# Beam Management in 5G:
A Stochastic Geometry Analysis

Sanket S. Kalamkar, François Baccelli,
Fuad M. Abinader Jr., Andrea S. Marcano Fani, and Luis G. Uzeda Garcia F.
Baccelli and S. S. Kalamkar are with INRIA-ENS, Paris, France. F. M. Abinader
Jr., A. S. Marcano Fani, and L. G. Uzeda Garcia are with Nokia Bell Labs,
Paris, France. (e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>luis.uzeda_garcia@nokia-bell-labs.com).
This work was carried out at the Laboratory of Information, Networking and
Communication Sciences (LINCS), Paris, France.This work has received funding
from the European Research Council (ERC) under the European Union’s Horizon
2020 research and innovation programme grant agreement number 788851.Part of
this work has been accepted in the 2020 IEEE Global Communications Conference
(GLOBECOM’20) [1].

###### Abstract

Beam management is central in the operation of beamformed wireless cellular
systems such as 5G New Radio (NR) networks. Focusing the energy radiated to
mobile terminals (MTs) by increasing the number of beams per cell increases
signal power and decreases interference, and has hence the potential to bring
major improvements on area spectral efficiency (ASE). This paper proposes a
first system-level stochastic geometry model encompassing major aspects of the
beam management problem: frequencies, antenna configurations, and propagation;
physical layer, wireless links, and coding; network geometry, interference,
and resource sharing; sensing, signaling, and mobility management. This model
leads to a simple analytical expression for the effective rate that the
typical user gets in this context. This in turn allows one to find the number
of beams per cell and per MT that maximizes the effective ASE by offering the
best tradeoff between beamforming gains and beam management operational
overheads and costs, for a wide variety of 5G network scenarios including
millimeter wave (mmWave) and sub-6 GHz. As part of the system-level analysis,
we define and analyze several underlying new and fundamental performance
metrics that are of independent interest. The numerical results discuss the
effects of different systemic tradeoffs and performance optimizations of
mmWave and sub-6 GHz 5G deployments.

###### Index Terms:

5G NR, beam management, handover, interference, Poisson point process,
stochastic geometry

##  I Introduction

The ever-increasing demand in capacity for mobile communications necessitates
new implementation approaches that can significantly boost data rates and the
area spectral efficiency (ASE) of mobile networks. One key enabler considered
in 5G [2] to face this demand is the use of the spectrum beyond the sub-6 GHz
frequencies, known as millimeter wave (mmWave). More specifically, 5G is
designed to make use of spectrum above 20 GHz, where bandwidth sizes of up to
400 MHz per carrier can be used to offer very high data rates (above 10 Gbps
peak rates) and increase the network capacity [3]; nevertheless, the sub-6 GHz
bands, with up to 100 MHz of bandwidth per carrier, are still needed to ensure
wide area coverage and data rates up to a few Gbps [4].

One of the key difficulties faced by 5G operation on mmWave frequencies is
their challenging propagation characteristics: they are subject to high path
loss, penetration loss, and diffraction loss, and are thus limited to short
distance links, typically of a few hundred meters. But what was once
considered a limitation makes nowadays mmWave a suitable candidate for small
cells deployments, which can be used for network densification and capacity
boosting.

To overcome the propagation challenges at mmWave frequencies, steerable
antenna arrays with a large number of antenna elements each can be used to
create highly directional beams that concentrate the transmitted energy to
achieve high gains and increase coverage. Also, mmWave communications must be
designed to operate under mobility conditions, covering users in LOS (Line-of-
Sight) and NLOS (Non-Line-of-Sight) at pedestrian and vehicular speeds. This
can become quite challenging since operation at mmWave frequencies is highly
sensitive to changes in the environment (e.g., moving of the hand/head,
passing cars). Thus, any mmWave-based wireless access system relies on
techniques such as adaptive beamforming, beam-tracking and fast beam-switching
to always maintain the best selected beam that maximizes the performance even
in the face of beam misalignment events due to intra-cell mobility and inter-
cell handovers. Although more important for mmWave frequencies, beamforming
techniques with directional narrow beams can also be used in sub-6 GHz
frequencies to improve the network performance.

###  I-A Motivation

The introduction of beamforming as a central feature in 5G New Radio (NR)
requires a series of beam management procedures to ensure efficient operation.
One of such procedures is the so-called beam selection (or, beam refinement).
Here the mobile terminal (MT) monitors beamformed reference signals (RSs)
periodically transmitted by the base station (BS) to identify and report to
the BS the best serving BS beam. Such RSs can be either synchronization
signal/physical broadcast channel blocks (SSBs), or channel state information
reference signals (CSI-RSs). Another such procedure is the beam failure
detection and recovery, where the MT monitors the radio link quality of the
serving BS beam. If the quality of the radio link is repeatedly below a
certain threshold, the BS triggers a procedure to quickly re-establish
communication on neighboring beams.

Although beam management procedures are necessary for beamformed 5G NR
operations, they also bring their own inherent costs in terms of signaling and
delay overheads. For instance, there should be one beamformed BS RS for each
beam in the beamset for a proper beam management operation. Hence, increasing
the beamset size increases the signaling overhead. Thus the network planning
for 5G NR deployments must take beam management into consideration. This
requires the understanding of existing systemic effects and tradeoffs for any
given MT mobility pattern, inter-site distance (also known as inter-BS
distance), and the number/width of the beams. For example, the adoption of a
BS beamset containing a large number of narrow beams during the network
planning phase improves the signal-to-interference-plus-noise ratio (SINR) due
to the increased signal power and reduced interference. But it may also lead
to more frequent service interruption due to, e.g., beam switching delays
(i.e., between the moment the BS RSs are measured and reported and the moment
the BS decides to make the beam switch) and beam misalignment, which in the
end may degrade the network performance.

System-level simulations can be crucial to evaluate all such tradeoffs and
determine the best combination of parameters for a given network deployment.
But given the complex nature of beamforming techniques and beam management
procedures in 5G NR, such simulations can be very time-consuming and
expensive. This paper proposes a mathematical framework that permits a system-
level analysis of beam management in 5G, leads to crisp insights into the
associated tradeoffs, and has the potential to make beam management efficient.
We believe that this mathematical framework will complement and help guide
complex system-level simulations.

###  I-B Contributions

This work makes the following contributions:

  1. 1.

We develop a first mathematical optimization framework for the beam management
problem in 5G. Using the well-established tools from stochastic geometry [5,
7, 8, 6], our proposed optimization yields the number of beams at the BS and
the MT that maximizes the effective ASE of downlink transmissions to the
directional MT.

  2. 2.

The optimization permits a system-level analysis of 5G NR deployment based
solely on a small set of input parameters. In particular, it takes into
account the average speed of the MTs, mobility-induced beam misalignment,
geometry-based beam selections during BS handovers and beam reselections
within a cell, and time overheads associated with these beam (re)selections.
For instance, given a mix of omnidirectional and directional MTs, a mix of MT
speeds (pedestrians, bikes, and cars), a mix of geometries (some cells are
bigger, other smaller, with the MTs either close or far from the serving BS),
a mix of fading and blockages present in the network, our proposed
optimization framework enables the identification of the number of beams that
provides the best effective ASE, namely the best “sum rate” for MTs of all
types in a large domain.

  3. 3.

As part of the calculation of the effective ASE at the system level, we define
and evaluate many new and fundamental performance metrics. These metrics are
of independent interest and are given in Table I with their equation numbers
in the paper and their interpretation.

  4. 4.

While the proposed framework is generic and can cover a variety of related
beam-centric systems such as IEEE 802.11ax/Wi-Fi 6, we give numerical results
to evaluate a 5G NR-compliant radio access network (RAN) operating in a dense
urban macro/picocell scenario, for both sub-6 GHz and mmWave frequencies. The
results reveal the inter-dependencies between system parameters and provide
insights into the associated tradeoffs.

TABLE I: New Performance Metrics Performance Metric | Formula |  Interpretation of the Metric
---|---|---
Probability of beam misalignment | (2) and (3) |  The probability of beam misalignment between the beams of the BS and the MT due to the mobility of the MT between two consecutive beam measurement events.
Intensity of geometry-based beam reselection | (16) |  Frequency of geometry-based beam switching within the cells.
Geometry-based time-of-stay | (18) |  Average time spent by the MT within the main lobe of a beam.
Effective intensity of beam reselection | (17) |  Frequency of beam reselections within a cell by taking into account the geometry and the system design. It determines the time overhead associated with beam reselection.
Effective time-of-stay | (19) |  Average duration of time intervals during which the MT has selected any given serving BS beam, which is essential to, e.g., evaluate whether the MT will have enough time to perform MT beam refinement and link adaptation before data transmission.
Probability of a beam switch | (22) |  At a cell boundary, it is the probability that the MT has to switch panels/beams. This allows one to predict the no beam switching event, which can help reduce the number of beam measurements and the associated time overhead.
Time overheads | (20), (21), (23) |  The time overheads are associated with the MT mobility and reference signal measurements in 5G that are necessary for inter-cell handovers and beam switching within the cell. The overheads determine the effective time available for data transmission.

###  I-C Related Work

Stochastic geometry has been widely used to model and analyze several aspects
of wireless networks including cellular networks [5, 7, 8, 6]. A stochastic
geometry framework is also popular to model mmWave cellular networks and
evaluate key performance aspects such as initial access, success probability,
the average rate of the typical user, and the effect of blockages [9, 10, 11,
12].

The study of handovers due to user mobility in non-beam-centric cellular
networks and their effect on various performance metrics such as throughput is
a rich area of research [13, 14, 15]. Most of the works have focused on BS
handovers (or, equivalently, cell handovers) [16, 17, 21, 19, 20, 18], and
they ignore the effect of these handovers on beam-based communications. The
work in [22] is probably the closest one to our work. This work studies both
initial beam selections during BS handovers and beam reselections within a
cell and their associated overheads in a beam-centric mmWave cellular network.
The work in [22] obtains analytical expressions of inter-cell and inter-beam
handover rates based on which the ASE is calculated subject to overheads due
to these handovers. But, it considers only noise-limited scenarios and ignores
interference from other BSs, while our work considers both noise and
interference in the analysis. Second, our work also studies the effect of
handovers on optimizing the number of beams during network planning phase,
while [22] assumes a fixed number of beams. Third, unlike [22], we consider
blockages and beam misalignment due to mobility. Finally, we also consider the
directional reception at MTs and its effect on beam management. Overall, there
is a lack of understanding on how network planning decisions (such as BS
beamset and inter-site distance), beam management procedures (with its
associated overhead) and MT mobility affects system-level performance in the
presence of interference, fading, blockages, and beam misalignment error. The
present paper can be seen as an attempt to incorporate all this above in a
system-level model.

