# Outage Probability of Millimeter Wave Cellular Uplink with Truncated Power
Control

Oluwakayode Onireti, Lei Zhang, Ali Imran, and Muhammad Ali Imran,

###### Abstract

In this paper, using the stochastic geometry, we develop a tractable uplink
modeling framework for the outage probability of both the single and multi-
tier millimeter wave (mmWave) cellular networks. Each tier’s mmWave base
stations (BSs) are randomly located and they have particular spatial density,
antenna gain, receiver sensitivity, blockage parameter and pathloss exponents.
Our model takes account of the maximum power limitation and the per-user power
control. More specifically, each user, which could be in line-of-sight (LOS)
or non-LOS to its serving mmWave BS, controls its transmit power such that the
received signal power at its serving BS is equal to a predefined threshold.
Hence, a truncated channel inversion power control scheme is implemented for
the uplink of mmWave cellular networks. We derive closed-form expressions for
the signal-to-interference-and-noise-ratio (SINR) outage probability for the
uplink of both the single and multi-tier mmWave cellular networks.
Furthermore, we analyze the case with a dense network by utilizing the
simplified model, where the LOS region is approximated as a fixed LOS disc.
The results show that imposing a maximum power constraint on the user
significantly affects the SINR outage probability in the uplink of mmWave
cellular networks.

###### Index Terms:

mmWave, power control, stochastic geometry, truncated channel inversion,
uplink communication.

##  I Introduction

A fundamental requirement for the 5G-and-beyond mobile networks is the radical
increase in data rate. Recent studies have identified massive multiple-input-
multiple-output (MIMO), extreme network densification, and increased bandwidth
as the key technologies toward meeting this requirement [1]. The millimeter
wave (mmWave) frequencies (ranging from $30-300~{}\mathrm{GHz}$) offers a
large available bandwidth thus, making them attractive for the 5G mobile
networks [1, 2, 3]. Meanwhile, the mmWave band has long been considered ill-
suited for the cellular communication due to the excessive pathloss and the
poor penetration through materials such as concrete and water. Recent capacity
studies and survey measurement on mmWave technologies in [3, 4, 5, 6] have
shown its great promise for 5G urban small cell deployments. The recent
advances in low-power CMOS RF circuit and the smaller wavelength associated
with the band have further substantiated this promise. The later also makes it
viable to have more miniaturized antennas within the same physical area of the
transmitter and receiver [2, 6]. Further, with a large antenna array, the
mmWave network can apply beamforming at the transmit and receive sides to
provide array gain which compensates for the pathloss[7]. The directionality
gained from beamforming will lead to a reduction in interference [5]. Hence,
mmWave spectrum holds great potential for providing the high data rate
(Gigabits range) expected in the upcoming 5G cellular networks [8].

Modeling and analysis of cellular networks by using stochastic geometry have
recently received significant attention due to its high accuracy and
tractability. In this approach, the network topology is abstracted to a point
process for ease of modeling and analysis. Earlier works in this area were
mainly focused on the conventional ultra-high-frequency (UHF) cellular
networks [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]. In the pioneering work
on using stochastic geometry for cellular networks [9], it was shown that
stochastic model provides a lower bound to real cellular deployment. The work
in [9] was based on the downlink of cellular networks with the single slope
pathloss model. This has been extended by considering a multi-slope pathloss
model in [10, 11], the multi-tier cellular networks in [12, 13, 14], the
single tier uplink cellular networks in [15] and the multi-tier uplink
cellular networks in [16, 17, 18].

The stochastic geometry framework developed for the UHF networks do not
directly apply to the mmWave networks due to blockage effects that they suffer
from and the much different pathloss model. Furthermore, directional
beamforming is fundamental in the design of the mmWave cellular networks. The
authors in [8, 20, 21, 22, 23, 24] have analyzed the mmWave cellular networks
by using the stochastic geometry framework with the blockage effect, realistic
pathloss model and the beamforming gain incorporated in their model. In
particular, [8] pioneered the research work on the downlink of mmWave cellular
networks by leveraging on an earlier work in [25], which characterized the
blockage parameter by some random distribution. Furthermore, the proposed
analytical framework in [8] also captures the significant difference between
the non-line-of-sight (NLOS) and line-of-sight (LOS) pathloss characteristics.
The work in [8] has been extended to the downlink multi-tier mmWave cellular
networks in [20], base station (BS) cooperation in [21] and the uplink single-
tier mmWave cellular networks in [22, 23, 24].

The uplink analysis for both the conventional UHF cellular networks and the
mmWave cellular networks are deemed to be quite involved as compared to the
downlink analysis due to the per-user power control and the correlation among
the interferers [16]. The former is due to the fact that an interfering user
could even be closer to a reference mmWave BS than the user that is tagged to
the reference mmWave BS. Furthermore, regarding mmWave cellular networks,
measurements showed that mmWave signals propagate with a pathloss exponent of
2 in LOS paths and a much higher pathloss exponent with additional shadowing
in NLOS paths [4, 3]. This poses a further challenge since the difference in
pathloss exponents could results in excessive interference from NLOS users
when the per-user power control is implemented. Hence, power control must be
implemented with a constraint on the maximum user transmit power in order to
mitigate the interference. On the other hand, the correlation among the
interferer results from the implementation of orthogonal allocation scheme
that does not allow for a reuse of a channel resource within the same cell,
i.e., the coupling of the mmWave BS and served user-per channel point
processes [15, 16]. To ensure analytical tractability, various generative
models have been proposed in [15, 16, 17, 22, 23, 24] to approximate the
spatial distribution of interferers in the uplink of UHF and mmWave cellular
networks.

In this paper, we present a stochastic geometry framework for modeling and
analyzing the uplink in single-tier and multi-tier mmWave cellular networks.
Similar to the earlier works in this area [15, 16, 17, 22, 23, 24], we rely on
some approximation so as to maintain analytical tractability. Notably, we
partially ignore the correlation among the interfering users. Our model
captures the correlation between the interfering users and the reference
mmWave BS, which serves the typical user, but it ignores the correlation among
the interfering users. As evidence from [15, 16, 17], this approximation holds
true for the uplink of both the single and multi-tier UHF cellular networks.
The accuracy of this approach is validated for both the single and multi-tier
mmWave cellular networks via Monte-Carlo simulations. We here extend the work
in [24] which is based on a single-tier mmWave network and does not take into
account the maximum transmit power of the user. Our proposed framework takes
into account the limitation in the user transmit power, the per-user power
control and the cutoff threshold for the power control. We compare the
findings of our analysis with that from [16] and [24]. The comparison reveals
that our analysis provides several new insights that can be leveraged for
designing the mmWave networks more accurately. The main contributions of this
work are summarized as follows.

  * •

We present a stochastic geometry framework for the signal-to-interference-and-
noise-ratio (SINR) outage probability in the uplink of a single-tier mmWave
cellular networks, which is generic and serving as a foundation extends to the
multi-tier mmWave cellular networks. The model takes into account the
limitation in the transmit power of the user, and the network defined per-user
power control and the cutoff threshold. Closed-form expressions are derived
for the SINR outage probability.

  * •

We present the asymptotic dense network analysis of the SINR outage
probability in both the single and multi-tier mmWave cellular networks. The
asymptotic analysis leverage on approximating an intricate LOS function as a
step function.

  * •

The analytical derivations are verified via Monte-Carlo based simulations.
Results show that the maximum power constraint significantly affects the SINR
outage probability. Further, contrary to the SINR outage of UHF networks,
which is non-increasing in the cutoff threshold, the SINR outage probability
in mmWave networks could increase over some range of cutoff thresholds for
some mmWave BS density, LOS and NLOS pathloss exponents, and blockage
parameter.

The rest of the paper is organized as follows. The system model of the uplink
of a multi-tier mmWave cellular network with truncated channel inversion power
control is presented in Section II. In Section III, the uplink modeling
framework for a single-tier mmWave cellular network is presented. In Section
IV, we generalize the developed framework for the multi-tier mmWave cellular
networks. In Section V, we utilize a simplified system model to analyze the
asymptotic behavior and performance in dense mmWave networks. Numerical and
simulation results are presented in Section VI. Finally, Section VII concludes
the paper.

##  II System Model

###  II-A Network Model

We consider the uplink of a $K$-tier mmWave cellular network and focus on the
SINR experienced by outdoor users served by outdoor mmWave BSs. Each tier’s
BSs are randomly located and they have particular spatial density, antenna
gain, receiver sensitivity, blockage parameter and pathloss exponents. The
outdoor BSs of each tier are spatially distributed in $\mathbb{R}^{2}$
according to an independent homogeneous Poisson point process (PPP)
$\Theta_{k}$ with density $\lambda_{k}$. The users locations (before
association) are assumed to form a realization of homogeneous PPP $\Phi$ with
density $\lambda_{u}$. It is assumed that the density of the users is high
enough such that each BS will have at least one user served per channel. Each
BS serves a single user per channel, which is randomly selected from all the
users located in its Voronoi cell by using a round-robin scheduler. As in [15,
16, 19, 17, 22, 23, 24], we assume that the active users also form PPP even
after associating just one user per BS. Note that this approximation only
partly ignores the correlation imposed by the system model, i.e., the coupling
of the BS and served user-per channel resource point processes. The
correlation between the reference mmWave BS and the typical user is captured
in the derivation of the outage probability in Sections III and IV.

Each tier in the mmWave network is characterized by a non-negative blockage
constant $\beta_{k}$ for $k\in\\{1,\ldots,K\\}$. The parameter $\beta_{k}$ is
determined by the average size and density of blockages in that tier and where
the average LOS range is given by $\frac{1}{\beta_{k}}$ [25, 8, 21]. The
probability of a communication link in the $k^{th}$ tier with length $r$ being
a LOS is $\mathbb{P}(\mathrm{LOS}_{k})=e^{-\beta_{k}r}$, while the probability
of a link being NLOS is
$\mathbb{P}(\mathrm{NLOS}_{k})=1-\mathbb{P}(\mathrm{LOS}_{k})$. The LOS and
NLOS links of the $k^{th}$ tier have different pathloss exponents denoted by
$\alpha_{L}^{k}$ and $\alpha_{N}^{k}$, respectively, $\forall
k\in\\{1,\ldots,K\\}$.

###  II-B Receiver Sensitivity and Truncated Outage

We assume that all users have an equal maximum transmit power $P_{u}$.
Furthermore, all mmWave BS in the $k^{th}$ tier have the same receiver
sensitivity which is denoted by $\rho_{\min}^{k}$. The received signal at the
mmWave BS must be greater than the receiver sensitivity $\rho_{\min}^{k}$ for
successful transmission in the uplink channel. Hence, each user (with either
LOS or NLOS link to its serving mmWave BS) associated with the $k^{th}$ tier
adjusts its transmit power such that the average received signal at its
serving mmWave BS is equal to a predefined threshold $\rho_{o}^{k}$, where
$\rho_{o}^{k}>\rho_{\min}^{k}$. Moreover, as a result of the maximum transmit
power constraint, users utilize a truncated channel inversion power control
scheme, where the transmitters compensate for the pathloss in the link to the
receiver to keep the average received signal power to the threshold
$\rho_{o}^{k}$. Any user-mmWave BS connection that requires a transmit power
that exceeds $P_{u}$ for the pathloss inversion will not be established,
hence, such a connection experiences a truncation outage [16].

