# Scalable Device-to-Device Communications For Frequency Reuse $>>$ 1

Daniel Verenzuela and Guowang Miao D. Verenzuela is with the Division of
Communication Systems in the Department of Electrical Engineering (ISY),
Linköping University (LiU), Linköping, Sweden e-mail:
<EMAIL_ADDRESS>Miao is with the Department of Communication
Systems, Royal Institute of Technology (KTH), Stockholm, Sweden e-mail:
<EMAIL_ADDRESS>project was supported by KTH ICT-TNG and was carried out
as a thesis project of D. Verenzuela at KTH.

###### Abstract

Proximity based applications are becoming fast growing markets suggesting that
Device-to-Device (D2D) communications is becoming an essential part of future
mobile data networks. We propose scalable admission and power control methods
for D2D communications underlay cellular networks to increase the reuse of
frequency resources and thus network capacity while maintaining QoS to all
users. The aim of the proposed methods is to maximize the number of D2D links
under QoS constraints, therefore maximizing network frequency reuse, while
considering different levels of complexity and available channel state
information (CSI) in a multi-cell environment. Numerical results show that by
using D2D and the proposed multi-cell interference coordination and low power
transmission method, the network spectral efficiency can be increased by as
much as ten times, while low outage probability can be assured to provided QoS
for all users.

###### Index Terms:

device-to-device communication, ICIC, multi-cell, admission control, transmit
power control, spectral efficiency.

##  I Introduction

Frequency reuse enables reusing the same frequency in cellular networks to
support simultaneous communication demands on the same frequency and thus
improves network capacity and spectral efficiency (SE). In the past decades,
the global demand on mobile data traffic has increased considerably. The
increase in network frequency reuse has been the driving force to boost
network capacity to accommodate the traffic demand. It is expected that global
mobile data traffic will continue its remarkable growth in the coming decades,
most of which come from short distance and indoor communications [1]. Thus new
techniques need to be developed to further increase network frequency reuse
and the network capacity. Device-to-Device (D2D) communications has been
proposed to improve the performance of cellular networks by allowing devices
to communicate directly without relaying traffic to the base station (BS).
This feature provides the means of increasing both spectral and energy
efficiency, reducing delay, improving coverage and supporting new proximity
based applications [2]. In addition, D2D enables a low cost solution for
reusing frequency in short distance communications and thus is a good
candidate technology to push the frequency reuse of cellular networks to far
greater than one. Besides, from the service demand perspective, the emergence
of new applications based on geographical proximity is becoming a fast growing
market [3] suggesting that D2D communications will become an important part of
future mobile data networks. Thus studying the scalability of D2D
communications is of paramount importance to accommodate future traffic
demands.

Allowing D2D links to share the resources with cellular user equipments (CUEs)
creates two levels of networks. The first level is the primary cellular
network that is comprised of the communications between CUEs and their
respective base stations (BSs). The secondary network level is composed of the
simultaneous D2D links that share the resources of the primary cellular
network. The main idea is to reutilize the resources of the primary cellular
network as much as possible while minimizing the impact on its performance.
For this reason we assume that D2D links will only share the uplink resources
of the primary cellular network. Notice that when downlink resources are
shared the D2D links may cause strong interference towards the CUEs whereas in
the case of sharing uplink resources the interference caused by D2D links
affects the BS, and this interference can be controlled by the BS [2]. The
sharing of resources between the two network levels allows for higher
frequency reuse and SE. However as the reutilization of resources becomes
higher the interference levels may increase to a point where the performance
of both cellular and D2D networks is seriously degraded. Thus one of the main
limitations on the scalability of D2D communications is the interference
management.

When D2D links are added to the network, two main levels of interference are
found:

  1. 1.

Interference caused by the cellular network

     * •

From CUEs towards other BSs (inter-cell interference).

     * •

From CUEs towards D2D links.

  2. 2.

Interference caused by the D2D network

     * •

From D2D links towards the BSs.

     * •

From D2D links towards other D2D links.

In the first level we mainly have the inter-cell interference. In the uplink
of the last generation cellular networks the resources within each cell are
allocated orthogonally resulting in zero intra-cell interference. However the
resources are shared by several cells causing inter-cell interference between
the CUEs and BSs of different cells. This problem is well known and there has
been important research done in the last years, e.g., in the case of the
uplink of Long Term Evolution (LTE) networks the authors in [4] analyze the
impact of inter-cell interference coordination (ICIC) while considering the
effects of power limitations, radio resource management functions, fast
retransmissions, power control and adaptive modulation and coding methods. The
work [5] proposes an adaptive soft frequency reuse method that decreases
inter-cell interference improving the average throughput per user. In [6] the
authors propose an interference aware joint scheduling method based on
proportional fairness. The work [7] studies the problem of resource allocation
considering the impact of inter-cell interference while maintaining a
frequency reuse of one. The works [8] and [9] perform an evaluation the LTE
open loop fractional power control (OFPC) and the closed loop power control
respectively considering the impact of inter-cell interference while giving an
insight to the proper configuration of the design parameters.

The second effect of the first level interference refers to the impact of
cellular communications on the D2D network. Since we assume the reuse of
uplink resources, the CUEs may cause strong interference towards D2D links.
However given that the cellular network is considered to have higher priority
we do not consider modifying its functionality to accommodate the D2D network,
rather the opposite. In addition, a D2D pair can always switch to the cellular
mode if it is receiving too much interference from CUEs. Thus in this study we
want to focus on the effects caused by the D2D network.

In the second level of interference we consider two main impacts. First the
D2D links cause aggregated interference towards the BSs that may compromise
the QoS of the uplink of CUEs. Secondly the D2D links may cause strong
interference among each other limiting their QoS and reducing the SE of the
network. Thus a careful design to deal with the second level interference is
needed to obtain the most benefits of D2D communications and assure the QoS of
both CUEs and D2D links.

To solve the problem of coordinating the interference, dynamic resource
allocation algorithms have been presented in [10], allowing an increase of
capacity. Also joint resource block (RB) scheduling [11] and interference
avoidance methods [12] have been proposed to increase SE and reduce harmful
interference respectively. Power control techniques have also been proposed to
coordinate the interference, the work [13] conducts a comparative study of LTE
power control techniques applied to D2D communications. A double threshold
power control algorithm is proposed in [14] to maximize the system throughput.
The authors in [15] proposed a continuous fuzzy logic power control method to
limit the interference and improve the QoS of D2D links. In [16] the capacity
of the system is studied under cooperative and non-cooperative interference
coordination methods.

However in all of these works the number of D2D links that share the resources
with the CUEs is always fixed or set. The network capacity can indeed be
dramatically improved by allowing more D2D links in the network as it
increases frequency reuse by reusing the same frequency more times for sending
different data streams. The capacity in terms of the maximum number of D2D
links that system can support has only been considered by few studies [17, 18,
19]. In [17] a greedy heuristic resource allocation algorithm to maximize the
number of D2D links in a single-cell is presented. The results show that by
allowing more D2D links to share the same resources a considerable increase in
SE can be achieved, however in this work power control is not considered and
full channel state information (CSI) is assumed which may not be scalable in
practical implementations. The study [18] proposes and evaluates a series of
distributed power control algorithms in D2D communications. However the
implementation of the admission control procedures in terms of signaling is
not mentioned and the impact on network scalability is not fully addressed.

In [19], we have studied the feasibility of admitting several D2D pairs to
share the resources of a CUE considering only the number of D2D links in a
single-cell. Optimal and suboptimal solutions are proposed to coordinate the
interference with the goal of maximizing the number of active D2D links in the
system. The results show that allowing several D2D links to share the same
recourses with CUEs without any limitation is indeed possible and it increases
network SE.

It is worth mentioning that the studies [17, 18, 19] are all focused on a
single-cell scenario neglecting the impact of inter-cell interference. However
in real applications this impact can be significant and needs to be
considered. Another important remark mentioned in a comprehensive survey for
D2D communications [20] is that in most available literature the existing
interference coordination solutions assume full CSI to be known at the BS.
However this is usually not practical due to the signaling overhead,
especially in multi-cell environments.

