# Freezing and Thawing of Entanglement in Lossless Multiparty Systems

X.-F. Qian 1<EMAIL_ADDRESS>C. Qu 1 J.H. Eberly 2 1Center for Quantum
Science and Engineering, and Department of Physics, Stevens Institute of
Technology, Hoboken, New Jersey 07030, USA
2Center for Coherence and Quantum Optics, and Department of Physics &
Astronomy, University of Rochester, Rochester, NY 14627, USA

###### Abstract

Entanglement freezing has been demonstrated existing in various noisy
decoherence mechanisms. Here we explore its universality by investigating
freezing behavior in a lossless multiparty system, i.e., an $N$-site optical
lattice (or equivalently spin-chain) system. Interesting phenomenon of
permanent entanglement freezing is found for the infinite-size case
($N\rightarrow\infty$). As the multiparty system size reduces to finite $N$,
the interesting behavior of dynamical entanglement “thawing” start to emerge.
Especifically, alternative appearances of “freezing” and “thawing” can be
identified as time evolves. Our results may provide useful guidance to
entanglement control in quantum tasks.

Introduction: Correlations can be said to be the backbone of mathematical
physics. Quantum entanglement is a special form of multi-party correlation,
known for decades as a powerful non-classical resource, essential for quantum
events such as teleportation and for inclusion in processes designed to assist
quantum communication and quantum computing NC2000 ; Horodecki2009RMP .
Coherence itself, as a correlation feature definable in any linear vector
space, has also been identified as a resource (see Streltsov2017RMP ).

A newly discovered measure of correlation OlZur ; HendVedr  named quantum
discord is now regarded as a generalization of entanglement in the following
surprising way. Some mixed states that have nonzero discord, previously
considered as classical states because entanglement is absent, have
nevertheless been reported to be available for non-classical computational
speed-up Datta .

Equally unusually, for certain quantum systems, even if entanglement has
already decayed to zero, their quantum discord can persist for a long time
before decaying suddenly Mazzo . This temporary steady persistence is labeled
as the “freezing” of quantum discord. It has been studied together with
“thawing”, the equally sudden onset of discord’s decay (see Maziero2009PRA ;
Pinto ; Cianciar ). Experimental observations confirm the existence of these
sudden transitions between freezing and thawing Silva ; Paula . Further study
has shown that entanglement and coherence can also be subject to freezing
AliRau ; Bromley2015PRL ; WuXu ; Ding .

All of these studies have been concerned with freezing in the active presence
of one or another decoherence mechanism. However, in most quantum tasks
information is often processed in finite-size multiparty network systems
NC2000 ; Kimble2008Nature . A natural question arises: is freezing a universal
phenomenon for all size systems? In this report we explore entanglement
freezing in a lossless $N$-site optical lattice system. Besides freezing, the
interesting complementary phenomenon of entanglement “thawing” is also
identified. In addition, the size of the system (determined by $N$) is playing
a crucial role in controlling proportionally the length of the freezing time
interval. Our result refutes the widespread impression that entanglement
freezing and thawing occur as a consequence of an available decoherence, and
illustrates their universality in lossless multiparty systems.

For the convenience of discussion, we define a “freezing” process as a
“stable” behavior within a time interval where the physical quantity such as
entanglement $E(t)$ is approximately independent of time with $\partial
E(t)/\partial t\approx 0$ or the fluctuation is relatively negligible
comparing to the (relative) value of the quantity, i.e., $\sigma(E)/\Delta
E\ll 1$.

Multiparty optical lattice context: Now we describe a multiparty context where
the track of entanglement information flow is under consideration. We study a
scenario where a system of interest (a qubit $A$) is interacting with a
multiparty network (an optical lattice chain $B$). The most general initial
state (at the time of interest $t=0$) of a qubit is an arbitrary mixed state,
i.e.,

$\rho_{A}=\left(\begin{array}[]{cc}\rho_{ee}&\rho_{eg}\\\
\rho_{ge}&\rho_{gg}\end{array}\right).$ (1)

Here the qubit is described in its excited and ground state basis,
$|e\rangle$, $|g\rangle$. We ascribe without loss of generality that this
mixedness is caused by interaction of previously interacting partners during a
preparation process that may include the physical qubit host, experimental
device, previous environment, etc., which we lump together and label as the
marginal party $M$. Then this unspecified marginal $M$ naturally purifies the
system state, i.e.,
$|\psi_{AM}\rangle=\cos\theta|e\rangle\otimes|m_{1}\rangle+\sin\theta|g\rangle\otimes|m_{2}\rangle$,
where $\langle m_{1}|m_{1}\rangle=\langle m_{2}|m_{2}\rangle=1$ describe
normalized marginal states. Here $\cos\theta=\sqrt{\rho_{ee}}$,
$\sin\theta=\sqrt{\rho_{gg}}$ and $\langle
m_{1}|m_{2}\rangle=\alpha=\rho_{eg}/\sqrt{\rho_{ee}\rho_{gg}}$ charactering a
generic overlap determined by the arbitrary density matrix (1).

After the qubit system is prepared, it is then sent to participate in certain
tasks run by a quantum network partner $B$, which can be reasonably treated as
separable (due to no previous interaction) with the system. Thus the overall
initial state can be described as

$|\psi_{ABM}(0)\rangle=(\cos\theta|e\rangle\otimes|m_{1}\rangle+\sin\theta|g\rangle\otimes|m_{2}\rangle)|\phi_{0}\rangle,$
(2)

where $|\phi_{0}\rangle$ is the initial quantum net work state.

The total Hamiltonian is described as

$H=H_{A}+H_{B}+H_{I}+H_{M},$ (3)

where $H_{A}$, $H_{B}$ and $H_{M}$ are the Hamiltonians of qubit system $A$,
quantum network $B$ and marginal $M$ respectively. $H_{I}$ denotes the
currently active interaction after $t=0$, between qubit $A$ and its quantum
network partner $B$. Here we consider the multiparty network system $B$ as $N$
ultracold atoms confined in an optical lattice formed by several standing-wave
laser beams. In the nearest neighbor consideration, such a general Hamiltonian
(see Ref. Duan2003PRL ) can be written as

$\displaystyle H_{B}$ $\displaystyle=$ $\displaystyle-\sum_{\langle ij\rangle
s}\left(t_{\mu s}a_{is}^{\dagger}a_{js}+{\rm H.c.}\right)$ (4)
$\displaystyle+$
$\displaystyle\frac{1}{2}\sum_{i,s}U_{s}n_{is}\left(n_{is}-1\right)+U_{\uparrow\downarrow}\sum_{i}n_{i\uparrow}n_{i\downarrow},$

where the summation index $\langle ij\rangle$ indicates nearest neighbor
interactions, $a_{is}$ are annihilation operators,
$n_{is}=a_{is}^{\dagger}a_{is}$ are number operators, and the spin parameter
$s=\uparrow,\downarrow$ denotes two relevant internal states of each atom.
Here $t_{\mu s}$ and $U_{s}$ ($U_{\uparrow\downarrow}$) are tunneling and on-
site interaction energies depending on laser-created potentials, spin
orientations, and lattice cubic geometry directions $\mu=x,y,z$. In the
controlled parameter regime $t_{\mu s}\ll U_{s},U_{\uparrow\downarrow}$ and
$\langle n_{i\uparrow}\rangle+\langle n_{i\downarrow}\rangle\approx 1$
(representing the single-occupation insulating phase), the above Hamiltonian
(4) is equivalent to an antiferromagnetic or ferromagnetic Heisenberg spin XXZ
model Duan2003PRL .

