* [37] Polizzotto, C.  Unified thermodynamic framework for nonlocal/gradient continuum theories.  _European Journal of Mechanics-A/Solids_ 22, 651–668 (2003).
  * [38] Chung, E. T., Efendiev, Y., Leung, W. T. & Vasilyeva, M.  Nonlocal multicontinua with representative volume elements. Bridging separable and non-separable scales.  _Computer Methods in Applied Mechanics and Engineering_ 377, 113687 (2021).
  * [39] Oden, J. T. & Reddy, J. N.  _An introduction to the mathematical theory of finite elements_ (Courier Corporation, 2012).
  * [40] Pang, G., Chen, W. & Sze, K.-Y.  Gauss–Jacobi-type quadrature rules for fractional directional integrals.  _Computers & Mathematics with Applications_ 66, 597–607 (2013).
  * [41] Mannan, S., Parameswaran, V. & Basu, S.  Stiffness and toughness gradation of bamboo from a damage tolerance perspective.  _International Journal of Solids and Structures_ 143, 274–286 (2018).
  * [42] Sebaa, N., Fellah, Z. E. A., Lauriks, W. & Depollier, C.  Application of fractional calculus to ultrasonic wave propagation in human cancellous bone.  _Signal Processing_ 86, 2668–2677 (2006).
  * [43] Nair, S., Jokar, M. & Semperlotti, F.  Nonlocal acoustic black hole metastructures: Achieving broadband and low frequency passive vibration attenuation.  _Mechanical Systems and Signal Processing_ 169, 108716 (2022).
  * [44] Sidhardh, S. & Ray, M.  Inclusion problem for a generalized strain gradient elastic continuum.  _Acta Mechanica_ 229, 3813–3831 (2018).
  * [45] Seleson, P., Gunzburger, M. & Parks, M. L.  Interface problems in nonlocal diffusion and sharp transitions between local and nonlocal domains.  _Computer Methods in Applied Mechanics and Engineering_ 266, 185–204 (2013).
  * [46] Rahimi, Z., Rezazadeh, G. & Sumelka, W.  A non-local fractional stress–strain gradient theory.  _International Journal of Mechanics and Materials in Design_ 16, 265–278 (2020).


# Spectrally resolved linewidth enhancement factor of a semiconductor
frequency comb

Nikola Opačak<EMAIL_ADDRESS>Institute of Solid State
Electronics, TU Wien, Vienna, Austria  Florian Pilat  Institute of Solid State
Electronics, TU Wien, Vienna, Austria  Dmitry Kazakov  Institute of Solid
State Electronics, TU Wien, Vienna, Austria  John A. Paulson School of
Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts
USA  Sandro Dal Cin  Institute of Solid State Electronics, TU Wien, Vienna,
Austria  Georg Ramer  Institute of Chemical Technologies and Analytics, TU
Wien, Vienna, Austria  Bernhard Lendl  Institute of Chemical Technologies and
Analytics, TU Wien, Vienna, Austria  Federico Capasso  John A. Paulson School
of Engineering and Applied Sciences, Harvard University, Cambridge,
Massachusetts USA  Benedikt Schwarz<EMAIL_ADDRESS>Institute
of Solid State Electronics, TU Wien, Vienna, Austria  John A. Paulson School
of Engineering and Applied Sciences, Harvard University, Cambridge,
Massachusetts USA

###### Abstract

The linewidth enhancement factor (LEF) has recently moved into the spotlight
of research on frequency comb generation in semiconductor lasers. Here we
present a novel modulation experiment, which enables the direct measurement of
the spectrally resolved LEF in a laser frequency comb. By utilizing a phase-
sensitive technique, we are able to extract the LEF for each comb mode. We
first investigate and verify this universally applicable technique using
Maxwell-Bloch simulations and then present the experimental demonstration on a
quantum cascade laser frequency comb.

Semiconductor lasers are compact, electrically pumped and provide substantial
broadband gain. They are recently gaining vast attention due to a wide range
of applications that utilize their coherence properties, such as high-
precision spectroscopy Hänsch (2006). Their asymmetric gain spectrum
additionally sets them apart from other laser types, where the lasing
transition takes place between two discrete levels. Following the Kramers-
Kronig relations, an asymmetric gainshape results in a dispersion curve of the
refractive index that has a non-zero value at the gain peak, where the
emission frequency of a free-running laser lies Yariv (1989). As a
consequence, a remarkable property of semiconductor lasers is that both the
refractive index and the optical gain change simultaneously with the varying
carrier population Osinski and Buus (1987). This was quantified with the
linewidth enhancement factor (LEF), also called the $\alpha$-factor, defined
by Henry as the ratio of changes of the modal index and gain at the gain peak
Henry (1982). Many unique properties of semiconductor lasers can be traced
back to the non-zero value of this factor. The LEF was first introduced in the
1980s to describe the broadening of the semiconductor laser linewidth Henry
(1982); Vahala and Yariv (1983) beyond the Schawlow-Townes limit Schawlow and
Townes (1958). Furthermore, the LEF determines the dynamics of semiconductor
lasers as it describes the coupling between the amplitude and phase of the
optical field Agrawal and Dutta (1995); Gray and Agrawal (1992). In lasers
with fast gain recovery times, the LEF was recently connected to the onset of
a giant Kerr nonlinearity Opačak  _et al._ (2021) and frequency modulated
combs Opačak and Schwarz (2019). It was shown that the LEF can lead to a low-
threshold multimode instability and frequency comb formation Piccardo  _et
al._ (2020); Meng  _et al._ (2020). Appropriate values of the LEF were
predicted to result in the emission of solitons in active media Columbo  _et
al._ (2020). The precise knowledge of the LEF represents a key point in
understanding many astonishing features of semiconductor lasers and
subsequently controlling them.

The physical origin of the LEF is explained through the asymmetric gain
spectrum of semiconductor lasers. In interband lasers, this is due to the
opposite curvatures of the valence and conduction bands in k-space Agrawal and
Dutta (1995), which yield LEF values around 2-7 Osinski and Buus (1987). In
intersubband lasers, the states have similar curvatures, so the gain asymmetry
originates from the non-parabolicity Liu  _et al._ (2013), counter-rotating
terms Pereira (2016), and Bloch gain Opačak  _et al._ (2021). Measured values
range from -0.5 to 2.5 Aellen  _et al._ (2006); Jumpertz  _et al._ (2016);
Hangauer and Wysocki (2015); von Staden  _et al._ (2006); Piccardo  _et al._
(2020, 2020).

An established technique for extracting subthreshold values of the LEF is the
Hakki-Paoli method by measuring the gain and wavelength shift Hakki and Paoli
(1975); Henning and Collins (1983). Above threshold values can be inferred
from the measurements of the linewidth broadening Henry (1982) and the phase
noise Henry (1983). Other methods are based on the analysis of the locking
regimes induced by optical injection from a master laser Liu  _et al._ (2001),
or on the optical feedback and the characterization of the self-mixing signal
Yu  _et al._ (2004); Fan  _et al._ (2015). Harder et al. provided a study of
the laser’s response under a modulation of the injection current and were able
to extract the LEF value Harder  _et al._ (1983). A modified experiment which
includes heterodyning allowed a direct measurement of the LEF Aellen  _et al._
(2006). However, all of the mentioned techniques have one substantial
limitation – they do not resolve the spectral behavior of the LEF. Most
methods rely on single-mode operation, which is either achieved in Fabry-Pérot
lasers very close to threshold or using distributed feedback (DFB) laser. In
DFBs, the lasing wavelength is detuned from the exact gain peak, thus
introducing an uncertainty in the LEF measurement.

Figure 1:  Intensity modulation (IM) and frequency modulation (FM) of a
single-mode laser and the LEF extraction.  (a) Modulation sidebands and their
beating signals $B_{\pm}$ with the center mode in the case of a pure IM or FM
(top). The beatings are represented in the complex plane. A mixture of both IM
and FM is analysed below for two values of the IM-FM phase shift $\theta$. (b)
Time traces of the intensity and frequency. Analytic curves given by Eq. (1)
(red dash-dotted lines) are fitted to the numeric time traces (blue solid
lines), obtained from a simulation Opačak and Schwarz (2019). The frequency is
modulated around the lasing frequency $f_{0}$ and the time is normalized to
the modulation period $T_{\mathrm{mod}}$, with
$f_{\mathrm{mod}}=3~{}\mathrm{GHz}$. The phase shift between the frequency and
intensity modulations is $\theta\approx\pi/2$. (c) Calculated LEF of a
simulated single-mode laser. The lasing frequency is swept in steps around the
gain peak frequency $f_{0}$. The extracted values match the LEF from an
analytic model (solid lines).

In this work, we introduce a novel measurement technique, which enables the
direct and spectrally resolved measurement of the LEF of a semiconductor laser
operating in the frequency comb regime. It builds upon the modulation
experiment by Harder et al. Harder  _et al._ (1983) and enables the extraction
of the LEF for each individual comb mode from a single measurment. This is
made possible by the shifted-wave interference Fourier transform spectroscopy
(SWIFTS) Burghoff  _et al._ (2015). Each comb mode, together with its
neigboring modulation sidebands, produces beatings. SWIFTS allows the
spectrally resolved measurement of both the amplitude and phase of these
beatings. We first lay down the theoretical foundations of our method. The
analytic values of the LEF are subsequently compared with the values extracted
from a numerical model of both a single-mode and multimode laser. This is
followed by the experimental demonstration, where the method is employed on a
quantum cascade laser frequency comb.

