# A Joint Optimization Approach for Power-Efficient Heterogeneous OFDMA Radio
Access Networks

Gabriel O. Ferreira1, André F. Zanella2,3, Stefanos Bakirtzis4,6, Chiara
Ravazzi1, Fabrizio Dabbene1, Giuseppe C. Calafiore5, Ian Wassel4, Jie Zhang6,
Marco Fiore2 This work was supported by the European Commission through the
Horizon 2020 Framework Programme, H2020-MSCA-ITN-2019, MSCAITN-EID, Proposal
No. 860239, BANYAN. The work of Stefanos Bakirtzis is supported by the Onassis
Foundation and the Foundation of Education and European Culture.
{chiara.ravazzi, fabrizio.dabbene<EMAIL_ADDRESS>{andre.zanella<EMAIL_ADDRESS><EMAIL_ADDRESS>{jie.zhang<EMAIL_ADDRESS><EMAIL_ADDRESS>1IEIIT
CNR Institute, Italy, 2IMDEA Networks Institute, Spain, 3Universidad Carlos
III de Madrid, Spain, 4University of Cambridge, UK, 5Politecnico di Torino,
Italy, and CECS, VinUniversity, Vietnam, 6Ranplan Wireless Network Design Ltd,
UK.

###### Abstract

Heterogeneous networks have emerged as a popular solution for accommodating
the growing number of connected devices and increasing traffic demands in
cellular networks. While offering broader coverage, higher capacity, and lower
latency, the escalating energy consumption poses sustainability challenges. In
this paper a novel optimization approach for OFDMA heterogeneous networks is
proposed to minimize transmission power while respecting individual users’
throughput constraints. The problem is formulated as a mixed integer geometric
program, and optimizes at once multiple system variables such as user
association, working bandwidth, and base stations transmission powers.
Crucially, the proposed approach becomes a convex optimization problem when
user-base station associations are provided. Evaluations in multiple realistic
scenarios from the production mobile network of a major European operator and
based on precise channel gains and throughput requirements from measured data
validate the effectiveness of the proposed approach. Overall, our original
solution paves the road for greener connectivity by reducing the energy
footprint of heterogeneous mobile networks, hence fostering more sustainable
communication systems.

###### Index Terms:

Heterogeneous mobile networks, OFDMA, energy efficiency, geometric
optimization, resource allocation.

##  I Introduction

The efficiency of Radio Access Networks (RAN) is of ever-growing importance to
guarantee the combined performance and sustainability of mobile networks. The
increasing user traffic demands, the growing number of connected devices and
objects, the emergence of new classes of services with very strict Quality of
Service (QoS) requirements, and the surging heterogeneity of applications
jointly set very high expectations and standards for the operation of 6G RANs
[1]. As a consequence, RAN optimization is a subject that has received
substantial attention from the scientific community, and a wide range of
problems have been explored in this area, including how to ensure energy
savings via spectral efficiency, transmission power minimization, base
stations (BS) deployments, and user association, among others.

Optimizing the many variables that characterize modern RANs is especially
challenging in ultra-dense heterogeneous networks (HetNet), where macro,
micro, and femto cells are jointly deployed within a relatively small
geographical area in order to provide high wireless capacity and enhanced QoS
to the local end terminals [2]. These dense and hierarchical deployments are
today a reality, especially in populated urban centers: Figure 1 shows an
example of such a RAN configuration observed in a production-grade mobile
network, which includes overlapping macro and micro cells providing coverage
to the city center of a major European metropolis. In these heterogeneous RAN
infrastructures, a lack of optimization of transmission powers, resource
allocations, and assignments between users and BSs might lead to high
inefficiency, such as high transmission powers and low QoS for users [3, 4].

In heterogeneous cellular infrastructure scenarios, not only the
configurations of the RAN is especially complex, but, if not performed
properly, it also exposes the mobile network operator to high expenditures in
terms of energy costs that risk to make the whole technology not viable from
an economic viewpoint. The problem is exacerbated in newer generation of Radio
Access Technologies (RAT) that are increasingly demanding from an energy
perspective: for instance, 5G New Radio is known to be two to three times more
energy-consuming than its 4G equivalent [5], due to a need to face much higher
traffic volumes per user. The problem is such that over 90% of leading mobile
network operators have expressed concerns about the rise in energy-induced
operating expenses (OPEX) [6]. It is thus unsurprising that many operators and
equipment vendors are investing significant effort on developing solutions to
make RANs more energy prudent [7].

In this paper, we contribute to the endeavour of making dense, heterogeneous
RAN deployments greener, by proposing and efficiently solving a novel joint
optimization problem that allows minimizing transmission power in OFDMA
networks composed of different types of cells, while meeting individual users’
throughput requirements. Unlike many previous works in the field, our
optimization approach does not rely on iterative techniques that, in general,
can only guarantee local minima through a computationally demanding search,
whilst their convergence highly depends on the initial condition selection. In
addition, our problem formulation allows considering multiple variables
simultaneously, e.g., user association, resource allocation, and transmission
power, instead of dividing the optimization problem into smaller ones in a
ceteris paribus manner, as discussed in Section II.

We achieve the targets above by deriving original piecewise concave
approximations of the Shannon-Hartley formula, and consequently applying a
variable transformation that enables posing the optimization problem into a
mixed-integer geometric program (MIGP) form. A differentiating aspect of our
study is also that, unlike previous works that rely on simplistic evaluation
scenarios, we demonstrate the effectiveness of the proposed optimization
framework in dependable settings where up to 800 users generate demands
modeled after mobile network traffic measurements and served by real-world
heterogeneous RAN deployments, in the presence of wireless channel
characteristics reproduced via a high-performance signal propagation solver.

The remainder of the manuscript is organized as follows: in Section II we
present the related works, while the target scenario, its constraints, and the
problem we aim to solve are analyzed in Section III. The mixed integer
geometric program is formulated in Section IV, and the realistic example is
exploited in Section V. Finally, conclusions and future works are discussed in
Section VI.

Figure 1: Real-world topology of an operational heterogeneous network deployed
in a neighborhood of a large European city.

##  II Related work

The approaches proposed in the literature to date in order to tackle the
HetNet resource allocation problem can be grouped into three classes: (i)
iterative/sequential-based, (ii) convex optimization-based, and (iii) neural
network-based. We expound on these approaches separately in the next
subsections.

###  II-A Iterative/Sequential approaches

Two types of iterative/sequential approaches can be distinguished. In the
first one, the non-convex function is linearized and solved approximately
around a point of the optimization space. Then, based on the solution, a new
point of the optimization space is selected and this process is repeated until
convergence is achieved. The second solves the problem in ceteris paribus
manner, splitting the problem variables into two subsets of constant and non-
constant parameters. Then, a solution to a simplified version of the
optimization problem is retrieved considering only the impact of the non-
constant parameters. Consequently, the parameters of the two sets are
interchanged, i.e., those that were previously constant are now variables and
vice versa, and a solution for the new problem is computed. Again, this
process is repeated until convergence to a local minimum is achieved.