###  II-C Beamforming Gain

For analytical tractability, we assume that all users and BSs are equipped
with directional antennas with a sectorized gain pattern. The main lobe gain,
side lobe gain and beamwidth of the users are $G_{u}^{\max}$, $G_{u}^{\min}$
and $\zeta_{t}$, respectively, while the corresponding parameters of the
$k^{th}$ tier BS antennas are $G_{bk}^{\max}$, $G_{bk}^{\min}$ and
$\zeta_{rk}$, respectively. We consider that based on channel estimation, the
reference BS in the $j^{th}$ tier and the typical user adjust their beam
steering angles to achieve the maximum array gains. As a result of this, the
total directivity gain of the desired signal is
$\mathcal{G}_{j}=G_{bj}^{\max}G_{u}^{\max}$. Since the underlying PPP is
isotropic in $\mathbb{R}^{2}$, we model the beam directions of the interfering
link as a uniform random variable on $[0,2\pi]$. Further, the directivity gain
in the interference link $G_{l}^{j}$ (interference experienced at the
reference BS in the $j^{th}$ tier) can be approximated as discrete random
variable whose probability distribution is given as $a_{v}^{j}$ with
probability $b_{v}^{j}~{}(v\in\\{1,2,3,4\\})$ [25], where $a_{v}^{j}$ and
$b_{v}^{j}$ are defined in Table I.

TABLE I: Probability Mass Function of the Directivity Gain in an Interference Link of the $j^{th}$ Tier [25] v | 1 | 2 | 3 | 4
---|---|---|---|---
$a_{v}^{j}$ | $G_{bj}^{\max}G_{u}^{\max}$ | $G_{bj}^{\max}G_{u}^{\min}$ | $G_{bj}^{\min}G_{u}^{\max}$ | $G_{bj}^{\min}G_{u}^{\min}$
$b_{v}^{j}$ | $\frac{\zeta_{rj}\zeta_{t}}{4\pi^{2}}$ | $\frac{\zeta_{rj}}{2\pi}(1-\frac{\zeta_{t}}{2\pi})$ | $(1-\frac{\zeta_{rj}}{2\pi})\frac{\zeta_{t}}{2\pi}$ | $(1-\frac{\zeta_{rj}}{2\pi})(1-\frac{\zeta_{t}}{2\pi})$

In general, the $k^{th}$ tier is characterized by a set of parameters
$\mathcal{V}_{k}$ whose element include the $k^{th}$ tier’s BS density
$\lambda_{k}$, blockage parameter $\beta_{k}$, cutoff threshold
$\rho_{o}^{k}$, main lobe gain $G_{bk}^{\max}$, side lobe gain
$G_{bk}^{\min}$, beamwidth $\zeta_{rk}$, LOS pathloss exponent
$\alpha_{L}^{k}$ and its NLOS pathloss exponent $\alpha_{N}^{k}$ such that
$\mathcal{V}_{k}=\\{\lambda_{k},\beta_{k},\rho_{o}^{k},G_{bk}^{\max},G_{bk}^{\min},\zeta_{rk},\alpha_{L}^{k},\alpha_{N}^{k}\\},\forall
k=1,\ldots,K$.

##  III Uplink of Single-Tier mmWave Cellular Networks

In this section, we develop our framework model for the uplink of a single-
tier mmWave cellular network. In particular, we present the mmWave
transmission power analysis and the SINR outage probability analysis.

###  III-A mmWave Transmission Power Analysis

Considering the mmWave cellular network with the truncated channel inversion
scheme, each user, which could be in LOS or NLOS to its serving mmWave BS will
transmit with different power in order to invert the pathloss towards its
serving BS. As a result of the truncation channel inversion, not all users
will be able to communicate in the uplink channel111Note that for all the
parameters in single tier network, we have removed the subscript/superscript
$k$ used to distinguish the $k^{th}$-tier parameters in a multi-tier network..
In particular, LOS and NLOS users located at a distance greater than
$(Pu/\rho_{o})^{1/\alpha_{L}}$ and $(Pu/\rho_{o})^{1/\alpha_{N}}$,
respectively, from their associated BS are unable to communicate in the uplink
direction as a result of insufficient transmit power [16]. Hence, in addition
to the fact that the whole user set is divided into a subset of LOS and NLOS
users based on their association with their serving mmWave BS, the LOS and
NLOS user sets are further divided into a non-overlapping subset of active
users and inactive users. The distribution of the transmit power of a typical
user is obtained from the following theorem. In mmWave cellular networks with
truncated channel inversion power control and cutoff threshold $\rho_{o}$, the
probability distribution function (PDF) of the transmit power of a typical
user in the uplink is given by

$f_{P}(p)=\frac{\lambda(p)e^{-\Lambda(p)}}{\int_{0}^{P_{u}}\lambda(y)e^{-\Lambda(y)}\mathrm{d}y},~{}~{}~{}~{}0\leq
p\leq P_{u}$ (1)

where

$\displaystyle\lambda(p)=\frac{2\pi\lambda}{\alpha_{L}\rho_{o}^{2/\alpha_{L}}}p^{\frac{2}{\alpha_{L}}-1}e^{-\beta\left(\frac{p}{\rho_{o}}\right)^{\frac{1}{\alpha_{L}}}}+\frac{2\pi\lambda}{\alpha_{N}\rho_{o}^{2/\alpha_{N}}}p^{\frac{2}{\alpha_{N}}-1}\left(1-e^{-\beta\left(\frac{p}{\rho_{o}}\right)^{\frac{1}{\alpha_{N}}}}\right)$
(2)

and

$\displaystyle\Lambda(p)$
$\displaystyle=\frac{2\pi\lambda}{\beta^{2}}\left(1-e^{-\beta\left(\frac{p}{\rho_{o}}\right)^{\frac{1}{\alpha_{L}}}}\left(1+\beta\left(\frac{p}{\rho_{o}}\right)^{\frac{1}{\alpha_{L}}}\right)\right)+\pi\lambda\left(\frac{p}{\rho_{o}}\right)^{\frac{2}{\alpha_{N}}}$
(3)
$\displaystyle-\frac{2\pi\lambda}{\beta^{2}}\left(1-e^{-\beta\left(\frac{p}{\rho_{o}}\right)^{\frac{1}{\alpha_{N}}}}\left(1+\beta\left(\frac{p}{\rho_{o}}\right)^{\frac{1}{\alpha_{N}}}\right)\right).$

The $\delta^{th}$ moment of the transmit power is thus obtained as

$\mathbb{E}\left[P^{\delta}\right]=\int_{0}^{P_{u}}p^{\delta}f_{P}(p)\mathrm{d}p,$
(4)

where $f_{P}(p)$ is given in (1).

Further, the truncation outage probability, which is the probability that a
user experience outage due to insufficient power, is expressed as

$\mathcal{O}_{p}=e^{-\Lambda(Pu)},$ (5)

where ${\Lambda(p)}$ is given in (3).

###### Proof:

See Appendix -A. ∎

Note that $1-e^{-y}(1+y)$ is strictly increasing in $y$ for $y>0$, hence the
first term of $\Lambda(p)$ in (3) is greater than the third term and
$\Lambda(p)$ is strictly positive for all BS density $\lambda$, blockage
parameter $\beta$, cutoff threshold $\rho_{o}$ and pathloss exponent
$\alpha_{N}\geq\alpha_{L}>0$. Consequently, increasing the cutoff threshold
$\rho_{o}$ leads to increase in the truncation outage probability as long as
$\alpha_{N}\geq\alpha_{L}$. In other words, the higher the cutoff threshold
the poorer the mmWave network performance in terms of the truncation outage.
As we will show in the later section, a low cut-off threshold could actually
deteriorate the mmWave network performance in terms of the SINR outage
probability. Hence, it is essential to manage the trade-off between SINR
outage and truncation outage probabilities using the cutoff threshold.

On another note, by expanding (3), it can be seen that increasing the blockage
parameter value $\beta$ leads to a reduction in $\Lambda(p)$ for fixed mmWave
BS density $\lambda$, cutoff threshold $\rho$, and pathloss exponent
$\alpha_{N}>\alpha_{L}$. Thus, the truncation outage probability also
increases with increasing blockage parameter $\beta$. Increasing the blockage
parameter implies decreasing the average LOS range and hence we have more NLOS
paths requiring a much higher transmit power to meet the receiver sensitivity
requirement.

Regarding the expectation of the user transmission power, i.e., the average
user transmission power, it is not straightforward to gain insights. However,
from (2), we expect the plot of the average transmit power to be characterized
from the LOS-based average transmit power and the NLOS-based average transmit
power. We validate this observation later in the numerical results section.

###  III-B SINR Outage Probability

For an active typical user, the SINR at its connected BS (termed as the
reference mmWave BS) can be written as

$SINR=\frac{\rho_{o}|g_{o}|^{2}G_{b}^{\max}G_{u}^{\max}}{\sigma^{2}+\sum_{z\in\mathcal{Z}}P_{z}|g_{z}|^{2}G_{z}L(D_{z})},$
(6)

where the useful signal power (normalized by $G_{b}^{\max}G_{u}^{\max}$) is
equal to $\rho_{o}|g_{o}|^{2}$ due to the truncated channel inversion power
control, $\mathcal{Z}$ is the set of interfering users, $L(D_{z})$ is the
pathloss from the interfering users to the reference mmWave BS, $\sigma^{2}$
is the noise power, $G_{z}$ is the directivity gain on an interfering link and
$g_{z}$ is the small-scale fading which follows a Nakagami distribution with
parameter $N$. The SINR outage probability $\mathcal{O}_{s}$ is the
probability that the instantaneous SINR experienced at the reference mmWave BS
is less than the target SINR $\theta$, i.e.
$\mathcal{O}_{s}=\mathbb{P}(SINR<\theta)$. Given that the average received
signal at the reference mmWave BS (normalized by the directivity gain
$G_{b}^{\max}G_{u}^{\max}$) is equal to the cutoff threshold $\rho_{o}$. The
SINR outage probability can be computed as

$\mathbb{P}(SINR\leq\theta)=\mathbb{P}\\{\rho_{o}|g_{o}|^{2}\mathcal{G}\leq\theta(\sigma^{2}+I_{L}+I_{N})\\},$
(7)

where $I_{L}$ and $I_{N}$ are the interference strength from LOS and NLOS
users, respectively and $\mathcal{G}=G_{b}^{\max}G_{u}^{\max}$. Noting that
$|g_{o}|^{2}$ is normalized gamma random variable with parameter $N$, we have
the following approximation

$\displaystyle\mathbb{P}$
$\displaystyle\\{|g_{o}|^{2}\leq\theta(\sigma^{2}+I_{L}+I_{N})/(\rho_{o}\mathcal{G})\\}$
(8)
$\displaystyle\overset{(a)}{\approx}1-\left(1-\mathbb{E}\left[\left(1-e^{-\frac{\eta\theta\left(\sigma^{2}+I_{L}+I_{N}\right)}{\rho_{o}\mathcal{G}}}\right)^{N}\right]\right)$
$\displaystyle\overset{(b)}{=}1-\sum_{n=1}^{N}\\!\left(-1\right)^{n+1}\binom{N}{n}\mathbb{E}\left[e^{-\frac{n\eta\theta\left(\sigma^{2}+I_{L}+I_{N}\right)}{\rho_{o}\mathcal{G}}}\right]$
$\displaystyle=1-\sum_{n=1}^{N}\\!\left(-1\right)^{n+1}\\!\binom{N}{n}\\!e^{-sn\sigma^{2}}\mathcal{L}_{I_{L}}(sn)\mathcal{L}_{I_{N}}(sn),$

where $s=\frac{\eta\theta}{\rho_{o}\mathcal{G}}$, $\eta=N(N!)^{-\frac{1}{N}}$,
$(a)$ follow from the fact that $\lvert g_{0}\rvert^{2}$ is a normalized gamma
random variable with parameter $N$ and the fact that for a constant
$\gamma>0$, the probability $\mathbb{P}(\lvert g_{0}\rvert^{2}<\gamma)$ is
tightly upper bounded by $\left[1-\exp\left(-\gamma
N\left(N!\right)^{-\frac{1}{N}}\right)\right]^{N}$ [26]. Further, the
expectation is with respect to $I_{L}$ and $I_{N}$. $(b)$ follows from the
binomial theorem and the assumption that $N$ is an integer, and
$\mathcal{L}_{I_{L}}$ and $\mathcal{L}_{I_{N}}$ denote the Laplace transforms
of the random variables $I_{L}$ and $I_{N}$, respectively.