In the presence of a non-zero value of the LEF, a sinusoidal modulation of the
bias current results in both an intensity modulation (IM) and a frequency
modulation (FM) of the laser output Agrawal and Dutta (1995); Harder  _et al._
(1983). Similarly to the work done in Zhu and Cassidy (1997), the light
emitted by a single-mode laser with an average intensity $I_{0}$ is described
with an electric field of:

$\displaystyle\begin{split}E(t)=&\sqrt{I_{0}}\sqrt{1+m\cos{(2\pi
f_{\mathrm{mod}}t+\phi)}}\\\ &\times\cos{\big{(}2\pi f_{0}t+\beta\sin{(2\pi
f_{\mathrm{mod}}t+\phi+\theta)}\big{)}},\end{split}$ (1)

where $f_{0}$ is the lasing frequency, $f_{\mathrm{mod}}$ is the modulation
frequency, $m$ and $\beta$ are the IM and FM indices, respectively. We include
an additional arbitrary phase shift $\phi$ with respect to the current
modulation and an FM-IM phase shift $\theta$. Following Eq. (1), the frequency
is modulated as $f=f_{0}+\beta f_{\mathrm{mod}}\cos(2\pi
f_{\mathrm{mod}}t+\phi+\theta)$.

Figure 2:  LEF of a simulated multimode semiconductor laser.  The normalized
intenisty spectrum is depicted on the top. The spectrally resolved LEF is
shown on the bottom. Red dots represent the LEF for each lasing mode,
calculated using Eq. (4). The solid blue line represents the LEF from the
analytic model, given by the Eq. (B3) in the Appendix.  Figure 3:
Experimental data of a SWIFTS measurement with LEF evaluation.  (a) Sketch of
the experimental setup: Fourier transform infrared spectrometer (FTIR), beam
splitter (BS), quantum cascade laser (QCL), bias voltage (DC), RF generator
(RF), local oscillator (LO), quantum well infrared photodetector (QWIP), low-
noise amplifier (LNA). (b) Measured intensity spectrum of the QCL and
extracted LEF values for each mode (red dots) with a fit (blue dashed line).
(c) Intensity (top), beating intensity (middle) and beating phase spectrum as
obtained from the SWIFTS measurement for the highlighted grey area in (b).

Using the Jacobi-Anger expansion Abramowitz and Stegun (1965), Eq. (1) can be
written as a Fourier series:

$\displaystyle
E(t)=\sqrt{I_{0}}\sum_{n=-\infty}^{+\infty}E_{n}\exp{\big{(}2\pi(f_{0}+nf_{\mathrm{mod}})t\big{)}},$
(2)

as is derived in the Supplementary Material. Under the assumption of a weak
modulation strength ($m,\beta\ll 1$), the complex beating signals $B_{\pm}$
between the centerline $E_{0}$ and its first modulation sidebands $E_{\pm 1}$
can be written as:

$\displaystyle\begin{split}B_{+}&=E_{1}E^{*}_{0}=\mathrm{e}^{i(\phi+\theta)}\Big{(}\frac{\beta}{2}+\frac{m}{4}\mathrm{e}^{-i\theta}\Big{)}\\\
B_{-}&=E_{0}E^{*}_{-1}=\mathrm{e}^{i(\phi+\theta)}\Big{(}-\frac{\beta}{2}+\frac{m}{4}\mathrm{e}^{-i\theta}\Big{)}.\end{split}$
(3)

The extraction of the modulation indices $m$ and $\beta$ is possible from Eq.
(3), if both the amplitudes and phases of $B_{\pm}$ are known. Furthemore,
this is also valid in the case of a multimode laser, where each mode $k$
produces beatings $B_{k,\pm}$ with its neigboring modulation sidebands. With
the knowledge of the modulation indices, the spectral LEF for each mode can
finally be calculated directly as Harder  _et al._ (1983); Osinski and Buus
(1987); Aellen  _et al._ (2006); Hangauer and Wysocki (2015):

$\displaystyle\mathrm{LEF}_{k}=2\frac{\beta_{k}}{m_{k}}=\bigg{|}\frac{B_{k,+}-B_{k,-}}{B_{k,+}+B_{k,-}}\bigg{|},$
(4)

The modulation sidebands of a single-mode laser are sketched in Fig. 1a (top)
in the cases when only IM or FM is present. The corresponding beating signals
$B_{\pm}$ are plotted below in the complex plane, following the analysis of
Eq. (3). They are in-phase with each other in the case of a pure IM, and anti-
phase ($\pi$ phase-shifted) for a perfect FM. In a modulated semiconductor
laser, a mixture of IM and FM is always present. The beating signals in this
case are sketched on the bottom of Fig. 1a for two exemplary values of the FM-
IM phase shift $\theta=0$ and $\pi/2$, which is unknown a priori. This
highlights why a phase-sensitive technique is required to measure the
amplitudes and phases of $B_{\pm}$.

In Fig. 1b, we show the instantaneous intensity and frequency of a
semiconductor single-mode laser biased above threshold with a small
superimposed sinusoidal modulation. The results are obtained from a numerical
time-domain model of the laser based on the Maxwell-Bloch formalism Opačak and
Schwarz (2019). The analytical model given by Eq. (1) is fitted to the
numerical time traces of both the instantaneous intensity and frequency. The
IM-FM phase shift approaches ${\pi}/{2}$ Hangauer  _et al._ (2014); Hangauer
and Wysocki (2015), as can be seen from Fig. 1b. Fig. 1c shows the calculated
LEF of a simulated single-mode laser, whose frequency was tuned in discrete
steps around the constant gain peak frequency $f_{0}$. We show the results for
three different values of the LEF at the gain peak $\alpha=0.5$, $2$ and $4$,
represented by green triangles, red squares and blue circles, respectively.
The extracted LEF values from the numerical model follow the LEF obtained from
the analytic model, which is plotted with solid lines. The analytic model is
given by Eq. (B3) in the Appendix.

In the following we will discuss how to employ this technique to a frequency
comb. The modulation of the current induces modulation sidebands around each
comb mode. By finding the beatings $B_{k,\pm}$ of each mode with its
corresponding neighboring sidebands, one can obtain the LEF for each comb
mode. Fig. 2 shows an intensity spectrum of a simulated laser in a multimode
comb regime. The extracted spectrally resolved LEF is plotted with red dots
below, together with the LEF from an analytic model of the laser gain medium
(Eq. (B3) in the Appendix), represented with the blue solid line. The slight
deviations can be attributed to coherent mechanisms that couple the frequency
comb modes.

In experiments, the amplitudes and phases of the beatings $B_{k,\pm}$ can be
measured most elegantly using SWIFTS. The experimental setup is shown in Fig.
3a. SWIFTS employs a Fourier-transform infrared (FTIR) spectrometer in order
to spectrally resolve all individual beatings. The modulation of the laser is
measured in amplitude and phase using a fast photodetector and a lock-in
amplifier to obtain the SWIFTS interferogram (see Fig. C1 in the Appendix).
The beatings are obtained using the Fourier transformation. We extended our
previous SWIFTS setup Hillbrand  _et al._ (2018) with a custom-built high-
resolution FTIR spectrometer ($\approx$500 MHz), in order to resolve the
narrowly spaced beatings. The setup consists of a Newport Optical Delay Line
Kit (using DL325), a broadband mid-infrared beamsplitter, a temperature
stabilized HeNe-Laser and a Zürich Instruments HF2LI lock-in amplifier for the
acquisition of the intensity and SWIFTS interferograms of the measured laser,
as well as the interferogram of the HeNe-laser. We operate the FTIR
spectrometer in rapid scan mode and employ the Hilbert-transform to obtain the
mirror delay from the HeNe interferogram.

The QCL, which was used in this measurement, is a $3.5\,$mm long ridge laser
emitting at around $8\,\mathrm{\mu}$m. It consists of a long gain section and
a shorter modulation section to efficiently couple RF signals to the laser.
This feature, however, is not a necessity for the proposed technique. The
laser was operated in a free-running frequency comb state with a repetition
frequency of 12.2 GHz. The frequency of the weak modulation was chosen to be
sufficiently lower at 9.593 GHz and a demodulation frequency of 9.570 GHz was
set on the local oscillator. An RF-optimized quantum well infrared
photodetector (QWIP), cooled to 78 K, was used for light detection. The
intensity spectrum (Fig. 3b) shows a frequency comb spanning over 75 modes. A
zoom of the grey-shaded area can be seen in Fig. 3c. The weak modulation
sidebands are below the noise of the intensity spectrum (top). Nevertheless,
the beatings, produced by the laser modes and their weak sidebands, can be
precisely measured using SWIFTS in both the amplitudes and the phases (middle
and bottom, respectively). Applying Eq. 4 to the beating amplitudes and
phases, we extract the LEF for each individual comb mode, as displayed in Fig.
3b (bottom). The extracted spectrally resolved LEF follows the expected
characteristic shape (see Fig. 1c and 2). Its value matches the prediction
from recent theoretical work, which pinpointed the origin of the LEF to the
Bloch gain Opačak  _et al._ (2021).

In conclusion, we present a novel technique to directly measure the spectrally
resolved LEF of a running semiconductor laser frequency comb. The measurement
concept is first verified using elaborate numerical simulations of a modulated
semiconductor laser. There, an excellent agreement is observed with the
expected spectral LEF from the analytic model. The experimental demonstration
was performed on a multimode QCL frequency comb, while the technique itself is
universal. It will allow to extract the spectral LEF in frequency combs based
on any type of a semiconductor laser. The LEF governs many coherent processes
in a running semiconductor lasers among which is the frequency comb regime.
Its precise knowledge will provide a better fundamental understanding of the
light evolution that will promote further technological advancements.

This work was supported by the Austrian Science Fund (FWF) within the projects
”NanoPlas” (P28914-N27) and by the European Research Council (ERC) under the
European Union’s Horizon 2020 research and innovation programme (Grant
agreement No. 853014).

