Joint Resource Block and Power Allocation for Interference Management in Device to Device Underlay Cellular Networks: A Game Theoretic Approach

Abstract

This paper addresses the problem of joint resource block (RB) and uplink transmission power allocation in a Device to Device (D2D) underlay cellular network via a game theoretic approach. In contrast to the majority of previous research works in this area, the proposed framework aims mainly at interference mitigation and energy efficiency. We do not restrict a priori the number of D2D pairs which can reuse a RB allocated to a cellular link. However, this is determined dynamically by the objective of interference minimization. To address the resource allocation problem under consideration and deal with its inherent complexity, a two-step distributed approach is proposed. At first the RB allocation process is modeled as an exact potential game which is shown to minimize the total interference in the network. Second a non-cooperative game theoretic model for the uplink transmission power allocation process is proposed. For both stages of the proposed framework, efficient and distributed algorithms for the computation of the desired Nash Equilibrium (NE) point of each game are introduced. The efficiency of the overall proposed framework is evaluated through modeling and simulation, while comparative numerical results are presented that demonstrate the superiority of our methodology against other state of the art approaches.

Full text

Joint Resource Block and Power Allocation for Interference
Management in Device to Device Underlay Cellular
Networks: A Game Theoretic Approach
Georgios Katsinis
1
&Eirini Eleni Tsiropoulou
2
&Symeon Papavassiliou
1
#Springer Science+Business Media New York 2016
Abstract This paper addresses the problem of joint resource
block (RB) and uplink transmission power allocation in a
Device to Device (D2D) underlay cellular network via a game
theoretic approach. In contrast to the majority of previous
research works in this area, the proposed framework aims
mainly at interference mitigation and energy efficiency. We
do not restrict a priori the number of D2D pairs which can
reuse a RB allocated to a cellular link. However, this is deter-
mined dynamically by the objective of interference minimiza-
tion. To address the resource allocation problem under con-
sideration and deal with its inherent complexity, a two-step
distributed approach is proposed. At first the RB allocation
process is modeled as an exact potential game which is shown
to minimize the total interference in the network. Second a
non-cooperative game theoretic model for the uplink trans-
mission power allocation process is proposed. For both stages
of the proposed framework, efficient and distributed algo-
rithms for the computation of the desired Nash Equilibrium
(NE) point of each game are introduced. The efficiency of the
overall proposed framework is evaluated through modeling
and simulation, while comparative numerical results are
presented that demonstrate the superiority of our methodology
against other state of the art approaches.
Keywords 5G networks .Game theory .Interference
management .D2D underlay cellular networks
1 Introduction
The dramatic growth of mobile data services driven by wire-
less Internet and smart devices has triggered the investigation
of fifth generation (5G) for the next generation mobile tele-
communications [1–4]. The realization of the forthcoming of
5G mobile networks demands that traditional performance
indicators, such as network capacity and spectral efficiency
need to be continuously improved. Furthermore, a wider va-
riety of communication modes and applications need to be
provided to enhance user experience. Device to Device
(D2D) communications has been considered as one of the
key characteristics of 5G mobile networks [4,5]. It emerged
as an add-on communication paradigm to the modern wireless
cellular networks [6] drawing significant attention in the in-
dustry for its potential to improve system performance, en-
hance user experience and expand cellular applications [7].
D2D communications is a direct type of communication be-
tween two mobile terminals of a wireless cellular network.
The benefits of D2D communications mainly stem from the
proximity of the connected devices, the possibility of efficient
radio resources reuse and the single hop nature of the D2D
links [8,9].
D2D communications can take place either overlaying or
underlaying a cellular network. In the overlay case, D2D com-
munications use dedicated resources (e.g. resource blocks
(RBs) that represent combinations of frequency subcarriers
and symbols in the time domain), while in the underlay case
*Georgios Katsinis
gkatsinis@netmode.ntua.gr
Eirini Eleni Tsiropoulou
eetsirop@utdallas.edu
Symeon Papavassiliou
papavass@mail.ntua.gr
1
School of Electrical & Computer Engineering, National Technical
University of Athens (NTUA), Heroon Polytechniou 9,
15780 Zografou, Greece
2
Erik Jonsson School of Engineering and Computer Science,
University of Texas at Dallas, Richardson, TX 75080, USA
Mobile Netw Appl
DOI 10.1007/s11036-016-0764-y

