# Statistical Analysis and Optimization of a 5th-Percentile User Rate
Constrained Design for FFR/SFR-aided OFDMA-based Cellular Networks

Jan García-Morales, Guillem Femenias, and Felip Riera-Palou J. García-Morales,
G Femenias and F Riera-Palou are with the Mobile Communications Group,
University of the Balearic Islands, Palma 07122, Illes Balears, Spain (e-mail:
{jan.garcia,guillem.femenias,felip.riera}@uib.es).

###### Abstract

Interference mitigation strategies are deemed to play a key role in the
context of the next generation (B4G/5G) of multi-cellular networks based on
orthogonal frequency division multiple access (OFDMA). Fractional and soft
frequency reuse (FFR, SFR) constitute two powerful mechanisms for inter-cell
interference coordination (ICIC) that have been already adopted by emerging
cellular deployments as an efficient way to improve the throughput performance
perceived by cell-edge users. This paper presents a novel optimal 5th-
percentile user rate constrained design (R5pD) for FFR/SFR-based networks
that, by appropriately dimensioning the center and edge regions of the cell,
rightly splitting the available bandwidth among these two areas while
assigning the corresponding transmit power, allows a tradeoff between cell
throughput performance and fairness to be established. To this end, both the
cumulative distribution function of the user throughput and the average
spectral efficiency of the system are derived assuming the use of the
ubiquitous proportional fair scheduling policy. The analytical framework is
then used to obtain numerical results showing that the novel proposed design
clearly outperforms previous schemes in terms of throughput fairness control
due to a more rational compromise between average cell throughput and cell-
edge ICIC.

###### Index Terms:

OFDMA cellular networks, fractional frequency reuse, soft frequency reuse,
optimal FFR/SFR designs.

##  I Introduction

Modern cellular communications standards such as Long-Term Evolution (LTE) and
LTE-Advanced (LTE-A) have adopted the orthogonal frequency division multiple
access (OFDMA) as the downlink radio interface [1], [2]. Furthermore, it is
envisaged that OFDMA will keep playing a major role in 5G cellular networks
[3]. The physical layer of an OFDMA network relies on the division of a
wideband frequency-selective channel into multiple orthogonal narrowband
frequency-flat channels that, when operated in a time-slot manner, define a
time/frequency grid whose elements (resource blocks (RBs)) serve as the basic
resource allocation unit. Owing to the RB orthogonality, no intracell
interference arises, however, since most cellular deployments nowadays rely on
aggressive universal frequency reuse schemes, users situated in the cell-edge
areas are prone to experience large levels of inter-cell interference (ICI).
In order to address the performance degradation ICI brings along while still
pursuing high spectral efficiencies, a plethora of ICI coordination (ICIC)
techniques have been investigated [4], among which fractional and soft
frequency reuse (FFR, SFR) and variants thereof, have achieved widespread use
thanks to the excellent compromise they provide between overall spectral
efficiency and cell-edge performance.

In FFR-based cellular systems a low frequency reuse factor is used for the
cell-center users, less affected by ICI, and a larger frequency reuse factor
is selected for the cell-edge users, much more exposed to ICI. SFR is a
variation of FFR whereby the central region of each cell is also allowed to
employ the frequency resources allocated to the edges of the neighboring
cells, thus the whole system bandwidth is reused in every cell [5].
Furthermore, different transmit powers can be allocated to the central and
edge RBs in SFR-based cellular systems, with the aim of increasing the signal-
to-interference-plus-noise ratio (SINR) experienced by those users located far
away from the BS. Summarizing, FFR/SFR-based frequency reuse schemes provide
compound reuse factors usually lying between 1 and 3, thus sacrificing
spectral efficiency in favor of some degree of fairness between cell-center
and cell-edge users. One of the major issues when dealing with FFR/SFR-based
ICI mitigation strategies is the design of long-term resource allocation
algorithms aiming at the optimization of throughput-related utility functions
with constraints on the degree of fairness between users located throughout
the cell [3, 6]. In order to tackle these optimization problems, it is very
convenient to obtain mathematically tractable analytical models for both the
system and user throughput. Unfortunately, the analytical throughput models
used to design the long-term FFR/SFR-based resource allocation algorithms are
fully related to the short-term scheduling rules used to select the set of
users on each time/frequency/energy resource, thus making the derivation of
these analytical models a challenging task.

It is well known that, regardless of the ICIC technique employed, channel-
aware schedulers are able to exploit the inherent system multiuser diversity
by allocating each RB, on a slot-by-slot basis, to users enjoying _favourable_
channel conditions (i.e., users experiencing high SINR values), and thus
leading to a significant enhancement of network spectral efficiency. Amongst
many scheduling rules, the proportional fair (PF) scheduler [7] has been
successfully deployed in the latest generations of wireless networks, owing to
the excellent trade-off it offers between fairness and spectral efficiency.
Hence, the obtention of exact mathematically tractable analytical models for
the system and user throughput of FFR/SFR-aided OFDMA-based cellular networks
under PF scheduling is a challenging aim of significant interest when
designing optimal long-term resource allocation strategies.

###  I-A Background work

Several research studies have focused on the theory of stochastic geometry,
where the BSs are distributed using Poisson Point Processes (PPP) [8, 9, 10]
or hard-core PPPs [11]. However, the use of stochastic geometry to
characterize cellular layouts hinders to accurately model the consequences of
using ICIC techniques such as FFR/SFR when considering regularly-planned
cellular arrangements.

In contrast to the above background work, which relies on the use of
stochastic geometry to model the cellular environment, the work in [12] does
indeed account for the regular deployment using an FFR-aided OFDMA-based
network. Unfortunately, the analytical framework proposed by Fan Jin _et al._
in [12] is limited to the use of resource allocation schemes based on round
robin (RR) scheduling. In [13], the authors investigate the cell capacity and
the cell-edge performance in terms of user satisfaction. They present an
extensive comparison between FFR and SFR, but using again the RR scheduling
rule. Similar approaches, lacking the consideration of channel-aware
scheduling policies, are also proposed by Assaad in [14], Najjar _et al._ in
[15], Kumar _et al._ in [16], Hambebo _et al._ [17] and Hung-Bin Chang _et
al._ in [18] to optimize FFR-based network parameters. A generalized unifying
analytical framework suitable for both FFR and SFR while also considering the
uplink has been recently presented in [19] but again, as the previous
aforementioned works, neglecting the role of the scheduler. Closer to the
problem addressed in this manuscript, Sagkriotis _et al._ in [20] target the
optimization of an FFR system in terms of maximum capacity and power
efficiency when assuming the use of a PF scheduler. However, the proposed
design does not incorporate any constraints on the cell-edge performance and
thus, it cannot provide QoS guarantees to the _worst_ users in the network.

Some of the limitations of this background work have been partially overcome
in our contributions [6] and [21], where an analytical framework was developed
that allows the evaluation of the cell throughput when using opportunistic
maximum SINR (MSINR) and PF schedulers. However, our previous works considered
only the case of _strict_ FFR and, moreover, the proposed designs were based
on average cell-center and cell-edge metrics, thus neglecting a complete
statistical characterization of the throughput [2]. Similarly, [22] presents
an analytical performance model for FFR when performing chunk-based (rather
than subcarrier-based) resource allocation but again focusing on the
maximization of the average spectral efficiency and thus neglecting any
minimum service guarantees to the worst users in the network. More recently
Wang et al. in [23] have proposed an extension of the FFR scheme by allowing
more than one reuse outer layers in an attempt to improve the performance on
cell-edge region but their design does not consider the incorporation of
explicit QoS requirements for the users in this region and furthermore, the
effects of fast fading and the use of channel-aware schedulers have both been
neglected. Remarkably, most 5G defining proposals (e.g., [24]) specify key
performance indicators (KPIs) whose targets must be accomplished by at least a
certain percentage of users, regardless of their position. To this end, a full
statistical characterization of the throughput is required. In [25], we
presented some preliminary results of our effort to close this gap. In that
work, we proposed an analytical framework allowing the obtention of
statistical models of the average user throughput, but we did not consider the
optimal designs required to maximize the cell throughput while guarantying a
certain degree of quality-of-service (QoS) throughout the cell.

We note at this point, that the focus of this work is on single-tier networks.
Extending the analytical framework proposed here to heterogeneous setups by
combining results presented in [21] with multi-tier specific ICIC techniques
such as those based on almost blank subframes [26, 27, 28], constitutes an
interesting avenue for further research.