## References

  * Hänsch (2006) T. W. Hänsch, Reviews of Modern Physics 78, 1297 (2006).
  * Yariv (1989) A. Yariv,  _Quantum electronics, 3rd edition_ (Wiley, 1989).
  * Osinski and Buus (1987) M. Osinski and J. Buus, IEEE Journal of Quantum Electronics 23, 9 (1987).
  * Henry (1982) C. Henry, IEEE Journal of Quantum Electronics 18, 259 (1982).
  * Vahala and Yariv (1983) K. Vahala and A. Yariv, IEEE Journal of Quantum Electronics 19, 1096 (1983).
  * Schawlow and Townes (1958) A. L. Schawlow and C. H. Townes, Physical Review 112, 1940 (1958).
  * Agrawal and Dutta (1995) G. P. Agrawal and N. K. Dutta,  _Semiconductor Lasers_ (Springer US, 1995).
  * Gray and Agrawal (1992) G. Gray and G. Agrawal, IEEE Photonics Technology Letters 4, 1216 (1992).
  * Opačak  _et al._ (2021) N. Opačak, S. Dal Cin, J. Hillbrand, and B. Schwarz, arXiv:2102.08167v1 (2021).
  * Opačak and Schwarz (2019) N. Opačak and B. Schwarz, Physical Review Letters 123 (2019), 10.1103/physrevlett.123.243902.
  * Piccardo  _et al._ (2020) M. Piccardo, B. Schwarz, D. Kazakov, M. Beiser, N. Opačak, Y. Wang, S. Jha, J. Hillbrand, M. Tamagnone, W. T. Chen, A. Y. Zhu, L. L. Columbo, A. Belyanin, and F. Capasso, Nature 582, 360 (2020).
  * Meng  _et al._ (2020) B. Meng, M. Singleton, M. Shahmohammadi, F. Kapsalidis, R. Wang, M. Beck, and J. Faist, Optica 7, 162 (2020).
  * Columbo  _et al._ (2020) L. Columbo, M. Piccardo, F. Prati, L. A. Lugiato, M. Brambilla, A. Gatti, C. Silvestri, M. Gioannini, N. Opačak, B. Schwarz, and F. Capasso, arXiv:2007.07533v2 (2020).
  * Liu  _et al._ (2013) T. Liu, K. E. Lee, and Q. J. Wang, Optics Express 21, 27804 (2013).
  * Pereira (2016) M. F. Pereira, Applied Physics Letters 109, 222102 (2016).
  * Aellen  _et al._ (2006) T. Aellen, R. Maulini, R. Terazzi, N. Hoyler, M. Giovannini, J. Faist, S. Blaser, and L. Hvozdara, Applied Physics Letters 89, 091121 (2006).
  * Jumpertz  _et al._ (2016) L. Jumpertz, F. Michel, R. Pawlus, W. Elsässer, K. Schires, M. Carras, and F. Grillot, AIP Advances 6, 015212 (2016).
  * Hangauer and Wysocki (2015) A. Hangauer and G. Wysocki, IEEE Journal of Selected Topics in Quantum Electronics 21, 74 (2015).
  * von Staden  _et al._ (2006) J. von Staden, T. Gensty, W. Elsäßer, G. Giuliani, and C. Mann, Optics Letters 31, 2574 (2006).
  * Hakki and Paoli (1975) B. W. Hakki and T. L. Paoli, Journal of Applied Physics 46, 1299 (1975).
  * Henning and Collins (1983) I. Henning and J. Collins, Electronics Letters 19, 927 (1983).
  * Henry (1983) C. Henry, IEEE Journal of Quantum Electronics 19, 1391 (1983).
  * Liu  _et al._ (2001) G. Liu, X. Jin, and S. Chuang, IEEE Photonics Technology Letters 13, 430 (2001).
  * Yu  _et al._ (2004) Y. Yu, G. Giuliani, and S. Donati, IEEE Photonics Technology Letters 16, 990 (2004).
  * Fan  _et al._ (2015) Y. Fan, Y. Yu, J. Xi, G. Rajan, Q. Guo, and J. Tong, Applied Optics 54, 10295 (2015).
  * Harder  _et al._ (1983) C. Harder, K. Vahala, and A. Yariv, Applied Physics Letters 42, 328 (1983).
  * Burghoff  _et al._ (2015) D. Burghoff, Y. Yang, D. J. Hayton, J.-R. Gao, J. L. Reno, and Q. Hu, Optics Express 23, 1190 (2015).
  * Zhu and Cassidy (1997) X. Zhu and D. T. Cassidy, Journal of the Optical Society of America B 14, 1945 (1997).
  * Abramowitz and Stegun (1965) M. Abramowitz and I. A. Stegun,  _Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables_ (Courier Corporation, 1965).
  * Hangauer  _et al._ (2014) A. Hangauer, G. Spinner, M. Nikodem, and G. Wysocki, Optics Express 22, 23439 (2014).
  * Hillbrand  _et al._ (2018) J. Hillbrand, A. M. Andrews, H. Detz, G. Strasser, and B. Schwarz, Nature Photonics 13, 101 (2018).

##  Appendix A Theoretical background

It is known from literature that the width of the laser line is due to the
fluctuations of the phase of the optical field Henry (1982). These
fluctuations partly come from the spontaneous emission events within the laser
active medium, which alter the phase and the intensity of the laser field.
This, together with the resonator losses, defines the Schawlow-Townes limit of
the laser linewidth Schawlow and Townes (1958). However, semiconductor lasers
are known to possess a much larger linewidth Osinski and Buus (1987). This was
explained as a consequence of an asymmetric spectral shape of the available
optical gain. Following the Kramers-Kronig relations, an asymmetric gain shape
will induce a non-zero value of the refractive index dispersion line at the
position of the gain peak. Since the gain depends on the carrier density, the
refractive index will change as the carrier density changes. The change of the
refractive index in a limited amount of time will enforce an additional phase
shift and an additional broadening of the linewidth which is much greater than
the one induced by the spontaneous emission. The dynamic coupling of the
refractive index and gain is described simply by the ratio of their respective
carrier-induced changes. This ratio is known as the linewidth enhancement
factor (LEF), or the $\alpha$-factor. In this work we will reveal for the
first time a measurement technique that is able to extract the value of LEF
over the whole spectrum of an operating laser.

We will start by analyzing the response of a single-mode semiconductor laser
biased above threshold with a superimposed weak modulation. The bias current
$J(t)$ is modulated sinusoidally as:

$\displaystyle\begin{split}J(t)=J_{0}+{\Delta}J\cos{(2\pi
f_{\mathrm{mod}}t)},\end{split}$ (5)

where $f_{\mathrm{mod}}$ is the modulation frequency. Due to the non-zero LEF
in semiconductor lasers, a modulation of the current will result in the
modulation of both the laser intensity and frequency. The optical field that
is emitted by the laser is the defined as:

$\displaystyle\begin{split}E(t)=&\sqrt{I_{0}}\sqrt{1+m\cos{(2\pi
f_{\mathrm{mod}}t+\phi)}}\\\ &\times\cos{\big{(}2\pi f_{0}t+\beta\sin{(2\pi
f_{\mathrm{mod}}t+\phi+\theta)}\big{)}},\end{split}$ (6)

where $I_{0}$ is the average intensity, $f_{0}$ is the lasing frequency, and
$m$ and $\beta$ are the intensity modulation (IM) and frequency modulation
(FM) indices, respectively. We include an additional arbitrary phase shift
$\phi$ with the respect to the current modulation and an FM-IM phase shift
$\theta$. Following Eq. (6), the frequency is modulated as $f=f_{0}+\beta
f_{\mathrm{mod}}\cos(2\pi f_{\mathrm{mod}}t+\phi+\theta)$. For the following
calculation it is of great use the complex field notation and assume weak
modulation ($m,\beta\ll 1$):

$\displaystyle\begin{split}E(t)=&\sqrt{I_{0}}{\big{(}1+\frac{m}{2}\cos{(2\pi
f_{\mathrm{mod}}t+\phi)}\big{)}}\\\ &\times\mathrm{e}^{i2\pi
f_{0}t+i\beta\sin{(2\pi f_{\mathrm{mod}}t+\phi+\theta)}}.\end{split}$ (7)

Since the modulation of $E(t)$ is a periodic function with frequency
$f_{\mathrm{mod}}$, we can expand Eq. (7) in an infinite Fourier series of
harmonic terms that oscillate at frequencies
$f_{0}+nf_{\mathrm{mod}},n\in\mathbb{Z}$. It follows:

$\displaystyle
E(t)=\sqrt{I_{0}}\sum_{n=-\infty}^{+\infty}E_{n}\exp{\big{(}2\pi(f_{0}+nf_{\mathrm{mod}})t\big{)}},$
(8)

where the complex amplitudes $E_{n}$ are to be determined. This is feasible by
employing the Jacobi-Anger expansion Abramowitz and Stegun (1965):

$\displaystyle\mathrm{e}^{ix\sin{y}}=\sum_{n=-\infty}^{+\infty}J_{n}(x)~{}\mathrm{e}^{iny},$
(9)

where $J_{n}(x)$ is the $n$-th Bessel function of the first kind. Combining
Eqs. (7) and (9), using the identity $J_{-n}(x)=(-1)^{n}J_{n}(x)$ for
$n\in\mathbb{Z}$, and keeping only the orders $n=-2,-1,0,1,2$ yields:

$\displaystyle\begin{split}E(t)&=\sqrt{I_{0}}~{}\mathrm{e}^{i2\pi
f_{0}t}\Big{(}1+\frac{m}{4}\mathrm{e}^{i(2\pi f_{\mathrm{mod}}t+\phi)}\\\
&+\frac{m}{4}\mathrm{e}^{-i(2\pi
f_{\mathrm{mod}}t+\phi)}\Big{)}\Big{[}J_{2}(\beta)~{}\mathrm{e}^{-i2{(2\pi
f_{\mathrm{mod}}t+\phi+\theta)}}\\\ &-J_{1}(\beta)~{}\mathrm{e}^{-i{(2\pi
f_{\mathrm{mod}}t+\phi+\theta)}}+J_{0}(\beta)\\\
&+J_{1}(\beta)~{}\mathrm{e}^{i{(2\pi
f_{\mathrm{mod}}t+\phi+\theta)}}+J_{2}(\beta)~{}\mathrm{e}^{i2{(2\pi
f_{\mathrm{mod}}t+\phi+\theta)}}\Big{]}.\end{split}$ (10)

