# Reliability and User-Plane Latency Analysis of mmWave Massive MIMO for
Grant-Free URLLC Applications

Joao V. C. Evangelista, Georges Kaddoum, , Zeeshan Sattar J. V. C. Evangelista
and G. Kaddoum were with the Department of Electrical Engineering, Ecole de
Technologie Supérieure, Montreal, QC, H3C 1K3 CA, e-mail: joao-victor.de-
<EMAIL_ADDRESS>and<EMAIL_ADDRESS>Z. Sattar
was with Ericsson Canada, Ottawa, ON, K2K 2V6 CA, email:
<EMAIL_ADDRESS>
###### Abstract

5G cellular networks are designed to support a new range of applications not
supported by previous standards. Among these, ultra-reliable low-latency
communication (URLLC) applications are arguably the most challenging. URLLC
service requires the user equipment (UE) to be able to transmit its data under
strict latency constraints with high reliability. To address these
requirements, new technologies, such as mini-slots, semi-persistent scheduling
and grant-free access were introduced in 5G standards. In this work, we
formulate a spatiotemporal mathematical model to evaluate the user-plane
latency and reliability performance of millimetre wave (mmWave) massive
multiple-input multiple-output (MIMO) URLLC with reactive and $K$-repetition
hybrid automatic repeat request (HARQ) protocols. We derive closed-form
approximate expressions for the latent access failure probability and validate
them using numerical simulations. The results show that, under certain
conditions, mmWave massive MIMO can reduce the failure probability by a factor
of $32$. Moreover, we identify that beyond a certain number of antennas there
is no significant improvement in reliability. Finally, we conclude that mmWave
massive MIMO alone is not enough to provide the performance guarantees
required by the most stringent URLLC applications.

###### Index Terms:

URLLC, massive MIMO, Spatiotemporal, latency, millimeter wave

##  I Introduction

The 3rd generation partnership project (3GPP) has identified three distinct
use cases for 5G new radio (NR) and beyond cellular networks based on their
different connectivity requirements: enhanced mobile broadband (eMBB), massive
machine-type communication (mMTC) and ultra reliable low-latency communication
(URLLC) [1]. Since the inception of the idea of 5G NR, it has been argued that
its main revolution is a change of paradigm from a smartphone-centric network
to a network capable of satisfying the requirements of diverse services, such
as machine-to-machine and vehicle-to-vehicle communications [2]. The URLLC
scenario targets applications that require high reliability and low latency,
such as augmented reality (AR), virtual reality (VR), vehicle-to-everything
(V2X), critical internet of things (cIoT), industrial automation and
healthcare. According to the use cases defined in [3], the main key
performance indicator (KPI) to be satisfied in URLLC applications is the
latent access failure probability, which incorporates the reliability and
latency requirements needed in such applications. The requirements for URLLC
applications vary from $1-10^{-5}$ transmission reliability to transmit $32$
bytes of data with a user-plane latency of less than $1$ ms to a $1-10^{-5}$
reliability to transmit $300$ bytes with a user-plane latency of between $3$
and $10$ ms, depending on the application [3].

lte and prior networks were not designed with such constraints in mind.
Scheduling in long-term evolution (LTE) follows a grant-based approach, where
the user equipment (UE) must request resources in a $4$-step random access
(RA) procedure before transmitting data [4]. In the best-case scenario, it
takes at least $10$ ms for a UE to start transmitting its payload.

Therefore, new mechanisms were introduced into the 5G NR specification to
support the latency requirements of URLLC applications. Firstly, a flexible
numerology was proposed, introducing the concept of a mini-slot that last as
little as $0.125$ ms [5], in contrast to the $1$ ms minimum slot duration on
LTE, enabling fine-grained scheduling of network resources [6]. Secondly, the
introduction of semi-persistent scheduling (SPS) of grants [7, 8], where some
of the networkś resource blocks are periodically reserved for URLLC
applications, thereby avoiding the grant request procedure. Despite the
efforts, not all URLLC applications have a periodic traffic pattern and are
therefore unable to benefit greatly from SPS. Additionally, some services
require low latency and reliable transmission to transmit small sporadic
packets. With that in mind, both the standards committee [9] and researchers
have put a lot of effort to investigate grant-free transmission, where the UEs
transmit their payload directly in the RA channel. This culminated with the
introduction of the $2$-step RA procedure introduced in Release 16 [9]. The
$2$-step RA procedure follows a grant-free approach, where instead of waiting
for a dedicated channel to be assigned by the network, it transmits its data
directly into the RA channel and waits for feedback from the network [10].

Moreover, massive multiple input multiple output (MIMO) is a fundamental part
of 5G NR [11, 12, 13]. It provides performance gains by improving diversity
against fading and, along with advanced signal processing techniques, can
provide directivity to transmission/reception, mitigating interference between
spatially uncorrelated UEs [14]. The performance enhancements provided by MIMO
are essential to ensure the reliability and the low latency required by URLLC
applications. In conjunction with massive MIMO comes millimeter wave (mmWave)
transmission. Due to its small wavelength, mmWave antennas can be packed into
massive arrays, making it a key enabler of massive MIMO systems which
attracted a significant interest on the topic [15, 16, 17, 18, 19, 20].
However, mmWave propagation comes with its own challenges due to the severe
propagation loss experienced by electromagnetic signals in this frequency
range.

In this paper, we develop a spatiotemporal analytical model to evaluate the
performance of mmWave massive MIMO communication systems for URLLC
applications. We use tools from stochastic geometry and probability theory to
evaluate and compare system performance metrics by deriving closed-form
approximate expressions for its latent access failure probability under
different hybrid automatic repeat request (HARQ) protocols.

###  I-A Related Work

In [21], the authors propose a queueing model to compare the throughput
performance of packet-based (grant-free) and connection-based (grant-based)
random access. They conclude that packet-based systems with sensing can
achieve greater throughput than connection-based one for small packet
transmissions. In [22, 23], the optimization of grant-free access networks is
investigated. The former considers the dynamic optimization of HARQ and
scheduling parameters with non-orthogonal multiple access (NOMA), while the
former considers the distributed link adaptation problem. Both papers
formulate the respective optimization tasks as multiagent reinforcement
learning (MARL) problems. The probability of success and the area spectral
efficiency of a grant-free sparse code multiple access (SCMA) system is
evaluated in [24, 25], in an mMTC context, using stochastic geometry. However,
none of the works consider the temporal aspects of the system, which are
crucial to analyze the latency and reliability of URLLC service. In [26], the
probability of success of grant-free RA with massive MIMO in the sub-6 GHz
band is investigated, and analytical expressions are derived for conjugate and
zero-forcing beamforming. Despite its contribution, the authors do not
evaluate the systemś temporal behavior, which is fundamental to characterize
URLLC service’s performance. Moreover, due to its distinct propagation
characteristics, this model is unsuitable for mmWave frequency bands.

The authors in [27] evaluate the scalability of scheduled uplink (grant-based)
and random access (grant-free) transmissions in massive internet of things
(IoT) networks, although they frame the problem through a revolutionary
spatiotemporal framework, fusing stochastic geometry and queueing theory. They
conclude that grant-free transmission offers lower latency, however, it does
not scale well to a massive number of devices. In our work, we show that using
massive MIMO base stations is a viable solution to address the scalability
issues of grant-free transmission without sacrificing its latency, rendering
it particularly suitable for URLLC applications. In [28], the authors use a
similar spatiotemporal model to characterize the performance of different RA
schemes with respect to the probability of a successful preamble transmission
in a grant-based massive IoT system. They conclude that a backoff scheme
performs close to optimally in diverse traffic conditions. In [29], the
authors perform system-level simulations of a grant-free URLLC network under
different HARQ configurations, and compare it to a baseline grant-based
system. They conclude that grant-free systems provide significantly lower
latency at the $1-10^{-5}$ reliability level. The same scenario is evaluated
in [30], however, the authors characterize system performance analytically,
using a stochastic geometry-based spatiotemporal model. This paper identifies
the suitability of each HARQ scheme for different network loads and received
power levels.

