u+R_{\mathrm{s}})^{-2\alpha}\right),\text{
and }$ (50) $\displaystyle
N(t,\,u)=-Q\left(\frac{1}{\alpha}+\frac{1}{2},t^{2}(u+R_{\mathrm{s}})^{-2\alpha},t^{2}u^{-2\alpha}\right)+\tau^{2/\alpha}Q\left(\frac{1}{\alpha}+\frac{1}{2},0,t^{2}\tau^{2}(u+R_{\mathrm{s}})^{-2\alpha}\right)$
(51)

Now, we can substitute

$\displaystyle t$ $\displaystyle\rightarrow s/{(\lambda\pi)}^{\alpha/2}$
$\displaystyle u$ $\displaystyle\rightarrow v/\sqrt{\lambda\pi}\ $ (52)

in (49), (50) and (51) to get the desired result.

##  Appendix C Proof of Lemma 2

From (23), the Laplace transform of desired signal power $S$ conditioned that
the nearest BS located at $\mathbf{X}_{0}$ is given as:

$\displaystyle\mathcal{L}_{S|\,\mathbf{X}_{0}}(s)=\mathbb{E}_{S\,|\,\mathbf{X}_{0}}\left[e^{-sS}\right]$
$\displaystyle=\mathbb{E}_{\\{\beta_{i},\mathbf{X}_{i}\\}|\,\mathbf{X}_{0}}\left[\exp\left(-s\beta_{0}{X_{0}}^{-\alpha}-\sum_{\mathbf{X}_{i}\in\Phi\cap\mathcal{B}{(\mathbf{o},X_{0}+R_{\mathrm{s}})}\setminus\mathbf{X}_{0}}s\beta_{i}{X_{i}}^{-\alpha}\mathbbm{1}\left(M_{i}=M_{0}\right)\right)\right]$

which is similar to (41) except the fact that the summation in the last term
is over only those points that satisfy an additional condition $M_{i}=M_{0}$.
From the independent thinning theorem, these points also form a PPP with
density $\lambda\mathbb{P}\left[M_{i}=M_{0}\right]=\lambda p$. Now using the
PGFL of this PPP, we get

$\displaystyle\mathcal{L}_{S|\,\mathbf{X}_{0}}(s)$
$\displaystyle=\frac{1}{1+sX_{0}^{-\alpha}}\exp\left(-2\pi\lambda
p\int_{X_{0}}^{X_{0}+R_{\mathrm{s}}}\frac{sr^{-\alpha}}{1+sr^{-\alpha}}r\mathrm{d}r\right).$
(53)

Now, from (24), the Laplace transform of sum interference is

$\displaystyle\mathcal{L}_{I}(s)=\mathbb{E}_{I}\left[e^{-sI}\right]$
$\displaystyle=\mathbb{E}_{I}\left[\exp\left(-\sum_{\mathbf{X}_{i}\in\Phi\cap\mathcal{B}(0,X_{0}+R_{\mathrm{s}})\setminus\mathbf{X}_{0}}s\beta_{i}{{X_{i}}}^{-\alpha}\mathbbm{1}\left(M_{i}\neq
M_{0}\right)-\sum_{\mathbf{X}_{j}\in\Phi\cap\mathcal{B}(0,X_{0}+R_{\mathrm{s}})^{\complement}}s\beta_{j}{X_{j}}^{-\alpha}\right)\right]$
$\displaystyle\stackrel{{\scriptstyle(a)}}{{=}}\exp\left(-2\pi\lambda(1-p)\int_{X_{0}}^{X_{0}+R_{\mathrm{s}}}\left(1-\mathbb{E}_{\beta}\left[e^{-s\beta
r^{-\alpha}}\right]\right)r\mathrm{d}r-2\pi\lambda\int_{X_{0}+R_{\mathrm{s}}}^{\infty}\left(1-\mathbb{E}_{\beta}\left[e^{-s\beta
r^{-\alpha}}\right]\right)r\mathrm{d}r\right)$
$\displaystyle\overset{(b)}{=}\exp\left(-2\pi\lambda(1-p)\int_{u}^{X_{0}+R_{\mathrm{s}}}\frac{sr^{-\alpha}}{1+sr^{-\alpha}}r\mathrm{d}r-2\pi\lambda\int_{X_{0}+R_{\mathrm{s}}}^{\infty}\frac{sr^{-\alpha}}{1+sr^{-\alpha}}r\mathrm{d}r\right),$
(54)

where (a) is due to the probability generating functional of homogeneous PPP
and independent thinning theorem and (b) is due to MGF of exponentially
distributed $\beta_{i}$’s.