As mentioned earlier, the location of the interfering users do not create a
PPP as a result of the correlation among the users from the channel assignment
process. The interfering users are thus better modeled by using soft-core
processes, which can capture such correlation [27]. However, most soft-core
processes lack analytical tractability [28, 29], thus making the expression
for the Laplace transforms of the aggregate LOS and NLOS interference,
$\mathcal{L}_{I_{L}}$ and $\mathcal{L}_{I_{N}}$, respectively, unobtainable.
Hence, we approximate the location of the interfering users with a PPP. Note
that the approximation has been shown to be accurate for the UHF network when
the correlation among the interfering nodes and the reference receiver is
captured [15, 16]. Our model here also captures this correlation. The accuracy
of our assumption will be verified later through simulations. Based on the PPP
approximation, and the independent and identical distributed transmit power
for the set of interfering users in the uplink channel, the SINR outage
probability can be obtained from the following theorem.

The SINR outage probability in the uplink of single-tier mmWave cellular
networks with truncated channel inversion power control with cutoff threshold
$\rho_{o}$ can be expressed as

$\mathcal{O}_{s}\\!=\\!1-\sum_{n=1}^{N}\left(-1\right)^{n+1}\binom{N}{n}\exp\left(-\frac{\eta
n\theta\sigma^{2}}{\rho\mathcal{G}}-Q_{n}-V_{n}\right)$ (9)

where

$\displaystyle
Q_{n}=2\pi\lambda\sum_{v=1}^{4}\\!b_{v}q_{v}^{\frac{2}{\alpha_{L}}}\int_{\mathcal{A}}^{\infty}\int_{0}^{P_{u}}\mathcal{F}\left(N,\frac{y^{-\alpha_{L}}}{N}\right)\exp\left({-\beta\left(q_{v}p\right)^{\frac{1}{\alpha_{L}}}y}\right)yP^{\frac{2}{\alpha_{L}}}f_{P}(p)\mathrm{d}p\mathrm{d}y,$
(10) $\displaystyle
V_{n}=2\pi\lambda\sum_{v=1}^{4}\\!b_{v}q_{v}^{\frac{2}{\alpha_{N}}}\int_{\mathcal{B}}^{\infty}\int_{0}^{P_{u}}\mathcal{F}\left(N,\frac{y^{-\alpha_{N}}}{N}\right)\left(1-\exp\left({-\beta\left(q_{v}p\right)^{\frac{1}{\alpha_{N}}}y}\right)\right)yP^{\frac{2}{\alpha_{N}}}f_{P}(p)\mathrm{d}p\mathrm{d}y,$
(11)

where $\eta=N(N!)^{-\frac{1}{N}}$, $\mathcal{G}=G_{u}^{\max}G_{b}^{\max}$,
$\mathcal{A}=\left(\frac{\eta n\theta
a_{v}}{\mathcal{G}}\right)^{-\frac{1}{\alpha_{L}}}$,
$\mathcal{B}=\left(\frac{\eta n\theta
a_{v}}{\mathcal{G}}\right)^{-\frac{1}{\alpha_{N}}}$, $q_{k}=\frac{\eta n\theta
a_{v}}{\rho_{o}\mathcal{G}}$,
$\mathcal{F}(N,y)=1-\frac{1}{\left(1+y\right)^{N}}$, $a_{v}$ and $b_{v}$ are
the antenna directivity parameters defined in Section II and $f_{P}(p)$ is
defined in (1)

###### Proof:

See Appendix -B ∎

Though this approximates the SINR outage probability, we find that the
expression compares very well with the simulation results in Section VI-1.
Furthermore, the expression here captures the user maximum power constraint
contrary to the prior result on the uplink of single tier mmWave networks in
[24], which is based on an unbounded power constraint. The maximum power
constraint is very important in the uplink power control of mmWave network due
to the significant difference in the LOS and NLOS pathloss exponent. We show
the impact of the maximum power constraint in a single-tier network later in
Section VI-2.

##  IV Uplink of Multi-Tier mmWave Cellular Networks

In this section, we extend our developed model for the uplink of a single-tier
mmWave cellular network to the uplink of a multi-tier mmWave cellular network.
As mentioned earlier, the $k^{th}$ tier is characterized by a set
$\mathcal{V}_{k}=\\{\lambda_{k},\beta_{k},\rho_{o}^{k},G_{bk}^{\max},G_{bk}^{\min},\zeta_{rk},\alpha_{L}^{k},\alpha_{N}^{k}\\},\forall
k=1,\ldots,K$. First, we present the distribution of transmit power for the
multi-tier mmWave cellular networks. Afterward, we derive its SINR outage
probability.

###  IV-A Distribution of the Transmit Power in the Uplink of Multi-tier
mmWave Cellular Networks

Similar to the single-tier network, given the cutoff threshold for the
$k^{th}$ tier $\rho_{o}^{k}$, LOS and NLOS users located at distances greater
than $(P_{u}/\rho_{o}^{k})^{1/\alpha_{L}^{k}}$ and
$(P_{u}/\rho_{o}^{k})^{1/\alpha_{N}^{k}}$, respectively, from their nearest
mmWave BS are unable to communicate in the uplink direction due to
insufficient transmit power. The distribution of the transmit power of a
typical user associated with the $j^{th}$ tier is obtained from the following
theorem In a $K$-tier mmWave cellular network with truncated channel inversion
power control where the $k^{th}$ tier is distinguished by the set
$\mathcal{V}_{k}$, $\forall k=1,\ldots,K$, i.e., its density $\lambda_{k}$,
blockage parameter $\beta_{k}$, cutoff threshold $\rho_{o}^{k}$, antenna
parameters, $G_{bk}^{\max}$, $G_{bk}^{\min}$ and $\zeta_{rk}$, LOS pathloss
exponent $\alpha_{L}^{k}$ and its NLOS pathloss exponent $\alpha_{N}^{k}$, the
PDF of the transmit power of a typical active user in the uplink of the
$j^{th}$ tier is given by

$f_{P_{j}}(p)=\frac{\sum_{k=1}^{K}\overline{\lambda}_{k}(p)}{1-e^{-\sum_{a=1}^{K}{\Lambda_{a}\left(\frac{P_{u}}{\rho_{o}^{j}}\right)}}}{e^{-\sum_{b=1}^{K}{\Lambda_{b}\left(\frac{p}{\rho_{o}^{j}}\right)}}}$
(12)

where

$\overline{\lambda}_{k}(p)=\frac{2\pi\lambda_{k}}{\alpha_{L}^{k}{\rho_{o}^{j}}^{2/\alpha_{L}^{k}}}p^{\frac{2}{\alpha_{L}^{k}}-1}e^{-\beta_{k}\left(\frac{p}{\rho_{o}^{j}}\right)^{\frac{1}{\alpha_{L}^{k}}}}+\frac{2\pi\lambda_{k}}{\alpha_{N}^{k}{\rho_{o}^{j}}^{2/\alpha_{N}^{k}}}p^{\frac{2}{\alpha_{N}^{k}}-1}\left(1-e^{-\beta_{k}\left(\frac{p}{\rho_{o}^{j}}\right)^{\frac{1}{\alpha_{N}^{k}}}}\right),$
(13)
$\Lambda_{k}(y)=\frac{2\pi\lambda_{k}}{\beta_{k}^{2}}\left(1-e^{-\beta_{k}{y}^{\frac{1}{\alpha_{L}^{k}}}}\left(1+\beta_{k}{y}^{\frac{1}{\alpha_{L}^{k}}}\right)\right)+\pi\lambda_{k}{y}^{\frac{2}{\alpha_{N}^{k}}}-\frac{2\pi\lambda_{k}}{\beta_{k}^{2}}\left(1-e^{-\beta_{k}{y}^{\frac{1}{\alpha_{N}^{k}}}}\left(1+\beta_{k}{y}^{\frac{1}{\alpha_{N}^{k}}}\right)\right)$
(14)

and $y$ is a dummy variable in (14). The $\delta^{th}$ moment of the transmit
power of a user in the $j^{th}$ tier is given as

$\mathbb{[}P_{j}^{\delta}]=\int_{0}^{P_{u}}{\frac{p^{\delta}\sum_{k=1}^{K}\overline{\lambda}_{k}(p)}{1-e^{-\sum_{a=1}^{K}{\Lambda_{a}\left(\frac{P_{u}}{\rho_{o}^{j}}\right)}}}{e^{-\sum_{b=1}^{K}{\Lambda_{b}\left(\frac{p}{\rho_{o}^{j}}\right)}}}}\mathrm{d}p$
(15)

Further, the truncation outage probability in the uplink of mmWave cellular
networks for the $j^{th}$ tier can be obtained as

$\mathcal{O}_{p}^{j}=e^{-\sum_{k=1}^{K}{\Lambda_{k}\left(\frac{P_{u}}{\rho_{o}^{j}}\right)}}.$
(16)

###### Proof:

See Appendix -C ∎

###  IV-B SINR Outage Probability

For an active typical user, the SINR at its connected BS in the $j^{th}$ tier
(termed as the reference mmWave BS) can be written as

$SINR_{j}=\frac{\rho_{o}^{j}|g_{o}|^{2}\mathcal{G}_{j}}{\sigma^{2}+\sum_{k=1}^{K}\sum_{z\in\mathcal{Z}_{k}}P_{z}|g_{z}|^{2}G_{z}^{j}L(D_{z})},$
(17)

where the useful signal power (normalized by $\mathcal{G}_{j}$) is equal to
$\rho_{o}^{j}|g_{o}|^{2}$ due to the truncated channel inversion power
control, $\mathcal{Z}_{k}$ is the set of interfering users associated with a
BS in the $k^{th}$ tier, $L(D_{z})$ is the pathloss from the interfering users
to the reference BS, $\sigma^{2}$ is the noise power, $G_{z}^{j}$ is the
directivity gain on an interfering link and $g_{z}$ is the small-scale fading
which follows a Nakagami distribution with parameter $N$. We consider that
each of the tiers has its own SINR threshold which is represented by
$\theta_{k}$. Further, the average received signal at any of the BSs in the
$j^{th}$ tier is equivalent to the cutoff threshold of the $j^{th}$ tier
represented by $\rho_{o}^{j}$.

The SINR outage probability of the $j^{th}$ tier can be expressed as

$\mathbb{P}(SINR_{j}\leq\theta_{j})=\mathbb{P}\left\\{\rho_{o}^{j}|g_{o}|^{2}\mathcal{G}_{j}\leq\theta_{j}\left(\sigma^{2}+\sum_{k=1}^{K}I_{L}^{k}+\sum_{k=1}^{K}I_{N}^{k}\right)\right\\},$
(18)

where $I_{L}^{k}$ and $I_{N}^{k}$ are the aggregate interference from LOS and
NLOS users of the $k^{th}$ tier, respectively. Note that $I_{L}^{j}$ and
$I_{N}^{j}$ represents the LOS and NLOS co-tier interference, respectively,
and $I_{L}^{k}$ and $I_{N}^{k}$ $\forall k\neq j$ denotes the LOS and NLOS
cross-tier interference, respectively. Similar to the single tier case, noting
that $|g_{o}|^{2}$ is normalized gamma random variable with parameter $N$, we
have the following approximation