Stochastic geometry has become the de facto tool for analyzing large networks
[31, 32, 33], and has been successfully used to investigate the performance of
MIMO systems for a while now [34, 35, 36, 37]. In [38], a unified stochastic
geometric mathematical model for MIMO cellular networks with retransmission is
proposed. In [39], a stochastic geometry-based analytical model for the
performance of downlink mmWave NOMA systems is developed. The authors propose
two random beamforming methods that are able to reduce system overhead while
providing performance gains for BSs with a large number of antennas.

We seek to answer the following main questions that are to the best of our
knowledge missing from the current literature:

  * •

How do we formulate a tractable spatiotemporal model to investigate the
reliability and latency of URLLC applications powered by BSs equipped with
massive antenna arrays operating on mmWave frequencies?

  * •

What closed-form analytical expressions can we derive for the latent access
failure probability in this scenario?

  * •

What are the performance gains obtained from increasing the number of antennas
at the BS, and what are the limitations?

###  I-B Contributions

This paper makes three major contributions:

  * •

We formulate a mathematical model to evaluate the performance of mmWave
massive MIMO on uplink grant-free URLLC networks with HARQ. This model uses
stochastic geometry to capture the spatial configuration of the UEs and the
BSs, a mmWave channel model, and probability theory to obtain the temporal
characteristics necessary to evaluate the performance of URLLC applications.

  * •

We derive closed-form approximate expressions for the latent access failure
probability using reactive and $K$-repetition HARQ schemes. To the best of our
knowledge, no previous works has presented closed-form analytic expressions
for this key performance measure of URLLC applications in a mmWave massive
MIMO communication system.

  * •

We analyze the system performance for an extensive range of scenarios,
identifying the gains and limitations provided by using the mmWave spectrum
together with a massive number of antennas at the BS, and identify the
scenarios that benefit the most from these two technologies.

###  I-C Notation and Organization

Italic Roman and Greek letters denote deterministic and random variables,
while bold letters denote deterministic and random vectors. The capital Greek
letter $\Phi$ denotes a point process and $x\in\Phi$ represents a point
belonging to said process. The notation $\Phi(A)$, where $A\in\mathbb{R}^{d}$,
is the counting process associated with $\Phi$ [40]. Notice that we overload
the meaning of $\Phi$ so that it can signify a point process, a counting
measure or a set depending on the context. ${n\choose
k}=\frac{n!}{k!\left(n-k\right)!}$ is the binomial coefficient of $n$ choose
$k$.

The uniform, complex normal and binomial distributions are represented by
$Uniform(a,b)$, $\mathcal{CN}(\mu,\sigma^{2})$ and $Binomial(p)$,
respectively. The vector $\mathbf{x}^{H}$ is the Hermitian transpose of vector
$\mathbf{x}$. The function $\mathbb{P}(\cdot)$ denotes the probability of the
event within parentheses. The notation $\mathbf{1}\left\\{\cdot\right\\}$
denotes the indicator function, which is equal to one whenever the event
within curly braces is true and zero otherwise.

This work is divided into five sections and an appendix. In Section I, we
introduce the contents of the manuscript and contextualize within the relevant
literature. In Section II, we present a mathematical model to characterize the
performance of the grant-free mmWave massive MIMO system in URLLC
applications. In Section III, we derive the latent access failure probability
of the proposed system using reactive and $K$-repetition HARQ protocols. In
Section IV, we show the results of the system simulation. We use these results
to validate the analytical derivations, investigate the system’s performance
for an extensive range of parameters and finish it by interpreting the results
in the context of URLLC applications. In Section V, we summarize our findings
and present our conclusions. Finally, in the appendix, we show the detailed
proofs of the lemmas and theorems required by the derivations in the paper.

##  II System Model

Figure 1: Comparison of the transmission procedure in grant-free and grant-
based systems.

In most cellular applications, uplink transmissions use a dedicated resource
(frequency, time or a MIMO spatial layer) previously assigned by the network
to transmit their data payload. Thus, when an UE receives new data, it must
send a request for the network to schedule a resource. With dedicated
resources, each UE can utilize the wireless channel to its full capacity, thus
maintaining good quality of service (QoS). TIn 5G NR networks, the schedule
request consists of four steps, illustrated in Figure 1:

  * •

The UE randomly selects one of the available preambles and transmits it on the
physical random access channel (PRACH).

  * •

The BS transmits a random access request (RAR), acknowledging receipt of the
preamble and time-alignment commands.

  * •

The UE and BS exchange contention resolution messages (messages 3 and 4) that
are used to identify possible collisions arising from two different devices
transmitting the same preamble.

  * •

If the grant request is successful, the UE transmits its payload on the
physical uplink shared channel (PUSCH).

This grant-based scheme is efficient for applications that need to use the
channel multiple times to transmit large amounts of data (e.g., video
streaming) or data that’s being continuously generated (e.g., voice). However,
in some URLLC applications, UEs sporadically generate data that need to be
transmitted reliably and with low latency, such as cIoT and sensors for
industrial automation. In such scenarios, the time spent on the schedule
request renders grant-based schemes inefficient. A more suitable alternative
is to transmit the data directly on the PRACH and thereby avoidall the
overhead involved in requesting a grant, as illustrated in Fig. 1.
Nonetheless, with grant-free transmission comes the possibility of collisions
whenever two UEs randomly select the same preambles. Therefore, HARQ is used
to ensure the reliability and robustness of grant-free transmission. harq
consists of using feedback information from the BS so the UE can retransmit
packets that were not successfully received. Despite this, it can be quite
challenging to scale grant-free networks because wireless resources are finite
and expensive. To this end, massive MIMO and beamforming can be applied to
reduce the interference of spatially uncorrelated UEs and thereby increasing
the reliability of the system.

In this section, we discuss the spatial model of the network, the mmWave
channel model, the BS receiver beamforming procedure and the different HARQ
schemes used.

###  II-A Physical Layer Model

Stochastic geometry and the theory of random point processes has proven to be
able to accurately model the spatial distribution of modern cellular network
deployments [41]. Therefore, we consider a cell of radius $R$ consisting of a
BS, equipped with $K$ antennas, located at the origin. We model the spatial
location of the single-antenna UEs according to a homogeneous Poisson point
process (HPPP) [40], denoted by $\Phi_{U}$ with intensity $\lambda_{U}$.
Furthermore, the distance between the $i$-th UE, $x_{i}\in\Phi_{U}$, and the
BS is given by $\left\lVert x_{i}\right\rVert$. Both the distance from the UE
to the BS and its normalized angle from the BS are uniformly distributed
random variables [40], $\left\lVert x_{i}\right\rVert\sim Uniform(0,R)$ and
$\theta_{i}\sim Uniform(-1,1)$, respectively.

