# Influence of accretion disk on the optical appearance of the Kazakov-
Solodukhin black hole

Yu-Xiang Huang1, Sen Guo∗2, Yu-Hao Cui3, Qing-Quan Jiang∗1, Kai Lin∗3 1School
of Physics and Astronomy, China West Normal University, Nanchong 637000,
People’s Republic of China
2Guangxi Key Laboratory for Relativistic Astrophysics, School of Physical
Science and Technology, Guangxi University, Nanning 530004, People’s Republic
of China
3Hubei Subsurface Multi-scale Imaging Key Laboratory, School of Geophysics and
Geomatics, China University of Geosciences, Wuhan 430074, People’s Republic of
China<EMAIL_ADDRESS>
###### Abstract

The optical characteristics of the Kazakov-Solodukhin (KS) black hole (BH)
surrounded by a thin disk is investigated. By utilizing the ray-tracing
technique, we derive the the direct and secondary images to illustrate the
influence of observational inclination angle ($\theta_{0}$) and quantum
correction parameter ($a$) on the observable properties. Our calculation
involves determining the effective potential, light deflection, and azimuthal
angle of the BH, as well as the radiation flux of the accretion disk. By
assuming three function models, we determine that the exponential function is
the most suitable for describing the relationship between the observed flux
and the quantum deformation parameter. Our research reveals that the
observable characteristics of a BH are affected not only by the shape of the
accretion disk, but also by its spatiotemporal properties and the observer’s
inclination angle. We observe that a BH encircled by an optically thick
accretion disk yields different results from those of an optically thin disk.

  * Oct. 2023

##  1 Introduction

Black hole (BH), which is regarded as the most mysterious compact object in
the Universe, have received a large amount of observational evidence to date.
The Event Horizon Telescope (EHT) has been instrumental in capturing the
images of the supermassive BH at the center of the Messier 87∗ (M87∗) galaxy
[1]. Additionally, the EHT published findings on the magnetic field structure
of the surrounding accretion disk in 2019, illustrating that the magnetically
arrested accretion disk is consistent with the predictions of general
relativistic magnetohydrodynamic (GRMHD) models [2]. Subsequently, the EHT
captured the inaugural horizon-scale radio observations image of the
Sagittarius A∗ (Sgr A∗) positioned at the heart of the Milky Way, and various
imaging and modeling analyses have supported the image being primarily
composed of a bright, thick ring with a diameter of $51.8{\pm}2.3$ ${\rm{\mu
as}}$ [3]. Given these discoveries, it is reasonable to hypothesize that the
accretion disk surrounding the BH could be influencing the observational
characteristics of the BH shadow.

If the accretion rate of a BH surpasses that of other celestial bodies in the
galaxy, it can lead to the formation of an active galactic nucleus (AGN)
through standard, radiation-efficient, thin disk accretion at the Eddington
limit. Research on BH accretion disks dates back to the 1970s, the Shakura-
Sunyaev model of the accretion disk describes it as geometrically thin and
optically thick, providing a description of its geometric thickness and
optical properties [4]. By considering the thin accretion disk model, Luminet
obtained the direct and secondary images of the accretion disk through semi-
analytic method, demonstrating that the simulated photograph is consistent
with BHs of any mass accreting matter at any moderate rate [5]. Notably, this
method uses elliptic integrals for analysis. Utilizing the Luminet’s method,
Gyulchev $et~{}al.$ analyzed the lensing properties of the Janis-Newman
solution, as well as the stable circular geodesics in the equatorial plane.
They obtained the optical appearance and the apparent radiation flux of a thin
accretion disk around the static Janis-Newman-Winicour naked singularity [6].
Furthermore, Paul $et~{}al.$ employed two distinct numerical methods to
generate images of thin accretion disks situated in the background of Kerr-
like and Teo class wormholes [7]. These results were extended to a rotating
traversable wormhole of Teo class and the Schwarzschild BH pierced by a cosmic
string [8, 9]. BH accretion disk can account for the spectral characteristics
of many AGNs, particularly the redshift of the $Fe$ $K\alpha$ line. In Tanaka
$et~{}al.$’s research, it was shown that the iron emission is influenced by
gravitational redshift [10]. Cunningham used the ray-tracing techniques to
study the accretion disk around the Kerr black hole, and found that different
observation angles have a greater impact on the observed radiation redshift
[11]. Based on the consideration of the thin accretion disk surrounding the
Schwarzschild BH, the image of the disk can be accurately characterized using
ray-tracing numerical methods and radiation transfer [12]. Extensive research
on the image of the disk in various BH modified gravity contexts has also been
discussed in literature [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23].

The investigation of singularities constitutes a vital area of focus in the
field of general relativity (GR). It is not only one of the earliest
applications of the energy condition, but also functions as a benchmark for
assessing the completeness of GR. The existence of singularities results in
the incompleteness of geodesics, which in turn highlights the need for quantum
gravity to address this pressing issue [24]. Investigations have been
conducted on the quantum effects in the vicinity of singular points, event
horizons, and spherically symmetric gravitational collapses [25, 26]. The
findings indicate that quantum effects may potentially prevent the emergence
of intrinsic singularities at the origin. Extensive studies have been
conducted on the quantum effects on BH spacetime in various contexts [27, 28,
29, 30, 31]. Hooft investigated quantum gravity within the semi-classical
method framework and discovered that BH production accompanied by the coherent
emission of real gravitons occurs at energies significantly higher than the
Planck mass [32]. On the other hand, the BH solution in the context of quantum
gravity remains an open question. The deformation situation of the
Schwarzschild BH solution caused by matter field were investigated by Kazakov
$et~{}al.$, who argue that this quantum correction solution could be describe
the quantum fluctuations in the field of matter [33]. Based on this
groundbreaking exploration, Konoplya utilized the computation of quasinormal
modes to further investigate this BH solution and discovered that turning on
quantum deformation results in a decrease in the radius of the BH’s shadow
[34]. Xu $et~{}al.$ investigated the weak and strong deflection gravitational
lensing by Kazakov-Solodukhin (KS) BH and reported observable effects [35].
Additionally, KS BH has been used in many other areas of research [36, 37, 38,
39].

It remains currently unclear whether quantum corrections, an important aspect
of theoretical BH solutions, can impact the observable properties of BHs. To
investigate this issue, Peng $et~{}al.$ examined the shadow of a quantum-
corrected Schwarzschild BH in the context of an optically and geometrically
thin disk. Their study demonstrated that due to quantum corrections, the
photon sphere and critical curve violate the general inequality of
asymptotically flat BHs adhering to the zero energy condition in Einstein’s
gravity. These violations result in observable changes in the appearance of
the accretion disk around the BH [40]. In this analysis, we extend the study
by examining the optical appearance of a KS BH with a geometrically thin and
optically thick accretion disk. Note that our’s work is an extension of [40],
with the goal of providing a more comprehensive understanding of the possible
optical appearance of the target BH. Specifically, we analyze the direct and
secondary images of the accretion disk and investigate the impact of the
deformation parameter on the radiation flux. Additionally, we consider the
distribution of redshift and flux in the images.

The paper is structured as follows. Section 2 discusses the effective
potential of the KS BH and limits its deformation parameter using EHT
observations. In Section 3, we analyze the orbits of the direct and secondary
images of the KS BH and investigate the unit flux distribution of radiation
energy on a disk, as well as the functional relationship between its flux and
the deformation parameter. Section 4 considers the red shift into its radiant
energy flux and presents its empirical function. Finally, we conclude in
Section 5.