TABLE II: Notation Symbol |  Definition | Symbol |  Definition
---|---|---|---
$\lambda$ |  BS intensity | $v$ |  MT speed
$f_{\rm c}$ |  Carrier frequency | $c$ |  Speed of light
$2^{n}$ |  Number of BS beams | $2^{k}$ |  Number of MT beams
$\varphi_{n}$ |  Beamwidth at the BS | $\varphi_{k}$ |  Beamwidth at the MT
$G_{{\rm m},n}$, $G_{{\rm s},n}$ |  Main and side lobe gains at the BS |  $G_{{\rm m},k}$, $G_{{\rm s},k}$ |  Main and side lobe gains at the MT
$p_{\rm bm}^{\rm BS}$ |  Beam misalignment probability at the BS | $p_{\rm bm}^{\rm MT}$ |  Beam misalignment probability at the MT
$\tau$ |  SSB periodicity | $R_{\rm c}$ |  LOS ball radius
$\alpha_{\rm L}$ |  LOS path-loss exponent | $\alpha_{\rm N}$ |  NLOS path-loss exponent
$W$ |  Bandwidth | $N_{0}$ |  Noise power density
$P$ |  Transmit power | $\sigma^{2}$ |  Noise power
$\beta$ |  SINR threshold | $Q_{\rm max}$ |  Maximum SINR
$\mathcal{R}$ |  Ergodic Shannon rate | $\mathcal{R}_{\rm eff}$ |  Effective ASE
$T_{\rm c}$ |  Cell handover overhead | $T_{\rm{b}}$ |  Beam reselection overhead
$\mu_{\rm{s,c}}$ |  Linear intensity of BS handovers | $\mu_{\rm c}$ |  Time intensity of BS handovers
$\mu_{\rm{s,b}}$ |  Linear intensity of geometry-based beam reselections | $\mu_{\rm t,b}$ |  Time intensity of geometry-based beam reselections
$\mu_{\rm b}$ |  Effective intensity of beam reselections | $T_{\rm o}$ |  Average time overhead per unit time
$T_{\rm st}$ |  Geometry-based time-of-stay | $T_{\rm eff,st}$ |  Effective time-of-stay
$p_{\rm sw}$ |  Probability of a beam switch | $p_{\rm s}$ |  Success probability of a transmission

###  I-D Organization of the Paper

Section II gives the models for network geometry, beamforming, channel, and
blockages and introduces the mobility-induced beam misalignment. Section III
calculates simple analytical expressions for the ergodic Shannon rate for
sub-6 GHz and mmWave deployments. Section IV discusses the beam selection
during an inter-cell handover and calculates the intensity of beam
reselections within the cell. This section also calculates the time overheads
associated with these beam (re)selections. Section V gives the final
optimization problem of maximizing the effective ASE. Section VI provides the
network setup with the values of network parameters. Sections VII and VIII
discuss several systemic tradeoffs for omnidirectional and directional MTs.
Section IX discusses briefly some new performance metrics mentioned in Table
I. Section X gives conclusions and future directions. Table II provides the
notation used in the paper.

Figure 1: A snapshot of a Poisson cellular network with directional
beamforming to an MT. Here, each BS has $2^{3}=8$ beams, i.e., $n=3$. Triangle
: Base station (BS), red square : MT location, brown circle : Geometry-based
beam reselection location, yellow circle : BS handover location, solid black
lines : Cell boundaries, dashed lines : Beam boundaries, and dotted line :
Trajectory of the MT.

##  II System Model

###  II-A Network Model

As shown in Fig. 1, we consider a downlink cellular network, where the BS
locations are modeled as a homogeneous Poisson point process (PPP)
$\Phi\subset\mathbb{R}^{2}$ with intensity $\lambda$. We assume that a
directional MT moves on a randomly oriented straight line with speed $v$.
Without loss of generality, thanks to the isotropy and the stationarity of the
PPP [5], this line of MT motion can be considered to be along the $x$-axis and
passing through the origin. We assume that each BS always has an MT to serve.
Also, a BS serves one MT at a time per resource block.111The analysis in this
paper can be extended to the case where a resource block at a BS is shared by
multiple MTs. The MT associates itself with the nearest BS. Such an
association results in BS cells forming a Poisson-Voronoi tessellation as
shown in Fig. 1. Let $X_{0}\in\Phi$ denote the location of the serving BS of
the MT at a given time. Without loss of generality, we can focus on the MT
that is located at the origin at that time. After averaging over the PPP, this
MT becomes the typical MT.

###  II-B Beamforming Model

A BS at location $X\in\Phi$ uses directional beamforming to communicate with
the typical directional MT located at the origin. As shown in Fig. 1, we
approximate the actual antenna pattern by a sectorized one, where each sector
corresponds to the main lobe of one beam of the BS. For simplicity, we assume
that each BS has $2^{n}$ beams with $n\in\mathbb{N}$, which corresponds to
$2^{n-1}$ beam boundaries.222Here, a beam boundary is a line segment that
connects two points on the cell boundary and passes through the location of
the BS, as shown in Fig. 1. Hence, the beamwidth for a BS is

$\displaystyle\varphi_{n}$
$\displaystyle=\frac{2\pi}{2^{n}}=\frac{\pi}{2^{n-1}}.$ (1)

We focus on a simple antenna pattern model where the main lobe is restricted
to the beamwidth. Both the main and side lobe gains depend on the number
$2^{n}$ of beams. In particular, the antenna gain $G_{n}$ is expressed as

$\displaystyle G_{n}(\psi)=\begin{cases}G_{{\rm
m},n}&\text{if}~{}|\psi|\leq\varphi_{n}/2\\\ G_{{\rm
s},n}&\text{otherwise},\end{cases}$

where $G_{{\rm m},n}$ and $G_{{\rm s},n}$ denote the antenna gains within the
main lobe and the side lobe, respectively, as a function of the number $2^{n}$
of beams, and $\psi$ is the angle off the boresight direction.

The beamforming model at the MT is the same as the one at the BS, where the
number of beams at the typical MT is $2^{k}$ with $k\in\mathbb{N}$. Thus the
antenna gain $G_{k}$ at the typical MT is expressed as

$\displaystyle G_{k}(\psi)=\begin{cases}G_{{\rm
m},k}&\text{if}~{}|\psi|\leq\varphi_{k}/2\\\ G_{{\rm
s},k}&\text{otherwise},\end{cases}$

where $\varphi_{k}=\frac{\pi}{2^{k-1}}$ is the beamwidth at the typical MT.

###  II-C Beam (Re)Selection

As shown in Fig. 1, the MT may cross several cell and beam boundaries as it
moves on a straight line. At a cell boundary, a BS handover occurs. Thus, a
new beam at the next serving BS and potentially at the MT need to be selected
during a BS handover.

Within the cell as well, the MT moves from the main lobe of one beam to that
of another beam. Thus, a new beam has to be reselected at the BS and
potentially at the MT as well in order for the MT to stay connected with the
best serving BS beam.333The best beam is the beam of the serving BS that has
the MT in its main lobe. For example, as shown in Fig. 2, at time $t$, the
beam represented by the solid blue line is the best beam, while at time
$t^{\prime}$, the beam represented by the dashed pink line is the best beam.
Hence, the best beam selection is purely geometry-based and ignores
reflections of signals from the BS in order to maintain the focus of the paper
on the most fundamental aspects of beam management. Ideally, the herein called
geometry-based beam reselection would take place, where in order for the MT to
stay connected with the main lobe of the best beam (as defined in Footnote 3),
it should reselect the beam at a beam boundary. Such a location of beam
reselection is marked by “brown circle” in Fig. 2. However, due to 5G NR
system characteristics the herein called measurement-based beam reselection
takes place, where beam reselection events only occur after beam measurement
events of SSB bursts [23], which are transmitted periodically by the BS with
period $\tau$.

Hence the overall beam reselection process considers both the geometry- and
measurement-based beam reselection events. However, the mismatch between the
locations of geometry- and measurement-based beam reselections may lead to the
beam misalignment, which we discuss in the following subsection using Fig. 2.

Figure 2: Mobility-induced beam misalignment. Ideally, the beam reselection
should happen at the beam boundary for the MT to stay connected to the main
lobe of the best beam. But, since the SSB (a beam-formed reference signal)
location might not coincide with the location of ideal beam reselection at the
beam boundary, the MT at time $t^{\prime}$ is still connected to the side lobe
of the previously selected beam at time $t$, which results in a beam
misalignment.

###  II-D Mobility-induced Beam Misalignment

Beam misalignment occurs when the main lobes of the serving BS and the typical
MT are not aligned with each other. Specifically, during an SSB, at the
serving BS, the beam that has the MT within its main lobe (i.e., the best
beam) is selected for communication. We call this beam the reference beam (see
Fig. 2). Due to the mobility of the MT, if the duration between two
consecutive SSBs is sufficiently long, it is possible that the MT has moved
out of the main lobe of the reference beam without selecting the new beam at
the beam boundary (i.e., without performing geometry-based beam
reselection).444The duration between two consecutive SSBs is a system design
parameter. It this duration is very small, we get the geometry-based beam
reselection. Thus, the MT is still connected to the reference beam via side
lobe until the next SSB when the new reference beam is selected. Such a
mobility-induced beam misalignment at the BS reduces the signal strength at
the MT as it then lies within the side lobe of the reference beam selected
during the previous SSB.

Similar to the beam misalignment at the BS, there can be beam misalignment at
the typical MT. During an SSB, the beam that is pointing towards the serving
BS is selected at the MT to receive from the serving BS. Due to the mobility
of the MT between two consecutive SSBs, it is possible that the beam selected
at the MT during the previous SSB is no more pointing its main lobe towards
the serving BS. As a result, the MT receives from the serving BS in the side
lobe of the beam that was selected earlier in the previous SSB. In the worst-
case of beam misalignment, the side lobe of the selected beam at the serving
BS is aligned with the side lobe of the selected beam at the typical MT.