D2D communications share common resources (e.g. RBs)
with the residual cellular network [7].
Efficient resource management, power control and interfer-
ence coordination are of extremely high importance in cellular
networks. They optimize the usage of network resources and it
is expected to play a key role in the advancement of 5G net-
works [10]. In this paper we address the radio resource man-
agement problem within a D2D underlay cellular network
targeting mainly at interference mitigation, power and spec-
trum efficiency.
Throughout the rest of the paper we characterize as D2D
links the pairs of D2D transmitters and receivers that commu-
nicate directly to each other. Likewise, we characterize as
cellular links the traditional links of a cellular network be-
tween a cellular transmitter and the serving Base Station (BS).
1.1 Related work
In a D2D underlay cellular network every user of the network
has the flexibility of deciding its optimal mode of communi-
cation. This can be either cellular or D2D mode of communi-
cation, establishing a cellular or a D2D link respectively [9].
This can be determined based on the received signal strength,
the interference situation or the distance between transmitter
and receiver [11].
The problem of efficient resource blocks (RBs) assignment
and transmission power allocation in D2D underlay cellular
networks has been recently studied in the literature [8,12,13].
In most of these problems it is assumed that the problem of
mode selection (cellular or D2D) has already been addressed,
i.e. at every time slot a certain number of D2D and cellular
links have already been determined. Within this framework
the problem of optimal RB and power allocation to the
existing links arises.
The approaches for solving the latter problem can be char-
acterized as either centralized or distributed. Centralized ap-
proaches as [12–16] typically formulate a mixed integer pro-
gramming problem [17] where the sum of the defined utility
functions of every link in the cell is optimized. In the majority
of the above cases the utility functions represent every link’s
achieved transmission rate. In general, the utility functions of
each link are mathematical functions [18] that express every
link’s desire to use certain resources i.e. RBs and transmission
power. The decision variables of the above formulated prob-
lems are the allocated RBs and the transmitted power values.
In [12] the authors consider the problem of searching for
feasible RB and power allocations, satisfying both the cellular
and the D2D links’Quality of Service (QoS) requirements.
They identify the non-convexity of the problem and propose
an alternative method in order to find sub-optimal solutions. In
[13] the authors analyze the optimal sum rate maximization in
the simple scenario of only one cellular and one D2D link. In
[14] the authors analyze the problem of throughput
maximization for the D2D links while protecting the QoS of
the cellular links. The problem is solved analytically and the
authors propose two suboptimal algorithms for a more com-
putationally efficient approach. In [15] the authors formulate
the above problem in both the downlink and uplink cases and
propose efficient greedy algorithms to address it. Specifically,
they allocate the RB of the cellular link with the highest chan-
nel gain to the D2D link with lowest channel gain between the
corresponding cellular transmitter and D2D link’s receiver.
This approach is computationally efficient but not optimal,
and restricts the number of D2D links capable of sharing a
RB to one. In [16] the authors propose a 3-step approach,
composed by an admission control stage, a power control
stage and an optimal resource allocation stage. In all of the
above stagesit is assumed that a RB can be shared by onlyone
cellular and D2D link.
The centralized approaches assume that the BS is capable
of having access at each time slot to the full channel state
information (CSI). This is not always feasible and/or practical
and most of the time requires significant signaling overhead
between the connected devices (cellular users, D2D users and
BS). The complexity of these problems is NP-hard and thus
the difficulty to solve them exactly raises intolerably in sce-
narios with a large number of cellular and D2D links.
Consequently, alternative approaches have gained merit.
Those approaches combine the advantages of distributed op-
eration with minimum necessary information exchange be-
tween the connected devices while protecting the links of
the network [19–22].
Towards this direction, related research work has been per-
formed in [23–25] using concepts from auction theory and
non-cooperative games. Specifically, in [23] the authors pro-
pose an auction-based theoretic mechanism for the downlink
resource (only RB) allocation case in order to maximize the
system sum rate. The complexity of this problem is known to
be NP-hard. Although the authors propose a reverse iterative
approach to reduce the complexity of the problem, it still re-
mains exponential with respect to the number of the RBs. In
[24] the authors propose a two-step (i.e. a combined auction
based and game theoretic) approach for the joint optimization
of RB and power resources. Since the complexity of solving
the problem exactly is NP-hard they propose an iterations-
restricted algorithm in order to provide approximate solutions.
They focus on the energy efficiency (i.e. rate to power ratio) of
the proposed solution, without being able to directly take into
account the signal-to-interference-plus-noise ratio (SINR) re-
quirements of cellular and D2D links. Instead a maximum
communication distance between D2D transmitter and
receiver is assumed. Finally, in [25] the authors pro-
pose a Stackelberg game model for the resource allo-
cation problem in the joint space of spectrum and
transmission power providing SINR protection only
for the cellular links.
Mobile Netw Appl

1.2 Paper contribution & outline
In this paper we consider the uplink of a D2D underlay cellu-
lar network using a SC-FDMA scheme according to 3GPP
LTE Advanced specifications [26]. We investigate the prob-
lem of optimal joint resource block and power allocation to-
wards interference mitigation. We assume each user’sequip-
ment (UE) device has already chosen its mode of communi-
cation, that is cellular or D2D.
A two-step approach is proposed in order to deal with the
complexity of the joint problem of RB and power allocation in
a D2D underlay cellular network. At the first stage an inter-
ference minimization approach for all the links in the network
is introduced. The problem of RB allocation is formulated as a
non-cooperative game [27] which is proven to be an exact
potential game [28]. In this case one or more NE points may
exist and coincide with the points that maximize the potential
function. Furthermore, distributed better or best response al-
gorithms exist which converge to one of the NE points of the
game [28].
The problem of computing one of the desirable NE points
of the game belongs to the Polynomial Local Search (PLS)
complexity class [29]. This means that best response algo-
rithms which compute one of the NE points may require in
some cases exponential number of iterations with respect to
the input of the problem. However polynomial algorithms
exist which compute ε-approximate Nash equilibria of the
corresponding game [30]. Nevertheless, we demonstrate in
section 7 (Fig. 3) that the required number of iterations re-
quired by the proposed best response algorithm in order to
converge to the desired NE point is quite low.
At the second stage we propose a game theoretic approach
for the power control problem taking into account every UE’s
SINR requirements. The best response function of the game is
standard [31] and thus there is a distributed best response
algorithm which converges to the unique fixed point of the
game after a finite number of iterations. In practice, as it is
shown in section 7 (Fig. 5) a relatively low number of itera-
tions is required to converge to the desired NE point. The
proposed NE best response algorithm is completely distribut-
ed and autonomous, i.e. its computation is performed in par-
allel by every UE of the network having access only to local
information (i.e. the information accessible to every UE of the
cell.) The solution is given in the joint space of RBs and
transmission power values since we search sequentially and
iteratively for a NE point for both games.
Our work differentiates from the previous ones in the fol-
lowing points:
&The first stage of our approach, i.e. the RB allocation
process, is modeled as an exact potential game. The opti-
mization goal is the minimization of the total interference
in the network in contrast to previous works where the
authors focus on the throughput maximization of the net-
work. Consequently, our approach is directly targeting
interference mitigation and combined with appropriate
power control results in energy efficient solutions, achiev-
ing high transmission rates under low transmission power.
&We do not restrict a priori the number of D2D pairs that
can reuse a RB allocated to a cellular link. This is deter-
mined dynamically by the objective of interference mini-
mization. Thus a RB reserved by a cellular link can be
potentially reused multiple times by different D2D links
in order to increase spectrum efficiency.
&The second stage of our approach is the SINR aware up-
link transmission power allocation process which is
modeled as a non-cooperative game. In this case, every
link has a utility function which can be computed taking
into account only local information. The utility functions
of each link in the network depend only on the transmis-
sion power of each link and on the sensed interference at
every RB that a link uses. This fact promotes a realistic,
distributed and autonomous implementation of the NE
computation algorithms.
&Capitalizing on respective properties stemming from the
nature of the games considered (i.e. exact potential game
and use of the standard function property) we propose best
response algorithms for both games which are shown to
satisfactorily converge to the desired NE point of each
game after a finite number of iterations. It is noted that
each step of the best response algorithm of the first stage
improves not only every link’s utility but also the total
network utility of all the links in the system. This is done
leveraging the desirable and socio-aware properties of the
exact potential games [28]. For the second stage of our
approach the used best response algorithm converges to
the unique fixed point of the game after a relatively low
number of iterations. This is due to the standard function
property [31] of the used best response function. These
arguments are also verified numerically in section 7.
&Finally, the performance of the proposed framework is
compared against other related state of the art approaches
(i.e. [15,24] and its superiority in terms of increase in the
achieved data rate and decrease in the corresponding con-
sumed power is demonstrated.
The outline of the remaining of the paper is as follows. In
section 2 we present the system model, while in section 3 the
formulation of the interference minimization problem in the
joint space of RBs and uplink transmission power values is
presented. In sections 4 and 5 the first stage of our approach,
i.e. the RB allocation is studied and analyzed. In section 6 the
power allocation process for every available RB is described
through a game theoretic formulation. In section 7 the perfor-
mance of our approach is evaluated through modeling and
simulation, while it is compared against the corresponding
Mobile Netw Appl