As an illustration, we consider a chain type of network, i.e, all atoms are
aligned in a chain with each atom interacting only with its two nearest
neighbors. Then the ultracold atom at one end of the chain is treated as qubit
$A$ and the remainder of the lattice-trapped atoms as the interacting network
$B$. Such a scenario can be understood as a prepared atom $A$ in site 1 that
is placed to initiate a quantum task with the assistance of the interacting
lattice chain $B$. We take the case $U_{s}=2U_{\uparrow\downarrow}$ and keep
the leading order in $t_{\mu s}/U_{\uparrow\downarrow}$ through the
Schrieffer-Wolff transformation Schrieffer-Wolff  (see also in Duan2003PRL ;
Hewson1997 ). Then the Hamiltonian $H_{AB}=H_{A}+H_{B}+H_{I}$ can be described
as

$\displaystyle H_{AB}$ $\displaystyle=$
$\displaystyle\eta_{\mu\perp}(\sigma_{A}^{+}\sigma_{1}^{-}+\sigma_{A}^{-}\sigma_{1}^{+})$
(5) $\displaystyle+$
$\displaystyle\sum_{i=1}^{N}\eta_{\mu\perp}(\sigma_{i}^{+}\sigma_{i+1}^{-}+\sigma_{i}^{-}\sigma_{i+1}^{+}).$

Here $\sigma^{\pm}=\sigma^{x}\pm i\sigma^{y}$ are usual raising and lowering
Pauli matrices with
$\sigma^{x}=a_{\uparrow}^{\dagger}a_{\downarrow}+a_{\downarrow}^{\dagger}a_{\uparrow}$,
$\sigma^{y}=-i(a_{\uparrow}^{\dagger}a_{\downarrow}-a_{\downarrow}^{\dagger}a_{\uparrow})$,
and
$\eta_{\mu\perp}=t_{\mu\uparrow}t_{\mu\downarrow}/2U_{\uparrow\downarrow}$.
The structure of such an ultracold atom optical lattice model (5) is
equivalent to that of an XY spin model.

We now assume that all the $N$ lattice sites in interaction partner $B$ are
initially in their lower energy state, i.e., in the eigenstates (which we
denote as $|0_{i}\rangle$) of the number operators $n_{i\downarrow}$. Then the
overall system $B$ initial state is given as
$|\phi_{0}\rangle=|0_{1}\rangle...|0_{i}\rangle...|0_{N}\rangle\equiv|{\bf
0}\rangle$.

The eigenstates and eigenenergies of the above Hamiltonian (5) can be obtained
as Qian2005PRA

$\displaystyle|k\rangle$ $\displaystyle=$
$\displaystyle\sqrt{\frac{2}{N+2}}\sum_{j=1}^{N}\sin\left(\frac{jk\pi}{N+2}\right)|{\bf
1}_{j}\rangle,$ (6) $\displaystyle E_{k}$ $\displaystyle=$ $\displaystyle
2\eta_{\mu\perp}\cos\left(\frac{k\pi}{N+2}\right),$ (7)

where $k\in\left[1,N\right]$ and {$|{\bf 1}_{j}\rangle$} are the states of the
multi-site system $B$ indicating the $j$-th atom is in the spin up state while
all the remaining ones are spin down.

Then one can straightforwardly obtain the time dependent state through the
unitary operator $U(t)=\sum_{k=1}^{N}e^{-iE_{k}t}|k\rangle\langle k|$ from the
initial state (2), i.e.,

$\displaystyle|\psi_{ABM}(t)\rangle$ $\displaystyle=$
$\displaystyle\cos\theta|m_{1}(t)\rangle\Big{(}c_{e}(t)|{\bf
0}\rangle|e\rangle+\sum_{j=1}^{N}c_{j}(t)|{\bf 1}_{i}\rangle|g\rangle\Big{)}$
(8) $\displaystyle+$ $\displaystyle\sin\theta|m_{2}(t)\rangle|{\bf
0}\rangle|g\rangle,$

where we have defined the time dependent coefficients

$\displaystyle c_{e}(t)$ $\displaystyle=$
$\displaystyle\sum_{k=1}^{N+1}\frac{2e^{-iE_{k}t}}{N+2}\sin\left(\frac{k\pi}{N+2}\right)\sin\left(\frac{k\pi}{N+2}\right),$
$\displaystyle c_{j}(t)$ $\displaystyle=$
$\displaystyle\sum_{k=1}^{N+1}\frac{2e^{-iE_{k}t}}{N+2}\sin\left(\frac{k\pi}{N+2}\right)\sin\left(\frac{j+1}{N+2}k\pi\right)$
(9)

with $c_{e}(0)=1$, $c_{j}(0)=0$ and $j=1,2,...,N$.

Entanglement freezing and thawing: Since atom $A$ is the key initiation qubit
and the multi-site optical lattice chain $B$ is an integrated assisting
system, it is natural to consider entanglements focused on these two parties
respectively. Based on the time-dependent multiparty state (8), we analyze two
dynamical entanglements, i.e., $K_{A}(t)$ between qubit $A$ and the $BM$
remainder, and $K_{B}(t)$ between the optical lattice chain $B$ and the $AM$
remainder. The multi-dimensional pure state nature of state (8) suggests the
optimal measure to adopt is Schmidt weight Grobe ; Eberly2006 , which is
defined as

$K=\frac{1}{\sum_{j}\lambda^{2}_{j}},$ (10)

where $\lambda_{j}$ are the eigenvalues of the reduced density matrix
$\rho_{Q}$ with $Q=A,B$ by tracing out the remaining parties.

The reduced density matrix of the system $A$ can be obtained as $\rho_{A}={\rm
Tr}_{BM}|\psi_{ABM}(t)\rangle\langle\psi_{ABM}(t)|$, i.e.,

$\rho_{A}=\left(\begin{array}[]{cc}\cos^{2}\theta|c_{e}(t)|^{2}&\sin\theta\cos\theta
c_{e}(t)\alpha^{*}\\\ \sin\theta\cos\theta
c^{*}_{e}(t)\alpha&\sin^{2}\theta+\cos^{2}\theta\sum_{i=1}^{N}|c_{i}(t)|^{2}\end{array}\right).$
(11)

Then one can achieve the dynamical entanglement as

$\displaystyle K_{A}(t)=\frac{2}{1+[1-f(t)]^{2}+2\sin^{2}\theta
f(t)|\alpha|^{2}},$ (12)

where $f(t)=2\cos^{2}\theta|c_{e}(t)|^{2}$.

Similarly, one can achieve the reduced density matrix for party $B$ and
consequently compute the Schmidt weight $K_{B}(t)$ as

$\displaystyle K_{B}(t)=\frac{2}{1+(1-g(t))^{2}+2\sin^{2}\theta
g(t)|\alpha|^{2}},$ (13)

where $g(t)=2\cos^{2}\theta\sum_{j=1}^{N}|c_{j}(t)|^{2}$.

Figure 1: Entanglement freezing and thawing behaviors of Schmidt weights
$K_{A}(t)$ and $K_{B}(t)$. Panels (a), (b), and (c) illustrate entanglement
dynamics for $N\rightarrow\infty$, $N=25$, and $N=40$ respectively. The dashed
black and solid blue lines denote correspondingly $K_{A}(t)$ and $K_{B}(t)$,
and the $x$ axis is in units of $\eta_{\mu\perp}t$. Permanent freezing is
phenomena is identified for both entanglements when $N\rightarrow\infty$.
Entanglement thawing/defreezing occurs in the finite $N$ cases at
approximately periodically at times $\eta_{\mu\perp}t=k(N+2)$ with arbitrary
integer $k$. In all cases, the first freezing period starts at approximately
$\eta_{\mu\perp}t=2$.

In both cases, there are two effective number of eigenvalues of the reduced
density matrices. Thus both $K_{A}(t)$ and $K_{B}(t)$ take values between 1
and 2, where 1 means no entanglement and 2 stands for maximum entanglement
Qian2018NJP .

To analyze entanglement freezing behavior of our lossless multiparty scenario
in comparison with the noisy decoherence mechanism, we first consider the
closest case of the two, i.e., when the lattice chain system contains infinite
number of atoms with $N\rightarrow\infty$. Interestingly, one can show in this
limiting case, the time-dependent amplitude reduces to a compact form

$c_{e}(t)=\frac{J_{1}(2\eta_{\mu\perp}t)}{\eta_{\mu\perp}t},$ (14)

where $J_{1}(.)$ is the standard Bessel functions of the first kind. The
detailed proof of the this relation is given in Appendix A, see also a similar
analysis in PrattEberly .

Then the two dynamical entanglements $K_{A}(t)$, $K_{B}(t)$, in this limiting
case $N\rightarrow\infty$, are simple functions of the Bessel function
$J_{1}(2\eta_{\mu\perp}t)$. Their behaviors with respect to the time unit
$\eta_{\mu\perp}t$ are illustrated in Fig. 1 (a). Permanent freezing behavior
can be identified for both entanglements, which is similar to the results
obtained with noisy decoherence mechanisms in previous studies AliRau ;
Bromley2015PRL ; WuXu . To prove analytically the freezing behavior, one will
need to show the vanishing of the derivatives, i.e., $\frac{\partial
K_{A}(t)}{\partial t}\approx 0$ and $\frac{\partial K_{B}(t)}{\partial
t}\approx 0$ which is done in detail in Appendix. B.

