# Energy Efficient Power Allocation for Device-to-Device Communications
Underlaid Cellular Networks Using Stochastic Geometry

Negar Zabetian, Abbas Mohammadi, and Meysam Masoudi N. Zabetian and A.
Mohammadi are with the Department of Electrical Engineering, Amirkabir
University of Technology (Tehran Polytechnic), Tehran, Iran. Meysam Masoudi is
with the School of Electrical Engineering and Computer Science, KTH Royal
Institute of Technology, Stockholm, Sweden

###### Abstract

In this paper, we study an energy efficiency maximization problem in uplink
for D2D communications underlaid with cellular networks on multiple bands.
Utilizing stochastic geometry, we derive closed-form expressions for the
average sum rate, successful transmission probability, and energy efficiency
of cellular and D2D users. Then, we formulate an optimization problem to
jointly maximize the energy efficiency of D2D and cellular users and obtain
optimum transmission power of both D2D and cellular users. In the optimization
problem, we guarantee the QoS of users by taking into account the success
transmission probability on each link. To solve the problem, first we convert
the problem into canonical convex form. Afterwards, we solve the problem in
two phases, energy efficiency maximization of devices and energy efficiency
maximization of cellular users. In the first phase, we maximize the energy
efficiency of D2D users and feed the solution to the second phase where we
maximize the energy efficiency of cellular users. Simulation results reveal
that significant energy efficiency can be attained e.g., 10% energy efficiency
improvement compared to fix transmission power in high density scenario.

###### Index Terms:

Device-to-device communication, Energy efficiency, Stochastic geometry,
Successful transmission probability

##  I Introduction

With the unprecedented demand of mobile traffic, the fifth generation (5G) of
mobile networks are anticipated to support 1000 times more data traffic [1].
This requirement urges the development of new technologies, and among them,
device-to-device (D2D) communication is considered as one of the key
technologies that aid 5G networks to achieve their goals. In D2D
communication, proximate devices can directly communicate with each other
under the control of base stations (BSs) [2]. Due to the proximity of devices,
D2D communication has the advantage of providing services with higher data
rate, spectral efficiency (SE), energy efficiency (EE) with lower power
consumption and latency [3]. Despite the appealing potential gains of D2D
communication, sharing the same spectrum with cellular users induce new
challenges into the network such as interference management. D2D
communications can operate on either licensed band (in-band) or unlicensed
band (out-band). In the literature, in-band communication is classified into
two categories, namely, underlay and overlay communications. In the overlay
scenario, the system reserves dedicated parts of the spectrum for D2D links
and cellular users use another part of the spectrum to avoid interference
between D2D users and cellular users. However, in this case, SE is less than
underlay mode since the spectrum is underutilized. In the underlay
communication, D2D devices reuse the cellular spectrum and hence create
interference on cellular users [4]. Therefore, there is a need to utilize
efficient resource allocation approaches to take advantage of D2D benefits and
mitigate the challenges of using such technology.

In the literature, there are bulks of studies investigating the efficient
resource allocation to benefit from the advantages of D2D communications while
diminishing the performance degradation of the cellular system due to new
sources of interference. In particular, power control is one of the promising
approaches that mitigates interference [5]. In [6] a power control scheme is
proposed to manage the cross-tier interference between cellular and D2D users.
The authors in [7] proposed an interference-aware algorithm for power control
in D2D communications underlaying cellular networks. In [8], a single-cell D2D
underlay cellular network is considered. A centralized and distributed
algorithms are proposed to find the users’ optimal power transmission. A
dynamic power control for D2D communication underlaying uplink multi-cell
network is investigated in [9] considering interference mitigation. However
power control is a key technique to manage the interference, both the energy
efficiency (EE) and QoS are still influenced by the interference in the
network.

Energy efficiency (EE) is a metric quantifying the efficiency of resource
utilization. In fact, EE not only brings considerable economic benefits into
the network but also can be interpreted as concerning about the environment
[10]. Extensive research studies have been devoted to the energy efficiency of
the system. EE maximization in D2D communication underlaying cellular network
in cloud radio access network is studied in [1]. A closed-form expression for
EE is derived in [11] by utilizing stochastic geometry tool. In this study, EE
is maximized by optimizing transmit power and density of base station (BS) in
cellular networks. An energy efficient power control algorithm is proposed in
[12] to share resources among the cellular and D2D users. Joint power and
subcarrier allocation is proposed in [13, 14] for energy efficient D2D
communication on multiple bands and single-cell scenario. EE and SE trade-off
in a single-cell D2D communication scenario is investigated in [15] by a
distributed resource allocation scheme. The optimal power control for energy
efficiency of D2D communication underlaying cellular networks is analyzed in
[16] where only a single band is considered. In [17], EE maximization problem
with a constraint on SE is investigated in D2D communication underlay cellular
network on multiple bands. A power allocation algorithm is proposed to obtain
optimum power of D2D users. However, they did not consider the power
allocation for cellular users. The authors in [18] formulated an energy-
efficient resource allocation problem considering multiple bands; however the
QoS is limited to the minimum required rate for cellular users. The D2D
communication underlaying cellular networks on multiple bands is considered in
[19], and the derivative-based algorithm is applied to maximize the energy
efficiency of D2D users, however they did not optimize the cellular users
transmission power. In this study, the authors derived the outage transmission
probability as QoS metric using Stochastic geometry.

Stochastic geometry is a powerful tool to model and analyze D2D networks and
provides closed-form expressions for several metrics such as average sum rate
and successful transmission probability [20, 21, 22]. In [20], the authors
model the D2D users distribution by Poisson point process (PPP). In [21], the
outage probability for a cellular user and a D2D receiver is obtained. In
[22], a power control scheme is proposed to improve the outage probability in
the network.

According to the best of our knowledge, there are few studies in the
literature obtaining optimum transmission power of D2D users in the cellular
networks where users are modeled based on stochastic geometry on multiple
bands. The authors in [19] modeled the users by stochastic geometry on
multiple bands and maximized the EE of D2D users. However, they neglect the
impact of cellular users’ transmission power. In this paper, we bring up an
effective solution to maximize the energy efficiency of cellular network
underlaid D2D communication in multiple bands and derived optimum power in
each band. Because the channel fading coefficient can be varying in different
bands. Also, there are different densities for D2D and cellular users in each
band and the number of users which are allowed to transmit is different. Due
to the fluctuation of resources and the evolving radio and network conditions,
having multiple bands is more practical in future 5G wireless networks, and it
can consider the dynamic behavior of a system. Since the wireless channel
parameters such as channel fading may vary, having multiple bands and
adjustable power transmission on each, can enhance the energy efficiency of
the network. Moreover, due to having multiple bands, D2D users do not
interfere on the cellular users and devices in other bands, so, the complexity
of interference management becomes affordable, and the performance of D2D
communications is more acceptable.