This study proposes admission control, inter-cell interference coordination,
and power allocation methods for D2D communications underlay cellular networks
to increase the network frequency reuse far beyond a factor of one by enabling
as many simultaneous D2D links to communicate at the same time as possible.
The proposed methods aim at maximizing the number of active D2D links subject
to QoS constraints in a multi-cell environment of any size while considering
three cases: no CSI, partial CSI or full CSI available. In the first case we
calculate a theoretical upper bound for the number of D2D links and implement
a blind admission control (BAC) method considering average QoS requirements
for CUE and D2D links. For the second case we present a distributed admission
control (DAC) method to further improve the performance where each D2D pair
decides its mode based on transmission power constraints to assure QoS for all
users, the DAC methods is compatible with LTE standard and can be easily
implemented in 5G LTE systems. In the last case we present an optimal
admission control (OAC) method, which is based on exhaustive search and serves
as the performance benchmark.

In our analysis we bound the number of D2D links based on approximate SINR
expressions by considering the expected value of interfering terms. An
alternative approach could be to use chance constrained programming to
establish QoS constraints as outage probabilities [21]. However, this problem
formulation is not always tractable and may be expensive to solve in terms of
computational resources. Our approach yields tractable and practical results
that can be easily applied in current wireless networks. In particular, both
BAC and DAC are based in algorithms that have extremely low signaling overhead
and can be implemented in a multi-cell network regardless of its size.
Therefore they constitute highly scalable solutions that can be easily
implemented in the next generation networks, e.g., LTE Release 13 and onward.

The remainder of this paper is presented as follows: section II depicts the
system model;In section III a statistical model for the interference is
introduced; section IV presents the analysis conducted to obtain the BAC
method; section V shows the analysis conducted to obtain DAC method; section
VI presents the formulation of the OAC method; section VII depicts the
numerical results and analysis; finally section VIII concludes our work.

##  II System Model

Consider a multi-cell system where a number of D2D pairs are available in each
cell. The D2D pairs and CUEs are randomly distributed in all cells. Our goal
is to find the maximum number of D2D pairs that can communicate at the same
time, thus maximizing the frequency reuse of the network. We focus on a single
RB scenario where only one CUE is considered in each cell. We assume that
resource allocation for the primary cellular network has already been
implemented. We define $\phi_{xk}\in\\{0,1\\}$ $\big{(}\forall
x\in\\{1,\,...,\,N\\},\,\forall k\in\\{1,\,...,\,\hat{N}_{x}\\}\big{)}$, as a
binary random variable that indicates the state of each link $k$ in cell $x$.
The parameter $N$ corresponds to the number of cells in the system and
$\hat{N}_{x}$ is the number of available D2D links in cell $x$. When
$\phi_{xk}=1$ the D2D link is active, otherwise $\phi_{xk}=0$. The frequency
reuse factor is the rate at which the same frequency can be used in the
network. Considering D2D, the frequency reuse factor of a cell $x$ is given by
$K=\sum_{k}^{\hat{N}_{x}}\phi_{xk}+1$. To maximize the network frequency reuse
$K$, the objective is equivalent to maximizing
$\sum_{x}^{N}\sum_{k}^{\hat{N}_{x}}\phi_{xk}$, that is, finding the maximum
number of D2D links that can be in active communications and their
corresponding transmission power levels while the QoS could be assured to all
active users.

The maximum level of interference that can be tolerated in the system is given
by the signal-to-interference plus noise ratio (SINR) constraints for the CUEs
and D2D communications, depicted in (1a) and (1b) respectively. We also
consider an upper bound for the transmission power of D2D links shown in (1c).

$\Gamma_{x0}=\,\frac{P_{x0}G_{x0x0}}{I_{x0}^{D2D}\,+\,I_{x0}^{CUE}\,+\,\mathcal{N}_{BS}}\,\geq\,\gamma_{x0}^{th},$
(1a)
$\Gamma_{xk}\,=\,\frac{\phi_{xk}P_{xk}G_{xkxk}}{I_{xk}^{D2D}+I_{xk}^{CUE}+\mathcal{N}_{D}}\,\geq\,\gamma_{xk}^{th},$
(1b) $\normalsize{\phi_{xk}P_{xk}\,\leq\,P_{D}^{max},}$ (1c)

$\forall x\in\\{1,\,...,\,N\\}\;\forall k\in\\{1,\,...,\,\hat{N}_{x}\\}.$

The terms $I_{x0}^{D2D}$ and $I_{x0}^{CUE}$ correspond to the interference
received at the BS of cell $x$, from the D2D links and CUEs respectively.
Similarly $I_{xk}^{D2D}$ and $I_{xk}^{CUE}$ correspond to the interference
received at the D2D link $k$ of cell $x$ from other D2D links and CUEs
respectively. $\mathcal{N}_{BS}$ and $\mathcal{N}_{D}$ are the noise power at
the BS and D2D links receivers respectively. $P_{x0}$ corresponds to the
transmission power from the CUE at cell $x$, $P_{xk}$ is the power of the
transmitting device of D2D pair $k$ in cell $x$ and $P_{D}^{max}$ is the
maximum transmission power of D2D links. $\gamma_{x0}^{th}$ and
$\gamma_{xk}^{th}$ represent the target SINR of the CUE uplink and the D2D
link $k$ in cell $x$ respectively.

To describe the channel gains the following nomenclature is implemented:
$G_{abij}$ corresponds to the path gain from the transmitter $b$ in cell $a$
to the receiver $j$ in cell $i$. Note that in all variables, CUE and BSs are
indexed as $0$ and D2D users are indexed with non-negative integer numbers. In
equations (1a) and (1b), $G_{x0x0}$ corresponds to the channel gain between
the CUE and the BS of cell $x$, while $G_{xkxk}$ corresponds to the channel
gain within the D2D pair $k$ in cell $x$, i.e., between the D2D transmitter
and receiver of the same D2D pair. Thus we define the interference terms as:

$\normalsize{I_{x0}^{D2D}=\sum_{i=1}^{N}\sum_{j=1}^{\hat{N}_{i}}\phi_{ij}P_{ij}G_{ijx0},}$
(2a) $\normalsize{I_{x0}^{CUE}=\sum_{\begin{subarray}{c}i=1\\\ i\neq
x\end{subarray}}^{N}P_{i0}G_{i0x0},}$ (2b)
$\normalsize{I_{xk}^{D2D}=\sum_{i=1}^{N}\sum_{j=1}^{\hat{N}_{i}}\phi_{ij}P_{ij}G_{ijxk}-\phi_{xk}P_{xk}G_{xkxk},}$
(2c) and $\normalsize{I_{xk}^{CUE}=\sum_{i=1}^{N}P_{i0}G_{i0xk},}$ (2d)

$\forall x\in\\{1,\,...,\,N\\}\;\forall k\in\\{1,\,...,\,\hat{N}_{x}\\}.$

Here we can see that the terms $I_{x0}^{D2D}$ and $I_{xk}^{D2D}$ correspond to
the second level of interference related to the impact of the D2D network,
where $I_{x0}^{D2D}$ represents the interference caused by D2D links to BSs
and $I_{xk}^{D2D}$ the interference among different D2D links. These are the
main effects that we want to control in order to provide scalable interference
coordination methods for D2D communications.

On the other hand the terms $I_{x0}^{CUE}$ and $I_{xk}^{CUE}$ correspond to
the first level of interference that is related to the impact of the cellular
network, where $I_{x0}^{CUE}$ depicts the inter-cell interference and
$I_{xk}^{CUE}$ the interference caused by CUEs towards D2D links. Since we are
not interested in modifying the functionality of the cellular networks these
terms are not part of the design variables for the interference coordination
methods. Thus we redefine the SINR target for CUEs as:

$\gamma_{x0}^{th}=\frac{\Gamma_{x0}^{i}}{\delta}=\frac{P_{x0}G_{x0x0}}{\left(I_{x0}^{CUE}+\mathcal{N}_{BS}\right)\delta},\;\forall\delta\in\\{\mathbb{R}^{+};\delta>1\\},$
(3)

where $\Gamma_{x0}^{i}$ is the SINR of CUEs before D2D links are added to the
system. The parameter $\delta$ corresponds to the desired ratio between the
CUE’s SINR before and after D2D links are added, i.e., the SINR loss of CUEs
due to the interference caused by D2D links. This definition allows a clear
evaluation of the impact of D2D links to the CUEs uplink. The parameter
$\delta$ can be broadcasted by the BS or sent to a D2D link when it is newly
established. In our analysis we assume a fixed target SINR for all D2D links
defined as $\gamma_{D}$ in order to implement fairness in the admission
control of D2D links.

##  III Statistical Interference Model

To develop the interference coordination methods we assume different levels of
CSI to be available. For the low complexity solutions, we need to establish
statistical models for the interference and channel gains in order to account
for the unavailable CSI.