##  2 The effective potential of KS BH

The KS BH metric can be written as [33]

${\rm d}s^{2}=-f(r){\rm d}t^{2}+\frac{1}{f(r)}{\rm d}r^{2}+r^{2}{\rm
d}\theta^{2}+r^{2}\sin^{2}\theta{\rm d}\phi^{2},$ (1)

where $f(r)$ is the metric potential,

$f(r)=\frac{\sqrt{r^{2}-a^{2}}}{r}-\frac{2M}{r},$ (2)

in which $M$ represents the BH mass. The parameter $a$ arises from the
effective two-dimensional dilaton gravity and serves as a deformation
parameter. When $a$ is equal to zero, Eq. (2) simplifies the Schwarzschild BH.
The horizon radius of this BH can be expressed as

$r_{\rm H}=\sqrt{a^{2}+4M^{2}}.$ (3)

Note that the curvature of the two-dimensional sphere $r=a$ diverges in a
four-dimensional image, and the extremal value problem presented in the above
equation can be resolved [33].

In order to study the optical appearance of the KS BH, we must know how
photons move around it. The motion of photons can be represented by the Euler-
Lagrange equation

$\frac{{\rm d}}{{\rm
d}\lambda}\Big{(}\frac{\partial{\rm\mathcal{L}}}{\partial\dot{x}^{\rm\mu}}\Big{)}=\frac{\partial{\rm\mathcal{L}}}{\partial
x^{\rm\mu}},$ (4)

where $\lambda$ is the affine parameter and $\dot{x}^{\mu}$ is the four-
velocity of the photon. If we only consider the motion of photons on the
equatorial plane, the Lagrange equation can be re-expressed as

$\mathcal{L}=-\frac{1}{2}g_{\rm\mu\nu}\frac{{\rm d}x^{\rm\mu}}{{\rm
d}\lambda}\frac{{\rm d}x^{\rm\nu}}{{\rm d}\lambda}=0.$ (5)

The generalized momentum $p_{\rm\mu}$ of a particle can be represented as
follows:

$p_{\rm\mu}=\frac{\partial{\rm\mathcal{L}}}{\partial{\dot{x}^{\rm\mu}}}=g_{\rm\mu\nu}\dot{x}^{\rm\nu},$
(6)

which leads to the derivation of four equations of motion for a particle
having energy $E$ and angular momentum $L$, i.e.,

$\displaystyle p_{\rm t}=g_{\rm tt}\dot{t}=-E,$ (7) $\displaystyle
p_{\rm\phi}=g_{\rm\phi\phi}\dot{\phi}=-L,$ (8) $\displaystyle p_{\rm r}=g_{\rm
rr}\dot{r},$ (9) $\displaystyle
p_{\rm\theta}=g_{\rm\theta\theta}\dot{\theta}.$ (10)

Therefore, the motion equation of the null geodesic can be written as

$\frac{p_{\rm t}^{2}}{f(r)}-\frac{p_{\rm\phi}^{2}}{r^{2}}=\frac{(p^{\rm
r})^{2}}{f(r)}.$ (11)

Using the Eq. (11), the radial component of the null geodesic is derived, i.e.

$p^{\rm r}={\pm}E\sqrt{1-\frac{b^{2}}{r^{2}}f(r)},$ (12)

where $b\equiv L/E$ is the impact parameter, the symbol “$\pm$” indicates the
direction of photon motion, either clockwise ($+$) or counterclockwise ($-$).
The effective potential of the KS BH is obtained, we have

${\rm V}_{\rm eff}=r^{4}\Bigg{(}\frac{1}{b^{2}}-\frac{f(r)}{r^{2}}\Bigg{)}.$
(13)

If the effective potential satisfies ${\rm V}_{\rm eff}=\frac{1}{b_{\rm
c}^{2}}$ and ${\rm V}_{\rm eff}^{\prime}=0$, the stability region of the
photon orbit, namely the photon ring, can be determined based on the effective
potential. For the KS BH, the photon ring radius and critical impact parameter
can be expressed as

$\displaystyle r_{\rm ph}=\sqrt{\frac{3(\sqrt{9+2{x}^{2}}+3+{x}^{2})}{2}}M,$
(14) $\displaystyle b_{\rm
c}=\sqrt{\frac{\sqrt{27(\sqrt{9+2{x}^{2}}+3+{x}^{2})^{3}}}{2\sqrt{9+{x}^{2}+3\sqrt{9+2{x}^{2}}}-4\sqrt{2}}}M,$
(15)

in which ${x}=\frac{a}{M}$ is a dimensionless parameter.

Note that the KS BH shadow radius is affected by the quantum deformation
parameter $a$. In terms of observation, the BH shadow diameter $d_{\rm sh}$
can be measured with the EHT data. As we all known that the angular size of
the shadow of M87∗ is $\delta=(42{\pm}3)$ ${\rm{\mu as}}$, its distance is
$D=16.8_{-0.7}^{+0.8}$ ${\rm Mpc}$, and BH mass is $M=(6.5{\pm}0.9)\times
10^{9}M_{\odot}$. Using this observation data, the diameter of its shadow can
be calculated $d_{M87^{*}}=\frac{D\delta}{M}\simeq 11.0\pm 1.5$ [41, 42].

For Sgr A∗ BH, the EHT not only measured the angular size of the emission ring
as $\delta_{\rm d}=(51.8{\pm}2.3)$ ${\rm{\mu as}}$ but also estimated the
shadow angular $\delta=(48.7{\pm}7)$ ${\rm{\mu as}}$. The distance and BH mass
of Sgr A∗ are also given separately as $D=(8.15{\pm}0.15)$ ${\rm kpc}$ and
$M=(4.0_{-0.6}^{+1.1})\times 10^{6}M_{\odot}$ [3]. Therefore, the diameter of
its shadow $d_{\rm SgrA^{*}}$ could be got. Based on the aforementioned
observation data, we constrain $a$ with EHT observations. As illustrated in
Fig. 1, the result of KS BH is consistent with that derived from the EHT
observations within the observational uncertainty. The results indicate that
using the $1\sigma$ and $2\sigma$ confidence intervals of the $d_{\rm
M87^{*}}$, the deformation parameter can be constrained as $a\leq 2.51$ within
$1\sigma$ and $a\leq 3.39$ within $2\sigma$. When considering the confidence
intervals of $d_{\rm sgrA^{*}}$, it can be constrained as $a\leq 0.55$ within
$1\sigma$ and $a\leq 1.27$ within $2\sigma$.

Fig 1. Shadow diameter of the KS BH as a function of the deformation
parameter. The green and brown shaded regions represent the regions of
$1\sigma$ and $2\sigma$ confidence intervals, respectively, with respect to
the M87∗ and SgrA∗ observations.

Then, we derive the deflection angle of light moving around the KS BH. Based
on Eqs. (7) - (11), we have

$\Omega(u)\equiv\Big{(}\frac{{\rm d}u}{{\rm
d}\phi}\Big{)}^{2}=2Mu^{3}+\frac{1}{b^{2}}-u^{2}\sqrt{1-a^{2}u^{2}},$ (16)

where the $u$ is defined as $u\equiv 1/r$. If the impact parameter satisfies
$b>b_{\rm c}$, the light rays will be deflected by the gravity effect, but
will not be absorbed into the BH. When the impact parameter satisfies
$b<b_{\rm c}$, light rays are drawn into BH in this situation. By using the
Cardano formula, the Eq. (16) can be re-written as follows:

$\displaystyle 2Mu^{3}+\frac{1}{b^{2}}-u^{2}\sqrt{1-a^{2}u^{2}}\equiv 2MG(u)$
$\displaystyle=2M(u-u_{1})(u-u_{2})(u-u_{3}).$ (17)

Here, the cubic polynomial $G(u)$ has two positive roots and one negative
root, satisfying $u_{1}\leq 0<u_{2}<u_{3}$. Following the Ref. [5], these
three roots can be represented by introducing a parameter, that is
$Q^{2}\equiv(P-2M)(P+6M)$. Utilizing the periastron distance $P$ (a given
periastron distance $P$ can be used to determine the corresponding impact
parameter $b$ at infinity), we have