###  I-B Contributions of the paper

In this paper, different optimal designs for a downlink OFDMA-based multi-
cellular network are studied and compared. To this end, an analytical
framework is presented allowing the performance evaluation of both SFR and FFR
schemes using a PF scheduling policy. The main contributions of this paper can
be summarized as follows:

  * •

A novel analytical optimization approach, which we term 5th-percentile user
rate constrained design (R5pD), is introduced whereby a cell throughput-based
utility function maximization is conducted subject to a constrained 5th-
percentile user rate111It should be noted that the focus of this paper is on
the 5th-percentile since that is the percentage typically specified in most
curent standards and future proposals. Nevertheless, the framework developed
here is generally applicable to any $x$th-percentile constrained design..
Guarantying a specified value for the 5th-percentile user rate is one of the
most important technical challenges in current and emerging cellular systems
trying to provide uniform QoS levels throughout the cell [2].

  * •

In order to deal with the R5pD-based design, a global characterization of
throughput is introduced for both SFR- and FFR-aided OFDMA-based cellular
networks using PF scheduling. This analytical framework is used to derive
mathematically tractable expressions for important metrics such as the
cumulative distribution function (CDF) of the average user throughput and the
average system spectral efficiency.

  * •

Furthermore, results for the R5pD-based strategy are compared to those
obtained using well-known optimal designs, namely, the fixed spectrum/power
factor design (FxD) and the QoS-constrained design (QoScD), whose optimization
criteria strike a balance between the cell-edge and cell-center throughput
leading to different network performance behaviors. Moreover, the proposed
framework allows an exhaustive comparison between FFR and SFR to be carried
out that reveals under what conditions one frequency reuse technique is
preferable over the other.

Although results are obtained for an FFR/SFR-aided deployment, this analytical
framework opens the door to the theoretical spectral efficiency evaluation of
OFDMA-based cellular networks using more sophisticated ICIC techniques such as
adaptive frequency reuse or network MIMO, as well as to the assessment of
cellular multi-tier networks where the macro-cellular network is underlaid by
different tiers of pico- and femto-cellular BSs [29, 21].

The rest of the paper is organized as follows. In Section II the system model
under consideration in the context of FFR/SFR-aided OFDMA-based networks is
introduced. The novel optimal R5pD-based design is presented in Section III.
The statistical characterization of both the users throughput and the average
cell throughput are derived in Section IV. Extensive analytical and simulation
results, for each reuse scheme and using the PF scheduler, are provided in
Section V. Finally, the main outcomes of this paper are recapped in Section
VI.

##  II FFR/SFR system model

###  II-A Network model

Figure 1: FFR/SFR-aided OFDMA network topology.

This work focuses on the downlink of an FFR/SFR-aided OFDMA-based multi-
cellular system that, as shown in Fig. 1, has been modeled as a regular
tessellation of hexagonal cells whose serving BSs are located at their
centers222This paper assumes the use of omnidirectional antennas at the BSs
leaving for further work the issues related to antenna sectorization.. Towards
the implementation of FFR/SFR, a user categorization is first conducted. In
particular, users whose average SINR is above a prescribed threshold
$\Gamma_{th}$ are deemed as central ones, whereas the rest, those whose
average SINR is below $\Gamma_{th}$, are considered to be edge users.
Subsequently, non-overlapping frequency bands are allocated to cell-center and
cell-edge users. The frequency bands are usually designed in such a way that a
low frequency reuse factor can be used to serve the central users and a higher
reuse factor can be used for the edge users. Throughout this paper, and
without loss of generality, a frequency reuse factor of 3 is assumed for cell-
edge users whereas central users are assumed to employ a reuse factor of 1
(universal). Furthermore, and for the sake of analytical tractability, it is
also assumed that a circumference of radius $R_{th}$ (threshold distance)
delimitates the cell-center and cell-edge regions. Finally, the central cell
constituting the cell/BS of interest is approximated by a circle of the same
area as the corresponding hexagon (see Fig. 1). This implies that if the side
of the regular hexagon is denoted by $R_{h}$, the radius of the circular cell
is $R_{m}=R_{h}\sqrt{3\sqrt{3}/2\pi}$.

The full system bandwidth is exploited by means of a set $\mathcal{F}_{T}$ of
$N_{\text{RB}}$ RBs (or orthogonal subbands), each made of $N_{sc}$ adjacent
subcarriers and whose bandwidth, $B_{\text{RB}}$, is small enough so that all
subcarriers in a subband can be safely assumed to experience frequency flat
fading. When using FFR, the set $\mathcal{F}_{T}$ is divided into a subset
$\mathcal{F}_{C}$ of cell-center RBs and a subset
$\mathcal{F}_{T}\backslash\mathcal{F}_{C}$ of cell-edge RBs. We define the
number of RBs allocated to the cell-center area to be $N_{C}=\rho
N_{\text{RB}}$, where $0{\leq\rho\leq}1$ is the spectrum allocation factor.
The set $\mathcal{F}_{T}\backslash\mathcal{F}_{C}$ is then segmented into
three equal parts, namely $\mathcal{F}_{E1}$, $\mathcal{F}_{E2}$ and
$\mathcal{F}_{E3}$, of size $N_{E}=(1-\rho)N_{\text{RB}}/3$, that are assigned
to cell-edge users while taking care that adjacent cells will use different
subsets of RBs. Note that $N_{C}=N_{\text{RB}}-3N_{E}$ must be a non-negative
integer value less or equal than $N_{\text{RB}}$ and thus,
$N_{E}\in\\{0,1,\ldots,\lfloor N_{\text{RB}}/3\rfloor\\}$ and $\rho$ can only
take values in the set

$\mathcal{S}_{\rho}=\left\\{\frac{N_{\text{RB}}-3\lfloor
N_{\text{RB}}/3\rfloor}{N_{\text{RB}}},\frac{N_{\text{RB}}-3(\lfloor
N_{\text{RB}}/3\rfloor-1)}{N_{\text{RB}}},\ldots,1\right\\},$ (1)

where $\lfloor x\rfloor$ denotes the floor operator. When relying on SFR, the
set $\mathcal{F}_{T}$ is divided into three subsets of $N_{\text{RB}}/3$ RBs
that, with a slight abuse of notation, will also be denoted by
$\mathcal{F}_{E1}$, $\mathcal{F}_{E2}$ and $\mathcal{F}_{E3}$. On any given
cell, one of the RB subsets is allocated to the cell-edge area, thus
$N_{E}=N_{\text{RB}}/3$, while the remaining RBs are assigned to the cell-
center area, that is, $N_{C}=2N_{\text{RB}}/3$.

The positions of the users at a given instant are assumed to conform to a
stationary PPP of normalized intensity $\lambda$ (measured in users per area
unit). As a consequence, the probability distribution of the number
$M_{\mathcal{S}}$ of users falling within any spatial region $\mathcal{S}$ of
area $A_{\mathcal{S}}$ adheres to a Poisson distribution, thus implying

$Pr\\{M_{\mathcal{S}}=k\\}=\frac{(\lambda A_{\mathcal{S}})^{k}e^{-\lambda
A_{\mathcal{S}}}}{k!}.$ (2)

###  II-B Channel model

The downlink channel includes both a distance-dependent path loss and a small-
scale fading333As it is typically done in the background literature (see [8,
12, 30, 31]), and for analytical simplicity, only pathloss and small scale
fading are taken into account in this paper while leaving the incorporation of
large scale fading (shadowing) to future work. Note that in this case, as the
cellular networks under consideration are always limited by inter-cell
interference, average SINR- and distance-based strategies used to classify
users as cell-center or cell-edge are virtually equivalent.. The instantaneous
SINR experienced by user $u$ in the cell of interest, also denoted as cell 0,
on any of the $N_{sc}$ subcarriers belonging to the $n$th RB during the period
$t$ can be expressed as

$\gamma_{u,n}(t)=\frac{P_{n}L(d_{0,u})|H_{0,u,n}(t)|^{2}}{N_{0}\Delta
f+I_{u,n}(t)},$ (3)

where, under the assumption of uniform power allocation, $P_{n}$ is the power
allocated per subcarrier, $d_{b,u}$ represents the distance between BS $b$ and
user $u$, $L(d_{b,u})$ is the path loss component characterizing the link
between the $b$th BS and the $u$th user, $H_{b,u,n}(t)\sim\mathcal{CN}(0,1)$
is the frequency response444Note that, since the proposed scheduling and
resource allocation framework are rooted on capacity formulas optimized on an
RB-by-RB basis, the possible frequency correlation among RBs is irrelevant,
thus making unnecessary the specification of any frequency correlation
profile. resulting from the small-scale fading channel linking the $b$th BS to
user $u$ on the $n$th RB during scheduling period $t$, $N_{0}$ is the noise
power spectral density, $\Delta f=B_{\text{RB}}/N_{sc}$ is the subcarrier
bandwidth, and $I_{u,n}(t)$ corresponds to the interference term.