At a given time, the beam misalignment probability $p_{\rm bm}^{\rm{BS}}$ at
the BS depends on the speed $v$ of the MT, the duration $\tau$ between two
SSBs, and the average distance between two consecutive BS’s beam boundaries.
Here, we propose a simple yet effective formula that captures the effects of
these parameters on the probability of beam misalignment at the BS due to MT
mobility in a snapshot of the network. We evaluate the probability $p_{\rm
bm}^{\rm{BS}}$ that the MT is outside the main lobe of the reference beam by
the:

$\displaystyle p_{\rm bm}^{\rm{BS}}=1-\exp\left(-\frac{v\tau}{1/\mu_{\rm
s,b}}\right),$ (2)

where $1/\mu_{\rm s,b}=\frac{\pi}{2^{n}\sqrt{\lambda}}$ is the average
distance between two consecutive BS’s beam boundaries, with $\mu_{\rm s,b}$
being the linear intensity of beam boundary crossings given in Theorem 2 in
Section IV-B. We can interpret (2) as

$\displaystyle 1-p_{\rm bm}^{\rm{BS}}=\mathbb{P}(T>\tau),$

where $T$ is an exponential random variable with mean $\frac{1}{v\mu_{\rm
s,b}}$, with $T$ interpreted as the time between two consecutive SSBs. Note
that $\frac{1}{v\mu_{\rm s,b}}$ is the average time needed by the typical MT
to travel between two consecutive BS’s beam boundaries. Hence, the average
time between two SSBs is equal to the average time taken by the typical MT to
travel between two consecutive BS’s beam boundaries.

As in the case of beam misalignment on BS’s side, we use a similar formula to
calculate the probability $p_{\rm bm}^{\rm MT}$ of beam misalignment on MT’s
side as

$\displaystyle p_{\rm bm}^{\rm MT}=1-\exp\left(-\frac{v\tau}{1/\mu_{\rm
s,m}}\right)=1-\exp\left(-\frac{2^{k}\sqrt{\lambda}v\tau}{\pi}\right),$ (3)

where $1/\mu_{\rm s,m}=\frac{\pi}{2^{k}\sqrt{\lambda}}$ is the average
distance between two consecutive MT’s beam boundaries.

Based on the beam misalignment at both the serving BS and the typical MT, the
antenna gain $G_{0}$ of the serving BS is given by

$\displaystyle G_{0}(n,k)=\begin{cases}G_{{\rm m},n}G_{{\rm
m},k}&\mathrm{w.p.}~{}\left(1-p_{\rm bm}^{{\rm BS}}\right)\left(1-p_{\rm
bm}^{{\rm MT}}\right)\\\ G_{{\rm m},n}G_{{\rm
s},k}&\mathrm{w.p.}~{}\left(1-p_{\rm bm}^{{\rm BS}}\right)p_{\rm bm}^{{\rm
MT}}\\\ G_{{\rm s},n}G_{{\rm m},k}&\mathrm{w.p.}~{}p_{\rm bm}^{{\rm
BS}}\left(1-p_{\rm bm}^{{\rm MT}}\right)\\\ G_{{\rm s},n}G_{{\rm
s},k}&\mathrm{w.p.}~{}p_{\rm bm}^{{\rm BS}}p_{\rm bm}^{{\rm MT}}.\end{cases}$
(4)

###  II-E Blockage and Channel Model

In mmWave bands, the presence of blockages leads to LOS and NLOS propagation
between the MT and a BS. We adopt a LOS ball model [9] to capture the effect
of blockages. The propagation between a BS and the typical MT separated by
distance $d$ is LOS if $d<R_{\rm c}$ where $R_{\rm c}$ is the maximum distance
for LOS propagation. The LOS and NLOS channel conditions induced by the
blockage effect are characterized by different path-loss exponents, denoted by
$\alpha_{\rm L}$ and $\alpha_{\rm N}$, respectively. As shown in [24], typical
values of these path-loss exponents are $\alpha_{\rm L}\in[1.8,2.5]$ and
$\alpha_{\rm N}\in[2.5,4.7]$.

The channel follows Rayleigh fading with unit mean power gain.555Although the
LOS mmWave channels are better modeled by Nakagami-$m$ fading, Rayleigh fading
allows one much better analytical tractability. Also, simulation results in
Section VIII-B (see Figs. 6 and 7) show that Rayleigh and Nakagami fading
yield the same trends in terms of effective ASE. Let $h_{X}$ denote the
channel power gain from the BS at location $X\in\Phi$ to the MT. Note that,
the random variables $h_{X}$ are i.i.d. exponential random variables with unit
mean, i.e., $h_{X}\sim\exp(1)$. Let $|X|$ denote the distance between a BS at
$X\in\Phi$ and the typical MT located at the origin. We consider the standard
power-law path-loss model with path-loss function

$\displaystyle l(X)=\begin{cases}K|X|^{-\alpha_{\rm
L}}&\mathrm{if}~{}|X|<R_{\rm c}\\\ K|X|^{-\alpha_{\rm
N}}&\mathrm{if}~{}|X|\geq R_{\rm c},\end{cases}$ (5)

where $K=\left(\frac{c}{4\pi f_{\rm c}}\right)^{2}$ with $c$ the speed of
light and $f_{\rm c}$ the carrier frequency.

On the other hand, the signals in sub-6 GHz bands experience more scattering
and diffraction against to blocking and reflection in mmWave bands. Thus the
effect of blockages is insignificant. Hence, when analyzing performance in
sub-6 GHz bands, one may neglect blockages. This is equivalent to a special
case of mmWave bands where one assumes only NLOS propagation between the
typical MT and all BSs with the path-loss exponent between 2.5 and 4.7.

###  II-F Signal-to-Interference-Noise Ratio (SINR)

When the typical MT is associated with the BS located at $X_{0}\in\Phi$ with
antenna gain $G_{0}$ as in (4), the SINR at the typical MT is given by

$\displaystyle\mathsf{SINR}_{n,k}=\frac{PG_{0}(n,k)h_{X_{0}}l(|X_{0}|)}{\sigma^{2}+I_{n,k}},$
(6)

where $P$ is the transmit power of the BS. Also, $\sigma^{2}=WN_{0}$ is the
noise power, where $W$ and $N_{0}$ are the bandwidth and the noise spectral
density, respectively. In the denominator of (6), $I_{n,k}$ is the
interference power at the typical MT given by666The framework can also be
extended to the scenario where there is a frequency reuse among BSs as
described in [6].

$\displaystyle
I_{n,k}=\sum_{X\in\Phi\setminus\\{X_{0}\\}}PG_{X}(n,k)h_{X}l(|X|),$ (7)

where $G_{X}(n,k)$ is the antenna gain of an interfering BS located at
$X\in\Phi$.

Since the beams of all BSs are oriented towards their respective MTs, the
direction of arrivals between interfering BSs and the typical MT is
distributed uniformly in $[-\pi,\pi]$. Thus, the gain $G_{X}$ of an
interfering BS at $X\in\Phi$ is an i.i.d. discrete random variable with the
probability mass function as:

$\displaystyle G_{X}(n,k)=\begin{cases}G_{{\rm m},n}G_{{\rm
m},k}&\mathrm{w.p.}~{}p_{\rm mm}=\frac{\varphi_{n}\varphi_{k}}{4\pi^{2}}\\\
G_{{\rm m},n}G_{{\rm s},k}&\mathrm{w.p.}~{}p_{\rm
ms}=\frac{\varphi_{n}(2\pi-\varphi_{k})}{4\pi^{2}}\\\ G_{{\rm s},n}G_{{\rm
m},k}&\mathrm{w.p.}~{}p_{\rm
sm}=\frac{(2\pi-\varphi_{n})\varphi_{k}}{4\pi^{2}}\\\ G_{{\rm s},n}G_{{\rm
s},k}&\mathrm{w.p.}~{}p_{\rm
ss}=\frac{(2\pi-\varphi_{n})(2\pi-\varphi_{k})}{4\pi^{2}},\end{cases}$ (8)

where $\varphi_{n}=\frac{\pi}{2^{n-1}}$ and $\varphi_{k}=\frac{\pi}{2^{k-1}}$
are the beamwidths of the BS and the typical MT, respectively.

##  III Ergodic Shannon Rate

The downlink ergodic Shannon rate at the typical MT is given as

$\displaystyle\mathcal{R}(n,k)=W\mathbb{E}\left[\log\left(1+\mathsf{SINR}_{n,k}\right)\right],$

where $\mathbb{E}(\cdot)$ denotes expectation.

######  Definition 1 (Success Probability)

The success probability $p_{\rm s}$ of the typical MT is the probability that
the SINR at the typical MT exceeds a predefined threshold. Mathematically,

$\displaystyle p_{\rm
s}(n,k,\beta)\triangleq\mathbb{P}(\mathsf{SINR}_{n,k}>\beta),$

where $\beta>0$ is the predefined SINR threshold, which also parametrizes the
transmission rate.

###### Lemma 1

Let

$\displaystyle F(\alpha_{\rm S},\alpha_{\rm I},w,G_{0})$
$\displaystyle\triangleq 2\pi\lambda\left(1-\frac{p_{\rm mm}}{1+\frac{\beta
r^{\alpha_{\rm S}}G_{{\rm m},n}G_{{\rm m},k}}{G_{0}w^{\alpha_{\rm
I}}}}-\frac{p_{\rm ms}}{1+\frac{\beta r^{\alpha_{\rm S}}G_{{\rm m},n}G_{{\rm
s},k}}{G_{0}w^{\alpha_{\rm I}}}}-\frac{p_{\rm sm}}{1+\frac{\beta
r^{\alpha_{\rm S}}G_{{\rm s},n}G_{{\rm m},k}}{G_{0}w^{\alpha_{\rm
I}}}}-\frac{p_{\rm ss}}{1+\frac{\beta r^{\alpha_{\rm S}}G_{{\rm s},n}G_{{\rm
s},k}}{G_{0}w^{\alpha_{\rm I}}}}\right)w.$ (9)