$u_{1}=\frac{P-2M-Q}{4MP},~{}~{}u_{2}=\frac{1}{P},~{}~{}u_{3}=\frac{P-2M+Q}{4MP},$
(18)

and

$b^{2}=\frac{1}{\frac{P-2M}{P^{3}}+u^{2}(\sqrt{1-a^{2}u^{2}}-1)}.$ (19)

Hence, the bending angle of light can be expressed as

$\psi(u)=\sqrt{\frac{2}{M}}\int_{0}^{u_{2}}\frac{{\rm
d}u}{\sqrt{(u-u_{1})(u-u_{2})(u-u_{3})}}-\pi.$ (20)

Obviously, it is difficult to directly integrate the above equation, so we
refer to the semi analytical method of Luminet and convert this equation into
the elliptic integral, we can obtain the following equation:

$\psi(u)=\sqrt{\frac{2}{M}}\Bigg{(}\frac{2F(\Psi_{1},k)}{\sqrt{u_{3}-u_{1}}}-\frac{2F(\Psi_{2},k)}{\sqrt{u_{3}-u_{1}}}\Bigg{)}-\pi,$
(21)

where $\Psi_{1}=\frac{\pi}{2}$,
$\Psi_{2}=\sin^{-1}\sqrt{\frac{-u_{1}}{u_{2}-u_{1}}}$, and
$k=\sqrt{\frac{u_{2}-u_{1}}{u_{3}-u_{1}}}$. With the help of Eq. (18), the
total change of bending angle of the KS BH is calculate, i.e.,

$\psi(u)=2\sqrt{\frac{P}{Q}}\Big{(}K(k)-F(\Psi_{2},k)\Big{)}-\pi,$ (22)

in which $K(k)$ is the complete elliptic integrals of the first kind.

##  3 Trajectory and radiation intensity of the KS BH

In this section, we consider the KS BH, which is surrounded by an optically
thick and geometrically thin accretion disk. We investigate the direct and
secondary images of the KS BH using the Luminet method. Furthermore, to
clearly describe this model, we provide a simple observation diagram (see Fig.
2). For a given emitter $M$ with the coordinates $(r,\varphi)$, the observer
can observe it at point $m$ on the observation plane $(b,\alpha)$. By changing
the position of the emitter $M$, we can obtain the direct image with the
coordinates $(b^{(d)},\alpha)$ and the secondary image with coordinates
$(b^{(s)},\alpha+\pi)$. (These are described in detail in Ref. [5]).

Fig 2. The coordinate system is indicated by Ref. [5].

###  3.1 Ray trajectory of the KS BH

The angle of deflection required to reach the observer from point $M$ is
$\gamma$, and the observer’s inclination angle is denoted as $\theta_{0}$. By
using the spherical triangle sine theorem, we have

$\cos\alpha=\cos\gamma\sqrt{\cos^{2}\alpha+\cot^{2}\theta_{0}}.$ (23)

We can use elliptic integral to rewrite Eq. (2) to get the direct image of
accretion disk around the KS BH

$\displaystyle\gamma$
$\displaystyle=\frac{1}{\sqrt{2M}}\int_{0}^{1/r}\frac{1}{\sqrt{G(u)}}{\rm d}u$
(24) $\displaystyle=2\sqrt{\frac{P}{Q}}\Big{(}F(\zeta_{\rm
r},k)-F(\zeta_{\rm\infty},k)\Big{)},$

where $F(\zeta_{\rm r},k)$ and $F(\zeta_{\rm\infty})$ are the elliptical
integrals, and there are $k^{2}=\frac{Q-P+6M}{2Q}$, $\sin^{2}\zeta_{\rm
r}=\frac{Q-P+2M+4MP/r}{Q-P+6M}$, and
$\sin^{2}\zeta_{\rm\infty}=\frac{Q-P+2M}{Q-P+6M}$. Thus, we can write $r$ as a
function of $\alpha$ and $P$, i.e.,

$\frac{1}{r}=\frac{P-2M-Q}{4MP}+\frac{Q-P+6M}{4MP}sn^{2}\Big{(}\frac{\gamma}{2}\sqrt{\frac{Q}{P}}+F(\zeta_{\rm\infty},k)\Big{)}.$
(25)

Thinking of this equation, the iso-radial curves for a given angle
$\theta_{0}$ is derived. For the $(n+1)th$ order image of the KS BH accretion
disk, the Eq. (24) satisfies

$2n\pi-\gamma=2\sqrt{\frac{P}{Q}}\Big{(}2K(k)-F(\zeta_{\rm
r},k)-F(\zeta_{\rm\infty},k)\Big{)},$ (26)

where $K(k)$ is the complete elliptic integral.

Fig 3. The direct (red line) and secondary (blue line) images of KS BH
accretion disk on the observation angles. The lines from inside out correspond
to $r=6.2M$, $r=10M$, and $r=15M$. The observation angles from left to right
are $\theta_{0}=17^{\circ}$, $\theta_{0}=53^{\circ}$, and
$\theta_{0}=80^{\circ}$, respectively. Top Panel – the deformation parameter
$a=0.1$ and Bottom Panel – the deformation parameter $a=0.5$. The BH mass is
taken as $M=1$.

Figure 3 depicts the direct and secondary images of photon orbits around the
KS BH for different values of the deformation parameter. In the images
presented, photons emitted in a direction above the equatorial plane are
represented by the red line, while the blue lines correspond to secondary
images generated by photons moving in a direction below the equatorial plane.
It can be observed that the radius of the photon orbit decreases as the value
of $a$ decreases. Moreover, our research results are in line with those of
previous studies, indicating that the observation angle significantly
influences the results obtained. This implies that the outcomes would be
entirely different if viewed from different perspectives during observation.

###  3.2 Radiation flux

The radiation flux of a thin accretion disk is discussed in this section. The
expression of radiation flux is given in [43, 44]

$F=-\frac{\dot{M}}{4\pi\sqrt{-g}}\frac{\Omega_{,\rm r}}{(E-\Omega
L)^{2}}\int_{r_{\rm in}}^{r}(E-\Omega L)L_{,\rm r}{\rm d}r,$ (27)

where $\dot{M}$ is the mass accretion rate, $g$ is the metric determinant, and
$r_{\rm in}$ represents the inner edge of the accretion disk. The $E$,
$\Omega$ and $L$ denote the energy, angular momentum, and angular velocity,
respectively. For a static spherically symmetric BH, the metric can be
expressed as ${\rm d}s^{2}=g_{\rm tt}{\rm d}t^{2}+g_{\rm\phi\phi}{\rm
d}\phi^{2}+g_{\rm rr}{\rm d}r^{2}+g_{\rm\theta\theta}{\rm d}\theta^{2}$. The
$E$, $L$ and $\Omega$ are represented as

$\displaystyle E=-\frac{g_{\rm tt}}{\sqrt{-g_{\rm
tt}-g_{\rm\phi\phi}\Omega^{2}}},$ (28) $\displaystyle
L=\frac{g_{\rm\phi\phi}\Omega}{\sqrt{-g_{\rm tt}-g_{\rm\phi\phi}\Omega^{2}}},$
(29) $\displaystyle\Omega=\frac{{\rm d}\phi}{{\rm d}t}=\sqrt{-\frac{g_{\rm
tt,r}}{g_{\rm\phi\phi,r}}}.$ (30)

Utilizing the above equations, the radiant energy flux over the disk is
obtained, we have

$\displaystyle F=$
$\displaystyle\frac{-\dot{M}(6Mh^{3}+4a^{2}r^{2}-3a^{4})}{4\sqrt{2}\pi
r^{4}h^{2}(3a^{2}-2r^{2}+6Mh)\sqrt{\frac{2Mr^{2}+a^{2}(-2M+h)}{r^{5}-a^{2}r^{3}}}}$
$\displaystyle\int_{r_{\rm
in}}^{r}\frac{-2M(6a^{4}-6a^{2}r^{2}+r^{4})+12M^{2}h^{3}+3a^{4}h}{r^{2}h^{2}(3a^{2}-2r^{2}+6Mh)\sqrt{\frac{4Mr^{2}+2a^{2}(-2M+h)}{r^{5}-a^{2}r^{3}}}}{\rm
d}r,$ (31)

where $h\equiv\sqrt{r^{2}-a^{2}}$.