In the particular case of FFR, the ICI can be obtained as

$I_{u,n}(t)=\sum_{b\in\Phi_{n}}P_{n}L\left(d_{b,u}\right)|H_{b,u,n}(t)|^{2},$
(4)

where $P_{n}=P_{T}/(N_{sc}(N_{C}+N_{E}))$, with $P_{T}$ denoting the transmit
power available at the BS, and $\Phi_{n}$ represents the RB-dependent set of
interfering BSs. In the SFR configuration, the transmit power allocated per
cell-edge subcarrier $P_{n}^{E}=P_{T}/(N_{sc}(\beta N_{C}+N_{E}))$ is larger
than the corresponding one allocated per cell-center subcarrier
$P_{n}^{C}=P_{T}/(N_{sc}(N_{C}+N_{E}/\beta))$, where $P_{n}^{C}=\beta
P_{n}^{E}$, with $0{\leq\beta\leq}1$ denoting the power control factor. Hence,
in (3) it holds that $P_{n}=P_{n}^{E}$ for a cell-edge user and
$P_{n}=P_{n}^{C}$ for a cell-center user. Hence, the SFR interference term is
given by

$\begin{split}I_{u,n}(t)=&\sum_{b\in\Phi_{n}^{C}}P_{n}^{C}L\left(d_{b,u}\right)|H_{b,u,n}(t)|^{2}\\\
+&\sum_{b\in\Phi_{n}^{E}}P_{n}^{E}L\left(d_{b,u}\right)|H_{b,u,n}(t)|^{2}\textrm{
},\end{split}$ (5)

where $\Phi_{n}^{C}$ and $\Phi_{n}^{E}$ denote the sets of interfering BSs
affecting cell-center and cell-edge users, respectively.

##  III Optimal R5pD-based design

In this section we propose a 5th-percentile user rate constrained design for
an FFR/SFR-aided OFDMA-based multi-cellular network. The proposed optimization
strategy aims at determining the size of the reuse scheme-related spatial,
frequency and power partitions maximizing the average cell throughput while
satisfying the corresponding operator-defined system constraints. The
parameters used to pose this novel optimal design are the distance threshold
ratio $\omega\triangleq R_{th}/R_{m}$, the spectrum allocation factor
$\rho\triangleq N_{C}/N_{\text{RB}}$ and the power control factor
$\beta\triangleq P_{n}^{C}/P_{n}^{E}$. The appropriate selection of these
parameters significantly affects the average cell throughput (measured in bps)
that can be expressed as

$\overline{\eta}(\omega,\zeta)=\overline{\eta}^{C}(\omega,\zeta)+\overline{\eta}^{E}(\omega,\zeta),$
(6)

where $\overline{\eta}^{A}(\omega,\zeta)$ represents the average throughput in
the cell-region $A$, being $A$ a token used to represent either the cell-
center region $C$ or the cell-edge region $E$ and $\zeta$ represents a token
that can be either $\rho$ when using FFR or $\beta$ using SFR.

Under R5pD, the constraint of the 5th-percentile user rate $R5\%$ is used.
This is an important fairness metric to consider, since modern cellular
networks are increasingly required to provide high data-rate and also
guaranteed QoS throughout the cell. In fact, 5G initiatives explicitly state
requirements for the 5th-percentile rate [3]. The $R5\%$ (measured in bps)
reflects the throughput achieved by the worst users in the system, typically
located at the cell-edge. In general, the $x$th-percentile of users rate
$Rx\%$ is defined as the $x$%-tile of the users throughput CDF, that is,
$F_{\overline{\eta}_{u}}(Rx\%)=x/100$. The optimization problem using R5pD can
then be formulated as

$\begin{split}(\omega^{*},\zeta^{*})=&\arg\max_{\begin{subarray}{c}\frac{R_{0m}}{R_{m}}{\leq\omega\leq}1\\\
0\leq\zeta\leq
1\end{subarray}}\overline{\eta}^{C}(\omega,\zeta)+\overline{\eta}^{E}(\omega,\zeta),\\\
&\textup{ subject to }F^{-1}_{\overline{\eta}_{u}}(0.05)\geq v_{o}\end{split}$
(7)

where $R_{0m}$ denotes the minimum distance of a user from its serving BS and
$v_{o}$ represents the throughput requirement.

Hence, using R5pD allows the control of a more equitable distribution of both
spectrum and transmit power that can result in a higher degree of fairness
between cell-center and cell-edge users, at the cost of reducing the average
cell throughput with respect to the well-known FxD-based design [30, 31, 6].
It is worth pointing out that the optimization problem (7) can be efficiently
solved using standard software optimization packages (e.g., Matlab). In
following sections the analytical framework allowing the theoretical
calculation of the performance metrics used to pose the optimization problem
in (7) is developed.

##  IV Characterization of the throughput

###  IV-A Statistical characterization of the users throughput

By definition, the CDF characterizing the average throughput allocated to user
$u$ in region $A$ can be expressed as
$F_{\overline{\eta}^{A}_{u}}(v)=Pr\\{\overline{\eta}_{u}^{A}\leq v\\}$. Due to
the Poissonian distribution of users in a cell, the number of users in the
cell of interest, denoted as $M_{0}$, is a non-negative integer random
variable. For a particular event $M_{0}=k$, with
$k\in\\{0,1,\ldots,\infty\\}$, the random variable $k_{A}$, representing the
number of users in cell region $A$, can only take values in the set
$\\{0,1,\ldots,k\\}$. Furthermore, for a particular value of $k_{A}$ there are
a total of ${k\choose k_{A}}$ combinations of users in which $k_{A}$ of them
are located in region $A$ and the remaining $(k-k_{A})$ are located in the
tagged cell but outside region $A$. As all previous events are mutually
exclusive and its probabilities sum to unity, the total probability theorem
can be applied to obtain

$\begin{split}F_{\overline{\eta}^{A}_{u}}(v)&=\sum_{k=1}^{\infty}Pr\\{M_{0}=k\\}\sum_{k_{A}=1}^{k}{k\choose
k_{A}}P_{A}^{k_{A}}\\\
&\times\left(1-P_{A}\right)^{k-k_{A}}\left[\frac{k_{A}}{k}Pr\\{\overline{\eta}_{u}^{A}(k_{A})\leq
v|k_{A}\\}\right],\end{split}$ (8)

where $P_{A}$ is the probability that a user is located in cell-region $A$ and
$\overline{\eta}_{u}^{A}(k_{A})$ represents the average throughput experienced
by a user $u$ located in cell-region $A$ conditioned on the event that there
are $k_{A}$ users in that region. As users are assumed to be uniformly
distributed in the cell, then

$P_{A}=\frac{(R_{U}^{A})^{2}-(R_{L}^{A})^{2}}{R_{m}^{2}-R_{0m}^{2}},$ (9)

where $R_{L}^{A}$ and $R_{U}^{A}$ are the lower and upper radii of the
circumferences delimiting cell-region $A$.

Now, defining $M_{A}$ as a non-negative integer random variable representing
the number of users in the cell-region $A$, the average throughput experienced
by a user $u$ located in cell-region $A$ when $M_{A}=k$, can be expressed as

$\begin{split}\overline{\eta}^{A}_{u}(k)&=\mathbb{E}_{d_{0,u}}\left\\{N_{A}\textup{
}\overline{\eta}^{A}_{u,n}(k,d)\right\\}\\\
&=N_{A}\int_{R_{L}^{A}}^{R_{U}^{A}}\overline{\eta}^{A}_{u,n}(k,d)f_{d_{0,u}}(d)\textrm{
}\mathrm{d}d,\end{split}$ (10)

where $\overline{\eta}_{u,n}^{A}(k,d)$ is the scheduler-dependent average
throughput experienced by a user $u$ on the $n$th RB in cell-region $A$,
$N_{A}$ is the number of RBs allocated to cell-region $A$ and $f_{d_{0,u}}(d)$
is the probability density function (PDF) of the random variable $d_{0,u}$.