Due to path loss attenuation, the signal received from UEs located further
from the BS is “drowned” by the signal from closer users transmitting with the
same power, also known as the near-far problem. Uplink power control is
fundamental to deal with this issue. We consider that the UEs utilize path
loss inversion power control [42], with received power threshold $\rho$, where
each user controls its transmit power such that the average received power at
its associated BS is $\rho$, by selecting their transmit powers as
$p_{i}=\rho\left\lVert x_{i}\right\rVert^{\alpha}$, where $\alpha$ is the path
loss exponent. We assume that there are $N_{S}$ subcarriers reserved for
grant-free URLLC transmissions and $N_{P}$ orthogonal preambles. Thus, at each
transmission time interval (TTI), the active UEs select a subcarrier and
preamble randomly from the $N_{S}$ available subcarriers and $N_{P}$ available
preambles. Moreover, we assume that at $t=0$, one packet arrives to the
transmitting queue of each UE. Therefore, the HPPP of active users $\Phi_{A}$
on a specific subcarrier is obtained by thinning $\Phi_{U}$ [40] and its
effective intensity at $t=0$ is given by

$\lambda_{A}=\frac{\lambda_{U}}{N_{S}}.$ (1)

Massive MIMO technology and mmWave frequencies are intrinsically connected.
Even though one does not imply the other, they complement each other really
well. The former requires large antenna arrays, and the size of such arrays is
proportional to the targeted wavelength. Moreover, mmWave antennas must be
really small to operate in such large frequencies, therefore, a larger number
of them are necessary to gather enough energy. In this work, we consider that
the BS is equipped with a massive uniform linear array (ULA) containing $K\gg
1$ antennas operating at mmWave frequencies, while the UEs possess a single
antenna. The channel vector between user $i$ and the bs is given by

$\mathbf{h}_{i}=\sqrt{K}\left[\underbrace{\frac{g_{i,0}\mathbf{a}(\theta_{i}^{0})}{\sqrt{\left\lVert
x_{i}\right\rVert^{\alpha_{LOS}}}}}_{\text{LOS
component}}+\underbrace{\frac{\underset{j=1}{\overset{J}{\sum}}g_{i,j}\mathbf{a}(\theta_{i}^{j})}{\sqrt{\left\lVert
x_{i}\right\rVert^{\alpha_{NLOS}}}}}_{\text{NLOS components}}\right],$ (2)

where $g_{i,j}\sim\mathcal{CN}(0,1)$ is the complex gain on the $j$-th path
and $\theta^{j}_{i}$ is the normalized direction of the $j$-th path. We assume
that the complex gains of different paths are independent. $\alpha_{LOS}$ and
$\alpha_{NLOS}$ denote the path loss exponent of the line-of-sight (LOS) and
non-line-of-sight (NLOS) paths, respectively. The vector

$\mathbf{a}(\theta)=\frac{1}{\sqrt{K}}\begin{bmatrix}1&e^{-j\pi\theta}&\dots&e^{-j\pi(K-1)\theta}\end{bmatrix}^{T}$
(3)

denotes the phase of the signal received by each antenna. Due to high
penetration losses suffered by mmWave signals, the LOS path has a dominant
effect on channel gain, being $20$ dB larger than the NLOS in some cases [39,
37]. Hence, we can safely approximate $\mathbf{h}_{i}$ as

$\mathbf{h}_{i}\approx\sqrt{K}\frac{g_{i}\mathbf{a}(\theta_{i})}{\sqrt{\left\lVert
x_{i}\right\rVert^{\alpha}}}$ (4)

for mathematical tractability. Additionally, to avoid cluttering the notation,
we drop the subscripts denoting different paths and distinguishing LOS and
NLOS variables.

Due to the dominant effect of the LOS link, the channel model also needs to
consider a blockage model to determine the probability that the LOS path
between the UE the BS is obstructed. To model the effects of blockage, we
adopt the model proposed in [43]. This model is obtained by assuming that the
obstructing building and structures form an HPPP with random width, length and
orientation. So, let $LOS$ be the set of los UEs; then, the probability that
user $x_{i}$ has a LOS link is given by

$\mathbb{P}\left(x_{i}\in LOS\right)=\exp\left(-\beta\left\lVert
x_{i}\right\rVert\right),$ (5)

where $\beta$ is directly proportional to the density, and the average width
and length of obstructing structures. This model nicely captures the
exponentially vanishing probability of having a LOS link the further you move
away from the BS, and can be easily fitted to real urban scenarios.

#### Signal Model

At each TTI, the active users transmit an information signal $s_{i}$ such that
$\left\lvert s_{i}\right\rvert=1$. Therefore, the vector of the signal
received at the BS is given by

$\mathbf{y}=\underset{x_{i}\in\Phi_{A}}{\sum}\mathbf{1}\left\\{x_{i}\in
LOS\right\\}\sqrt{\rho}\mathbf{h}_{i}s_{i}+\mathbf{n},$ (6)

where $\mathbf{n}\sim\mathcal{CN}\left(0,\sigma_{2}\mathbf{I}\right)$ is a
circularly symmetric complex Gaussian random variable representing additive
white Gaussian noise (AWGN).

To successfully recover the data transmitted by a given user, the BS must be
able to accurately estimate its channel response.

######  Definition 1 (Preamble Collision).

Preamble collision event, denoted by $C$, happens when two or more devices
transmit the same preamble on the same subcarrier.

We assume that the BS is able to perfectly estimate the UE channel response
$\mathbf{h}_{i}$ whenever there is no preamble collision. Then, the BS
performs conjugate beamforming to separate the intended user’s signal from
those of the other interfering UEs by multiplying the received signal by the
Hermitian transpose of the intended user channel response. Therefore the
recovered signal of the intended user, the $j$-th user, is

$\displaystyle y_{j}$ $\displaystyle=$
$\displaystyle\mathbf{1}\left\\{x_{j}\in
LOS\right\\}\mathbf{1}\left\\{\bar{C}\right\\}\sqrt{\rho}\mathbf{h}_{j}^{H}\mathbf{h}_{j}s_{j}+$
$\displaystyle\underset{x_{i}\in\Phi_{A}\setminus\left\\{x_{j}\right\\}}{\sum}\mathbf{1}\left\\{x_{i}\in
LOS\right\\}\sqrt{\rho}\mathbf{h}_{j}^{H}\mathbf{h}_{i}s_{i}+\tilde{n}_{j},$

where $\bar{C}$ is the event when user $j$ does not experience preamble
collision and $\tilde{n}_{j}\sim\mathcal{CN}\left(0,\sigma^{2}\right)$ is a
linear combination of the noise vector, which is a Gaussian distributed random
variable. Therefore, the signal-to-interference-plus-noise-ratio (SINR)
experienced by user $j$ is given by

$\displaystyle\mathrm{SINR}$ (8) $\displaystyle=$
$\displaystyle\frac{\mathbf{1}\left\\{x_{j}\in
LOS\cap\bar{C}\right\\}\rho\left\lvert\mathbf{h}_{j}^{H}\mathbf{h}_{j}\right\rvert^{2}}{\underset{x_{i}\in\Phi_{A}\setminus\left\\{x_{j}\right\\}}{\sum}\mathbf{1}\left\\{x_{i}\in
LOS\right\\}\rho\left\lvert\mathbf{h}_{j}^{H}\mathbf{h}_{i}\right\rvert^{2}+\sigma^{2}}$
$\displaystyle=$
$\displaystyle\frac{\mathbf{1}\left\\{TX_{j}\right\\}\rho\left\lvert
g_{j}\right\rvert^{2}\left\lvert\mathbf{a}\left(\theta_{j}\right)^{H}\mathbf{a}\left(\theta_{i}\right)\right\rvert^{2}}{I+\sigma^{2}},$

where $TX_{j}=\\{x_{j}\in LOS\cap\bar{C}\\}$ is the probability that user $j$
has a LOS link and does not suffer from preamble collision, and
$I=\underset{x_{i}\in\Phi_{A}\setminus\left\\{x_{j}\right\\}}{\sum}\mathbf{1}\left\\{x_{i}\in
LOS\right\\}\rho\left\lvert
g_{i}\right\rvert^{2}\left\lvert\mathbf{a}\left(\theta_{j}\right)^{H}\mathbf{a}\left(\theta_{i}\right)\right\rvert^{2}$
is the interference from the other UEs. Moreover, the beamforming gain,
$\left\lvert\mathbf{a}\left(\theta_{j}\right)^{H}\mathbf{a}\left(\theta_{i}\right)\right\rvert^{2}$,
can be expressed as [44]

$\begin{split}\left\lvert\mathbf{a}\left(\theta_{j}\right)^{H}\mathbf{a}\left(\theta_{i}\right)\right\rvert^{2}&=F_{K}\left(\frac{\pi}{2}\left(\theta_{i}-\theta_{j}\right)\right)\\\
&=\frac{1}{K}\left\lvert\frac{\sin\left(\frac{K\pi}{2}\left(\theta_{i}-\theta_{j}\right)\right)}{\sin\left(\frac{\pi}{2}\left(\theta_{i}-\theta_{j}\right)\right)}\right\rvert^{2},\end{split}$
(9)

where $F_{K}(x)$ is the Fejer kernel [45], with $F_{K}(0)=K$.