###  III-A Interference Model

Consider a victim receiver $v$ surrounded by $\tilde{N}$ devices, we define
the aggregated interference received at $v$ as:

$\normalsize{I_{v}\,=\,\sum_{i=1}^{\tilde{N}}P_{tx_{vi}}G_{vi},}$ (4)

where $P_{tx_{vi}}$ is the transmission power of an interfering device $i$ and
$G_{vi}$ is the channel gain between $v$ and $i$. To find a statistical model
for the interference we assume that the interfering devices are randomly
distributed within a given area $A$, as shown in Fig 1. Thus the channel gains
can be represented as a random variable $G_{vi}$. We also assume that the
interfering devices have the same transmission power $P_{tx_{vi}}=P_{tx}\leq
P_{max}$, where $P_{max}$ is the maximum transmission power allowed for the
devices.111Note that this assumption is only made for the estimation of the
aggregated interference. In the proposed interference coordination methods
power control is implemented. Thus we can define an expected value for the
aggregated interference within $A$ as:

$\normalsize{\mathbb{E}[I_{v}]\,=\,\tilde{N}_{A}\,A\,P_{tx}\,\mathbb{E}[G_{vi}],}$
(5)

where $\tilde{N}_{A}$ is the density of interfering devices per unit area.

Figure 1: Interference Model.

In order to obtain reasonable values for $\tilde{N}_{A}$ the area $A$ needs to
be finite, which means that in practical applications there will be
interfering devices outside of $A$, as shown in Fig 1. We define
$A=\pi(d_{w})^{2}$ as a circular interference area around $v$ where $d_{w}$ is
the maximum distance between $v$ and an interfering device. Notice that the
interference caused by the devices outside of $A$ is negligible compared to
the one caused by the users inside due to the path loss attenuation.

To determine the value of $d_{w}$ we assume that if the received power from an
interfering device is lower than a threshold, then its effect can be
neglected. Notice that the received power of the interfering devices depends
also on their transmission power and this is meant to be used as a design
variable in the later analysis. Thus we calculate $d_{w}$ considering the
maximum transmission power that is allowed $P_{max}$ so that the interference
area $A$ is obtained for the worst interference scenario. Then we consider an
interferer $w$ that is located at the edge of the interference area and we
define $d_{w}$ as:

$\normalsize{\begin{array}[]{c}P_{max}\mathbb{E}[G_{vw}]\leq\mathcal{N}_{v},\\\\[5.69054pt]
d_{vw}\geq\text{\Large{$\left(\frac{P_{max}c_{v}\mathbb{E}[|h_{vw}|^{2}]}{\mathcal{N}_{v}}\right)^{1/\alpha_{v}}$}}=d_{w},\end{array}}$
(6)

where $\mathcal{N}_{v}$ is the noise power at the victim receiver and $d_{vw}$
is the distance between devices $v$ and $w$. We define the channel gain
between $v$ and $w$ as:

$\normalsize{G_{vw}=c_{v}d_{vw}^{-\alpha_{v}}|h_{vw}|^{2},}$ (7)

where $c_{v}$ refers to a propagation constant and $\alpha_{v}$ is the path
loss exponent. The effects of fast fading are represented by $|h_{vw}|^{2}$.

###  III-B Channel Gain Model

If CSI is not available we can model the channel gain between two devices $v$
and $i$ as a random variable $G_{vi}$, defined in (7), where $d_{vi}$ and
$h_{vi}$ are independent random variables. Thus we can calculate the expected
value of $G_{vi}$ as:

$\normalsize{\mathbb{E}[G_{vi}]=c_{v}\mathbb{E}[d_{vi}^{-\alpha_{v}}]\mathbb{E}[|h_{vi}|^{2}].}$
(8)

Notice that in our analysis we want to establish statistical models of the
channel gains. So consider the channel to be invariant during the period of
interest, thus we assume $\mathbb{E}[|h_{vi}|^{2}]=1$.

We also assume device $v$ to be located at a fixed point and device $i$ to be
positioned randomly following a circular distribution around $v$. Thus the
probability density function of $d_{vi}$ is given by a triangular distribution
depicted as:

$\normalsize{f_{d_{vi}}(x)=\Bigg{\\{}\begin{array}[]{cl}\frac{\text{\normalsize{$2x$}}}{\text{\normalsize{$(d_{max})^{2}$}}}&\text{if}\quad
d_{min}\leq x\leq d_{max},\\\\[8.53581pt] 0&\text{otherwise}.\end{array}}$ (9)

Combining (8) and (9) we have that
$\forall\alpha_{v}\in\\{\mathbb{R}^{+};\alpha_{v}>2\\}$.

$\normalsize{\begin{array}[]{l}\mathbb{E}[G_{vi}]=c_{v}\text{\large{$\int_{\\!d_{min}}^{d_{max}}x^{-\alpha_{v}}f_{d_{vi}}$}}(x)dx\\\\[14.22636pt]
\qquad\quad=c_{v}\text{\large{$\int_{\\!d_{min}}^{d_{max}}\frac{2x^{(1-\alpha_{v})}}{(d_{max})^{2}}$}}dx\\\\[14.22636pt]
\qquad\quad=\text{\large{$\frac{2c_{v}\left(d_{min}^{-(\alpha_{v}-2)}-\>d_{max}^{-(\alpha_{v}-2)}\right)}{(d_{max})^{2}\left(\alpha_{v}-2\right)}$}}.\end{array}}$
(10)

This result is applied to model all channel gains considered in this
investigation. Note also that for practical applications the probability
density function of $d_{vi}$ can be changed to match real user distributions.

##  IV Blind Admission control

In practical applications obtaining CSI is not always possible because of high
signaling overhead. Particularly if we consider the case of D2D
communications, having CSI from every D2D link in the system would
considerably increase the signaling overhead. Thus we present the BAC method
where no CSI is necessary. Admission control here refers to letting a pair of
devices in proximity communicate in the D2D mode.

In the BAC method each BS independently estimates an upper bound for the
number of D2D links that can be active in its cell based on average
constraints for the QoS of CUEs and D2D links. Then the active D2D links are
selected randomly from the available ones within the cell. The transmission
power of D2D links is obtained by applying the channel inversion power control
algorithm which allows for a fixed received power at the receiving device.
Thus the transmission power of a given D2D link $k$ in a cell $x$ is depicted
as:

$P_{xk}=\frac{P_{r_{D}}}{G_{xkxk}}\,\leq\,P_{D}^{max},\begin{array}[]{c}\forall
x\in\\{1,\,...,\,N\\}\\\ \forall
k\in\\{1,\,...,\,\tilde{N}_{x}\\}\end{array},$ (11)

where $P_{r_{D}}$ is the received power for all D2D links which is calculated
and broadcasted by the BS. Notice that the channel gain between devices of the
same pair $G_{xkxk}$ is known to them but unknown to the BS.

To obtain an upper bound for the number of D2D links first we assume
$\phi_{xk}=1\forall x,k$ and calculate the expected value of interference
constrains (1a) and (1b) combined with (3) and (11). As a result we have:

$\frac{P_{x0}G_{x0x0}}{\mathbb{E}[I_{x0}^{D2D}]\,+\,\mathbb{E}[I_{x0}^{CUE}]\,+\,\mathcal{N}_{BS}}\,\geq\,\frac{\Gamma_{x0}^{i}}{\delta},$
(12a)
$\frac{P_{r_{D}}}{\mathbb{E}[I_{xk}^{D2D}]+\mathbb{E}[I_{xk}^{CUE}]+\mathcal{N}_{D}}\,\geq\,\gamma_{D},$
(12b)

$\forall x\in\\{1,\,...,\,N\\}\;\forall k\in\\{1,\,...,\,\hat{N}_{x}\\}.$

Since no CSI is available we consider the channel gains to be random
variables, then by applying the interference model described in section III-A
we can define the interference terms that depend on D2D links as:

$\mathbb{E}[I_{x0}^{D2D}]\,=\,\tilde{N}_{A}\,A_{C}\,P_{r_{D}}\,\frac{\mathbb{E}[G_{D2D-BS}]}{\mathbb{E}[G_{D2D}]},$
(13)
$\begin{array}[]{c}\mathbb{E}[I_{xk}^{D2D}]\,=\,\left(\tilde{N}_{A}\,A_{D}-1\right)\,P_{r_{D}}\,\frac{\mathbb{E}[G_{D2D-I}]}{\mathbb{E}[G_{D2D}]},\\\
\forall x\in\\{1,\,...,\,N\\}\;\forall
k\in\\{1,\,...,\,\hat{N}_{x}\\}.\end{array}$ (14)

The term $G_{D2D}$ is a random variable that represents the channel gain
between two devices of the same D2D pair. Similarly $G_{D2D-BS}$ describes the
channel gain between D2D transmitters and the BS. $G_{D2D-I}$ represents the
channel gain between D2D transmitters and a given D2D receiver from different
D2D pairs. $A_{D}$ and $A_{C}$ are the interference areas for D2D links and
CUEs constraints respectively. $\tilde{N}_{A}$ represents the density of
active D2D links per unit area. Note that in (14) the number of D2D links is
subtracted by one because there needs to be more than one D2D link in $A_{D}$
to cause interference.