The success probability $p_{\rm s}(n,k,\beta)$ is

$\displaystyle p_{\rm s}(n,k,\beta)$ $\displaystyle=\left(1-p_{\rm bm}^{\rm
BS}\right)\left(1-p_{\rm bm}^{\rm MT}\right)q_{\rm s}(n,k,\beta,G_{{\rm
m},n}G_{{\rm m},k})+\left(1-p_{\rm bm}^{\rm BS}\right)p_{\rm bm}^{\rm
MT}q_{\rm s}(n,k,\beta,G_{{\rm m},n}G_{{\rm s},k})$ $\displaystyle+p_{\rm
bm}^{\rm BS}\left(1-p_{\rm bm}^{\rm MT}\right)q_{\rm s}(n,k,\beta,G_{{\rm
s},n}G_{{\rm m},k})+p_{\rm bm}^{\rm BS}p_{\rm bm}^{\rm MT}q_{\rm
s}(n,k,\beta,G_{{\rm s},n}G_{{\rm s},k}),$ (10)

where

$\displaystyle q_{\rm s}(n,k,\beta,G_{0})$ $\displaystyle=\int_{0}^{R_{\rm
c}}f_{R}(r)\exp\left(-\frac{\beta r^{\alpha_{\rm
L}}\sigma^{2}}{PKG_{0}}\right)\exp\left(-\int_{r}^{R_{\rm c}}F(\alpha_{\rm
L},\alpha_{\rm L},w,G_{0})\mathrm{d}w-\int_{R_{\rm c}}^{\infty}F(\alpha_{\rm
L},\alpha_{\rm N},w,G_{0})\mathrm{d}w\right)\mathrm{d}r$
$\displaystyle+\int_{R_{\rm c}}^{\infty}\exp\left(-\frac{\beta r^{\alpha_{\rm
N}}\sigma^{2}}{PKG_{0}}-\int_{r}^{\infty}F(\alpha_{\rm N},\alpha_{\rm
N},w,G_{0})\mathrm{d}w\right)f_{R}(r)\mathrm{d}r$ (11)

with $f_{R}(r)=2\pi\lambda re^{-\lambda\pi r^{2}}$ where $R$ is the distance
to the nearest BS.

###### Proof:

The proof is given in Appendix A. ∎

###### Lemma 2

For sub-6 GHz band, the success probability $p_{\rm s}$ is given by (10) with

$\displaystyle q_{\rm
s}(n,k,\beta,G_{0})=\int_{0}^{\infty}f_{R}(r)\exp\left(-\frac{\beta
r^{\alpha_{\rm
L}}\sigma^{2}}{PKG_{0}}\right)\exp\left(-\int_{r}^{\infty}F(\alpha_{\rm
N},\alpha_{\rm N},w,G_{0})\mathrm{d}w\right)\mathrm{d}r.$ (12)

###### Proof:

The desired expression of the success probability is obtained by substituting
$R_{\rm c}=\infty$ and replacing $\alpha_{\rm L}$ with $\alpha_{\rm N}$ in
(11). ∎

###### Theorem 1

Imposing the limit $Q_{\rm max}$ on the maximum achievable SINR stemming from
RF imperfections and modulation schemes, the ergodic Shannon rate per unit
time is

$\displaystyle\mathcal{R}(n,k)=W\int_{0}^{Q_{\rm max}}\frac{p_{\rm
s}(n,k,z)}{z+1}~{}\mathrm{d}z,$ (13)

where $p_{\rm s}(n,k,z)$ is given by (11).

###### Proof:

The proof follows directly from

$\displaystyle\mathcal{R}(n,k)$
$\displaystyle=W\int_{0}^{\infty}\mathbb{P}(\log(1+\min(\mathsf{SINR}_{n,k},Q_{\rm
max}))>z)~{}\mathrm{d}z$ $\displaystyle=W\int_{0}^{Q_{\rm max}}p_{\rm
s}(n,k,e^{z}-1)~{}\mathrm{d}z$ $\displaystyle=W\int_{0}^{Q_{\rm
max}}\frac{p_{\rm s}(n,k,z)}{z+1}~{}\mathrm{d}z.$ (14)

∎

##  IV Beam (Re)Selection

###  IV-A Beam Selection during BS Handovers

When the MT performs a BS handover, a beam is selected with the new BS as well
as the typical MT. We know from [25] that, for the Poisson-Voronoi
tessellation, the linear intensity of cell boundary crossings, i.e., BS
handovers, is $\mu_{\rm s,c}=\frac{4\sqrt{\lambda}}{\pi}$. Hence, the time
intensity of BS handover (or, equivalently, beam selection) is

$\displaystyle\mu_{\rm c}=\frac{4\sqrt{\lambda}}{\pi}v.$ (15)

###  IV-B Beam Reselection Within the Cell

We now calculate the average number of beam reselections the typical MT
performs per unit length and the time intensity of beam reselection, i.e., the
average rate of beam reselections. As discussed in Section II-C, the beam
reselection events are a combination of geometry- and measurement-based beam
reselection events.

Recall that a geometry-based beam reselection occurs at a beam boundary within
the Voronoi cell of a BS. In Fig. 1, the locations of geometry-based beam
reselections are denoted by brown circles.

######  Theorem 2 (Intensity of geometry-based beam reselection)

For $2^{n}$ beams at a BS and PPP of intensity $\lambda$, the linear intensity
$\mu_{\rm s,b}$ of geometry-based beam reselection for the typical MT moving
on a straight line with speed $v$ is $\frac{2^{n}\sqrt{\lambda}}{\pi}$, while
the time intensity $\mu_{\rm t,b}$ is

$\displaystyle\mu_{\rm t,b}(n)=\frac{2^{n}\sqrt{\lambda}}{\pi}v.$ (16)

###### Proof:

The proof is given in Appendix B. ∎

This theorem is valid when integrating over all motion directions of the MT
chosen uniformly at random, and it does not hold for a fixed direction of MT
motion.

###### Corollary 1

When taken into account the average number of geometry-based beam reselections
that are skipped between two consecutive SSBs, the effective time intensity of
beam reselection becomes

$\displaystyle\mu_{\rm b}(n)=\frac{1}{\max\left(\tau,\frac{1}{\mu_{\rm
t,b}(n)}\right)},$ (17)

where $\tau$ is the SSB periodicity.

###### Corollary 2

The geometry-based time-of-stay $T_{\rm st}$ of the MT defined as the average
time spent by the MT within the main lobe of a beam is simply the inverse of
the time intensity $\mu_{\rm t,b}$, i.e.,

$\displaystyle T_{\rm st}=\frac{1}{\mu_{\rm
t,b}}=\frac{\pi}{2^{n}\sqrt{\lambda}v}.$ (18)

###### Corollary 3

The effective time-of-stay $T_{\rm eff,st}$ defined as the average time for
which the MT is connected with a beam of the BS is

$\displaystyle T_{\rm eff,st}=\frac{1}{\mu_{\rm
b}(n)}=\max\left(\tau,\frac{1}{\mu_{\rm t,b}(n)}\right).$ (19)

###### Remark 1

The effective time-of-stay allows one to evaluate whether the MT will have
enough time to perform MT beam refinement and link adaptation before data
transmission. Also, for a small SSB periodicity $\tau$, the effective time-of-
stay is equal to the geometry-based time-of-stay.

###  IV-C Time Overheads due to Beam (Re)Selection

This subsection builds on the previous subsections on analytical results for
beam (re)selections and incorporates the practical time overheads that arise
from the 5G NR design.

BS handovers and beam reselections may result in significant overheads in
terms of additional delay required for periodically conducting the SSB
measurement and report which are required for the SSB beam sweeping and beam
alignment procedures. Such an overhead in time reduces the overall time
available for data transmissions, in turn reducing the ergodic Shannon rate.

The two components that contribute to the time overhead are:

  1. 1.

The time $T_{\mathrm{c}}$ for executing the simultaneous cell and beam
reselection procedures during each BS handover at cell boundary crossing. This
includes the periodic SSB measurement, receiver processing time for the SSBs,
and the handover interruption time due to cell switch and radio resource
control (RRC) reconfiguration.

  2. 2.

The time $T_{\mathrm{b}}$ for beam alignment after each beam reselection
within the cell of a BS, which includes the periodic SSB measurement and the
receiver processing time for the SSBs.

There is also a scaling factor for those SSB measurement delays, associated
with the way the MT implements the SSB measurements, which is intrinsically
related to its number of antenna panels and their associated baseband
processing chains. In particular, the 3GPP specifies several multi-panel user
equipment assumptions (MPUE-Assumption) depending on the number of baseband
processing chains and the number of antenna panels at the MT side [27].777Note
that, in this paper, we use the terms the multi-panel MT and the directional
MT interchangeably in that a panel at the MT corresponds to a directional
beam.

For instance, MPUE-Assumption #1 covers the case when the MT has multiple
antenna panels but only a single baseband processing chain to execute SSB
measurements on them. This requires that the MT iterates over each single
antenna panel—one at a time—during SSB bursts for conducting SSB measurements.
This iterative process incurs additional delays for executing SSB measurements
over all MT antenna panels. On the other hand, MPUE-Assumption #3 covers the
case where the MT has multiple antenna panels but each antenna panel has its
own baseband processing chain. Thus the MT is capable of conducting SSB
measurements on all antenna panels in a single SSB burst, hence resulting in a
smaller measurement delay in comparison to MPUE-Assumption #1. In short, the
total average overhead per unit time depends on whether each of $2^{k}$ beams
(interchangeably, antenna panels) at the typical MT has its own baseband
processing chain or not. We discuss three possible cases as follows:

####  IV-C1 Limited Overhead Case

This case corresponds to MPUE-Assumption #3, where each of the $2^{k}$ beams
at the typical MT has its own baseband processing chain for independently
conducting SSB measurements. Thus the SSB beam measurement can be done in all
the beams of the MT simultaneously. In this case, the total average overhead
per unit time is

$\displaystyle T_{\mathrm{o}}(n)=\mu_{\rm b}(n)T_{\mathrm{b}}+\mu_{\rm
c}T_{\mathrm{c}}.$ (20)

Note that, in spite of $2^{k}$ beams at the typical MT, the average overhead
per unit time is the same as if the MT is omnidirectional. But such a gain
comes at the cost of increased hardware complexity at the MT due to the need
of a baseband processing chain for each beam.

####  IV-C2 Full Overhead Case

This case corresponds to MPUE-Assumption #1, where there is a single baseband
processing unit that needs to be shared by all of the $2^{k}$ beams at the
typical MT. Thus the SSB measurement for each of the $2^{k}$ beams is done
sequentially. In this case, the total average overhead per unit time is

$\displaystyle T_{\mathrm{o}}(n)=2^{k}\left(\mu_{\rm
b}(n)T_{\mathrm{b}}+\mu_{\rm c}T_{\mathrm{c}}\right).$ (21)

Here, the average overhead per unit time is $2^{k}$ times the one in the
limited overhead case, but with an advantage of significantly less hardware
complexity at the MT.