performances of other related state of the art approaches.
Finally, section 8 concludes the paper.
2 System model
We consider the uplink of a D2D underlay cellular network,
adopting SC-FDMA scheme. We assume a single cell envi-
ronment with multiple UEs and one evolved NodeB (eNB)
located at the center of the cell (Fig. 1). The UEs and the eNB
are equipped with a single omni-directional antenna. The sys-
tem contains two types of UEs, cellular UEs (CUEs) and D2D
UEs where we assume that their communication modes have
already been selected. The CUEs communicate through the
eNB.
The D2D UEs communicate directly to each other in pairs
where the two D2D UEs of every D2D pair are adequately
close to each other, thus forming a D2D link.
We assume that the eNB has a certain number of orthogo-
nal RBs (i.e. KRBs) to be shared. Every CUE occupies a
different RB, therefore there is no interference among cellular
links given the orthogonality of the RBs. Additionally we
assume that the D2D links share the available RBs with at
least one cellular link for reasons of spectrum efficiency. As
a result, we consider that Lcellular links and DD2D links
exist in this single cell. For reasons of simplicity in the fol-
lowing analysis we assume that every cellular link occupies
one RB from the available bandwidth Band all the available
RBs are allocated to the cellular links. The analysis however
can be extended to a multiple RB allocation per UE. It should
be noted that due to the RB reuse (among one cellular link and
multiple D2D links it could be that L+D>K). Finally, it is
clarified that it is possible that multiple D2D pairs may reuse
the same RB, if interference conditions permit.
We define the SINR at the receiver of link i-eithercellular
or D2D - for RB cassigned to this link, as follows:
γc
i¼Gc
ii⋅Pc
i
X
j≠i
δj;cGc
ji⋅Pc
jþσ2ð1Þ
where idenotes the transmission link, c∈Cis the RB assigned
to this link, Cis the set of available RBs, G
ii
c
denotes the
channel gain of the itransmission link at RB c,G
ji
c
is the
channel gain between the transmitter of jlink and the receiver
of ilink at RB c,P
i
c
is the transmission power of link iat RB c,
σ
2
is the thermal background noise power at the receiver of
each link and δ
i,c
is the indicator variable of the usage of the c
RB from the ilink. This means that δ
i,c
= 1 if link ioccupies
RB cotherwise δ
i,c
is zero.
We also assume that every link has a certain minimum
SINR requirement representing the QoS prerequisites of the
corresponding user. This SINR requirement should be explic-
itly taken into account in order to protect both the cellular and
the D2D links from intolerable created interference from other
co-channel links.
The achievable data rate for every link ican be expressed
by the Shannon formula as follows:
Rc
i¼W⋅log 1 þγc
i
 ð2Þ
Above it is assumed that link ioccupies RB cand the
bandwidth of each RB is W.Incasethatlinkiis assigned more
RBs, the total achievable rate of link iwill be given as the sum
of the individual rates from every selected RB i.e.:
Ri¼X
c∈C
δi;c⋅Rc
i¼X
c∈C
δi;c⋅W⋅log 1 þγc
i
 ð3Þ
3 Problem formulation
In this section we formulate the problem of optimal joint RB
and power allocation in a D2D underlay cellular network in a
single cell. The objective of this problem is to minimize the
total interference of the links in the network. The cellular links
occupy different RBs while the D2D links share RBs with the
cellular links and among each other. Thus interference is
added to every link occupying a RB which is shared among
more than one link (i.e. one cellular and potentially multiple
Fig. 1 System model of uplink D2D underlay cellular network
Mobile Netw Appl