We now further analyze the effect of the system size (i.e., the finite number
of ultracold atoms $N$ in the multi-site optical lattice chain $B$) on
entanglement freezing. The specific behaviors of both $K_{A}(t)$ and
$K_{B}(t)$ are illustrated in detail in Fig. 1 for two cases $N=25,40$. The
permanent freezing behavior in (a) disappears and is replaced with alternative
occurrences of both freezing and thawing. The first freezing and first thawing
occur approximately at

$t_{freeze}=\frac{2}{\eta_{\mu\perp}}\quad{\rm and}\quad
t_{thaw}=\frac{N+4}{\eta_{\mu\perp}}$ (15)

respectively. The analytical derivation of the freezing and thawing timings
are given in detail in Appendix C.

Surprisingly, from comparing the three plots in Fig. 1, one notes that the
dynamical entanglements $K_{A}(t)$, $K_{B}(t)$ for finite $N$ are almost
identical to those of the infinite $N$ case before the first thawing happens
at $\eta_{\mu\perp}t=N+4$. This can be confirmed by comparing the significant
terms in the Taylor expansion of the time-dependent amplitude $c_{e}(t)$ with
respect to time $t$ for both finite and infinite $N$ (see detailed analysis in
Appendix D). Such a behavior demonstrates directly that entanglement freezing
is preserved for multiparty lossless systems, and it is not a unique
phenomenon of noisy decoherence mechanisms. The first occurrence of freezing
is independent of the size of the multiparty system but the length of the
freezing time interval is however strongly controlled by the size $N$ as shown
in (15).

Furthermore, the alternative occurrence of freezing and thawing has an
approximate period,

$T=\frac{N+2}{\eta_{\mu\perp}},$ (16)

which is proportional to the size $N$ of the system. A detailed derivation of
this period is also given in Appendix C. From this result, one notes
entanglement thawing happens when $\eta_{\mu\perp}t=k(N+2)+2$ with arbitrary
integer $k$. These properties are demonstrated in the dynamical plots in Fig.
1.

Summary and discussion: In conclusion, we have identified the persistence of
entanglement freezing dynamics in a lossless multiparty system hat consists of
$N$-site ultracold atoms in an optical lattice chain. This indicates the
universality of this entanglement dynamical phenomenon in both lossy and
lossless contexts. When the size of the multiparty system is infinitely large,
permanent freezing behavior is retrieved just as in lossy decoherence
scenarios. Although the freezing phenomenon exists for finite size (finite
$N$) lossless contexts, its behavior has to be balanced by the alternative
occurrence of entanglement thawing or defreezing.

Entanglement freezing in lossless multiparty system can be very useful for
quantum information tasks, most of which require a relatively stable behavior
of entanglement. This will largely release the strict restrictions, thus
providing huge margins, on the timing of information processing and
computation. The obtained specific size-dependent freezing and thawing time
parameters, $t_{freeze},t_{thaw},T$ may provide useful guidance to quantum
control.

We expect entanglement freezing and thawing dynamics to be found in various
other physical contexts.

Acknowledgement: We acknowledge discussions with Y. Ding and S.B Xie, as well
as financial support from NSF grants PHY-1505189, INSPIRE PHY-1539859, and
Stevens Institute of Technology.

## References

  * (1) M.A. Nielsen and I.L. Chuang, Quantum Computation and Quantum Information (Camb. Univ. Press, 2000).
  * (2) Ryszard Horodecki, Paweł Horodecki, Michał Horodecki, and Karol Horodecki, “Quantum entanglement”, Rev. Mod. Phys. 81, 865 (2009).
  * (3) Alexander Streltsov, Gerardo Adesso, and Martin B. Plenio, “Colloquium: Quantum coherence as a resource”, Rev. Mod. Phys. 89, 041003 (2017).
  * (4) H. Ollivier and W. H. Zurek, ”Quantum Discord: A Measure of the Quantumness of Correlations”, Phys. Rev. Lett. 88, 017901 (2001).
  * (5) L. Henderson and V. Vedral, “Classical, quantum and total correlations”, J. Phys. A 34, 6899 (2001).
  * (6) A. Datta, A. Shaji, and C. M. Caves, “Quantum Discord and the Power of One Qubit”, Phys. Rev. Lett. 100, 050502 (2008).
  * (7) L. Mazzola, J. Piilo, and S. Maniscalco, “Sudden Transition between Classical and Quantum Decoherence”, Phys. Rev. Lett. 104, 200401 (2010).
  * (8) J. Maziero, L. C. Ćeleri, R. M. Serra, and V. Vedral, “Classical and Quantum Correlations under Decoherence”, Phys. Rev. A 80, 044102 (2009).
  * (9) J. P. G. Pinto, G. Karpat, and F. F. Fanchini, “Sudden change of quantum discord for a system of two qubits”, Phys. Rev. A 88, 034304 (2013).
  * (10) M. Cianciaruso, T. R. Bromley, W. Roga, R. Lo Franco, and G. Adesso, “Universal freezing of quantum correlations within the geometric approach”, Scientific Reports 5, 10177 (2015).
  * (11) I. A. Silva, D. Girolami, R. Auccaise, R. S. Sarthour, I. S. Oliveira, T. J. Bonagamba, E. R. deAzevedo, D. O. SoaresPinto, and G. Adesso, Phys. Rev. Lett. 110, 140501 (2013).
  * (12) F. M. Paula, et al., Phys. Rev. Lett. 111, 250401 (2013).
  * (13) M. Ali and A. R. P. Rau, Phys. Rev. A 90, 042330 (2014).
  * (14) T.R. Bromley, M. Cianciaruso, and G. Adesso, Phys. Rev. Lett. 114 210401 (2015).
  * (15) W. Wu and J.-B. Xu, Journal of Physics B: Atomic, Molecular and Optical Physics 49, 115502 (2016).
  * (16) Y. Ding, S-B Xie and J.H. Eberly, Phys. Rev. A 103, 032418 (2021).
  * (17) J.H. Kimble, “The quantum internet”, Nature 453, 1023-1030 (2008).
  * (18) L.-M. Duan, E. Demler, and M. D. Lukin, “Controlling Spin Exchange Interactions of Ultracold Atoms in Optical Lattices”, Phys. Rev. Lett. 91, 090402 (2003).
  * (19) J. R. Schrieffer and P. A. Wolff, “Relation between the Anderson and Kondo Hamiltonians”, Phys. Rev. 149, 491 (1966).
  * (20) A.C. Hewson, “The Kondo Problem to Heavy Fermions”, (Cambridge University Press, Cambridge, England, 1997).
  * (21) See for an example in X.-F. Qian, Y. Li, Y. Li, Z. Song, and C. P. Sun, Phys. Rev. A 72, 062329 (2005).
  * (22) R. Grobe, K. Rza̧zewski and J. H. Eberly, J. Phys. B 27, L503 (1994).
  * (23) J. H. Eberly, Laser Phys. 16, 921 (2006).
  * (24) A similar analysis can be found in X.-F. Qian, M.A. Alonso, J.H. Eberly, New J. Phys. 20 063012 (2018).
  * (25) J.S. Pratt and J.H. Eberly, Phys. Rev. B 64, 195314 (2001).

##  I Appendix

###  I.1 Bessel function analysis

Here we now analyze the behavior of the time dependent coefficient $c_{e}(t)$
given in equation (23) in the limiting case when $N\rightarrow\infty$. At the
large $N$ case, one can transfer the summation into an integral by defining

$\frac{1}{N+2}=dy\text{, \ }\frac{k}{N+2}=y,$ (17)

which leads to the integration expression

$\displaystyle c_{e}\left(t\right)$ $\displaystyle=$ $\displaystyle
2\int_{\frac{1}{N+2}}^{\frac{N+1}{N+2}}dy\exp\left[-i2\eta_{\mu\perp}t\cos\left(y\pi\right)\right]\sin^{2}\left(y\pi\right)$
(18) $\displaystyle=$
$\displaystyle-\frac{2}{\pi}\int_{0}^{1}\exp\left[-i2\eta_{\mu\perp}t\cos\left(y\pi\right)\right]\sin\left(y\pi\right)d\cos\left(y\pi\right)$
$\displaystyle=$
$\displaystyle-\frac{2}{\pi}\int_{1}^{-1}\exp\left[-i2\eta_{\mu\perp}tx\right]\sqrt{1-x^{2}}dx$
$\displaystyle=$
$\displaystyle\frac{1}{\eta_{\mu\perp}t}J_{1}(2\eta_{\mu\perp}t)$

where $x=\cos\left(y\pi\right)$.