A key disadvantage of the full overhead case is that the beam measurement is
done during a BS handover even if the current serving BS and the next serving
BS lie in the coverage of the same MT beam. Thus we propose a probabilistic
beam switch case that uses the network geometry, namely, BS locations to
reduce the overhead compared to the full overhead case.

####  IV-C3 Probabilistic Beam Switch

The idea is to predict the possibility of a beam switch during a BS handover
since a beam switch is necessary if and only if the current and the next
serving BSs lie in the coverage of different MT beams. The case of $2$ beams
at the MT is depicted in Fig. 3. Exploiting the properties of the PPP that
models BS locations, the probability that the current and the next serving BSs
lie in different beams can be calculated [26]. Specifically, when there are
$2^{k}$ beams at the typical MT with main lobes oriented in uniformly at
random directions and covering all directions, the probability of a beam
switch is [26, Corollary 3]

$\displaystyle p_{\rm sw}=\frac{\pi}{2^{k}}\sin\frac{2^{k}}{\pi},\qquad k\geq
1.$ (22)

Alternatively, only $p_{\rm sw}$ fraction of BS handovers require a beam
switch. When there are $2$, $4$, or $8$ beams at the MT, the probability of a
beam switch during a BS handover is $0.637$, $0.9$, and $0.975$, respectively.
As a result, the total average overhead per unit time can be given by

$\displaystyle T_{\mathrm{o}}(n)=2^{k}\mu_{\rm b}(n)T_{\mathrm{b}}+(p_{\rm
sw}2^{k}+1-p_{\rm sw})\mu_{\rm c}T_{\mathrm{c}}.$ (23)

Figure 3: Probabilistic beam switching with two beams (panels) at the MT. The
triangles depict the BSs located according to a PPP. The solid black lines
denote cell boundaries. A “cross” denotes the BS handover location of the MT.
$\rm{P1}$, $\rm{P2}$, and $\rm{P3}$ denote randomly oriented panels of the MT
pertaining to each BS handover location, where each of the two beams covers
the left and right halves separated by the panel. For instance, the BSs
$X_{1}$ and $X_{2}$ lie in the coverage of two different beams, i.e., on the
opposite side of the panel $\rm{P1}$. Thus there will be a beam switch during
the handover from $X_{1}$ to $X_{2}$. On the contrary, $X_{2}$ and $X_{3}$ lie
in the coverage of the same beam (the left beam). Thus there is no need of a
beam switch during the handover from $X_{2}$ to $X_{3}$.

##  V Effective Area Spectral Efficiency

For each of the three overhead cases discussed in Section IV-C, the effective
ASE per unit time is

$\displaystyle\mathcal{R}_{\mathrm{eff}}(n,k)=\lambda(1-T_{\rm
o}(n))^{+}\mathcal{R}(n,k),$

where $\mathcal{R}(n,k)$ is given by (13) and $(A)^{+}=\max(0,A)$.

For a given $k$, our objective is to find the integer $n$ that maximizes the
effective ASE per unit time, i.e.,

$\displaystyle
n_{*}=\arg\max_{n\in\mathbb{N}}~{}\mathcal{R}_{\mathrm{eff}}(n,k).$

The value of $n_{*}$ can easily be found by a linear search.

##  VI Network Setup

To validate our proposed model, we consider a $5$G NR-compliant radio access
network (RAN) operating in a dense urban macro/pico cell scenario. A summary
of the model parameters for FR1 (sub-$6~{}\mathrm{GHz}$) and FR2 (above
$6~{}\mathrm{GHz}$) network deployments is provided in Table III.

TABLE III: Network parameter values for FR1 and FR2 deployments [4, 3] Parameter | FR1 | FR2
---|---|---
Carrier frequency ($f_{\rm c}$) | $3.5~{}\mathrm{GHz}$ | $28~{}\mathrm{GHz}$
Bandwidth ($W$) |  $100~{}\mathrm{MHz}$ per carrier |  $400~{}\mathrm{MHz}$ per carrier
Noise density ($N_{0}$) | $-174~{}\mathrm{dBm/Hz}$ | $-174~{}\mathrm{dBm/Hz}$
Transmit power ($P$) | $43~{}\mathrm{dBm}$ | $36~{}\mathrm{dBm}$
Beam reselection overhead ($T_{\rm b}$) | $23~{}\mathrm{ms}$ | $23~{}\mathrm{ms}$
Cell handover overhead ($T_{\rm c}$) | $43~{}\mathrm{ms}$ | $43~{}\mathrm{ms}$
SSB periodicity ($\tau$) | $20~{}\mathrm{ms}$ | $20~{}\mathrm{ms}$
MT speed ($v$) | $[3,30,120]~{}\mathrm{km/h}$ | $[3,30]~{}\mathrm{km/h}$
Inter-site distance (ISD) | $[250,500,1000]~{}\mathrm{m}$ | $[75,125,250{]}~{}\mathrm{m}$
Maximum SINR ($Q_{\rm max}$) | $30~{}\mathrm{dB}$ | $30~{}\mathrm{dB}$
Path loss exponent ($\alpha$) | 3.5 |  $\alpha_{\rm L}=1.9$, $\alpha_{\rm N}=3.5$
Blockage model | Implicit (NLOS) | LOS ball
LOS ball radius ($R_{\rm c}$) | $-$ | $75~{}\mathrm{m}$

To explore the best potential of $5$G NR networks in operational bands FR1 and
FR2, we use the maximum bandwidth allowed as per $5$G NR Release 15[4, 3],
namely $100~{}\mathrm{MHz}$ per carrier for FR1 and $400~{}\mathrm{MHz}$ per
carrier for FR2. For FR1, we assume NLOS conditions. Hence we set the value of
path loss exponent $\alpha=\alpha_{\rm N}$ = 3.5. For FR2, when inside the LOS
ball we assume $\alpha=\alpha_{\rm L}$ = 1.9, and we take $\alpha=\alpha_{\rm
N}$ = 3.5 outside the LOS ball.

As the inter-site (or, equivalently inter-BS) distances (ISDs) for FR1 are
expected to be larger due to lower frequency-dependent path loss, we choose a
transmission power $P$ = $43~{}\mathrm{dBm}$. For FR2, we can expect smaller
ISDs due to (a) higher attenuation loss at higher carrier frequencies, and (b)
because massive MIMO and high beamforming gains imposes challenges in terms of
RF exposure and EMF limitations. Thus we decrease $P$ to $36~{}\mathrm{dBm}$.
We assume a noise level $N_{0}$ of -174 $\mathrm{dBm/Hz}$, in order to
represent a realistic scenario that is not limited by noise.

An important parameter is the intensity $\lambda$ of BSs, which is related to
the average cell size and the ISD. For an intensity $\lambda$ of BSs, the
average cell size is $1/\lambda$. The average cell radius is defined as the
radius $r_{\rm cell}$ of a ball having the same average area as the cell:
$1/\lambda$. Then, the average ISD is $2r_{\rm cell}=2/(\sqrt{\pi\lambda})$.
Hence we simply adjust the value of the intensity $\lambda$ of BSs to
represent different ISD scenarios. For FR1 bands, we analyze the following
ISDs: $1000~{}\mathrm{m}$, $500~{}\mathrm{m}$, and $250~{}\mathrm{m}$. As for
FR2 bands, the effect of propagation attenuation requires us to decrease the
ISD. Hence we consider the following ISDs: $250~{}\mathrm{m}$,
$125~{}\mathrm{m}$, and $75~{}\mathrm{m}$.

One important network planning decision is the number of beams per BS and per
MT. In our model, this is related to the choice of the values for $n$ and $k$,
respectively. The main and side lobe gains depend on the number $2^{n}$ and
$2^{k}$ of beams. Specifically, an increase in the number of beams increases
the main lobe gain and decreases the side lobe gain. Thus, without loss of
generality, we assume the following antenna gains. The main lobe gain is
$G_{{\rm m},n}=2^{n}$ at a BS ($G_{{\rm m},k}=2^{k}$ at the MT), while the
side lobe gain is $G_{{\rm s},n}=\frac{1}{2^{n}}$ at a BS ($G_{{\rm
s},k}=\frac{1}{2^{k}}$ at the MT).888As seen from previous sections, our model
is amenable to different modeling of antenna gain patterns.

##  VII Numerical Results: Omnidirectional MT

We now demonstrate how the proposed model can be used for system-level
evaluation of the effect of different system parameters on 5G NR network
deployments over FR1 and FR2 for the omnidirectional MT for different MT
speeds $v$ and ISDs (in Figs. 4a, 4b, and 5). In Section VIII, we will
consider the case of the directional MT that has $2^{k}$, $k\geq 1$ beams. The
discussions in this section pertain to Figs. 4a, 4b, and 5.

###  VII-A Effect of the Number of Beams at BSs

In both FR1 and FR2 deployments, for a given ISD, as $n$ (and hence the number
$2^{n}$ of beams) increases, the beamforming gain increases and the
interference from interfering BSs decreases due to narrower beams. But, at the
same time, for a given $v$, the typical MT performs more frequent beam
reselections due to a smaller beamwidth, increasing the time overhead. The
negative impact of the increased time overhead gradually dominates the
beamforming gain. Also, an increase in $n$ increases the probability of beam
misalignment at the BS (given by (2)) as the MT is more likely to move into
the beamwidth of another beam between two consecutive SSB measurements. As a
result of this tussle between the increased beamforming gain, increased time
overhead due to beam handovers, and the increased probability of beam
misalignment, we observe that there is a value $n_{*}$ of $n$ that maximizes
the effective ASE.

(a) FR1: Dashed line: $v=3~{}\rm{km/h}$, Dashed-dotted line:
$v=30~{}\rm{km/h}$, and Solid line: $v=120~{}\rm{km/h}$.  (b) FR2: Dashed
line: $v=3~{}\rm{km/h}$ and Solid line: $v=30~{}\rm{km/h}$.

Figure 4: Effect of average inter-site distance (ISD) on the effective ASE for
different $n$.

###  VII-B Effect of the MT Speed

With an increase in $v$, the MT crosses beam and cell boundaries more
frequently resulting in a higher number of beam reselections and BS handovers,
respectively. Also, the probability of beam misalignment increases. Hence,
with an increase in $v$, the optimal value $n_{*}$ decreases, and for a fixed
$n$, the effective ASE decreases.