Applying (13) and (14) to (12a) and (12b) allows us to obtain two statistical
upper bounds of the density of D2D links per unit area that can be accepted in
the system. One is obtained from the CUE constraint $\tilde{N}_{C}^{UB}$ and
another from the D2D constraint $\tilde{N}_{D}^{UB}$, depicted as:

$\tilde{N}_{A}\leq\frac{\mathbb{E}[G_{D2D}]I_{C}}{P_{r_{D}}A_{C}\mathbb{E}[G_{D2D-BS}]}=\tilde{N}_{C}^{UB},$
(15a)
$\normalsize{\text{\small{$\tilde{N}_{A}\leq\left(\frac{\mathbb{E}[G_{D2D}]}{\mathbb{E}[G_{D2D-I}]\gamma_{D}}+1-\frac{\mathbb{E}[G_{D2D}]I_{D}}{P_{r_{D}}\mathbb{E}[G_{D2D-I}]}\right)\frac{1}{A_{D}}=\tilde{N}_{D}^{UB}$}},}$
(15b) where
$\normalsize{I_{C}=\left(\delta-1\right)\left(\mathbb{E}[I_{x0}^{CUE}]+\mathcal{N}_{BS}\right),}$
(15c)
$\normalsize{I_{D}=\left(\mathbb{E}[I_{xk}^{CUE}]+\mathcal{N}_{D}\right),}$
(15d)

$\forall x\in\\{1,\,...,\,N\\}\;\forall k\in\\{1,\,...,\,\hat{N}_{x}\\}.$

Figure 2: Max. density of D2D links vs received power $P_{r_{D}}$ for
$\delta=3$[dB], $\gamma_{D}=10$[dB]. In a circular area of radius $R=400$ [m]
and with $d_{D2D}$ between 10 [m] and 50 [m].

These upper bounds depend on the received power of D2D links $P_{r_{D}}$, Fig
2 depicts a numerical example of $\tilde{N}_{C}^{UB}A$ and
$\tilde{N}_{C}^{UB}A$ as the upper bounds for the number of D2D links in a
circular area $A$ of radius $R$. The term $d_{D2D}$ is a random variable that
represents the distance between the transmitter and receiver of a given D2D
pair. $\tilde{N}_{C}^{UB}$ is monotonically decreasing with respect to
$P_{r_{D}}$. $\tilde{N}_{D}^{UB}$ increases until it reaches a saturation
point, which means that after certain value of $P_{r{D}}$, the density of D2D
links per unit area cannot be increased for a D2D target SINR $\gamma_{D}$.

The overall upper bound for the density of D2D links is given by the minimum
between $\tilde{N}_{C}^{UB}$ and $\tilde{N}_{D}^{UB}$ in order to satisfy both
QoS constraints (12a) and (12b). Thus it is possible to find $\hat{P}_{r_{D}}$
where $\tilde{N}^{UB}=\tilde{N}_{C}^{UB}=\tilde{N}_{D}^{UB}$ so that the
density of D2D links is maximized. Solving for $\tilde{N}^{UB}$, the upper
bound for the density of D2D links per unit area is:

$\normalsize{\tilde{N}^{UB}=\frac{I_{C}\left(\frac{\mathbb{E}[G_{D2D}]}{\gamma_{D}}+\mathbb{E}[G_{D2D-I}]\right)}{A_{D}\mathbb{E}[G_{D2D-I}]I_{C}+A_{C}\mathbb{E}[G_{D2D-BS}]I_{D}}}$
(16)

where

$\normalsize{\hat{P}_{r_{D}}=\frac{\mathbb{E}[G_{D2D}]\left(A_{D}\mathbb{E}[G_{D2D-I}]I_{C}+A_{C}\mathbb{E}[G_{D2D-BS}]I_{D}\right)}{A_{C}\mathbb{E}[G_{D2D-BS}]\left(\frac{\mathbb{E}[G_{D2D}]}{\gamma_{D}}+\mathbb{E}[G_{D2D-I}]\right)}.}$
(17)

$\forall x\in\\{1,\,...,\,N\\}\;\forall k\in\\{1,\,...,\,\hat{N}_{x}\\}.$

It is worth mentioning that as shown in (15c) and (15d), $I_{C}$ and $I_{D}$
depend on $\mathbb{E}[I_{x0}^{CUE}]$ and $\mathbb{E}[I_{xk}^{CUE}]$
respectively. These terms can be calculated by applying the interference model
described in section III-A. In this case the density of users per unit area is
known and corresponds to $1/A_{cl}$, where $A_{cl}$ is the area of the cell,
given that there is only one CUE per cell.

From equation (16) we can see that the expected values of the channel gains
determine the maximum density of D2D links per unit area that can be allowed
in the system. By implementing the model found in section III-B we define
these terms as:

$\normalsize{\mathbb{E}[G_{D2D}]=\frac{2c_{d}\left(d_{D2D_{min}}^{-(\alpha_{d}-2)}-\>d_{D2D_{max}}^{-(\alpha_{d}-2)}\right)}{(d_{D2D_{max}})^{2}\left(\alpha_{d}-2\right)},}$
(18a)
$\normalsize{\mathbb{E}[G_{D2D-BS}]=\frac{2c_{0}\left(d_{D2D-BS_{min}}^{-(\alpha_{0}-2)}-\>d_{D2D-BS_{max}}^{-(\alpha_{0}-2)}\right)}{(d_{D2D-BS_{max}})^{2}\left(\alpha_{0}-2\right)},}$
(18b)
$\normalsize{\mathbb{E}[G_{D2D-I}]=\frac{2c_{d}\left(d_{D2D-I_{min}}^{-(\alpha_{d}-2)}-\>d_{D2D-I_{max}}^{-(\alpha_{d}-2)}\right)}{(d_{D2D-I_{max}})^{2}\left(\alpha_{d}-2\right)},}$
(18c)

where the distance between the D2D transmitter and receiver of the same pair
is a random variable within $[d_{D2D_{min}},d_{D2D_{max}}]$. The distance
between the D2D links and the BS is randomly distributed in the interval
$[d_{D2D-BS_{min}},d_{D2D-BS_{max}}]$. The distance between D2D transmitters
and receivers of different D2D pairs is randomly distributed within
$[d_{D2D-I_{min}},d_{D2D-I_{max}}]$. The term $\alpha_{0}$ corresponds to the
path loss exponent for the channel between devices and the BS, whereas
$\alpha_{d}$ corresponds to the path loss exponent for the channel between
devices. The term $c_{0}$ refers to a propagation constant for the channel
between devices and the BS, and $c_{d}$ corresponds to a propagation constant
for the channel between devices.

Notice that the maximum limit for the distribution of the distances
$d_{D2D-BS_{max}}$ and $d_{D2D-I_{max}}$ are given by the definition of the
interference area (see section III-A, eq: 6). In practical applications more
sophisticated spatial distributions of users can be obtained in order to have
more accurate values for the expectations of the channel gains.

At this point we are able to estimate the maximum number of D2D links that can
be allowed in the system without considering CSI. In order to implement this
result each BS estimates independently the number of D2D links that can be
active in its cells as:

$\normalsize{\tilde{N}_{x}=\min\\{\lfloor\tilde{N}^{UB}A_{cl_{x}}\rfloor,\,\hat{N}_{x}\\},\forall
x\in\\{1,\,...,\,N\\}.}$ (19)

The term $A_{cl_{x}}$ is the area of cell $x$ and $\hat{N}_{x}$ is the number
of available D2D links in the cell. Once the BS calculates the number of
active D2D links $\tilde{N}_{x}$, it simply selects them randomly from the
available ones and broadcasts the received power parameter $\hat{P}_{r_{D}}$
for the power control of D2D links, which can be done in the admission control
of D2D communications.