Relying on the knowledge of the instantaneous SINRs experienced by all users
$q\in\mathcal{M}_{A}$, the PF scheduler allocates RB $n\in\mathcal{F}_{A}$ to
the user $u\in\mathcal{M}_{A}$ satisfying

$u=\arg\max_{q\in\mathcal{M}_{A}}\\{w_{q}(t)\gamma^{A}_{q,n}(t)\\},$ (11)

where $\mathcal{M}_{A}$ is the set used to index all users in cell-region $A$,
and $w_{q}(t)=1/\mu_{q}(t)$ is the weighting (prioritization) coefficient for
user $q$, which can be computed using a moving average of the short-term CSI
over a window of $W$ scheduling periods as

$\mu_{q}(t)=\left(1-\frac{1}{W}\right)\mu_{q}(t-1)+\sum_{n\in\mathcal{F}_{A}}\iota_{q,n}(t)\frac{\gamma^{A}_{q,n}(t)}{W},$
(12)

with $\iota_{q,n}(t)$ being the indicator function of the event that user $q$
during scheduling period $t$ is scheduled on RB $n$.

It is shown in [32] that, for large values of $W$ and after the PF scheduler
becomes stable, $\mu_{q}(t)$ exhibits very little variation with $t$ and thus,
it can be harmlessly approximated by its statistical expectation555In
particular, and as it has been duly checked via simulation results, the
approximation of constant PF weights is valid for window lengths as low as
$W=50$ scheduling periods., that is,
$\mu_{q}(t)\simeq\mathbb{E}\\{\mu_{q}(t)\\}\triangleq\overline{\mu}_{q}$.
Hence, user $u\in\mathcal{M}_{A}$ will be scheduled on subcarrier
$n\in\mathcal{F}_{A}$ whenever

$\varphi^{A}_{u,n}(t)>\varphi^{A}_{\max,u,n}(t)\triangleq\max_{\begin{subarray}{c}q\in\mathcal{M}_{A}\\\
q\neq u\end{subarray}}\left\\{\varphi^{A}_{q,n}(t)\right\\},$ (13)

where $\varphi^{A}_{q,n}(t)\triangleq\gamma^{A}_{q,n}(t)/\overline{\mu}_{q}$.
According to the previous definition of the PF scheduler, the instantaneous
throughput experienced by a user $u$ on the $n$th RB in cell-region $A$,
conditioned on the event that there are $M_{A}=k$ users and on the set of
distances $\boldsymbol{d}=\\{d_{0,u}\\}_{\forall\,u\in\mathcal{M}_{A}}$, can
be evaluated as666Note that since the channel is assumed to be stationary,
from this point onwards the time dependence (i.e., (t)) of all the variables
will be dropped unless otherwise stated.

$\begin{split}&\eta_{u,n}^{A}(\gamma_{u,n}|k,\boldsymbol{d})\\\
&\qquad=\begin{cases}\qquad\qquad
0,&\varphi^{A}_{\max,u,n}>\frac{\gamma^{A}_{u,n}}{\overline{\mu}_{u}}\\\
B_{\text{RB}}\log_{2}(1+\gamma^{A}_{u,n}),&\varphi^{A}_{\max,u,n}\leq\frac{\gamma^{A}_{u,n}}{\overline{\mu}_{u}}\end{cases}\end{split}$
(14)

and the average throughput experienced by a user $u$ on the $n$th RB in cell-
region $A$ can be obtained as

$\begin{split}\overline{\eta}_{u,n}^{A}(k,\boldsymbol{d})&=B_{\text{RB}}\int_{0}^{\infty}\log_{2}(1+x)f_{\gamma^{A}_{u,n}|d_{0,u}}\left(x|d\right)\\\
&\quad\qquad\times
F_{\varphi^{A}_{\max,u,n}|\boldsymbol{d}}\left(\frac{x}{\overline{\mu}_{u}}\,\Bigr{|}\,\boldsymbol{d}\right)\mathrm{d}x\\\
&=B_{\text{RB}}\int_{0}^{\infty}\log_{2}(1+x)f_{\gamma^{A}_{u,n}|d_{0,u}}\left(x|d\right)\\\
&\quad\qquad\times\prod_{\begin{subarray}{c}q\in\mathcal{M}_{A}\\\ q\neq
u\end{subarray}}F_{\varphi^{A}_{q,n}|d_{0,q}}\left(\frac{x}{\overline{\mu}_{u}}\,\Bigr{|}\,d_{0,q}\right)\mathrm{d}x,\end{split}$
(15)

where $F_{\varphi^{A}_{\max,q,n}|\boldsymbol{d}}\left(x|\boldsymbol{d}\right)$
is the conditional CDF of $\varphi^{A}_{\max,q,n}$ conditioned on the set of
distances $\boldsymbol{d}$, and $f^{A}_{\gamma_{q,n}|d_{0,q}}(x|d_{0,q})$ and
$F^{A}_{\varphi_{q,n}|d_{0,q}}(x|d_{0,q})$ are used to denote, respectively,
the PDF of $\gamma^{A}_{q,n}$ and the CDF of $\varphi^{A}_{q,n}$ conditioned
on $d_{0,q}$.

Under the large $W$ assumption, $\overline{\mu}_{q}$ tends to be proportional
to $\mathbb{E}\\{\gamma^{A}_{q,n}(t)\\}$ and thus, the random variable
$\varphi^{A}_{q,n}(t)=\gamma^{A}_{q,n}(t)/\overline{\mu}_{q}$, as shown by Liu
and Leung in [32], tends to be unrelated to the pathloss component.
Consequently, it can be safely assumed that the conditional random variables
$\\{\varphi^{A}_{q,n}|d_{0,q}\\}_{\forall\,q\in\mathcal{M}_{A}}$ are
independent and identically distributed (i.i.d.). That is, given the positions
of users in region $A$, it is assumed that on each subcarrier $n$ in region
$A$ the users are statistically equivalent in terms of the scheduling metrics.
Then expression (15) simplifies to

$\begin{split}\overline{\eta}_{u,n}^{A}(k,d)&=B_{\text{RB}}\int_{0}^{\infty}\log_{2}(1+x)f_{\gamma^{A}_{u,n}|d_{0,u}}\left(x|d\right)\\\
&\quad\qquad\times\left(F_{\gamma^{A}_{u,n}|d_{0,u}}(x|d)\right)^{k-1}\mathrm{d}x,\end{split}$
(16)

where the conditional CDF $F_{\gamma^{A}_{u,n}|d_{0,u}}(x|d)$ of the
instantaneous SINR experienced by user $u$ on any subcarrier of the $n$th RB
can be obtained as shown in the Appendix.

Let us now define the distance
$d(k,v)\triangleq\\{d:\overline{\eta}_{u}^{A}(k,d)=v\\}$. Since
$\overline{\eta}_{u}^{A}(k,d)$ is a decreasing function of $d$ on the interval
$\left[R_{L}^{A},R_{U}^{A}\right]$, then

$\begin{split}Pr&\\{\overline{\eta}_{u}^{A}(k)\leq v|k\\}=Pr\\{d_{0,u}\geq
d(k,v)\\}\\\ &=\left\\{\begin{split}&\textup{ }\textup{ }\textup{ }\textup{
}\textup{ }\textup{ }1\textup{ }\textup{ }\textup{ }\textup{ }\textup{
}\textup{ }\textup{ }\textup{ }\textup{ }\textup{ }\textup{ }\textup{
},\textup{ }d(k,v)\leq R_{L}^{A}\\\
&\frac{(R_{U}^{A})^{2}-{d(k,v)}^{2}}{(R_{U}^{A})^{2}-(R_{L}^{A})^{2}},\textup{
}R_{L}^{A}\leq d(k,v)\leq R_{U}^{A}\\\ &\textup{ }\textup{ }\textup{ }\textup{
}\textup{ }\textup{ }0\textup{ }\textup{ }\textup{ }\textup{ }\textup{
}\textup{ }\textup{ }\textup{ }\textup{ }\textup{ }\textup{ }\textup{
},\textup{ }d(k,v)\geq R_{U}^{A}\end{split}\right..\end{split}$ (17)

Thus, substituting (17) in (8), the users throughput CDF of FFR/SFR-aided
OFDMA-based multi-cellular networks can be finally expressed as

$F_{\overline{\eta}_{u}}(v)\triangleq Pr\\{\overline{\eta}_{u}\leq
v\\}=F_{\overline{\eta}^{C}_{u}}(v)+F_{\overline{\eta}^{E}_{u}}(v).$ (18)

###  IV-B Cell throughput analysis

The downlink average throughput experienced in the cell-region $A$ of the
FFR/SFR-aided OFDMA-based multi-cellular network can be derived as

$\begin{split}\overline{\eta}^{A}(\omega,\zeta)=&\sum_{k=1}^{\infty}Pr\\{M_{0}=k\\}\sum_{k_{A}=1}^{k}{k\choose
k_{A}}P_{A}^{k_{A}}\\\
&\times\left(1-P_{A}\right)^{k-k_{A}}\left[k_{A}\textup{
}\overline{\eta}_{u}^{A}(k_{A})\right].\end{split}$ (19)