Figure 5: Comparison between FR1 and FR2 deployments for different ISDs with
$v=30~{}\rm{km/h}$.

###  VII-C Effect of the ISD

Note that a smaller ISD means a denser cellular network. As the ISD increases,
the average cell size increases. Thus, BSs need to increase the number $2^{n}$
of beams for a higher beamforming gain (equivalently, a smaller beamwidth) and
smaller interference. Hence, the value of $n_{*}$ increases with ISD. For a
smaller $n$, the time overhead associated with beam reselections is small.
Hence, a smaller ISD results in a higher effective ASE due to a smaller
distance between the typical MT and its serving BS. This dominates the
negative effects of increased interference power due to a denser deployment of
BSs and increased time overhead due to more frequent BS handovers. But, for a
large $n$, both beam reselections and BS handovers happen more frequently for
a smaller ISD, and the network with a larger ISD achieves a higher effective
ASE.

###  VII-D Comparison of FR1 and FR2 Deployments

When comparing beamwidths for FR1 with those for FR2 given the same $n$ value,
we can expect that, even though the angular beamwidth for a given $n$ is the
same, the linear width of the beams will be larger in FR1 than that in FR2 due
to a usually larger cell radius in the former.999Although nothing prohibits
small ISDs for FR1 (e.g., Wi-Fi operation in $2.4~{}\mathrm{GHz}$ and
$5~{}\mathrm{GHz}$), pico/femto cell deployment models are the best fit for
mmWave deployments. Thus, it is expected that MTs operating in FR1 are less
penalized due to overheads associated with beam reselections and BS handovers.
Also, due to a larger ISD, the distance-based path loss when operating in FR1
is higher than when operating in FR2. Hence for the same value of $n$, we can
expect that the interference in FR1 is smaller than that in FR2 despite a bit
higher transmit power in FR1. This indicates that the additional received
interference from a higher number of beams will not be as intense in FR2 as in
FR1. As shown in Fig. 5, this behavior is captured by our model, where the
optimal value $n_{*}$ for the ISD $=250~{}\mathrm{m}$ in FR1 ($n_{*}=8$) is
higher than that in FR2 ($n_{*}=6$) with ISD $=75~{}\rm{m}$ or in FR2 with ISD
$=125~{}\rm{m}$ ($n_{*}=7$).

Also, as Fig. 5 shows, when we compare the performances of FR1 and FR2
deployments for the same ISD, we observe an interesting tradeoff. Smaller ISDs
(e.g., $75~{}\rm{m}$) benefit FR2 irrespective of the value of $n$ as, for a
given critical LOS distance $R_{\rm c}$, it is more likely that the serving BS
has LOS propagation to the MT (so smaller path loss) boosting signal power.
Recall that, in FR1, even the serving BS always has NLOS propagation to the
MT. As a result, the impact of increased time overheads with $n$ on the
effective ASE is less in FR2 than that in FR1. As the ISD increases, the
probability of serving BS lying outside the LOS ball increases, in turn,
reducing signal power significantly in FR2. Thus, when combined with higher
path loss at higher frequencies in FR2, for higher values of $n$, the impact
of time overheads is relatively higher in FR2 than FR1. Note that, as a result
of the tussle between competing effects, the values of $n_{*}$ in FR1 and FR2
are close to each other for the same ISD.

Overall, the numerical results discussed in this section validate the claim
that the proposed model allows for a system-level evaluation of network
planning decisions in beam-based access networks such as $5$G NR.

Note that, although the discussions on the effect of number of beams at the
BS, MT speed, and the ISD on the effective ASE have considered until now an
omnidirectional MT, they are also valid for the case of directional MT.

##  VIII Numerical Results: Directional MT

We now study the effect of multiple beams at the typical MT and associated
systemic tradeoffs. We will also compare the performance of the directional MT
with that of the omnidirectional MT. Since it is expected that multiple beams
at the MT will be used mostly in FR2 (mmWave) due to the need of directional
communications, we restrict our discussion to FR2.

(a) Rayleigh fading (b) Nakagami-$4$ fading

Figure 6: Effect of the number $2^{k}$ beams at the typical MT on the
effective ASE. $v=30~{}\rm{km/h}$ and $\rm ISD=75$ m.

(a) Rayleigh fading (b) Nakagami-$4$ fading

Figure 7: Effect of the number $2^{k}$ beams at the typical MT on the
effective ASE. $v=30~{}\rm{km/h}$ and $\rm ISD=250$ m.

###  VIII-A Effect of the Number of Beams at the MT

Figs. 6 and 7 depict, for different ISDs and fading models, the systemic
tradeoffs associated with the directional MT and the effect of the number of
baseband processing chains discussed in Section IV-C. Recall that, as given by
(20), (21), and (23), the time overheads for the limited overhead (LO) case
and the omnidirectional MT are the same, and are smaller when compared to the
probabilistic overhead (PO) and full overhead (FO) cases, with the FO case
leading to the highest time overhead. Specifically, Figs. 6 and 7 compare the
performance of the directional MT with $2$ ($k=1$) and $4$ ($k=2$) directional
beams with the omnidirectional MT. For the LO case, the additional beamforming
gain due to multiple directional beams at the MT (in addition to the
beamforming gain at the BS) dominates consistently over the omnidirectional MT
case, as the time overhead due to multiple beams at the MT remains constant
for the LO case irrespective of the number $2^{k}$ of beams at the MT. In
other words, for all values of $n$, the LO case with 4 beams at the MT
performs best, and the LO case with 2 beams outperforms the omnidirectional MT
case. Although the probability of beam misalignment at the MT increases with
$k$ (see (3)), the beamforming gain dominates this performance deteriorating
factor.

On the other hand, for the FO case, the time overhead increases proportionally
to both the number $2^{k}$ of beams at the MT as well as the number $2^{n}$ of
beams at the BS (see (21)). Thus, after a certain value of $n$, the
beamforming gain at the MT is eventually dominated by the increased time
overhead and the probability of beam misalignment at both the MT and the BS.
For instance, as shown in Fig. 6a, for $2^{5}=32$ beams at the BS, the case
with $4$ beams at the MT achieves better effective ASE compared to the case
with $2$ beams at the MT, and the trend reverses from $2^{6}=64$ beams onwards
at the BS. Also, for the FO case, when the number $2^{n}$ of beams at the BS
is large enough (e.g., 32 beams in Fig. 6a), the advantage of the directional
communication at the MT vanishes due to the increased time overhead, and the
omnidirectional reception at the MT leads to the better effective ASE. The
optimal value of $n$ that maximizes the effective ASE reduces with an increase
in $k$ in order to balance between the positive effect of the beamforming and
the negative impact of the increased time overhead and the increased
probability of beam misalignment. For a given number of beams at the MT, i.e,
for fixed $k$, the PO case offers an advantage over the FO case due to a
smaller number of beam switching at the MT for the given set of network
parameters.

To determine the effect of fading on the performance, we compare two fading
models: Rayleigh and Nakagami-$m$. For the latter, we assume that the LOS
propagation is characterized by Nakagami-$4$ fading. As shown in Figs. 6 and
7, both fading models present the same performance trend in the effective ASE
for different number of beams at the BS and the MT, ISDs, and the overhead
cases. This confirms the interest of our mathematical results based on
Rayleigh fading.

As Figs. 6 and 7 together show, the effect of the ISD for the directional MT
is the same as that for the omnidirectional MT (discussed in Section VII-C).
With respect to the directional MT case, the comparison of Figs. 6 and 7
reveals that the crossover point, where the FO case with 4 beams at the MT
($k=2$) starts performing worse compared to the FO case with 2 beams at the MT
($k=1$) and to the omnidirectional MT, shifts towards right with an increase
in ISD as the influence of the beamforming gain increases due to larger
average cell sizes.

(a) Limited overhead case (b) Full overhead case

Figure 8: Three-dimensional plot of the effective ASE against the number
$2^{n}$ beams at the BS and the number $2^{k}$ beams at the MT.
$v=30~{}\rm{km/h}$ and $\rm ISD=200$ m.

Fig. 8 gives a general view of the effect of directional communications from
the perspectives of both the BS and the MT. Specifically, it gives the optimal
number of beams at the BS as well as the MT using a 2-D numerical
optimization. For the LO case, the optimal number of beams at the MT is
$2^{8}=256$, while it is $2^{6}=64$ at the BS. This is due to the fact that,
for the LO case, the time overhead does not increase with the number of beams
at the MT. Thus, it is better to increase the number of beams at the MT
instead of at the BS to achieve sufficient beamforming gain. However, the
negative impact of increased probability of beam misalignment eventually
dominates, which reduces the effective ASE after increasing the number of
beams beyond $256$ at the MT.101010As of now, having $256$ beams at the MT may
be infeasible, but it may be a reality in the future, especially for MTs that
are not smartphones, such as flying drones and vehicles. On the other hand, as
Fig. 8b shows, the optimal number of beams at the MT reduces to $4$ for the FO
case as a result of the proportional increase in the time overhead with the
number of beams at the MT.

Figure 9: Heterogeneous vs. homogeneous networks. $v=30~{}\rm{km/h}$ and $\rm
ISD=200$ m. $p_{0}=0.6$, $p_{1}=0.3$, and $p_{2}=0.1$.

###  VIII-B Network Performance for the Mix of MTs

Fig. 9 compares a homogeneous network consisting of only MTs with $4$ beams
($k=2$) and a heterogeneous network consisting of a mix of MTs traveling at
the same speed of $v=30~{}\rm{km/h}$. For the heterogeneous network, $60\%$ of
the MTs are omnidirectional ($p_{0}=0.6$), $30\%$ of the MTs are with $2$
beams ($p_{1}=0.3$), and $10\%$ of the MTs are with $4$ beams ($p_{2}=0.1$).
The performance for the mix of the MTs can be obtained by weighing the ergodic
Shannon rate given by (13) by the probabilities $p_{0}$, $p_{1}$, and $p_{2}$.