D2D links). Then the problem can be formulated as a mixed
integer programming problem as follows
minPX
C
c¼1X
LþD
i¼1
δi;c⋅Ic
ið4Þ
Subject to
0≤pc
i≤pmax
i;∀c∈C;i∈L∪Dð4aÞ
γc
i≥γtar
i;∀i∈L∪D;∀c∈Cð4bÞ
where Ic
i¼∑
j≠i
δj;c⋅Gc
ji⋅Pc
jþσ2is the total received interfer-
ence at the receiver of link iat the occupied RB c.
The above problem aims at minimizing the total interfer-
ence for all links in the cell under the conditions stated in (4a)
and (4b). Specifically, (4a) refers to the feasible interval of the
power transmission values and (4b) concerns the targeted
(minimum) SINR requirement of link ifor every available
RB in the cell.
The above problem is a mixed integer programming prob-
lem. It is known that the complexity of this type of problems is
NP-hard [17]. For this reason, either approximation methods
for obtaining suboptimal solutions of the problem or exact
methods for certain instances of the problem can be used to
solve the above problem efficiently. Also this problem is cen-
tralized in nature which means that the eNB will be in charge
of solving it. The eNB should have knowledge of the full
channel state information (CSI) among all links within the
cell. Most of the times this requirement is not practically fea-
sible or at least its implementation is quite inefficient. In order
to tackle this issue in this paper we propose a two-step distrib-
uted approach, composed by a RB allocation stage and a
SINR aware power control stage.
4 RB allocation
Let us consider the previously introduced in section 2 single
cell architecture where we assume that in total ND2D and
cellular links coexist. Every link of the cell is considered as
a player of a non-cooperative game G. This game is denoted
by G={N,S,U}whereNis the set of the players i.e. the set of
the links inside the cell, Sis the set of the strategies of the
players where S=[S
1
,S
2
,…,S
|N|
] and each strategy set S
i
=C
is the action set of RBs for every player iof the game i.e.
C={c
1
,c
2
,…,c
K
}. U=[U
1
,U
2
,…,U
|N|
] denotes the set of
the utility functions of the players of the game and each utility
function is given by the following equation:
Uisi;s−i
ðÞ¼T1
isi;s−i
ðÞþT2
isi;s−i
ðÞ ð5Þ
where s
i
=c
i
∈C,s
‐i
∈S
−i
are the strategies sets of link iand
all the other links.
T1
isi;s−i
ðÞ¼−X
N
j≠i;j¼1
pjGcj
ji fc
j;ci
 ð6Þ
denotes the interference sensed by link iat RB c
i
and Gcj
ji is the
channel gain between the transmitter of the link jand the
receiver of link iat RB c
j
chosen by link jwhile f(c
j
,c
i
)is
defined as follows:
fc
j;ci

¼1if cj¼ci
0otherwise
 ð7Þ
Similarly:
T2
isi;s−i
ðÞ¼−X
N
j≠i;j¼1
piGci
ijfc
i;cj
 ð8Þ
denotes the interference created by link iat RB c
i
.Fromnow
on and throughout the whole paper we omit the upper index c
j
from Gcj
ji for the ease of notation without changing the mean-
ing of the symbol. We can prove the following.
Lemma 1: The above formulated game G={N,S,U}is
proven to be an exact potential game in the strategy space
of the RBs with the following exact potential function
Pot si;s−i
ðÞ¼
X
N
i¼1
aT1
isi;s−i
ðÞþ1−aðÞT2
isi;s−i
ðÞ

ð9Þ
where a∈(0, 1).
Proof: See Appendix I.
The main characteristic property of the exact potential
games is the following:
Pot si;s−i
ðÞ−Pot s0
i;s−i

¼Uisi;s−i
ðÞ−Uis0
i;s−i

∀i∈N;∀si;s0
i∈Si
ð10Þ
From the above property it follows that every link’sdiffer-
ence in utility due to strategy change can be mapped via a
common potential function Pot(s
i
,s
−i
) to a difference in the
common potential function for the same strategy change. This
is a highly important property showing the socially aware
strategy changes of each player of the potential game.
It should be noted that potential games are a class of
non-cooperative games with the following desirable
properties. Those are the existence of at least one NE
point [27], the finite improvement path property [28]
and the convergence of either a better or a best re-
sponse algorithm to one of the NE points of the game
Mobile Netw Appl

[28]. The NE point of a strategic game is a fixed sta-
tionary point of the game (i.e. set of the players’strat-
egies) where none player has the incentive to change
his strategy without a change of strategy from any other
player. In mathematical terms this means that a vector
of strategies s*=(s
i
*
,s
−i
*
)isaNEpointifU
i
(s
i
*
,s
−i
*
)≥
U
i
(s
i
,s
−i
*
)∀i∈N,∀s
i
∈S
i
Towards computing the desirable NE point of the game G,
in the next section a sequential best response algorithm is
presented which is proven to converge to one of the NE points
of the game Gafter a finite number of steps, given that it is an
exact potential game. It is noted that initially during the RB
allocation stage (i.e. first stage of our approach) we consider
that transmission power remains stable for all links of the
network at a random feasible level (e.g. at each link’smaxi-
mum transmission power level).
5 Nash equilibrium computation (RB allocation)
As shown before, the previously formulated game is an exact
finite potential game. However, the synchronous best re-
sponse algorithm is not guaranteed to converge [20]. We pro-
pose a sequential version of the best response algorithm which
is proven to converge to one of the NE points of the game G
[28]. In this case a coordination mechanism is needed so as to
preserve the order of every acting player.
We assume that every cellular link has already chosen a
different RB and every D2D link enters the game in a sequen-
tial fashion. The D2D links compete for every potential RB
under the constraints of their transmission power values. At
each step of the algorithm every link identifies the RB which
maximizes its utility function defined in (5). Given the strate-
gies of the other links, the utility function of every link de-
pends only on the selection of every potential RB as follows
where C
1
is the sensed interference from D2D link icaused by
the cellular co-channel link to link i,D
1
is the total interference
caused by the other co-channel D2D links, C
2
is the caused
interference from D2D link ito the co-channel cellular link and
D
2
is the caused interference to the other co-channel D2D links.
Based on the above expressions it is noted that the utility ac-
counts for both the interference that will be caused to link iby
other potential links (either cellular or D2D) sharing the same
RB, as well as the interference that link iwould cause to others if
thesameRBisused.
We focus on a distributed NE computation mechanism
using only local information, available at every UE. The first
term T
i
1
(s
i
,s
−i
) of the utility function is the sensed interference
at the selected RB from the D2D link i. The second term
T
i
2
(s
i
,s
−i
) is the caused interference from the D2D link ito
the other links who share the same RB. In general, this term
cannot be computed in a distributed way and requires heavy
coordination between the BS and the interfered/co-channel
links. In this paper in order to overcome this problem, we
propose an alternative practical way to obtain it.
We assume that every link knows its location (i.e. its coor-
dinates) and the BS can send via a broadcast message the
locations and the RBs assignments of only the cellular links.
From this information every link can calculate the sensed in-
terference from each cellular link (i.e. C
1
in the previous equa-
tion) and can distinguish the cellular from the D2D sensed
interference (i.e. C
1
from D
1
).
The caused interference can also be separated into two parts.
The first part C
2
refers to the interference caused to the co-
12
1,1,
11 2 2
(, ) (, ) (, )
(,) (, )
ii i i i i i i
NN
jji ji iij i j
jijjij
sensed Interference caused Interference
Us Tss T ss
pG f c c pG f c c
CD CD
-i
s
(11)
Mobile Netw Appl