Along these lines, in [8], the authors studied the problem of joint user
association and resource allocation in heterogeneous networks, assuming that
the transmission power at all time is the maximum allowed. A relaxation for an
NP-Hard optimization problem was developed, where the new convex formulation
is used to obtain upper bounds to the network aggregated throughput. The
authors then propose an association rule based on users’ signal to
interference and noise ratio (SINR) levels and prove its efficacy by comparing
the optimal values obtained for the relaxed convex problem with the one when
the assignments are obtained with the association rule. Through experiment,
the authors show that the optimal values are tight and conclude that their
association technique is a simple but yet effective rule.

Optimization problem for orthogonal frequency division multiple access (OFDMA)
heterogeneous networks is studied in [9] for downlink transmissions. The work
proposes a joint optimization for user association, sub-channel allocation,
and power allocation. The problem is divided into two: first, the powers are
fixed and a solution for users association and sub-channel allocation is
obtained with the Hungarian algorithm; subsequently, the variables of the
first sub-problem are fixed, and a power allocation problem is solved with the
difference of two convex functions approximation method. Then, these problems
are solved alternately to obtain a local optimal solution. [10] proposes
analytical solutions for association problems subject to backhaul constraints
and network topology, allowing each user equipment (UE) at a given location to
determine its optimal connecting BS. Subsequently, an iterative algorithm to
reach the optimal load of the BS inside the heterogeneous network is derived.

In [11], energy efficiency (EE) resource allocation for OFDMA heterogeneous
networks with dense deployment is studied. The proposed algorithms have as
constraints minimum throughput for delay sensitive users and fairness for
delay tolerant users, in addition to the ones imposed by the OFDMA technology.
The resource block and power allocations problems, which are non-convex and
with combinatorial characteristics, are tackled with three different
approaches, named: mixed-integer programming formulation, time-sharing
formulation, and sparsity inducing formulation. All of them lead to non-convex
problems, which is circumvented using successive convex approximation (SCA)
methods, obtaining the problem solution through iterative procedures.

###  II-B Convex Optimization problems

Convex Optimization-based approaches aim at transforming and posing the
optimization problem in a convex form, for which a global solution can be
retrieved efficiently. For instance, [12] formulates a convex optimization
problem based on geometric programming (GP) to solve the problem of
simultaneous routing and power allocation in code-division multiple access
(CDMA) wireless data networks. GP is also used in [13], where the problems of
minimizing transmitter power subject to outage probability and minimizing
outage probability subject to transmission power constraints are written in
such a format. In that work, an outage happens when the quality of service of
a user, measured in terms of signal to interference ratio (SIR), is less than
a given threshold value. Many of the works discussed in the previous topic
also apply convex optimization techniques, however solving the problem
multiple times through successive approximation methods.

###  II-C Neural Networks

Neural networks, known to be universal approximators of a very large class of
functions, are also applied to resource/power allocation problems. In [14],
beamforming coordination and sub-carriers/power allocation problems are
addressed with the neural network based algorithm transfer learning multi-
agent deep Q-network (TL-MADQN). The algorithm is able to extract knowledge
from pre-trained agents (BS or channel gains) and adapt to new networks
environments, selecting the most appropriate allocation strategy. In [15], the
problem of joint association and resource allocation to maximize the system
throughput is formulated as a mixed integer quadratically constrained
quadratic program. The authors decompose the problem into two, and the sub-
part associated to the binary variables is solved with a neural network (NN)
approach. The proposed NN is based on encoder-decoder architecture and makes
use of the well-known long-short term memory (LSTM) and pointer network
mechanism.

There are other strategies used to solve optimization problems related to
heterogeneous cellular networks, such as Graph Theory based, stochastic
optimization, and Lyapunov drift plus penalty technique [16, 17, 18]. For a
better overview on 5G resource allocation for heterogeneous networks the
reader may refer to the Survey in [19], where the authors present the taxonomy
and strategies applied to such scenarios.

##  III Problem Formulation

Let us consider a scenario where each of $n$ users shall be connected to one
base station (BS) among $N$ possible. The downlink transmission rate of a user
$i$ connected to a BS $j$ can be calculated with the Shannon-Hartley Theorem
[20]:

$r_{ij}=x_{ij}B_{j}\log_{2}(1+S_{ij}(\mathcal{P})),$ (1)

with:

  * •

$P_{j}$: transmission power (W) of one resource block (RB) in BS $j$. Let
$\mathcal{P}\doteq[P_{1},\ldots,P_{N}]$.

  * •

$r_{ij}$: throughput (bits/s) of user $i$ connected to BS $j$.

  * •

$B_{j}$: bandwidth (Hz) of BS $j$.

  * •

$x_{ij}$: resources of BS $j$ assigned to user $i$. Hence, $x_{ij}B$ can be
seen as the working bandwidth assigned by BS $j$ to user $i$. We denote by
$x\in[0,1]^{n\times N}$ the matrix of the elements $x_{ij}$, with
$x_{ij}\in[0,1]$.

  * •

$S(\mathcal{P})$: Signal-to-Interference-Noise-Ratio. It is the ratio of the
power of the wanted signal to the sum of noise and interfering powers of the
other BSs, defined as:

$S_{ij}(\mathcal{P})=\frac{P_{j}g_{ij}}{\sigma^{2}+\sum_{k\neq
j}P_{k}g_{ik}},$ (2)

where $\sigma^{2}$ denotes the noise power.

Note that in equation (2), $P_{j}$ is the power of the BS to which the user is
connected and $P_{k}$, $k\neq j$ are the powers of the remaining ones.
$g_{ij}$ and $g_{ik}$ are the channel gains and their estimation will be
further discussed.

###  III-A Constraints

One of the requirements of the problem is that a user must be connected to
just one BS. To specify this constraint, we introduce binary variables
$z_{ij}\in\\{0,1\\},$ $i\in[n]$, $j\in[N]$ as

$z_{ij}=\begin{cases}\text{1, if user $i$ is connected to BS $j$,}\\\ \text{0,
otherwise.}\\\ \end{cases}$

$z\in\\{0,1\\}^{n\times N}$ is the matrix with elements $z_{i,j}$.

  1. 1.

Each user must be connected to just one BS

$\sum_{j=1}^{N}z_{ij}=1,\quad i\in[n]$ (3)

where $[n]$ denotes $\\{1,\ldots,n\\}$.

  2. 2.

A BS $j$ cannot provide more resources than it has available, that is:

$\sum_{i=1}^{n}x_{ij}\leq 1,\quad j\in[N].$ (4)

  3. 3.