$\displaystyle\mathbb{P}$
$\displaystyle\left\\{|g_{o}|^{2}\leq\theta_{j}\left(\sigma^{2}+\sum_{k=1}^{K}I_{L}^{k}+\sum_{k=1}^{K}I_{N}^{k}\right)/\left(\rho_{o}^{j}\mathcal{G}_{j}\right)\right\\}$
(19)
$\displaystyle=1-\left(1-\mathbb{E}\left[\left(1-e^{-\frac{\eta\theta_{j}\left(\sigma^{2}+\sum_{k=1}^{K}I_{L}^{k}+\sum_{k=1}^{K}I_{N}^{k}\right)}{\rho_{o}^{j}\mathcal{G}_{j}}}\right)^{N}\right]\right)$
$\displaystyle=1-\sum_{n=1}^{N}(-1)^{n+1}\binom{N}{n}\mathbb{E}_{\Phi}\left[e^{-\frac{n\eta\theta_{j}\left(\sigma^{2}+\sum_{k=1}^{K}I_{L}^{k}+\sum_{k=1}^{K}I_{N}^{k}\right)}{\rho_{o}^{j}\mathcal{G}_{j}}}\right]$
$\displaystyle=1-\sum_{n=1}^{N}(-1)^{n+1}\binom{N}{n}e^{-sn\sigma^{2}}\prod_{k=1}^{K}\mathcal{L}_{I_{L}^{k}}(sn)\prod_{k=1}^{K}\mathcal{L}_{I_{N}^{k}}(sn),$

where $s=\frac{\eta\theta_{j}}{\rho_{o}^{j}\mathcal{G}_{j}}$,
$\eta=N(N!)^{-\frac{1}{N}}$. In addition, the interfering users do not
constitute a PPP while the transmit power of the interfering users are also
correlated as in the case of the single tier. However, in order to keep
analytical tractability, we approximate the interfering users process as a PPP
while also ignoring the correlations in the transmit power of interfering
users. The SINR outage probability of a typical user in the $j^{th}$ tier of
mmWave cellular networks can be obtained from the following theorem. In a
$K-$tier mmWave network with truncated channel inversion power control where
each tier is distinguished by its density $\lambda_{k}$, cutoff threshold
$\rho_{o}^{k}$, blockage parameter $\beta_{k}$, antenna parameters,
$G_{bk}^{\max}$, $G_{bk}^{\min}$ and $\zeta_{rk}$, LOS pathloss exponent
$\alpha_{L}^{k}$ and NLOS pathloss exponent $\alpha_{N}^{k}$, the SINR outage
probability of a typical user in the $j^{th}$ tier is given by

$\mathcal{O}_{s}^{j}=1-\sum_{n=1}^{N}\left(-1\right)^{n+1}\binom{N}{n}\exp\left(-\frac{\eta
n\theta_{j}\sigma^{2}}{\rho_{o}^{j}\mathcal{G}_{j}}-\sum_{k=1}^{K}\left(Q_{n}^{k}+V_{n}^{k}\right)\right)$

where

$\displaystyle
Q_{n}^{k}=2\pi\lambda_{k}\sum_{v=1}^{4}b_{v}^{j}{q_{v}^{j}}^{\frac{2}{\alpha_{L}^{j}}}\times\int_{\mathcal{A}_{j}}^{\infty}\int_{0}^{P_{u}}\mathcal{F}\left(N,\frac{y^{-\alpha_{L}^{j}}}{N}\right)e^{-\beta_{j}\left(q_{v}^{j}p\right)^{\frac{1}{\alpha_{L}^{j}}}y}yp^{\frac{2}{\alpha_{L}^{j}}}f_{P_{k}}(p)\mathrm{d}p\mathrm{d}y,$

$\displaystyle
V_{n}^{k}=2\pi\lambda_{k}\sum_{v=1}^{4}\\!b_{v}^{j}{q_{v}^{j}}^{\frac{2}{\alpha_{N}^{j}}}\times\int_{\mathcal{B}_{j}}^{\infty}\int_{0}^{P_{u}}\mathcal{F}\left(N,\frac{y^{-\alpha_{N}^{j}}}{N}\right)\left(1-e^{-\beta_{j}\left(q_{v}^{j}p\right)^{\frac{1}{\alpha_{N}^{j}}}y}\right)yp^{\frac{2}{\alpha_{N}^{j}}}f_{P_{k}}(p)\mathrm{d}p\mathrm{d}y,$

$\eta=N(N!)^{-\frac{1}{N}}$, $\mathcal{A}_{j}=\left(\frac{\eta
n\theta_{j}a_{v}^{j}\rho_{o}^{k}}{\rho_{o}^{j}\mathcal{G}_{j}}\right)^{-\frac{1}{\alpha_{L}^{j}}}$,
$\mathcal{B}_{j}=\left(\frac{\eta
n\theta_{j}a_{v}^{j}\rho_{o}^{k}}{\rho_{o}^{j}\mathcal{G}_{j}}\right)^{-\frac{1}{\alpha_{N}^{j}}}$,
$\mathcal{F}(N,y)=1-\frac{1}{\left(1+y\right)^{N}}$, $q_{v}^{j}=\frac{\eta
n\theta_{j}a_{v}^{j}}{\rho_{o}^{j}\mathcal{G}_{j}}$, $a_{v}^{j}$ and
$b_{v}^{j}$ are the antenna directivity parameters defined in Section II and
$f_{P_{k}}(p)$ is defined in (12).

###### Proof:

See Appendix -D ∎

It has been shown in [8] that the LOS probability function can be approximated
by a step function in a dense mmWave network. Hence, in the next section, we
propose to simplify our uplink system model and the subsequent analysis by
using a step function approximation of the LOS probability function as well.

##  V Analysis of the Uplink of Dense mmWave Networks

In this section, we present the analysis for the uplink of a dense mmWave
cellular network. The motivation for the dense network analysis is based on
the fact that mmWave cellular networks must be dense in order to achieve its
forecasted gain [8]. Here we approximate the LOS probability
$\mathbb{P}(\mathrm{LOS}_{k})$ by using a step function such that the LOS
probability $\mathbb{P}(\mathrm{LOS}_{k})$ is taken to be $1$ when the link is
within a circular disc $\mathcal{B}(0,R_{B})$ centered at the reference mmWave
BS and $0$ when outside the disc. Next, we present uplink signal to
interference ratio (SIR) distribution for the dense multi-tier mmWave
networks, which we later degrade to the single tier scenario.

###  V-A Outage Analysis in the Uplink of Dense Multi-tier mmWave Networks

The dense mmWave network will be interference limited with mainly LOS
interferers limiting its performance. Hence, we ignore both the noise power
and the NLOS interfering users in the analysis. Further, according to [3], the
signal power from LOS interferers are nearly deterministic, hence we also
ignore the small-scale fading. Consequently, the SIR at the BS that the
typical user connects to in the $j^{th}$ tier can be expressed from (17) as

$SIR_{j}=\frac{\rho_{o}^{j}\mathcal{G}_{j}}{\sum_{k=1}^{K}\sum_{z\in\mathcal{Z}_{k}\cap\mathcal{B}(0,R_{B})}P_{z}G_{z}^{j}L(D_{z})}.$
(20)

As mentioned earlier, for the dense deployment, the LOS interferers are
dominant and the SINR outage probability in the $j$ tier can thus be
approximated as

$\mathbb{P}(SIR_{j}\leq\theta_{j})=\mathbb{P}\left\\{\rho_{o}^{j}\mathcal{G}_{j}\leq\theta_{j}\sum_{k=1}^{K}I_{L}^{k}\right\\},$
(21)

where
$I_{L}^{k}=\sum_{z\in\mathcal{Z}_{k}\cap\mathcal{B}(0,R_{B})}P_{z}G_{z}L(D_{z})$
is the interference power received at the reference BS from users connected to
BSs in the $k^{th}$ tier. Note that the average receive signal at the
reference BS normalized by the directivity gain $\mathcal{G}_{j}$ is
equivalent to the cutoff threshold $\rho_{o}^{j}$. The SINR outage probability
can be approximated as

$\displaystyle\mathbb{P}\\{\rho_{o}^{j}\mathcal{G}_{j}<\theta_{j}\sum_{k=1}^{K}I_{L}^{k}\\}$
$\displaystyle\overset{(a)}{\approx}\mathbb{P}\left\\{h<\frac{\theta_{j}\sum_{k=1}^{K}I_{L}^{k}}{\rho_{o}^{j}\mathcal{G}_{j}}\right\\}$
$\displaystyle\overset{(b)}{=}1-\left(1-\mathbb{E}_{\Phi_{L}}\left[\left(1-e^{-\frac{\eta\theta_{j}\sum_{k=1}^{K}I_{L}^{k}}{\rho_{o}^{j}\mathcal{G}_{j}}}\right)^{L}\right]\right)$
$\displaystyle=1-\sum_{l=1}^{L}\binom{L}{l}(-1)^{l+1}\prod_{k=1}^{K}\mathcal{L}_{I_{L}^{k}}(sl)$
(22)

where the dummy variable $h$ in $(a)$ is used to denote normalized gamma
variable with parameter $L$. Note that the distribution of the normalized
gamma variable converges to an identity when its parameters tend to infinity,
$(b)$ follows from [26] such that the probability $\mathbb{P}(\lvert
h\rvert^{2}<\gamma)$ is tightly upper bounded by $\left[1-\exp\left(-\gamma
N\left(N!\right)^{-\frac{1}{N}}\right)\right]^{N}$ and
$s=\frac{\eta\theta_{j}}{\rho_{o}^{j}\mathcal{G}_{j}}$. In the following
theorem, we summarize the main result for the SIR outage distribution in a
multi-tier network. The SINR outage probability in the $j^{th}$ tier of a
$K$-tier mmWave cellular network with truncated channel inversion power
control can be approximated as

$\displaystyle\overline{\mathcal{O}}_{s}^{j}$
$\displaystyle=1-\sum_{n=l}^{L}(-1)^{l+1}\binom{L}{l}\prod_{k=1}^{K}\exp\Bigg{(}\int_{0}^{Pu}\bigg{(}\pi\lambda_{k}\left(\frac{p}{\rho_{o}^{k}}\right)^{\frac{2}{\alpha_{L}^{j}}}-\lambda_{0}^{k}$
(23)
$\displaystyle+\frac{2\pi\lambda_{k}}{\alpha_{L}^{j}}\sum_{v=1}^{4}\left(\frac{\eta
l\theta_{j}a_{v}^{j}}{\rho_{o}^{j}\mathcal{G}_{j}}\right)^{\frac{2}{\alpha_{L}^{j}}}b_{v}^{j}p^{\frac{2}{\alpha_{L}^{j}}}\left(\Gamma\left(\frac{-2}{\alpha_{L}^{j}},\frac{\eta
l\theta_{j}a_{v}^{j}p}{\rho_{o}^{j}\mathcal{G}_{j}R_{B}^{\alpha_{L}}}\right)-\Gamma\left(\frac{-2}{\alpha_{L}^{j}},\frac{\eta
l\theta_{j}a_{v}^{j}\rho_{o}^{k}}{\rho_{o}^{j}\mathcal{G}_{j}}\right)\right)\bigg{)}f_{P_{k}}(p)\mathrm{d}x\Bigg{)}$

where $\lambda_{0}^{k}=\lambda_{k}\pi R_{B}^{2}$,
$\Gamma(a,b)=\int_{b}^{\infty}t^{a-1}e^{-t}\mathrm{d}t$ is the upper
incomplete gamma function, $a_{v}^{j}$ and $b_{v}^{j}$ are the antenna
directivity parameters defined in Section II, $\eta=L(L!)^{-\frac{1}{L}}$, $L$
is used in the approximation and $f_{P_{k}}(p)$ is defined as

$f_{P_{k}}(p)=\frac{\sum_{c=1}^{K}\frac{2\pi\lambda_{c}p^{\frac{2}{\alpha_{L}^{c}}-1}}{\alpha_{L}^{c}\left(\rho_{o}^{k}\right)^{\frac{2}{\alpha_{L}^{c}}}}}{1-e^{-\sum_{a=1}^{K}\pi\lambda_{a}\left(\frac{P_{u}}{\rho_{o}^{k}}\right)^{\frac{2}{\alpha_{L}^{a}}}}}e^{-\sum_{b=1}^{K}\pi\lambda_{b}\left(\frac{p}{\rho_{o}^{k}}\right)^{\frac{2}{\alpha_{L}^{b}}}}$
(24)

###### Proof:

See Appendix -E. ∎

For a single-tier dense mmWave network, the approximation of SINR outage
probability can be obtained from the following corollary.