We derived the closed-form expressions for the average sum rate (ASR) and
successful transmission probability (STP) for both the cellular users and D2D
users. Then, we formulated a mixed integer nonlinear programming (MINLP)
optimization problem to maximize the energy efficiency of both cellular and
D2D users while considering the QoS of both cellular and D2D users. To solve
the problem, first, we transformed the MINLP problem into a convex form
utilizing the technique of changing the optimization variable. We have also
broken down the problem into two sub-problems. To address the primary problem,
we solve these sub-problems iteratively. The first sub-problem maximizes the
EE of D2D users and finds the optimal transmission power of D2D users. By
using the solution of the first sub-problem, the second problem finds the
optimal transmission power of cellular users to maximize the EE. As a
reference case, we compared our study with [19] which has similar system model
and assumptions to our paper.

The remainder of the paper is organized as follows: The system model is
presented in Section II. In Section III, the analytic studies and problem
formulations are described. The solution methodology is proposed in Section IV
followed by Section V where the numerical results are discussed. Finally, in
Section VI the concluding remarks are presented.

Notation: $\mathbb{P(.)}$, $\mathbb{E}(x)$ and $\exp(.)$ represent the
probability, expectation value of random variable $x$ and exponential
function, respectively. $\rm\Gamma(.)$ expresses the gamma function with the
form $\rm\Gamma(z)=\int\limits_{0}^{\infty}{{t^{z-1}}{e^{-t}}}dt$ and
${\mathcal{L}_{x}}\left(s\right)$ stands for the Laplace transform of $x$.

##  II system model

In this study, we investigate a cellular network underlaid with D2D
communications in the uplink direction as depicted in Fig. 1. In this system,
the set of active cellular users in each band share the same bandwidth with
the set of active D2D users in that band. Cellular and D2D users are
distributed over the network with homogeneous PPP distribution. As can be seen
in Fig. 1, all cellular users and devices are distributed on a two-dimensional
plane $\mathbb{R}^{2}$. In the rest of this paper, superscript C and D are
used to denote cellular and D2D users. We denote the total bandwidth by $W$,
and we break it down to $M$ sub-bands. The bandwidth of each sub-band is
$W_{i}$ where $i=1,2,\dots,M$. Since the wireless channel parameters such as
channel fading may vary, having multiple bands and adjustable power
transmission on each, can enhance the energy efficiency of the network. We
assume that the signals in different bands do not interfere with each other.
Both cellular and D2D users can utilize these sub-bands with density
${\lambda_{c,i}}$ and ${\lambda_{d,i}}$ on each band, respectively. In each
band, there are different densities for D2D and cellular users. If the more
bandwidth is allocated to the users, the more users are allowed to transmit,
and the density of users become greater. We consider that the channel model
has path loss and small-scale fading which is modeled by Rayleigh distribution
with unit mean. Therefore, channel gain between users has an exponential
distribution which is denoted by $g$. The received power of both devices and
cellular users are obtained by considering large- and small scale fading as
follows.

${P_{r}}={P_{t}}g{r^{-\alpha}},$ (1)

where $\alpha$ is path loss exponent, $r$ is the distance between receiver and
transmitter and $P_{t}$ is the transmission power.

Figure 1: A representation of D2D communication underlaying cellular network.

According to Palm theorem [23], for Poisson point process, each point has a
similar probability of being chosen. Therefore, we have randomly chosen point
or typical point. According to Slivnyak’s theorem [23], conditioning on a
point does not influence the distribution of the rest of the process. Thus, we
can condition on having a typical D2D receiver and a typical base station for
D2D and cellular communication at the center on $\mathbb{R}^{2}$, respectively
and analyze the performance of the whole network.

The users on the same band can interfere with each other while they have no
interference on other bands. Assume the signal which is received at the D2D
receiver in $i$-th band is denoted by $Y_{d,i}$ and calculated as,

$\begin{array}[]{*{20}{l}}{{Y_{d,i}}=\overbrace{{s_{d,i}}{h_{d,00}}R_{d,00,i}^{-\frac{\alpha}{2}}}^{Signal{\kern
1.0pt}from{\kern 1.0pt}typical{\kern 1.0pt}D2D{\kern
1.0pt}user}+\underbrace{\sum\limits_{j\in{\phi_{c,i}}}{{s_{c,i}}{h_{c,j0}}R_{c,j0,i}^{-\frac{\alpha}{2}}}}_{Interference{\kern
1.0pt}from{\kern 1.0pt}cellular{\kern
1.0pt}users}}{+\overbrace{\sum\limits_{\ell\in{\phi_{d,i}}}{{s_{d,i}}{h_{d,\ell
0}}R_{d,\ell 0,i}^{-\frac{\alpha}{2}}}}^{Interference{\kern 1.0pt}from{\kern
1.0pt}D2D{\kern 1.0pt}users}+n},\end{array}$ (2)

where ${\phi_{d,i}}$ and ${\phi_{c,i}}$ denote the set of active D2D and
cellular users in $i$-th band, ${s_{d,i}}$ and ${s_{c,i}}$ represent
information signals of D2D and cellular receiver in $i$-th band, respectively.
${h_{d,00}}$ and ${R_{d,00,i}}$ are the channel coefficient and the distance
between the typical D2D receiver and the corresponding transmitter in the
$i$-th band, respectively. ${h_{c,j0}}$ and ${R_{c,j0,i}}$ represent the
channel coefficient and the distance between the $j$-th cellular transmitter
and the typical D2D receiver in the $i$-th band, respectively. Similarly,
${h_{d,\ell 0}}$ and ${R_{d,\ell 0,i}}$ are the channel coefficient and the
distance between the $\ell$-th D2D transmitter and the typical D2D receiver in
the $i$-th band, respectively.