Fig 4. The intrinsic flux of the source $F$ of the accretion disk of the KS BH
is demonstrated at the inclination angles of the observer
$\theta_{0}=17^{\circ}$, $53^{\circ}$, and $80^{\circ}$. The inner edge of the
disk at $r_{in}=r_{isco}$, and the outer edge of the disk is at $r=15M$. Top
Panel: the deformation parameter $a=0.1$. Bottom Panel: the deformation
parameter $a=0.5$. The BH mass is taken as $M=1$.

Figure 4 depicts the radiation intensity of the accretion disk. It is evident
that an increase in the deformation parameter $a$ causes a notable decrease in
the flux ($F$) for large values. The main reason for this phenomenon is that
the strength of the gravitational field decreases with the increase of $a$,
which leads to the change of the motion speed of the matter on the accretion
disk around the BH, resulting in the change of temperature and other
properties, and ultimately the reduction of radiant flux. Additionally, the
figure shows that with an increase in the tilt angle, the separation between
the direct and secondary images becomes more pronounced, and the direct image
takes on a more prominent “hat-like” shape. In order to visually demonstrate
the relationship between radiation flux and deformation parameter, we assume
three functional models. These are _Case 1:_ Fourier function, _Case 2:_
Exponential function, and _Case 3:_ Multinomial function. The red dots
represent the accurate values of $F$ as a function of $a$. As $a$ increases,
the value of $F$ decreases, while the slope of the curve increases with the
increase of $a$. Additionally, we fit this curve using three hypothetical
functional models and find that the exponential function, under certain fixed
parameters, is more consistent with the results (see Fig. 5).

Fig 5. Fitting of fourier function (the black solid line), exponential
function(the bule solid line), and multinomial function (the green solid line)
to the accurate value of the intrinsic flux of the source $F$ and the
deformation parameter $a$.

##  4 Observable features of the thin disk

###  4.1 Observed flux

The gravity of a black hole affects matter on the accretion disk at different
distances, resulting in varying degrees of gravitational red shift.
Furthermore, due to the rotation of the disk, matter on the plane
perpendicular to the rotation axis experiences different rates and directions
of motion, leading to different degrees of red shift or blue shift. In the
left half of the disk, the contribution of Doppler blueshift caused by the
disk’s rotation can exceed the gravitational red shift, resulting in a
blueshift for the corresponding photons. In this scenario, we can calculate
the observed flux $F_{\rm obs}$ of the KS BH surrounding the accretion disk.
Following the Ref. [5], the observed flux $F_{\rm obs}$ satisfies

$F_{\rm obs}=\frac{F}{(1+z)^{4}}.$ (32)

When a photon is discharged from a source particle in orbit around a BH, its
energy can be expressed as the dot product of the photon’s 4-momentum vector
$p$ and the 4-velocity vector $\mu$ of the emitting particle. We have

$E_{\rm em}=p_{\rm t}\mu^{\rm t}+p_{\rm\phi}\mu^{\rm\phi}=p_{\rm t}\mu^{\rm
t}\Bigg{(}1+\Omega\frac{p_{\rm\phi}}{p_{\rm t}}\Bigg{)},$ (33)

where $p_{\rm t}$ and $p_{\rm\phi}$ represent the photon 4-momentum. For a
distant observer, $p_{\rm t}/p_{\rm\phi}$ is nothing but the impact parameter
of the photon relative the z-axis. According Eq. (23) and trigonometric
function relationship

$\sin\theta_{0}\cos\alpha=\cos\gamma\sin\beta,$ (34)

one can obtain

$\frac{p_{\rm t}}{p_{\rm\phi}}=b\sin\theta_{0}\sin\alpha.$ (35)

Thus, the redshift factor is

$1+z=\frac{E_{\rm em}}{E_{\rm obs}}=\frac{1+b\Omega\cos\beta}{\sqrt{-g_{\rm
tt}-2g_{\rm t\phi}-g_{\rm\phi\phi}}}.$ (36)

Using the KS BH metric, the above equation can be re-written as

$1+z=\frac{2+b\sqrt{\frac{4Mr^{2}+2a^{2}(-2M+\sqrt{r^{2}-a^{2}})}{r^{5}-a^{2}r^{3}}}\sin\alpha\sin\theta_{0}}{\sqrt{\frac{4r^{2}(\sqrt{r^{2}-a^{2}}-3M)-6a^{2}(\sqrt{r^{2}-a^{2}}-2M)}{r^{3}-a^{2}r}}}.$
(37)

Fig 6. Redshift distribution (curves of constant redshift $z$) of the
accretion disk of the KS BH at the inclination angles of the observer
$\theta_{0}=17^{\circ}$, $53^{\circ}$, and $80^{\circ}$. The inner edge of the
disk at $r_{in}=r_{isco}$, and the outer edge of the disk is at $r=15M$. Top
Panel: the deformation parameter is $a=0.1$ and Bottom Panel: the deformation
parameter is $a=0.5$. The BH mass is taken as $M=1$.

Figure 6 illustrates the redshift images of the accretion disk with different
parameter values. It is apparent that both red and blue shifts are present
simultaneously. Moreover, as the observation angle $\theta_{0}$ increases
gradually, the range of the redshift also gradually expands, which is
consistent with the findings reported in [11]. Due to the small impact of the
change in the deformation parameter $a$ on the redshift $z$, the change in the
redshift image in the figure is not significant.

Fig 7. The observed flux $F_{\rm obs}$ of the KS direct image with the
inclination angles of the observer $\theta_{0}=17^{\circ}$, $53^{\circ}$, and
$80^{\circ}$. The inner edge of the disk at $r_{\rm in}=r_{isco}$, and the
outer edge of the disk is at $r=15M$. Top Panel: the deformation parameter
$a=0.1$ and Bottom Panel: the deformation parameter $a=0.5$. The BH mass is
taken as $M=1$.

Fig 8. The observed flux $F_{\rm obs}$ of the KS secondary image with the
inclination angles of the observer $\theta_{0}=17^{\circ}$, $53^{\circ}$, and
$80^{\circ}$. The inner edge of the disk at $r_{\rm in}=r_{\rm isco}$, and the
outer edge of the disk is at $r=15M$. Top Panel: the deformation parameter
$a=0.1$ and Bottom Panel: the deformation parameter $a=0.5$. The BH mass is
taken as $M=1$.

Figures 7 and 8 depict the observed flux for the direct and secondary images,
respectively. It is evident that when the inclination angle of the observer
($\theta_{0}$) increases, the flux ($F_{\rm obs}$) becomes increasingly
asymmetric. And at the same inclination angle, the increase of deformation
parameter $a$ also leads to a decrease in flux. These phenomenons can be
attributed to several factors. As $a$ increases, the gravitational field of
the BH decreases, leading to changes in the velocity and density distribution
of matter as it moves toward BH, which affects the amount of energy needed for
light to escape from the BH, thus leading to the radiant flux distribution
observed by distant observer. Secondly, due to the rotation of the accretion
disk and the increase of the observation angle, the influence of the direction
of light movement on the observer increases. Since the light on the left side
of the disk moves towards the observer, while the right side moves away from
the observer, the observed flux on the left side is larger. In summary, the
observed radiant flux is affected by both gravitational redshift and Doppler
effect. Additionally, an increase in the deformation parameter ($a$) results
in a larger range of flux decreases.