The SINR outage probability in the $j^{th}$ tier of a $K$-tier mmWave cellular
network with truncated channel inversion power control can be approximated as

$\displaystyle\overline{\mathcal{O}}_{s}=1-$
$\displaystyle\sum_{n=l}^{L}(-1)^{l+1}\binom{L}{l}\exp\Bigg{(}\int_{0}^{Pu}\bigg{(}\pi\lambda\left(\frac{p}{\rho_{o}}\right)^{\frac{2}{\alpha_{L}}}-\lambda_{0}$
(25)
$\displaystyle+\frac{2\pi\lambda}{\alpha_{L}}\sum_{v=1}^{4}\left(\frac{\eta
l\theta
a_{v}}{\rho_{o}\mathcal{G}}\right)^{\frac{2}{\alpha_{L}}}b_{v}p^{\frac{2}{\alpha_{L}}}\left(\Gamma\left(\frac{-2}{\alpha_{L}},\frac{\eta
l\theta
a_{v}p}{\rho_{o}\mathcal{G}R_{B}^{\alpha_{L}}}\right)-\Gamma\left(\frac{-2}{\alpha_{L}},\frac{\eta
l\theta a_{v}}{\mathcal{G}}\right)\right)\bigg{)}f_{P}(p)\mathrm{d}p\Bigg{)}$

where $\lambda_{0}=\lambda\pi R_{B}^{2}$,
$\Gamma(a,b)=\int_{b}^{\infty}t^{a-1}e^{-t}\mathrm{d}t$ is the upper
incomplete gamma function, $a_{v}$ and $b_{v}$ are the antenna directivity
parameters defined in Section II, $\eta=L(L!)^{-\frac{1}{L}}$, $L$ is the
number of terms used in the approximation and $f_{P}(p)$ is defined as

$f_{P}(p)=\frac{2\pi\lambda
p^{\frac{2}{\alpha_{L}}-1}e^{-\pi\lambda\left(\frac{p}{\rho_{o}}\right)^{\frac{2}{\alpha_{L}}}}}{\alpha_{L}\rho_{o}^{\frac{2}{\alpha_{L}}}\left(1-e^{-\pi\lambda\left(\frac{P_{u}}{\rho_{o}}\right)^{\frac{2}{\alpha_{L}}}}\right)}.$
(26)

###### Proof:

The proof follows directly from proof of the multi-tier mmWave cellular
networks and is omitted here. ∎

##  VI Numerical Results

In this section, we present numerical results to illustrate our analytical
findings for both the single-tier and two-tier mmWave cellular networks.
Unless otherwise stated, we set the BS densities
$\lambda_{1}=10~{}\mathrm{BS/km^{2}}$ and $\lambda_{2}=2\lambda_{1}$, the
maximum transmit power $P_{u}=1~{}\mathrm{W}$,
$\sigma^{2}=-110~{}\mathrm{dBm}$, the tier blockage parameters
$\beta_{1}=0.0071$ and $\beta_{2}=0.0143$ with corresponding pathloss exponent
$\alpha_{L}^{1}=2,\alpha_{N}^{1}=4,\alpha_{L}^{2}=2.9,\alpha_{N}^{2}=5$, the
Nakagami fading parameter $N=3$. Further, unless otherwise stated, the antenna
parameters of the first and second tier BSs are equivalent such that
$G_{b1}^{\max}=G_{b2}^{\max}=\mathrm{7~{}dB},~{}G_{b1}^{\min}=G_{b2}^{\max}=\mathrm{-10~{}dB}$
and $\zeta_{b1}=\zeta_{b2}=30^{\circ}$, while that of the users are assumed to
be characterized with
$G_{u}^{\max}=\mathrm{7~{}dB,}~{}G_{u}^{\min}=\mathrm{-10~{}dB}$ and
$\zeta_{u}=90^{\circ}$. In addition, we have utilized the system parameters of
the first tier for the single-tier network results.

####  VI-1 Accuracy of Analysis

Figure 1: Comparison of the analytical results with simulation in single-tier
mmWave newtorks for $\beta_{1}=0.0071$, $\alpha_{L}^{1}=2$ and
$\alpha_{N}^{1}=4$.  Figure 2: Comparison of the analytical results with
simulation in a two-tier mmWave newtork for
$\beta_{1}=0.0071,\beta_{2}=0.0143$, $\alpha_{L}^{1}=2$,
$\alpha_{N}^{1}=4,\alpha_{L}^{2}=2.9,\alpha_{N}^{2}=5$ and BS densities
$\lambda_{1}=10\mathrm{BS/km^{2}}$ and $\lambda_{2}=2\lambda_{1}$. Red and
blue dashed-lines represent mmWave networks 1 and 2, respectively.

In Figs. 1 and 2, we verify our derivation by plotting the analytical and
simulation results for the single-tier and two-tier mmWave cellular networks,
respectively. The results show that our derived analytical model accurately
captures the SINR outage probability for both the single-tier and multi-tier
mmWave cellular networks. Hence, our derived model finds great application in
mmWave multi-tier network where each tier can be identified via its BS
density, blockage parameter and corresponding LOS and NLOS pathloss exponents,
receiver sensitivity, and the BS antenna gain. Note that this validation is
essential since the cumulative distribution of the SINR is based on the
assumption that the active user constitute a PPP and that the transmit powers
of the users are independent. Further, independent LOS probability was
assumed. The approach in this paper, however, captures the correlation between
the location of the reference mmWave BS and that of the interfering users. It
also captures the correlation between the typical user’s (served by the
reference mmWave BS) transmit power and the interfering users’ transmit
powers.

####  VI-2 Comparison with the case without maximum power constraint

In Fig. 3, we compare our analysis with the one presented in [24], which does
not incorporate the maximum user power constraint. Note that the analysis in
[24] is for the single-tier, and hence the comparison presented in Fig. 3 is
also based on a single tier. As it can be seen, the maximum power constraint
significantly affects the SINR outage probability. The figure shows that the
SINR outage probability derived in [24] does not vary with $P_{u}$, since the
maximum power constraint is ignored in [24].

Figure 3: Effect of the maximum transmit power in a single-tier mmWave newtork
for $\theta_{1}=20~{}\mathrm{dB},\beta_{1}=0.0071$, $\alpha_{L}^{1}=2$,
$\alpha_{N}^{1}=4$ and BS densities $\lambda=10~{}\mathrm{BS/km^{2}}$.

####  VI-3 Effect of the BS Density

Figure 4: Effect of the BS density on the SINR outage probability for
$\lambda_{1}=10,100~{}\mathrm{BS/km^{2}}$, $\beta_{1}=0.0071$,
$\alpha_{L}^{1}=2$ and $\alpha_{N}^{1}=4$ in a single tier network .

In Fig. 4, we plot the SINR outage probability for the uplink of a single-tier
mmWave networks with truncated channel inversion power control for SINR
threshold $\theta=20,25$ and $30\mathrm{dB}$, and BS densities
$\lambda_{1}=10,100~{}\mathrm{BS/km^{2}}$. It can be seen that the SINR outage
probability of mmWave deviates from that of the UHF network presented in [16].
More specifically, four sections can be identified from the plot for the BS
density $\lambda_{1}=10~{}\mathrm{BS/km^{2}}$: 1) a decrease in SINR outage
probability can be seen for the cutoff threshold $\rho_{o}^{1}$ ranging from
$-100$ to $-50$ dBm with a slow descent region observed for $\rho_{0}^{1}$
ranging from $-85$ to $-75$ dBm; 2) a fairly stable outage probability can be
observed for $\rho_{o}^{1}$ ranging from $-50$ to $-31$ dBm; 3) an increase in
SINR outage probability can be seen for $\rho_{o}^{1}$ ranging from $-31$ to
$-18$ dBm, and 4) a decrease in SINR outage probability can be seen for
$\rho_{o}^{1}$ ranging from $-18$ to $0$ dBm. This observation is as a result
of the large difference in the pathloss exponent of the LOS and NLOS
propagation path, with each having its dominance region which also depends on
the BS density and blockage parameter. The latter specifies the LOS range.
Further, the receiver sensitivity also specifies the density of active LOS and
NLOS users and consequently, the interference received at the reference BS. It
can also be observed from Fig. 4 that for the same SINR threshold, increasing
the BS density leads to an increase in the SINR outage probability.

####  VI-4 Effect of the Blockage

Figure 5: Effect of the blockage parameter on the SINR outage probability for
$\beta_{1}=0.0071,0.003$, $\lambda_{1}=10~{}\mathrm{BS/km^{2}}$,
$\alpha_{L}^{1}=2$ and $\alpha_{N}^{1}=4$ in a single tier network.

In Fig. 5, we show the effect of blockages on the SINR outage probability.
Based on the LOS probability function $e^{-\beta_{1}r}$, a lower $\beta_{1}$
yields a larger number of LOS interfering user. Hence, the interference power
increases when $\beta_{1}$ is lowered leading to a higher SINR outage
probability for a lower $\beta_{1}$, as it can be seen in Fig. 5.

####  VI-5 Truncation Outage Probability

Fig. 6 compares the truncation outage probability for the uplink of mmWave and
UHF cellular networks for BS density $\lambda=1,10$ and
$100~{}\mathrm{BS/km^{2}}$. The truncation outage probability of UHF networks
has been defined in [16]. It can be seen that similar to the UHF case,
increasing the cutoff threshold increases the outage probability since more
users are unable to communicate due to insufficient transmit power. These
results are in line with the insights previously drawn on equations (3) and
(5). Furthermore, for BS densities $\lambda=1,10$, the truncation outage of
mmWave networks experience a slow growth region as the cutoff threshold
increases before its saturation contrary to UHF networks, which does not
experience a slow growth region. The slow growth region is due to the
difference in the truncation outage probability for LOS and NLOS links at a
given cutoff threshold. Meanwhile, for a high BS density of $\lambda=100$, the
truncation outage probability of mmWave converges to that of UHF with
$\alpha=2$ since more paths becomes LOS as the BS density increases. As
expected, Fig. 6 shows that the truncation outage of mmWave networks reduces
with as the BS density increases. This observation is due to the shortening of
the average link lengths as the BS density is increased.

Figure 6: Comparison of the truncation outage probability of mmWave and UHF
cellular networks for BS density $\lambda=1,10$ and
$100~{}\mathrm{BS/km^{2}}$. $\beta_{1}=0.0071$, $\alpha_{L}^{1}=2$ and
$\alpha_{N}^{1}=4$ in the mmWave network.

Fig. 7 shows the effect of blockages on the truncated outage probability. As
the average line of sight, which is proportional to the inverse of the
blockage parameter value $\frac{1}{\beta}$, increases the truncation outage
probability reduces as much lower transmit power is required to meet the
receiver sensitivity requirement when the density of blockages and the average
size of blockages are much lower. These results are also in line with the
insights previously drawn on equations (3) and (5).

Figure 7: Effect of the blockage parameter on the truncation outage
probability in mmWave network for $\alpha_{L}^{1}=2$, $\alpha_{N}^{1}=4$, BS
density $\lambda=1,10$ and $100~{}\mathrm{BS/km^{2}}$.

####  VI-6 Total Outage Probability

Figure 8: Total outage probability in a single tier mmWave network with
$\theta_{1}=20$ dB, $\beta_{1}=0.0071$ $\alpha_{L}^{1}=2$, $\alpha_{N}^{1}=4$
and BS density $\lambda=1,10$ and $100~{}\mathrm{BS/km^{2}}$.

In Fig. 8, we show the tradeoff introduced by the cutoff threshold
$\rho_{o}^{1}$ on the total outage probability, which is defined as
$\mathcal{O}_{t}=\mathcal{O}_{p}+(1-\mathcal{O}_{p})\mathcal{O}_{s}$ [16], for
the single tier mmWave network. It can be observed that $\rho_{o}^{1}$ tunes
the tradeoff between the truncation and SINR outage probabilities and there
exists a cutoff threshold $\rho_{o}^{1\star}$ that minimizes the total outage
probability in the single-tier network. Further, the SINR probability
dominates the total outage probability at lower cut-off threshold while the
truncation outage probability dominates the total outage probability at high
values of the cutoff threshold $\rho_{o}^{1}$.