##  V Distributed Admission Control

In the implementation of D2D communications there is a certain amount of CSI
that is already available in the system. Thus we present the distributed
admission control (DAC) method that makes use of the available information to
better coordinate the interference between D2D links and CUEs.

The DAC method is based on a distributed algorithm where the D2D pairs decide
independently their active status and their transmission power by adding a
limited amount of semi-static signaling overhead. The main idea is that every
D2D pair decides their own active status and transmission power based on
general information parameters that are broadcasted by the BSs, e.g, number of
active D2D links in the cells, path loss towards the BS, area of the cells,
etc. In contrast with the BAC method where the admission control is done at
random, this solution is able to adapt the admission and transmission power of
each D2D link to better coordinate the interference.

In order to implement this method we use the same constraints defined in
previous sections to derive an upper and lower bound for the transmission
power of D2D links based on the QoS of CUEs and D2D links. Then each D2D link
decides its active status depending on the feasibility of its transmission
power constraints, i.e., being able to assure QoS for itself while maintaining
the aggregated interference to the CUEs below a threshold.

To calculate the constraints for the transmission power of D2D links, we need
a statistical estimation of the interference scenario given that CSI is
limited. Thus we make use of the interference model found in section III-A
which can be applied at the D2D pairs if the BSs broadcast the numbers of
active D2D links in their cells.

In D2D communications underlay cellular networks the BS plays a role in the
discovery procedure, hence we can assume that the active status of each D2D
pair can be known to its serving BS. Thus the BS would have information about
the number of active D2D links in their respective cells.

To illustrate the implementation of the DAC method, let us assume a D2D pair
$k$ in a cell $x$, denoted by $D2D_{xk}$, that needs to decide its active
status. Since each D2D link makes an independent decision with limited CSI,
$D2D_{xk}$ assumes the same transmission power for all D2D links
$P_{ij}=P_{D_{xk}},\>\forall i\in\\{1,\,...,\,N\\},\forall
j\in\\{1,\,...,\,\hat{N}_{i}\\}$.222Note that this assumption is only made to
estimate the interference that each D2D link would received from other D2D
links. However, in the implementation of the DAC method the transmission power
of each D2D pair is different and decided in a distributed manner (see (27)
and (28b)).

Initially we define an upper and lower bound for the transmission power of a
D2D pair as $P_{D_{xk}}^{UB}$ and $P_{D_{xk}}^{LB}$ respectively. Then we
compare the two sets $[-\infty,P_{D_{xk}}^{UB}]$ and
$[P_{D_{xk}}^{LB},\infty]$. If their intersection is a non-empty set,
$D2D_{xk}$ is active $\phi_{xk}=1$, otherwise $\phi_{xk}=0$. This rule allows
$D2D_{xk}$ to evaluate the feasibility of its link given that the upper bound
limits the interference to the CUE uplink and the lower bound assures the QoS
of $D2D_{xk}$ link. Notice that our objective is to maximize the number of
active D2D links while assuring QoS to all users, thus $D2D_{xk}$ should only
be in active mode if the two power sets intersect.

To obtain the upper bound first we redefine the term $I_{x0}^{D2D}$, found in
(2a), as:

$\normalsize{I_{x0}^{D2D}=\phi_{xk}P_{xk}G_{xkx0}+\hat{I}_{x0}^{D2D}=\phi_{xk}P_{D_{xk}}G_{xkx0}+\hat{I}_{x0}^{D2D},}$
(20)

where $\hat{I}_{x0}^{D2D}$ corresponds to the aggregated interference caused
by active D2D links to the BS of cell $x$ ($BS_{x}$). Since $D2D_{xk}$ does
not have CSI to calculate $\hat{I}_{x0}^{D2D}$, we consider it to be a random
variable. Thus we can calculate its expected value by applying the model found
on section III-A. As a result we have:

$\normalsize{\mathbb{E}[\hat{I}_{x0}^{D2D}]=\frac{\tilde{N}_{x}}{A_{cl_{x}}}\,A_{x0}\,P_{D_{xk}}\,\mathbb{E}[G_{D2D-BS}],}$
(21)

where $A_{x0}$ is the interference area and $A_{cl_{x}}$ is the area of cell
$x$. The term $\mathbb{E}[G_{D2D-BS}]$ is the expected value of the channel
gain between active D2D links and BS.

Finally we obtain a statistical upper bound for the transmission power of D2D
links $P_{D_{xk}}^{UB}$ by combining the expected value of (1a) and (1c) with
(21), thus $P_{D_{xk}}^{UB}$ is defined as:

$\normalsize{P_{D_{xk}}^{UB}=\min\left\\{\frac{\text{\normalsize{$\hat{I}_{x0}^{th}$}}}{\text{\normalsize{$G_{xkx0}+\frac{\tilde{N}_{x}}{A_{cl_{x}}}A_{x0}\mathbb{E}[G_{D2D-BS}]$}}},P_{D}^{max}\right\\},}$
(22a)
$\normalsize{\hat{I}_{x0}^{th}=\left(I_{x0}^{th}-\mathbb{E}[I_{x0}^{CUE}]-\mathcal{N}_{BS}\right),}$
(22b)

$\forall x\in\\{1,\,...,\,N\\}\;\forall k\in\\{1,\,...,\,\hat{N}_{x}\\}.$

where $G_{xkx0}$ corresponds to the channel gain between $D2D_{xk}$ and
$BS_{x}$, which can be obtained by monitoring the downlink reference signals.
The term $I_{x0}^{CUE}$ is considered to be a random variable and can be
estimated by applying the interference model of section III-A.

The parameter $\hat{I}_{x0}^{th}$ is the amount of interference that the D2D
links can cause to the $BS_{x}$ so that the QoS of the CUE can be assured. The
term $I_{x0}^{th}$ is given by the definition of the CUE QoS based on the SINR
loss, thus we have:

$\normalsize{I_{x0}^{th}=\frac{P_{x0}G_{x0x0}}{\gamma_{x0}^{th}}=\frac{\delta
P_{x0}G_{x0x0}}{\Gamma_{x0}^{i}},\;\forall x\in\\{1,\,...,\,N\\},}$ (23)

however this information is not available at the D2D links in normal
conditions, thus we assume it is broadcasted by $BS_{x}$.

Note that $P_{D_{xk}}^{UB}$ can be easily implemented in LTE-A systems as the
existing power control for D2D communications in LTE-A ensures the
interference from D2D communications to the serving BSs to be at fixed
tolerable levels, which are configured by the BSs [23]. The BSs can configure
the parameters in the power control formula so that the power control value is
the one in (22a) to serve as the upper bound.

To obtain the lower bound for the transmission power we consider the
constraint (1b), where the term $I_{xk}^{D2D}$ represents the interference
from active D2D links to $D2D_{xk}$. Similarly to the previous analysis we can
estimate this term as:

$\normalsize{\mathbb{E}[I_{xk}^{D2D}]=\frac{\tilde{N}_{xd}}{A_{dk}}\,A_{xk}\,P_{D_{xk}}\,\mathbb{E}[G_{D2D-I}].}$
(24)

Here the parameter $A_{xk}$ is the interference area and
$\mathbb{E}[G_{D2D-I}]$ is the expected value of the channel gain between an
interfering D2D link (within $A_{xk}$) and $D2D_{xk}$. To estimate the number
of active D2D links per unit area in the surroundings of $D2D_{xk}$ we assume
that the cells can be divided into three sectors, which is highly common in
practical applications. Thus $BS_{x}$ can know the number of active D2D links
in each sector and this could be broadcasted to the users. $\tilde{N}_{xd}$
represents the sum of active D2D links in the three sectors that are closer to
$D2D_{xk}$ and $A_{dk}$ is the area enclosed by such sectors, as illustrated
in Fig. 3, a simple way of estimating $\tilde{N}_{xd}$ and $A_{dk}$ is:

$\tilde{N}_{xd}=\tilde{N}_{xc}+\tilde{N}_{ya}+\tilde{N}_{zb},$ (25)
$A_{dk}=A_{xc}+A_{ya}+A_{zb}.$ (26)

More advanced estimators can be used to obtain more accurate results.

Figure 3: Information obtained from three closest sectors of pair $D2D_{xk}$.