Comparison of Eqs. (8) and (10) yields the expressions for the complex
amplitude of the centerline $E_{0}$ and its first modulation sidebands $E_{\pm
1}$:

$\displaystyle\begin{split}E_{0}&=1\\\ E_{\pm 1}&=\mathrm{e}^{\pm
i(\phi+\theta)}\Big{(}\pm
J_{1}(\beta)+\frac{m}{4}\big{(}J_{2}(\beta)\mathrm{e}^{\pm
i\theta}+J_{0}(\beta)\mathrm{e}^{\mp i\theta}\big{)}\Big{)}.\end{split}$ (11)

We can now also employ the weak modulation approximation so that
$J_{0}(\beta)\approx 1$, $J_{1}(\beta)\approx\frac{\beta}{2}$, and
$J_{2}(\beta)\approx 0$ holds. By finding the beatings between the centerline
and its first sidemodes, we can directly extract the modulation indices $m$
and $\beta$. The beating signals $B_{\pm 1}$ are defined as:

$\displaystyle\begin{split}B_{1}&=E_{1}E^{*}_{0}=\mathrm{e}^{i(\phi+\theta)}\big{(}\frac{\beta}{2}+\frac{m}{4}\mathrm{e}^{-i\theta}\big{)}\\\
B_{-1}&=E_{0}E^{*}_{-1}=\mathrm{e}^{i(\phi+\theta)}\big{(}-\frac{\beta}{2}+\frac{m}{4}\mathrm{e}^{-i\theta}\big{)}.\end{split}$
(12)

The LEF can be calculated from the modulation indices, as is known from
literature Harder  _et al._ (1983); Osinski and Buus (1987); Aellen  _et al._
(2006); Hangauer and Wysocki (2015):

$\displaystyle\mathrm{LEF}=2\frac{\beta}{m}.$ (13)

Combining Eqs. (12) and (13) finally yields the LEF as a function of the
beating signals:

$\displaystyle\mathrm{LEF}=\bigg{|}\frac{B_{1}-B_{-1}}{B_{1}+B_{-1}}\bigg{|},$
(14)

##  Appendix B Analytical expression for the LEF based on the complex
susceptibility

The spectral response of the laser active medium is fully described with its
complex optical susceptibility $\chi=\chi_{R}+i\chi_{I}$. Assuming that the
gain varies linearly with the carrier density, we can define $\chi(\omega)$ as
Agrawal and Dutta (1995):

$\displaystyle\chi(\omega)=\frac{\mu^{2}\Delta
N}{\varepsilon_{0}\hbar}\frac{(1+i\alpha)^{2}}{\omega-\omega_{0}-\frac{i}{T_{2}}},$
(15)

where $\omega$ is the angular frequency, $\omega_{0}=2\pi f_{0}$, $\mu$ is the
dipole matrix element, $\Delta N$ is the carrier density, and $T_{2}$ is the
dephasing time. The value of the LEF at the gain peak is $\alpha$. A non-value
$\alpha$-factor describes the asymmetric gainshape of the semiconductor
lasers.

The LEF is defined as the ratio of the carrier-induced change of the
refractive index and gain. As the optical gain is described by $\chi_{I}$ and
the refractive index is defined with $\chi_{R}$, we can write
$\mathrm{LEF}=-(\nicefrac{{\partial\chi_{R}}}{{\partial
N}})/(\nicefrac{{\partial\chi_{I}}}{{\partial N}})$. Under the assumption that
both the real and imaginary part of the susceptibility are linear functions of
the carrier density, we can remove the derivatives in the definition of the
LEF Agrawal and Dutta (1995):

$\displaystyle\mathrm{LEF}(\omega)=-\frac{\chi_{R}(\omega)}{\chi_{I}(\omega)}.$
(16)

Insertion of Eq. (15) in Eq. (16) yields the definition of the spectrally
resolved LEF as a function of the value at the gain peak $\alpha$, angular
frequency $\omega$ and the dephasing time $T_{2}$, which describes the
gainwidth. It is obtained:

$\displaystyle\mathrm{LEF}(\omega)=\frac{2\alpha-(1-\alpha^{2})(\omega-\omega_{0})T_{2}}{(1-\alpha^{2})+2\alpha(\omega-\omega_{0})T_{2}}.$
(17)

It is trivial to confirm that the LEF at the gain peak is equal to the
inserted $\alpha$-factor – $\mathrm{LEF}(\omega_{\mathrm{peak}})=\alpha$,
where $\omega_{\mathrm{peak}}=\omega+\alpha/T_{2}$ is the frequency at the
gain peak.

##  Appendix C Additional experimental data

Figure C1:  Experimental data of a SWIFTS measurement: Normalized intensity
interferogram (top), Normalized SWIFTS X and Y quadrature interferograms
(middle, bottom) with a zoom around the second burst indicated by the arrow
(right).

The recorded intensity interferogram as well as the X and Y trace of the
SWIFTS measurement can be seen in Fig. C1. The custom-built FTIR spectrometer
allows for a large mirror displacement, resulting in a substantially high
resolution in frequency space to clearly distinguish the beatings of the
central modes with the weak modulation sidebands.


# Towards Energy Efficiency in RAN Network Slicing

Hnin Pann Phyu  Département de Génie Logiciel et TI
École de Technologie Supérieure (ÉTS)
Montreal, Canada
<EMAIL_ADDRESS>Diala Naboulsi  Razvan Stanica  Département de
Génie Logiciel et TI
École de Technologie Supérieure (ÉTS)
Montreal, Canada
<EMAIL_ADDRESS>Univ Lyon, INSA Lyon, Inria, CITI
Villeurbanne, France
<EMAIL_ADDRESS>Gwenael Poitau This work was supported by the
National Natural Sciences and Engineering Research Council of Canada (NSERC)
through research grant RGPIN-2020-06050 and by the CHIST-ERA ECOMOME project,
through the Fonds de Recherche du Québec – Nature et Technologies (FRQNT).
Dell Technologies
Montreal, Canada
<EMAIL_ADDRESS>
###### Abstract

Network slicing is one of the major catalysts to turn future telecommunication
networks into versatile service platforms. Along with its benefits, network
slicing is introducing new challenges in the development of sustainable
network operations. In fact, guaranteeing slices requirements comes at the
cost of additional energy consumption, in comparison to non-sliced networks.
Yet, one of the main goals of operators is to offer the diverse 5G and beyond
services, while ensuring energy efficiency. To this end, we study the problem
of slice activation/deactivation, with the objective of minimizing energy
consumption and maximizing the users quality of service (QoS). To solve the
problem, we rely on two Multi-Armed Bandit (MAB) agents to derive decisions at
individual base stations. Our evaluations are conducted using a real-world
traffic dataset collected over an operational network in a medium size French
city. Numerical results reveal that our proposed solutions provide
approximately 11-14% energy efficiency improvement compared to a configuration
where all the slice instances are active, while maintaining the same level of
QoS. Moreover, our work explicitly shows the impact of prioritizing the energy
over QoS, and vice versa.

###### Index Terms:

5G, Network Slicing, Energy Efficiency, QoS

##  I Introduction

The telecommunication industry accounts for approximately 2% of total global
carbon emissions [1]. By 2030, 8% of the projected global electricity demand
will come from the information and communications technology sector as a
whole, even in the best case scenario [2]. Energy consumption will continue
increasing in beyond 5G and 6G networks, where computationally intensive
services will be largely deployed. Although 5G equipment is more energy
efficient than 4G [3], with the data traffic volume increasing tremendously
along with 5G services, overall energy consumption will increase too. In fact,
the energy consumption of a 5G base station is three times higher than that of
a 4G base station, when both are considered at a full load [4].

5G is envisioned to serve a wide variety of services, with heterogeneous
traffic, through network slicing [5]. This is done by forming, on one physical
network, multiple virtual networks on a per-service basis, i.e., slices. That
said, slices requirements need to be met, including performance isolation.
Guaranteeing these requirements and the additional virtualization layer come
with some overhead, which produces higher energy consumption with respect to
non-sliced network deployments [6]. One of the key objectives in the field is
to offer this service differentiation, while reducing the associated CO2
emissions. Indeed, energy efficiency in networks is no longer an option but a
necessity. When delving into this topic, we observe that, today, the highest
amount of energy is consumed in the radio access network (RAN), approximately
70% of the overall network energy utilization [7].

To deal with this, several research works consider base station sleep schemes
to further optimize the energy consumption in 5G networks [8, 9]. While such
techniques show effective results, they are more challenging to be applied
directly in the case of multi-services network slicing environments. That is
mainly because slice instances can exhibit quite different temporal traffic
patterns. Completely shutting down or putting the entire base station into
sleep mode could notoriously impact the quality of service (QoS) of users in
specific slice instances. This motivates us to introduce a new approach, in
which slice instances are dynamically activated and deactivated, according to
their traffic patterns, thereby enhancing the overall base station energy
efficiency. However, deactivating some slice instances to minimize the energy
consumption can potentially degrade the QoS of users. On the other hand,
activating all slices all the time, so as to maximize QoS, significantly
increases energy consumption. Accordingly, the energy minimization objective
shall be coupled with a QoS maximization objective [10].

To manage the trade-off between the two objectives, operators may consider
using an EcoSlice, which is a slice instance with bare minimum resources and
network functions. By that, it incurs much lower energy consumption than
typical slice instances. The EcoSlice is up and running $24/7$ to provide a
bare-minimum service. In some conditions, e.g., low traffic demand, operators
may switch the users of other slices to this specific EcoSlice, without a
significant QoS impact.