channel cellular link, while the second one D
2
refers to the inter-
ference caused to other co-channel D2D links. The interference
caused to the D2D links can be quite well approximated from the
interference sensed from the D2D links since the distance of the
transmitter and the receiver of a D2D link is quite low (10–50 m)
[32]. For all practical purposes we can assume D
1
≃D
2
.
The interference C
2
caused to the cellular links (i.e. to the BS)
is common for every potential RB since we consider a single cell
topology where only one BS exists (i.e. one common receiver for
every cellular link). The interference caused to the BS can be
estimated knowing the distance between the UE and the BS (its
UE knows its location and the BS broadcasts its location). Based
on the previous arguments it is concluded that every link can
estimate in a distributed way not only the sensed but also the
caused interference. The latter is quite important since it enables a
practical distributed implementation of the NE computation.
Following the above discussion, Algorithm 1 for the RB
allocation process is presented. In this case every D2D link
enters the game and chooses the RB which optimizes his util-
ity function. This is repeated for all D2D links sequentially
until a NE point is achieved. Due to the exact potential game
formulation, it is ensured that the following algorithm will
converge to the NE. Moreover, every step of the convergence
procedure will improve not only every player’s utility function
but also the total network utility function.
6 SINR aware power allocation
In this section we propose a game theoretic model for the SINR
aware power allocation process within a D2D underlay cellular
network. We assume that the problem of RB allocation has al-
ready been solved, as dictated in the previous section, and thus
every link occupies a specific RB. The problem of uplink trans-
mission power allocation for every co-shared RB, while
attempting to protect both the cellular and the D2D links in terms
of SINR, can be modeled as a non- cooperative game.
Let this game be G
c
=[N
c
,A,U]whereN
c
is the number of
the players (i.e. the cellular and the D2D links) who have
chosen RB c,A¼A1;A2;…;ANc
½, where Α
i
= [0, p
i
max
]is
the set of the strategies of player i,andU¼
U1;U2;…;UNc
½denotes the set of the utility functions of
all players of the game G
c
where
Uipi;p−i
ðÞ¼−ciγi−γtar
i

2ð12Þ
Lemma 2: Every game G
c
=[N
c
,A,U] has at least one NE
point.
Proof
The above can be easily proven since G
c
is defined on a
convex strategy space (i.e. the closed set of uplink
transmission power values) and the utility function of every
player is a concave function of his strategy. According to the
theorem of Debreu, Glicksberg and Fan [27]gameG
c
has at
least one NE point.
In order to find the desirable NE point for every game G
c
,
we propose a best response algorithm (i.e. Algorithm 2) for
every game G
c
. Every playerof the game (i.e. either cellular or
D2D link) executes Algorithm 2 and thus will choose an up-
link transmission power value from the feasible set
Α
i
=[0,p
i
max
], which maximizes his utility function.
Corollary 1: The power transmission value which maxi-
mizes one player’s utility function is proven to be p*
i
¼min pmax
i;γtar
i
Ii
Gii
no
where Ii¼∑
j≠i
Gji⋅pjþσ2is the re-
ceived interference at the receiver of link i.
Proof
In order to find the maximum of the concave utility func-
tion defined in (12), we should search for the critical point of
it, i.e.
∂Uipi
ðÞ
∂pi
¼0⇒γtar
i−γi¼0⇒γi¼γtar
ið13Þ
Let p
i
req.
= arg max U
i
(p
i
,p
−i
)
If a feasible value of transmission power exists, such that
condition (13) holds true, then this value can be determined as
follows:
GiiPi
X
j≠i
Gji⋅pjþσ2¼γtar
i⇒Pi¼γtar
i⋅X
j≠i
Gji⋅pjþσ2
Gii
If p
i
max
≥P
i
then we can conclude that preq:
i¼argmaxUi
pi;p−i
ðÞ¼γtar
i
∑
j≠i
Gji⋅pjþσ2
Gii ¼γtar
i
Ii
Gii else p
i
req.
=argmax
U
i
(p
i
,p
−i
)=p
i
max
since for p
i
≤p
i
max
⇒γ
i
≤γ
i
tar
because the
first derivative ∂Uipi
ðÞ
∂pi>0 in this range of power values.
It is concluded that the utility function of player iis non-
decreasing in the closed set [0, p
i
max
], thus it is maximized at
the maximum value of his transmission power.
Therefore, in any case
preq:
i¼min pmax
i;γtar
i
Ii
Gii
 ð14Þ
Based on Corollary 1 and Algorithm 2 every player of the
game G
c
will choose either to transmit at a power level which
Mobile Netw Appl