By calculating the expected value of (1b) and combining it with (24) we can
obtain an statistical lower bound for the transmission power of D2D links as:

$\normalsize{P_{D_{xk}}^{LB}=\frac{\text{\normalsize{$\left(\mathbb{E}[I_{xk}^{CUE}]+\mathcal{N}_{D}\right)\gamma_{D}$}}}{\text{\normalsize{$G_{xkxk}-\left(\gamma_{D}\frac{\tilde{N}_{xd}}{A_{dk}}A_{xk}\mathbb{E}[G_{D2D-I}]\right)$}}},}$
(27)

$\forall x\in\\{1,\,...,\,N\\},\forall k\in\\{1,\,...,\,\hat{N}_{x}\\}.$

The term $G_{xkxk}$ corresponds to the channel gain between the transmitter
and receiver of $D2D_{xk}$ which is obtained from the discovery procedure. The
parameter $I_{xk}^{CUE}$ corresponds to the interference caused by CUEs
towards $D2D_{xk}$ and can be estimated by applying the interference model of
section III-A.

Finally the decision of $D2D_{xk}$ to be active is given by:

$\normalsize{\phi_{xk}=\Bigg{\\{}\begin{array}[]{l}1\;\text{if}\;P_{D_{xk}}^{LB}\leq
P_{D_{xk}}^{UB}\\\\[5.69054pt]
0\;\text{if}\;P_{D_{xk}}^{LB}>P_{D_{xk}}^{UB}\end{array},}$ (28a)
$\normalsize{\begin{array}[]{c}\text{\normalsize{$P_{xk}=\phi_{xk}P_{D_{xk}}^{LB}$}},\\\\[5.69054pt]
\forall x\in\\{1,\,...,\,N\\},\forall
k\in\\{1,\,...,\,\hat{N}_{x}\\}.\end{array}}$ (28b)

Notice that if $D2D_{xk}$ is in active mode, the transmission power is set as
lower bound. This is done because the lower bound is calculated by considering
the estimation of the interference between different D2D links. Thus
increasing the transmission power above this level would result in higher
interference between D2D links limiting the density of active D2D pairs. As a
result we select the active transmission power as the lower bound to minimize
the interference and maximize the density of D2D links.

In this solution the BS needs to broadcast a limited number of parameters that
are common to all D2D links, thus the amount of signaling overhead introduced
is significantly lower compared to a solution where CSI needs to be exchanged
between the BS and each D2D link. In Appendix VIII we give a detailed summary
of the parameters that need to be calculated by the BS and D2D links to apply
the algorithm.

Figure 4: Implementation of the DAC method

(a) At $D2D_{xk}$.

(b) At $BS_{x}$.

Figure 5: Flow chart of DAC method implementation.

To illustrate the main concept of the DAC method Fig. 4 depicts the roles of a
D2D pair and its serving BS. The BS keeps track of the number of active D2D
links, calculates the amount of interference that the CUE uplink can tolerate
(defined as $\hat{I}_{x0}^{th}$) and other necessary parameters so that the
D2D links can calculate their transmission power constraints. At the same time
the D2D pairs receive the parameters, calculate their transmission power
constraints and notify the BS after their active status is decided.

Fig. 5a presents a flow chart of the specific steps of the DAC algorithm at
$D2D_{xk}$. First all necessary information is collected from the BS, then
$D2D_{xk}$ calculates the necessary parameters to estimate the interference
conditions and the transmission power constraints. Finally $D2D_{xk}$ decides
its active status and notifies the BS of its decision. Similarly Fig. 5b
depicts the necessary steps taken by $BS_{x}$ to support the DAC method. First
$BS_{x}$ collects the active status of D2D links and calculates the number of
active D2D links in the cell and in each sector. Then $BS_{x}$ calculates
$\hat{I}_{x0}^{th}$ for the QoS of CUE and $\mathbb{E}[P_{tx_{CUE}}]$ for the
expectation of the interference caused by CUEs (see Appendix VIII), finally it
broadcasts all necessary parameters or sends them in D2D discovery messages.

##  VI Optimal Admission Control

In this section, we present the optimal admission control assuming complete
CSI is available. This serves as the performance benchmark and it is based on
the same approach found in [19]. The goal is to maximize the frequency reuse,
i.e., the number of active D2D links in the system while providing QoS to CUEs
and D2D links. So a mixed integer programming (MIP) optimization problem is
formulated as follows:

$\normalsize{\max_{\left(\begin{subarray}{c}\phi_{ij}\in\left\\{0,1\right\\}\\\
P_{ij}\in\mathbb{R}^{+}\end{subarray}\right)}\left\\{\sum_{i=1}^{N}\sum_{j=1}^{\hat{N_{i}}}\phi_{ij}\right\\},}$
(29)

Subject to:

$\Gamma_{x0}=\,\frac{P_{x0}G_{x0x0}}{I_{x0}^{D2D}\,+\,I_{x0}^{CUE}\,+\,\mathcal{N}_{BS}}\,\geq\,\gamma_{x0}^{th},$
(30a)
$\Gamma_{xk}\,=\,\frac{\phi_{xk}P_{xk}G_{xkxk}+\left(1-\phi_{xk}\right)M_{xk}}{I_{xk}^{D2D}+I_{xk}^{CUE}+\mathcal{N}_{D}}\,\geq\,\gamma_{xk}^{th},$
(30b) $\normalsize{\phi_{xk}P_{xk}\,\leq\,P_{D}^{max},}$ (30c)

$\forall x\in\\{1,\,...,\,N\\}\;\forall k\in\\{1,\,...,\,\hat{N}_{x}\\}$.

$M_{xk}$ is a set parameter defined in such a way that when a D2D link is
inactive $\phi_{xk}=0$, its SINR constraint is always satisfied. By doing this
we avoid the issue of having an unfeasible optimization problem because of
inactive D2D links. The constraint for $M_{xk}$ is given by:

$\normalsize{\begin{array}[]{l}M_{xk}\geq\gamma_{xk}^{th}\Big{(}P_{D}^{max}\left(\sum_{i=1}^{N}\sum_{j=1}^{\hat{N}_{i}}G_{ijxk}-G_{xkxk}\right)\\\
\qquad\quad\;+\,P_{C}^{max}\sum_{i=1}^{N}G_{i0xk}\,+\,\mathcal{N}_{D}\Big{)},\end{array}}$
(31)

where $P_{C}^{max}$ represents the maximum transmission power of CUEs. The
optimization variables are the state of D2D links $\phi_{xk}$ and their
transmission power $P_{xk}$.

The MIP optimization problem depicted in (29) and (30) cannot be solved
directly because the constraint (30c) is nonlinear, given that $\phi_{xk}$ is
a binary variable. To obtain a linear constraint we define:

$\normalsize{\tilde{P}_{xk}=\phi_{xk}P_{xk}\,\leq\,P_{D}^{max}.}$ (32)

Furthermore the constraints (30a) and (30b) are also nonlinear, thus we
combine (32) and (2) to rewrite the optimization problem as:

$\normalsize{\max_{\left(\begin{subarray}{c}\phi_{ij}\in\left\\{0,1\right\\}\\\
\tilde{P}_{ij}\in\mathbb{R}^{+}\end{subarray}\right)}\left\\{\sum_{i=1}^{N}\sum_{j=1}^{\hat{N_{i}}}\phi_{ij}\right\\},}$
(33)

Subject to:

$\sum_{i=1}^{N}\sum_{j=1}^{\hat{N}_{i}}\tilde{P}_{ij}G_{ijx0}\leq\sum_{i=1}^{N}P_{i0}A_{ix0}^{C}-\mathcal{N}_{BS},$
(34a)
$\sum_{i=1}^{N}\sum_{j=1}^{\hat{N}_{i}}\tilde{P}_{ij}A_{ijxk}^{D}+\phi_{xk}M_{xk}\leq
B_{xk}^{D},$ (34b) $\normalsize{\tilde{P}_{xk}\,\leq\,P_{D}^{max},}$ (34c)

$\forall x\in\\{1,\,...,\,N\\}\;\forall k\in\\{1,\,...,\,\hat{N}_{x}\\},$

where

$\normalsize{A_{ix0}^{C}=\Bigg{\\{}\begin{array}[]{l}\frac{\text{\normalsize{$G_{x0x0}$}}}{\text{\normalsize{$\gamma_{x0}^{th}$}}}\;\;\,\text{if}\;i=x\\\
-G_{i0x0}\;\text{if}\;i\neq x\end{array},}$ (35a)
$\normalsize{A_{ijxk}^{D}=\Bigg{\\{}\begin{array}[]{l}\text{\normalsize{$-G_{xkxk}$}}\;\;\,\text{if}\;\left(i=x\>\text{and}\>j=k\right)\\\
\text{\normalsize{$\gamma_{xk}^{th}G_{ijxk}$}}\;\text{if}\;\left(i\neq
x\;\text{or}\;\;\>j\neq k\right)\end{array},}$ (35b)
$\normalsize{B_{xk}^{C}=M_{xk}-\gamma_{xk}^{th}\left(\sum_{i=1}^{N}P_{i0}G_{i0xk}+\mathcal{N}_{D}\right).}$
(35c)

The optimization problem found in (33), (34) and (35) is a mixed integer
linear programming (MILP) problem and its solution can be found by using
exhaustive search algorithms.