Substituting the expressions (16) and (10) in (19), the average throughput in
the cell-region A can be simplified to

$\begin{split}&\overline{\eta}^{A}(\omega,\zeta)=\frac{2B_{\text{RB}}N_{A}\textup{log}_{2}e}{(R_{U}^{A})^{2}-(R_{L}^{A})^{2}}\int_{R_{\textup{L}}^{A}}^{R_{\textup{U}}^{A}}\int_{0}^{\infty}\frac{1-\Psi(x,y)}{1+x}y\,\mathrm{d}x\,\mathrm{d}y,\end{split}$
(20)

where

$\Psi(x,y)\triangleq
e^{-\pi\lambda(R_{m}^{2}-R_{0m}^{2})P_{A}\left[1-F_{\gamma^{A}_{u,n}|d_{0,u}}(x|y)\right]}.$
(21)

TABLE I: Network parameters

Parameter | Value
---|---
Cell radius | 500 m
Minimum distance between BS and users | 35 m
Antenna configuration | SISO
Transmit power of the BS | 46 dBm
Antenna gain at the BS | 14 dBi
Power spectral density of noise | -174 dBm/Hz
Receiver noise figure | 7 dB
Total bandwidth | 20 MHz
Subcarrier spacing | 15 kHz
FFT size | 2048
Occupied subcarriers (including DC) | 1201
Guard subcarriers | 847
Number of resource blocks | 100
Subcarriers per RB | 12
Path loss model |  $15.3+37.6\log_{10}(d)$ dB
Scheduling policy | proportional fair

##  V Numerical results

In order to validate the proposed analytical framework while also providing
valuable design guidelines, the 19-cell network shown in Fig. 1 is considered,
where the cell of interest is surrounded by two rings of interfering BSs. As
stated in previous sections, users are distributed over the coverage area
using a PPP of normalised intensity $\lambda$. For the sake of presentation
clarity, although the analytical framework has been developed using the
normalised intensity of the PPP and to ease the visual interpretation, results
in this section will be shown as a function of the average number of users per
cell $M\triangleq\pi\lambda\left(R_{m}^{2}-R_{0m}^{2}\right)$. The main system
parameters used to generate both the analytical and simulation results have
been particularized using numerology drawn from the downlink specification of
an LTE/LTE-A network and have been summarized in Table I. Monte Carlo
simulations have been performed using a Matlab™-based environment.

###  V-A Background designs

The proposed R5pD-based design will be compared to optimization designs
previously proposed in the literature. In particular, two benchmark designs
will be explored:

  * •

Fixed spectrum/power factor design (FxD): With this design, the use of a fixed
frequency partition is considered when the FFR scheme is employed or a fixed
power control when the SFR scheme is used. This design typically leads to high
spectral efficiencies at the cost of user fairness [30, 31, 6].

  * •

Quality-constrained Design (QoScD): Under this design the optimization problem
is constrained in order to ensure that the cell-edge throughput is, at least,
a fixed fraction of the cell-center throughput. Consequently, a certain
fairness degree (i.e., a certain QoS for the cell-edge users) is enforced
between cell-center and cell-edge users while maximizing the area spectral
efficiency [12, 6].

Owing to the fixed spectrum-allocation factor design in FFR, the spectrum
allocation factor is selected to be $\rho=\rho_{o}$ (with, typically,
$\rho_{o}\simeq 1/2$). Consequently, the parameter $\omega$ is the only one
that remains to be optimized, thus allowing the optimization problem to be
formulated as

$\omega^{*}=\arg\max_{\frac{R_{0m}}{R_{m}}{\leq\omega\leq}1}\overline{\eta}^{C}(\omega,\rho_{o})+\overline{\eta}^{E}(\omega,\rho_{o}).$
(22)

When the SFR scheme is used with the fixed power-control factor design, the
power factor is fixed to $\beta=\beta_{o}$ and now, the optimization problem
can be formally defined as

$\omega^{*}=\arg\max_{\frac{R_{0m}}{R_{m}}{\leq\omega\leq}1}\overline{\eta}^{C}(\omega,\beta_{o})+\overline{\eta}^{E}(\omega,\beta_{o}).$
(23)

In the QoScD approach, a QoS requirement $\varrho$ is specified enforcing that
the guaranteed average cell throughput per-edge user is at least a fraction
$\varrho$ of the average cell throughput per-center user. Hence, the system
parameters are adjusted to trade the data rates required by the cell-center
users against those of the cell-edge users. The corresponding optimization
problem can be formulated as

$\begin{split}(\omega^{*},\zeta^{*})=&\arg\max_{\begin{subarray}{c}\frac{R_{0m}}{R_{m}}{\leq\omega\leq}1\\\
\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }0\leq\zeta\leq
1\end{subarray}}\overline{\eta}^{C}(\omega,\zeta)+\overline{\eta}^{E}(\omega,\zeta),\\\
&\textup{ subject to
}\tau_{u}^{E}(\omega,\zeta)\geq\varrho\,\tau_{u}^{C}(\omega,\zeta),\end{split}$
(24)

where
$\tau_{u}^{A}(\omega,\zeta)=\frac{\overline{\eta}^{A}(\omega,\zeta)}{MP_{A}}$
is the per-user average throughput in cell-region $A$.

(a) R5pD

(b) FxD

(c) QoScD

Figure 2: Average cell throughput versus distance threshold ratio for
different optimal designs under both FFR and SFR.

###  V-B Proposed R5pD-based design and Comparison

Figure 2 presents the average cell throughput as a function of the distance
threshold ratio $\omega$ under both FFR and SFR reuse schemes. Illustrating
the system behavior for different optimal designs, analytical and simulation
results are provided using an average number of users per cell $M=32$. The
particular shape of the curves shown in this figure is a direct consequence of
the trade-off between the average throughput values attained at both the cell-
edge and cell-center regions as a function of the distance threshold ratio
$\omega$ and the optimal spectrum or power allocation factor $\zeta^{*}$. The
first point to highlight is the very good match between analytical and
simulation results, thus validating the analytical framework that has been
developed in Section IV.

The results, in Fig. 2(a), have been obtained for the R5pD-based design using
two different values of the $R5\%$ constraint, corresponding to low and high
cell-edge user performance requirements. For each value of $\omega$, the value
of $\zeta$ (which is equal to $\rho$ when using FFR and equal to $\beta$ for
SFR) maximizing the average cell throughput has been used. The optimal pairs
$(\omega^{*},\zeta^{*})$ in the graphs are indeed the solutions to the problem
posed in (7). As expected, increasing the $R5\%$ requirement enforces a higher
cell-edge user performance at the cost of decreasing the average cell
throughput for both reuse schemes. However, the behavior of both frequency
reuse schemes when facing high and low cell-edge user performance requirements
is markedly different. First of all, it can be observed that while the average
cell throughput of an FFR-based system increases a lot when decreasing the
fairness requirement, the average cell throughput of an SFR-based system seems
to have very low sensitivity to variations in the 5th-percentile user rate
constraint. Furthermore, and related to the previous statement, for a system
with a high 5th-percentile user rate constraint (e.g., 0.7 Mbps) the SFR
strategy clearly outperforms the FFR scheme, whereas for a system with a low
5th-percentile user rate constraint (e.g., 0.1 Mbps) the SFR scheme is
outperformed by the FFR strategy777Note that, in contrast to what has been
considered in our paper, Novlan _et al._ in [9], one of the most cited works
in the stochastic geometry-based FFR/SFR analytical frameworks, implicitly
assume the use of an over-simplistic round robin (RR) scheduling rule and a
uniform distribution of users of sufficient density such that all available
sub-bands are utilized, a condition that cannot be guaranteed in realistic
deployments serving a reduced number of users and can eventually lead to a
non-negligible performance degradation due to the _waste_ of resources when
allocating sub-bands to unpopulated FFR/SFR-defined cellular regions. However,
despite the differences between the assumptions considered in both research
works, it is remarkable that the results just shown are consistent with one of
the most important results derived in [9]. In particular, as stated in [9,
Section V.C], defining $\beta$ as the quotient between the powers allocated to
cell-edge and cell-center resources, “…at low values of $\beta$ [leading to
low 5th-percentile user rates], SFR provides lower coverage probability
compared to Strict FFR. However, …if $\beta$ is sufficiently large [leading to
high 5th-percentile user rates], SFR will surpass strict FFR …”.. The
explanations sustaining these interesting but somewhat counterintuitive
results are:

  * •

From the results presented in Fig. 2(a) it is quite obvious that, for a given
average number of users per cell and irrespective of the frequency reuse
scheme, the optimal distance threshold ratio $\omega$ is rather insensitive to
the variations of the $R5\%$ requirement. Thus, the only parameter playing a
central role when optimizing the resource allocation is either $\rho$ in the
FFR-based systems or $\beta$ in the SFR-based schemes. Since the network under
study is to a large extent an interference-limited cellular system, varying
$\beta$ has a very limited effect on the SINR experienced by the served users
and, as a consequence, changing the value of $\beta$ has also very limited
effects on the average cell throughput. This can be straightforwardly deduced
from the mild variations of the optimal average cell throughput the
modifications of the $R5\%$ requirement produce. In contrast, adapting the
number of subcarriers allocated to both cell-center and cell-edge in
accordance to the $R5\%$ constraint can produce large modifications of the
average cell throughput. This is because the subcarriers that are not needed
to satisfy the throughput requirements of the cell-edge users can be
transferred to the cell-center users in order to increase the average cell
throughput.