####  VI-7 Dense Network Simulation

Figure 9: Dense network approximation of the SINR outage probability in a
single-tier mmWave network with $\alpha_{L}^{1}=2$, $\alpha_{N}^{1}=4$,
$R_{B}=200~{}\mathrm{m}$ and $\lambda_{0}=100$.

Figs. 9 and 10 shows the numerical results based on the dense network
approximations in Section V. In particular, Fig. 9 compares the dense network
approximation of the SINR outage probability of a single-tier mmWave network
given in Theorem V-A with the exact expression in Theorem III-B. For the dense
network approximation, we take the radius of the LOS disc $R_{B}$ to be equal
to $200~{}\mathrm{m}$ and a relative BS density $\lambda_{0}=100$ where
$\lambda_{0}=\lambda_{1}\pi R_{B}^{2}$. The dense network approximation
becomes more accurate as $L$ increases. Further, Fig. 10, compares the multi-
tier dense network approximation in Theorem V-A with the exact expression in
Theorem IV-B while focusing on a two-tier network. The first tier’s BS density
is obtained from $\lambda_{0}=\lambda_{1}\pi R_{B}^{2}$ while the second
tier’s BS density $\lambda_{2}=2\lambda_{1}$ and $L$ equals $10$. It can be
seen that similar to the single-tier network, the dense network approximation
of the SINR outage probability is also fairly accurate for the multi-tier
network.

Figure 10: Dense network approximation of the SINR outage probability in a
two-tier mmWave network with $\alpha_{L}^{1}=2$, $\alpha_{N}^{1}=4$,
$\alpha_{L}^{2}=2.9$, $\alpha_{N}^{2}=5$, $R_{B}=200~{}\mathrm{m}$,
$\lambda_{0}=100$, $\lambda_{2}=2\lambda_{1}$,
$G_{b1}^{\max}=10~{}\mathrm{dB}$ and $G_{b2}^{\max}=7~{}\mathrm{dB}$.

####  VI-8 Average User Transmit Power

In Fig. 11, we plot the average transmit power of the users against the cutoff
threshold $\rho_{0}^{1}$ for the single-tier mmWave network. It can be
observed that for the case with BS density $\lambda=1$, the average transmit
power increases with the cutoff threshold for $\rho_{o}^{1}$ ranging from
$-100$ to $-75$ dBm and it then falls for $\rho_{o}^{1}$ ranging from $-75$ to
$-40$ dBm. Afterwards average transmit power then rises with the cutoff
threshold until its saturation. Note that increasing $\rho_{o}^{1}$ the user
need to transmit a higher power to invert the pathloss and maintain a high
threshold at the serving BS [16]. However, each user is constrained to a
maximum power $P_{u}$. Hence, a user becomes inactive when its transmit power
requirement exceeds $P_{u}$. An initial increase in $\rho_{o}^{1}$ increases
the transmit power of all users and hence the first increase in the mean
transmit power. A point is reached where the density of active NLOS users
starts to decrease with increasing cutoff threshold since the maximum power
constraint cannot be satisfied and hence the reduction in the mean transmit
power. The large discrepancy between the pathloss exponent of the LOS and NLOS
users also means a large difference in the transmit power. However, a cutoff
threshold is reached where the active LOS user starts to dominate since most
NLOS users are inactive, thus leading to an increase in the mean transmit
power till its saturation value given bg
$\displaystyle\lim_{\rho_{o}^{1}\rightarrow\infty}\mathbb{E}[P_{k}]=\frac{2}{\min(\alpha_{L}^{1},\alpha_{N}^{1})+2}P_{u}$.
For dense deployment such as $\lambda=100$, most of the paths are LOS and the
transmit power is non-decreasing with $\rho_{o}$ in this case.

Figure 11: Average transmit power in a single tier mmWave network.

##  VII Conclusions

In this paper, we have presented a stochastic geometry based framework to
analyze the SINR outage probability in the uplink of both the single and
multi-tier mmWave cellular networks with truncation channel inversion power
control. The framework incorporates the effect of blockages, the per-user
power control as well as the maximum power limitations of the users. Further,
each user controls its transmit power such that the received signal at its
serving BS is equal to predefined cutoff threshold. Based on the proposed
framework, we derived accurate expressions of the truncation outage
probability and SINR outage probability for the uplink of both the single and
multi-tier mmWave cellular networks. Numerical results show that contrary to
the conventional ultra-high-frequency networks there exists a slow growth
region for the truncated outage probability. Furthermore, increasing the
cutoff threshold does not necessarily lead to a reduction in the SINR outage
probability of the mmWave networks.

###  -A Proof of Theorem III-A

Given that the pathloss exponent $\alpha$ is a random variable which takes the
values $\alpha_{L}$ and $\alpha_{N}$ with probability $e^{-\beta r}$ and
$1-e^{-\beta r}$, respectively, where $r$ is the length of the link. Because
the transmit power of a typical user $P_{s}=\rho_{o}r_{s}^{\alpha_{s}}$ for
$s=\\{L,N\\}$, $r_{s}\in\Phi$ is a function mapping $r_{s}$ in
$\mathbb{R}^{2}$ to $P_{s}$ in $\mathbb{R}$, according to the Mapping theorem
[28, Thm 2.34], $P_{s}$ forms an inhomogeneous PPP with the intensity measure
$\Lambda(P)=\Lambda_{L}(P)+\Lambda_{N}(P)$, where

$\Lambda_{L}(P)=\frac{2\pi\lambda}{\beta^{2}}\left(1-e^{-\beta\left(\frac{P}{\rho_{o}}\right)^{\frac{1}{\alpha_{L}}}}\left(1+\beta\left(\frac{P}{\rho_{o}}\right)^{\frac{1}{\alpha_{L}}}\right)\right),$
(27)
$\Lambda_{N}(P)=\pi\lambda\left(\frac{P}{\rho_{o}}\right)^{\frac{2}{\alpha_{N}}}-\frac{2\pi\lambda}{\beta^{2}}\left(1-e^{-\beta\left(\frac{P}{\rho_{o}}\right)^{\frac{1}{\alpha_{N}}}}\left(1+\beta\left(\frac{P}{\rho_{o}}\right)^{\frac{1}{\alpha_{N}}}\right)\right)$
(28)

[21] and the intensity function $\lambda(P)=\frac{\partial\Lambda(P)}{\partial
P}=\lambda_{L}(P)+\lambda_{N}(P)$, where

$\lambda_{L}=\frac{2\pi\lambda}{\alpha_{L}\rho_{o}^{\frac{2}{\alpha_{L}}}}P^{\frac{2}{\alpha_{L}}-1}e^{-\beta\left(\frac{P}{\rho_{o}}\right)^{\frac{1}{\alpha_{L}}}}$
(29)

and

$\lambda_{N}=\frac{2\pi\lambda}{\alpha_{N}\rho_{o}^{\frac{2}{\alpha_{N}}}}P^{\frac{2}{\alpha_{N}}-1}\left(1-e^{-\beta\left(\frac{P}{\rho_{o}}\right)^{\frac{1}{\alpha_{N}}}}\right).$
(30)

Therefore, the PDF of $P$ is given by

$f_{P}(p)=\frac{\lambda(p)e^{-\Lambda(p)}}{\int_{0}^{P_{u}}\lambda(y)e^{-\Lambda(y)}\mathrm{d}y},~{}~{}~{}~{}0\leq
p\leq P_{u}$ (31)

The PDF of $P$ has been normalized as a result of the truncated channel
inversion power control. Furthermore, the $\eta^{th}$ moment of $P$ is thus
given by $\int_{0}^{P_{u}}p^{\eta}f_{P}(p)$ and the theorem is obtained.

###  -B Proof of Theorem III-B

Noting that the average interference received from any interfering user
(normalized by $\mathcal{G}$) is strictly less than $\rho_{o}$. Consequently,
the sum interference received at the reference BS from LOS interferers can be
expressed from (6) as

$I_{L}=\\!\\!\\!\\!\\!\\!\sum_{u_{z}\in\Phi_{L}\backslash\\{o\\}}\\!\\!\\!\\!\\!\\!\mbox{1{1}}\\!\\!\left({P_{z}\|u_{z}\|^{-\alpha_{L}}}<\rho_{o}\\!\right)\\!\\!P_{z}G_{z}|g_{z}|^{2}|u_{z}\|^{-\alpha_{L}}$
(32)

where $\Phi_{L}$ is a PPP of LOS interferers, and $\mbox{1{1}}(.)$ is an
indicator function which takes the values of one when $(.)$ is true and zero
otherwise. Consequently, the Laplace transform of the aggregate interference
from LOS interferer $\mathcal{L}_{I_{L}}$ in (8) can be computed as

$\displaystyle\mathcal{L}_{I_{L}}$
$\displaystyle=\mathbb{E}_{\Phi_{L}}\left[e^{-snI_{L}}\right]$ (33)
$\displaystyle=\\!\mathbb{E}_{\Phi_{L}}\\!\\!\left[e^{-sn\sum_{u_{z}\in\Phi_{L}\backslash\\{o\\}}\\!\\!\mbox{1{1}}\\!\left({P_{z}\|u_{z}\|^{-\alpha_{L}}}<\rho_{o}\\!\right)\\!P_{z}G_{z}|g_{z}|^{2}|u_{z}\|^{-\alpha_{L}}}\right]$
$\displaystyle\overset{(b1)}{=}\\!\mathbb{E}_{P_{z},g_{z},G_{z}}\\!\\!\\!\left[\\!\prod_{u_{z}\in\Phi_{L}\backslash\\{o\\}}\\!\\!\\!\\!\\!\\!\\!\\!\\!e^{-sn\mbox{1{1}}\left(\\!\|u_{z}\|>\left(\\!\frac{P_{z}}{\rho_{o}}\\!\right)^{\frac{1}{\alpha_{L}}}\\!\right)\\!P_{z}G_{z}|g_{z}|^{2}|u_{z}\|^{-\alpha_{L}}}\right]$
$\displaystyle\overset{(b2)}{=}e^{\left(\\!-2\pi\lambda\sum_{v=1}^{4}\\!b_{v}\\!\int_{\left(\frac{P}{\rho_{o}}\right)^{\frac{1}{\alpha_{L}}}}^{\infty}\\!\\!\mathbb{E}_{P,g}\left[\left(1-e^{-sna_{v}Pgr^{-\alpha_{L}}}\right)\right]re^{-\beta
r}\mathrm{d}r\right)}$
$\displaystyle\overset{(b3)}{=}e^{\\!\left(\\!-2\pi\lambda\\!\sum_{v=1}^{4}\\!b_{v}\\!\\!\int_{\left(\frac{P}{\rho_{o}}\right)^{\frac{1}{\alpha_{L}}}}^{\infty}\\!\\!\\!\mathbb{E}_{P}\\!\left[\\!\left(\\!1-\\!\frac{1}{\left(1+sna_{v}Pr^{-\alpha_{L}}/N\right)^{N}}\\!\right)\\!\right]re^{-\beta
r}\mathrm{d}r\\!\right)}$ $\displaystyle=\prod_{v=1}^{4}e^{-2\pi\lambda
b_{v}\int_{\left(\frac{P}{\rho_{o}}\right)^{\frac{1}{\alpha_{L}}}}^{\infty}\mathbb{E}_{P}\left[\left(1-\frac{1}{\left(1+sna_{v}Pr^{-\alpha_{L}}/N\right)^{N}}\right)re^{-\beta
r}\right]\mathrm{d}r}$
$\displaystyle\overset{(b4)}{=}\\!\\!\prod_{v=1}^{4}\\!\\!e^{\\!-2\pi\lambda
q_{v}^{\frac{2}{\alpha_{L}}}\\!b_{v}\\!\int_{\mathcal{A}}^{\infty}\\!\\!\\!\int_{0}^{P_{u}}\\!\left(\\!\\!1\\!-\frac{1}{\left(\\!1+\frac{y^{-\alpha_{L}}}{N}\right)^{N}}\\!\\!\right)\\!yP^{\frac{2}{\alpha_{L}}}\\!e^{-\beta\left(q_{v}P\right)^{\frac{1}{\alpha_{L}}}\\!\\!y}\\!f_{P}\mathrm{d}P\mathrm{d}y}$
$\displaystyle=e^{-Q_{n}},$

where $(b1)$ follows from the independence of $\Phi_{L}$, $g_{z}$, $G_{z}$ and
$P_{z}$, $(b2)$ follows from the probability generation functional (PGFL) of
the PPP [9] and the independence of the interference link directivity gain
$G_{z}$ with probability distribution $G_{z}=a_{v}$ with probability $b_{v}$,
$(b3)$ follows from from computing the moment generating function of a gamma
random variable $g$, $(b4)$ is obtained by changing the variables
$y={r}/{\left(sna_{v}P\right)^{\frac{1}{\alpha_{L}}}}$ while $f_{P}$ is given
in (1). Further,
$\mathcal{A}=\left(sna_{v}\rho_{o}\right)^{-\frac{1}{\alpha_{L}}}$ and
$q_{v}=sna_{v}$. Similarly, the for the NLOS interfering links,
$\mathcal{L}_{I_{N}}$can be computed as