##  VII Numerical analysis

To evaluate the performance of the proposed interference coordination methods
we conduct extensive Monte-Carlo simulations. We consider 7 circular cells of
radius $R$ where BSs are located at the center of the cells. In order not to
underestimate the interference conditions we only collect data from the center
cell. In each realization we generate one CUE uniformly distributed per cell,
where the distance to the BS is within $d_{CUE}\in[d_{min},R]$. Also $\hat{N}$
D2D pairs are generated following an uniform distribution where the distance
between devices of the same pair is randomly selected within
$d_{D2D}\in[D_{min},D_{max}]$. The channel model accounts for path loss and
log-normal shadow fading implemented according to 3GPP specifications [22].
Notice that our analysis is not sensitive to the specific density of users in
the system. The reason for this is that first, the density of D2D nodes is not
of high importance given that most of them may be silent, what it is important
is the maximum density of D2D nodes that can be active at the same time in the
network, which is determined by the admission control algorithms. Second,
since we have included log-normal shadow fading in our channel model, the
received powers and interference terms will not be tightly dependent on the
user distribution given that the log-normal distribution has quite long tails.
Table VII summarizes the main parameters used in the simulations.

TABLE I: Simulation parameters.

Description |  Representation and Value
---|---
Radius |  $R=$400 [m]
Noise power |  $\mathcal{N}_{D},\mathcal{N}_{BS}=-$174[dBm/Hz]
RB bandwidth |  $B_{w}=$180 [kHz]
Carrier frequency |  $f_{c}=$2 [GHz]
Max. transmission power |  $P_{D}^{max},P_{C}^{max}=$23 [dBm]
Min. distance between the BS and the users. |  $d_{min}=$10 [m]
D2D distance ($d_{D2D}$) bounds |  $D_{min}=$ 10 [m]
$D_{max}=$ 40 [m] |
Number of cells |  $N=$ 7
Available D2D pairs per cell |  $\hat{N}=$ 10
Path loss coeff. (user to BS) |  $c_{0}=-$30.55 [dB]
Path loss coeff. (user to user) |  $c_{d}=-$28.03 [dB]
Path loss exp. (user to BS) |  $\alpha_{0}=$ 3.67
Path loss exp. (user to user) |  $\alpha_{d}=$ 4
Monte-Carlo realizations |  $5000$

The MILP optimization problem is solved by using the optimization software
MOSEK implemented in MATLAB.

To present a comparative analysis with previous solutions we introduce two
single-cell methods provided in [19]. The first is an optimization problem
formulated with full CSI within each cell and the second is a centralized
solution where a statistical upper bound for the number of active D2D links is
derived assuming no available CSI. The initial solution is denoted as peak
interference constraint (PIC) in a single-cell “PIC. Single-cell” and the
latter as average interference constraint (AIC) in a single-cell “AIC. Single-
cell”. These two solutions were developed for a single-cell scenario, thus we
implement them independently in each cell.

The transmission power of CUEs is given by the LTE OFPC [9] depicted as:

$\begin{array}[]{c}P_{x0_{\text{dBm}}}\,=\,P_{0_{\text{dBm}}}-\alpha_{p}G_{x0x0_{\text{dB}}},\\\\[5.69054pt]
P_{0_{\text{dBm}}}=\alpha_{p}\left(\gamma_{CUE_{\text{dB}}}^{th}+\mathcal{N}_{BS_{\text{dB}}}\right)+\left(1-\alpha_{p}\right)\left(P_{C_{dBm}}^{max}\right),\end{array}$
(36)

where $\alpha_{p}$ is the path loss compensation factor and
$\gamma_{CUE_{\text{dB}}}^{th}$ is the open loop target signal-to-noise ratio
(SNR). We assume that the transmission powers of CUEs are given and cannot be
modified by the proposed interference coordination methods.

Fig. 6 depicts the behavior of the theoretical upper bound for the number of
D2D links, in a circular area if radius R, derived in (16) versus the target
CUE SINR loss and maximum distance between devices within the same D2D pair.
In Fig. 5a as $\delta$ increases the number of D2D links also increases until
it reaches a point of saturation. This occurs because as $\delta$ increases
the constraint regarding the QoS of CUEs becomes less strict and more D2D
links can be admitted in the system. However after a certain point no more D2D
links can be admitted due to the interference caused by D2D links towards each
other. Fig. 5b shows that as the distance $d_{D2D_{max}}$ becomes higher the
number of D2D links decreases exponentially, thus the most gain of having D2D
links is obtained when the D2D transmitter and receiver of the same pair are
in proximity to each other. This result shows that the distance between the
transmitter and receiver of the same D2D pair and the tolerable performance
loss of CUEs play a key role in the maximum number of D2D links that the
system can support.

(a) Max. Nr. D2D links vs Target CUE SINR Loss $\delta$ for $d_{D2D_{max}}=40$
[m].

(b) Upper bound Nr. D2D links vs D2D max. distance $d_{D2D_{max}}$ for
$\delta=3$ [dB].

Figure 6: Upper Bound of number of D2D links, for $\gamma_{D}=8$ [dB], in a
circular area of radius $R$.

Now we compare the overall performance of the different interference
coordination methods for a low target CUE SINR loss ($\delta=2$ [dB]) and a
high D2D SINR target ($\gamma_{D}=16$[dB]). These conditions represent the
ideal case of D2D communications where the D2D links achieve high data rates
while causing the least amount of disturbance towards the QoS of CUEs.

(c) CDF of Nr of D2D links with QoS. (d) CDF of SE. Figure 7: CDF of overall
performance for $\gamma_{D}=$ 16 [dB] $\delta=$ 2 [dB].

Fig. LABEL:fig:sfig:CDF_CUE_loss_PC and Fig. LABEL:fig:sfig:CDF_D2D_SINR_PC
show the empirical cumulative distribution function (CDF) of the SINR loss of
CUEs and the SINR of active D2D links respectively. For the single-cell
methods, the SINR of D2D links is mostly below the required target
$\gamma_{D}$ and there is a significant CUE SINR loss when compared to the
multi-cell solutions. Fig. 6c depicts the empirical CDF of the number of
active D2D links with QoS and Fig. 6d shows the empirical CDF of the SE of the
system. We see that the single-cell solutions neglect the impact of inter-cell
interference when coordinating the admission and power control of D2D links,
thus perform poorly.

For the BAC method we can see that the SINR of D2D links is quite high and for
over 80% of devices the required target is fulfilled, furthermore the CUE SINR
loss is well below the required target for 80% of the CUEs. However the number
of D2D links with QoS and the SE are quite limited. This occurs because the
BAC method does not control the individual admission of each D2D link, thus
when their target SINR $\gamma_{D}$ is high the number of D2D links that can
be active is low limiting the scalability and SE.

In the case of the DAC method we can see that the SINR of D2D links is more
balanced around the required target with over 70% of devices above it and the
CUE SINR loss is quite low, assuring the respective target for over 90% of the
CUEs. Furthermore we can see a significant improvement in the number of D2D
links combined with high values of SE.

Let the outage probability be the probability of not achieving the target SINR

$P_{out_{xk}}=Prob\left\\{\Gamma_{xk}<\gamma_{xk}^{th}\right\\}.$ (37)

For the results shown in Fig. VII we see that the outage probability assured
by each of the evaluated methods is different, thus it would be interesting to
compare the performance when the QoS offered by all methods is the same. To
achieve this we calculate the 5 percentile of the SINR of D2D links which
gives us the different values of SINR for which all methods provide a 5% D2D
outage probability. Similarly we consider the 95 percentile of the CUE SINR
loss that shows the different values of CUE SINR losses for which a 5% CUE
outage probability is assured by all methods.