Each user $i$ has a minimum required throughput $t_{i}$ (bits/s):

$r_{ij}\geq t_{i}.$ (5)

where $t\in\mathbb{R}_{\geq 0}^{n}$ is the vector with the elements $t_{i}$.
Combining (1) with equation in (3), the constraint (5) can be rewritten in the
following equivalent form:

$\sum_{j=1}^{N}\left[x_{ij}B_{j}\log_{2}(1+S_{ij}(\mathcal{P}))\right]z_{ij}\geq
t_{i},\quad i\in[n].$ (6)

###  III-B Optimization Problem

The optimization problem that minimizes the BS transmission powers and meets
the constraints above is written as

$\displaystyle\min_{x,z,P}\hskip 14.22636pt\sum_{j=1}^{N}P_{j}$ (7)
$\displaystyle\text{s.t.}\,x\in[0,1]^{n\times N}$
$\displaystyle\,z\in\\{0,1\\}^{n\times N}$ $\displaystyle\,\mathcal{P}\geq 0$
$\displaystyle\text{constraints \eqref{eq:y}, \eqref{eq:x} and
\eqref{eq:r2}}.$

Note that the above optimization problem is usually very hard to solve, since
it entails a combination of binary and continuous variables. More importantly,
constraint (6) is highly non convex, and also contains a product of continuous
and binary variables, to deal with the requirement of assigning a user to just
one BS. To tackle this last issue, we make use of a standard optimization
technique known as the Big-M Method [21, 22].

######  Lemma 1 (Big-M trick)

Let $y_{j}\doteq\log P_{j}$, $\overline{z}_{ij}\doteq 1-z_{ij}$ for all $i,j$,
and define

$f(x_{ij},y)\doteq x_{ij}B_{j}\log_{2}(1+S_{ij}(y)),$ (8)

where with a slight abuse of notation we let $S_{ij}(y)\doteq S_{ij}(P(y))$.
Then, the set of constraints in (3) and (6) can be rewritten as follows

$\displaystyle f(x_{ij},y)\geq t_{i}-M\overline{z}_{ij},\quad
i\in[n],j\in[N],$ (9) $\displaystyle\sum_{j=1}^{N}\overline{z}_{ij}=N-1,\quad
i\in[n]$ (10)

where $M$ is a sufficiently large constant.

Proof:  The constraint in (9) is inactive when $\overline{z}_{ij}=1$, since
${f}(x_{ij},y)\geq-M$ is always satisfied if $M$ is chosen large enough. On
the contrary, $\overline{z}_{ij}=0$ activates the restriction and the user’s
throughput requirement is respected. Constraint (10) forces each user to be
connected to only one BS (user $i$ is connected to $j$ if and only if
$\overline{z}_{ij}=0$). From (10), one can write

$\displaystyle\sum_{j=1}^{N}1-{z}_{ij}=N-1,\quad\sum_{j=1}^{N}{z}_{ij}=1,\quad
i\in[n],$

repeating constraint in (3). By multiplying (9) with $z_{ij}$ and summing over
$j$, we have

$\displaystyle\sum_{j=1}^{N}{f}(x_{ij},y)z_{ij}\geq
t_{i}\sum_{j=1}^{N}z_{ij}-M\sum_{j=1}^{N}{\overline{z}_{ij}}z_{ij},\quad
i\in[n],$ $\displaystyle\sum_{j=1}^{N}{f}(x_{ij},y)z_{ij}\geq t_{i}.$

which is the constraint in (6).

##  IV Mixed Integer Geometric programming

The proposed approach is based on a piecewise concave approximation of the
throughput constraint followed by variables transformation, leading to a mixed
integer geometric program.

###  IV-A Piecewise Concave Approximation

The piecewise concave approximation approach is a method used to approximate a
function with a set of $m$ simpler concave and monotonically increasing
functions. Here the approach is employed to achieve a lower approximation of
function $\log_{2}(1+\gamma)$, where $\gamma>0$ represents the signal-to-
interference-plus-noise ratio (SINR). By dividing the range of $\gamma$ into
intervals, the original function is approximated using a family of $m$ simpler
concave functions, each chosen to transform $\log_{2}(1+\gamma)$ into a log-
sum-exp form. This lower approximation ensures that the resulting piecewise
concave function captures the essential behavior of the original function
within each interval, providing a computationally efficient and tractable
representation for optimization and mathematical analysis. More specifically,
we consider the following family of functions

$\mathfrak{G}=\\{\varphi\in
C^{\infty}(0,\infty):\varphi_{a,b}(\gamma)=a\gamma^{b},\text{with
}a>0,b\in(0,1)\\}$

###### Proposition 1

For any $\varphi\in\mathfrak{G}$ we have

  1. 1.

$\varphi(0)=0$;

  2. 2.

$\lim_{\gamma\rightarrow 0}\varphi^{\prime}(0)=+\infty$;

  3. 3.

$\varphi$ is monotonically increasing in $(0,+\infty)$;

  4. 4.

$\varphi$ is a concave function in $(0,+\infty)$.

Proof:  Points 1-2 can be directly verified. Regarding points 3-4, the first
and second derivatives of $\varphi(\gamma)$ have the following properties

$\displaystyle\varphi^{\prime}(\gamma)=\frac{ab}{\gamma^{1-b}}>0,\hskip
2.84544pt\varphi^{\prime\prime}(\gamma)=\frac{ab(b-1)}{\gamma^{2-b}}<0.$

Therefore, the function $\varphi$ is monotonically increasing and concave in
$(0,+\infty)$.

###### Proposition 2

The following relations hold.

  1. 1.

Let $\gamma_{r}>0$, then for
$a=\log_{2}(1+\gamma_{\mathrm{r}})/\gamma_{\mathrm{r}}$ we have

$\begin{cases}\varphi_{a,1}(\gamma)=a\gamma\leq\log_{2}(1+\gamma),&\gamma\in[0,\gamma_{r}],\\\
\varphi_{a,1}(\gamma)=a\gamma\geq\log_{2}(1+\gamma),&\gamma\in(\gamma_{r},\infty).\end{cases}$
(11)

  2. 2.

For any $\gamma_{\text{l}},\gamma_{\text{r}}\in(0,\infty)$ with
$\gamma_{\text{r}}>\gamma_{\text{l}}$ there exists
$\varphi_{a,b}\in\mathfrak{G}$ such that

$\begin{cases}\varphi_{a,b}(\gamma)\geq\log_{2}(1+\gamma),&\gamma\in[0,\gamma_{\text{l}}),\\\
\varphi_{a,b}(\gamma)\leq\log_{2}(1+\gamma),&\gamma\in[\gamma_{\text{l}},\gamma_{\text{r}}],\\\
\varphi_{a,b}(\gamma)>\log_{2}(1+\gamma),&\gamma\in(\gamma_{\text{r}},\infty).\end{cases}$
(12)

Moreover, the parameters $a,b$ of this function may be computed by solving for
$a$ and $b$ the following:

$\begin{cases}\varphi_{a,b}(\gamma_{l})=a\gamma_{l}^{b}=\log_{2}(1+\gamma_{l})\\\
\varphi_{a,b}(\gamma_{r})=a\gamma_{r}^{b}=\log_{2}(1+\gamma_{r}).\end{cases}$
(13)

Proof:  Statement 1) is straightforward to verify since

$a=\log_{2}(1+\gamma_{\mathrm{r}})/\gamma_{\mathrm{r}}<\log_{2}(\mathrm{e})=\lim_{\gamma\rightarrow
0}\log_{2}(1+\gamma)^{\prime}.$

Let now $\gamma_{l},\gamma_{r}\in(0,\infty)$ with $\gamma_{r}>\gamma_{l}$ and
$\varphi_{a,b}\in\mathfrak{G}$, the system (13) consists of a pair of
equations with two unknown variables and admits a unique solution
$(a^{\star},b^{\star})$. It can be easily verified that $\gamma_{l}$ and
$\gamma_{r}$ are the only contact points between
$\varphi_{a^{\star},b^{\star}}$ and $\log_{2}(1+\gamma)$. Finally, notice that
since $\varphi(0)=\log_{2}(1+0)=0$ and

$\lim_{\gamma\rightarrow 0}\varphi^{\prime}(0)>\lim_{\gamma\rightarrow
0}\log_{2}(1+\gamma))^{\prime},$

then the first inequality in (12) is true in a neighborhood of the origin.