$n$ is additive white Gaussian noise (AWGN) with zero mean and variance
$N_{0}$. The signal to noise plus interference ratio (SINR) of typical D2D
receiver in $i$-th band is given by

$SIN{R_{d,i}}=\frac{{{g_{d,00}}R_{d,00,i}^{-\alpha}}}{{{I_{d,c0,i}}+{I_{d,d0,i}}+\frac{N_{0}}{{P_{d,i}}}}}.$
(3)

The interference which is introduced by cellular users to the typical D2D
receiver is given by

${I_{d,c0,i}}=\sum\limits_{j\in{\Phi_{c,i}}}{\frac{{{P_{c,i}}}}{{{P_{d,i}}}}{g_{c,j0}}R_{c,j0,i}^{-\alpha}},$
(4)

where ${P_{d,i}}$ and ${P_{c,i}}$ are transmission power of the D2D and
cellular users in the $i$-th band.

Also, the interference from D2D users to the typical D2D receiver in the
$i$-th band is given by

${I_{d,d0,i}}=\sum\limits_{\ell\in{\Phi_{d,i}}}{{g_{d,\ell 0}}R_{d,\ell
0,i}^{-\alpha}}.$ (5)

Similarly, the received signal by the typical base station is given by

$\begin{array}[]{l}{Y_{c,i}}={s_{c,i}}{h_{c,00}}R_{c,00,i}^{-\frac{\alpha}{2}}+\sum\limits_{j\in{\phi_{c,i}}}{{s_{c,i}}{h_{c,j0}}R_{c,j0,i}^{-\frac{\alpha}{2}}}\begin{array}[]{*{20}{c}}{}\hfil&{+\sum\limits_{\ell\in{\phi_{d,i}}}{{s_{d,i}}{h_{d,\ell
0}}R_{d,\ell 0,i}^{-\frac{\alpha}{2}}}+n.}\end{array}\end{array}$ (6)

The SINR of the typical BS in $i$-th band is as follows.

$SIN{R_{c,i}}=\frac{{{g_{c,00}}R_{c,00,i}^{-\alpha}}}{{{I_{c,c0,i}}+{I_{c,d0,i}}+\frac{N_{0}}{{P_{c,i}}}}},$
(7)

The interference from cellular users to the typical BS in $i$-th band is shown
by

${I_{c,c0,i}}=\sum\limits_{j\in{\Phi_{c,i}}}{{g_{c,j0}}R_{c,j0,i}^{-\alpha}}.$
(8)

Also, the interference from D2D users to the typical BS in $i$-th band is
shown by

${I_{c,d0,i}}=\sum\limits_{\ell\in{\Phi_{d,i}}}{\frac{{{P_{d,i}}}}{{{P_{c,i}}}}{g_{d,\ell
0}}R_{d,\ell 0,i}^{-\alpha}}.$ (9)

Similar to other studies, i.e., [20], [19], in the rest of this study, the
performance of the system is studied under the interference-limited regime,
where the interference, which is introduced by spectrum sharing between large
number of users, is dominating the noise power. Hence, the impact of thermal
noise can be neglected and the SINR expression is converted to signal to
interference ratio (SIR).

##  III Problem Formulation

In this section, we intend to maximize the energy efficiency of the network.
To achieve this goal, first, we introduce the total energy efficiency (EE) as
the ratio of average sum rate (ASR) to the total power consumption [19]. Then,
we use the closed-form expressions for the ASR using the success probability
formula in [19]. Afterwards, we formulate the energy efficiency maximization
as an optimization problem.

###  III-A Users Success Probability

Total energy efficiency of the network is defined as,

$EE=\frac{{ASR}}{P_{t}},$ (10)

where $ASR$ and $P_{t}$ are average sum rate and total transmission power of
all users. To calculate $P_{t}$, the power consumption per unit area in $i$-th
band is considered which is expressed as ${\lambda_{d,i}}{P_{d,i}}$ and
${\lambda_{c,i}}{P_{c,i}}$ for D2D and cellular users, respectively.
$ASR_{d,i}$ is defined as ${\lambda_{d,i}}{\bar{R}_{d,i}}$. To compute ASR, we
use the lower bounds on the average rates of cellular and D2D users [24].

$\begin{array}[]{l}{{\bar{R}}_{c,i}}={\rm\mathbb{E}}\left[{W_{i}}{{{\log}_{2}}\left({1+SIR_{c,i}}\right)}\right]=\mathop{\sup}\limits_{{T_{c,i}}\geq
0}{W_{i}}\log_{2}(1+{T_{c,i}})\mathbb{P}(SI{R_{c,i}}\geq{T_{c,i}}),\\\
{{\bar{R}}_{d,i}}={\rm\mathbb{E}}\left[{W_{i}}{{{\log}_{2}}\left({1+SIR_{d,i}}\right)}\right]=\mathop{\sup}\limits_{{T_{d,i}}\geq
0}{W_{i}}\log_{2}(1+{T_{d,i}})\mathbb{P}(SI{R_{d,i}}\geq{T_{d,i}}),\end{array}$
(11)

where ${T_{c,i}}$ and ${T_{d,i}}$ are SIR thresholds for cellular and D2D
users in $i$-th band.

Theorem 1. The successful transmission probability for typical D2D receiver in
$i$-th band is given by

$\mathbb{P}(SI{R_{d,i}}\geq{T_{d,i}})=\exp\bigg{\\{}{\bigg{.}{-{\varsigma_{d,i}}\Big{[}{{\lambda_{d,i}}+{\lambda_{c,i}}{{\big{(}{\frac{{{P_{c,i}}}}{{{P_{d,i}}}}}\big{)}}^{\frac{2}{\alpha}}}}\Big{]}}\bigg{\\}}}\bigg{.},$
(12)

where ${\lambda_{d,i}}$ and ${\lambda_{c,i}}$ are the density of D2D and
cellular users in $i$-th band, respectively. Moreover,
${\varsigma_{d,i}}\buildrel\Delta\over{=}\pi{T_{d,i}}^{\frac{2}{\alpha}}R_{d,00,i}^{2}\rm\Gamma\big{(}{1+\frac{2}{\alpha}}\big{)}\rm\Gamma\big{(}{1-\frac{2}{\alpha}}\big{)}$
[19].