Figure 8: CUE outage probability vs 5 % D2D links SINR, for $\delta=$ 2 [dB].

Fig. 8 depicts the CUE outage probability for different values of the 5
percentile of the SINR of D2D links to illustrate the effect of the QoS of D2D
links on the QoS of CUEs. We can see that the BAC method has a concave shape
behavior as the SINR of D2D links increases. This is caused by the selection
of the target received power of the D2D links ($\hat{P}_{r_{D}}$) done in the
BAC method. Initially as the SINR of D2D links increases more interference is
caused to the CUEs increasing their outage, then for higher D2D SINR values
less D2D links are allowed to be active which in turn reduces the outage
probability of CUEs. For the DAC method we see that for a wide range of D2D
SINR values the outage probability of CUEs is maintained at acceptable levels.

At this point we have seen the performance of the proposed methods in terms of
the QoS that they can provide to D2D links and CUEs. Now we would like to get
a more comprehensive view of the performance of the system when a specific QoS
is assured by all methods for all users. For this we present 3D graphs where
on one axis we place the 5 percentile of the SINR of D2D links and on the
other axis we put the 95 percentile of the CUE SINR loss. By making this
selection of axes we can evaluate the performance for a 5% outage probability
of both CUEs and D2D links. Notice that the OAC method is not shown in the
figures as it does not have outage probability since it corresponds to an
optimal solution where all QoS requirements are assured for all active users.

Fig. 9 shows the performance of the number of D2D links with QoS and Fig. 10
depicts the SE for all the evaluated methods. We see that for the DAC method
as the SINR of D2D links grows, the number of active pairs with QoS decreases,
which causes a quasi-concave behavior for the SE. When the SINR of D2D links
is low there are more D2D links active in the system but their data rates are
low as well, then as the SINR of D2D links increases the data rates also
increase, resulting in higher SE. However as the SINR of D2D links increases
the number of D2D links decays which causes the overall SE to drop regardless
of the SINR values of D2D links. As the CUE SINR loss increases more D2D links
can be admitted in the system which in turn increments the SE, however after a
certain point the number of D2D links reaches a saturation point along with
the SE. This means that the interference between different D2D links is
becoming a more and more important limiting factor in the density of active
D2D links.

Figure 9: Nr of active D2D links with QoS vs 5% SINR of active D2D links and
95% CUE SINR loss. Figure 10: Spectral efficiency [bps/Hz] vs 5% SINR of
active D2D links and 95% CUE SINR loss.

Compared to the other methods, the DAC method provides a significant
improvement in D2D density and SE for a wide range of D2D SINR with a low CUE
SINR loss. Thus the DAC method is a low complexity solution suitable for
practical D2D communications.

##  VIII Conclusion

We have conducted a comprehensive study on the admission control, interference
coordination, and power control for scalable D2D communications underlay
cellular networks. While there have been sophisticated ICIC techniques to
handle the first level interference among cellular users, our study shows that
it is of paramount importance to develop new interference coordination
techniques to solve the second level interference issues, i.e., those from D2D
to cellular communications and those among D2D links themselves. We have
developed three interference coordination methods, i.e. OAC, DAC and BAC to
maximize the network frequency reuse, while assuring QoS to all users.

The results show that the best performance is achieved by the OAC followed by
the DAC and finally the BAC method. Thus we have seen that as the CSI becomes
available the performance of the interference coordination is increased,
however the complexity and signaling overhead is also increased. Thus, a
trade-off must be found in order to provide a practical solution that can be
deployed in real systems.

The DAC method has been proven to be a good candidate for practical
applications. This method provides good SE and scalability for a wide range of
QoS requirements for both D2D links and CUEs and can be easily implemented in
5G LTE-A systems. We can conclude that proper design of D2D communications can
dramatically increase the network frequency reuse and thus network capacity,
especially for short distance communications. In addition, it is possible to
achieve this with low complexity interference coordination and low power
transmission methods without affecting existing cellular communications.

[Parameters of the DAC method]

Tables II depicts all additional parameters that are calculated at the BS and
broadcasted to the D2D links to support the DAC method. Similarly table III
illustrates all additional parameters that the D2D links calculate in order to
obtain their transmission power constraints.

TABLE II: Parameters broadcasted by the BS. Parameter |  Description
---|---
|  $\mathbb{E}[I_{x0}^{CUE}]=$
---
$\frac{A_{x0}^{C}}{A_{cl_{x}}}\mathbb{E}[P_{tx_{CUE}}]\mathbb{E}[G_{CUE-BS-
out}]$
Expected value of inter-cell interference caused by CUEs towards $BS_{x}$.
|  $\mathbb{E}[P_{tx_{CUE}}]=$
---
$\mathbb{E}\\!\left[\\!\left(\\!\frac{\gamma_{CUE}^{th}\mathcal{N}_{BS}}{G_{CUE-
BS-
in}}\\!\right)^{\\!\\!\alpha_{p}}\\!\\!(P_{C}^{max})^{(1-\alpha_{p})}\\!\right]$
Expected value of the transmission power of CUEs.
|  $\mathbb{E}[G_{CUE-BS-in}]=$
---
$\frac{2c_{0}\left(d_{CUE-
BS_{min}}^{-(\alpha_{0}-2)}-R^{-(\alpha_{0}-2)}\right)}{R^{2}\left(\alpha_{0}-2\right)}$
Expected value of the channel gain between a CUE and its serving BS.
|  $\mathbb{E}[G_{CUE-BS-out}]=$
---
$\frac{2c_{0}\left(R^{-(\alpha_{0}-2)}-d_{CUE-
BS_{max}}^{-(\alpha_{0}-2)}\right)}{(d_{D2D-BS_{max}})^{2}\left(\alpha_{0}-2\right)}$
Expected values of the channel gain between an interfering CUE and a given BS.
$d_{CUE-BS_{max}}=\left(\frac{P_{C}^{max}c_{0}}{\mathcal{N}_{BS}}\right)^{1/\alpha_{0}}$ |  Max. distance between an interfering CUE and a given BS.
$A_{x0}^{C}=\pi(d_{CUE-BS_{max}})^{2}-\pi R^{2}$ |  Interference area for $BS_{x}$ as the receiver, when CUEs are the source of interference.
TABLE III: Parameters calculated by the D2D links. |  $\mathbb{E}[I_{xk}^{CUE}]=$
---
$\frac{A_{xk}^{C}}{A_{cl_{x}}}\,\,\mathbb{E}[P_{tx_{CUE}}]\,\mathbb{E}[G_{CUE-D}]$
Expected value of interference caused by CUEs towards $D2D_{xk}$.
|  $\mathbb{E}[G_{CUE-D}]=$
---
$\frac{2c_{d}\left(d_{CUE-D_{min}}^{-(\alpha_{d}-2)}\\!-d_{CUE-
D_{max}}^{-(\alpha_{d}-2)}\right)}{(d_{CUE-
D_{max}})^{2}\left(\alpha_{d}-2\right)}$
Expected value of the channel gain between a given D2D link and a CUE.
$d_{CUE-D_{max}}\\!=\\!\left(\frac{P_{C}^{max}c_{d}}{\mathcal{N}_{D}}\right)^{\frac{1}{\alpha_{d}}}$ |  Max. distance between a given CUE and a D2D link.
$A_{xk}^{C}=\pi(d_{CUE-D_{max}})^{2}$ |  Interference area for $D2D_{xk}$ as a receiver, when CUEs are the source of interference.
$A_{x0}=\pi(d_{D2D-BS_{max}})^{2}$ |  Interference area for $BS_{x}$ as the receiver, when D2D links are the source of interference.
$d_{D2D-BS_{max}}\\!=\\!\left(\\!\frac{P_{D}^{max}c_{0}}{\mathcal{N}_{BS}}\\!\right)^{\frac{1}{\alpha_{0}}}$ |  Max. distance between an interfering D2D link and a given BS.
$A_{xk}=\pi(d_{D2D-I_{max}})^{2}$ |  Interference area for $D2D_{xk}$ as the receiver, when D2D links are the source of interference.
$d_{D2D-I_{max}}=\left(\frac{P_{D}^{max}c_{d}}{\mathcal{N}_{D}}\right)^{\frac{1}{\alpha_{d}}}$ |  Max. distance between an interfering D2D link and another given D2D pair.

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