##  Appendix D Proof of Theorem 2

Using Lemma 2, we get

$\displaystyle\mathcal{L}_{S|\,X_{0}=u}(-jt)=\frac{1+jtu^{-\alpha}}{1+t^{2}u^{-2\alpha}}\exp\left(-2\pi\lambda
p\int_{u}^{u+R_{\mathrm{s}}}\frac{-jtr^{-\alpha}}{1-jtr^{-\alpha}}rdr\right)$
(55)
$\displaystyle\mathcal{L}_{I|\,X_{0}=u}(jt\tau)=\exp\left(-2\pi\lambda\left[(1-p)\int_{u}^{u+R_{\mathrm{s}}}\frac{jt\tau
r^{-\alpha}}{1+jt\tau
r^{-\alpha}}r\,\mathrm{d}r+\int_{u+R_{\mathrm{s}}}^{\infty}\frac{jt\tau
r^{-\alpha}}{1+jt\tau r^{-\alpha}}r\,\mathrm{d}r\right]\right).$ (56)

Substituting the above values in (11), we get

$\displaystyle\mathbb{P}\left(S>(I+\sigma^{2})\tau|\,X_{0}=u\right)$
$\displaystyle=$
$\displaystyle\frac{1}{2}+\frac{1}{\pi}\int_{0}^{\infty}\frac{1}{t}\mathsf{Im}\left[\left(\frac{1+jtu^{-\alpha}}{1+t^{2}u^{-2\alpha}}\right)e^{-jt\tau\sigma^{2}}\cdot\right.$
$\displaystyle\left.\exp\left(-2\pi\lambda\left(\int_{u+R_{\mathrm{s}}}^{\infty}\frac{jt\tau
r^{-\alpha}}{1+jt\tau
r^{-\alpha}}r\,\mathrm{d}r+(1-p)\int_{u}^{u+R_{\mathrm{s}}}\frac{jt\tau
r^{-\alpha}}{1+jt\tau r^{-\alpha}}r\,\mathrm{d}r\right.\right.\right.$
$\displaystyle\indent\left.\left.\left.+p\int_{u}^{u+R_{\mathrm{s}}}\frac{-jtr^{-\alpha}}{1-jtr^{-\alpha}}r\,\mathrm{d}r\right)\right)\right]\mathrm{d}t$
$\displaystyle=$
$\displaystyle\frac{1}{2}+\frac{1}{\pi}\int_{0}^{\infty}\frac{1}{t}\mathsf{Im}\left[\left(\frac{1+jtu^{-\alpha}}{1+t^{2}u^{-2\alpha}}\right)e^{-jt\tau\sigma^{2}}\cdot\right.$
$\displaystyle\left.\exp\left(-2\pi\lambda\left(\int_{u}^{u+R_{\mathrm{s}}}\frac{p\,t^{2}r^{-2\alpha+1}}{1+t^{2}r^{-2\alpha}}\mathrm{d}r\,\int_{u+R_{\mathrm{s}}}^{\infty}\frac{t^{2}\tau^{2}r^{-2\alpha+1}}{1+t^{2}\tau^{2}r^{-2\alpha}}\mathrm{d}r\,+\,\int_{u}^{u+R_{\mathrm{s}}}\frac{(1-p)t^{2}\tau^{2}r^{-2\alpha+1}}{1+t^{2}\tau^{2}r^{-2\alpha}}\mathrm{d}r\right.\right.\right.$
$\displaystyle\left.\left.\left.+\,j\left[\int_{u+R_{\mathrm{s}}}^{\infty}\frac{t\tau
r^{-\alpha+1}}{1+t^{2}\tau^{2}r^{-2\alpha}}\mathrm{d}r\,+\,\int_{u}^{u+R_{\mathrm{s}}}\frac{(1-p)\,t\tau
r^{-\alpha+1}}{1+t^{2}\tau^{2}r^{-2\alpha}}\mathrm{d}r\,-\,\int_{u}^{u+R_{\mathrm{s}}}\frac{p\,tr^{-\alpha+1}}{1+t^{2}r^{-2\alpha}}\mathrm{d}r\right]\right)\right)\right]\mathrm{d}t$
(57)

where the last step is obtained using multiplication of conjugate terms and
rearranging into the real and imaginary parts. Now, if we define