is sufficient for his SINR requirement or to transmit at his
maximum power level. Following the analysis in [31]we
can prove that the above best response function (14)isstan-
dard. This implies that the NE of the above game is unique and
the best response algorithm converges to this unique fixed
point after a finite number of iterations. At this point we
should note for reasons of clarity that every link participates
in one game for the RB that it has already chosen.
7 Performance analysis
In this section we initially study the performance and opera-
tional characteristics of our proposed two-step distributed ap-
proach via modeling and simulation (subsection 7.1), while
afterwards its performance is compared against other state of
the art approaches in terms of achievable transmission data
rates and consumed transmission powers (subsection 7.2).
7.1 Performance evaluation of the proposed two-step
approach
We consider a single cell topology similar to the one shown in
Fig. 2. The BS is located at the center of a cell of radius 300 m.
Cellular links are placed randomly in two different concentric
circles around the BS (i.e. at 100 m and 200 m away from the
BS). The distances between D2D transmitter and receiver are
assumed randomly distributed in the range between 10 and
50 m. The numbers in Fig. 2indicatively represent every
link’s id, while different symbols are adopted for representing
cellular transmitters, D2D transmitters and D2D receivers.
Taking into account the large scale fading effects, the channel
gains between transmitter and receiver can be modeled as
gij ¼l
d4
ij
where dis the distance between the transmitter of link
iand the receiver of link jand lis a scaling constant. We assume
that each of the cellular links has been assigned a RB and the
D2D links reuse the RBs of the cellular links. Also each of the
D2D links is assumed to use one RB. Unless otherwise ex-
plicitly indicated, in the following we consider a scenario with
8 RBs, 8 cellular links each one assigned a different RB, and
24 D2D links that will be sharing all the available RBs. We
also assume that every cellular link’s maximum transmission
power is 2 Watts, while the corresponding maximum trans-
mission power for the D2D transmitter is 10
-1
Watts.
In order to reveal and understand the benefits and operation
characteristics of the proposed approach better in the following
we present gradually and compare the achieved performance
results under the following cases: a) the application of the first
stage algorithm only (i.e. RB allocation only), b) the two stage
framework that combines both RB and power allocation (in the
following we refer to as two stage single run), and c) an en-
hanced version of our two stage proposed framework where it
is applied in an iterative manner (in the following we refer to as
two stage iterative). The latter essentially extends the single run
operation of the proposed two stage approach to an iterative
implementation as follows: the transmission power vector, ob-
tained in the second stage of the single run case (i.e. SINR
aware power control) is used as an input to the RB allocation
algorithm 1 (i.e. first stage); then this process can be repeated
and executed iteratively until convergence with respect to both
RB and power allocation is obtained. It should be noted that for
the first case we assume fixed transmission power for all links
equal to the maximum power level. For the second case, we
assume that each link initially transmits with its maximum
power in order to perform the RB allocation, and then calcu-
lates its optimal transmission power based on the SINR aware
power control (i.e. Algorithm 2).
We initially evaluate the operation of the application of the
first stage only (RB allocation process) of our proposed frame-
work, aiming at interference mitigation. With reference to the
topology under consideration in Fig. 2(8 RBs, 8 cellular links
and 24 D2D links) we assume that every link of the network
has been assigned a RB as it is presented in Table 1.Following
the operation of the first stage of our framework (i.e.
Algorithm 1) the D2D links change their initial RB selection
in order to obtain the NE of the game of RB allocation aiming
at interference mitigation. The resulting RB allocation after
the execution of this stage is shown in Table 2.
-300 -200 -100 0100 200 300
-300
-200
-100
0
100
200
300
1
2
3
4
5
6
7
8
9
10
11
12
13
14 15
16
17
18
19
20
21
22 23
24
25
26
27
28
29
30 31
32
X - Coordinate
etanidrooC - Y
Cellular transm itter
D2D transmitter
D2D receiver
Fig. 2 Single cell topology with 8 cellular and 24 D2D links
Tabl e 1 Initial RB allocation of all links of the network
RBs 12345678
linkid12345678
169 101112131415
24 17 18 19 20 21 22 23
Mobile Netw Appl

We observe that the D2D links choose RBs occupied by
cellular links which are located quite far from them. This is
performed in order to reduce the interference caused by/to
cellular links to/from D2D links. Same reasoning applies with
respect to the reuse of the same RB among D2D links. This is
due to the key objective of our problem formulation in the RB
allocation, of minimizing the total network interference, as
demonstrated later in this section.
In Fig. 3the evolution of the total sum of the utility func-
tion of every link in the network with respect to the number of
iterations of the Algorithm 1 is shown. We observe that the
sum of the utilities in the network is always increasing during
the procedure of the NE convergence following the theoretical
analysis presented in section 4 and the finite improvement
path property of potential games [28]. We note that each iter-
ation refers to every step of the UE’s RB update process. We
observe that a quite low number of iterations is needed in
order to converge to the required NE point. The BS needs to
make a decision regarding the RB allocation at every time slot
(i.e. one time slot ≃0.5 msec), or for practical purposes for a
short window of time slots. The reduced number of iterations
and therefore convergence time is a highly desirable charac-
teristic of our algorithm.
In Fig. 4we show that the application of Algorithm 1
(green line) improves (compared to the initial RB allocation
–blue line) the achieved SINR for all D2D links. A minimal
SINR degradation for some of the cellular links is observed
(with reference to Fig. 4cellular link IDs are 1-8 while D2D
link IDs are 9–32). Those occur due to the significant decrease
in the total sensed interference of all links (as observed in
Fig. 6below).
After the execution of Algorithm 1 and taking as an input
the RB allocation presented in Table 2we apply the SINR
aware power control approach (i.e. Algorithm 2). In Fig. 5
we present the convergence of the transmission power (in
dBW) for all links that have been allocated RB 7. This is the
worst case for the scenario under consideration, starting from
the maximum possible power values as initial ones. It is noted
that in practice even in that case approximately fifty iterations
are sufficient to achieve convergence. Considering a more
Beducated^scenario, the initial transmission power values
are based on the corresponding values calculated in the previ-
ous time-slot. This way the number of iterations and the cor-
responding convergence time can be further decreased.
Next in Fig. 6we present the sensed interference (in dBW)
of each link of the network for the following three cases: a) the
initial RB allocation b) the application of the first stage algo-
rithm only, and c) the two-stage single run case (Algorithm 1
and Algorithm 2). The latter demonstrates the additional ben-
efits achieved by the application of the two stage framework
that combines both RB and power allocation. We consider that
every link in the network has a desirable SINR requirement/
010 20 30 40 50 60 70
-2.5
-2
-1.5
-1
-0.5
0x 10
-8
Iterati ons
seitilitUfomuSlatoT
Fig. 3 Total sum of the utilities of all links in the network
0 5 10 15 20 25 30 35
-20
-10
0
10
20
30
40
50
Link id
)Bd(RNIS
Initi al (Rand om)
First Stage (RB allocation only)
Two stage - Single R un (RB + Power allocation)
Fig. 4 Achieved SINR at link’s receiver versus the link id
020 40 60 80 100
-60
-50
-40
-30
-20
-10
0
10
Iterations
)WBd(rewoP
Cellular link 7
D2D link 9
D2D link 11
D2D link 23
Fig. 5 Transmission power vs iterations for RB 7
Tab l e 2 Final RB allocation of all links of the network after the
execution of Algorithm 1 (first stage)
RBs 12345678
linkid12345678
21 15 16 17 13 14 9 10
27 30 26 18 19 24 11 12
32 31 20 22 29 23 25
28
Mobile Netw Appl