###### Proof.

By using (3), we have

$\begin{array}[]{l}\mathbb{P}\big{(}SI{R_{d,i}}\geq{T_{d,i}}\big{)}\\\
=\mathbb{P}\big{(}{\frac{{{g_{d,00}}R_{d,00,i}^{-\alpha}}}{{{I_{d,c0,i}}+{I_{d,d0,i}}}}\geq{T_{d,i}}}\big{)}\\\
=\mathbb{P}\Big{(}{{g_{d,00}}\geq{T_{d,i}}R_{d,00,i}^{\alpha}\big{(}{{I_{d,c0,i}}+{I_{d,d0,i}}}\big{)}}\Big{)}\\\
\mathop{=}\limits^{(a)}\mathbb{E}\Big{(}{\prod\limits_{\ell\in{\phi_{d,i}}}{\exp\big{(}{-{T_{d,i}}R_{d,00,i}^{\alpha}{g_{d,\ell
0}}R_{d,\ell
0,i}^{-\alpha}}\big{)}}}\Big{)}\times\mathbb{E}\Big{(}{\prod\limits_{j\in{\phi_{c,i}}}{\exp\big{(}{-{T_{d,i}}R_{d,00,i}^{\alpha}\frac{{{P_{c,i}}}}{{{P_{d,i}}}}{g_{c,j0}}R_{c,j0,i}^{-\alpha}}\big{)}}}\Big{)}\\\
={\mathcal{L}_{{I_{d,d0,i}}({g_{d,\ell
0}})}}({T_{d,i}}R_{d,00,i}^{\alpha}){\mathcal{L}_{{I_{d,c0,i}}({g_{c,j0}})}}({T_{d,i}}R_{d,00,i}^{\alpha}),\end{array}$
(13)

where (a) is due to exponential distribution and independence of cellular and
D2D channel gains. Considering Laplace transform and stochastic geometry
theorem [23], we have

$\begin{array}[]{l}{\mathcal{L}_{{I_{d,d0,i}}({g_{d,\ell
0}})}}({T_{d,i}}R_{d,00,i}^{\alpha})=\exp\Big{[}{-{\lambda_{d,i}}\int\limits_{0}^{\infty}{E\big{(}{{g_{d,\ell
0}}}\big{)}}\big{(}{1-{e^{-{T_{d,i}}R_{d,00,i}^{\alpha}{r^{-\alpha}}}}}\big{)}dr}\Big{]}\\\
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=\exp\Big{[}{-{\lambda_{d,i}}\pi
T_{d,i}^{\frac{2}{\alpha}}R_{d,00,i}^{2}\rm\Gamma\big{(}{1+\frac{2}{\alpha}}\big{)}\rm\Gamma\big{(}{1-\frac{2}{\alpha}}\big{)}}\Big{]},\\\
{\mathcal{L}_{{I_{d,c0,i}}({g_{c,j0}})}}({T_{d,i}}R_{d,00,i}^{\alpha})=\exp\Big{[}{-{\lambda_{c,i}}{{\left({\frac{{{P_{c,i}}}}{{{P_{d,i}}}}}\right)}^{\frac{2}{\alpha}}}\pi
T_{d,i}^{\frac{2}{\alpha}}R_{d,00,i}^{2}\rm\Gamma\big{(}{1+\frac{2}{\alpha}}\big{)}\rm\Gamma\big{(}{1-\frac{2}{\alpha}}\big{)}}\Big{]},\end{array}$
(14)

Substituting (14) in (13) concludes the proof. ∎

Theorem 2. The STP for typical base station in $i$-th band is obtained as,

$\mathbb{P}(SI{R_{c,i}}\geq{T_{c,i}})=\exp\bigg{\\{}{\bigg{.}{-{\varsigma_{c,i}}\Big{[}{{\lambda_{c,i}}+{\lambda_{d,i}}{{\big{(}{\frac{{{P_{d,i}}}}{{{P_{c,i}}}}}\big{)}}^{\frac{2}{\alpha}}}}\Big{]}}\bigg{\\}}}\bigg{.},$
(15)

where
${\varsigma_{c,i}}\buildrel\Delta\over{=}\pi{T_{c,i}}^{\frac{2}{\alpha}}R_{c,00,i}^{2}\rm\Gamma\big{(}{1+\frac{2}{\alpha}}\big{)}\rm\Gamma\big{(}{1-\frac{2}{\alpha}}\big{)}$.

###### Proof.

The proof is obtained similarly to Theorem 1 [19]. ∎

By substituting, (12) and (15) in (11) ASR of cellular users and devices are
obtained. Therefore, EE of cellular and D2D users in $i$-th band are given by,

$\begin{array}[]{l}E{E_{c,i}}=\frac{{AS{R_{c,i}}}}{{{\lambda_{c,i}}{P_{c,i}}}}=\frac{{{W_{i}}}}{{{P_{c,i}}}}\log_{2}\big{(}1+{T_{c,i}}\big{)}\exp\bigg{\\{}{\left.{-{\varsigma_{c,i}}\Big{[}{{\lambda_{c,i}}+{\lambda_{d,i}}{{\big{(}{\frac{{{P_{d,i}}}}{{{P_{c,i}}}}}\big{)}}^{\frac{2}{\alpha}}}}\Big{]}}\right\\}}\bigg{.},\\\
E{E_{d,i}}=\frac{{AS{R_{d,i}}}}{{{\lambda_{d,i}}{P_{d,i}}}}=\frac{{{W_{i}}}}{{{P_{d,i}}}}\log_{2}\big{(}1+{T_{d,i}}\big{)}\exp\bigg{\\{}{\left.{-{\varsigma_{d,i}}\Big{[}{{\lambda_{d,i}}+{\lambda_{c,i}}{{\big{(}{\frac{{{P_{c,i}}}}{{{P_{d,i}}}}}\big{)}}^{\frac{2}{\alpha}}}}\Big{]}}\right\\}}\bigg{.}\end{array}$
(16)

###  III-B Energy Efficiency Optimization Problem

In the previous subsection, we defined energy efficiency based on the success
probability of cellular and D2D communications. Since the interference between
cellular users and devices, has an impact on the energy efficiency of each
user, it is more practical to study the EE of the entire cellular network. In
what follows, we will define energy efficiency maximization problems for D2D
and cellular users, separately and find optimal power of D2D and cellular
users.

####  III-B1 D2D Users Energy Efficiency

In this subsection, we formulate an optimization problem to maximize the total
energy efficiency of D2D users. EE optimization problem for D2D users is
expressed as follows.