Specifically, for the LO case, the homogeneous network of MTs with $4$ beams
suffers from a loss in the effective ASE if other types of MTs are to be added
in that network. This is an expected result as the MTs with $2$ beams and the
omnidirectional MTs achieve a smaller beamforming gain, while the time
overhead due to multiple beams at the MT remains the same irrespective of the
type of the MT. On the other hand, for the FO case, the design of the network
is more complicated. In particular, Fig. 9 depicts how the number of beams at
the BS must be adjusted depending on the composition of the network. For
instance, when the network consists of only MTs with $4$ beams, having a small
number beams at the BS is beneficial from the effective ASE point of view. On
the contrary, if the network is heterogeneous as per the aforementioned
composition, having a large number of beams at the BS is beneficial.

Figure 10: Overall performance for the mix of MTs. $\rm ISD=200$ m.
$p_{0}=0.6$, $p_{1}=0.3$, and $p_{2}=0.1$. $q_{3}=0.6$ and $q_{30}=0.4$.

Fig. 10 captures the overall performance of the network consisting of a mix of
MTs where $60\%$ of MTs move at $3~{}{\rm km/h}$ ($q_{3}=0.6$), while $40\%$
move at $30~{}{\rm km/h}$ ($q_{30}=0.4$). For each type of MT in terms of
speed, $60\%$ of the MTs are omnidirectional ($p_{0}=0.6$), $30\%$ of the MTs
have $2$ beams ($p_{1}=0.3$), and $10\%$ of the MTs are with $4$ beams
($p_{2}=0.1$). In such a network, $2^{9}=512$ beams are required at the BS to
maximize the effective ASE for the LO case, while $2^{8}=256$ beams are
sufficient for the FO case. Note that, despite of large time overhead in the
FO case compared to the LO case, the optimal effective ASE achieved in the
former case ($137.9$ nats/s/Hz/$\rm{km}^{2}$) is close to that in the latter
case ($142.3$ nats/s/Hz/$\rm{km}^{2}$). Recalling that the FO case has reduced
hardware complexity compared to the LO case, Fig. 10 implies that the network
operator might be able to provide almost the same quality of service to MTs at
reduced hardware complexity for certain network compositions.

##  IX Study of Some New Performance Metrics

In this section, we investigate how new performance metrics obtained in this
paper and given in Table I behave. These metrics are of independent interest.

###  IX-A Probability of Beam Misalignment at the BS

Fig. 11 depicts how the probability of beam misalignment $p_{\rm bm}^{\rm BS}$
at the BS behaves with the ISD, MT speed $v$, and the SSB periodicity $\tau$
as a function of $2^{n}$ beams at the BS. As the network becomes denser with a
decrease in the ISD, $p_{\rm bm}^{\rm BS}$ increases since the average cell
size decreases, which in turn increases the chances that the MT moves in the
coverage of the new beam between two consecutive SSBs without performing a
beam reselection (Fig. 11a). Thus, for a dense 5G NR network, such as with ISD
= $25~{}\rm{m}$, having a large number beams at the BS might not offer much
advantage as the beamforming gain is quickly dominated by the negative impact
of the beam misalignment as $n$ increases. Also, the faster the MT moves, the
higher are the chances that the MT moves in the coverage of the new beam
between two consecutive SSBs without performing a beam reselection (Fig. 11b).
Finally, Fig. 11c shows that increasing the SSB periodicity results in a
longer duration between two consecutive SSB bursts, which hence increases
$p_{\rm bm}^{\rm BS}$ at the BS. A similar analysis could be performed to
analyze the effect of the ISD, MT speed, and the SSB periodicity on the
probability of beam misalignment at the MT.

(a) ISD = 25, 75, 250, 500, 1000 m (left to right), $v$ = 30 km/h, $\tau$ = 20
ms. (b) $v$ = 3, 30, 120, 250 km/h (right to left), ISD = 200 m, $\tau$ = 20
ms. (c) $\tau$ = 5, 10, 20, 40, 80, 160 ms (right to left), ISD = 200 m, $v$ =
30 km/h.

Figure 11: Effect of the ISD, MT speed, and the SSB periodicity on the
probability of beam misalignment at the BS. Figure 12: Time overhead per unit
time vs. the number of beams at the BS. $v=30~{}\rm{km/h}$, $k=1$ (i.e., 2
beams at the MT), $\tau=20~{}\rm{ms}$.

###  IX-B Time Overheads

As discussed in Section IV-C, the time overheads due to beam selections during
BS handovers and beam reselections due to the intra-cell mobility of the MT
crucially depend on the number of baseband processing units at the MT. Fig. 12
depicts the effect of the number of beams at the BS on the time overhead. For
smaller ISD values, the average cell size is smaller, which leads to more
frequent BS handovers and beam reselections within the cell. This in turn
increases the time overhead per unit time. Due to the same reason, the
difference between the full overhead case and the probabilistic overhead case
is larger at a smaller ISD. Thus the latter case is expected to bring more
gains in terms of the time for data transmissions in ultradense cellular
networks.

When the number of beams at the BS is smaller (say $2^{3}=8$), the difference
between the three overhead cases is smaller. Thus they all lead to almost the
same effective ASE as seen in Figs. 6 and 7. But the cases of full and
probabilistic overhead have lesser hardware complexity due to a single
baseband processing unit compared to two units in the limited overhead case.
On the other hand, for a large number of beams at the BS, the time overhead in
the limited overhead case is much smaller, which may justify having a higher
hardware complexity.

##  X Conclusions and Future Directions

This paper has presented a first system-level mathematical framework for beam
management in $5$G RAN. This framework allows one to capture essential
scenario characteristics (e.g., operational frequency, blockage
characteristics), technological features (e.g., beamforming configuration,
delay overheads), network deployment choices (e.g., ISD, carrier bandwidth),
and traffic scenarios (e.g., mix of MTs). Also, it leads to the definition of
several new key performance metrics proper to 5G NR RANs, and stochastic
geometry was used to derive closed-form expressions of each of them. These
expressions are then aggregated in a global formula allowing one to evaluate
the effective area spectral efficiency of this 5G RAN. This effective area
spectral efficiency is the Shannon rate per Hertz averaged out over all users
of a large network, when taking into account both the pros of beam
densification in terms of improved SINR, and its cons in terms of the handover
overheads that are incurred by mobile users when they switch beams or cells.
For a dense urban macro/pico cell scenario in sub-6 GHz and mmWave bands, the
framework accurately captures the systemic tradeoffs between ISDs, path loss,
interference, and signaling overhead due to beam management. This also shows
that the stochastic geometry model can be effective for conducting system-
level analysis of beam-based radio access networks such as $5$G NR.

In the future, the model can be improved to incorporate sectorized BSs with
antenna panels and their effects on beam shaping/gains, MT blockage models,
and others. Further, our model can be extended to capture other tradeoffs
involved in designing a beam-based RAN. For instance, the average beam time-
of-stay, i.e., the time a user remains with a given beam before switching to
another beam, decreases with an increase in the number of beams in the
beamset, which impacts the available time for performing channel estimation,
link adaptation, and power control. Also, decreasing the periodicity interval
in which beamformed reference signal resources are monitored may improve the
effectiveness and response time of beam refinement, but at the same time it
results in reduced spectral efficiency due to the control overhead.

##  Appendix A Proof of Lemma 1

The success probability is

$\displaystyle p_{\rm s}(n,k,\beta)$
$\displaystyle\triangleq\mathbb{P}(\mathsf{SINR}_{n,k}>\beta).$ (24)

Based on whether the serving BS lies within the LOS ball of radius $R_{\rm c}$
or not, we can write the success probability $p_{\rm s}$ as

$\displaystyle p_{\rm s}(n,k,\beta)$ $\displaystyle=\int_{0}^{R_{\rm
c}}\mathbb{P}(\mathsf{SINR}_{n,k}>\beta\mid r)f_{R}(r)\mathrm{d}r+\int_{R_{\rm
c}}^{\infty}\mathbb{P}(\mathsf{SINR}_{n,k}>\beta\mid r)f_{R}(r)\mathrm{d}r,$
(25)

where $f_{R}(r)=2\pi\lambda r\exp(-\lambda\pi r^{2})$ is the probability
density function of the distance $R$ of the typical MT to the nearest BS. When
$0<r<R_{\rm c}$, we have

$\displaystyle\mathbb{P}(\mathsf{SINR}_{n,k}>\beta\mid r)$
$\displaystyle=\mathbb{P}\left(\frac{PKG_{0}h_{X_{0}}r^{-\alpha_{\rm
L}}}{N_{0}+I_{n,k}}>\beta\,\middle|\,r\right)$
$\displaystyle=\mathbb{P}\left(h_{X_{0}}>\frac{\beta r^{\alpha_{\rm
L}}(N_{0}+I_{n,k})}{PKG_{0}}\,\middle|\,r\right)$
$\displaystyle=e^{-\frac{\beta r^{\alpha_{\rm
L}}N_{0}}{PKG_{0}}}\mathbb{E}\left[\exp\left(-\frac{\beta r^{\alpha_{\rm
L}}I_{n,k}}{PKG_{0}}\right)\right]$ (26)

for a given $G_{0}$ which is a random variable with the probability mass
function given by (4).

We have

$\displaystyle\mathbb{E}\left[\exp\left(-\frac{\beta r^{\alpha_{\rm
L}}I_{n,k}}{PKG_{0}}\right)\right]$
$\displaystyle=\mathbb{E}\left[\exp\left(-\frac{\beta r^{\alpha_{\rm
L}}\sum_{X\in\Phi\setminus\\{X_{0}\\}}PG_{X}h_{X}l(|X|)}{PKG_{0}}\right)\right]$
$\displaystyle\overset{(\mathrm{a})}{=}\mathbb{E}\left[\prod_{X\in\Phi\setminus\\{X_{0}\\}}\mathbb{E}\left(\exp\left(-\frac{\beta
r^{\alpha_{\rm L}}G_{X}h_{X}l(|X|)}{KG_{0}}\right)\right)\right]$
$\displaystyle\overset{(\mathrm{b})}{=}\mathbb{E}\left[\prod_{X\in\Phi\setminus\\{X_{0}\\}}\mathbb{E}\left(\frac{1}{1+\frac{\beta
r^{\alpha_{\rm L}}l(|X|)G_{X}}{KG_{0}}}\right)\right]$
$\displaystyle\overset{(\mathrm{c})}{=}\mathbb{E}\left[\prod_{X\in\Phi\setminus\\{X_{0}\\}}\left(\frac{p_{\rm
mm}}{1+\frac{\beta r^{\alpha_{\rm L}}l(|X|)G_{{\rm m},n}G_{{\rm
m},k}}{KG_{0}}}+\frac{p_{\rm ms}}{1+\frac{\beta r^{\alpha_{\rm
L}}l(|X|)G_{{\rm m},n}G_{{\rm s},k}}{KG_{0}}}\right.\right.$
$\displaystyle\left.\left.+\frac{p_{\rm sm}}{1+\frac{\beta r^{\alpha_{\rm
L}}l(|X|)G_{{\rm s},n}G_{{\rm m},k}}{KG_{0}}}+\frac{p_{\rm ss}}{1+\frac{\beta
r^{\alpha_{\rm L}}l(|X|)G_{{\rm s},n}G_{{\rm s},k}}{KG_{0}}}\right)\right],$
(27)

where $(\mathrm{a})$ follows from the i.i.d. nature of fading random
variables, $(\mathrm{b})$ follows from averaging over the channel power gains
on interfering channels, and $(\mathrm{c})$ follows from the probability mass
function of the gain $G_{X}$ (given by (8)) of an interfering BS at
$X\in\Phi$.