Figure 2: Fejer kernel value for normalized angles of arrival varying from
$-0.2$ to $0.2$.

A useful property of the Fejer kernel is that [45]

$\underset{K\rightarrow\infty}{\lim}\int_{\delta\leq\left\lvert
x\right\rvert\leq\pi}F_{K}(x)dx=0,$ (10)

meaning that for an asymptotically large value of $K$, the interference for
the signals not aligned with beam angle $\theta_{j}$ goes to zero. Fig. 2
illustrates this property by plotting the Fejer kernel for increasing values
of $K$.

###  II-B HARQ Schemes

HARQ protocols determine how transmitters and receivers exchange information
about successful packet reception, by transmitting an acknowledgement (ACK)
signal, and how UEs retransmit in the event of failure, which is signaled by
the transmission of a negative acknowledgement (NACK) signal. They are
especially important to ensure reliability in grant-free transmission. The
HARQ protocol used also impacts the overall latency of the system. Hence, in
this paper, we investigate the performance of the massive MIMO URLLC network
under two distinct HARQ protocols.

With respect to transmissions latency, the HARQ protocols investigated have a
few aspects in common. First, the UE spends $T_{A}$ TTIs to process a newly
arrived packet. As soon as the packet is processed, it spends $T_{TX}$ TTIs
transmitting it. Upon receipt of the packet, the BS spends $T_{DP}$ TTIs to
process it and $T_{F}$ TTIs to send feedback and for it to reach the UE. Once
the UE receives the feedback signal, it takes $T_{UP}$ TTIs to process it. We
consider that the transmit and feedback time already take into account the
propagation delay between the transmitter and receiver. Without loss of
generality, we assume that $T_{A}=T_{TX}=T_{DP}=T_{UP}=1$ TTI. Another concept
shared between different HARQ protocols is the round-trip time (RTT), which
consists of the time it takes from the start of a transmission by the UE to
the end of processing of the feedback signal, either ACK or NACK, the UE
received from the BS.

Figure 3: An illustration of a couple of reactive HARQ protocol round trips.
Figure 4: An illustration of a couple of $K$-repetition HARQ protocol round
trips.

####  II-B1 Reactive Scheme

The reactive HARQ protocol is the more straight-forward one of the two
considered in this paper. The UE attempts to transmit one packet and waits for
feedback from the BS. Once the feedback is processed, it either attempts to
retransmit the same packet if it got a NACK signal or sits idle until a new
packet arrives. This protocol is illustrated in Fig. 3, which shows the
processing times and signal exchange between the UE and the BS. Under the
assumptions considered in this paper, the reactive RTT $T^{reac}_{RTT}$ is
given by

$T^{reac}_{RTT}=4\text{ TTIs}.$ (11)

From (11), the user-plane latency of the $m$-th HARQ round-trip is

$T^{reac}(m)=T_{A}+4m\text{ TTIs}.$ (12)

####  II-B2 $K$-Repetition Scheme

To increase the reliability and robustness of each transmission attempt, the
$K$-repetition HARQ protocol repeats the same packet $K_{rep}$ times on each
attempt. Therefore, the only way a transmission attempt fails is if each of
the $K_{rep}$ transmissions fail, which translates into an increased
reliability of the overall system. However, feedback on the transmission
attempt is sent only after the last repetition is processed by the BS. So,
there is a tradeoff between enhancing the reliability of each transmission and
increasing the latency of a transmition attempt. Fig. 4 shows two
$K$-repetition round-trip transmissions, where the first transmission fails
and the second is successful. The RTT of the $K$-repetition HARQ protocol is

$T^{K_{rep}}_{RTT}=K_{rep}+3\text{ TTIs}.$ (13)

Therefore, the total latency of $m$ $K$-repetition transmissions is given by

$T^{K_{rep}}(m)=T_{A}+mT^{K_{rep}}_{RTT}=1+m(K_{rep}+3)\text{ TTIs}.$ (14)

##  III System Analysis

The main requirement of URLLC applications is to reliably keep the user-plane
latency below an application-dependent latency constraint. We begin this
section by unambiguously defining what we mean by reliably and user-plane
latency.

######  Definition 2 (User-Plane Latency).

User-plane latency is the time spent between the arrival of a packet to the
UE’s queue and the successful processing of an ACK signal received from the
BS.

######  Definition 3 (Latent Access Failure Probability Requirement).

Latent access failure probability $\mathcal{P}_{F}(T\leq\tau)$, where $T$ is
the user-plane latency and $\tau$ is the latency constraint, is the
probability that the UE data cannot be successfully decoded.

Therefore, the QoS requirement of URLLC applications can be stated as

$\mathcal{P}_{F}(T\leq\tau)\leq\epsilon,$ (15)

where $\tau$ is the latency constraint and $\epsilon$ is the minimum
reliability, and both are application-dependent. Thus, to satisfy the QoS
requirement, the probability that an UE cannot transmit its data before $\tau$
must be bounded by $\epsilon$. Typically, $\tau$ varies between $1$ and $10$
ms and $\epsilon$ varies between $10^{-5}$ and $10^{-6}$ depending on the
URLLC application.

Let $M$ be the maximum number of retransmissions under the latency constraint
$\tau$. Moreover, notice that some of the UEs will transmit successfully
earlier than others, and if the UE’s transmission queue stays idle, the
interference levels in distinct retransmissions are different. Therefore, the
latent access failure probability is a function of the fraction of active
users at the $m$-th retransmission ($\mathcal{A}_{m}$), the probability that
the $m$-th retransmission is successful ($\mathcal{P}_{m}$( and the maximum
number of retransmissions ($M$), as given by [30]

$\mathcal{P}_{F}(T\leq\tau)=\begin{cases}1,&\text{, if }M=0\\\
1-\underset{m=1}{\overset{M}{\sum}}\mathcal{A}_{m}\mathcal{P}_{m}&\text{, if
}M\geq 1,\end{cases}$ (16)

where $\mathcal{A}_{m}$ is

$\mathcal{A}_{m}=\begin{cases}1,&\text{, if }m=1\\\
1-\underset{i=1}{\overset{m-1}{\sum}}\mathcal{A}_{i}\mathcal{P}_{i}&\text{, if
}m\geq 2.\end{cases}$ (17)

Given the expressions for $\mathcal{P}_{m}$, the latent access failure
probability is obtained by iteratively computing (17) and (16).

In the rest of this section, we derive closed-form expressions for
$\mathcal{P}_{m}$ under the reactive and $K$-repetition HARQ protocols,
denoted by $\mathcal{P}^{reac}_{m}$ and $\mathcal{P}^{Krep}_{m}$,
respectively. To do so, we use stochastic geometric analysis to obtain the
probability of success of a randomly chosen user $x_{0}$, herein the typical
user. From Slivnyak’s theorem [46], the performance of the typical user in an
HPPP is representative of the average user’s performance.