Moreover, due to the fact that $\varphi(\gamma)$ and $\log_{2}(1+\gamma)$ are
monotonically increasing and they intercept at $\gamma_{l}$ and $\gamma_{r}$,
together with the facts that $\log_{2}(1+\gamma)=o(\varphi)$ as
$\gamma\rightarrow\infty$ we conclude (12).

With the help of Proposition 2, we show that a given interval
$[0,\tilde{\gamma}]$ can always be divided in $m$ subintervals with extremes
$0=\gamma_{1},\gamma_{2},\ldots,\gamma_{m},\gamma_{m+1}=\tilde{\gamma}$ and
find $m$ approximations ${g_{d}}(\gamma)=a_{d}\gamma^{b_{d}}$, $d\in[m]$. The
impact of the hyper-parameter $m$ is shown in Figure 2 (a). It is clear that
$m=2$ leads to a very conservative approximation, since its respective
functions strongly underestimate the original one, which could make the
problem infeasible. On the contrary, already $m=5$ generates a far better
approximation compared to both $m=2$ and
$\log_{2}(1+\gamma)\simeq\log_{2}(\gamma)$. Figure 2 (b) shows the intervals
where each of the functions are active. Furthermore, one can see that if the
active constraint is respected for a given $\gamma$, the inactive ones are
simultaneously respected, since they are always overestimated. As a
consequence, solving the optimization problem considering only the active
function and its respective interval is equivalent to solving for all
$\varphi_{a,b}(\gamma)$ with $d\in[m]$ simultaneously regardless the value of
$\gamma$. Note also the fact that the first function is linear. Due to the
first inequality in equation (12), the function used to approximate the users’
throughput around the origin must be linear, i.e, $b=1$, since we cannot
overestimate the throughput capacity of a user.

(a)  (b)

Figure 2: (a) Impact of $m$ and (b) constraints active intervals.

######  Remark 1 (Interval length $\tilde{\gamma}$ and selection)

our approach is based on piecewese concave approximations inside an interval
$[0,\tilde{\gamma}]$. If $\tilde{\gamma}$ is very small, the problem may
become infeasible since the set of possible solutions is very conservative.
Conversely, a large $\tilde{\gamma}$ leads to a high value of $m$ due to the
aforementioned reasons. Despite the fact this might not affect the optimal
value (considering a proper approximation), it increases the computational
cost since each $d\in[m]$ represents a constraint in the optimization problem.
Therefore, $\tilde{\gamma}$ should be set properly, so the problem is feasible
and $m$ is not very large. Note that in case of an infeasible problem (which
could happen by not respecting the constraints or the interval
$[0,\tilde{\gamma}]$), both $m$ and $\tilde{\gamma}$ can be increased. The
active interval selection for each $\varphi_{a,b}(\gamma)$ does not need to
follow a specific rule. For instance, one could just divide the approximation
interval into $m$ ones of equal length.

The following result collects all the previous reasoning.

###### Theorem 1

Given $m$ subintervals between $0$ and $\tilde{\gamma}$, define $m$
approximating functions as follows

$\hat{f_{d}}(q_{j},u_{ij})\doteq\log\left(\frac{\sigma^{2}}{g_{ij}}e^{-q_{j}-\frac{u_{ij}}{b_{d}}}+\sum_{k\neq
j}\frac{g_{ik}}{g_{ij}}e^{q_{k}-q_{j}-\frac{u_{ij}}{b_{d}}}\right),$ (14)

for $d\in[m]$. Then, constraint in (6) is implied by the following set of
(possibly more conservative) $m$ convex constraints

$\hat{f_{d}}(q_{j},u_{ij})\leq\frac{\log\left(\frac{B_{j}a_{d}}{t_{i}}\right)}{b_{d}}+M\bar{z}_{ij},\quad
d\in[m],i\in[n],j\in[N].$ (15)

Moreover, the conservatism of this approximation vanishes for $m\to\infty$.

Proof:  By using the proposed approximation, inequality (6) can be written as

$r_{ij}\simeq x_{ij}B_{j}a_{d}\left(\frac{P_{j}g_{ij}}{\sigma^{2}+\sum_{k\neq
j}P_{k}g_{ik}}\right)^{b_{d}}\geq t_{i},\quad d\in[m].$ (16)

Then, with some mathematical manipulations and taking the $\log$ of both
sides, we have

$a_{d}\left(\frac{\sigma^{2}}{g_{ij}}P_{j}^{-1}x_{ij}^{-\frac{1}{b_{d}}}+\sum_{k\neq
j}\frac{g_{ik}}{g_{ij}}P_{k}P_{j}^{-1}x_{ij}^{-\frac{1}{b_{d}}}\right)^{-b}\geq\frac{t_{i}}{B_{j}},$
$-\log\left(\frac{\sigma^{2}}{g_{ij}}P_{j}^{-1}x_{ij}^{-\frac{1}{b_{d}}}+\sum_{k\neq
j}\frac{g_{ik}}{g_{ij}}P_{k}P_{j}^{-1}x_{ij}^{-\frac{1}{b_{d}}}\right)\geq\frac{\log\left(\frac{t_{i}}{B_{j}a_{d}}\right)}{b_{d}}.$

With the variables transformation $P_{j}=e^{q_{j}}$ and $x_{ij}=e^{u_{ij}}$,
it is possible to write

$-\log\left(\frac{\sigma^{2}}{g_{ij}}e^{-q_{j}-\frac{u_{ij}}{b_{d}}}+\sum_{k\neq
j}\frac{g_{ik}}{g_{ij}}e^{q_{k}-q_{j}-\frac{u_{ij}}{b_{d}}}\right)\geq\frac{\log\left(\frac{t_{i}}{B_{j}a_{d}}\right)}{b_{d}},$

then we have

$\displaystyle\hat{f_{d}}(q_{j},u_{ij})\leq\frac{\log\left(\frac{B_{j}a_{d}}{t_{i}}\right)}{b_{d}},$
(17)

which is constraint in equation (15) before the Big-M Method. It should be
noticed that constraint in (17) is a log-sum-exp function, which is convex.