$\displaystyle\mathcal{L}_{I_{N}}$
$\displaystyle=\mathbb{E}_{\Phi_{N}}\left[e^{-snI_{N}}\right]$ (34)
$\displaystyle=\\!\prod_{v=1}^{4}\\!e^{\\!-2\pi\lambda
q_{v}^{\frac{2}{\alpha_{N}}}\\!b_{v}\\!\int_{\mathcal{B}}^{\infty}\\!\\!\int_{0}^{P_{u}}\\!\left(\\!1-\frac{1}{\left(1+\frac{y^{-\alpha_{N}}}{N}\right)^{N}}\\!\right)\\!yP^{\frac{2}{\alpha_{N}}}Z(y,p)f_{P}\mathrm{d}P\mathrm{d}y}$
$\displaystyle=e^{-V_{n}},$

where $\mathcal{B}=\left(sna_{v}\rho\right)^{-\frac{1}{\alpha_{N}}}$ and
$Z(y,p)=\left(1-e^{-\beta\left(q_{v}P\right)^{\frac{1}{\alpha_{N}}}y}\right)$.

###  -C Proof of Theorem IV-A

Given that
$y_{k}=\displaystyle\min_{m_{k}\in\Phi_{k}}(||u-m_{k}||^{\alpha_{s}^{k}})$ is
used to select the serving BS in the $k^{th}$ tier. Then
$f_{y_{k}}(y)=\lambda_{k}(y)e^{-\Lambda_{k}(y)}$ where

$\displaystyle\lambda_{k}(y)=\frac{2\pi\lambda_{k}}{\alpha_{L}^{k}}y^{\frac{2}{\alpha_{L}^{k}}-1}e^{-\beta_{k}{y}^{\frac{1}{\alpha_{L}^{k}}}}+\frac{2\pi\lambda_{k}}{\alpha_{N}^{k}}y^{\frac{2}{\alpha_{N}^{k}}-1}\left(1-e^{-\beta_{k}{y}^{\frac{1}{\alpha_{N}^{k}}}}\right)$
(35)

and

$\displaystyle\Lambda_{k}(y)=\frac{2\pi\lambda_{k}}{\beta_{k}^{2}}\left(1-e^{-\beta_{k}{y}^{\frac{1}{\alpha_{L}^{k}}}}\left(1+\beta_{k}{y}^{\frac{1}{\alpha_{L}^{k}}}\right)\right)+\pi\lambda_{k}{y}^{\frac{2}{\alpha_{N}^{k}}}-\frac{2\pi\lambda_{k}}{\beta_{k}^{2}}\left(1-e^{-\beta_{k}{y}^{\frac{1}{\alpha_{N}^{k}}}}\left(1+\beta_{k}{y}^{\frac{1}{\alpha_{N}^{k}}}\right)\right).$
(36)

Noting that the user connects to the BS that provides the maximum average
received signal, we can follow the same approach for the multi-tier UHF
network in [16]. The typical user $u$ connects to the reference BS from the
$j^{th}$ tier, then $y_{j}=\displaystyle\min_{k}(y_{k})$. The transmit power
of the typical user connected to the reference user in the $j^{th}$ tier is
given by $P_{j}=\rho_{o}^{j}\displaystyle\min_{k}(y_{k})$ where $P_{j}\leq
P_{u}$. Consequently, the cumulative distribution function (CDF) of the
transmit power can be expressed as

$F_{P_{j}}(p)=\frac{1-e^{-\sum_{k=1}^{K}{\Lambda_{k}\left(\frac{p}{\rho_{o}^{j}}\right)}}}{1-e^{-\sum_{k=1}^{K}{\Lambda_{k}\left(\frac{P_{u}}{\rho_{o}^{j}}\right)}}}$
(37)

and the PDF of the transmit power is given as

$\displaystyle f_{P_{j}}(p)$
$\displaystyle=\frac{\mathrm{d}F_{P_{j}}(p)}{\mathrm{d}p}$ (38)
$\displaystyle=\frac{\sum_{k=1}^{K}\overline{\lambda}_{k}(p)}{1-e^{-\sum_{a=1}^{K}{\Lambda_{a}\left(\frac{P_{u}}{\rho_{o}^{j}}\right)}}}{e^{-\sum_{b=1}^{K}{\Lambda_{b}\left(\frac{p}{\rho_{o}^{j}}\right)}}}$

where

$\overline{\lambda}_{k}(p)=\frac{2\pi\lambda_{k}}{\alpha_{L}^{k}{\rho_{o}^{j}}^{2/\alpha_{L}^{k}}}y^{\frac{2}{\alpha_{L}^{k}}-1}e^{-\beta_{k}\left(\frac{y}{\rho_{o}^{j}}\right)^{\frac{1}{\alpha_{L}^{k}}}}+\frac{2\pi\lambda_{k}}{\alpha_{N}^{k}{\rho_{o}^{j}}^{2/\alpha_{N}^{k}}}y^{\frac{2}{\alpha_{N}^{k}}-1}\left(1-e^{-\beta_{k}\left(\frac{y}{\rho_{o}^{j}}\right)^{\frac{1}{\alpha_{N}^{k}}}}\right)$
(39)

and $\Lambda_{c}(.)$ is given in (36)

###  -D Proof of Theorem IV-B

Noting that the average interference received from any interfering user from
the $k^{th}$ tier is less than $\rho_{o}^{k}$. The sum interference received
at the reference BS in the $j^{th}$ tier from LOS users in the $k^{th}$ can be
expressed as

$I_{L}^{k}=\sum_{u_{z}\in\Phi_{L}^{k}\backslash\\{o\\}}{\mbox{1{1}}\left(P_{zk}||u_{z}||^{-\alpha_{L}^{j}}<\rho_{o}^{k}\right)P_{zk}G_{z}|g_{z}|^{2}||u_{z}||^{-\alpha_{L}^{j}}},$
(40)

where $\Phi_{L}^{k}$ is a PPP of LOS interfering users from the $k^{th}$ tier.
The indicator function is used to capture the correlation among the location
of the interfering users and the location of the reference BS. Hence, the
Laplace transform of the aggregate interference from LOS users in the $k^{th}$
tier received by the reference mmWave BS in the $j^{th}$ tier
$\mathcal{L}_{I_{L}^{k}}$ can be computed as

$\displaystyle\mathcal{L}_{I_{L}^{k}}$
$\displaystyle=\mathbb{E}_{\Phi_{L}^{k}}\left[e^{-snI_{L}^{k}}\right]$ (41)
$\displaystyle=\mathbb{E}_{\Phi_{L}^{k}}\left[e^{-sn\sum_{u_{z}\in\Phi_{L}^{k}\backslash\\{o\\}}{\mbox{1{1}}\left(P_{zk}||u_{z}||^{-\alpha_{L}^{j}}<\rho_{o}^{k}\right)P_{zk}G_{z}|g_{z}|^{2}||u_{z}||^{-\alpha_{L}^{j}}}}\right]$
$\displaystyle=\exp\left(-2\pi\lambda_{k}\sum_{v=1}^{4}b_{v}^{j}\int\limits_{\left(\frac{P_{k}}{\rho_{o}^{k}}\right)^{\frac{1}{\alpha_{L}^{j}}}}^{\infty}\mathbb{E}_{P_{k},g}\left[\left(1-e^{-sna_{v}^{j}P_{k}gr^{-\alpha_{L}^{j}}}\right)\right]re^{-\beta_{j}r}\mathrm{d}r\right)$
$\displaystyle=\exp\left(-2\pi\lambda_{k}\sum_{v=1}^{4}b_{v}^{j}\int\limits_{\left(\frac{P_{k}}{\rho_{o}^{k}}\right)^{\frac{1}{\alpha_{L}^{j}}}}^{\infty}\mathbb{E}_{P_{k}}\left[\left(1-\frac{1}{\left(1+sna_{v}^{j}P_{k}r^{-\alpha_{L}^{j}}/N\right)^{N}}\right)re^{-\beta_{j}r}\right]\mathrm{d}r\right)$
$\displaystyle=e^{\left(-2\pi\lambda_{k}\sum_{v=1}^{4}b_{v}^{j}{q_{v}^{j}}^{\frac{2}{\alpha_{L}^{j}}}\int\limits_{\left(sna_{v}^{j}\rho_{o}^{k}\right)^{-\frac{1}{\alpha_{L}^{j}}}}^{\infty}\int\limits_{0}^{P_{u}}\left(1-\frac{1}{\left(1+\frac{y^{-\alpha_{L}^{j}}}{N}\right)^{N}}\right)yP_{k}^{\frac{2}{\alpha_{L}^{j}}}e^{-\beta_{j}\left(q_{v}^{j}P_{k}\right)^{\frac{1}{\alpha_{L}^{j}}}y}f_{P_{k}}\mathrm{d}P\mathrm{d}y\right)}$
$\displaystyle=e^{-Q_{n}^{k}}$

Similarly, the Laplace transform of the aggregate interference from NLOS users
in the $k^{th}$ tier received by the reference BS in the $j^{th}$ tier
$\mathcal{L}_{I_{N}^{k}}$ can be expressed as

$\displaystyle\mathcal{L}_{I_{N}^{k}}$
$\displaystyle=\mathbb{E}_{\Phi_{N}^{k}}\left[e^{-snI_{N}^{k}}\right]$ (42)
$\displaystyle=e^{\left(-2\pi\lambda_{k}\sum_{v=1}^{4}b_{v}^{j}{q_{v}^{j}}^{\frac{2}{\alpha_{N}^{j}}}\int\limits_{\left(sna_{v}^{j}\rho_{o}^{k}\right)^{-\frac{1}{\alpha_{N}^{j}}}}^{\infty}\int\limits_{0}^{P_{u}}\left(1-\frac{1}{\left(1+\frac{y^{-\alpha_{N}^{j}}}{N}\right)^{N}}\right)yP_{k}^{\frac{2}{\alpha_{N}^{j}}}\left(1-e^{-\beta_{j}\left(q_{v}^{j}P_{k}\right)^{\frac{1}{\alpha_{N}^{j}}}y}\right)f_{P_{k}}\mathrm{d}P\mathrm{d}y\right)}=e^{-V_{n}^{k}}$