  * •

For a high $R5\%$ requirement, the fact that the SFR-based networks reuse the
whole set of available subcarriers in each cell of the system, with the
consequent increase in potentially achievable spectral efficiency, provides a
clear advantage of this frequency reuse scheme in front of FFR. However, for a
low $R5\%$ requirement, the FFR-based scheme can take advantage of the
flexibility provided by its adaptive frequency allocation capability to
increase the average cell throughput while decreasing the throughput
experienced by the cell-edge users.

For comparative purposes, the average cell throughput obtained when using FxD
and QoScD are also presented in Figs. 2(b) and 2(c), respectively. Each value
of $\omega^{*}$ leading to the maximum average cell throughput illustrated in
Fig. 2(b) is indeed the solution to either problem (22), for FFR, or problem
(23), for SFR. As the value of $\beta_{o}$ increases, the optimal average cell
throughput of SFR-aided systems improves due to the higher SINRs experienced
by the cell-center users, although this comes at the cost of harming the cell-
edge throughput due to the increased ICI. In fact, decreasing the power
control factor $\beta_{o}$ causes the cell throughput of SFR to tend to that
of FFR for $\rho_{o}=1/2$. The most important disadvantage of FxD-based
schemes in front of strategies based on the proposed R5pD is that the former
are unable to adaptively provide QoS guarantees to the users served by the
cellular network. Figure 2(c) shows the behavior of the QoScD strategy for
different QoS constraints. Quality factors of $\varrho=0.02$ and $\varrho=0.2$
have been considered, corresponding to low and high fairness requirements
between cell-center and cell-edge users, respectively. Irrespective of the
applied reuse scheme, increasing the quality factor $\varrho$, and similar to
the behavior related to the $R5\%$ constraint, enforces a higher degree of
fairness among cell-center and cell-edge users at the cost of decreasing the
average cell throughput. Again, for each value of $\omega$, the value of
$\zeta$ maximizing the average cell throughput has been used and the pairs
$(\omega^{*},\zeta^{*})$ are now the solutions to problem (24). In this case,
QoScD-based schemes can apparently provide similar fairness related control
capabilities as those provided by the R5pD-based strategies. However, note
that the QoS constrained design is based on performance metrics that, in
contrast to R5pD, are not usually employed in neither the evaluation of
current LTE/LTE-A cellular networks nor the specification of envisaged 5G
systems.

(a) FFR ($R5\%=0.7$ Mbps)

(b) FFR

(c) SFR

Figure 3: CDF of the per-region user throughput and CDF of the average user
throughput for R5pD under both FFR and SFR.

(a) FxD, FFR/SFR

(b) QoScD, FFR

(c) QoScD, SFR

Figure 4: CDF of the average user throughput for FxD and QoScD under both FFR
and SFR.

Having established the accuracy of the proposed analytical method and for the
sake of clarity, results shown in Fig. 3, depict only the theoretical curves
notwithstanding the fact that they have been duly checked via simulation.
Figure 3(a) shows the CDF of the average user throughput (blue curve with
circle markers) for the FFR case and using the constraint $R5\%~{}=~{}0.7$
Mbps. Since the pattern observed in this CDF curve will recurrently appear in
most graphs shown in this section, it is worth explaining the causes of its
shape. To this end, in Fig. 3(a) the CDF curves for cell-edge only (green
curve with triangle markers) and cell-center only (red curve with square
markers) users’ throughput are also depicted. The dashed vertical line signals
the throughput value for which the cell-edge CDF reaches probability one. That
is, there is no cell-edge user experiencing a throughput higher than this
value. The solid vertical line pinpoints the throughput value for which the
cell-center CDF departs from zero. That is, there is no cell-center user
experiencing a throughput lower than this value. Hence, the cell-edge CDF
conditions the average user CDF for low throughput values whereas the cell-
center CDF conditions the average user CDF for high throughput values. As the
throughput value pinpointed by the solid line is higher that the one signaled
by the dashed line, the average user throughput CDF shows a plateau whose
width is determined by the disparity between the throughput values experienced
by cell-edge and cell-center users and its constant value represents the
percentage of cell-edge users in the system. Logically, given the uniform user
distribution, the percentage of users in each cell region is determined by the
optimal distance threshold value (see Fig. 2(a)) and thus, the higher the
value of $\omega^{*}$ the lower the level of the plateau.

Figures 3(b) and 3(c) represent the CDF of the average user throughput under
FFR and SFR, respectively, when considering different values of the $R5\%$
constraint. For the sake of clarity, a zoomed region focusing on the lower
values of the curves is also shown where it can easily be checked how the
throughput values for which $F_{\overline{\eta}_{u}}=0.05$ coincide with the
corresponding $R5\%$ constraints. The first remarkable point to notice is that
as the optimal distance thresholds for SFR are slightly smaller than their FFR
counterparts (see Fig. 2(a)) and consequently, the plateau values observed for
SFR-based networks are somewhat larger than those obtained when using FFR-
based frequency reuse schemes. Another interesting fact to highlight is that a
lower $R5\%$ requirement leads to a larger disparity between cell-center and
cell-edge users’ throughput as it can be inferred from the wider plateau
width, particularly noticeable for the FFR schemes. Wider plateaus imply
larger cell-center throughput in particular and larger average cell throughput
in general as it can be corroborated by looking at the results presented in
Fig. 2(a). Notice how in SFR-based systems the differences in the plateau
widths are much more modest than in FFR-based networks and correspondingly, as
it has been mentioned previously, the differences they experience in optimal
average cell throughput are barely influenced by the $R5\%$ requirement.

In order to compare R5pD-based results with those obtained under the benchmark
designs, Fig. 4 shows the CDF of the average user throughput for FxD and QoScD
under both FFR and SFR frequency reuse schemes and assuming an average number
of users per cell $M=32$. As shown in Fig. 4(a), results for FFR/FxD have been
obtained using two fixed spectrum allocation factors, namely, $\rho=1/2$ and
$\rho=1/4$, and results for SFR/FxD have been obtained using two fixed power
allocation factors, namely, $\beta_{0}=1/2$ and $\beta_{0}=1/4$. These results
reveal, first, that the obtained $R5\%$ values are somewhere in between 0.5
and 1 Mbps, however, these values cannot be controlled by design. Second, in
all cases the plateau level is located around 0.9, that is, approximately
$90\%$ of the users are located in the cell-edge whose optimal area, as was
shown in Fig. 2(b), is barely dependent on the frequency reuse strategy (FFR
vs SFR) and/or fixed allocation factor ($\rho_{0}$ or $\beta_{0}$). Finally,
the plateau widths are very sensitive to variations of $\rho_{0}$ for the FFR-
based networks and fairly insensitive to variations of $\beta_{0}$ for the
SFR-based systems, thus corroborating the conclusions extracted from Fig. 2(b)
in which we have stated that the optimal average throughput values in FFR-
based networks are very sensitive to changes in $\rho_{0}$ and they are barely
sensitive to changes in $\beta_{0}$ in SFR-based cellular infrastructures.

Similarly, Figs. 4(b) and 4(c) represent the CDF of the average user
throughput for FFR/QoScD and SFR/QoScD, respectively, using two quality
factors, namely, $\varrho=0.02$ and $\varrho=0.2$. First of all, it can be
observed that increasing the quality factor generates an increase in the
$R5\%$ value but, again, its particular value cannot be controlled by design.
Furthermore, varying the quality factor requirement drastically modifies the
shape of the CDF curves, with a large decrease in both the level and the width
of the plateau as $\varrho$ increases from 0.02 to 0.2. This serves to
corroborate results presented in Fig. 2(c), where it was observed that
changing the quality factor produces large variations in both the optimal
distance threshold ratio $\omega^{*}$ and the optimal average cell throughput.
Noticeably, the use of high quality factors, which is equivalent to enforcing
strong throughput fairness requirements between cell-center and cell-edge
users, causes a decrease in the disparity between cell-center and cell-edge
users’ throughput, which is revealed by the practical disappearance of the
plateau in the corresponding CDF curves.