###  III-A Reactive HARQ

The maximum number of HARQ transmissions following the reactive HARQ protocol
with the delay constraint $\tau$ is given by

$M^{reac}=\left\lceil\frac{\tau-1}{T^{reac}_{RTT}}\right\rceil=\left\lceil\frac{\tau-1}{4}\right\rceil.$
(18)

The first step in deriving an expression for the latent access failure
probability is to obtain the probability that the $m$-th reactive
retransmission is successful ($\mathcal{P}^{reac}_{m}$).

Let $\Phi_{I}=\\{x_{i}\lvert
x_{i}\in\Phi_{A}\setminus\left\\{x_{0}\right\\}\cap x_{i}\in LOS_{m}\\}$ be
the set of users interfering with the typical user’s transmission on the
$m$-th RTT. Notice that due to the exponentially decreasing probability of a
LOS link with the increase in distance, $\Phi_{I}$ is a non-HPPP with density
$\lambda_{I}(x)=\lambda_{A}\exp\left(-\beta\left\lVert x\right\rVert\right)$.
The mean measure of $\Phi_{I}$, the average number of points in a given area,
is obtained as

$\displaystyle\Lambda\left(b(0,r)\right)$ $\displaystyle=$ $\displaystyle
E\left[\Phi_{I}\left(b(0,r)\right)\right]=\int_{\mathbb{R}^{2}}\lambda_{I}(x)dx$
$\displaystyle=$ $\displaystyle 2\pi\lambda_{A}\int_{0}^{r}\exp\left(-\beta
r\right)rdrd\theta$ $\displaystyle=$
$\displaystyle\frac{2\pi\lambda_{A}}{\beta^{2}}\left[1-\exp\left(-\beta
r\right)\left(1+\beta r\right)\right],$

where $b(0,r)$ is a $2$-dimensional ball with radius $r$ that is centered at
the origin. Now, let $N_{m}$ be a random variable denoting the number of users
that interfere with the typical user on the $m$-th retransmission. From
(III-A), the probability that there are $n$ interferers in the cell with
radius R is derived as

$\displaystyle\mathbb{P}\left(N_{m}=n\right)$ $\displaystyle=$
$\displaystyle\frac{\left[\mathcal{A}^{reac}_{m}\Lambda\left(b(0,R)\right)\right]^{n}}{n!}$
$\displaystyle\times\exp\left(-\mathcal{A}^{reac}_{m}\Lambda\left(b(0,R)\right)\right).$

######  Lemma 1.

If $K\gg 1$, the probability that the $m$-th reactive HARQ retransmission of
the typical user conditioned on the events that the typical user does not
experience preamble collision, has a LOS link and is affected by $n$
interferers can be approximated as

$\displaystyle\mathbb{P}\left(\mathrm{SINR}_{m}\geq\gamma\left\lvert
TX_{0},N_{m}=n\right.\right)$ $\displaystyle\approx$
$\displaystyle\underset{n^{\prime}=0}{\overset{n}{\sum}}{n\choose
n^{\prime}}\left(\frac{2}{K}\right)^{n^{\prime}}\left(1-\frac{2}{K}\right)^{n-n^{\prime}}$
$\displaystyle\times\exp\left(-\frac{\gamma}{\rho
K}\right)\left[\frac{\tanh^{-1}\left(\sqrt{\frac{\gamma}{1+\gamma}}\right)}{\sqrt{\gamma\left(1+\gamma\right)}}\right]^{n^{\prime}},$

where $n^{\prime}$ is the number of interferers within the primary lobe of the
beam directed at the typical user.

###### Proof.

See Appendix A. ∎

After deriving the expressions for the probability of having $N_{m}$ users
interfere with retransmission $m$ in (III-A) and the conditional probability
of success obtained in Lemma 1, the success probability can be obtained as
follows:

######  Theorem 1.

The probability that the $m$-th reactive harq retransmission is successfully
decoded is

$\displaystyle\mathcal{P}_{m}^{reac}$ $\displaystyle=$
$\displaystyle\underset{n=0}{\overset{\infty}{\sum}}$
$\displaystyle\mathbb{P}(N_{m}=n)\mathbb{P}\left(\left.\bar{C}\right.\lvert
N_{m}=n\right)$ $\displaystyle\times\mathbb{P}(x_{0}\in LOS_{m})$
$\displaystyle\times\mathbb{P}$
$\displaystyle(\mathrm{SINR}_{m}\geq\gamma\left\lvert TX_{0},N_{m}=n\right.),$

where $\mathbb{P}(N_{m}=n)$ is the probability that there are $n$ interferers
in the cell and is given by (III-A). The probability of no preamble collision
is given by

$\mathbb{P}\left(\left.\bar{C}\right\lvert
N_{m}=n\right)=\left(1-\frac{1}{Ns}\right)^{n}.$ (23)

And finally, the probability that the typical user has a LOS link to the BS is

$\displaystyle\mathbb{P}\left(x_{i}\in LOS_{m}\right)$ $\displaystyle=$
$\displaystyle\frac{1}{R}\int_{0}^{R}\exp\left(-\beta\left\lVert
x_{i}\right\rVert\right)d\left\lVert x_{i}\right\rVert$ (24) $\displaystyle=$
$\displaystyle\frac{1}{\beta R}\left[1-\exp\left(-\beta R\right)\right].$

###### Proof.

The proof is straight forward if the conditional probability obtained in Lemma
1 is averaged out. ∎

From the results of Theorem 1, the latent access failure probability can be
easily obtained by iteratively computing

$\mathcal{P}_{F}(T\leq\tau)=\begin{cases}1\text{, if }M^{reac}=0\\\
1-\underset{m=1}{\overset{M^{reac}}{\sum}}\mathcal{A}^{reac}_{m}\mathcal{P}^{reac}_{m}\text{,if
}M^{reac}\geq 1.\end{cases}$ (25)

###  III-B $K$-Repetition HARQ

In the $K$-repetition HARQ system, the RTT lasts from when the UE transmits
the first repetition until it receives the ACK/NACK feedback signal. Thus,
under delay constraint $\tau$, the maximum number of retransmissions is

$M^{K_{rep}}=\left\lceil\frac{\tau-1}{T^{K_{rep}}_{RTT}}\right\rceil=\left\lceil\frac{\tau-1}{K_{rep}+3}\right\rceil.$
(26)

Under the $K$-repetition HARQ, the same data is repeated $K_{rep}$ times for
every transmission attempt, and after the BS receives all the repetitions, it
sends either an ACK or a NACK signal depending whether any of the repetitions
sent in the transmission could be successfully decoded. Additionally, the UE
selects a new random subcarrier and preamble for the transmission of each
distinct repetition. To obtain a closed-form expression for the latent access
failure probability, we follow the same steps as were taken for the reactive
HARQ derivation.

######  Lemma 2.

If $K\gg 1$, the probability that the $m$-th $K$-repetition HARQ
retransmission of the typical user, conditioned on the event that the typical
user does not experience preamble collision, has a LOS link and is affected by
$n$ interferers can be approximated as (2), shown at the top of next page,

$\displaystyle\mathbb{P}\left(\underset{l=1}{\overset{K_{rep}}{\bigcup}}\mathrm{SINR}_{m,l}\geq\gamma\left\lvert\bar{C},x_{0}\in
LOS_{m},N_{m}=n\right.\right)\quad$
$\displaystyle\approx\underset{l=1}{\overset{K_{rep}}{\sum}}{K_{rep}\choose
l}\left(-1\right)^{l+1}\underset{n^{\prime}=0}{\overset{n}{\sum}}{n\choose
n^{\prime}}\left(\frac{2}{K}\right)^{n^{\prime}}\left(1-\frac{2}{K}\right)^{n-n^{\prime}}$
$\displaystyle\exp\left(-\frac{l\gamma}{\rho
K}\right)\left[\frac{\tanh^{-1}\left(\sqrt{\frac{\gamma}{1+\gamma}}\right)}{\sqrt{\gamma\left(1+\gamma\right)}}\right]^{ln^{\prime}}$
(27)

where the double subscript $m,l$ indicates the $l$-th repetition of the $m$-th
HARQ retransmission attempt.