###  4.2 Data analysis

To demonstrate the relationship between $F_{\rm obs}$ and $a$ more
intuitively, we use the same method that produced Fig. 5. Fig. 9 illustrates
that, under different observer angles $\theta_{0}$ ($\theta_{0}=17^{\circ}$,
$53^{\circ}$, and $80^{\circ}$). Obviously, the accurate value (indicated by
the red dotted line) reveals a gradual decrease in both the radiation flux of
the observer $F_{\rm obs}$ and the intrinsic flux of the source $F$ as the
deformation parameter $a$ increases in the range of 0 to 1. Furthermore, the
rate of decrease accelerates progressively.

Fig 9. Fitting of fourier function (the black solid line), exponential
function(the blue solid line), and multinomial function (the green solid line)
to the accurate value of the intrinsic flux of the observed flux $F_{\rm obs}$
and the deformation parameter $a$. Left Panel: The observer
$\theta_{0}=17^{\circ}$. Middle Panel: The observer $\theta_{0}=53^{\circ}$.
Right Panel: The observer $\theta_{0}=80^{\circ}$.

We assume that three functions are

$\displaystyle
F=a_{1}e^{-(\frac{a-a_{2}}{a_{3}})^{2}}+a_{4}e^{-(\frac{a-a_{5}}{a_{6}})^{2}},$
(38) $\displaystyle F=b_{1}+b_{2}\cos(ab_{3})+b_{4}\sin(ab_{5}),$ (39)
$\displaystyle
F=c_{1}a^{6}+c_{2}a^{5}+c_{3}a^{4}+c_{4}a^{3}+c_{5}a^{2}+c_{6}a+c_{7},$ (40)

where $a$, $b$, and $c$ with subscripts are the coefficient in the fitting
function. Table 1 presents the numerical relationship between the intrinsic
flux of the source ($F$), the observed flux ($F_{\rm obs}$), and the
deformation parameter ($a$) at various angles ($\theta_{0}$). We fit these
data with three different functions assumed above: a Fourier function
(represented by the black solid line), an exponential function (represented by
the blue solid line), and a multinomial function (represented by the green
solid line).

The Mean Squared Error (MSE) is a commonly used metric to assess the
discrepancy between predicted and actual values in different prediction
models. By calculating the squared differences between predicted and true
values, the MSE provides a reliable measure of the accuracy of predictive
models. In the present investigation, we have computed the MSE between the
three functions and the true value, in order to determine the best fitting
function. As shown in Table 2, the exponential function has an extremely small
magnitude of MSE in all cases, indicating that it is the closest to the
theoretical results. Conversely, the magnitudes of the Fourier and multinomial
functions are relatively large, suggesting that their results deviate from the
accurate value. Based on these results, we consider the exponential function
as an empirical function for the radiation flux and the deformation parameter,
and the parameter values for its fitting formula under various circumstances
are presented in Table 3. Our analysis revealed that the exponential function
can effectively describe the functional relationship between the observed flux
and quantum deformation parameter, providing new insights into the influence
of quantum correction on the observation results. Our results could serve as a
valuable tool for understanding quantum gravity.

Table 1: Numerical relationship between observed flux and the deformation parameter for $\alpha=\frac{\pi}{2}$ and $r$=15 $a$ | ${F}$ | ${F_{\rm obs}(\theta_{0}=17^{\circ})}$ | ${F_{\rm obs}(\theta_{0}=53^{\circ})}$ | ${F_{\rm obs}(\theta_{0}=80^{\circ})}$
---|---|---|---|---
$0.05$ | $0.833439$ | $0.405174$ | $0.262100$ | $0.207945$
$0.10$ | $0.832546$ | $0.404692$ | $0.261774$ | $0.207748$
$0.15$ | $0.831055$ | $0.403887$ | $0.261231$ | $0.207242$
$0.20$ | $0.828963$ | $0.402758$ | $0.260469$ | $0.206625$
$0.25$ | $0.826263$ | $0.401303$ | $0.259702$ | $0.205831$
$0.30$ | $0.822950$ | $0.399519$ | $0.258286$ | $0.204858$
$0.35$ | $0.819014$ | $0.397403$ | $0.256860$ | $0.203705$
$0.40$ | $0.814445$ | $0.394951$ | $0.255209$ | $0.202652$
$0.45$ | $0.809232$ | $0.392158$ | $0.253330$ | $0.200852$
$0.50$ | $0.803361$ | $0.389020$ | $0.251220$ | $0.199147$
$0.55$ | $0.796816$ | $0.385529$ | $0.248876$ | $0.197254$
$0.60$ | $0.789582$ | $0.381680$ | $0.246294$ | $0.19517$
$0.65$ | $0.781638$ | $0.377464$ | $0.243469$ | $0.192891$
$0.70$ | $0.772964$ | $0.372875$ | $0.240398$ | $0.194150$
$0.75$ | $0.763537$ | $0.367903$ | $0.237075$ | $0.187737$
$0.80$ | $0.753332$ | $0.362539$ | $0.233494$ | $0.184852$
$0.85$ | $0.742322$ | $0.356772$ | $0.229650$ | $0.181760$
$0.90$ | $0.730477$ | $0.350590$ | $0.225536$ | $0.178452$
$0.95$ | $0.717766$ | $0.343983$ | $0.221145$ | $0.174924$
Table 2: MSE of Fourier functions, exponential functions, and polynomial functions with accurate values for $\alpha=\frac{\pi}{2}$ and $r$=15 $-$ | ${F}$ | ${F_{\rm obs}(\theta_{0}=17^{\circ})}$ | ${F_{\rm obs}(\theta_{0}=53^{\circ})}$ | ${F_{\rm obs}(\theta_{0}=80^{\circ})}$
---|---|---|---|---
$FourierFunction$ | $6.1196\times 10^{-5}$ | $2.46304\times 10^{-5}$ | $8.91666\times 10^{-6}$ | $8.25137\times 10^{-6}$
$ExponentialFunction$ | $4.64208\times 10^{-9}$ | $1.02525\times 10^{-9}$ | $7.94073\times 10^{-10}$ | $8.33254\times 10^{-10}$
$MultinomialFunction$ | $9.21466\times 10^{-5}$ | $2.67287\times 10^{-5}$ | $7.06738\times 10^{-6}$ | $9.61621\times 10^{-6}$
Table 3: Coefficient value of exponential fitting function under different conditions for $\alpha=\frac{\pi}{2}$ and $r$=15 $-$ | ${a_{1}}$ | ${a_{2}}$ | ${a_{3}}$ | ${a_{4}}$ | ${a_{5}}$ | ${a_{6}}$
---|---|---|---|---|---|---
${F}$ | $0.8336$ | $0.01004$ | $2.561$ | $-0.02833$ | $1.349$ | $0.4081$
${F_{\rm obs}(\theta_{0}=17^{\circ})}$ | $0.4053$ | $0.007409$ | $2.451$ | $-0.01798$ | $1.456$ | $0.4693$
${F_{\rm obs}(\theta_{0}=53^{\circ})}$ | $0.2622$ | $0.004475$ | $2.422$ | $-0.0102$ | $1.407$ | $0.4711$
${F_{\rm obs}(\theta_{0}=80^{\circ})}$ | $0.208$ | $0.01325$ | $2.341$ | $-0.007433$ | $1.383$ | $0.4006$

Fig 10. Numerical simulation image of the KS BH is demonstrated at the
inclination angles of the observer $\theta_{0}=17^{\circ}$, $53^{\circ}$, and
$80^{\circ}$. The inner edge of the disk at $r_{\rm in}=r_{\rm isco}$, and the
outer edge of the disk is at $r=15M$. Top Panel: the deformation parameter
$a=0.1$. Bottom Panel: the deformation parameter $a=0.5$. The BH mass is taken
as $M=1$.