$\begin{array}[]{l}\mathop{\max}\limits_{{P_{d,i}}}\begin{array}[]{*{20}{c}}{}\hfil&{E{E_{d}}}\end{array}=\sum\limits_{i=1}^{M}{E{E_{d,i}}}\\\
s\cdot t\cdot\,\,\,\,\,\,(1)\,\sum\limits_{i=1}^{M}{{P_{d,i}}\leq{P_{d}}}\\\
\begin{array}[]{*{20}{c}}{}\hfil&{\,\,\,\,\,\,\,\,\,\,\,\,(2)\,0\leq{P_{d,i}}\leq{P_{d,i,\max}}}\end{array}\\\
\begin{array}[]{*{20}{c}}{}\hfil&{\,\,\,\,\,\,\,\,\,\,\,\,(3)\,1-\mathbb{P}(SI{R_{c,i}}\geq{T_{c,i}})}\end{array}\leq{\theta_{c,i}}\\\
\begin{array}[]{*{20}{c}}{}\hfil&{\,\,\,\,\,\,\,\,\,\,\,\,(4)\,1-\mathbb{P}(SI{R_{d,i}}\geq{T_{d,i}})}\end{array}\leq{\theta_{d,i}},\end{array}$
(17)

In (17), $EE_{d,i}$ is the energy efficiency of D2D users in band $i$. The
first constraint ensures that the total transmission powers of D2D users over
all bands do not exceed the maximum possible value which is denoted by
${P_{d}}$. The second constraint ensures that the power of D2D users in each
band cannot exceed an allowable threshold indicated by ${P_{d,i,\max}}$. To
satisfy the quality of service of users, the outage probability of both
cellular and D2D users should be less than predefined thresholds which are
denoted by ${\theta_{c,i}}$ and ${\theta_{d,i}}$, respectively. According to
the quality of service constraints and objective function, the problem is not
convex.

####  III-B2 Cellular Users Energy Efficiency

In the previous subsection, when maximizing energy efficiency, we assumed a
constant power for cellular users. Since cellular users are operating in the
same band as D2D users, there is more room to enhance the energy efficiency of
the entire network by optimizing the power consumption of cellular users. In
this subsection, we calculate the optimal transmission power of cellular users
in each band. Energy efficiency optimization problem for cellular users is
formulated as follows.

$\begin{array}[]{l}\mathop{\max}\limits_{{P_{c,i}}}\begin{array}[]{*{20}{c}}{}\hfil&{E{E_{c}}}\end{array}=\sum\limits_{i=1}^{M}{E{E_{c,i}}}\\\
s\cdot t\cdot\,\,\,\,\,\,(1)\,\sum\limits_{i=1}^{M}{{P_{c,i}}\leq{P_{c}}}\\\
\begin{array}[]{*{20}{c}}{}\hfil&{\,\,\,\,\,\,\,\,\,\,\,\,(2)\,0\leq{P_{c,i}}\leq{P_{c,i,\max}}}\end{array}\\\
\begin{array}[]{*{20}{c}}{}\hfil&{\,\,\,\,\,\,\,\,\,\,\,\,(3)\,\sum\limits_{i=1}^{M}{{P_{d,i}}\leq{P_{d}}}}\end{array}\\\
\begin{array}[]{*{20}{c}}{}\hfil&{\,\,\,\,\,\,\,\,\,\,\,\,(4)\,0\leq{P_{d,i}}\leq{P_{d,i,\max}}}\end{array}\\\
\begin{array}[]{*{20}{c}}{}\hfil&{\,\,\,\,\,\,\,\,\,\,\,\,(5)\,1-\mathbb{P}(SI{R_{c,i}}\geq{T_{c,i}})}\end{array}\leq{\theta_{c,i}}\\\
\begin{array}[]{*{20}{c}}{}\hfil&{\,\,\,\,\,\,\,\,\,\,\,\,(6)\,1-\mathbb{P}(SI{R_{d,i}}\geq{T_{d,i}})}\end{array}\leq{\theta_{d,i}},\end{array}$
(18)

where $EE_{c,i}$ is the energy efficiency of cellular users in band $i$. The
total transmission power of cellular users over all bands and transmission
power of cellular users in each band should be less than predefined thresholds
which are denoted by ${P_{c}}$ and ${P_{c,i,\max}}$, respectively. Moreover,
D2D users consume more power to coordinate the interference which is produced
by cellular communication. Thus, we have constraints for the power of devices
besides cellular users’ power constraints. Also, we have the quality of
service constraints similar to the optimization problem of D2D users.

##  IV Solution Methodology

In this section, we aim at addressing the optimization problem. First, we
transform the optimization problems into the canonical convex form by a
technique of changing the variable. Then, we solve the problem of D2D and then
the corresponding results are given as an input to the cellular users’
optimization problem.

###  IV-A D2D Users Energy Efficiency

To transform the problem defined in (17), we define a new optimization
variable, ${x_{i}}$, as,

$\displaystyle{x_{i}}\buildrel\Delta\over{=}\exp\Big{(}{{\varsigma_{d,i}}{\lambda_{c,i}}{{\big{(}{\frac{{{P_{c,i}}}}{{{P_{d,i}}}}}\big{)}}^{\frac{2}{\alpha}}}}\Big{)}.$
(19)

Therefore, by inserting $x_{i}$ into (12), (15) and (16), the optimization
problem defined in (17) can be rewritten as follows.

$\begin{array}[]{l}\mathop{\max}\limits_{{x_{i}}}\,\,\,E{E_{d}}=\sum\limits_{i}{\frac{{{W_{i}}\log_{2}\big{(}1+{T_{d,i}}\big{)}\exp\big{(}{-{\varsigma_{d,i}}{\lambda_{d,i}}}\big{)}{{\big{(}{\ln{x_{i}}}\big{)}}^{{}^{\frac{\alpha}{2}}}}}}{{{P_{c,i}}{x_{i}}{{\big{(}{{\varsigma_{d,i}}{\lambda_{c,i}}}\big{)}}^{{}^{\frac{\alpha}{2}}}}}}}\\\
s\cdot
t\cdot\,(1)\,\sum\limits_{i}{{P_{c,i}}{{\left({\frac{{{\varsigma_{d,i}}{\lambda_{c,i}}}}{{\ln{x_{i}}}}}\right)}^{{}^{\frac{\alpha}{2}}}}}\leq{P_{d}}\\\
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(2)\,{P_{c,i}}{\big{(}{\frac{{{\varsigma_{d,i}}{\lambda_{c,i}}}}{{\ln{x_{i}}}}}\big{)}^{{}^{\frac{\alpha}{2}}}}\leq{P_{d,i,\max}}\\\
\begin{array}[]{*{20}{c}}{}\hfil&{\,\,\,\,\,\,\,\,(3)}\end{array}{x_{i}}\leq\frac{{\exp\big{(}{-{\varsigma_{d,i}}{\lambda_{d,i}}}\big{)}}}{{1-{\theta_{d,i}}}}\\\
\begin{array}[]{*{20}{c}}{}\hfil&{\,\,\,\,\,\,\,\,(4)}\end{array}{x_{i}}\geq\exp\big{(}{\frac{{-{\varsigma_{d,i}}{\lambda_{c,i}}}}{{\frac{{ln(1-{\theta_{c,i}})}}{{{\varsigma_{c,i}}{\lambda_{d,i}}}}+\frac{{{\lambda_{c,i}}}}{{{\lambda_{d,i}}}}}}}\big{)}.\end{array}$
(20)

Theorem 3. $E{E_{d,i}}$ is concave on the interval $({t_{1}},{t_{2}})$ and
convex on the interval $(1,{t_{1}})\cup({t_{2}},+\infty)$, where
${t_{1,2}}\buildrel\Delta\over{=}\exp(\frac{{3\alpha\mp\sqrt{{\alpha^{2}}+16\alpha}}}{8})$.