In this regard, we study the problem of slice activation/deactivation, with
the objective of minimizing the energy consumption while satisfying the user
QoS. The contributions of our work are twofold. First of all, we propose two
different approaches for solving the problem, namely a Deep Contextual MAB
(DCMAB) algorithm and a Thompson Sampling Contextual (Thompson-C) algorithm.
These approaches allow to derive solutions dynamically over time, while
considering traffic patterns of individual slice instances deployed at a base
station. Moreover, our proposed agents enable operators to navigate users to
and from an EcoSlice, if their requested slice instance is
activated/deactivated. Second, we evaluate the performance of the proposed
approaches and their computational cost using a real-world traffic dataset.

The rest of the paper is organized as follows. Section II discusses the
related work of energy efficiency in network slicing. Then, the network model
and problem statement are laid out in Section III. In Section IV, we present
the detailed design of our proposed solutions. We articulate the results in
Section V and conclude the paper in Section VI.

##  II Related Work

With the aim of enabling energy efficiency, several research works consider
optimizing the allocation of network slice resources (i.e., radio, CPU,
transmission bandwidth and power) in the different domains (i.e., RAN, Edge
Computing (EC), Core Network (CN) and end-to-end network). At the RAN slicing
level, [11] combines deep learning (DL) and reinforcement learning (RL) on a
distributed framework to efficiently allocate radio and transmission power
resources over base stations. They use stacked and bidirectional Long-Short
Term Memory (SBiLSTM) to predict the per slice resources demand on a large
time scale and rely on asynchronous advantage actor-critic (A3C) to allocate
resources to users on a small time scale. Their proposed framework achieves
higher energy efficiency than baselines using static power allocation.

In [12], the authors optimize the energy consumption and computation cost in a
network slicing based Cloud-RAN (C-RAN) setting, using a twin-delayed double-Q
soft Actor-Critic (TDSAC) approach. Their agent performs the up/down scaling
of computing and beamforming power resources. Their work outperforms other
baseline RL models in terms of overall network energy and computing cost.
Similarly, [6] designs a slice energy consumption model based on the C-RAN
architecture. An optimisation problem is solved per-slice, with the objective
of minimizing the overall network energy cost, jointly considering
communication and computation resources. This approach improves energy
efficiency over a baseline focusing only on radio resources.

Focusing on CN slicing, the authors in [13] formulate a security-aware network
slicing optimization problem to enhance the energy efficiency of CN nodes.
They limit themselves to static resource allocation. Their proposed solution
provides more power savings than a greedy approach.

Considering an obvious trade-off, it is sensible to couple QoS maximization
and energy consumption minimization. Therefore, focusing on end-to-end network
slicing, the authors in [14] aim to maximize the energy efficiency while
respecting service level agreement (SLA) constraints. To this end, they rely
on statistical federated learning (stFL). Their federated local agents
coordinate and predict per slice network metrics, without transferring
datasets to a central unit, and largely outperform other federated learning
and centralised solutions.

While prior works attempt to achieve the energy efficiency as well as ensuring
QoS, we believe there is still room to further optimize the energy efficiency
by switching off some of the underutilized slice instances, under some
conditions. As of our knowledge, there is no contemporary work studying this
problem. In this light, we introduce the RAN slice activation/deactivation
problem with the aim of minimizing energy consumption and maximizing QoS. To
this end, we rely on the fully decentralized state-aware MAB approaches to
enable decisions for slice instances at each base station, while considering
the impact on energy and QoS factors.

##  III System Model and Problem Statement

We study the problem of slice instance activation and deactivation at
individual base stations with the aim of minimizing energy consumption and
maximizing QoS. We thus explicitly lay out the system model, energy
consumption and user QoS model, needed as part of our defined problem. We then
formalise the main objective of our problem. We formulate the latter as a
Markov decision process (MDP), and re-design it after that as a contextual MAB
problem.

###  III-A System Model

We consider a time-slotted system. Accordingly, we define $\tau$ as the slice
activation/deactivation interval (SADI), where slices are active or inactive
continuously over the period of that interval. Accordingly,
activation/deactivation decisions are made at the end of every $\tau$, for the
upcoming $\tau+1$. The period of the SADI is defined based on the operator
policies. Besides, we define $T$ as a time interval of interest, such that
$\tau\in T$. Figure 1 illustrates the time frame consideration in our proposed
model. In this example figure, we consider four different types of slices:
three of them, denoted as Enhanced Mobile Broadband (eMBB), Ultra-Reliable Low
Latency Communication (URLLC) and Massive Machine-Type Communication (mMTC),
provide services with different QoS levels and they can be
activated/deactivated as needed, as well as the EcoSlice which is always up
and running.

Figure 1: Slice Activation-Deactivation Interval (SADI) illustration.

We define $\mathcal{I}_{b}$ as a set of slice instances attached to base
station $b\in\mathcal{B}$. We define $\mathcal{U}_{b}^{\tau}$ as the set of
users served by base station $b$ at $\tau$. Accordingly, $U_{i,b}^{\tau}$ is
the set of users that can be served by slice instance $i$ of base station $b$
at $\tau$. Thus,
$\mathcal{U}_{b}^{\tau}=\bigcup_{i\in\mathcal{I}_{b}}U_{i,b}^{\tau}$. Each
slice instance $i\in\mathcal{I}_{b}$ is characterized by a specific QoS class
identifier (QCI) [15] and its energy consumption $E_{i,b}^{\tau}$.

Without loss of generality, we consider user delay as an indicator of the QoS,
but any other metric could be easily integrated instead. In the optimal slice
instance activation scheme, underutilized slice instances are switched off to
save energy when certain conditions are met. Consequently, the user-perceived
delay is prone to deteriorate. In this light, we deliberately study the impact
of optimal slice instance activation on both energy and delay. Hence, we
define $\delta_{i}$ as a predefined achievable delay for a slice instance $i$.
We also consider an EcoSlice instance $i_{e}\in\mathcal{I}_{b}$, to which
users are switched when their requested slice instances are inactive. The
EcoSlice instance $i_{e}$ also has a predefined achievable delay
$\delta_{i_{e}}$ and its energy consumption over ${\tau}$ is denoted as
$E_{i_{e},b}^{\tau}$.

At every $\tau$, each user $u\in\mathcal{U}_{b}^{\tau}$ makes a specific
request, characterized by a delay requirement $d_{u}^{\tau}$ and a traffic
flow demand $l_{u}^{\tau}$. Each base station $b$ has a set of possible
configurations $\mathcal{K}_{b}$. Each configuration $k\in\mathcal{K}_{b}$
implies activation/deactivation decisions for some slices. In detail, $k$
contains $\left\\{c_{i}^{\tau}|i\in\mathcal{I}_{b}\right\\}$. Here,
$c_{i}^{\tau}=1$ if slice instance $i$ is active at $b$ during $\tau$, and 0
otherwise. Needless to say, $c_{i_{e}}^{\tau}=1$ for any $\tau$.

###  III-B Energy Consumption Model

We define the function $f(.)$ to refer to the overall energy consumption of a
base station. This energy model can be further fine-tuned based on specific
operator resource management policies and activated RAN energy saving
features. It is composed of the energy consumption resulting from its
individual deployed slice instances $E_{i,b}^{\tau}$ and the static energy
consumption of the base station $P_{b}^{static}$ (i.e cooling and circuit
power):

$f_{b}(c_{i}^{\tau},\mathcal{I}_{b})=\sum_{i\in\mathcal{I}_{b}}c_{i}^{\tau}\cdot
E_{i,b}^{\tau}+P_{b}^{static}$ (1)

with

$E_{i,b}^{\tau}=\rho_{i,b}^{\tau}\psi_{i}P_{b}^{dynamic}+\psi_{i}P_{b}^{fixed}$
(2)

As indicated in the above equation, the energy consumption of slice instances
for a base station $b$ encompasses a load-dependent power consumption
component, $P_{b}^{dynamic}$, and a load-independent power consumption
component, $P_{b}^{fixed}$. Specifically, as the name implies
$P_{b}^{dynamic}$ depends on the traffic load of the base station. It is worth
stressing that a slice instance consumes some power even at zero load traffic,
in order to run its corresponding network functions. Thereupon,
$P_{b}^{fixed}$ is independent of the traffic load, but related to the energy
consumption of associated network functions of specific slice instances.

We note here that not all slice instances are designed equally [16]. Their
required network functions and signalling traffic are different [17]. For
instance, an URLLC service potentially consumes higher energy, because it
requires specific network functions to offer high reliability and very low
latency [18]. In short, the more stringent the latency requirements of the
service, the higher its consumed energy [19]. Regardless of their requirements
for energy, both mMTC and eMBB have flexibility in terms of latency. It is
fair to say that, even under the same amount of traffic load, the energy
consumption of each service is different. Meanwhile, the EcoSlice is deployed
with relatively low energy consumption. It is sensible to conclude that
different slice types consume different amounts of energy not only because of
their traffic portion but also because of their service attributes [20].