It is widely recognized that theoretical calculations of BH images can provide
valuable guidance for the observation of the Event Horizon Telescope (EHT),
which utilizes Very Long Baseline Interferometry (VLBI) technology to achieve
high-resolution imaging. Nonetheless, there exist some discrepancies between
the observation results of the EHT and the theoretical calculations of BH
images. Notwithstanding this, there is a close relationship between the two,
which not only enhances our understanding of the nature and properties of BHs
but also inspires further research in this field. In this study, we employed
numerical simulations to generate an accretion disk image of the KS BH.
Figure. 10 illustrates the predicted observation images of the KS BH under
various observation angles. It is worth noting that these predicted results
may provide valuable guidance for achieving the scientific goals of the EHT in
future research.

##  5 Conclusions

In this analysis, we investigate the optical radiation emitted by the Kazakov-
Solodukhin black hole in the context of an optically thick and geometrically
thin accretion disk. We derive the effective potential and deflection angle of
the KS BH using the four-momentum of photons. Our findings suggest that the
results obtained by Luminet’s semi-analytical method and full numerical
integration are in agreement. Additionally, we use EHT observations to
constrain the quantum correction parameter $a$, and our results indicate that
for M87, $a$ is constrained to be $a\leq 2.51$ within $1\sigma$ and $a\leq
3.39$ within $2\sigma$, while for Sgr A, $a$ is constrained to be $a\leq 0.55$
within $1\sigma$ and $a\leq 1.27$ within $2\sigma$.

Using elliptic integrals, we simulate the orbits and flux of the direct and
secondary images of the KS BH accretion disk under various observation angles.
Additionally, we observe that the radius of the photon orbit decreases as the
value of $a$ decreases. The significant impact of observation inclination on
observation results is also consistent with Cunningham’s research. For the
radiation flux of the accretion disk, our results show that an increase in the
deformation parameter $a$ leads to a notable reduction in the flux for large
values. As the tilt angle increases, the separation between the direct and
secondary images becomes more prominent, and the image takes on a more
distinct “hat-like” shape. We assume three hypothetical functional models to
fit the flux results, and it is observed that the exponential function is more
consistent with the results under specific fixed parameters.

The radiation emitted from the accretion disk surrounding a BH is influenced
by both gravitational redshift and Doppler effect. As a result, the radiation
spectrum observed by a remote observer may vary depending on their position
relative to the disk. We derive the redshift images of the accretion disk
under different parameter values. Our findings reveal that as the observation
angle $\theta_{0}$ increases, the range of the redshift expands gradually. The
change in the deformation parameter $a$ has a small impact on the
gravitational redshift $z$, and as a result, the change in the redshift image
in the figure is not significant. As the inclination angle of the observer
($\theta_{0}$) increases, the flux ($F_{\rm obs}$) exhibits increasingly
asymmetric. Additionally, an increase in the deformation parameter ($a$)
results in a larger range of flux decreases.

Finally, by fitting a functional model to the observed flux at various
observation inclinations, we find that the exponential function can
effectively describe the relationship between the observed flux and the
quantum deformation parameter. These findings provide new insights into the
impact of quantum correction on observation outcomes and can serve as a
valuable tool for understanding quantum gravity.

## Acknowledgments

This work is supported by the National Natural Science Foundation of China
(Grant No. 11805166) and the Sichuan Youth Science and Technology Innovation
Research Team (Grant No. 21CXTD0038).

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# Can Edge Computing fulfill the requirements of automated vehicular services
using 5G network ?

Wendlasida Ouedraogo  Telecom SudParis, Institut Polytechnique de Paris;
France<EMAIL_ADDRESS>Andrea Araldo  Telecom
SudParis, Institut Polytechnique de Paris; France;
<EMAIL_ADDRESS>Badii Jouaber  Telecom SudParis,
Institut Polytechnique de Paris; France; {firstname.lastname}@telecom-
sudparis.eu  Hind Castel  Telecom SudParis, Institut Polytechnique de Paris;
France<EMAIL_ADDRESS>Remy Grunblatt  Telecom
SudParis, Institut Polytechnique de Paris; France;
<EMAIL_ADDRESS>
###### Abstract

Communication and computation services supporting Connected and Automated
Vehicles (CAVs) are characterized by stringent requirements, in terms of
response time and reliability. Fulfilling these requirements is crucial for
ensuring road safety and traffic optimization. The conceptually simple
solution of hosting these services in the vehicles increases their cost
(mainly due to the installation and maintenance of computation infrastructure)
and may drain their battery excessively. Such disadvantages can be tackled via
Multi-Access Edge Computing (MEC), consisting in deploying computation
capability in network nodes deployed close to the devices (vehicles in this
case), such as to satisfy the stringent CAV requirements. However, it is not
yet clear under which conditions MEC can support CAV requirements and for
which services. To shed light on this question, we conduct a simulation
campaign using well-known open-source simulation tools, namely OMNeT++,
Simu5G, Veins, INET, and SUMO. We are thus able to provide a reality check on
MEC for CAV, pinpointing what are the computation capacities that must be
installed in the MEC, to support the different services, and the amount of
vehicles that a single MEC node can support. We find that such parameters must
vary a lot, depending on the service considered. This study can serve as a
preliminary basis for network operators to plan future deployment of MEC to
support CAV.

###### Index Terms:

5G Simulation; MEC; Connected and Automated Vehicles

##  I Introduction

In order to safely and efficiently perceive their surroundings, make real-time
decisions, and navigate complex environments, CAVs need some computational
resources, e.g., CPU, GPU, and RAM, to perform some algorithmic tasks.
Computational requirements may fluctuate depending on road traffic conditions.
Installing a large amount of computational resources to be ready to perform
peak computation at any moment is very costly and would make vehicles too
expensive, thus severely restricting the market for CAV. Moreover, a large
amount of computation consumes power, thus reducing vehicle autonomy or
requiring big, heavy, and expensive batteries. Cloud Computing partially
addresses this problem by offering offloading capabilities. However, CAVs have
stringent requirements regarding latency and bandwidth that can hardly be met
by Cloud Computing[1]. Hence, MEC emerges as an alternative capable of meeting
those requirements, thanks to deploying computational resources very close to
vehicles, i.e., in the base stations or in the Road Side Units (RSUs). Also,
the reliability, the high bandwidth, and the extensive coverage offered by 5G
networks, together with the integration of MEC, offer promising conditions for
a large penetration of CAVs. In this context, a question remains open:

_With which computational capacity should MEC nodes be equipped in order to
support the different CAV services?_

To answer this question, we first present a basic queueing theory model to
find the lower bounds on the required computational resources (§IV). We then
perform a simulation campaign (§V-VI). We simulate the realistic mobility of
vehicles using SUMO. Packet exchanges between vehicles and the MEC, as well as
CAV service computation, are simulated using essentially OMNeT++[2],
Simu5G[3], and Veins[4]. Our code, released in open
source111https://github.com/zazim13/Simu5G-MecBasedAV can be used by other
researchers as a base to study the feasibility of MEC-based CAV services.

We find that for some CAV services, such as remote driving and cooperative
sensing, the amount of required MEC resources is high, and only a few vehicles
can be supported at the same time by a MEC node, making MEC a difficult avenue
to follow. For other CAV services such as cooperative maneuver and awareness,
instead, MEC is more promising, being able to support a larger number of
vehicles.

##  II Background and motivation

CAVs are categorized into different levels of automation, as defined by the
Society of Automotive Engineers (SAE)[5]. These levels range from level 0 (no
automation) to level 5 (full automation). For this work, we consider scenarios
with a higher level of automation (from Level 3 to 5), which means that
vehicles can perform the majority of driving tasks, such as platooning,
collision avoidance, or cooperative sensing.

In these scenarios, vehicles communicate using vehicle-to-everything (V2X)
communications, which can be with other vehicles (vehicle-to-vehicle or V2V),
the infrastructure (vehicle-to-infrastructure or V2I), pedestrians (vehicle-
to-pedestrians or V2P), or the network (vehicle-to-network or V2N). Performing
in the edge node computation pertaining to the aforementioned driving tasks
relies on V2N communications, enabling communication with the MEC. In this
case, we talk about vehicle-to-edge and edge-to-vehicle communications.