###### Proof.

If the second derivative is non-positive or non-negative, the function is
concave or convex [25]. We have

$\begin{array}[]{l}\frac{{{\partial^{2}}E{E_{d,i}}}}{{\partial
x_{i}^{2}}}={A_{i}}\frac{{2{{\left({\ln{x_{i}}}\right)}^{\frac{\alpha}{2}}}-\frac{{3\alpha}}{2}{{\left({\ln{x_{i}}}\right)}^{\frac{\alpha}{2}-1}}+\left({\frac{\alpha}{2}-1}\right)\frac{\alpha}{2}{{\left({\ln{x_{i}}}\right)}^{\frac{\alpha}{2}-2}}}}{{{P_{c,i}}x_{i}^{3}}},\end{array}$
(21)

where
${A_{i}}\buildrel\Delta\over{=}\frac{{{W_{i}}\log_{2}(1+{T_{d,i}})\exp\left({-{\varsigma_{d,i}}{\lambda_{d,i}}}\right)}}{{{{\left({{\varsigma_{d,i}}{\lambda_{c,i}}}\right)}^{\frac{\alpha}{2}}}}}$.
$t_{1}$ and $t_{2}$ are the solution of
$\frac{{{\partial^{2}}E{E_{d,i}}}}{{\partial x_{i}^{2}}}=0$. The second
derivative is positive on the interval $(1,{t_{1}})\cup({t_{2}},+\infty)$ and
negative on the interval $({t_{1}},{t_{2}})$. Hence $E{E_{d,i}}$ is convex in
the former and concave in the latter interval. This completes the proof. ∎

The objective function is the summation of concave (convex) functions in
defined sets, and the problem is solved in the intersection of these feasible
sets. Therefore, the domain of the objective function is the intersection
between the domain of each ${EE_{d,i}}$. Also the constraints are convex.
Since the second derivative of constraints (1) and (2) are always positive,
and the constraints (3) and (4) are linear functions of ${x_{i}}$.
Consequently, the optimization problem is convex [25]. Therefore, to find the
optimal points, the problem is solved in the concave interval.

###  IV-B Cellular Users Energy Efficiency

Similar to the approach explained for D2D scenario, by substituting $x_{i}$ in
(16), the optimization problem defined in (18) can be rewritten as,

$\begin{array}[]{l}\mathop{\max}\limits_{{P_{c,i}}}\,\,E{E_{c}}=\sum\limits_{i=1}^{M}{\frac{{{W_{i}}\log_{2}(1+{T_{c,i}})\exp\left({-{\varsigma_{c,i}}{\lambda_{c,i}}}\right)}}{{{P_{c,i}}\exp\left({\frac{{{\varsigma_{d,i}}{\varsigma_{c,i}}{\lambda_{d,i}}{\lambda_{c,i}}}}{{\ln{x_{i}}}}}\right)}}}\\\
s\cdot t\cdot\,\,\,\,\,\,\,\,\,\,(1)\,\sum\limits_{i}{{P_{c,i}}}\leq{P_{c}}\\\
\begin{array}[]{*{20}{c}}{}\hfil&{}\hfil\end{array}\,\,\,\,\,\,\,\,\,\,\,\,\,(2)\,0\leq{P_{c,i}}\leq{P_{c,i,\max}}\\\
\begin{array}[]{*{20}{c}}{}\hfil&{}\hfil\end{array}\,\,\,\,\,\,\,\,\,\,\,\,\,(3)\,\sum\limits_{i}{{P_{c,i}}{{\left({\frac{{{\varsigma_{d,i}}{\lambda_{c,i}}}}{{\ln{x_{i}}}}}\right)}^{\frac{\alpha}{2}}}}\leq{P_{d}}\\\
\begin{array}[]{*{20}{c}}{}\hfil&{}\hfil\end{array}\,\,\,\,\,\,\,\,\,\,\,\,\,(4)\,{P_{c,i}}{\left({\frac{{{\varsigma_{d,i}}{\lambda_{c,i}}}}{{\ln{x_{i}}}}}\right)^{\frac{\alpha}{2}}}\leq{P_{d,i,\max}},\end{array}$
(22)

It is worth mentioning that after simplification, the QoS constraints in (18)
will be in terms of $x_{i}$ and hence the last two constraints in (18) can be
ignored at this problem.

The second derivative of $E{E_{c,i}}$ is given by,

$\begin{array}[]{l}\frac{{{\partial^{2}}E{E_{c,i}}}}{{\partial
P_{c,i}^{2}}}=\frac{{2{W_{i}}\log_{2}\big{(}1+{T_{c,i}}\big{)}\exp\big{(}{-{\varsigma_{c,i}}{\lambda_{c,i}}}\big{)}\exp\big{(}{-\frac{{{\varsigma_{d,i}}{\varsigma_{c,i}}{\lambda_{d,i}}{\lambda_{c,i}}}}{{\ln{x_{i}}}}}\big{)}}}{{P_{c,i}^{3}}}.\end{array}$
(23)

It can be seen that $\frac{{{\partial^{2}}E{E_{c,i}}}}{{\partial
P_{c,i}^{2}}}$ is always positive. Thus, $E{E_{c,i}}$ is convex everywhere.
Hence, $E{E_{c}}$ which is the summation of convex functions is convex as
well. Moreover, the constraints are linear functions of $P_{c,i}$ and thus the
optimization problem is convex.