(a) FFR Pareto front, $M=8$

(b) FFR Pareto front, $M=32$

(c) FFR Pareto front, $M=128$

(d) SFR Pareto front, $M=8$

(e) SFR Pareto front, $M=32$

(f) SFR Pareto front, $M=128$

Figure 5: Average cell throughput versus 5th-percentile user rate for R5pD
using different values of $M$ under both FFR and SFR.

(a) FFR scheme

(b) SFR scheme

Figure 6: Average cell throughput versus user load using different values of
5th-percentile user rate under both FFR and SFR.

Figure 5 shows the analytical and simulated average cell throughput as a
function of the 5th-percentile user rate constraint considering configurations
with different average number of users per cell. The results obtained through
simulation correspond to an extensive set of $(\omega,\zeta)$ values, while
analytical results are represented only for the optimal pairs
$(\omega^{*},\zeta^{*})$ and assuming, for each one of them, a different
$R5\%$ requirement. Using this setup, Figs. 5(a), 5(b) and 5(c), corresponding
to FFR, and Figs. 5(d), 5(e) and 5(f), obtained for SFR, provide a
comprehensive overview of the trade-offs that a network manager can establish
among contradictory objective functions such as the average cell throughput
and the 5th-percentile user rate. To this end, it is useful to examine these
results as solutions to a multiobjective optimization problem (MOP) [33] where
parameters ($\omega,\zeta$) lead to different performance compromises. A key
concept in MOPs is the optimal Pareto front, defined as the set of Pareto
optimal solutions (also known as non-dominated, Pareto efficient or non-
inferior solutions), defined as those for which none of the objective
functions can be improved without degrading some of the other ones. In Fig. 5,
the optimal Pareto front has been obtained using the analytical framework
described in previous sections by using, for each particular value of the 5th-
percentile user rate, the corresponding optimal pair $(\omega^{*},\zeta^{*})$.
Obviously, analytical solutions representing the Pareto front outperform the
rest of solutions obtained by simulation and, remarkably, the optimal
simulated values show a very good match with those representing the Pareto
front, thus validating once again the analytical framework proposed in this
paper. Note that for FFR, a discrete Pareto front has been represented, rather
than a continuous one as in the SFR case, because the optimal spectrum
allocation factor $\rho^{*}$ can only take values in the set
$\mathcal{S}_{\rho}$ in (1). In fact, for the specific system parameters used
in the simulation setups only the 5th-percentile of users rate represented by
the cross markers are implementable if our aim is to maximize the average cell
throughput. For example, for the $M=32$ users case (see Fig. 5(b)) setting a
target 5th-percentile user rate of, for instance, 1 Mbps provides an average
cell throughput of approximately 98 Mbps with a 5th-percentile user rate of
roughly 1.54 Mbps, which is obviously higher that the required target value.
Note that, due to the discrete set of subcarrier allocations, this is indeed
the only point of the Pareto front that actually fulfills the prescribed
$R5\%$ target. Furthermore, observe how, in contrast to what happens when
using FFR-based frequency reuse schemes, in SFR-based networks there is a wide
range of low 5th-percentile user rate values for which the average cell
throughput remains virtually constant. That is, decreasing the $R5\%$
requirement in SFR-based networks does not always lead to increasing the
global throughput performance of the network and this, as was previously
stated, is basically due to that B4G/5G cellular networks are, to a large
extent, interference-limited and hence, varying $\beta$ has a very limited
effect on the SINR experienced by the served users. This results in the fact,
clearly observable when comparing the optimal Pareto fronts depicted in Fig.
5, that the FFR-based frequency reuse schemes provide a better performance
than the SFR-based strategies for low values of the $R5\%$ constraint.

To gain further insight on the interrelation among the average cell
throughput, the 5th-percentile user rate and the number of active users in the
system, Fig. 6 depicts the average cell throughput versus user load under
different levels of $R5\%$ for both FFR- and SFR-based networks. Regarding the
FFR performance, it can be observed how increasing the $R5\%$ user rate
invariably leads to a lower average cell throughput as more resources need to
be allocated to the users experiencing poor channel conditions thus decreasing
the global throughput performance of the system. Increasing the load, up to a
certain number of users, produces an increase in the average cell throughput
thanks to the larger degree of multiuser diversity provided by the use of the
channel-aware PF scheduler. Beyond this number of users, which decreases when
increasing the $R5\%$ user rate, the amount of resources that need to be
allocated to the _worst_ users in the cell in order for them to achieve the
required $R5\%$ clearly masks the benefits produced by the corresponding
multiuser diversity and leads to a loss in average cell throughput.
Remarkably, as it can be observed in the graph corresponding to $R5\%=300$
Kbps (although not shown in the figure, for lower values of $R5\%$ the same
phenomenon would become visible for higher system loads), under very heavy
user loads the average cell throughput experiences a non-negligible increase.
This can be explained by the fact that the spectrum allocation factor can only
take values on the discrete set $\mathcal{S}_{\rho}$. There is a certain user
load for which the $R5\%$ constraint can be fulfilled by using
$\rho=(N_{\text{RB}}-3)/N_{\text{RB}}$, that is, there are still 3 RBs left
for use in the cell-edge. However, the admission of extra users causes, on one
hand, the optimal spectrum allocation factor to become 1, that is, the sudden
disappearance of the cell-edge, and on the other hand, the abrupt increase of
the optimal 5th-percentile user rate effectively provided to the _worst_ users
in the system (for instance, we can observe in Fig. 5(c) that a system with
$M=128$ users with a constraint $R5\%\geq 300$ Kbps effectively provides a
5th-percentile user rate of $\sim 490$ Kbps). The elimination of the cell-edge
should produce a very large increase in the average system throughput but the
increase of $R5\%$ translates to a decrease of this performance metric. The
combined effects of these rather radical changes produce the observed moderate
increase of the global throughput performance. The effects of user load in
SFR-based networks are very similar to those observed in the FFR case except
for the fact that, on one hand, due to the interference-limited character of
this scheme, the average cell throughput values that can be achieved for light
user loads are rather insensitive to the $R5\%$ user rate and, on the other
hand, given the continuous nature of the optimization parameters used in the
SFR-based networks, heavy user loads do not produce the rather
counterintuitive increase of average cell throughput shown by the FFR-based
systems. Nonetheless, it can be clearly observed that for a low $R5\%$
requirement and light to moderate user loads, the FFR-based scheme exploits
the flexibility provided by its adaptive frequency allocation characteristic
to increase the average cell throughput as, in this case, the amount of
resources that needs to be allocated to the edge users can be rather limited.
For a high $R5\%$ constraint and heavily loaded networks, however, the ability
of the SFR scheme to reuse all the subcarriers in each cell provides a clear
advantage of this frequency reuse scheme in front of FFR.

###  V-C R5pD-based design guidelines

Figure 7: Jain’s fairness index of the average user’s throughput versus
distance threshold ratio for different optimal designs under both FFR and SFR.

Having established the accuracy of the proposed model and in light of the
different performance behavior observed for FFR and SFR, it is appropriate to
single out some important facts that allow some practical insight to be gained
regarding the application of either SFR or FFR:

  * •

First and foremost, as already mentioned in Section VI.B, under the proposed
optimization framework of maximizing overall throughput subject to the
fulfilment of a minimum $R5\%$ throughput constraint, the use of FFR is
recommended when there are low $R5\%$ requirements, in contrast, SFR is the
preferred option for increasing values of the $R5\%$ constraint. Note that
this implicitly means that in setups where fairness plays an important role,
frequency reuse should take the form of SFR (in alignment with the findings in
[9]). In fact, and at a more general level, results shown so far have
demonstrated that setting a restriction on the $R5\%$ implicitly serves to
balance the cell-center and cell-edge performance.

  * •

Another important aspect worth taking into account in practical deployments is
the performance sensitivity to the different parameters and constraints. In
this respect, SFR allows a wider range of $R5\%$ throughput values while
barely affecting the overall system performance and, remarkably, the increase
of the $R5\%$ constraint induces a graceful degradation of the average cell
throughput.