###### Proof.

See Appendix B. ∎

With the result from Lemma 2, the probability that the $m$-th retransmission
attempt is successful can be obtained by averaging (2) over the conditional
random variables.

######  Theorem 2.

The probability that the $m$-th $K$-repetition harq retransmission is
successfully decoded is given by (28), shown at the top of next page,

$\displaystyle\mathcal{P}_{m}^{K_{rep}}=$
$\displaystyle\underset{n=0}{\overset{\infty}{\sum}}$
$\displaystyle\mathbb{P}(N_{m}=n)\mathbb{P}\left(\left.\bar{C}\right.\lvert
N_{m}=n\right)$ (28) $\displaystyle\mathbb{P}(x_{0}\in
LOS)\mathbb{P}\left(\underset{l=1}{\overset{K_{rep}}{\bigcup}}\mathrm{SINR}_{m,l}\geq\gamma\left\lvert\bar{C},x_{0}\in
LOS_{m},N_{m}=n\right.\right),$

where the closed-form expression for
$\mathbb{P}\left(\underset{l=1}{\overset{K_{rep}}{\bigcup}}\mathrm{SINR}_{m,l}\geq\gamma\left\lvert\bar{C},x_{0}\in
LOS_{m},N_{m}=n\right.\right)$ is derived on Lemma 2.

Given the analytical expression for the probability that the $m$-th
$K$-repetition HARQ retransmission is successfully received by the BS in
Theorem 2 and the fact that the probability that a randomly selected UE is
active can be computed from (17), the latent access failure probability is
derived as

$\mathcal{P}_{F}(T\leq\tau)=\begin{cases}1,\text{, if }M^{K_{rep}}=0\\\
1-\underset{m=1}{\overset{M^{K_{rep}}}{\sum}}\mathcal{A}^{K_{rep}}_{m}\mathcal{P}^{K_{rep}}_{m}\text{,}\\\
\text{if }M^{K_{rep}}\geq 1.\end{cases}$ (29)

##  IV Numerical Results and Discussion

In this section, we report the results of Monte-Carlo simulations of the
system model described in Section II. We use the simulation results to: a)
validate the closed-form analytical approximations derived in Section III b)
characterize the performance of the two HARQ protocols in the mmWave massive
MIMO scenario and c) discuss the insights provided by the analytical results.

At the beginning of each simulation instance, the users’ locations are
generated according to an HPPP inside a cell with radius $R=0.5$ km. At every
TTI:

  * •

The channel gain between the UEs and the BS located at the origin is generated
as an exponential random variable with unit mean.

  * •

All active UEs are determined to have either a LOS or NLOS link according to
the probability in (24), with $\beta=1$.

  * •

All active UEs select a random subcarrier from one of the $N_{S}=48$
subcarriers available.

  * •

All active UEs select a random preamble from one of the $N_{P}=64$ preambles
available.

  * •

The BS checks all UEs with LOS links on every subcarrier for preamble
collision.

  * •

The BS computes the dot product between the signal received and the conjugate
beam for all the UEs whose preambles have not collided. If the resulting SINR
is greater than $\gamma=-2$ dB, the transmission is successful, otherwise it
fails.

  * •

The BS sends an ACK feedback signal to the UEs whose transmission was
successful and a NACK feedback signal to those whose transmission attempt
failed. As the main goal of this work is to characterize grant-free uplink
performance, we assume that the feedback sent through the downlink channel is
error free.

  * •

All UEs move to a new location.

In accordance with 3GPP standards [47, 48], we consider a TTI mini-slot having
a duration of $0.125$ ms and a subcarrier spacing of $60$ kHz, which is a
configuration compatible with 5G NR frequency range 2 (FR2) operation, located
in the mmWave spectrum. We consider a noise figure of $-174$ dBm/Hz, a path
loss exponent of $\alpha=4$ and a received power threshold of $\rho=-130dBm$.

Figure 5: CCDF of the latent access failure probability for $\lambda_{U}=1000$
UE/$\text{km}^{2}$ for the reactive and $K$-repetition HARQ protocols with
$K_{rep}=2,4,8$. The plots in the figure show the results for $K=64$, $K=128$
and $K=256$ antennas. Figure 6: CCDF of the latent access failure probability
for $\lambda_{U}=5000$ UE/$\text{km}^{2}$ for the reactive and $K$-repetition
HARQ protocols with $K_{rep}=2,4,8$. The plots in the figure show the results
for $K=64$, $K=128$ and $K=256$ antennas.

Figs. 5 and 6 show the complementary cumulative distribution function (CCDF)
of the latent access probability for a user density of $\lambda_{U}=1000$
UE/$\text{km}^{2}$ and $\lambda_{U}=5000$ UE/$\text{km}^{2}$, respectively.
The three plots in each figure display the performance for $64$, $128$ and
$256$ antennas, from left to right. The behavior of the performance curves,
where the latent access failure probability remains constant for a period of
time and then drops on the following TTI, is due to the transmission
propagation time on the uplink and the feedback, and the processing times.
Fig. 5 depicts the performance with a moderate UE density scenario and shows
that the reactive HARQ protocol is the best option for strict delay
constraints, with $\tau\leq 6$ TTIs ($0.725$ ms), as there is no time for any
of the $K$-repetition configurations to finish their first round-trip. When
the first and second round-trips for $K_{rep}=2$ are completed, it has the
best performance in $6\leq\tau\leq 8$ TTIs and $11\leq\tau\leq 12$ TTI
intervals. From this point on, the best performance is dominated by
$K_{rep}=4$ and $K_{rep}=8$, with the best configuration being the one that
has more completed round-trips in under $\tau$ TTIs. A similar trend occurs
with a higher user density as shown in Fig. 6.

Tables I and II show the reduction in the latent access failure probability
upon increasing the number of antennas from $64$ to $256$. There is little
improvement for a delay constraint of $1$ ms in either scenario. In
applications with a moderate UE density and a delay constraint of $2$ ms or
more, we notice an average improvement of around $2$ across both HARQ
protocols investigated, while in applications with a higher user density, the
failure probability is reduced by as much as $32$ times for $K_{rep}=8$
repetitions and a delay constraint greater or equal to $3$ ms.

HARQ |  $T\leq 1$ ms |  $T\leq 2$ ms |  $T\leq 3$ ms
---|---|---|---
$K_{rep}=2$ | 1.25 | 1.59 | 1.982
$K_{rep}=4$ | 1 | 2.18 | 1.98
$K_{rep}=8$ | 1 | 2.49 | -
Reactive | 1.12 | 1.42 | 1.64
TABLE I: Latent access failure probability reduction in increasing from $64$ to $256$ antennas when $\lambda_{U}=1000$ UE/$\text{km}^{2}$ HARQ |  $T\leq 2$ ms |  $T\leq 3$ ms |  $T\leq 4$ ms
---|---|---|---
$K_{rep}=2$ | 1.54 | 6.81 | 13.69
$K_{rep}=4$ | 3.25 | 17.13 | -
$K_{rep}=8$ | 1.75 | 32.51 | 32.51
Reactive | 1.40 | 2.60 | 6.18
TABLE II: Latent access failure probability reduction in increasing from $64$
to $256$ antennas when $\lambda_{U}=5000$ UE/$\text{km}^{2}$

Nonetheless, notice from Figs. 5 and 6 that increasing the number of antennas
from $128$ to $256$ does not change the latency performance significantly.
Additionally, the increase in the number of antennas has a larger impact on
performance for the higher user density scenario shown in Fig. 6 than for the
moderate density one in Fig. 5. Later in this section, we discuss why this
happens and how to possibly address it.