From the probability generating functional (PGFL) of PPP, it follows that

$\displaystyle\mathbb{E}\left[\exp\left(-\frac{\beta r^{\alpha_{\rm
L}}I_{n,k}}{PKG_{0}}\right)\right]$
$\displaystyle=\exp\left(-2\pi\lambda\int_{r}^{\infty}\left(1-\frac{p_{\rm
mm}}{1+\frac{\beta r^{\alpha_{\rm L}}l(w)G_{{\rm m},n}G_{{\rm
m},k}}{KG_{0}}}-\frac{p_{\rm ms}}{1+\frac{\beta r^{\alpha_{\rm L}}l(w)G_{{\rm
m},n}G_{{\rm s},k}}{KG_{0}}}\right.\right.$
$\displaystyle\left.\left.-\frac{p_{\rm sm}}{1+\frac{\beta r^{\alpha_{\rm
L}}l(w)G_{{\rm s},n}G_{{\rm m},k}}{KG_{0}}}-\frac{p_{\rm ss}}{1+\frac{\beta
r^{\alpha_{\rm L}}l(w)G_{{\rm s},n}G_{{\rm
s},k}}{KG_{0}}}\right)w\mathrm{d}w\right)$
$\displaystyle=\exp\\!\left(\\!-\int_{r}^{R_{\rm c}}F(\alpha_{\rm
L},\alpha_{\rm L},w,G_{0})\mathrm{d}w-\int_{R_{\rm
c}}^{\infty}\\!F(\alpha_{\rm L},\alpha_{\rm N},w,G_{0})\mathrm{d}w\\!\right),$
(28)

where, in (28), the first integral corresponds to interfering BSs within the
LOS ball and the second integral to those outside the LOS ball with
$F(\alpha_{\rm S},\alpha_{\rm I},w,G_{0})$ given by (9).

Similarly, one can obtain an expression for
$\mathbb{P}(\mathsf{SINR}_{n}>\beta\mid r)$ when $R_{\rm c}\leq r<\infty$,
where the interfering BSs lie outside the LOS ball. By averaging over the
distance $R$ of the typical MT to its nearest BS, one gets (11). Finally,
averaging over the gain $G_{0}$ from the serving BS yields the final
expression of the success probability in (10).

##  Appendix B Proof of Theorem 2

The geometry-based beam reselections occur at beam boundaries of a BS. In Fig.
1, these locations of beam reselections are denoted by “brown circles”. Let
$\theta$ denote the angle of a beam boundary with respect to the direction of
the motion of the MT (see Fig. 13). This angle $\theta$ is distributed
uniformly at random in $[0,\pi]$. In Fig. 13, a ‘brown circle’ denotes a point
along the $x$-axis where the typical MT reselects the beam. Let
$\Psi\subset\mathbb{R}$ denote the point process of beam reselection events,
where the points of $\Psi$ are indicated by ‘brown circles’. We are interested
in calculating the average number of beam reselections the typical MT performs
per unit length, which is equivalent to calculating the intensity of $\Psi$.

Figure 13: At most one geometry-based beam reselection for a BS.
$\bigtriangleup$ : Base station locations, brown circle : Beam reselection
location, dashed lines : Beam boundaries, and dotted line : MT trajectory
along the $x$-axis.

Without loss of generality, consider the motion of the typical MT in the
interval $[0,\ell]$, which corresponds to the motion of the typical MT from
$(0,0)$ to $(\ell,0)$ along the $x$-axis. First, we calculate the average
number of beam reselections when there is at most one beam reselection
corresponding to a BS. This happens when there are two beams of the same size
for a BS. Using the result for two beams, we can obtain the intensity of beam
reselections corresponding $2^{n}$ beams at a BS.

During the interval $[0,\ell]$, the typical MT may move through the Voronoi
cells of multiple BSs. As shown in Fig. 14, let $\omega(X)$ denote the point
of the beam reselection corresponding to the BS located at $X\in\Phi$. The
event of beam reselection corresponding to a BS at $X\in\Phi$ occurs when the
following two events occur simultaneously:

  1. 1.

the point of the beam reselection lies in the Voronoi cell of the BS at
$X\in\Phi$, i.e., $\omega(X)\in V_{\Phi}(X)$ where $V_{\Phi}(X)$ is the
Voronoi cell of the BS located at $X\in\Phi$.

  2. 2.

the point of the beam reselection lies on the line connecting $(0,0)$ and
$(\ell,0)$, i.e, $\omega(X)\in[0,\ell]$.

Consequently, conditioning on $\theta$, the average of number of beam
reselections in $[0,\ell]$ is

$\displaystyle\mathbb{E}(\Psi[0,\ell]\mid\theta)=\mathbb{E}\left(\displaystyle\sum_{X\in\Phi}\boldsymbol{1}_{\omega(X)\in
V_{\Phi}(X)}\boldsymbol{1}_{\omega(X)\in[0,\ell]}\right),$

where $\boldsymbol{1}$ is the indicator function. Conditioning on the fact
that there is a BS at location $z\in\mathbb{R}^{2}$ and using the Campbell’s
theorem [5], it follows that

$\displaystyle\mathbb{E}(\Psi[0,\ell]\mid\theta)=\lambda\int_{\mathbb{R}^{2}}\mathbb{P}_{z}(\omega(z)\in
V_{\Phi}(z))~{}\boldsymbol{1}_{\omega(z)\in[0,\ell]}~{}\mathrm{d}z,$ (29)

where $\mathbb{P}_{z}(\cdot)$ denotes the Palm probability.

Figure 14: Strip $\mathcal{S}$ (shaded area) between two blue solid lines with
angle $\theta$ with the $x$-axis. Triangle : Base station location, brown
circle : Beam reselection location, and dashed line : Beam boundary.

Event $\boldsymbol{\omega(z)\in V_{\Phi}(z)}$: This event occurs when there is
no BS closer to the typical MT than the one at location $z$. This is
equivalent to the event that there is no BS in the ball of radius
$\|z-\omega(z)\|$ centered at $\omega(z)$. Since the BS point process is a PPP
with intensity $\lambda$, we have

$\displaystyle\mathbb{P}_{z}(\omega(z)\in V_{\Phi}(z))$
$\displaystyle=\exp\left(-\lambda\pi\|z-\omega(z)\|^{2}\right)$
$\displaystyle=\exp\left(-\frac{\lambda\pi y^{2}}{\sin^{2}\theta}\right),$
(30)

where $\|z-\omega(z)\|=\frac{y}{\sin\theta}$ since $z=(x,y)$.

Event $\boldsymbol{\omega(z)\in[0,\ell]}$: This event occurs if the BS at $z$
lies within the strip $\mathcal{S}$ between two lines passing through the
origin $(0,0)$ and $(\ell,0)$ at angle $\theta$ as shown in Fig. 14.
Equivalently, by representing $z$ in Euclidean coordinates as $z=(x,y)$, it
follows that

$\displaystyle\boldsymbol{1}_{\omega(z)\in[0,\ell]}~{}\mathrm{d}z=\boldsymbol{1}_{z\in\mathcal{S}}~{}\mathrm{d}z=\boldsymbol{1}_{(x,y)\in\mathcal{S}}~{}\mathrm{d}x\mathrm{d}y.$
(31)

The left line passing through the origin at an angle $\theta$ can be given as
$y=-x\tan\theta$, while the right line passing through $(\ell,0)$ at an angle
$\theta$ can be given as $y=(\ell-x)\tan\theta$. Thus, the BS at $z$ lies in
the strip $\mathcal{S}$ if

$\displaystyle-\infty<y<\infty~{}~{}~{}\text{and}~{}~{}~{}-\frac{y}{\tan\theta}\leq
x\leq\ell-\frac{y}{\tan\theta}.$ (32)

From (30), (31), and (32), we can express (29) as

$\displaystyle\mathbb{E}(\Psi[0,\ell]\mid\theta)$
$\displaystyle=\lambda\int_{y=-\infty}^{\infty}\mathrm{d}y\int_{x=-\frac{y}{\tan\theta}}^{\ell-\frac{y}{\tan\theta}}\exp\left(-\frac{\lambda\pi
y^{2}}{\sin^{2}\theta}\right)\mathrm{d}x$
$\displaystyle=\ell\left(\sqrt{\lambda}|\sin\theta|\right).$

Since $\theta$ is uniformly distributed in $[0,\pi]$, averaging over $\theta$
yields
$\mathbb{E}(\Psi[0,\ell])=\left(\frac{2\sqrt{\lambda}}{\pi}\right)\ell.$
Hence, for the case of two beams, the linear intensity of beam reselections is
$\frac{2\sqrt{\lambda}}{\pi}$.

For $2^{n}$ beams with $n\in\mathbb{N}$, there are $2^{n-1}$ possibilities of
beam reselections corresponding to the serving BS. In this case, we can obtain
the average number of beam reselections by superimposing the beam reselection
events corresponding to each beam boundary crossing considered earlier in this
proof. Hence, the linear intensity of beam reselections for $2^{n}$ beams at a
BS is

$\displaystyle\mu_{\rm
s,b}(n)=2^{n-1}\frac{2\sqrt{\lambda}}{\pi}=\frac{2^{n}\sqrt{\lambda}}{\pi}.$
(33)

Considering the speed $v$ of the MT, one gets the time intensity $\mu_{\rm
t,b}$ of geometry-based beam reselections as $\mu_{\rm
t,b}(n)=\frac{2^{n}\sqrt{\lambda}}{\pi}v.$

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