In this vein, $\psi_{i}$ denotes the power consumption impact factor of slice
instance $i$ on both $P_{b}^{dynamic}$ and $P_{b}^{fixed}$. We let the
operators define the value of $\psi_{i}$ based on their slice instances
configurations and analytics. Having said that, based on the prior
explanation, it is pragmatic to assume that URLLC has larger $\psi$ value than
eMBB and mMTC. Needless to say, $\psi$ value of the EcoSlice is the lowest.
Besides, $\rho_{i,b}^{\tau}$ is the traffic load portion of associated slice
instance $i$ of base station $b$ during ${\tau}$. For this, we can simply
obtain $\rho_{i,b}^{\tau}$ by dividing the traffic load over slice
$l_{i,b}^{\tau}$ by the total base station traffic load, as below:

$\rho_{i,b}^{\tau}=\frac{l_{i,b}^{\tau}}{\sum_{i\in I_{b}}l_{i,b}^{\tau}}$ (3)

###  III-C User QoS Model

As explained, our objective is coupled with ensuring the user QoS. In this
regard, we define the user satisfaction factor $\eta_{u}^{\tau}$ for each user
$u$ at $\tau$ associated to slice instance $i$ as follows:

$\eta_{u}^{\tau}=\begin{cases}1&\text{if}\ \delta_{i}\leq d_{u}^{\tau}\\\
0&\text{ otherwise }\end{cases},u\in
U_{i,b}^{\tau},i\in\mathcal{I}_{b},\tau\in T$ (4)

where $d_{u}^{\tau}$ is the delay requirement of user $u$ at time $\tau$.
Consequently, the average per slice QoS at base station $b$ is defined as:

$\eta_{i,b}^{\tau}=\frac{\sum_{u\in
U_{i,b}^{\tau}}\eta_{u}^{\tau}}{|U_{i,b}^{\tau}|}$ (5)

We then compute the average QoS of the base station $b$ considering all the
associated slice instances during $\tau$:

$\eta^{\tau}_{b}=\frac{\sum_{i\in\mathcal{I}_{b}}\eta_{i,b}^{\tau}}{|\mathcal{I}_{b}|}$
(6)

###  III-D Objective Function

Given the system model and utility models mentioned in the preceding sections,
the objective of our slice activation/deactivation problem can be expressed
as:

$max\
\sum_{\tau=1}^{T}\left[\frac{1}{f_{b}(c_{i}^{\tau},\mathcal{I}_{b})}+\eta^{\tau}_{b}\right]$
(7)

As one can see in Equation 7, the objective function is influenced by the
time-varying user demand and active slice instances. Thus, we believe that RL-
based approaches are best-suited for this problem, as they enable complex
decision-making without requiring an explicit modeling of the network
environment [21]. In what follows, we formulate the problem as MDP and
contextual MAB.

###  III-E Markov Decision Process (MDP)

An MDP is defined by a tuple $(\mathcal{S},\mathcal{A},P,\mathcal{R})$.
$\mathcal{S}$ denotes the set of states in the system. Representing states as
feature vectors helps the RL agent converge to a near-optimal reward value,
but one can also simplify a problem with a less complex state representation
(which might converge to a similar reward, with less computation). In this
light, we explore two definitions for a state $s$ in this problem: _i)_ energy
consumption and QoS of base station in the previous SADI:
$s=\left\\{f_{b}(c_{i}^{\tau-1},\mathcal{I}_{b}),\eta^{\tau-1}_{b}\right\\}$
and _ii)_ simply the SADI identifier: $s=\left\\{\tau\right\\}$. The reason
for the latter definition is that, since the user demand depends on time and
it shows a significant periodicity, a simple state representation based only
on the SADI might already contain enough information for the RL agent. Each
action $a\in\mathcal{A}$ is a configuration $k$, as defined in Section III-A,
and $\mathcal{A}$ is the set of possible configurations:
$\mathcal{A}=\mathcal{K}_{b}$.

In the MDP model,
$P:\mathcal{S}\times\mathcal{A}\times\mathcal{S}\rightarrow\left[0,1\right]$
captures the stochastic transition probability function to transition to
$s^{\prime}$ from state $s$ based on action $a$, with
$\sum_{s^{\prime}\in\mathcal{S}}P(s^{\prime}|s,a)=1$ for all $s\in\mathcal{S}$
and $a\in\mathcal{A}$. $P$ is unknown to an RL agent. However, it occurs that
if an agent knows the current state and the reward obtained in each iteration,
it can still converge to the optimal solutions through RL approaches [22].
Accordingly, the formulation of reward function is critical for the RL agent
to be able to learn the optimal policy and it usually boils down to the main
objectives of the problem. We therefore define reward $r\in\mathcal{R}$ as:

$r(s,a,s^{\prime})=\frac{1}{f_{b}(c_{i}^{\tau},\mathcal{I}_{b})}+\beta\cdot\eta^{\tau}_{b}$
(8)

As shown in Equation 8, our reward function is aligned with our objective
function. Besides, we define $\beta$ as a QoS impacting factor on the reward.
In short, the larger the $\beta$ value is, the more the QoS is emphasized with
respect to the energy consumption of the base station. If $\beta=1$, the
objective is to find the trade-off between energy and QoS.

Due to the nature of our defined problem, we observe two key points in our MDP
representation: _i)_ the transition probability of MDP can be simplified, such
as $P(s^{\prime}|s,a)\equiv P(s^{\prime}|a)$, where states are independent of
each other, and _ii)_ unlike typical MDP [12], our reward function depends
only on current state and action, but not on the successor state. With this,
the reward function could be simplified as $r(s,a,s^{\prime})\equiv r(s,a)$.
Based on these observations, we present an equivalent formulation as a state-
aware MAB in the following.

###  III-F Multi-armed Bandit (MAB)

In this section, we formulate our slice instance activation/deactivation
problem as a state-aware MAB. Formally, state-aware MAB is tupled with
$(\mathcal{S},\mathcal{A},\mathcal{R})$. Same as in our MDP model, we use the
two different state definitions for $s\in\mathcal{S}$: energy consumption and
QoS observed over a SADI, or simply the SADI identifier. Similarly, the set of
actions $a\in\mathcal{A}$ is the set of available configurations.

We then define the associated reward set $\mathcal{R}$. Since we consider the
state-aware MAB (i.e., involving multiple states), our reward distribution is
non-stationary, and changes based on the state $s$ (also called context in the
following). With this, the reward set can be defined as
$\mathcal{R}=\left\\{r(s,a)|\ a\in\mathcal{A},s\in\mathcal{S}\right\\}$. In
this regard, we rely on the same reward calculation as Equation 8. Needless to
say, the objective here is to maximize the expected reward $E\left[\sum
r(s,a)\right]$.

To evaluate our reward function, one standard approach is to compete with the
best-action benchmark. On the other hand, we compute the regret resulting from
not selecting the optimal action at each iteration. That said, one would
define the cumulative regret incurred by an agent over a total of $J$ time
steps as [23]:

$Regret(J)=\sum_{j=1}^{J}(r_{j}^{*}(s,a)-r_{j}(s,a_{j}))$ (9)

where $r_{j}^{*}(s,a)$ is the best-action benchmark at round $j$ and can be
obtained via $r_{j}^{*}(s,a)=\underset{a}{max}\ r_{j}(s,a)$, and $a_{j}$ is
the action selected by the agent in round $j$. It is worth mentioning that the
regret function is non-negative, as it compares the optimal reward to the
actual reward obtained by the agent.

##  IV Proposed Solution

In this section, we lay out our two fully decentralized approaches, based on
DCMAB and Thompson-C agents.

###  IV-A DCMAB Agent

As explained, we use two different context/state definitions and thus we have
two types of DCMAB agents: _i)_ DCMAB-EQ where the state is the overall energy
and QoS over the base station:
$s=\left\\{f_{b}(c_{i}^{\tau-1},\mathcal{I}_{b}),\eta^{\tau-1}_{b}\right\\}$,
and _ii)_ DCMAB-SADI when the state is the SADI: $s=\left\\{\tau\right\\}$.
Due to space limitation, we outline them together in Algorithm 1. However, we
make sure the main differences (occuring at Line 3 and Line 11 of Algorithm 1)
are clearly outlined.

Input : Probability of selecting a random action $\epsilon$, learning rate
$\alpha$, maximum time steps $J$

Output: $\widehat{R}(\tilde{w})$

1 Initialize $\tilde{w}$ randomly and $j=0$

2 repeat

3 Observe context/state: $s$ \- based on state definition
$s=\left\\{f_{b}(c_{i}^{\tau-1},\mathcal{I}_{b}),\eta^{\tau-1}_{b}\right\\}$
for DCMAB-EQ or $s=\left\\{\tau\right\\}$ for DCMAB-SADI

4 Predict Reward Distribution for each action:
$\left[\widehat{R}_{\tilde{w}}(a|s)\right]_{a\in A}$

5 if  _generate random probability: $rand()\ <\ \epsilon$_ then

6 Select a random action $a$

7

8 else

9 Select $a=\underset{a\in
A}{argmax}\left[\widehat{R}_{\tilde{w}}(a|s)\right]$

10

11 end if

12 Evaluate reward $r(a|s)$

13 Update new state $s^{\prime}$

14 Calculate the loss:
$\mathcal{L}(\tilde{w})\overset{\Delta}{=}(r(a|s)-\left[\widehat{R}_{\tilde{w}}(a|s)\right]_{a})^{2}$

15 Update the weights:
$\tilde{w}\leftarrow\tilde{w}-\alpha\bigtriangledown\mathcal{L}(\tilde{w})$

16 $j\leftarrow j+1$

17until  _$j >J$_;

18

Algorithm 1 DCMAB-EQ and DCMAB-SADI

The inputs of the DCMAB agent consist of the probability of selecting a random
action $\epsilon$, the learning rate $\alpha$ for the deep neural network
(DNN) model, and maximum time steps $J$ to train the DCMAB agent. The output
is a trained model, which can predict a reward distribution
$\widehat{R}(\tilde{w})$ for the available actions of the associated states.
With this, the algorithm begins by the initialization of the weights
$\tilde{w}$ with arbitrary values and the variable $j$ referring to an
iteration and starting with zero (Line 1). At each step, the agent observes a
context $s$: for DCMAB-EQ , $s$ is the overall energy consumption and QoS
factor and for DCMAB-SADI, $s$ is a SADI identifier (Line 3). Then, it
predicts the reward distribution for all the actions (Line 4) for a given
context. After that, an action is selected by considering the exploration and
exploitation tradeoffs (Line 5 - Line 9). Precisely, the random action is
selected with probability $\epsilon$, and otherwise the action giving the
maximum reward is selected. Next, the chosen action is applied to the defined
network environment, which, after the potential reconfiguration, returns an
actual reward (calculated using Equation 8) (Line 10). Accordingly, the new
state $s^{\prime}$ is obtained based on the action $a$ for DCMAB-EQ, or new
state $s^{\prime}$ is simply the next SADI (which is independent of the
previous action $a$) for DCMAB-SADI (Line 11). Then, the loss between the
predicted reward and the actual reward is computed for the purpose of model
training (Line 12). We rely on a gradient descent method to update the weights
matrix of the DNN with a learning rate $\alpha$ (Line 13). Then, we go for
another iteration (Line 14) and the above process is repeated for a maximum
number of time steps $J$ (Line 15).