Moving a large part of driving-related computation from vehicles to edge nodes
provides several advantages. Such advantages have been shown for platooning
[6], but they extend, more generally, to various CAV applications. Notably,
the MEC facilitates interoperability among diverse CAV hardware, enabling a
central controller to simplify collaboration between different vehicle types.
In addition, edge nodes can be made more resilient by appropriate redundancy
practices. If appropriately positioned, edge nodes can be less susceptible to
shadowing when communicating with vehicles. Finally, an edge node can
aggregate information from multiple vehicles, fostering enhanced cooperation
and facilitating global decision-making, as showcased in [7] and [8].

However, while the MEC provides improved latency compared to the Cloud, it
does introduce some latency in contrast to on-board computations. Studies such
as [9] suggest balancing the computational load between the edge node and the
Cloud, based on latency considerations. In our approach, we only focus on
leveraging the MEC, as our goal is to comprehend its limitations in supporting
CAV services.

##  III Vehicular services and architecture

###  III-A Vehicular applications

We consider the following CAV services, related to the high degree of
automation, characterized by stringent requirements, as described in [10].

  * •

_Remote driving_ allows vehicles to be driven by a human operator or an
application outside the vehicle.

  * •

_Cooperative sensing_ involves exchanging sensed data to enhance a vehicle’s
environmental perception.

  * •

_Cooperative maneuver_ consists of exchanging messages to synchronize
vehicles’ maneuvers, such as lane changing or platooning.

  * •

_Cooperative awareness_ : information exchange to inform vehicles about
relevant events, such as Emergency Vehicle Alerts, electronic emergency brake
signals, etc.

For these use cases, the acceptable requirements are provided in [10] and
[11]. Table I summarizes these requirements. The requirements pertain to end-
to-end latency, signifying the delay from the transmission of a message to its
reception at the application level, and reliability, indicating the
probability of successfully transmitting data within the end-to-end delay
threshold, as in [11].

TABLE I: Use cases requirements

Use cases | End-to-End Latency Threshold | Reliability
---|---|---
Remote driving | 20ms | 99%
Cooperative sensing | 10ms | 95%
Cooperative maneuver | 100ms | 99%
Cooperative awareness | 100ms | 95%

###  III-B Communication behavior

Our objective is to assess the feasibility of deploying applications in the
MEC to serve the use cases described in §III-A. Since our paper is not a study
on the internals of the aforementioned applications, we made a deliberate
decision not to implement each specific application in detail, which would be
time-consuming. Instead, we simulate a generic parametric application. By
adjusting the parameters to match the application characteristics (as in Table
III), we can mimic the behavior of the different applications at a high level,
only focusing on network-related characteristics. By mimicking, we mean
generating communications that such an application would perform, consuming
the same bandwidth, with the same data rate, and demanding the same processing
load. We consider two communication behaviors that represent the most distinct
services.

_Dissemination_. For cooperative applications, data such as cooperative
sensing, warning alerts, or traffic flow information needs to be shared with
nearby vehicles or RSUs. Our approach emphasizes vehicle-to-edge and edge-to-
vehicle communication, following the methodology in [6] for platooning
services. In this model, the edge node processes data, runs algorithms, and
disseminates results to relevant vehicles. We introduce the ’dissemination
radius’ parameter, creating a circle around the data-producing vehicle for
information sharing. Each vehicle wishing to cooperate sends its data to the
edge node, which processes it and shares the information in a unicast manner
with all vehicles within the dissemination circle. While broadcast is an
option, it complicates limiting dissemination to the specified circle. This
behavior is depicted in Fig. 1.

_Client-Server_. For remote driving service, the edge node is generally used
to offload certain tasks, such as object recognition and parking assistance
(as in [12]). In this paper, we assume that the edge node is completely in
charge of remotely operating cars. This means that remotely driven vehicles
send data (videos, LiDAR, or sensed data) to the edge node, which handles the
execution of the necessary tasks and sends the right commands, e.g., steering,
braking, and accelerating.

We choose a proactive behaviour for all applications, meaning that data are
periodically sent to the edge node then shared when needed, and deploying the
service is trigering the sending of data, no particular event is waited.

Figure 1: Communication behavior of type “Dissemination”.

##  IV Performance characterization

The total delay is composed of several components, as depicted in Fig. 2. Each
application running in the MEC (MecApp), related to one service and one
vehicle, is allocated a limited amount of processing resources. For the sake
of simplicity, we here only focus on the CPU deployed in an edge node. We
assume that the CPU of the edge node is allocated among several MecApps using
a fair sharing discipline [13]. A MecApp operates as a queueing system for
incoming packets. Queued packets are processed sequentially, with the
processing time determined by the allocated CPU and task complexity. The CPU
allocation directly influences both processing time and end-to-end delay, a
critical factor for meeting reliability requirements in the context of
automated vehicles.

If we aim to achieve a reliability of $R_{req}$, meaning that more than
$R_{req}$ percent of the packets should be successfully transmitted within the
end-to-end delay requirement $D_{req}$, we must ensure that for an end-to-end
delay $D$.

$P(D\leq D_{req})\geq R_{req}$ (1)

From §V-A2 the packet generation follows a Poisson process, and we also assume
that the arrival rate at the MecApp follows a Poisson distribution. In
addition, the processing time is assimilated to an exponential distribution.
Given that, each MecApp can be assimilated to a queueing system, we can
approximate these queues as M/M/1 queues. Consequently, the time taken by each
packet within the MecApp (queueing and processing delay) follows an
exponential distribution, with a parameter $\mu-\lambda$, where $\lambda$
represents the packet arrival rate at the MecApp and $\mu$, the processing
rate (in packets/s).

To ensure that the requirement can be met, it’s crucial that the time taken by
each packet within the MecApp $D_{mec}$ respects the condition (1) i.e. a
necessary condition to meeting the reliability requirement is :

$P(D_{mec}\leq D_{req})\geq R_{req}$ (2)

then we obtain:

$\mu\geq\lambda-\frac{ln(1-R_{req})}{D_{req}}$ (3)

where $R_{req}$ is the overall reliability requirement and $D_{req}$ is the
overall delay requirement.

Given that:

$\mu=\frac{CPU}{E(IPR)}$ (4)

Where $CPU$ is the amount of CPU (in instructions/s) allocated to the MecApp
and $IPR$ is defined in §V-A2. The minimum CPU that should be allocated to
each MecApp in order to respect the necessary condition (2) is:

$CPU_{min}=\left(\lambda-\frac{\ln(1-R_{req})}{D_{req}}\right)\cdot E(IPR)$
(5)

We compute these minimum CPU allocations regarding the requirements presented
in Table I. They are summarized in Table III. Observe a notable discrepancy
among the CPU required for different services.

Figure 2: Delay components

##  V Simulation environment

We use several open-source software and frameworks to perform our simulations.
Fig. 3 illustrates the diagram of interconnectivity within these frameworks.
Our work is mainly based on Simu5G [3], a popular open-source 5G simulation
library, because it provides 3GPP-compliant 5G New Radio Access and integrates
MEC features. Simu5G is based on the OMNeT++ framework [2] and integrates INET
[14], which is an open-source framework providing tools for communication
network simulation. In order to enable a vehicular network with realistic
mobility, we use SUMO[15]. SUMO allows us to generate road networks and
traffic demand, specifying routes and vehicles with realistic behavior. SUMO
provides an API [16] for interacting with the vehicles it generates, and Veins
(especially its subproject Veins_INET), another open-source vehicular network
simulation framework allows us to access this API within the OMNeT++
simulation environment and give to vehicles in Simu5G the realistic mobility
provided by SUMO. With these tools, we can simulate scenarios aligning with
our vehicle-to-edge and edge-to-vehicle communication approach.