###  IV-B MIGP optimization Problem

Let $\hat{P_{j}}$ be the maximum transmission power of a RB in BS $j$, defined
as the ratio between total transmission power of a BS and its number of RBs.
Hence, a Mixed-Integer Geometric Programming problem that solves optimization
problem (7) is

$\displaystyle\min_{\bar{z}_{ij},u_{ij},q_{j}}$
$\displaystyle\sum_{j=1}^{N}e^{q_{j}}$ (18) s.t.: $\displaystyle
e^{q_{j}}\leq\hat{P_{j}},\quad j\in[N],$ $\displaystyle e^{u_{ij}}\leq 1,\quad
i\in[n],j\in[N]$ $\displaystyle\sum_{i=1}^{n}e^{u_{ij}}\leq 1,\quad j\in[N],$
$\displaystyle\hat{f_{d}}(q_{j},u_{ij})\leq\frac{\log\left(\frac{B_{j}a_{d}}{t_{i}}\right)}{b_{d}}+M\bar{z}_{ij},\quad
d\in[m],$ $\displaystyle\hskip 142.26378pti\in[n],j\in[N]$
$\displaystyle\bar{z}_{ij}\in\\{0,1\\},\quad i\in[n],j\in[N],$
$\displaystyle\sum_{j=1}^{N}\bar{z}_{ij}=N-1,\quad i\in[n].$

Note that the above problem is a mixed-integer problem (since it contains the
binary variables $\bar{z}_{ij}$) which can be efficiently tackled with
available solvers, and that solves simultaneously the association and
assignment problems and the power optimization one. It should be noted that,
for given $\bar{z}_{ij}$, i.e. for given association, the problems becomes a
classical Geometric Program, i.e. a convex optimization problem for which an
optimal solution is always found. This is an important remark since, as we
will see in the next sections, there are situations in which simple assignment
rules appear to work rather efficiently. In such cases, our approach hence
becomes even faster and more scalable.

###  IV-C Sub-carrier assignments given $x$

In OFDMA networks, the wide bandwidth transmitted signal is divided into many
orthogonal narrow band sub-carriers, such that they do not interfere with each
other. Sets of these sub-carriers form a resource block, which can be
allocated to the users according to their throughput requirements (represented
by the variables $x_{ij}=[0,1]$). Therefore, after solving the aforementioned
optimization problem, we still need to assign $\rho_{ij}$ RBs to user $i$
connected to BS $j$ given the respective $x_{ij}$ computed via optimization
problem (18).

Since each RB is allocated exclusively to one user at a given instant,
$\rho_{ij}$ is clearly an integer variable. Ideally, if a user requires a non-
integer $\rho_{ij}$, one could just round such a value to its smallest greater
integer (ceil function). In practice this is not possible, since the number of
RBs is limited and the ceil rounding of all $\rho_{ij}$ would lead to more RBs
than the BS can provide. Instead, we propose to solve the optimization problem
where the users’ throughput requirements and the BSs available working
bandwidth are slightly increased and decreased, respectively. As a
consequence, some of the users would still respect the throughput requirements
even if $\rho_{ij}$ is rounded to its greatest smaller integer (floor
function) and some RBs could be allocated to those who do not satisfy such a
condition. The procedure to obtain $\rho_{ij}$ given $x_{ij}$ is presented in
Algorithm 1, with

  * •

$\delta_{1}$: parameter to increase users’ throughput requirements;

  * •

$\delta_{2}$: parameter to decrease BS working bandwidth;

  * •

$\lfloor\rho_{ij}\rfloor$, $\lceil\rho_{ij}\rceil$: floor($\rho_{ij}$) and
ceil($\rho_{ij}$), respectively;

  * •

$B_{0}$: bandwidth of one RB.

Algorithm 1 RB assignemtns given $x_{ij}$.

Increase/decrease throughput/working bandwidth:

$\hat{t}_{i}\leftarrow(1+\delta_{1})t_{i}$, $i\in[n]$

$\sum_{i=1}^{n}e^{u_{ij}}\leq 1-\delta_{2}$, $j\in[N]$

Solve MIGP in (18)

for $i\in[n]$

$j\leftarrow\arg\max(X(i,:))$

$\rho_{ij}\leftarrow\frac{\max(X(i,:))B}{B_{0}}$

if $\lfloor\rho_{ij}\rfloor B_{0}\log_{2}(1+S_{ij}(y))\geq t_{i}$

$\rho_{ij}\leftarrow\lfloor\rho_{ij}\rfloor$

else

$\rho_{ij}\leftarrow\lceil\rho_{ij}\rceil$

end if

end for

Note that one needs to certify if the number of RBs assigned to the users does
not exceed the maximum available. In case such a constraint is not respected,
$\delta_{1}$ and $\delta_{2}$ can be increased and decreased, respectively.
Practically, this means that more users respect the throughput requirement if
$\rho_{ij}\leftarrow\lfloor\rho_{ij}\rfloor$ and more RBs are available to
users that must use the ceil function.

##  V Application use cases

We demonstrate the applicability of the proposed algorithm in few HetNet
deployments, based on the real-world configuration of a production-grade RAN.
To this end, we first outline the process we followed to generate faithful
HetNet deployment replicas of an MNO in a major European city. Next, we
present how MIGP enables energy-efficient resource allocation in a HetNet,
benchmarking its performance against other available solutions. Finally, we
probe the scalability of MIGP and of the benchmarks, as the complexity of the
HetNet, the number of users, and their traffic demands increase.

###  V-A Realistic scenario generation

We generate our realistic scenarios based on the data obtained from the
production network of a leading MNO in a European capital city. The data
provided includes the location of the transmitting BS and the configuration of
their antennas, i.e., the antenna type (macro or micro cell) and steering
direction, the technology for that deployment (2G, 3G, 4G, or 5G NSA), and the
corresponding statistics about session-level traffic demands. This information
allows generating a high-fidelity digital twin of the actual network that
enables dependable assessments under realistic conditions.

For each scenario, $S$, abiding by the operational network topology we choose
a set of base stations, $BS_{j}$, composed of one or more antennas, and we
measure the number of sessions arriving on a 1-minute interval of an average
busy work day. This interval represents a moment where the network operates at
higher loads, with the number of connected users and their traffic demands
reaching their peak values. We do not access individual users data in order to
comply to privacy rules of the country where the network is located; instead,
our data is aggregated at BS level with 1 minute granularity. This means we
cannot simply replicate the session-level values, but need to rely on
statistics from the measured values.

For each scenario $S$, we estimate the number of session arriving at each
$BS_{j}$ during the time interval of a busy work day. From this, we calculate
the shares of sessions belonging to each mobile applications, obtaining a
heterogeneity of session demands in each scenario, since each mobile service
has its own characteristic traffic load. For each expected session arriving at
$BS_{j}$, we estimate traffic and duration (and thereby throughput) according
to the demands of the corresponding mobile service. These demands are
calculated based on the statistics collected at BS level across the network
over multiple months, which allows us to reliably estimate the session-level
traffic probability distribution function of each mobile service, which we use
to generate samples of traffic consumed by each session. In this regard, this
methodology not only respects the architecture of a major operational network
but also the expected density and spatial distribution of users, while
complying with user-privacy rules.

Each generated session is assigned to an individual user placed in a random
location, lying within the covered area of $s$-th HetNet topology, determined
initially by a Poisson random point process but with additional conditions to
respect the boundaries of the street layout of the topology (as to not
allocate too many users unrealistically inside buildings). We note that while
looking at the traffic and solving the problem for each $BS_{j}$, we consider
the full coverage area attributed to all 5G NSA-enabled antennas comprising
each $BS_{j}$.