###  IV-C Iterative Power Allocation

In previous subsections, we transformed the primary optimization problem into
a convex form. However, these problems are not independent and should be
solved jointly. This dependency is due to the impact of cellular users’
transmission power on the entire network energy efficiency. To jointly solve
the problem, first, the problem of D2D users’ energy efficiency, with fixed
transmission power of cellular users is solved. Afterward, using the solution
of the D2D optimization problem, we optimize the transmission power of
cellular users to maximize the network’s energy efficiency. Since these
problems must be solved consequently, we feed the result of cellular users’
power to the D2D optimization problem in the next iteration, and we continue
this procedure until the convergence criteria are met.

Algorithm 1 Iterative power allocation algorithm

1:Initialization:

2:$P_{d}$$\leftarrow 0$, $P_{c}$$\leftarrow 0$,
$\varepsilon_{d}$$\leftarrow{10^{-5}}$,
$\varepsilon_{c}$$\leftarrow{10^{-5}}$, $n\leftarrow 0$, $N_{max}\leftarrow
10$

3:while TRUE do

4:Phase I

5: Solve (20) for given $P_{c}$ and obtain $P_{d,cur}$

6: $\Delta_{d}$$=$$P_{d,cur}$$-$$P_{d}$ and $P_{d}$$=$$P_{d,cur}$

7:Phase II

8: Solve (22) for given $P_{d}$ and obtain $P_{c,cur}$

9: $\Delta_{c}$$=$$P_{c,cur}$$-$$P_{c}$ and $P_{c}$$=$$P_{c,cur}$

10: $n\leftarrow n+1$

11: if $n==N_{max}$ then

12: Break

13: end if

14: if $\Delta_{d}$ $\leq$ $\varepsilon_{d}$ and $\Delta_{c}$ $\leq$
$\varepsilon_{c}$ then

15: Break

16: end if

17:end while

The proposed algorithm is reviewed in Algorithm 1. The algorithm can be split
into two phases. The first phase is D2D power allocation and the second one is
cellular power allocation. In the first phase, we initialize the fixed values
of cellular users’ parameters, e.g., transmission power, in $i$-th band and
(20) is solved to find optimal power of D2D users in $i$-th band. Note that
$x_{i}$ is convertible to $P_{d,i}$ in each iteration. In the second phase,
the optimal power of cellular users is achieved by utilizing the D2D users’
power in the previous phase. In each iteration, ${\Delta_{c,i}}$ and
${\Delta_{d,i}}$ are computed, which are defined as
${\Delta_{c,i}}\buildrel\Delta\over{=}{{\rm{P}}_{{\rm{c,i,cur}}}}{\rm{-}}{{\rm{P}}_{{\rm{c,i}}}}$
and
${\Delta_{d,i}}\buildrel\Delta\over{=}{{\rm{P}}_{{\rm{d,i,cur}}}}{\rm{-}}{{\rm{P}}_{{\rm{d,i}}}}$.
${P_{c,i,cur}}$, ${P_{d,i,cur}}$ and ${P_{c,i}}$, ${P_{d,i}}$ are the optimum
power of cellular and D2D users in $i$-th band in the current iteration and
previous iteration, respectively. The iteration will continue until
${\Delta_{c,i}}\leq{\varepsilon_{c,i}}$ and
${\Delta_{d,i}}\leq{\varepsilon_{d,i}}$ or the maximum iteration number is
reached which is denoted by $N_{max}$. ${\varepsilon_{d,i}}$ and
${\varepsilon_{c,i}}$ are the maximum tolerance of D2D users’ power and
cellular users’ power in $i$-th band, respectively. According to [1], the
iterative optimization algorithm converges to the maximum EE if problems (20)
and (22) are feasible. For obtaining optimum power in each iteration, the
interior point method which is a numerical solver is used. For this purpose,
we use OPTI toolbox which is a free MATLAB toolbox for optimization [26].

###  IV-D Complexity Analysis

In this subsection, we investigate the computational complexity of the
proposed power allocation algorithm. In this algorithm, after initialization,
we have to solve the optimization problem (20). In this problem, in total we
have $M$ number of decision variables and $3M+1$ linear and convex constraints
where $M$ is total number of bands. Hence, the computational complexity of
solving this problem is $O\big{(}M^{3}(3M+1)\big{)}$ [27]. Similar to this
approach, solving (22) has the computational complexity of
$O\big{(}M^{3}(2M+1)\big{)}$. Other operations in the loop of power allocation
algorithm have the computational complexity of $O\big{(}1\big{)}$ and can be
ignored compared to complexity order of solving optimization problems.
Therefore, assuming that,$N_{max}$ is the maximum number of loop iterations,
the computational complexity of proposed algorithm is given by

$\displaystyle O\big{(}[M^{3}(3M+1)+M^{3}(2M+1)]\big{)}\approx
O\big{(}N_{max}M^{4}\big{)}$ (24)

##  V Numerical Results

In this section, the energy efficiency of the D2D and cellular users, namely
$EE_{c}$ and $EE_{d}$ is evaluated. According to Fig. 1, we consider an uplink
system where both cellular users and devices share the same spectrum with $M$
different bands. The simulation parameters are shown in Table 1, and since we
want to compare our results with a benchmark, we kept the simulation
parameters as is set in [19].

Fig. 2 represents the convergence behavior of Algorithm 1, for the
transmission power of cellular and D2D users on one band with
${\lambda_{d,ref}}=\,{10^{-4}}$ and ${\lambda_{c,ref}}=\,{15\times 10^{-6}}$.
Fig. 2 demonstrates that Algorithm 1 converges to the optimum value within a
few iterations. According to (11), the capacity is averaged over the
distribution of channel so the effect of channel randomness is eliminated and
due to less variation of channel, very few iterations are needed for our
algorithm to be converged.

Table I: Simulation parameter values Parameter | Value
---|---
$M$ | 5
$W_{i}$ | 20 MHz
$\alpha$ | 4
${\theta_{c,i}}$ | 0.05
${\theta_{d,i}}$ | 0.05
$\left[{{R_{d,00,1}},{R_{d,00,2}},...,{R_{d,00,5}}}\right]$ | $\left[{10,\,20,\,30,\,20,\,10}\right]\,m$
$\left[{{R_{c,00,1}},{R_{c,00,2}},...,{R_{c,00,5}}}\right]$ | $\left[{50,\,60,\,70,\,80,\,90}\right]\,m$
$\left[{{\lambda_{d,1}},{\lambda_{d,2}},...,{\lambda_{d,5}}}\right]$ | $\left[{10,\,1,\,10,\,10,\,10}\right]\,\times{\lambda_{d,ref}}$
$\left[{{\lambda_{c,1}},{\lambda_{c,2}},...,{\lambda_{c,5}}}\right]$ | $\left[{10,\,1,\,10,\,10,\,10}\right]\,\times{\lambda_{c,ref}}$
${P_{d,i,\max}}$ | 20 mW
${P_{c,i,max}}$ | 300 mW
${P_{c}}$ | 1 W
${\varepsilon_{d,i}}$ | ${10^{-5}}$
${\varepsilon_{c,i}}$ | ${10^{-5}}$
Figure 2: Power of users against number of iterations.