Input : Context data $\mathbb{C}=(d_{k})_{|T|\times|\mathcal{K}_{b}|}$, number
of dimensions of context/state vector $z=|T|$, $\varphi$, $M$, maximum time
steps $J$, $\sigma=M\sqrt{9zln(\frac{J}{\varphi})}$

Output: $\mathcal{N}\left(\hat{\mu},\sigma^{2}D^{-1}\right)$

1 Initialize $D=\mathbb{I}_{z}$ , $\hat{\mu}=(0)_{z}$ , $j=0$

2 repeat

3 Sample $\tilde{\mu}$ from Gaussian distribution
$\mathcal{N}\left(\hat{\mu},\sigma^{2}D^{-1}\right)$

4 Select action $a=\underset{k}{argmax}\ d_{k}^{\mathbb{T}}\tilde{\mu}$

5 Observe reward $r_{a}$

6 $D=D+d_{a}d_{a}^{\mathbb{T}}$

7 $\hat{\mu}=D^{-1}\left(d_{a}r_{a}\right)$

8 $j\leftarrow j+1$

9

10until  _$j >J$_;

11

Algorithm 2 Thompson-C

###  IV-B Thompson-C Agent

Unlike DCMAB, the Thompson-C agent adopts a statistical approach with the goal
of achieving a proper estimation of the posterior distribution of expected
reward for each action. The Thompson-C agent (Algorithm 2) operates as
follows. We note that we only consider the SADI identifier as context/state
for the Thompson-C agent. Accordingly, the inputs of the algorithm are the
context data for all the actions:
$\mathbb{C}=(d_{k})_{|T|\times|\mathcal{K}_{b}|}$, where $d_{k}$ is a context
vector for configuration $k$, the number of dimensions of the context vector
$z=|T|$, the tunable parameters $\varphi$ and $M$ (that can be tuned by the
operator as needed) and the maximum number of time steps $J$ to run the
Thompson-C. The output is a posterior distribution
$\mathcal{N}\left(\hat{\mu},\sigma^{2}D^{-1}\right)$ of having the optimal
parameter $\hat{\mu}$. For better understanding, $\hat{\mu}$ can be regarded
as a weighted vector for a z-dimensional context/state. The parameter $\sigma$
can be obtained via $\sigma=M\sqrt{9zln(\frac{J}{\varphi})}$.

The algorithm begins by setting the parameter $D$ as the identity matrix with
a z-dimensional vector. $\hat{\mu}$ is initialized with zeros as a
z-dimensional vector and $j=0$ (Line 1). In each time step $j$ (Line 2), the
Thompson-C agent samples a parameter $\tilde{\mu}$, from the posterior
distribution $\mathcal{N}\left(\hat{\mu},\sigma^{2}D^{-1}\right)$ (Line 3). It
then selects an action that yields the best sample (Line 4) and observes an
associated reward (Line 5). Finally, the parameters $D$ and $\hat{\mu}$ are
updated (Line 6 and Line 7). Then, we go for another iteration (Line 8). The
above process is repeated for a maximum number of time steps $J$ (Line 9).

##  V Evaluation

In this section, we evaluate the performance of our proposed solutions. We
start with a description of the dataset and simulation environment.
Afterwards, we explain the benchmark approaches and implementations that we
use. Finally, we discuss the overall results.

TABLE I: LIST OF PARAMETERS

Parameter Value $P_{b}^{static}$ [24] 18 Watts $P_{b}^{fixed}$ [25] 139 Watts
$P_{b}^{dynamic}$ [25] 742 Watts $\psi_{i}$ [Facebook, YouTube, Google,
EcoSlice] [1.2, 1.6, 1.4, 1] $\delta_{i}$ [Facebook, YouTube, Google,
EcoSlice] [10, 1, 15, 11] ms Number of users per slice [11-30] Users delay
requirement $d_{u}^{\tau}$ Facebook: [11-20]ms YouTube: [6-17]ms Google:
[16-25]ms Loss function MSE Learning rate $\alpha$ 0.001 $\beta$ [5,1,0.8]
Maximum episodes $J$ 1000 $\varphi$ 0.5 $M$ 0.01

###  V-A Dataset and Simulation Setup

((a)) Reward.

((b)) Regret.

Figure 2: Reward and regret obtained for different $\beta$ values.

We evaluate the proposed solutions using a real-world dataset collected from
the Orange 4G network, in Poitiers, France. The dataset includes mobile data
traffic demand of different mobile applications at a base station level. We
assume slice instances are deployed on an application-basis, i.e., one
application maps to one slice instance. More precisely, Facebook, YouTube, and
Google have been considered as three different types of slice instances
attached to each base station. It is worth mentioning that those three
different applications exhibit very different traffic demands, which we
considerate appropriate for the simulation of network slicing [26]. On the
user side, we assume stochastic delay requirements of users for each
application, as indicated in Table I.

The granularity of the dataset is 10 minutes, for 10 days in May 2019. With
this, for our simulation purposes, we consider $T$ is 10 days and SADI
$\tau=10\ minutes$. Thus $T$ includes 1440 $\tau$. We analyse 10 base stations
from the Poitiers city center, where different slice instances are associated.
We then apply our proposed solutions using an action set where each action
implies the activation/deactivation of one of the slices:
$\left[\text{Facebook, YouTube, Google, EcoSlice}\right]$.

Figure 3: Energy improvement over AllActive Figure 4: QoS based on different
$\beta$ values

###  V-B Benchmarks and Implementation Setup

We compare the performance of our proposed DCMAB and Thompson-C solutions with
three counterparts: Thompson Sampling Non-Contextual (Thompson-NC), AllActive
and Random. Unlike Thompson-C, no state/context information is considered in
Thompson-NC [27]. For AllActive, as the name implies it, all the slice
instances are active. And a random action is selected at each iteration for
the Random approach.

For the implementation, we rely on the Pytorch framework for the DCMAB agents.
All the models (i.e. DCMAB, Thompson-C, Thompson-NC, AllActive and Random) are
implemented using a Python environment and trained on the high-performance
Linux server provided by Digital Research Alliance of Canada. The DNN of a
DCMAB agent is composed of three fully-connected layers of 100 neurons,
followed by a RELU activation function for each. The detailed parameters are
summarized in Table I.

###  V-C Results

((a))

((b))

((c))

((d))

((e))

Figure 6: Comparisons of Thompson-C Vs Thompson-C (w/o EcoSlice)

####  V-C1 Overall Agents Performances

First of all, to fully comprehend the performance of our agents, we study the
trends of reward, regret, QoS and energy for $\beta=[5,1,0.8]$. We note that
agents focus on QoS when $\beta=5$, search for a trade-off when $\beta=1$, and
stress on energy when $\beta=0.8$. Therewith, we compare the reward trends of
our proposed solutions and their peers in Figure 2. The curves are smoothed by
averaging within a rolling window of 50 iterations.

In Figure 2a, the DCMAB-EQ and Thompson-C agents exhibit the superior reward,
even significantly better than AllActive, followed by DCMAB-SADI and Thompson-
NC. Random approaches failed our proposed solutions in every scenario.
Noticeably, AllActive shows inferior performance to that of the other agents
in general. We also perceive the same behavior for the regret trends for all
the agents in Figure 2b.

####  V-C2 Roles of Agents in Energy and QoS

We then explicitly verify if our proposed solutions are feasible for energy
optimization in RAN slicing by comparing them with the baselines. In this
regard, as depicted in Figure 3, compared with the AllActive strategy
(currently the standard approach), the energy improvement of Thompson-C is
approximately 24%, 18% and 14% respectively, for $\beta$ equals 0.8, 1 and 5.
As expected, the energy gain deteriorates when $\beta$ value increases. The
prior phenomenon holds for all the agents (except Random). Also, we saw a
great deal of energy gains over AllActive for DCMAB-EQ, DCMAB-SADI, Thompson-
NC and even Random as well. Overall, we notice the Thompson-C agent is quite
superior to others in terms of energy gain. Conversely, Thompson-NC has
slightly lower energy gains than the DCMAB agents for all $\beta$ values.

We are all aware that optimizing energy consumption means compromising QoS to
an extent. In this light, we visualize the QoS of all the agents in Figure 4.
We note that Thompson-C exhibits a comparable performance with its fellows in
terms of QoS, but a slightly lower performance when $\beta=0.8$. This is
linked to Thompson-C showing highest energy gain at $\beta=0.8$. It is
observed in Figure 4 that our proposed solutions satisfy almost 100% QoS at
$\beta=5$ (slightly lower for DCMAB-SADI). Therewith, it is at $\beta=5$ where
our agents make themselves stand out, as they deliver the same QoS as
AllActive, while providing significant improvement in energy efficiency.
Regardless of showing acceptable performance, the Thompson-NC agent shows
lower performance than our proposed solutions, comforting our modelling
choices. We can not help but stress that: _i)_ state-aware agents outperform
the one in which context/state is not considered, and _ii)_ a DNN approach is
constantly surpassed by a statistical approach.