###  I.2 Entanglement freezing for infinite N

In the following we compute the derivative of entanglement $K_{B}(t)$ respect
to time for the multi-site lattice chain system $B$. From the time dependent
entanglement (13), one has

$\displaystyle\frac{\partial K_{B}(t)}{\partial t}$ $\displaystyle=$
$\displaystyle\frac{-2}{[1+(1-g(t))^{2}+2\sin^{2}\theta
g(t)|\alpha|^{2}]^{2}}$ (19)
$\displaystyle\frac{\partial[1+(1-g(t))^{2}+2\sin^{2}\theta
g(t)|\alpha|^{2}]}{\partial t}$ $\displaystyle=$
$\displaystyle\frac{-2[-(1-g(t))g^{\prime}(t)+2\sin^{2}\theta
g^{\prime}(t)|\alpha|^{2}]}{[1+(1-g(t))^{2}+2\sin^{2}\theta
g(t)|\alpha|^{2}]^{2}}$ $\displaystyle=$
$\displaystyle\frac{2[(1-g(t))-2\sin^{2}\theta|\alpha|^{2}][-f^{\prime}(t)]}{[1+(1-g(t))^{2}+2\sin^{2}\theta
g(t)|\alpha|^{2}]^{2}}$ $\displaystyle=$
$\displaystyle\frac{8[2\sin^{2}\theta|\alpha|^{2}-1+g(t)][\cos^{2}\theta
c^{\prime}_{e}(t)]}{[1+(1-g(t))^{2}+2\sin^{2}\theta g(t)|\alpha|^{2}]^{2}},$

where we have used the fact $g^{\prime}(t)=-f^{\prime}(t)$ because
$g(t)=2\cos^{2}\theta|c_{g}(t)|^{2}=2\cos^{2}\theta(1-|c_{e}(t)|^{2})=2\cos^{2}\theta-f(t)$.

From the properties of Bessel function derivatives, i.e.,

$\frac{\partial J_{\nu}(x)}{\partial x}=\frac{\nu}{x}J_{\nu}(x)-J_{\nu+1}(x),$
(20)

where $\nu$ is the order of the Bessel function, one can straightforwardly
obtain the following derivative

$\frac{\partial c_{e}(t)}{\partial t}=\frac{\partial}{\partial
t}\frac{J_{1}(2\eta_{\mu\perp}t)}{\eta_{\mu\perp}t}=-\frac{J_{2}(\eta_{\mu\perp}t)}{\eta_{\mu\perp}t}\approx
0.$ (21)

When combined with the result obtained in (19), it is followed with

$\frac{\partial K_{B}(t)}{\partial t}\approx 0,$ (22)

for any large finite time. This directly proves the freezing behavior of
$K_{B}(t)$. The analysis for the qubit entanglement $K_{A}(t)$ is similar.

###  I.3 Freezing and thawing timings for finite $N$

We first analyze the period of entanglement freezing and thawing alternation.
Again, we focus on the time dependent amplitude $c_{e}(t)$

$\displaystyle c_{e}\left(t\right)$ $\displaystyle=$
$\displaystyle\frac{2}{N+2}\sum_{k=1}^{N+1}\exp\left[-iE_{k}t\right]\sin^{2}\left(\frac{k\pi}{N+2}\right),$
(23) $\displaystyle E_{k}$ $\displaystyle=$ $\displaystyle
2\eta_{\mu\perp}\cos\left(\frac{k\pi}{N+2}\right).$ (24)

It is noted in the summation of $c_{e}(t)$ the terms when $k$ is close to
$(N+2)/2$ make the largest contributions. Therefore, it is appropriate to
expand the eigen energies by taking the first order approximation as

$\displaystyle E_{k}$ $\displaystyle=$ $\displaystyle
2\eta_{\mu\perp}\cos\left(\frac{k\pi}{N+2}\right)\simeq
2\eta_{\mu\perp}\left(\frac{\pi}{2}-\frac{k\pi}{N+2}\right)$ (25)
$\displaystyle=\eta_{\mu\perp}\pi\left(1-\frac{2k}{N+2}\right),$

which is an equally spaced spectrum.

For such a spectrum, the revival time is inversely proportional to the common
factor of all eigen energies (simply given by the neighboring energy space
$\Delta E_{k}$) and can be obtained as

$T=\frac{2\pi}{\Delta
E_{k}}=\frac{2\pi}{2\eta_{\mu\perp}\pi}(N+2)=\frac{N+2}{\eta_{\mu\perp}}.$
(26)

This is exactly the period of the freezing-thawing alternation.

On the other hand, he time dependent coefficient can be rewritten as

$\displaystyle c_{e}(t)$ $\displaystyle\simeq$
$\displaystyle\sum_{k=1}^{N+1}\exp\left[\frac{-i\pi\eta_{\mu\perp}t(N+2-2k)}{N+2}\right]$
(27) $\displaystyle\times\frac{2}{N+2}\sin^{2}\left(\frac{k\pi}{N+2}\right)$

Entanglement freezing occurs when $c_{e}(t)=0$ corresponding to

$\eta_{\mu\perp}t=l(N+2)+2,$ (28)

where $l=0,1,2,3...$. Obviously, period information $T=(N+2)/\eta_{\mu\perp}$
is confirmed. Also the first freezing occurs at time $\eta_{\mu\perp}t=2$
independent of $N$.

###  I.4 Freezing behavior independent of $N$

In this section, we show why the behaviors at early times of entanglements
$K_{A}(t)$, $K_{B}(t)$ for arbitrary finite $N$ are almost identical to the
infinite $N$ case. We focus on the amplitude of the excited state as given in
Eq. (23) and perform a Taylor expansion with respect to $t$ to obtain

$\displaystyle c_{e}(t)$ $\displaystyle=$
$\displaystyle\sum_{n=0}^{\infty}\frac{2^{n+1}}{N+2}\frac{1}{n!}(-i\eta_{\mu\perp}t)^{n}$
(29)
$\displaystyle\bigg{[}\sum_{k=1}^{N+1}\cos^{n}\left(\frac{k\pi}{N+2}\right)\sin^{2}\left(\frac{k\pi}{N+2}\right)\bigg{]}.$

For $n=\text{odd}$, one has $c_{e}(t)=0$ because the term
$\cos^{n}(\theta)\sin^{2}(\theta)$, with $\theta=k\pi/(N+2)\in(0,\pi)$, is
asymmetric with respect to $\theta=\pi/2$. Only the even terms will survive
and one can always define $n=2m$ with $m=0,1,2,\ldots$ to achieve

$c_{e}(t)=\sum_{m=0}^{\infty}\frac{2^{2m+1}}{N+2}\frac{1}{(2m)!}(-\eta_{\mu\perp}^{2}t^{2})^{m}Y_{m},$
(30)

where

$Y_{m}=\sum_{k=1}^{N+1}\cos^{2m}(\theta)\sin^{2}(\theta).$ (31)

Then one can achieve the following new expression of the excited state
amplitude

$\displaystyle c_{e}(t)$ $\displaystyle=$
$\displaystyle\sum_{m=0}^{\infty}(-\eta_{\mu\perp}^{2}t^{2})^{m}\bigg{[}\underbrace{\frac{1}{m!(m+1)!}}_{\text{for
any
$m$}}+\underbrace{\frac{2(-7+m-8N-2N^{2})}{\Gamma(m-N)\Gamma(m+N+4)}}_{\text{when
$m>N$}}+\underbrace{\frac{2(-31+m-32N-8N^{2})}{\Gamma(m-2N-2)\Gamma(m+2N+6)}}_{\text{when
$m>2N+2$}}+\ldots\bigg{]}$ (32) $\displaystyle=$
$\displaystyle\underbrace{c_{e}(t)}_{\text{for
$N\rightarrow\infty$}}+\sum_{m=N+1}^{\infty}(-\eta_{\mu\perp}^{2}t^{2})^{m}\bigg{[}\underbrace{\frac{2(-7+m-8N-2N^{2})}{\Gamma(m-N)\Gamma(m+N+4)}}_{\text{when
$m>N$}}+\underbrace{\frac{2(-31+m-32N-8N^{2})}{\Gamma(m-2N-2)\Gamma(m+2N+6)}}_{\text{when
$m>2N+2$}}+\ldots\bigg{]}$

where $\Gamma(n)=(n-1)!$ is the normal Gamma function. Here we have used the
following series expansion identity

$\displaystyle\cos^{2m}(\theta)$ $\displaystyle=$
$\displaystyle\frac{1}{2^{2m}}\left(\begin{array}[]{c}2m\\\
m\end{array}\right)$ (38)
$\displaystyle+\frac{2}{2^{2m}}\sum_{l=0}^{m-1}\left(\begin{array}[]{c}2m\\\
l\end{array}\right)\cos\left[2(m-l)\theta\right].$

One notes from (32) that the first $N$ terms of the finite $N$ expansion is
exactly the same as the infinite $N$ (i.e., Bessel-relted function
$J_{1}(2\eta_{\mu\perp}t)/\eta_{\mu\perp}t$ as proved in Appendix A) case.
This leads to the result that their behaviors are almost identical for
$\eta_{\mu\perp}t<N+2$ due to the Taylor’s theorem for estimates of the
remainder. Therefore, one can conclude that for finite $N$ there is also
freezing behavior. Combining with the proof the periodicity of revivals, this
has confirmed the alternative appearances of freezing and thawing behaviors of
the entanglements $K_{A}(t)$ and $K_{B}(t)$ shown in Fig. 1 (b) and (c) of the
main text.