level of 10 dB. The achieved SINR results after the application
of the second stage of our framework (in the single run case)
are also presented in Fig. 4(red line). In combination with the
corresponding results in Fig. 6we observe that the interfer-
ence of every link decreases significantly due to the effective
application of the SINR aware power control part of our ap-
proach (second stage), while every link obtains its SINR
requirement.
Furthermore, in Fig. 7we present some comparative results
of the interference of all links of the network. We notice that
following the iterative approach the interference decreases
further. We also observed that no significant changes occurred
in the corresponding SINR values and all links obtained the
respective requirements.
Next, we demonstrate the performance of the proposed two
stage approach with respect to increasing number of D2D
links, while the number of cellular links and the number of
available RBs remain unchanged. Specifically, in Figs. 8and 9
we present the average interference per link as well as the
average achieved SINR while increasing the number of D2D
links. The corresponding results verify the efficiency of our
approach in case of increasing the number of D2D links.
The efficiency of our proposed NE point in the first step of
our approach can be determined by comparing the RB alloca-
tion at the concluding NE point with the optimal RB allocation
with respect to the total network utility sum. Due to the exact
potential game nature of the considered game, the NE points
of the game are local optimal points of the potential function
which equals to the minus of the total interference in the net-
work (i.e. the total network utility function). In this respect the
optimal point (RB allocation) is guaranteed to be a NE point of
the game. In order to estimate the efficiency of the proposed
NE point we examined several scenarios for multiple repeti-
tions, changing the initial starting RB allocation of Algorithm
1 and the order of the acting players. We concluded that many
different NE points exist but the total network utility value of
the point that our algorithm achieved did not differ significant-
ly from the corresponding value of the estimated optimal NE
point. Specifically, the total network utility values of the var-
ious NE points obtained from our approach ranged between −
0 5 10 15 20 25 30 35
-150
-149
-148
-147
-146
-145
-144
-143
-142
Link id
)WBd(ecnerefretnI
Two stage - Single Run ( RB + Power allocation)
Two stage - Iterative Run (RB + Power allocati on)
Fig. 7 Interference at link’s receiver versus link id
12 14 16 18 20 22 24 26 28 30
-150
-140
-130
-120
-110
-100
-90
-80
Number of D2D links
)WBd(ecnerefretnI
Initial (Random)
First Stage (RB allocation only)
Two stage - Single Run (RB + Power allocation)
Two stage - Iterative Run (RB + Power allocation)
Fig. 8 Average Interference versus number of D2D links
0 5 10 15 20 25 30 35
-150
-140
-130
-120
-110
-100
-90
-80
-70
Link id
)WBd(ecnerefretnI
Initial (Random)
First Stage (RB al location only)
Two stage - Single Run (RB + Power allocation)
Fig. 6 Interference at link’s
receiver versus link id
Mobile Netw Appl

2, 2 ⋅10
−10
and −1.67 ⋅10
−10
, and was always within 5 %
difference than the observed maximum value. Regarding the
second step of our approach the NE point of the power allo-
cation game is unique since the best response function is stan-
dard [31] and it is proven that the best response algorithm
converges to the unique fixed point of the power allocation
game.
7.2 Comparative results
In this subsection we compare the performance of our ap-
proach against two alternative state of the art approaches in
the literature [15,24]. The approach in [24]consistsoftwo
stages i.e. a RB allocation stage and a power control stage and
aims at improving the ratio of the rate to the total power con-
sumed by every UE. The approach introduced in [15]isa
greedy RB allocation approach targeting at total rate maximi-
zation. The authors allocate the RB of the cellular link with the
highest channel gain to the D2D link with the lowest channel
gain from the cellular transmitter to the D2D link’sreceiver.
For fairness in the comparison, regarding [15], we present
results where the approach of [15] is also enhanced with the
use of our proposed power control algorithm (i.e. Algorithm
2) or the power control algorithm introduced in [24] (in the
following we refer to them as BGreedy RA + Algorithm 2^
and BGreedy RA + PC of [24]^respectively). The key simu-
lation parameters of this scenario are the ones considered in
[24] and are presented in the Table 3.
The comparison has been performed in terms of both the
achieved total sum of rates per Hertz and total sum of trans-
mission power values. The corresponding results are shown in
Figs. 10 and 11, increasing the number of D2D links in the
network. Based on these results we clearly observe that our
approach outperforms in all scenarios both approaches in [24]
and [15], as well as their considered variations, as it achieves
higher transmission data rates while resulting in significantly
lower power consumption.
Tabl e 3 Simulations parameters and values
Parameters Values
Cell Radius 350 m
UE distribution randomly distributed
Number of cellular UEs 30
Number of RBs 30
Number of D2D pairs 6–30
Maximum UE transmit power 200 mW
Channel Bandwidth 180 kHz
5 10 15 20 25 30
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
Number of D2D links
Sum of Transmission Powers (W)
[24]
Greedy RA + PC of [24]
Two stage − Iterative Run
Greedy RA + Algorithm 2
Fig. 10 Total sum of transmission powers with respect to the number of
the D2D links in the network
12 14 16 18 20 22 24 26 28 30
10
15
20
25
30
35
Number of D2D links
)Bd(RNIS
Init ial (Rando m)
First Stage (RB allocation only)
Two stage - Single Run (RB + Power allocation)
Two stage - Iterative Run (RB + Power allocation)
Fig. 9 Average SINR versus
number of D2D links
Mobile Netw Appl