To evaluate the received signal distribution within the coverage area of each
$BS_{j}$ we employ a ray tracing simulator which allows us to carefully
emulate the electromagnetic wave propagation in the urban fabric [23, 24].
That enables us to quantify the interaction of the waves with objects in the
propagation environment and to have accurate estimates of the channel gains
$g_{i,j}$ required to evaluate equation (2). To conduct the site-specific ray
tracing simulations for each HetNet topology, we fetch from Open Street Maps
information related to the urban environment, such as the city layout and
building height, the terrain type, and the foliage. Then, we place antennas
and configure their properties according to the specifications provided by the
MNO. A HetNet topology with 1 macro ($BS_{3}$) and 4 micro BSs is shown in
Figure 3, depicting how the signal emitted by each BS propagates within the
urban fabric, and how the micro BSs compensate for the low signal level of the
macro cell at larger distances. It is worth mentioning that macro cell
$BS_{3}$ is composed of 3 directional antennas, while the remaining micro
cells contain a single antenna with an omnidirectional radiation pattern.

In total, we consider 5 scenarios to evaluate the optimization algorithms,
each one assuming different spatial and network complexities. Scenario $S_{1}$
is a downtown location that will be used in our comparative analysis in
Section V-B and V-C, containing 1 macro cell and 4 micro cells located in the
historical center in the downtown area of the city under consideration. The
remaining 4 scenarios are utilized in the scalability analysis in Section V-D,
where we expound on the resilience of MIGP at a variety of different HetNet
topologies spread throughout the city. Scenarios $S_{2}$ and $S_{3}$ build on
the HetNet operating at the same downtown region of $S_{1}$, but we increase
the number of micro BS to 7 and 8, respectively, to evaluate the scalability
of our solution in denser network configurations. Scenario $S_{4}$ represents
a less dense HetNet topology located by the river side of the city, consisting
of 1 macro and 2 micro cells. Finally, scenario $S_{5}$ is on an area at the
hill side of the city, composed of 3 macro and 2 micro cells. All scenarios
consider BSs that are 5G NSA-enabled.

The MIGP depends on the hyper-parameters presented in Table I. The utility of
each one was discussed on previous sections. The values of $a_{d}$ and $b_{d}$
allow a piecewise concave approximation of $\log_{2}(1+S_{ij}(\mathcal{P}))$
when $S_{ij}(\mathcal{P})\in[0,\hskip 2.84544pt513.85]$.

TABLE I: Optimization problem hyper-parameters. Hyper-parameter | Value
---|---
$\delta_{1}$ | 0.05
$\delta_{2}$ | 0.16
M | $10^{6}$
$a_{d}$, $d\in[m]$ | [1.408, 0.7330, 1.3150, 1.9061, 3.1232]
$b_{d}$, $d\in[m]$ | [1, 0.7821, 0.4201, 0.2589, 0.1697]

###  V-B Qualitative performance analysis

Initially, we demonstrate the applicability of the proposed framework to a
single HetNet deployment of scenario $S_{1}$, benchmarking the performance of
MIGP with other available approaches typically employed for user allocation.
To this end, we consider a 5G new radio (NR) access network operating at 3.5
GHz with 500 resource blocks, utilizing an OFDMA technique with 100 MHz of
bandwidth. As previously mentioned, $S_{1}$ consists of 1 macro and 4 micro
BSs that accommodate the throughput requirements of 400 users, bounded between
1.35 Kbps and 18.72 Mbps. Pushing for greener communications, via MIGP we
aspire to minimize the total transmission power of the HetNet. The noise power
is set to -174 dBm/Hz.

Figure 3: Channel gains provided by BSs and the full scenario for $S_{1}$.
Figure 4:  Users’ assignment after MIGP optimization for $S_{1}$.

The resulting assignment, Figure 4, shows a clear tendency of connecting the
users to the BS providing the highest channel gain. We also note on the map a
pattern of users being mostly around the outside of buildings, which is
amplified by the propagation loss experienced by 3.5MHz communication inside
buildings. As previously explained, the working bandwidth variables in $x$ are
relaxed to a continuous domain, but in practice we need integer numbers
$\rho_{ij}$ representing the number of resource blocks made available to a
user. Therefore, after solving the MIGP, such a conversion is done by using
Algorithm 1. The number of users $n_{j}$ connected to BS $j$, the number of RB
used in each BS, and the respective transmission powers are provided in Table
LABEL:tab:UA-RB. $P_{j}^{\text{Tot}}$ stands for the total transmission power
of BS $j$ (each RB consuming $P_{j}$ Watts).

TABLE II: Number of users connected to each BS for $S_{1}$. BS $j$ | $n_{j}$ | $\text{RB}_{j}$ |  $P_{j}$ (W) |  $P_{j}^{\text{Tot}}$ (W)
---|---|---|---|---
$j=1$ | 43 | 423 | $0.2894\times 10^{-3}$ | 0.1224
$j=2$ | 161 | 495 | $0.7117\times 10^{-3}$ | 0.3523
$j=3$ | 142 | 490 | $0.6535\times 10^{-3}$ | 0.3202
$j=4$ | 24 | 413 | $0.0038\times 10^{-3}$ | 1.5694 $\times 10^{-3}$
$j=5$ | 30 | 435 | $0.2151\times 10^{-3}$ | 0.0936
$\sum_{j=1}^{N}$ | 400 | 2256 | 0.0019 | 0.8901

###  V-C Comparison with other approaches

Since our proposed approach deals simultaneously with the variables $x,z$ and
$\mathcal{P}$, we can compare it with some standard procedures performed in
realistic network scenarios. As described in [8], the users’ association can
be done with simple rules, such as Received Signal Power or Maximum channel
gain, defined below.

  * •

Received Signal Power (RSP): a user $i$ is associated to the base station $j$
that maximizes the product between transmission power and channel gain, i.e.,
$j=\arg\max_{j}(g_{ij}P_{j})$.

  * •

Maximum channel gain (MCG): a user $i$ is associated to the base station $j$
that maximizes the channel gain between $i$ and $j$, that is,
$j=\arg\max_{j}(g_{ij})$.

Note that these simple association rules provide matrix $z$. Therefore our
problem becomes a geometric program (GP), which is convex. In Figure 5, we
used the MCG rule and refer to this situation as (GP + fixed $z_{ij}$). For
$S_{1}$, where each base station $j$ have $n_{j}$ users assigned via
association rule, the resources can be equally shared between the UE according
to

$x_{ij}=\begin{cases}\frac{1}{n_{j}},\hskip 2.84544pt\text{if}\hskip
5.69046ptz_{ij}=1,\hskip 5.69046pt\forall i\in[n],\hskip 5.69046pt\forall
j\in[N]\\\ \text{0, otherwise.}\\\ \end{cases}$

This situation is referred as (GP + fixed $z_{ij},x_{ij}$). We also compare
our approach with its early stage (without Big-M trick, piecewise concave
approximation, and variables transformation) given by optimization problem
(7). Since this is highly non-convex, finding its optimal solution is not
trivial and in general just local optimal is achieved. To deal with this
problem, we use the commercially available MIDACO (Mixed Integer Distributed
Ant Colony Optimization) solver, that is able to solve mixed integer non-
linear programs with discontinuities, non-convexity, and stochastic noise.
More details on MIDACO are given in [25].