As the approximation used to derive the results in Section III relies on $K\gg
1$, there is a gap between the analytical and simulation results when $K=64$
as, in this regime, the value of $F_{K}(x)$, and consequently the
interference, outside the main lobe are no longer negligible in comparison to
the gain on the main lobe.

Figure 7: The probability that an UE fails to transmit its packet under
$\tau=1$ ms for an user density ranging from $100$ UE/$\text{km}^{2}$. The
plots show the results for $K=64$, $K=128$ and $K=256$ antennas, respectively.
Figure 8: The probability that an UE fails to transmit its packet under
$\tau=3$ ms for an user density ranging from $100$ UE/$\text{km}^{2}$. The
plots show the results for $K=64$, $K=128$ and $K=256$ antennas, respectively.

Figs. 7 and 8 show how system reliability, i.e., in the probability of
transmission failure under the latency constraint $\tau$, scales with an
increase in user density for a latency constraint of $\tau=1$ ms and $\tau=3$
ms, respectively. In both figures the user density ranges from $100$ to $5000$
UE/$\text{km}^{2}$. Fig. 7 shows that that the combination of mmWave and
massive MIMO is not enough to satisfy the QoS requirement of URLLC
applications with the stricter delay constraint (failure probability below
$10^{-6}$). Also, in this latency range, the benefit from increasing the
number of BS antennas is rather small. For applications with less stringent
delay constraints, shown in Fig. 8, the $K$-repetition HARQ protocol with
$K_{rep}=4$ and $K_{rep}=8$ is able to support the URLLC QoS requirements.
Table III depicts the highest UE density that can be supported by each HARQ
and MIMO configuration. When $K_{rep}=4$, increasing the number of BS antennas
from $64$ to $128$ increases the supported user density by $11\%$ and
increasing the number from $128$ to $256$ increases it by only $7.5\%$. For
$K_{rep}=8$, those values are $14\%$ and $5\%$, respectively. It is worth
noting that as long as the QoS constraints are satisfied, it is desirable to
use the HARQ configuration with the least number of repetitions as possible in
order to save UE power.

HARQ |  $K=64$ antennas |  $K=128$ antennas |  $K=256$ antennas
---|---|---|---
$K_{rep}=2$ | - | - | -
$K_{rep}=4$ | 1800 UE/$\text{km}^{2}$ | 2000 UE/$\text{km}^{2}$ | 2150 UE/$\text{km}^{2}$
$K_{rep}=8$ | 2800 UE/$\text{km}^{2}$ | 3200 UE/$\text{km}^{2}$ | 3350 UE/$\text{km}^{2}$
Reactive | - | - | -
TABLE III: User density supported (failure probability below $10^{-6}$) by
each configuration. Figure 9: The impact of the number of antennas on the
latent access failure probability. The leftmost plot shows results for delay
constraint $\tau=1$ ms, while the rightmost for $\tau=3$ ms.

In Fig. 9, we show the impact of increasing the number of antennas on the
failure probability for a delay constraint of $\tau=1$ ms on the left and
$\tau=3$ ms on the right. From this figure, we can conclude that increasing
the number of antennas beyond $100$ for the configuration under consideration
($R=0.5$ km and $\beta=1$) has a decreasing impact on latency performance.
Moreover, the $K$-repetition HARQ protocol benefits more from an increased
number of antennas than the reactive HARQ protocol does. Also, the plot on the
right shows that the latency performance of applications with a moderate
latency constraint ($\tau=3$ ms) and higher user densities ($\lambda_{U}\geq
3000$ UE/$\text{km}^{2}$) is greatly improved by changing from traditional
MIMO to massive MIMO. This is explained by the capability of producing
narrower receiver beams on systems with a higher number of antennas.
Nevertheless, as of a certain point, the probability of having a LOS link
becomes the dominant bottleneck in reducing the latent access failure
probability. As the LOS link probability is unaffected by the number of
antennas, another measure must be taken to further reduce the latency. In the
system model formulated in this paper, one way to achieve this would be to
increase the BS deployment density, effectively decreasing the radius of the
cells.

##  V Conclusions

In this work, we formulated a model to analyze the latency and reliability of
mmWave massive MIMO URLLC applications using reactive and $K$-repetition HARQ
protocols. We used stochastic geometric spatiotemporal tools to derive closed-
form approximations of the system’s latent access failure probability. We
validated the analytical results using Monte-Carlo simulations, identifying
the limitations of our analytical results. Also, we investigated how the
system’s performance is impacted by the application’s latency constraint, the
density of UEs served by the system, and the number of antennas in the BS. We
concluded that:

  * •

Other than for extremely strict delay constraints ($\tau=0.625$ ms), the
$K$-repetition HARQ protocol is a better choice.

  * •

Increasing the number of BS antennas from $64$ to $256$ BS antennas can reduce
the latent access failure probability by a factor of 32 for the cell
configuration analyzed in the manuscript.

  * •

Massive MIMO’s interference reduction capability significantly improves the
reliability of systems with high user density and moderately improves the
performance of systems with low user density.

  * •

The increase in reliability from increasing the number of BS antennas beyond
$100$ is greatly reduced in the configuration investigated in Section IV, as
the probability of having a LOS link between the UE and the BS becomes the
main bottleneck.

  * •

Under the configurations investigated in this manuscript, the system can
support a UE density as high as $3350$ UE/$\text{km}^{2}$ for a URLLC
application with latency and reliability constraints of $3$ ms and $10^{-6}$,
respectively.

Overall, we can conclude that it is possible to increase the reliability of
URLLC applications by using mmWave massive MIMO, and when this technique is
combined with selecting reasonable configuration parameters, these two
techniques together can improve reliability under a strict latency constraint
($\tau=1$ ms) and can satisfy URLLC QoS requirements under a less strict
latency constraint ($\tau=3$ ms).

##  Appendix A Proof of Lemma 1

The probability of success in (1) can be expanded to

$\displaystyle\mathbb{P}\left(\mathrm{SINR}_{m}\geq\gamma\left\lvert
TX_{0},N_{m}=n\right.\right)$ (30) $\displaystyle=$
$\displaystyle\mathbb{P}\left\\{\left.\left\lvert
g_{0}\right\rvert^{2}\geq\frac{\gamma}{\rho
K}\left[\sigma^{2}+I\right]\right\rvert TX_{0},N_{m}=n\right\\}$
$\displaystyle=$ $\displaystyle\exp\left(-\frac{\gamma\sigma^{2}}{\rho
K}\right)\mathcal{L}_{I}\left(s\lvert TX_{0},N_{m}=n\right),$

where $s=\frac{\gamma}{\rho K}$,
$I=\underset{x_{i}\in\Phi_{I}}{\sum}\rho\left\lvert
g_{i}\right\rvert^{2}F_{K}\left(\frac{\pi}{2}\left(\theta_{0}-\theta_{i}\right)\right)$
is the interference on the typical user’s transmission and
$\mathcal{L}_{I}\left(s\lvert\bar{C},x_{0}\in LOS_{m},N_{m}=n\right)$ is the
Laplace transform of the interference conditioned on the user not experiencing
preamble collision, them having a LOS path, and there being $n$ interferers in
the cell.

The Laplace transform of the interference can be derived as (31), shown at the
top of the next page.