# A Novel OMNeT++-based Simulation Tool for Vehicular Cloud Computing in ETSI
MEC-compliant 5G Environments

Angelo Feraudo1, Alessandro Calvio1, and Paolo Bellavista1
1Department of Computer Science and Engineering, University of Bologna, Viale
Risorgimento 2, Bologna, Italy
<EMAIL_ADDRESS>

###### Abstract

Vehicular cloud computing is gaining popularity thanks to the rapid
advancements in next generation wireless communication networks. Similarly,
Edge Computing, along with its standard proposals such as European
Telecommunications Standards Institute (ETSI) Multi-access Edge Computing
(MEC), will play a vital role in these scenarios, by enabling the execution of
cloud-based services at the edge of the network. Together, these solutions
have the potential to create real micro-datacenters at the network edge,
favoring several benefits like minimal latency, real-time data processing, and
data locality. However, the research community has not yet the opportunity to
use integrated simulation frameworks for the easy testing of applications that
exploit both the vehicular cloud paradigm and MEC-compliant 5G deployment
environments. In this paper, we present our simulation tool as a platform for
researchers and engineers to design, test, and enhance applications utilizing
the concepts of vehicular and edge cloud. Our platform significantly extends
OMNet++ and Simu5G, and implements our ETSI MEC-compliant architecture that
leverages resources provided by far-edge nodes. In addition, the paper
analyzes and reports performance results for our simulation platform, as well
as provides a use case where our simulator is used to support the design,
test, and validation of an algorithm to distribute MEC application components
on vehicular cloud resources.

##  1 INTRODUCTION

Next generation wireless communication networks are advancing at a rapid pace,
leading to the development and prototyping of highly innovative application
scenarios. In this context, as the amount of connected vehicles rapidly
increases [Elektrobit, 2022], drivers can take advantage of a wide range of
distributed applications and services, very often cloud-based, as in other
vertical domains, including improved access to information and entertainment
features. However, it starts to be widely recognized that traditional client-
to-cloud architectures cannot meet the stringent requirements of many time-
sensitive Internet of Things (IoT) applications. Furthermore, with the
constant growth of intelligent cars, equipped with a large set of onboard
sensors [Dario Sabella et al., 2017], the amount of vehicle-generated data may
pose a significant challenge to the backbone network.

Edge Computing has emerged as a promising solution for such emerging
distributed environments, as it offers a more efficient and effective way to
handle the ever-increasing demand for computing and storage resources.
Moreover, edge computing has the potential to replace traditional cloud
solutions due to its potential to reduce network stress, decrease latency,
improve efficiency, and enable real-time data processing. To accelerate the
adoption of this paradigm, the European Telecommunications Standards Institute
(ETSI) has proposed the Multi-access Edge Computing (MEC) standard [ETSI,
2022a] for cloud micro-datacenters situated at the edge of Radio Access
Network (RAN). The MEC architecture offers a virtualized infrastructure that
orchestrates, distributes, and manages the life-cycle of MEC-compliant
applications enabling their execution as close as one step from the involved
datasources and/or users. In this way, MEC aims at becoming a key technology
for emerging radio networks (e.g., 5G), as it empowers legacy radio units with
cloud computing capabilities [Taleb et al., 2017].

Recently, Vehicular Cloud Computing (or more simply Vehicular Computing) has
been proposed as a promising paradigm to support distributed applications
designed according to the edge cloud approach [Gerla, 2012, Olariu et al.,
2011]. Vehicular Computing exploits the computing power locally available on
today’s vehicles to create cost-effective mobile clouds at the far-edge layer.
The formation of these dynamic clouds should be achieved autonomously by
vehicles, which share their resources among them and/or with nearby MEC nodes
to extend their virtualized resources for service execution.

Numerous research efforts have been devoted to developing architectures that
leverage the resources dynamically available on vehicles [Olariu, 2020]. To
enhance resource availability at the network edge, various works have proposed
to exploit the underutilized computational power of both stationary [Liu et
al., 2011, Li et al., 2019] and moving vehicles [Hou et al., 2016]. These
opportunistic resources can be utilized for diverse purposes to handle the
growing number of applications used in vehicular networks. For instance,
vehicles can serve as relay nodes [Liu et al., 2011] to improve network
connectivity, or as computing nodes [Huang et al., 2018, Rahman et al., 2020,
Ma et al., 2021] to reduce the impact of these applications on the performance
of edge nodes.

Moreover, some recentstudies have focused on providing simulation frameworks
[Ahmed et al., 2019] since practical experiments on vehicular network
environments are expensive and challenging. In fact, to validate their
proposals, some of the works in the literature [Cha et al., 2021, Ma et al.,
2021, Rahman et al., 2020, Feng et al., 2017] have utilized these simulation
frameworks mainly for i) generating vehicle traces and ii) simulating the
behavior of vehicular network protocols. To the best of our knowledge, there
is currently no simulation tool that offers a single vehicular computing-based
platform where researchers can design/test their algorithms and applications
while exploiting at the same time the vehicular cloud computing paradigm and
the MEC standard deployment environment.

In this paper, we propose a simulation tool that leverages the ETSI MEC
standard to incorporate the resources available at the edge of the network,
such as those provided by collaborating vehicles, into the cloud continuum
spectrum. The simulation tool implements an extended version of the ETSI
standard [Feraudo et al., 2023], which enhances MEC-compliant edge nodes with
resources from vehicles within a designated Area of Interest (AoI). The
resources provided by the nodes are registered in the edge resource pool and
can be accessed through standardized interfaces. To accommodate node mobility,
the extended MEC architecture supported by our simulation platform assists
devices during application migrations by triggering the user context transfer
to MEC applications running on nodes leaving the AoI through standard defined
API [ETSI, 2022b]. As an extension of the ETSI MEC standard, the modeled
architecture allows dealing with some of the primary challenges arising in
vehicular computing environments, such as integration with cloud resources and
enabling coexistence among heterogeneous technologies. Furthermore, it allows
dealing with resource volatility issues (i.e., nodes that dynamically
join/leave during service provisioning) via a standardized migration
mechanism. In particular, to build our simulation platform, we utilized the
OMNeT++ network simulator as the underlying framework and incorporated the
Simu5G library to model the 5G network and communications aspects; more
details about these tools will be provided in the next sections.

The remainder of the paper is structured as follows. Section 2 provides an
overview of the related background, which includes a description of our
previous preliminary work on which basis we have developed our simulation
platform. Section 3 presents the interactions and modules that we have
originally implemented to create a MEC-compliant vehicular computing
environment in a 5G network. We then evaluate the performance of our
simulation platform while performing resource management and by providing a
practical example of its usage in section 4.

Table 1: Table of acronyms for MEC elements

Abbreviation | Definition
---|---
AMS | Application Mobility Service
MEC | Multi-access Edge Computing
MEC-App | MEC Application
MEC-H | MEC Host
MEC-O | MEC Orchestrator
MEC-P | MEC Platform
MEC-PM | MEC Platform Manager
UALCMP | User Application LifeCycle Management Proxy
VI | Virtualisation Infrastructure
VIM | Virtualisation Infrastructure Manager

##  2 BACKGROUND

###  2.1 ETSI Multi-access Edge Computing

Figure 1: The extended MEC architecture, already presented in [Feraudo et al.,
2023]

Figure 1 illustrates the MEC reference architecture proposed by ETSI in [ETSI,
2022a]. It offers the opportunity to run contextualized MEC-compliant
applications within a virtualized and multi-tenant environment. The reference
architecture consists of two layers: the system level and the host level,
which together comprise all standard functional elements. The system level
works as an entry point for authorized users to request the execution of MEC
applications. It includes management-related functional elements, such as MEC
Orchestrator (MEC-O) and User Application LifeCycle Management Proxy (UALCMP),
as well as other components that are necessary for standard operation, e.g.,
Device Application. The MEC-O is the core functionality for layer management,
as it has a global view of the MEC-compliant edge nodes (MEC Hosts) present in
a particular area and serves as a controller for MEC application
distributions. By selecting the best MEC Host based on the requirements of the
requested services, the MEC-O triggers the instantiation and termination of
applications. The second management element, i.e., the UALCMP, acts as an
intermediary node between the user and the application. It supports
instantiation and termination requests from the users, which are forwarded to
the MEC-O. Device application operates on user devices and interacts with the
MEC system through the UALCMP.