In Fig. 3 and 4, we investigated the EE of D2D users with regards to the
density of D2D and cellular users. In these figures, we have solved the
optimization problem in (20), and the optimal power of D2D users in each band
is derived by OPTI toolbox. As a benchmark, we compared our results with [19],
where the problem of D2D energy efficiency in (17) is solved with a
derivative-based algorithm. It is worth mentioning that in their approach they
did not consider cellular users’ power and hence to compare our results;
first, we fix the transmission power of cellular users. Fig. 3 and 4 reveals
that in terms of EE, Algorithm 1 with fixed cellular users power transmission,
outperforms the algorithm in [19]. Later, we improve the EE even more by
enhancing cellular users’ transmission power.

Figure 3: Energy efficiency of D2D users versus ${\lambda_{d,ref}}$.  Figure
4: Energy efficiency of D2D users versus ${\lambda_{c,ref}}$.

In Fig. 3, total power of devices in all bands, $P_{d}$, is set to $60$ mW and
${\lambda_{c,ref}}$ is ${10^{-5}}$. It can be seen that by increasing
${\lambda_{d,ref}}$, EE increases at first and then declines. The STP
increases as the density of D2D users become sparser; thus ASR of D2D users
increases according to (11), and EE increases too. In higher densities, the
interference which is produced by spectrum sharing is more than ASR of D2D
users. So, the EE decreases. Also, by increasing the cellular users’
transmission power, the produced interference by cellular users increases and
EE decreases.

In Fig. 4, the simulation parameters are set as:
$\left[{{R_{d,00,1}},\,{R_{d,00,2}},\,{R_{d,00,3}},\,{R_{d,00,4}},\,{R_{d,00,5}}}\right]\,=\,{R_{d,ref}}\times\left[{1,2,3,2,1}\right]$,
$P_{d}=\,80\,mW$, ${\lambda_{d,ref}}=\,{10^{-4}}$ and $P_{c,i}=\,300\,mW$. As
can be seen in Fig. 4, when ${\lambda_{c,ref}}$ increases, the interference of
cellular users on devices increases and devices should use more power to
coordinate the interference. So, EE decreases. Also, by increasing the
distance between D2D users, EE decreases because the channel fading becomes
greater.

The cellular users’ transmission power has an impact on the EE of devices.
This impact is investigated in the following results. The comparison benchmark
is the case when (20) is solved numerically and the cellular users’
transmission power is fixed as $325\,mW$. In Fig. 5, we have depicted $EE_{d}$
versus density of devices. Fig. 5 shows that optimizing the transmission power
of cellular users and devices jointly can significantly improve the $EE_{d}$
since the destructive impact of cellular users on the D2D users is now
mitigated.

In Fig. 6, we investigated the impact of cellular users density on $EE_{d}$.
As the density of cellular users increases, $EE_{d}$ decreases since cellular
users create more interference on D2D users. To keep the link quality, D2D
users should consume more power and as a result, $EE_{d}$ decreases. Algorithm
1 still outperforms the reference scenario; however, at very dense scenario
the gap is low since there is no room for enhancing the $EE_{d}$. Moreover, we
plotted the $EE_{d}$ for different $R_{d,ref}$, which is defined as, the
distance between a pair of D2D connection. It can be seen that higher
$R_{d,ref}$ results in lower $EE_{d}$ since this further distance means that
more power is required to meet QoS requirements. Finally, as can be seen in
Fig. 5 and Fig. 6, $EE_{d}$ is higher than $EE_{d}$ in Fig. 3 and Fig. 4 and
this highlights the impact of optimizing the cellular users’ transmission
power.

Figure 5: Energy efficiency of D2D users versus ${\lambda_{d,ref}}$ by
obtaining power of cellular and D2D users, iteratively.  Figure 6: Energy
efficiency of D2D users versus ${\lambda_{c,ref}}$ by obtaining power of
cellular users and devices, iteratively.  Figure 7: Energy efficiency of
cellular users versus ${\lambda_{c,ref}}$ by obtaining power of cellular users
and devices, iteratively.

In Fig. 7, EE of cellular users, $EE_{c}$, versus ${\lambda_{c,ref}}$ is
illustrated. We compare the $EE_{c}$ of Algorithm 1 with the $EE_{c}$ of [19].
To calculate the $EE_{c}$ in [19], we first solved (20) to achieve the D2D
users’ transmission power. Then, assuming that the cellular users’
transmission power is constant, $P_{c,i}=325$ $mW$, we substituted the power
of devices in (16), and $EE_{c}$ can be calculated accordingly. By increasing
${\lambda_{c,ref}}$, the interference created by cellular users increases and
hence $EE_{c}$ decreases. As can be seen in Fig. 7, we can achieve higher
energy efficiency by Algorithm 1 because the cellular users are operating with
their optimal transmission power.

##  VI Conclusions

In this study, the EE maximization of the D2D communication underlaying
cellular networks is investigated. First, we derived the closed-form
expressions for the successful transmission probability (STP) and average sum
rate (ASR) of cellular and D2D users. Then, we have formulated an optimization
problem that maximizes the energy efficiency of both cellular and D2D users as
MINLP problem. In the optimization problem, we considered STP as QoS of
cellular and D2D users. To solve the optimization problem, first, we converted
the primary optimization problem into a canonical convex form. Then, we break
down the problem into two sub-problems. The first sub-problem was devoted to
solving the problem of D2D users energy efficiency. Utilizing the result of
the first sub-problem, we maximized the energy efficiency of cellular users in
the second sub-problem. Our simulation results showed that our scheme
outperforms the existing solutions in the literature. Moreover, significant
enhancement in terms of energy efficiency is achieved considering optimizing
D2D and cellular users transmission power jointly. As a future work, band
selection can be considered along with power allocation to minimize the cross-
tier interference between D2D users and cellular users. Moreover, the optimal
density of D2D users and cellular in each band and the corresponding trade-off
analysis can be investigated.

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