In Figure 5, the optimal value considering the situations described above is
calculated for different numbers of users connected to the network during
$S_{1}$, keeping the % of sessions across mobile applications consistent to
respect traffic dynamics. The corresponding KWh consumption if the power
levels are kept constant for our network snapshot are also shown. The number
of users associated to each BS is in Figure 6.

Figure 5: Optimal value as a function of number of users in the network for
different approaches for $S_{1}$. Figure 6: Number of users connected to each
BS of $S_{1}$ for (a) OP(7) + MIDACO, (b) GP + fixed $z_{ij}$ and (c) MIGP.

From these results, a series of keypoints can be highlighted.

  1. 1.

GP+fixed $z_{ij}$ $\times$ MIGP: the assignments and the corresponding optimal
values obtained with MIGP and (GP + fixed $z_{ij}$) are almost the same for
this network. This happens because the BSs are directional and without large
cross-interference, as shown in Figure 3. Physically, one could say that the
BS deployment is good, which was expected since the data set is from a
realistic scenario from a major European city. Regarding the optimization
problem point of view, assigning the users according to the MCG rule makes the
problem considerably simpler, since it becomes purely convex (can be solved
globally and efficiently) and the number of problem variables is reduced,
since $z$ would be known in advance. However, there is no guarantee that using
the MCG rule in a network where the cross-interference between BSs is large
might be a good option. In such a situation, solving the MIGP is more
appropriate.

  2. 2.

OP (7)+MIDACO: as already discussed, the problem in its most natural format is
highly non-convex. We solved it with MIDACO considering that the stop
condition is true when the optimal value converges and the initial condition
is given by the lower bound of each variable. One may notice that the results
obtained with both MIGP and (GP + fixed $z_{ij}$) are considerably better in
terms of optimal value when compared to (OP (7) + MIDACO). Therefore, the
piecewise concave approximation followed by the variables transformation are
indeed efficient mathematical tools to pose the problem in a more suitable
format.

  3. 3.

GP+fixed $z_{ij},x_{ij}$: fixing the working bandwidth $x$ of users
considerably degrades the optimal value, since each one has specifics
throughput requirements and channel gains. Furthermore, for $n\geq 150$ the
problem becomes infeasible, therefore sharing such a resource equally between
users is not a good choice and $x$ should remain as problem variable. Note
that the approach of fixing $x$ is rather common in the literature [26].

###  V-D Scalability of MIGP across different network topologies

Finally, we test how MIGP performs across different network topologies, which
offer variations on the number of BS, density of the deployment, number of
users and propagation paths due to the geography of the city. We exemplify the
scenarios through Figure 7, which shows downtown scenario $S_{3}$ (over same
area as $S_{1}$ and $S_{2}$, but with more BS), river scenario $S_{4}$ and
hill scenario $S_{5}$.

Figure 8 compares the optimal values obtained with MIGP and OP (7)+MIDACO for
the 4 network scenarios, i.e., distinct throughput requirements, channel
gains, number of BS, and users. In all cases, the MIGP leads to better
results.

Figure 7: Channel gains and antenna placements for (a) dense downtown
deployment of scenario $S_{3}$, (b) the less dense river deployment of $S_{4}$
and (c) the hill deployment of $S_{5}$. Figure 8: Power consumption
$\sum_{j=1}^{N}P_{j}$ for different networks.

The results above show that MIGP, contrary to OP (7)+MIDACO, is able to
provide feasible solutions even for scenarios with considerable numbers of BSs
and users. Since the association variables are naturally binary, increasing
the number of BSs is computationally more expensive than increasing the number
of users. However, as previously explained, in environments with good BS
deployment (not large cross interference), the MIGP generally associates a
user with the BS providing the respective highest channel gain. When using
simple association rules, the problem becomes purely convex and can be solved
globally and efficiently.

##  VI Conclusion

This research paper introduces an innovative approach to address the challenge
of minimizing transmission power in OFDMA heterogeneous networks while
ensuring individual users’ throughput requirements are met. The proposed
method is formulated as a mixed integer geometric program, leveraging the
Big-M trick to handle the problem’s combinatorial nature. Additionally, a
piecewise concave approximation of the Shannon-Hartley Theorem is employed,
followed by variable transformation to tackle the problem’s non-convex
characteristics effectively. The optimization problem involves determining the
optimal users’ working bandwidth, base station (BS) transmission powers, and
users’ associations. Remarkably, unlike many existing methods, this approach
does not rely on successive or iterative algorithms and does not necessitate
prior knowledge of a feasible initial condition for the problem. Moreover, it
is shown that the problem becomes convex when simple users’ association rules
are employed. The effectiveness of the proposed optimization problem is
demonstrated through extensive testing in a highly realistic scenario with a
considerable number of users. Channel gains are calculated using a high-
performance propagation solver, ensuring a practical evaluation. Looking
ahead, we plan to extend our work to include the temporal aspect of the
network, accounting for user mobility, varying channel gains, positions, and
transmission power over time. In such dynamic scenarios, the challenge of
users’ handovers will be a critical consideration to address. Overall, this
paper presents a promising optimization approach for enhancing the energy
efficiency of heterogeneous networks while accommodating the increasing
demands of connected devices, paving the way for greener and more efficient
cellular networks in the future. We plan to share our code, so the MIGP can be
easily compared to other approaches in the literature.