The host level is situated at the edge of the network and is responsible for
providing and managing the virtualization platform where MEC applications are
deployed. It comprises the edge node that provides virtualized computational,
network, and storage resources, i.e., MEC-H. The MEC-H enables MEC application
deployment through the Virtualization Infrastructure (VI), which is also
responsible for managing the data plane among the network interfaces through
traffic policies. The MEC-H also includes a MEC Platform (MEC-P) that exposes
a service registry, thus enabling the supported applications to discover,
offer, and consume standard services. The defined standards include Radio
Network Information Service, Location Service, and Application Mobility
Service, among the others. Moreover, at this level, the Virtualization
Infrastructure Manager (VIM) and the MEC Platform Manager (MEC-PM) act as the
management-related functional elements and serve as access points for the
MEC-O to the lower layer. The VIM is responsible for managing and releasing
the virtualized resources and configuring the VI to run software images
appropriately.

In the recent document released by the ETSI group [ETSI, 2022a], the MEC
reference architecture has been updated with an additional functional element,
the MEC Federator. This element is especially noteworthy due to the increased
recognition that the standard has received, and it will serve as the core
element for federating multiple MEC systems that belong to different network
domains.

###  2.2 Vehicular Computing Paradigm

Many current and future applications for connected vehicles can take advantage
of edge resources, including infotainment and time-sensitive applications
related to traffic safety. The integration of vehicular networks with the MEC
infrastructure can be referred as Vehicular Edge Computing (VEC) [Liu et al.,
2021]. VEC aims to bring computational capabilities to the proximity of
vehicular users, allowing for services to be readily available via Vehicle-to-
Infrastructure (V2I) communications. Nevertheless, the ever-increasing number
of applications and businesses is starting to make the infrastructure unable
to efficiently satisfy their demands and possibly stringent requirements,
e.g., on latency. For this reason, the need has emerged to identify new
resources that can support the more traditional edge infrastructure.

In [Olariu et al., 2011] and [Gerla, 2012], the authors introduced the
Vehicular Cloud Computing concept relying on the idea that modern-day vehicles
come equipped with powerful on-board computers, ample storage, and an array of
sensing devices. In their work [Olariu et al., 2011], the authors define
Vehicular Computing as a collaborative way to share resources among vehicles
to solve problems that would otherwise require a significant amount of time
with a more traditional centralized architecture, in particular for context-
specific applications. In line with this, in [Gerla, 2012] the author states
that the vehicular cloud paradigm allows keeping the information gathered by
vehicle sensors locally and sharing it solely with other vehicles, as the
sheer volume of in-vehicle generated data can pose serious technical
challenges for the network infrastructure. According to these definitions, a
cloud of vehicles, i.e., a set of vehicles that serve as computation nodes for
a diverse range of services, can be formed anywhere on the road and their
onboard resources can be dynamically allocated to authorized users. Despite
its potential benefits, such a vehicular computing environment also poses
several challenges that must be addressed. These challenges include
distributed ownership, as each vehicle has a single owner responsible for
deciding whether to share onboard resources; high node mobility, which makes
it difficult to predict the vehicular residency times in the cloud even when
clouds are formed using resources of vehicles within a parking lot; device
heterogeneity, as vehicles are manufactured by different companies; security
and privacy.

To deal with some of the aforementioned challenges, we recently presented an
ETSI MEC-compliant architecture that enlarges the edge resource pool through
vehicular computational resources [Feraudo et al., 2023]. In the next section,
we provide additional information about this architecture, which serves as the
architectural basis for the simulation platform originally described in this
paper.

###  2.3 Using Vehicle Resources to Expand MEC Nodes

In our previous work [Feraudo et al., 2023], we consider the evidence that
present-day far-edge nodes, e.g., vehicles, hold sufficient computing and
storage resources to form micro-datacenters at the network edge. The proposal
exploits the edge node components of the ETSI MEC standard, i.e., MEC-H, to
leverage far-edge node resources as part of its resource pool. Hence, compared
to the traditional ETSI MEC architecture, the proposed extension encompasses
mechanisms capable of deploying and distributing applications on local and
remote resources by keeping an eye on resource volatility issues.

As illustrated in figure 1 (blue lines), each MEC-H defines an Area of
Interest (AoI) corresponding to the area within which far-edge device
resources are collected. The AoI may coincide with the base station coverage,
thus it depends on where the MEC-H is located, i.e., at either the network
edge (close to the base station) or the central data network (at aggregation
points). The latter location enables MEC-Hs to subscribe for multiple AoI, as
subscriptions may depend on the latency introduced by the network. Indeed, for
MEC-Hs physically collocated with the base station, subscribing to several
AoIs may result in huge costs in terms of latency, as the distance with other
base stations increases.

To model the resource acquisition procedure, our extended MEC architecture
introduces an external entity at the MEC system level running a Broker, which
represents the message broker of a publish-subscribe system. It allows MEC-H
subscriptions to the AoI and manages their notification whenever a new device
enters or leaves the area. The same entity runs a reward system encouraging
far-edge devices to lease their local virtualized resources, e.g., computing
power, and join the resource pool. It relies on a device-initiated scheme
requiring far-edge nodes to request available rewards contextualized to the
AoI. Thus, whenever a new device accepts the rewards indicated by the MEC-H
related to that area, it publishes the amount of resources that it is willing
to make available. In addition, once a device leaves the AoI, the MEC-H
receives the notification, removes the concerned resources from those
available in the pool, and starts the mobility procedure for the apps running
on that device.

By delving into slightly finer technical details, the VIM is the main MEC-H
internal entity to be affected during resource acquisition, as it is in charge
of administering the MEC-H resource pool and preparing the VI for the
deployment of MEC applications. Indeed, in our MEC extended architecture
[Feraudo et al., 2023], the VIM operates to handle a heterogeneous pool of
distributed resources. Specifically, once registered to the MEC-O, the VIM
specifies the content of interest to the Broker (content-based subscription)
corresponding to the AoI parameters (e.g., circle center and diameter) given
during its configuration. Thus, when the Broker notifies it of new device
resource acquisition, the VIM stores the endpoint information corresponding to
an external VI address and the resource capacity of that device. This allows
the VIM to keep track of the single contributions that each host brings in
terms of computational resources. In this context, it is beneficial to
differentiate between infrastructure resources and transient ones that are
added based on the availability of far-edge nodes in the vicinity.

This section overviewed the architecture that underlies the simulation
platform originally presented in this paper. Let us point out that, while the
presented architecture is designed to be versatile and to support different
types of edge nodes offering virtualized resources at the far-edge layer, the
current implementation of our simulation platform supports the exploitation of
only parked vehicles as far-edge nodes.

Figure 2: Simulation Tool Modules Structure

##  3 OUR SIMULATION PLATFORM FOR MEC-COMPLIANT VEHICULAR COMPUTING

The Primary objective of our original simulation tool is to provide a platform
for researchers and engineers to design, test, and enhance applications by
utilizing the concept of vehicular computing in a 5G environment. The proposed
framework works within the OMNeT++ event-based simulator and uses the well-
known Simu5G communication library [Nardini et al., 2020]. Specifically, we
developed the OMNeT++ modules inside the Simu5G project and reused some of the
ETSI MEC modules provided by this library.

In the following sections, we will examine the primary internal components of
our simulation platform and how they interact to enable the exploitation of
external resource infrastructure in MEC-compliant environments.

(a) Resource Acquisition Sequence Diagram

(b) Resource Releasing Sequence Diagram

Figure 3: Sequence Diagram device-initiated scheme

###  3.1 Simulation Tool: Interactions

This sub/section describes the procedures supported for MEC-enabled resource
management for vehicular cloud computing, identified on the basis of the MEC
extended architecture presented in Section 2. Each procedure requires a
specific set of interactions to be completed effectively. These interactions
involve both MEC standard and non-standard components, which must be carefully
considered to ensure that the necessary resources are acquired, released, and
allocated.