8Conclusions
D2D communications has been considered as one of the
key features of 5G mobile networks. They are able to
alleviate the huge infrastructure investment required to
deal with the exponential growth of mobile traffic and
improve local service flexibility. In this paper we pro-
pose a two stage game theoretic approach for the prob-
lem of joint RB and uplink transmission power alloca-
tion in a D2D underlay cellular network. It is noted that
we do not restrict a priori the number of D2D pairs that
can reuse a RB allocated to a cellular link. However,
this is determined dynamically by the objective of inter-
ference minimization. As a result, a RB reserved by a
cellular link can be potentially reused multiple times by
different D2D links, in order to increase spectrum
efficiency.
In the first step of our approach a RB allocation pro-
cess is proposed that minimizes the total interference of
all the links in the network. In the second step we apply a
distributed and autonomous game theoretic approach for
the uplink transmission power control problem of all links
in the network. At this step every link will transmit at the
desired power level in order to satisfy its SINR require-
ment if it is feasible. Otherwise, it will transmit at its
maximum power in order to obtain a SINR level as close
as possible to its desired level. The performance effective-
ness of our proposed framework is evaluated through
modeling and simulation. The corresponding numerical
results confirmed that our framework achieves to mitigate
the interference of every link in the network while every
link satisfies its SINR requirement. Finally, comparative
numerical results are presented that demonstrate that our
methodology outperforms other state of the art approaches
in terms of sum of rates per bandwidth unit and total
transmission power consumption.
Appendix I
Proof of Lemma 1
The potential function of game Gcan be written as:
Pot si;s−i
ðÞ¼
X
N
i¼1
aT1
isi;s−i
ðÞþ1−aðÞT2
isi;s−i
ðÞ

¼
¼X
N
i¼1
−aX
N
j≠i;j¼1
pjGji fc
j;ci

−1−aðÞ
X
N
j≠i;j¼1
piGijfc
i;cj

"#
¼
−aX
N
j≠i;j¼1
pjGji fc
j;ci

−1−aðÞ
X
N
j≠i;j¼1
piGijfc
i;cj

X
N
k≠i;k¼1
−aX
N
j≠k;j¼1
pjGjk fc
j;ck

−1−aðÞ
X
N
j≠k;j¼1
pkGkjfc
k;cj

"#
¼
¼−aX
N
j≠i;j¼1
pjGji fc
j;ci

−1−aðÞ
X
N
j≠i;j¼1
piGijfc
i;cj

þX
N
k≠i;k¼1h−apiGik fc
i;ck
ðÞ−aX
N
j≠k;j≠i;j¼1
pjGjk fc
j;ck

−
−1−aðÞpkGki fc
k;ci
ðÞ−1−aðÞ
X
N
j≠i;j≠k;j¼1
pkGkjfc
k;cj

i
Pot si;s−i
ðÞ¼−aX
N
j≠i;j¼1
pjGji fc
j;ci

−1−aðÞ
X
N
j≠i;j¼1
piGijfc
i;cj

þX
N
k≠i;k¼1h−apiGik fc
i;ck
ðÞ−1−aðÞpkGki fc
k;ci
ðÞ
iþ
X
N
k≠i;k¼1h−aX
N
j≠k;j≠i;j¼1
pjGjk fc
j;ck

−1−aðÞ
X
N
j≠i;j≠k;j¼1
pkGkjfc
k;cj

i
Let Qs
−i
ðÞ¼ ∑
N
k≠i;k¼1
½−a∑
N
j≠k;j≠i;j¼1
pjGjk fc
j;ck

−1−aðÞ∑
N
j≠i;j≠k;j¼1
pkGkjf
ck;cj

Þ
Then:
Pot si;s−i
ðÞ¼−a∑
N
j≠i;j¼1
pjGji fc
j;ci

−1−aðÞ∑
N
j≠i;j¼1
piGijfc
i;cj

∑
N
k≠i;k¼1
½−apiGik fc
i;ck
ðÞ−1−aðÞpkGki fc
k;ci
ðÞþQs
−i
ðÞ¼−aþ1−aðÞðÞ
∑
N
j≠i;j¼1
pjGji fc
j;ci

−1−aðÞþaðÞ∑
N
j≠i;j¼1
piGijfc
i;cj

þQ
s−i
ðÞ¼−∑
N
j≠i;j¼1
pjGji fc
j;ci

−∑
N
j≠i;j¼1
piGijfc
i;cj

þQs
−i
ðÞ
If link ichanges strategy from s
i
to s
i
'
then from (9)wehave
Pot s0
i;s−i

¼−X
N
j≠i;j¼1
pjGji fc
j;c0
i

−X
N
j≠i;j¼1
piGijfc
0
i;cj

þQs
−i
ðÞ
5 10 15 20 25 30
100
120
140
160
180
200
220
240
Number of D2D links
Sum of Rates per Hertz (bps/Hz)
[24]
Greedy RA + PC of [24]
Two stage − Iterative Run
Greedy RA + Algorithm 2
Fig. 11 Total sum of rates per Hz with respect to the number of the D2D
links in the network
Mobile Netw Appl

Then follows:
Pot si;s−i
ðÞ−Pot s0
i;s−i

¼
¼−X
N
j≠i;j¼1
pjGji fc
j;ci

−X
N
j≠i;j¼1
piGijfc
i;cj

þQs
−i
ðÞ
−−
X
N
j≠i;j¼1
pjGji fc
j;c0
i

−X
N
j≠i;j¼1
piGijfc
0
i;cj

þQs
−i
ðÞ
!
¼Uisi;s−i
ðÞ−Uis0
i;s−i

∀i∈N;∀si;s0i∈Si
The above formulated game Gis an exact potential game
with the exact potential function Pot(s
i
,s
−i
). ■
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