$\displaystyle M^{\prime}(t,u)=$ $\displaystyle 2{\alpha
t^{-\alpha/2}}\left[\int_{u}^{u+R_{\mathrm{s}}}\frac{p\,t^{2}r^{-2\alpha+1}}{1+t^{2}r^{-2\alpha}}\mathrm{d}r\right.$
$\displaystyle\left.+\int_{u+R_{\mathrm{s}}}^{R}\frac{t^{2}\tau^{2}r^{-2\alpha+1}}{1+t^{2}\tau^{2}r^{-2\alpha}}\mathrm{d}r+\int_{u}^{u+R_{\mathrm{s}}}\frac{(1-p)t^{2}\tau^{2}r^{-2\alpha+1}}{1+t^{2}\tau^{2}r^{-2\alpha}}\mathrm{d}r\right],$
(58) $\displaystyle N^{\prime}(t,u)=$ $\displaystyle
2\alpha{t^{-\alpha/2}}\left[\int_{u+R_{\mathrm{s}}}^{R}\frac{t\tau
r^{-\alpha+1}}{1+t^{2}\tau^{2}r^{-2\alpha}}\mathrm{d}r\right.$
$\displaystyle+\left.\int_{u}^{u+R_{\mathrm{s}}}\frac{(1-p)\,t\tau
r^{-\alpha+1}}{1+t^{2}\tau^{2}r^{-2\alpha}}\mathrm{d}r-\int_{u}^{u+R_{\mathrm{s}}}\frac{p\,tr^{-\alpha+1}}{1+t^{2}r^{-2\alpha}}\mathrm{d}r\right],$
(59)

(57) can be written as

$\displaystyle\mathbb{P}\left(S>(I+\sigma^{2})\tau|\,X_{0}=u\right)$
$\displaystyle=\frac{1}{2}+\frac{1}{\pi}\int_{0}^{\infty}\frac{1}{t}\mathsf{Im}\left[\left(\frac{1+jtu^{-\alpha}}{1+t^{2}u^{-2\alpha}}\right)\right.$
$\displaystyle\times\left.\exp\left(-2\pi\lambda{\color[rgb]{0,0,0}t^{2/\alpha}}M^{\prime}(t,u)-j2\pi\lambda{\color[rgb]{0,0,0}t^{2/\alpha}}N^{\prime}(t,u)-jt\tau\sigma^{2}\right)\right]\mathrm{d}t.$
(60)

Now, with some trivial manipulations and substituting (60) in (9), we get

$\displaystyle\mathrm{p_{c}}(\tau,\lambda)=$
$\displaystyle\frac{1}{2}+\frac{1}{\pi}\int_{0}^{\infty}\int_{0}^{\infty}2\pi\lambda
ue^{-\pi\lambda
u^{2}}\cdot\frac{1}{t}\cdot\left[\frac{1}{1+t^{2}u^{-2\alpha}}\right]\cdot
e^{-\frac{\pi}{\alpha}\lambda t^{2/\alpha}M^{\prime}(t,\,u)}$
$\displaystyle\times\left[tu^{-\alpha}\cos{\left(\frac{\pi}{\alpha}\lambda{t^{2/\alpha}}N^{\prime}(t,u)+\,t\tau\sigma^{2}\right)}-\sin{\left(\frac{\pi}{\alpha}\lambda{t^{2/\alpha}}N^{\prime}(t,u)+\,t\tau\sigma^{2}\right)}\right]\mathrm{d}t\mathrm{d}u.$
(61)

Further, the forms of $M$ and $N^{\prime\prime}$ can be simplified by trivial
manipulations and definition of incomplete Beta function in (58) and (59) to
get

$\displaystyle
M^{\prime}(t,u)=pQ\left(\frac{1}{\alpha},t^{2}(u+R_{\mathrm{s}})^{-2\alpha},t^{2}u^{-2\alpha}\right)$
$\displaystyle+\tau^{2/\alpha}(1-p)Q\left(\frac{1}{\alpha},(t\tau)^{2}(u+R_{\mathrm{s}})^{-2\alpha},(t\tau)^{2}u^{-2\alpha}\right)+\tau^{2/\alpha}Q\left(\frac{1}{\alpha},0,t^{2}\tau^{2}(u+R_{\mathrm{s}})^{-2\alpha}\right).$
(62)

and,

$\displaystyle
N^{\prime}(t,u)=-pQ\left(\frac{1}{\alpha}+\frac{1}{2},t^{2}(u+R_{\mathrm{s}})^{-2\alpha},t^{2}u^{-2\alpha}\right)$
$\displaystyle+\tau^{2/\alpha}(1-p)Q\left(\frac{1}{\alpha}+\frac{1}{2},(t\tau)^{2}(u+R_{\mathrm{s}})^{-2\alpha},(t\tau)^{2}u^{-2\alpha}\right)+\tau^{2/\alpha}Q\left(\frac{1}{\alpha}+\frac{1}{2},0,t^{2}\tau^{2}(u+R_{\mathrm{s}})^{-2\alpha}\right).$
(63)

Now, we can substitute

$\displaystyle t$ $\displaystyle\rightarrow s/{(\lambda\pi)}^{\alpha/2}$
$\displaystyle u$ $\displaystyle\rightarrow v/\sqrt{\lambda\pi}\ $ (64)

in (61), (62) and (63) to get the desired result.

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