#### Resource Acquisition

Figure 3(a) illustrates the steps required by parking vehicles to partake in
the resource acquisition procedure. According to the sequence diagram,
whenever a new vehicle accepts the rewards indicated by the MEC-H related to
that area, it publishes the amount of resources that wants to make available.
In addition to resource availability information, the POST request, step (3)
in the figure, contains data related to its location, thus allowing the Broker
to notify the appropriate MEC-H.

#### Resource Allocation

An example of interaction scheme for app instantiation on remote resources is
reported in figure 4. The MEC-O receives an instantiation request from the
UALCMP and starts resource/service discovery throughout all the MEC-Hs under
its management. Once the MEC-H has been chosen, the VIM receives an
instantiation request (step (7) in Fig.4) and starts the scheduling procedure.
As also described in section 2.3, when the Broker notifies the VIM for new
device resource acquisition, the VIM stores the endpoint information,
corresponding to an external VI address (see Fig. 2), and the resource
capacity of that device. Hence, the VIM is aware of the single contributions
that each device brings to correctly allocate its resources when needed. The
scheduling phase leads to the identification of a single resource
infrastructure, either local or remote, where to deploy the requested MEC
applications.

#### Resource Releasing

Figure 3(b) shows the interactions needed when devices leave the resource
pool. In such a scenario, once the corresponding MEC-H receives the
notification, it removes the concerned resources from those available in the
pool and starts the mobility procedure for the apps running on that device.
The migration event exploits the standard Application Mobility Service (AMS)
API [ETSI, 2022b] defined by ETSI, which currently supports app migrations in
environments encompassing multiple edge nodes. Nonetheless, the proposed
design in [Feraudo et al., 2023] requires MEC components to support intra-host
migration, e.g., MEC applications mobility from remote to local resource
infrastructure. In our simulation platform, we decided to deploy a MEC-
assisted application mobility for intra-host migrations, which implies MEC
components to trigger the user-context transfer. Thus, once the VIM receives a
notification involving nodes leaving the resource pool (step (3) Fig. 3(b)),
it looks for MEC applications running on that host, and creates a migration
event for each of them. This series of events serve as inputs for an extended
version of the Application Mobility Service, which creates an event chain
capable of initiating the intra-host migration of the MEC applications.

Figure 4: Resource Allocation Sequence Diagram

###  3.2 Simulation Tool: Modules

In the previous sub-section, we emphasized the involved MEC entities and their
interactions when supporting the usage of remote resources provided by
collaborating vehicles. To enable these resources to host MEC applications in
a simulated environment, we built each of the MEC entities as an application,
abstracting their workload from the underlying host. Specifically, to make the
extended MEC architecture more agile and flexible, our simulation platform
models MEC components as Inet Application.

Figure 2 illustrates the deployment of simulation modules that replicate the
scenario presented in our proposal (Fig. 1). As already mentioned, our
simulation tool currently considers cars as far-edge devices capable of
hosting MEC applications. To support this functionality, we introduce the car
module, which extends the New Radio User Equipment (NRUE) defined in Simu5G.
The car module acts as a wrapper for any 5G-enabled device, providing
computational resources (e.g., CPU, RAM, and storage) and running applications
that enable the car local resource infrastructure to host MEC-compliant
applications. In addition, the car module forwards management messages to
prepare and configure the local virtualization infrastructure received from
the MEC-H management entities. Specifically, the module runs the ClientResApp,
which requests and decides whether to accept rewards and handles resource
registration and release (section 3.1). Moreover, it executes the VIApp that
applies management instructions received from the MEC-H, to perform local
resource allocation and release. It should be noted that the VIApp module can
properly carry out its functionalities on any host having a resource
infrastructure, thus it might be utilized to allow any device to become part
of the MEC-H resource pool.

With the MEC-H now responsible for managing both local and remote resources,
the VIM takes on the critical role of scheduling, preparing, and releasing
both local and remote resources. To support this functionality, the VIM module
implemented in our simulation platform sets up the remote virtual
infrastructure (VI) to handle remote commands for allocating, relocating, and
terminating MEC applications. Moreover, the VIM supports interchangeable
scheduling algorithms to determine the optimal remote host for deploying
applications based on the surrounding environment.

According to our MEC extended architecture [Feraudo et al., 2023], far-edge
devices may leave the resource pool while running MEC applications. When a
vehicle hosting MEC applications exits a parking lot in our simulation
scenario, it triggers the migration procedure to prevent service interruption
and delay. This highlights the importance of designing a reliable migration
mechanism for MEC applications in vehicular computing environments. Our
framework supplies a module representing an extended version of the
Application Mobility Service that transparently addresses resource volatility
issues. As previously mentioned in Section 3.1, it supports MEC-assisted
intra-host migration. Thus, when an application instance is relocated, e.g.,
from a remote to a local resource infrastructure, the module requires the MEC
application to transfer the user context associated with the NRUE device it is
serving.

To summarize, the novel modules included in our simulation platform provide an
unprecedented solution for researchers to develop and test MEC-compliant
applications in a vehicular computing environment that leverages parked
vehicles as sources of virtualized resources, in particular of virtualized
processing. Our platform also facilitates the design of novel scheduling
algorithms for identifying a suitable host for deploying MEC applications:
researchers can evaluate various metrics during their design, such as the
average latency between a host and the central infrastructure, or the
probability of a node contributing further to the resource pool based on
historical data.

Figure 5: Resource acquisition and releasing protocol times Figure 6: Ceentral
Garage City of Arnhem entering and leaving vehicles

##  4 PERFORMANCE EVALUATION

In this section, we report about some relevant performance indicators measured
on top of our simulation platform for some examples of vehicular cloud
computing applications. Additionally, we show how our simulation platform can
be utilized to define and evaluate an algorithm that efficiently distributes
MEC applications on stationary vehicular resources.

Our experiments were conducted in a 5G standalone network environment with a
numerology index $\mu=2$. The network consists of a single MEC-H that is
connected to a gNB. The scenario involves a parking area located close to the
gNB. We implemented a basic reward scheme for the resource acquisition
procedure, which utilizes integer values accepted by all participating
vehicles. The experiments were carried out on a Linux Virtual Machine running
OMNeT++ having 16 CPUs and 64Gb of RAM.

Figure 7: Resource Allocation Time

###  4.1 Resource Management

As described in section 3.1, the resource management process involves the
operations required to ensure the correct acquisition, allocation, and release
of remote resources.

Figure 5 illustrates the time required by the protocols for collecting and
releasing resources from vehicles as they enter or leave the parking lot
within the MEC-H AoI. The join time shown in the figure indicates the time
interval for the MEC-H to recognize the availability of a new vehicle for MEC
application allocation (step (1)-(6) in Figure 3(a)). On the other hand, the
release time is the interval required by the MEC-H to remove the vehicle from
the resource pool (steps (1)-(4) in Figure 3(b)). The figure indicates that
the join time follows a growing trend ranging from 13 to 40 ms as the number
of cars participating in the resource acquisition procedure increases, whereas
the release time remains relatively constant (around 7 ms). The difference in
performance between the join and release times can be attributed to the
varying number of request/response messages generated by the two protocols. On
the one hand, the resource release process necessitates only a few messages to
exclude a vehicle effectively from the pool. On the other hand, the resource
acquisition process involves a series of request/response messages because the
device-initiated reward scheme mandates that the vehicle request available
rewards. However, it is unlikely for a large number of vehicles to enter a
parking lot simultaneously. Such a scenario may only occur during special
events like festivals or football matches. To support this claim, we analyzed
the data of three parking garages in the city of Arnhem, which is available on
the Open Parkeerdata
portal111https://parkeerdata.nl/opendata/arnhem/parkeergarages/transactiedata-
parkeergarages. Figure 6 depicts the average number of cars entering and
leaving the most used garage during rush hours. The figure clearly shows that
the number of parked vehicles reaches almost the maximum considered in our
test setup between 17:30 and 18:30. Moreover, it is important to note that the
peak of participating vehicles does not necessarily occur simultaneously
because data were sampled with 30-minute periodicity.

To analyze the time required to allocate MEC applications on remote resources,
we measured the delay introduced by the interactions between the VI and VIM
during steps (7)-(8) of the process in Fig. 4. The simulation involves
multiple UEs requesting MEC app execution and several parked cars belonging to
the MEC-H resource pool. MEC applications are evenly distributed on remote
nodes using a Round Robin scheduler. We repeated the simulation 10 times with
varying numbers of UEs and parked cars.