  * •

Note that $R5\%$ can effectively act as a crank trading off fairness and
performance with the former metric being measurable in a variety of manners.
Our interest in $R5\%$ is because most wireless communications standards
target this metric when setting QoS requirements. Nevertheless, this fact does
not preclude the examination of fairness through any other indicator. As an
example, Fig. 7 shows the Jain’s fairness index (JFI) of the throughput among
all users in the system with the value of R5% as parameter. As expected, an
increase in the $R5\%$ requirement leads to an increase in the JFI thus
confirming an improvement in fairness. In connection with the previous point,
SFR is observed to be rather insensitive in terms of JFI whereas FFR is far
more influenced by this performance metric.

  * •

Finally, it is worth mentioning that the discrete nature of the set
$\mathcal{S}_{\rho}$ in the case of FFR, makes this scheme somewhat more
restrictive when it comes to allocating resources to cell-center and cell-edge
when compared to the continuous center-edge subcarrier split employed in SFR.

##  VI Conclusion

This paper has presented and validated a novel analytical framework allowing
the design of optimal 5th-percentile user rate constrained FFR/SFR-aided
OFDMA-based multi-cellular networks using channel-aware PF scheduling. The
proposed optimal design, termed R5pD, addresses the maximization of a cell
throughput-based utility function subject to a constraint on the 5th-
percentile user rate, a typical target requirement in most modern cellular
systems. To deal with the R5pD-based optimization problem, an statistical
characterization of both per-user and system throughput has been analytically
derived and its performance has been thoroughly compared to two previous
proposals, namely, the fixed spectrum/power factor design (FxD) and the QoS-
constrained design (QoScD).

Regarding the R5pD-based strategy, it has been shown that irrespective of the
frequency reuse scheme and in contrast to the benchmark designs, it is able to
provide the prescribed 5th-percentile user rate at the cost of sacrificing
average cell throughput. Remarkably, it has been observed that the trade-offs
at which the FFR- and SFR-based schemes arrive between in terms of system
spectral efficiency and user throughput fairness are quite different. For high
5th-percentile user rate constraints, the SFR-based frequency reuse scheme
shows a clear advantage when compared to the FFR-based design. This
superiority is rooted on the fact that SFR strategies can reuse the whole set
of available subcarriers in each cell of the system. When facing low 5th-
percentile user rate constraints, however, frequency reuse schemes based on
FFR clearly outperform those based on SFR. A rather unexpected behavior
arising due to the superior flexibility shown by FFR-schemes in interference-
limited cellular networks.

Further work will focus on enhancing the analytical tools developed in this
paper to assess the performance of multi-tier cellular networks where a tier
of macro-BSs is underlaid by tiers of pico- and/or femto-BSs. Moreover, the
framework presented here will be expanded in order to explore new spectrum
sharing strategies currently under study in the B4G/5G standardization arena.

[Statistical distribution of the SINR] As stated in Section II, RBs occupy a
bandwidth $B_{\text{RB}}$ small enough to assume that all subcarriers in a
subband experience frequency flat fading. Moreover, the proposed scheduling
and resource allocation framework relies on capacity formulas optimized on an
RB-by-RB basis. Under these assumptions, $h_{b}=|H_{b,u,n}|^{2}$ is subject to
an exponential PDF $f_{h_{b}}(x)=e^{-x}u(x)$ (assuming a Rayleigh fading
channel linking the $b$th BS to user $u$ on the $n$th RB), where $u(x)$
denotes the unit step function, and its corresponding CDF expression can be
easily obtained as $Pr\\{h_{b}\leq x\\}=\left(1-e^{-x}\right)u(x)$.
Furthermore, it is not necessary to specify the frequency correlation
characteristics of the channel model used in either the analytical framework
or the simulation set-up.

The CDF of the instantaneous SINR $\gamma_{u,n}$ conditioned on the set of
small-scale fading gains $\boldsymbol{h}\triangleq\\{h_{b}\\}_{\forall b\neq
0}$ when the distances $\boldsymbol{d}\triangleq\\{d_{b,u}\\}_{\forall b}$
from all BSs $b$ are given, can be derived from (3) as

$\begin{split}F_{\gamma_{u,n}|\boldsymbol{d},\boldsymbol{h}}&(x|d,\boldsymbol{h})\triangleq
Pr\\{\gamma_{u,n}\leq x|d_{0,u},\boldsymbol{h}\\}\\\
&=Pr\left\\{h_{0}\leq\frac{\left(N_{0}\Delta
f+I_{u,n}\right)}{\bar{\gamma}_{0}}x|d_{0,u},\boldsymbol{h}\right\\}\\\
&=1-e^{-\frac{x\left(N_{0}\Delta f+I_{u,n}\right)}{\bar{\gamma}_{0}}},\qquad
x\geq 0,\end{split}$ (25)

where note that the distances in the set $\boldsymbol{d}$ can be written in
terms of the distance $d_{0,u}=d$ from the serving BS to the user $u$ and
furthermore, $\bar{\gamma}_{0}=P_{n}L(d)$ represents the average received
signal power at a distance $d$ from the tagged BS.

Now, using the above expression and averaging over the PDFs of the i.i.d.
random variables $\boldsymbol{h}$, the conditional CDF of the instantaneous
SINR $\gamma^{A}_{u,n}$ experienced by user $u$ located at distance
$d_{0,u}=d$ from the serving BS and in the cell-region $A$, for the FFR
scheme, can be obtained as

$\begin{split}&F_{\gamma^{A}_{u,n}|d_{0,u}}(x|d)\triangleq
Pr\\{\gamma^{A}_{u,n}\leq x|d_{0,u}\\}\\\
&=\int_{0}^{\infty}...\int_{0}^{\infty}\left(1-e^{-\frac{x\left(N_{0}\Delta
f+I_{u,n}\right)}{\bar{\gamma}_{0}}}\right)\prod_{i\in\Phi_{n}}f_{h_{i}}(h_{i})\mathrm{d}h_{i}\\\
&=1-e^{-\frac{xN_{0}\Delta
f}{\bar{\gamma}_{0}}}\int_{0}^{\infty}...\int_{0}^{\infty}e^{-\frac{x\left(\sum_{i\in\Phi_{n}}h_{i}\bar{\gamma}_{i}\right)}{\bar{\gamma}_{0}}}\prod_{i\in\Phi_{n}}e^{-h_{i}}\mathrm{d}h_{i}\\\
&=1-e^{-\frac{xN_{0}\Delta
f}{\bar{\gamma}_{0}}}\prod_{i\in\Phi_{n}}\frac{1}{1+\frac{x\bar{\gamma}_{i}}{\bar{\gamma}_{0}}}\textrm{
},\qquad x\geq 0,\end{split}$ (26)

where $f_{h_{i}}(h_{i})$ is the PDF of the variable $h_{i}=|H_{i,u,n}|^{2}$,
$\overline{\gamma}_{0}=P_{n}L(d)$ represents the average received signal and
$\overline{\gamma}_{i}=P_{n}L\left(d_{i,u}\right)$ is the average interfering
signal from each interfering BS.

Under SFR, the conditional CDF of the instantaneous SINR $\gamma^{A}_{u,n}$
can be expressed as

$\begin{split}F_{\gamma^{A}_{u,n}|d_{0,u}}(x|d)=&1-e^{-\frac{xN_{0}\Delta
f}{\bar{\gamma}_{0}}}\prod_{i\in\Phi_{n}^{C}}\frac{1}{1+\frac{x\bar{\gamma}_{i}}{\bar{\gamma}_{0}}}\\\
&\times\prod_{j\in\Phi_{n}^{E}}\frac{1}{1+\frac{x\bar{\gamma}_{j}}{\bar{\gamma}_{0}}},\qquad
x\geq 0,\end{split}$ (27)

where, now, the average received signal is
$\overline{\gamma}_{0}=P_{n}^{A}L(d)$, and the average interfering signals are
$\overline{\gamma}_{i}=P_{n}^{C}L\left(d_{i,u}\right)$ and
$\overline{\gamma}_{j}=P_{n}^{E}L\left(d_{j,u}\right)$. Again, $d_{i,u}$ and
$d_{j,u}$ can be written in terms of $d$.

## Acknowledgements

This work has been supported in part by the Agencia Estatal de Investigación
and Fondo Europeo de Desarrollo Regional (AEI/FEDER, UE) under project ELISA
(subproject TEC2014-59255-C3-2-R), Ministerio de Economia y Competitividad
(MINECO), Spain, and the Conselleria d’Educació, Cultura i Universitats
(Govern de les Illes Balears) under grant FPI/1538/2013 (co-financed by the
European Social Fund).

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