####  V-C3 Impact of EcoSlice

To grasp the benefits EcoSlice, we examine in Figure 6 the performance of the
Thompson-C agent with and witout an EcoSlice, in terms of reward, regret, QoS
and energy utilization. We select the Thompson-C algorithm here as it shows
the best results in most of the scenarios compared to its fellows. As we can
observe, Thompson-C demonstrates better performance under the different
metrics when compared to Thompson-C (w/o EcoSlice). Therefore, an EcoSlice
significantly enhances the overall energy efficiency of the network by
allowing operators to switch off the underutilized slice instances and yet
ensure QoS. Without the assistance of an EcoSlice, one can not reach the level
of energy efficiency that we have accomplished.

Figure 7: Computing time comparisons

####  V-C4 Computing Time Comparisons

The detailed computing time comparisons conducted on the Digital Research
Alliance of Canada servers are shown in Figure 7. Paying the price for its
performance, Thompson-C takes much longer computing time than all the other
agents. Despite Thompson-C being considered to be a better agent than DCMAB-
EQ, it is not a good option for the real-time decision-making process, at
least not with a SADI of 10 minutes, as we consider. On the other side, the
DCMAB agents, who also outperformed the baselines in terms of energy
efficiency, are on par with Thompson-NC in terms of computing power. There is
no single answer here, and Thompson-C can be a favourable solution for a
system without computing time constraints. However, all the solutions we
proposed in this work offer avenues for operators to optimize the energy
efficiency by slice activation/deactivation while controlling the impact on
QoS. Operators can also opt between different design choices by varying
$\beta$, based on their targets and limitations.

Figure 8: MSE comparisons of DCMAB-EQ Vs DCMAB-SADI

####  V-C5 MSE of DCMAB

Last but not least, we also compare the MSE of the two DNN-based solutions,
DCMAB-EQ and DCMAB-SADI, in Figure 8. As we can see, DCMAB-EQ has a stable
training process with lower MSE than DCMAB-SADI. This explains the superior
performance of DCMAB-EQ in the prior results, as DCMAB-EQ predicts better
reward distribution of associated actions. In Figure 8, MSE results are shown
for $\beta=5$ only, but we noticed similar behaviour for other tested values.

##  VI Conclusion

In this paper, we focus on the slice activation/deactivation problem, to
further enhance the energy efficiency in RAN slicing. To this end, we advocate
the state-aware MAB approaches (i.e., DCMAB and Thompson-C), where an agent
attempts to activate the optimal slice instances while providing guaranteed
QoS. More than anything else, we investigate the important aspect of the
compromise between energy consumption and user QoS. The results are derived
based on a real-world datasets and demonstrate that the MAB approach in
general, and DCMAB and Thompson-C in particular, are appropriate for the slice
activation/deactivation problem. They significanlty alleviate the energy
consumption at the base station level while ensuring a satisfactory QoS level.

## References

  * [1] B. Cubukcuoglu, “The Importance of Environmental Sustainability in Telecom Service Providers’ Strategy,” _Risk, Reliability and Sustainable Remediation in the Field of Civil and Environmental Engineering_ , pp. 249–254, 2022.
  * [2] N. Jones, “The Information Factories,” _Nature_ , no. 561, pp. 163–167, 2018.
  * [3] NGMN, “Green Future Networks - Network Energy Efficiency,” NGMN, Tech. Rep., 2021.
  * [4] Y. Fan, B. Wang, J. Wei, M. Tan, and H. Ran, “A Hierarchical Distributed Operational Framework for Renewables-Assisted 5G Base Station Clusters and Smart Grid Interaction,” _Frontiers in Energy Research_ , vol. 10, pp. 1–4, June 2022.
  * [5] X. Foukas, G. Patounas, A. Elmokashfi, and M. K. Marina, “Network Slicing in 5G: Survey and Challenges,” _IEEE Communications Magazine_ , vol. 55, no. 5, pp. 94–100, 2017.
  * [6] M. Masoudi, O. T. Demir, J. Zander, and C. Cavdar, “Energy-Optimal End-to-End Network Slicing in Cloud-Based Architecture,” _IEEE Open Journal of the Communications Society_ , vol. 3, pp. 574–592, April 2022.
  * [7] N. Piovesan, D. López-Pérez, A. D. Domenico, X. Geng, H. Bao, and M. Debbah, “Machine Learning and Analytical Power Consumption Models for 5G Base Stations,” _IEEE Communications Magazine_ , vol. 60, no. 10, October 2022.
  * [8] F. Han, Z. Safar, and K. J. Liu, “Energy-Efficient Base-Station Cooperative Operation with Guaranteed QoS,” _IEEE Transactions on Communications_ , vol. 61, no. 8, pp. 3505–3517, 2013.
  * [9] M. Feng, S. Mao, and T. Jiang, “Base Station On-Off Switching in 5G Wireless Networks: Approaches and Challenges,” _Wireless Communications_ , vol. 24, no. 4, p. 46–54, January 2017.
  * [10] A. Chatzipapas, S. Alouf, and V. Mancuso, “On the Minimization of Power Consumption in Base Stations using On/Off Power Amplifiers,” _Proc. IEEE Online Conference on Green Communications (GreenCom)_ , pp. 18–23, 2011.
  * [11] Y. Azimi, S. Yousefi, H. Kalbkhani, and T. Kunz, “Energy-Efficient Deep Reinforcement Learning Assisted Resource Allocation for 5G-RAN Slicing,” _IEEE Transactions on Vehicular Technology_ , vol. 71, no. 1, pp. 856–871, 2022.
  * [12] F. Rezazadeh, H. Chergui, L. Christofi, and C. Verikoukis, “Actor-Critic-Based Learning for Zero-touch Joint Resource and Energy Control in Network Slicing,” _Proc. IEEE International Conference on Communications (ICC)_ , 2021.
  * [13] O. Akin, U. C. Gulmez, O. Sazak, O. U. Yagmur, and P. Angin, “GreenSlice: An Energy-Efficient Secure Network Slicing Framework,” _Journal of Internet Services and Information Security_ , vol. 12, no. 1, pp. 57–71, 2022\.
  * [14] H. Chergui, L. Blanco, L. A. Garrido, K. Ramantas, S. Kuklinski, A. Ksentini, and C. Verikoukis, “Zero-Touch AI-Driven Distributed Management for Energy-Efficient 6G Massive Network Slicing,” _IEEE Network_ , vol. 35, no. 6, pp. 43–49, 2021.
  * [15] G. Sun, Z. T. Gebrekidan, G. O. Boateng, D. Ayepah-Mensah, and W. Jiang, “Dynamic Reservation and Deep Reinforcement Learning Based Autonomous Resource Slicing for Virtualized Radio Access Networks,” _IEEE Access_ , vol. 7, pp. 45 758–45 772, 2019.
  * [16] C. Marquez, A. Banchs, M. Gramaglia, C. Ziemlicki, M. Fiore, and Z. Smoreda, “Not All Apps Are Created Equal: Analysis of Spatiotemporal Heterogeneity in Nationwide Mobile Service Usage,” _Proc. 13th International Conference on emerging Networking EXperiments and Technologies (CoNEXT)_ , pp. 180–186, 2017.
  * [17] W. Guan, X. Wen, L. Wang, Z. Lu, and Y. Shen, “A Service-Oriented Deployment Policy of End-to-End Network Slicing based on Complex Network Theory,” _IEEE Access_ , vol. 6, pp. 19 691–19 701, 2018.
  * [18] A. K. Bairagi, M. S. Munir, M. Alsenwi, N. H. Tran, S. S. Alshamrani, M. Masud, Z. Han, and C. S. Hong, “Coexistence Mechanism between eMBB and uRLLC in 5G Wireless Networks,” _IEEE Transactions on Communications_ , vol. 69, no. 3, pp. 1736–1749, 2021.
  * [19] M. Bennis, M. Debbah, and H. V. Poor, “Ultrareliable and Low-Latency Wireless Communication: Tail, Risk, and Scale,” _Proceedings of the IEEE_ , vol. 106, no. 10, pp. 1834–1853, 2018.
  * [20] C. Marquez, M. Gramaglia, M. Fiore, A. Banchs, and X. Costa-Perez, “Resource Sharing Efficiency in Network Slicing,” _IEEE Transactions on Network and Service Management_ , vol. 16, no. 3, pp. 909–923, 2019.
  * [21] H. P. Phyu, D. Naboulsi, and R. Stanica, “Machine learning in network slicing—a survey,” _IEEE Access_ , vol. 11, pp. 39 123–39 153, 2023\.
  * [22] Y. Shoham and K. Leyton-Brown, _Multiagent Systems: Algorithmic, Game-Theoretic, and Logical Foundations_. Cambridge University Press, 2009, pp. 215–216.
  * [23] A. Slivkins, _Introduction to Multi-Armed Bandits_. Foundations and Trends in Machine Learning, 2019, pp. 5–6.
  * [24] M. M. Aftab Hossain, C. Cavdar, E. Bjornson, and R. Jantti, “Energy Saving Game for Massive MIMO: Coping with Daily Load Variation,” _IEEE Transactions on Vehicular Technology_ , vol. 67, no. 3, pp. 2301–2313, 2018.
  * [25] B. Debaillie, C. Desset, and F. Louagie, “A Flexible and Future-Proof Power Model for Cellular Base Stations,” in _IEEE 81st Vehicular Technology Conference (VTC Spring)_ , 2015, pp. 1–7.
  * [26] H. P. Phyu, D. Naboulsi, and R. Stanica, “Mobile Traffic Forecasting for Network Slices: A Federated-Learning Approach,” in _IEEE 33rd Annual International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC)_ , 2022, pp. 745–751.
  * [27] S. Agrawal and N. Goyal, “Near-Optimal Regret Bounds for Thompson Sampling,” _Journal of the ACM_ , vol. 64, no. 5, 2017.