Figure 7 illustrates that the delays associated with resource allocation
follow an exponential growth that depends on the number of MEC applications
deployed on parked cars. This is confirmed by the overlapping curves, which
indicate that the delay values remain relatively constant even when the number
of parked cars varies. It is worth mentioning that we simulated the worst-case
scenario, in which all UEs requested MEC app execution simultaneously, thus
leading to a substantial increase in network traffic. Despite this, the delay
caused by these interactions remained negligible, even when the number of
requests exceeded 300, with a delay of approximately 40ms.

###  4.2 A Custom Scheduler for Stationary Vehicular Resources

Figure 8: Comparison Scheduling Algorithms in terms of Migrations

To show an example of the usage of our simulation platform to assist
researchers in the design, evaluation, and assessment of new algorithms and
protocols, we have built and tested the performance of a custom scheduling
algorithm that distributes MEC applications on remote resources offered by
parked vehicles. It aims at minimizing the number of migrations generated by
vehicles leaving the parking lot while running applications.

To generate vehicle and user behaviors, we recreated the scenario utilized in
[Feraudo et al., 2023]. This approach involves constructing a series of
Poisson and Gaussian distributions using two real-world datasets. The Arnhem
dataset, already presented in the previous section, was used to model the
distributions describing vehicle entry and residency times in a parking lot.
The Bologna WiFi
dataset222https://opendata.comune.bologna.it/explore/dataset/iperbole-wifi-
affluenza/information/ provided information on user activities on Open WiFi
networks within the city of Bologna, which enabled the creation of
distributions mimicking the user behavior during each hour of the day. As in
our previous work, we assume that each vehicle that enters the parking lot
accepts the rewards proposed by the MEC-H, and each user requests triggers the
execution of a one-to-one MEC application.

We considered a simulated period of 24 hours of vehicle and user activities.
We run three simulations, one for each scheduling algorithm, namely best
first, round robin, and our custom algorithm. The performance of these
algorithms was evaluated based on the number of migrations they generated, as
this directly impacts the reliability of MEC applications. In other words, a
lower number of migrations is desirable for improved performance.

Figure 8 reports the associated performance results, by referring to the most
challenging case of the day hours with highest levels of user and vehicle
activity. The best first algorithm chooses the first available vehicle from
the pool that has sufficient resources to execute the application. However,
this approach can lead to a large number of migrations, as the selected
vehicles may leave the parking lot while running all the applications they are
capable of executing. In fact, the number of migrations exceeds 60 at the 12th
hour of the simulation. Conversely, the round-robin algorithm maintains a
steady number of migrations (around 7.42 in average)as the applications are
equally distributed on the vehicles belonging to the MEC-H resource pool. In
addition, by using our simulation platform, we have developed a custom
scheduler that relies on multiple Gaussian distributions by using means and
standard deviation produced after a pre-processing phase of the Arnhem
dataset, which generated the average occupancy time based on a 10-minute
interval sampling. Hence, for each vehicle that belongs to the resource pool,
the custom scheduler utilizes the time at which it joined the pool and the
aforementioned distributions to predict its residency time. It then assigns
the MEC application to the vehicle with the highest remaining residency time.
The results in the figure demonstrate how this algorithm can largely over-
perform the others in the considered application scenario: even if it could be
enhanced via more sophisticated machine learning techniques, already in its
simple current version it generates only around 20 migrations at the 18th hour
of the simulation.

##  5 CONCLUSIVE REMARKS

This paper originally presents a simulation platform capable of assisting
researchers in the development, evaluation, and assessment of vehicular
application algorithms and protocols that exploit vehicular cloud resources
accessed according to the standard ETSI MEC specifications. After presenting
our extended MEC architecture for these scenarios, the paper reports about the
design and implementation of our original simulation platform, with its
resource management modules and procedure interactions. In addition, this
paper originally describes a concrete example of how our simulation platform
can be utilized to design, test, and implement applications that exploit the
vehicular computing paradigm.

## REFERENCES

  * Ahmed et al., 2019 Ahmed, B., Malik, A. W., Hafeez, T., and Ahmed, N. (2019).  Services and simulation frameworks for vehicular cloud computing: a contemporary survey.  EURASIP Journal on Wireless Communications and Networking, 2019:1–21.
  * Cha et al., 2021 Cha, N., Wu, C., Yoshinaga, T., Ji, Y., and Yau, K.-L. A. (2021).  Virtual Edge: Exploring Computation Offloading in Collaborative Vehicular Edge Computing.  IEEE Access, 9:37739–37751.
  * Dario Sabella et al., 2017 Dario Sabella et al. (2017).  Toward fully connected vehicles: Edge computing for advanced automotive communications.
  * Elektrobit, 2022 Elektrobit (2022).  Connected vehicle software solutions.  Last Accessed: 2022-10-03.
  * ETSI, 2022a ETSI (2022a).  GS MEC 003 - V3.1.1.  https://www.etsi.org/deliver/etsi˙gs/MEC/001˙099/003/03.01.01˙60/gs˙mec003v030101p.pdf.
  * ETSI, 2022b ETSI (2022b).  GS MEC 021 - V2.2.1.  https://www.etsi.org/deliver/etsi˙gs/MEC/001˙099/021/02.02.01˙60/gs˙mec021v020201p.pdf.
  * Feng et al., 2017 Feng, J., Liu, Z., Wu, C., and Ji, Y. (2017).  Ave: Autonomous vehicular edge computing framework with aco-based scheduling.  IEEE Transactions on Vehicular Technology, 66(12):10660–10675.
  * Feraudo et al., 2023 Feraudo, A., Calvio, A., Bujari, A., and Bellavista, P. (2023).  A novel design for advanced 5g deployment environments with virtualized resources at vehicular and mec nodes.
  * Gerla, 2012 Gerla, M. (2012).  Vehicular Cloud Computing.  In Annual Mediterranean Ad Hoc Networking Workshop (Med-Hoc-Net), pages 152–155.
  * Hou et al., 2016 Hou, X., Li, Y., Chen, M., Wu, D., Jin, D., and Chen, S. (2016).  Vehicular fog computing: A viewpoint of vehicles as the infrastructures.  IEEE Transactions on Vehicular Technology, 65(6):3860–3873.
  * Huang et al., 2018 Huang, X., Yu, R., Liu, J., and Shu, L. (2018).  Parked Vehicle Edge Computing: Exploiting Opportunistic Resources for Distributed Mobile Applications.  IEEE Access, 6:66649–66663.
  * Li et al., 2019 Li, C., Wang, S., Huang, X., Li, X., Yu, R., and Zhao, F. (2019).  Parked vehicular computing for energy-efficient internet of vehicles: A contract theoretic approach.  IEEE Internet of Things Journal, 6(4):6079–6088.
  * Liu et al., 2021 Liu, L., Chen, C., Pei, Q., Maharjan, S., and Zhang, Y. (2021).  Vehicular edge computing and networking: A survey.  Mobile networks and applications, 26:1145–1168.
  * Liu et al., 2011 Liu, N., Liu, M., Lou, W., Chen, G., and Cao, J. (2011).  Pva in vanets: Stopped cars are not silent.  In 2011 Proceedings IEEE INFOCOM, pages 431–435.
  * Ma et al., 2021 Ma, C. et al. (2021).  Parking Edge Computing: Parked-Vehicle-Assisted Task Offloading for Urban VANETs.  IEEE Internet of Things Journal, 8(11):9344–9358.
  * Nardini et al., 2020 Nardini, G., Stea, G., Virdis, A., and Sabella, D. (2020).  Simu5g: A system-level simulator for 5g networks.  In Simultech, pages 68–80.
  * Olariu, 2020 Olariu, S. (2020).  A survey of vehicular cloud research: Trends, applications and challenges.  IEEE Transactions on Intelligent Transportation Systems, 21(6):2648–2663.
  * Olariu et al., 2011 Olariu, S., Khalil, I., and Abuelela, M. (2011).  Taking VANET to the Clouds.  International Journal of Pervasive Computing and Communications, 7(1):7–21.
  * Rahman et al., 2020 Rahman, F. H. et al. (2020).  Off-Street Vehicular Fog for Catering Applications in 5G/B5G: A Trust-Based Task Mapping Solution and Open Research Issues.  IEEE Access, 8:117218–117235.
  * Taleb et al., 2017 Taleb, T., Samdanis, K., Mada, B., Flinck, H., Dutta, S., and Sabella, D. (2017).  On multi-access edge computing: A survey of the emerging 5g network edge cloud architecture and orchestration.  IEEE Communications Surveys & Tutorials, 19(3):1657–1681.