Figure 3: Simulation frameworks diagram

###  V-A Simulation parameters

####  V-A1 Radio conditions

Simu5G offers a number of parameters to tune the 5G New Radio network. It also
provides a realistic channel model that supports fading, path loss, shadowing,
and attenuations. We choose to use macro base stations in an urban scenario.
Other parameters are summarized in Table II and are essentially taken from
[17].

TABLE II: Simulation radio parameters

Parameter name | Value
---|---
Number of gNBs | 1
Carrier frequency | 6GHz
Bandwidth | 80MHz(100 PRBs)
Numerology | 2
Fading (Jakes) + shadowing | enabled
gNB Tx power | 46 dBm
gNB antenna gain | 8dBi
gNB noise figure | 5dB
UE antenna gain | 0dBi
UE noise figure | 7dB
Path loss model | (3GPP - TR 36.873)
Blershift | 5

####  V-A2 Use cases specifications

Our generic application aims to mimic real-world application behaviors. To
achieve this objective, we studied these behaviors in the literature and
collected specifications such as distributions, values, and other metrics for
parameters like uplink or downlink bandwidth usage, data rates, and more.
These specifications are summarized in Table III.

It is generally assumed in the literature, as in [18], that the packet
generation process follows a Poisson distribution for IoT-based applications.
Then, assuming a constant data accumulation by IoT devices, the payload size
of each sent packet can be approximated as an exponential random distribution.

The processing time is directly related to the size of the packet to be
processed, which follows an exponential distribution, we then also assume that
the processing time itself is exponentially distributed. In Simu5G’s MEC
implementation, it is through instructions required by the packet being
processed that the processing time is tuned, as they are proportional. For
each use case, we consider that each packet requires certain tasks to be
performed. We introduce the parameter $IPR$ for Instructions Per Request i.e.
the number of instructions required per each request. We detailed in Table III
values of $IPR$ for each use case. We extract them from [12], where we can
find task instructions requirements for tasks related to automated vehicles.
Each use case is attributed with sufficient $IPR$ to handle essential tasks
such as steering control, alongside characteristic tasks corresponding to each
use case (e.g., object recognition for remote driving or parking assistance
for cooperative maneuvers). We use MIPS (Millions of Instructions Per Second)
and MI(Million Instructions) as a measure of computational capability and
number of instructions to align with Simu5G.

Once processed, an answer with the processed information is transmitted back
to the sending vehicle, and in the case of cooperative applications, to
neighboring vehicles as well. These transmitted packets include processed
information, such as steering control, alerts, acknowledgments, and more. In
[19], a downlink bandwidth of 0.25 MBit/s is assumed for self-driven cars,
with downlink packets containing steering control information. With a sending
rate of 100 messages per second in our remote driving scenarios, this implies
an answering rate of 100 messages per second. Consequently, we assume a
payload size of 313 bytes for each packet, considering it to be adequate to
accommodate all potential types of responses across various use cases.

Each use case is defined by a communication range, representing the area over
which V2X messages can be transmitted. We adopt a circle of dissemination
approach, where the radius of this circle determines the communication
perimeter. We assume that this radius follows a uniform distribution within
the specified range detailed in [10], as indicated in Table III.

TABLE III: Use cases specifications

Use cases | Uplink bandwidth (Mb/s) | Uplink rate (msg/s) | Uplink payload (Bytes) | Downlink payload (Bytes) |  IPR [12] | Dissemination radius | Min CPU(MIPS)
---|---|---|---|---|---|---|---
Remote driving |  32 [20] | Pois(100) | Exp(40000) |  313 [19] | Exp(500) | - | 165130
Cooperative sensing |  10 [10] |  Pois(100) [21] | Exp(12500) |  313 [19] | Exp(200) |  Uniform(0-200m) [10] | 79915
Cooperative maneuver |  1.3 [10] |  Pois(10) [6] | Exp(16250) |  313 [19] | Exp(500) |  Uniform(0-500m) [10] | 28026
Cooperative awareness | 0.12 |  Pois(10) [21] |  Exp(1500) [10] |  313 [19] | Exp(200) |  Uniform(0-500m) [10] | 7992

####  V-A3 Simulation scenario

In our simulations, we essentially tune two parameters: the CPU capacity
available at the edge, shared among the MecApps of the cars, and the number of
injected vehicles. The processor used and their computational capacities are
displayed in Tables IV. Each experiment corresponds to a simulation involving
a specific edge node processor and a certain number of injected vehicles.
Every experiment is repeated five times with different seeds (i.e., five
repetitions in OMNeT++), and each simulation lasts 180 seconds. The road map
corresponds to a district of Paris of 1 square kilometer.

TABLE IV: Processor Processing Speed Id | Processor | Processing Speed(MIPS)[22]
---|---|---
1 | AMD Ryzen Threadripper | 2356230
2 | AMD Ryzen 9 | 749070
3 | Intel Core i9-9900K | 412090
4 | Intel Core i5-11600K | 346350

##  VI Results

VehiclesMIPS10010040804040801001004080000100100000001001000000012345672356230749070412090346350
(a) Remote driving

VehiclesMIPS1001008020000100100200000100100000001001000000011020304050602356230749070412090346350
(b) Cooperative maneuver

VehiclesMIPS100100100100100100010010040000010010000000100100000001204060801001202356230749070412090346350
(c) Cooperative awareness

VehiclesMIPS10010040808000100100604020006060200000100800000012345672356230749070412090346350
(d) Cooperative sensing

Figure 4: Success rate of each Edge CPU

Figure 4 illustrates the success rates across various experiments, where the
success rate is defined as the percentage of repetitions meeting the
reliability requirement out of all experiment repetitions. The red limitation
denotes a threshold where CPU allocation meets the minimum requirements for
each MECApp as outlined in Table III. Simulations below this threshold fail to
meet the reliability criteria. However, simulations above this threshold show
improved performance but still encounter challenges despite sufficient CPU
allocation for each MECApp. This issue stems not only from delays inherent to
the MEC node performance, addressed by the condition in (2) which specifies
the minimum CPU allocation required for each MEC application, but also from
additional uplink and downlink delays. An increase in the number of vehicles
leads to reduced bandwidth per vehicle, consequently increasing the end-to-end
delay and highlighting the need for scaling edge nodes in proportion to
network resources.

With their speed, processors id3 and id4 can effectively handle scenarios such
as remote driving or cooperative sensing for up to 2 vehicles, awareness tasks
for 20 vehicles, and maneuvering tasks for 10 vehicles. However, even doubling
the processing speed with processor id2 does not significantly increase the
number of vehicles that can be managed, with only a modest improvement
observed in remote driving and cooperative sensing services. This underscores
the challenge of scaling up the number of vehicles through improving the MEC
node performance.

Moreover, our observations indicate that services with more stringent
requirements support fewer vehicles given the same edge node capacity.
Specifically, services like remote driving and cooperative sensing, which
demand lower end-to-end delays, can support fewer vehicles than cooperative
maneuver and awareness service. The latter services are therefore more
promising.

##  VII Conclusion

We conducted a simulation campaign using diverse tools to assess the limits of
integrating 5G networks and edge computing via the MEC paradigm to handle
various CAVs services requirements. Our focus includes evaluating the
feasibility of edge-based control, where MEC manages all required
computations. We show a basic queuing model to provision the minimum
computational capability to satisfy different MEC application requirements.
Our results highlight a discrepancy in required resources for specific CAV
services. This indicates that even if controlling CAVs from the edge is
feasible, the efficiency greatly varies depending on the service.
Additionally, we reveal challenges in improving such efficiency by scaling up
the edge, as the communication delay can hinder the correct operation of
vehicular services from the edge. A limitation of our approach is that we did
not conduct an in-depth study of the communication delay nor did we consider
cloud or vehicle computational resources. We plan to incorporate these factors
in our future works to provide a more comprehensive analysis.

##  VIII Acknowledgments

This work was carried out in the context of Beyond5G, a project funded by the
French government as part of the economic recovery plan, namely ”France
Relance”, and the investments for the future program.

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