###  -E Proof of Theorem V-A

The proof is based on the key assumptions in Appendix -B such that

$\displaystyle\mathcal{L}_{I_{L}^{k}}=\mathbb{E}_{\Phi_{L}^{k}}\left[-slI_{L}^{k}\right]=\mathbb{E}_{\Phi_{L}^{k}}\left[e^{-sl\sum_{u_{z}\in\Phi_{L}^{k}\cap\mathcal{B}(0,R_{B})\backslash\\{o\\}}{\mbox{1{1}}\left(P_{zk}||u_{z}||^{-\alpha_{L}^{j}}<\rho_{o}^{k}\right)P_{zk}G_{z}||u_{z}||^{-\alpha_{L}^{j}}}}\right]$
$\displaystyle\overset{(e1)}{=}\exp\left(\mathbb{E}_{P_{k}}\left[\sum_{v=1}^{4}-2\pi\lambda_{k}b_{v}^{j}\int_{\left(\frac{P_{k}}{\rho_{o}^{k}}\right)^{\frac{1}{\alpha_{L}^{j}}}}^{R_{B}}\left(1-e^{slP_{k}a_{v}^{j}r^{-\alpha_{L}^{j}}}\right)r\mathrm{d}r\right]\right)$
$\displaystyle\overset{(e2)}{=}\exp\left(\mathbb{E}_{P_{k}}\left[\pi\lambda_{k}\left(\left(\frac{P_{k}}{\rho_{o}^{k}}\right)^{\frac{2}{\alpha_{L}^{j}}}-R_{B}^{2}\right)+\sum_{v=1}^{4}\frac{2\pi\lambda_{k}b_{v}^{j}}{\alpha_{L}^{j}}\left(sla_{L}^{j}\right)^{\frac{2}{\alpha_{L}^{j}}}P_{k}^{\frac{2}{\alpha_{L}^{j}}}\int_{sla_{v}^{j}P_{k}R_{B}^{-\alpha_{L}^{j}}}^{sla_{v}^{j}\rho_{o}^{k}}\frac{e^{-w}}{w^{1+\frac{2}{\alpha_{L}^{j}}}}\mathrm{d}v\right]\right)$
$\displaystyle\overset{(e3)}{=}e^{\left(\int_{0}^{P_{u}}\left(\pi\lambda_{k}\left(\left(\frac{P_{k}}{\rho_{o}^{k}}\right)^{\frac{2}{\alpha_{L}^{j}}}-R_{B}^{2}\right)+\sum_{v=1}^{4}\frac{2\pi\lambda_{k}b_{v}^{j}}{\alpha_{L}^{j}}\left(sla_{L}^{j}\right)^{\frac{2}{\alpha_{L}^{j}}}P_{k}^{\frac{2}{\alpha_{L}^{j}}}\left(\Gamma\left(\frac{-2}{\alpha_{L}^{j}},\frac{\eta
l\theta_{j}a_{v}^{j}P_{k}}{\rho_{o}^{j}\mathcal{G}_{j}R_{B}^{\alpha_{L}^{j}}}\right)-\Gamma\left(\frac{-2}{\alpha_{L}^{j}},\frac{\eta
l\theta_{j}a_{v}^{j}}{\mathcal{G}_{j}}\right)\right)\right)f_{P_{k}}\mathrm{d}P_{k}\right)},$
(43)

where $(e1)$ follows from computing the PGFL of the PPP $\Phi_{L}^{k}$, $(e2)$
follow from a change of variable $w=sla_{v}^{j}P_{k}r^{-\alpha_{L}^{j}}$ and
$(e3)$ follows from the simplified PDF of the transmit power over the LOS
region. The PDF of the uplink transmit power in the $k^{th}$ tier of dense
mmWave networks can be obtained by noting that
$P_{k}=\rho_{o}^{k}r_{k}^{\alpha_{L}}$, where $r_{k}$ is the $k^{th}$ tier
uplink distance in the dense deployment which follows a Rayleigh distribution
$f_{r_{k}}(r)=2\pi\lambda re^{-\pi\lambda r^{2}},~{}0\leq r\leq\infty$.
Consequently, following the same approach in Appendix -C we obtain the PDF of
the $k^{th}$ tier transmit power as

$f_{P_{j}}(x)=\frac{\sum_{k=1}^{K}\frac{2\pi\lambda_{k}p^{\frac{2}{\alpha_{L}^{k}}-1}}{\alpha_{L}^{k}\left(\rho_{o}^{j}\right)^{\frac{2}{\alpha_{L}^{k}}}}}{1-e^{-\sum_{a=1}^{K}\pi\lambda_{a}\left(\frac{P_{u}}{\rho_{o}^{j}}\right)^{\frac{2}{\alpha_{L}^{a}}}}}e^{-\sum_{b=1}^{K}\pi\lambda_{b}\left(\frac{p}{\rho_{o}^{j}}\right)^{\frac{2}{\alpha_{L}^{b}}}}$
(44)

Finally, the approximation of the SINR outage probability in the uplink of a
multi-tier mmWave network given in (23) can be obtained by substituting (-E)
for (V-A) and with $\lambda_{0}=\lambda\pi R_{B}^{2}$.

## References

  * [1] J. Andrews, S. Buzzi, W. Choi, S. Hanly, A. Lozano, A. Soong, and J. Zhang, “What will 5G be?” _IEEE J. Sel. Areas Commun._ , vol. 32, no. 6, pp. 1065–1082, Jun. 2014.
  * [2] Z. Pi and F. Khan, “An introduction to millimeter-wave mobile broadband systems,” _IEEE Commun. Mag._ , vol. 49, no. 6, pp. 101–107, Jun. 2011\.
  * [3] T. Rappaport, S. Sun, R. Mayzus, H. Zhao, Y. Azar, K. Wang, G. Wong, J. Schulz, M. Samimi, and F. Gutierrez, “Millimeter wave mobile communications for 5G cellular: It will work!” _IEEE Access_ , vol. 1, pp. 335–349, 2013.
  * [4] T. Rappaport, F. Gutierrez, E. Ben-Dor, J. Murdock, Y. Qiao, and J. Tamir, “Broadband millimeter-wave propagation measurements and models using adaptive-beam antennas for outdoor urban cellular communications,” _IEEE Trans. Antennas Propag._ , vol. 61, no. 4, pp. 1850–1859, Apr. 2013\.
  * [5] S. Rangan, T. Rappaport, and E. Erkip, “Millimeter-wave cellular wireless networks: Potentials and challenges,” _Proc. IEEE_ , vol. 102, no. 3, pp. 366–385, Mar. 2014.
  * [6] M. Akdeniz, Y. Liu, M. Samimi, S. Sun, S. Rangan, T. Rappaport, and E. Erkip, “Millimeter wave channel modeling and cellular capacity evaluation,” _IEEE J. Sel. Areas Commun._ , vol. 32, no. 6, pp. 1164–1179, Jun. 2014\.
  * [7] W. Roh, J.-Y. Seol, J. Park, B. Lee, J. Lee, Y. Kim, J. Cho, K. Cheun, and F. Aryanfar, “Millimeter-wave beamforming as an enabling technology for 5G cellular communications: Theoretical feasibility and prototype results,” _IEEE Commun. Mag._ , vol. 52, no. 2, pp. 106–113, Feb. 2014.
  * [8] T. Bai and R. Heath, “Coverage and rate analysis for millimeter-wave cellular networks,” _IEEE Trans. Wireless Commun._ , vol. 14, no. 2, pp. 1100–1114, Feb. 2015.
  * [9] J. Andrews, F. Baccelli, and R. Ganti, “A tractable approach to coverage and rate in cellular networks,” _IEEE Trans. Commun._ , vol. 59, no. 11, pp. 3122–3134, Nov. 2011.
  * [10] X. Zhang and J. Andrews, “Downlink cellular network analysis with multi-slope path loss models,” _IEEE Trans. Commun._ , vol. 63, no. 5, pp. 1881–1894, May 2015.
  * [11] M. Ding, D. López-Pérez, G. Mao, P. Wang, and Z. Lin, “Will the area spectral efficiency monotonically grow as small cells go dense?” in _IEEE Globecom_ , San Diego USA, Dec 2015, pp. 1–7.
  * [12] H.-S. Jo, Y. J. Sang, P. Xia, and J. Andrews, “Heterogeneous cellular networks with flexible cell association: A comprehensive downlink SINR analysis,” _IEEE Trans. Wireless Commun._ , vol. 11, no. 10, pp. 3484–3495, Oct. 2012\.
  * [13] S. Singh, H. S. Dhillon, and J. G. Andrews, “Offloading in heterogeneous networks: Modeling, analysis, and design insights,” _IEEE Trans. Wireless Commun._ , vol. 12, no. 5, pp. 2484–2497, 2013.
  * [14] M. Di Renzo, A. Guidotti, and G. Corazza, “Average rate of downlink heterogeneous cellular networks over generalized fading channels: A stochastic geometry approach,” _IEEE Trans. Commun._ , vol. 61, no. 7, pp. 3050–3071, Jul. 2013.
  * [15] T. Novlan, H. Dhillon, and J. Andrews, “Analytical modeling of uplink cellular networks,” _IEEE Trans. Wireless Commun._ , vol. 12, no. 6, pp. 2669–2679, Jun. 2013.
  * [16] H. ElSawy and E. Hossain, “On stochastic geometry modeling of cellular uplink transmission with truncated channel inversion power control,” _IEEE Trans. Wireless Commun._ , vol. 13, no. 8, pp. 4454–4469, 2014.
  * [17] J. Zhang, L. Xiang, D. W. K. Ng, M. Jo, and M. Chen, “Energy efficiency evaluation of multi-tier cellular uplink transmission under maximum power constraint,” _IEEE Trans Wireless Commun_ , vol. 16, pp. 7092–7107, Nov 2017\.
  * [18] O. Onireti, A. Imran, M. Imran, and R. Tafazolli, “Energy efficient inter-frequency small cell discovery in heterogeneous networks,” _IEEE Trans. Veh. Technol._ , vol. 65, no. 9, pp. 7122–7135, Sep 2016.
  * [19] J. Andrews, A. K. Gupta, and H. S. Dhillon, “A primer on cellular network analysis using stochastic geometry,” Apr 2016, [Online]: Available: http://arxiv.org/abs/1604.03183.
  * [20] M. D. Renzo, “Stochastic geometry modeling and analysis of multi-tier millimeter wave cellular networks,” _IEEE Trans. Wireless Commun._ , vol. 14, no. 9, pp. 5038–5057, Sep 2015.
  * [21] D. Maamari, N. Devroye, and D. Tuninetti, “Coverage in mmwave cellular networks with base station co-operation,” _IEEE Trans Wireless Commun_ , vol. 15, no. 4, pp. 2981–2994, Apr 2016.
  * [22] C. Fang, B. Makki, and T. Svensson, “Coverage analysis for millimeter wave uplink cellular networks with partial zero-forcing receivers,” in _International Symposium on Modeling and Optimization in Mobile, Ad Hoc, and Wireless Networks (WiOpt)_. 1-6, May 2017.
  * [23] O. Onireti, A. Imran, and M. Imran, “Coverage analysis in the uplink of mmwave cellular network,” in _European Conference on Networks and Communications (EuCNC)_ , Jun 2017, pp. 1–6.
  * [24] O. Onireti, A. Imran, and M. A. Imran, “Coverage, capacity and energy efficiency analysis in the uplink of mmwave cellular networks,” _IEEE Trans. Veh. Technol._ , vol. 67, no. 5, pp. 3982–3997, May 2018.
  * [25] T. Bai, R. Vaze, and R. Heath, “Analysis of blockage effects on urban cellular networks,” _IEEE Trans. Wireless Commun._ , vol. 13, no. 9, pp. 5070–5083, Sep. 2014.
  * [26] H. Alzer, “On some inequalities for the incomplete Gamma function,” _Math. Comput.,_ , vol. 66, no. 218, pp. 771–778, Apr. 1997.
  * [27] A. Guo and M. Haenggi, “Spatial stochastic models and metrics for the structure of base stations in cellular networks,” _IEEE Trans. Wireless Commun._ , vol. 12, no. 11, pp. 5800–5812, Nov 2013.
  * [28] M. Haenggi, _Stochastic Geometry for Wireless Networks_. Cambridge University Press, 2013.
  * [29] H. ElSawy, E. Hossain, and M. Haenggi, “Stochastic geometry for modeling, analysis, and design of multi-tier and cognitive cellular wireless networks: A survey,” _IEEE Commun. Surv. Tutorials_ , vol. 15, no. 3, pp. 996–1019, Jul 2013.