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##  VII Biography Section

|  Gabriel O. Ferreira (Member, IEEE) received the B.S. degree in mechatronics
engineering from the Federal Center for Technological Education of Minas
Gerais, Brazil, in 2018 and his M.Sc. degree in Electrical Engineering from
the Federal University of São João del-Rei, Brazil, in 2020. He is currently
pursuing the Ph.D. degree with the Department of Electronics and
Telecommunication, Politecnico di Torino, Italy. His research interests
include modeling and forecasting mobile network traffic, network optimization,
and network closed-loop control. Mr Ferreira has received the Marie
Skłodowska-Curie Actions-Innovative Training Networks Fellowship.
---|---
|  Andre Felipe Zanella (Member, IEEE) is a PhD researcher at IMDEA Networks
Institute and UC3M. He received his bachelor’s and master’s degrees in
Telecommunications Engineering at the Federal University of Parana, in Brazil.
He’s currently a part of the Networks Data Science group, working on mobile
traffic analysis and remote sensing with network metadata, where his interests
lay in developing techniques that help solving social sciences problems using
information gathered by network operators.
---|---
|  Stefanos Bakirtzis (Member, IEEE) received the diploma degree in electrical
and computer engineering from the National Technical University of Athens,
Athens, Greece, in 2017, and the M.A.Sc. degree in electrical and computer
engineering from the University of Toronto, Toronto, ON, Canada, in 2020. He
is currently pursuing the Ph.D. degree with the Department of Computer Science
and Technology, University of Cambridge, Cambridge, U.K.,He is currently
working on the Big Data Analytics for Radio Access Networks (BANYAN) project
as a member of Ranplan Wireless with the University of Cambridge. His research
interests include wireless communication systems, network optimization,
wireless channel modeling, machine learning and artificial intelligence,
computational modeling, and stochastic uncertainty quantification. Mr.
Bakirtzis has received the Onassis Foundation Scholarship, the Foundation of
Education and European Culture Grant, and the Marie Skłodowska-Curie Actions-
Innovative Training Networks Fellowship.
---|---
|  Chiara Ravazzi (M’13) is a senior researcher at the Italian National
Research Council (CNR-IEIIT) and adjunct professor at the Politecnico di
Torino. She obtained the Ph.D. in Mathematical Engineering from Politecnico di
Torino in 2011. In 2010, she spent a semester as a visiting scholar at the
Massachusetts Institute of Technology (LIDS), and from 2011 to 2016, she
worked as a post-doctoral researcher at Politecnico di Torino (DISMA, DET).
She joined the Institute of Electronics and Information Engineering and
Telecommunications (IEIIT) of the National Research Council (CNR) in the role
of a tenured researcher (2017-2022). Furthermore, she served as an Associate
Editor for IEEE Transactions on Signal Processing from 2019 to 2023, and she
currently holds the same position for IEEE Transactions on Control Systems
Letters (since 2021) and the European Journal of Control (since 2023). She has
achieved the national scientific qualification as a second-tier (associate
professor) in the fields of Automatica (09/G1) and Telecommunications (09/F2).
---|---
|  Fabrizio Dabbene (Senior Member, IEEE) received the Laurea and the Ph.D.
degrees from Politecnico di Torino, Italy, in 1995 and 1999, respectively. He
is a Director of Research with the institute IEIIT of the National Research
Council of Italy (CNR), Milan, Italy, where he is the coordinator of the
Information and Systems Engineering Group. He has held visiting and research
positions with The University of Iowa, Iowa City, IA, USA, Penn State
University, University Park, PA, USA, and with the Russian Academy of
Sciences, Institute of Control Science, Moscow, Russia. He has authored or
coauthored more than 100 research papers and two books. Dr. Dabbene served as
an Associate Editor for Automatica during 2008–2014 and for the IEEE
TRANSACTIONS ON AUTOMATIC CONTROL during 2008–2012, and he is currently the
Senior Editor of the IEEE CONTROL SYSTEMS SOCIETY LETTERS. He was Elected
Member of the Board of Governors during 2014–2016 and he served as the
VicePresident for Publications during 2015–2016. He is currently chairing the
IEEE-CSS Italy Chapter.
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|  Giuseppe C. Calafiore (Fellow, IEEE) is a full professor at DET,
Politecnico di Torino, where he coordinates the Control Systems and Data
Science group. He is also a full professor at the College of Mechanical
Engineering at VinUniversity in Hanoi, Vietnam, and an associate fellow of the
IEIIT-CNR, Italy. Dr. Calafiore held visiting positions at the Information
Systems Laboratory (ISL), Stanford University, California, in 1995; at the
Ecole Nationale Supérieure de Techniques Avancées (ENSTA), Paris, in 1998; and
at the University of California at Berkeley, in 1999, 2003, 2007, 2017, 2018
and 2019, and 2021 where he lately taught a Master course on Financial Data
Science. He was a Senior Fellow at the Institute of Pure and Applied
Mathematics (IPAM), University of California at Los Angeles, in 2010. Dr.
Calafiore is the author of about 210 journal and conference proceedings
papers, and of eight books. He is a Fellow of the IEEE. He received the IEEE
Control System Society “George S. Axelby” Outstanding Paper Award in 2008. His
research interests are in the fields of convex optimization, identification
and control of uncertain systems, with applications ranging from finance and
economic systems to robust control, machine learning, and data science. Dr.
Calafiore has over twenty years of teaching experience in Master-level and
Ph.D. courses in the areas of Systems and Control Theory, Convex Optimization
and Machine Learning.
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|  Ian Wassell (Member, IEEE) received the B.Sc. and B.Eng. degrees from the
University of Loughborough, Loughborough, U.K., in 1983, and the Ph.D. degree
from the University of Southampton, Southampton, U.K., in 1990. He is
currently a University Associate Professor with the Computer Laboratory,
University of Cambridge, Cambridge, U.K., and has experience in excess of 25
years in simulation and design of radio communication systems gained via a
number of positions in industry and higher education. He has authored more
than 200 articles. His current research interests include wireless sensor
networks, cooperative wireless networks, propagation modeling, sparse
representation, and machine learning. Dr. Wassell is a member of the IET and a
Chartered Engineer.
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|  Jie Zhang (Senior Member, IEEE) has held the Chair in Wireless Systems at
the Department of Electronic and Electrical Engineering, University of
Sheffield, Sheffield, U.K., on a part-time basis, since January 2011. He is
also the Founder, a Board Director and the Chief Scientific Officer of Ranplan
Wireless, Cambridge, U.K., a public company listed on Nasdaq First North.
Ranplan Wireless produces a suite of world leading indoor and the first joint
indoor-outdoor radio access network planning tool – Ranplan Professional,
which is being used by all the top 5 telecom equipment vendors and the world’s
largest mobile operators, system integrators and research organizations. Along
with his students and colleagues, he has pioneered research in small cell and
heterogeneous network and published some of the landmark papers and book on
these topics, widely used by both academia and industry. Since 2010, he and
his team have also developed ground-breaking work in smart radio environment
and building wireless performance modelling, evaluation and optimization, the
key concepts of which were introduced in a paper titled “Fundamental Wireless
Performance of a Building”, IEEE Wireless Communications, 29(1), 2022. His
Google scholar citations are in excess of 8500 with an H-index of 40. He
received a Ph.D. degree in industrial automation from East China University of
Science and Technology, Shanghai, China, in 1995. Prior to joining the
University of Sheffield, he had studied/worked with Imperial College London,
Oxford University, and University of Bedfordshire, reaching a status of Senior
Lecturer, Reader and Professor in 2002, 2005 and 2006, respectively.
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|  Marco Fiore (Senior Member, IEEE) is Research Associate Professor at IMDEA
Networks Institute and CTO at Net AI Tech Ltd. He received M.Sc. degrees from
University of Illinois at Chicago, IL, USA (2003), and Politecnico di Torino,
Italy (2004), a Ph.D. degree from Politecnico di Torino (2008), Italy, and a
Habilitation a Diriger des Recherches (HDR) from Universite de Lyon, France
(2014). He held tenured positions as Maitre de Conferences (Associate
Professor) at Institut National des Sciences Appliquees (INSA) de Lyon, France
(2009-2013), and Researcher at Consiglio Nazionale delle Ricerche (CNR), Italy
(2013-2019). He has been a visiting researcher at Rice University, TX, USA
(2006-2007), Universitat Politecnica de Catalunya (UPC), Spain (2008), and
University College London (UCL), UK (2016-2018). Dr. Fiore is a senior member
of IEEE, and a member of ACM. He was a recipient of a European Union Marie
Curie fellowship and a Royal Society International Exchange fellowship. He
leads the Networks Data Science group at IMDEA Networks Institute, which
focuses on research at the interface of computer networks, data analysis and
machine learning.
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