$\displaystyle\mathcal{L}_{I}\left(s\lvert TX_{0},N_{m}=n\right)$
$\displaystyle=$ $\displaystyle
E_{g_{i},\theta_{i}}\left\\{\exp\left[-s\underset{x_{i}\in\Phi_{I}}{\sum}\rho\left\lvert
g_{i}\right\rvert^{2}F_{K}\left(\frac{\pi}{2}\left(\theta_{0}-\theta_{i}\right)\right)\right]\right\\}$
(31) $\displaystyle=$
$\displaystyle\underset{x_{i}\in\Phi_{I}}{\prod}E_{g_{i},\theta_{i}}\left\\{\exp\left[-s\rho\left\lvert
g_{i}\right\rvert^{2}F_{K}\left(\frac{\pi}{2}\left(\theta_{0}-\theta_{i}\right)\right)\right]\right\\}$
$\displaystyle=$
$\displaystyle\underset{x_{i}\in\Phi_{I}}{\prod}E_{\theta_{i}}\left[\frac{1}{1+s\rho
F_{K}\left(\frac{\pi}{2}\left(\theta_{0}-\theta_{i}\right)\right)}\right].$

Unfortunately, obtaining a closed-form expression for the expectation in (31)
is not mathematically tractable. Therefore, we first obtain a suitable
approximation to the Fejer kernel. Due to the Fejer kernel property in (10),
as the number of antennas increases most of the energy is concentrated on the
main lobe as shown in Fig. 2. Hence, we choose to approximate it as

$F_{K}\left(x\right)\approx
f_{K}(x)=\begin{cases}-\frac{K^{3}x^{2}}{4}+K\text{, if
}x\in\left\\{-\frac{2}{K},\frac{2}{K}\right\\}\\\ 0\text{,
otherwise}.\end{cases}$ (32)

The quadratic approximation in (32) renders the derivation of the expectation
in (31) tractable. Furthermore, it ensures that $f_{K}(x)=0$ whenever
$x\notin\left(-\frac{2}{K},\frac{2}{K}\right)$, i.e., the contributions of the
signals arriving from directions outside of the main lobe to the interference
is zero, and that $F_{K}\left(0\right)=f_{K}(0)=K$. Hence,

$\displaystyle\underset{x_{i}\in\Phi_{I}\cap\theta_{i}\in\left(-\frac{2}{K},\frac{2}{K}\right)}{\prod}E_{\theta_{i}}\left[\frac{1}{1+s\rho
F_{K}\left(\frac{\pi}{2}\left(\theta_{0}-\theta_{i}\right)\right)}\right]$
$\displaystyle\overset{(a)}{=}\underset{x_{i}\in\Phi_{I}\cap\theta_{i}\in\left(-\frac{2}{K},\frac{2}{K}\right)}{\prod}\frac{\tanh^{-1}\left(\sqrt{\frac{\gamma}{1+\gamma}}\right)}{\sqrt{\gamma\left(1+\gamma\right)}}$
$\displaystyle\overset{(b)}{=}\underset{n^{\prime}=0}{\overset{n}{\sum}}{n\choose
n^{\prime}}\left(\frac{2}{K}\right)^{n^{\prime}}\left(1-\frac{2}{K}\right)^{n-n^{\prime}}$
$\displaystyle\times\left[\frac{\tanh^{-1}\left(\sqrt{\frac{\gamma}{1+\gamma}}\right)}{\sqrt{\gamma\left(1+\gamma\right)}}\right]^{n^{\prime}}$
(33)

where $(a)$ is obtained from $\int\frac{1}{1-x^{2}}dx=\tanh^{-1}(x)$. While
step $(b)$ comes from the fact that the interferers’ angles are uniformly
distributed, given that there are $n$ interferers in the cell, the number of
interferers within the typical user’s main lobe direction follows a binomial
distribution with $n^{\prime}\sim Binomial\left(\frac{2}{K}\right)$. Thus, the
conditional success probability can be obtained by summing the marginal
distribution weighted by $n^{\prime}$’s probability mass function (PMF). This
completes the proof.

##  Appendix B Proof of Lemma 2

In order for a $K$-repetition HARQ transmission attempt to be successful at
least one of the repetitions must be successfully decoded. Therefore, the
probability that the $m$-th retransmission attempt is successful, conditioned
on no preamble collisions, a LOS path and $n$ interferers, can be obtained as
the complement probability that all repetitions fail , as derived in (34),
shown at the top of the next page,

$\displaystyle\mathbb{P}\left(\underset{l=1}{\overset{K_{rep}}{\bigcup}}\mathrm{SINR}_{m,l}\geq\gamma\left\lvert
TX_{0},N_{m}=n\right.\right)$ $\displaystyle=$ $\displaystyle
1-\mathbb{P}\left(\underset{l=1}{\overset{K_{rep}}{\bigcap}}\mathrm{SINR}_{m,l}<\gamma\left\lvert
TX_{0},N_{m}=n\right.\right)$ (34) $\displaystyle\overset{(a)}{=}$
$\displaystyle
1-\underset{l=1}{\overset{K_{rep}}{\prod}}\left[1-\mathbb{P}\left(\mathrm{SINR}_{m,l}\geq\gamma\left\lvert
TX_{0},N_{m}=n\right.\right)\right]$ $\displaystyle\overset{(b)}{=}$
$\displaystyle
1-\left[1-\mathbb{P}\left(\mathrm{SINR}_{m}\geq\gamma\left\lvert
TX_{0},N_{m}=n\right.\right)\right]^{K_{rep}}$ $\displaystyle\overset{(c)}{=}$
$\displaystyle\underset{l=1}{\overset{K_{rep}}{\sum}}{K_{rep}\choose
l}\left(-1\right)^{l+1}$
$\displaystyle\mathbb{P}\left(\mathrm{SINR}_{m}\geq\gamma\left\lvert
TX_{0},N_{m}=n\right.\right)^{l}$

where step $(a)$ follows from the fact that the set of interferers is
different from one repetition to the next, as a new subcarrier is randomly
selected for every repetition by each UE, making the SINRs on distinct
repetitions mutually independent. Also, as the SINR of every repetition is
affected by an interferer process having the same intensity, the probability
of success of each repetition is equal, which justifies step $(b)$. Finally,
step $(c)$ is obtained from the binomial expansion of the power term. If we
work from (34) and follow the same steps derived in Appendix A, we obtain (2),
which completes the proof.

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  *[3GPP]: 3rd generation partnership project
  *[NR]: new radio
  *[eMBB]: enhanced mobile broadband
  *[mMTC]: massive machine-type communication
  *[URLLC]: ultra reliable low-latency communication
  *[AR]: augmented reality
  *[VR]: virtual reality
  *[V2X]: vehicle-to-everything
  *[cIoT]: critical internet of things
  *[KPI]: key performance indicator
  *[LTE]: long-term evolution
  *[UE]: user equipment
  *[RA]: random access
  *[SPS]: semi-persistent scheduling
  *[UEs]: user equipment
  *[MIMO]: multiple input multiple output
  *[mmWave]: millimeter wave
  *[HARQ]: hybrid automatic repeat request
  *[NOMA]: non-orthogonal multiple access
  *[MARL]: multiagent reinforcement learning
  *[SCMA]: sparse code multiple access
  *[IoT]: internet of things
  *[BSs]: base station
  *[BS]: base station
  *[QoS]: quality of service
  *[PRACH]: physical random access channel
  *[RAR]: random access request
  *[PUSCH]: physical uplink shared channel
  *[HPPP]: homogeneous Poisson point process
  *[TTI]: transmission time interval
  *[ULA]: uniform linear array
  *[LOS]: line-of-sight
  *[NLOS]: non-line-of-sight
  *[AWGN]: additive white Gaussian noise
  *[SINR]: signal-to-interference-plus-noise-ratio
  *[ACK]: acknowledgement
  *[NACK]: negative acknowledgement
  *[TTIs]: transmission time interval
  *[RTT]: round-trip time
  *[FR2]: frequency range 2
  *[CCDF]: complementary cumulative distribution function
  *[PMF]: probability mass function