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1
Multi-channel Resource Allocation towards Ergodic
Rate Maximization for Underlay Device-to-Device
Communications
Ruhallah AliHemmati, Member, IEEE, Min Dong, Senior Member, IEEE, Ben Liang, Senior Member, IEEE, Gary
Boudreau, Senior Member, IEEE, and S. Hossein Seyedmehdi
Abstract—In underlay device-to-device (D2D) communications,
a D2D pair reuses the cellular spectrum causing interference to
regular cellular users. Maximizing the performance of under-
lay D2D communications requires joint consideration for the
achieved D2D rate and the interference to cellular users. In
this work, we consider the D2D power allocation optimization
over multiple resource blocks (RBs), aiming at maximizing the
either the ergodic D2D rate or the ergodic sum rate of D2D and
cellular users, under the long-term sum-power constraint of the
D2D users and per-RB probabilistic signal-to-interference-and-
noise (SINR) requirements for all cellular users. We formulate
stochastic optimization problems for D2D power allocation over
time. The proposed optimization framework is applicable to both
uplink and downlink cellular spectrum sharing. To solve the
proposed stochastic optimization problems, we first convexify
the problems by introducing a family of convex constraints as a
replacement for the non-convex probabilistic SINR constraints.
We then present two dynamic power allocation algorithms: a
Lagrange dual based algorithm that is optimal but with a
high computational complexity, and a low-complexity heuristic
algorithm based on dynamic time averaging. Through simulation,
we show that the performance gap between the optimal and
heuristic algorithms is small, and effective long-term stochastic
D2D power optimization over the shared RBs can lead to
substantial gains in the ergodic D2D rate and ergodic sum rate.
Index Terms—Device-to-Device communications, ergodic re-
source allocation, power allocation.
I. INTRODUCTION
In D2D communications, two user equipments (UEs) di-
rectly communicate with each other without having the pay-
load traversed through the backhaul network. Due to its local
communications nature, D2D communication can be provided
with a lower cost than cellular communications. Furthermore,
D2D communications provides many benefits unavailable to
This work was supported in part by Ericsson Canada, in part by the
Natural Sciences and Engineering Research Council (NSERC) of Canada
Collaborative Research and Development Grant CRDPJ-466072-14, and in
part by NSERC Discovery Grants.
R. AliHemmati and B. Liang are with the Department of Electrical and
Computer Engineering, University of Toronto, Toronto, Ontario M5S 3G4,
Canada (e-mail: ruhallah.alihemmati@utoronto.ca; liang@ece.utoronto.ca).
M. Dong is with the Department of Electrical, Computer and Software
Engineering, University of Ontario Institute of Technology, Oshawa, Ontario
L1H 7K4, Canada (e-mail: min.dong@uoit.ca).
G. Boudreau and S. H. Seyedmehdi are with Ericsson Canada,
Ottawa, Ontario, Canada (e-mail: gary.boudreau@ericsson.com;
hossein.seyedmehdi@ericsson.com).
A preliminary version of this work [1] was presented at the 2016
IEEE International Workshop on Signal Processing Advances in Wireless
Communications (SPAWC), Edinburgh, UK, July 2016.
uncoordinated communications [2]–[4]. There are many cur-
rent and prospective applications for D2D communications.
For example, D2D has been proposed for use in LTE-based
public safety networks for its security and reliability [5]. Ad-
ditionally, D2D communications is necessary for the scenarios
where cellular transmission is not accessible [4].
To facilitate D2D communication, there are different chal-
lenges which should be addressed carefully. A survey on the
challenges and proposed solutions for D2D communications
can be found in [6]. In particular, sharing cellular recourses
between D2D and regular cellular users may cause intra-cell
and inter-cell interference. One possible option is to allocate
different resources for cellular and D2D communications, i.e.
overlay D2D communications. However, to achieve the highest
possible spectral efficiency, underlay D2D communications
has attracted more attention in the literature, where D2D and
cellular users within a cell share the same spectrum resource
and hence interfere with each other. In this paper we mainly
focus on underlay D2D communications.
Underlaying requires effective interference management and
resource sharing among all users. Many methods have been
presented in the literature to address these problems. For
example, Graph-based [7], [8] and game theoretic frameworks
[9]–[14] were considered. Power back-off approaches were
investigated in [15]–[17], and an interference cancelation
method was proposed in [18]. These works do not directly
address the optimization of spectrum resource and power
allocation in D2D communications.
Closer to our interest, resource and power optimization
methods have been proposed in [19]–[26] to maximize the
D2D rate, D2D-cellular sum rate, or power-rate efficiency. An
optimal power allocation solution for D2D users underlaying
cellular users in downlink transmission was given in [19].
The solutions in [19] were achieved without imposing any
constraint on the D2D power. In [20], a solution to encom-
pass mode selection, resource allocation, and power control
within a single framework was proposed. An energy efficient
power control design for resource sharing between cellular
and D2D users was proposed in [21]. The authors of [22]
investigated a weighted sum-rate maximization with multi-
carrier modulation for asynchronous D2D communications.
Performance bounds in the maximization of power efficiency
under signal-to-noise ratio (SNR) constraints were provided in
[23]. The authors of [24] and [25] proposed sub-optimal power
allocation solutions for D2D users in uplink transmission,
2
which divide the original problem into several easier sub-
problems. In [26], an optimal power allocation method based
on maximizing application-dependent weighted cell utility was
proposed.
However, the studies in [19]–[26] are incomplete and moti-
vate further study in the following two aspects of D2D commu-
nications. First, the methods proposed in [19]–[21], [23]–[26]
were designed for the simplified scenario where each D2D
node accesses only a single channel at a time. They cannot be
directly applied to the multi-channel scenario that is prevalent
in most practical systems, such as supporting multiple RBs
in an LTE network. Second, [20]–[26] considers only short-
term power constraints. Yet, D2D nodes are often powered by
batteries with limited energy storage capacity, which directly
corresponds to long-term D2D power constraints. Further-
more, long-term D2D power allocation on individual RBs can
give probabilistic guarantees on the interference from D2D
transmitters to cellular users over the shared RBs. These are
important characteristics of D2D communications that require
further investigation beyond [19]–[26]. In [27] and [28], we
solved the D2D-cellular sum-rate maximization problem over
multiple RBs. However, the solution was short-term with
regard to power and SINR constraints.
In this work, in a multi-channel communication environ-
ment, we aim to either maximize the ergodic D2D rate or the
ergodic D2D-cellular sum rate by optimizing the power alloca-
tion of the D2D users, under the long-term power constraint on
the D2D users and per-RB probabilistic SINR constraints for
all cellular users. The combination of long-term power and
SINR constraints with multi-channel communications leads
to a complicated non-convex stochastic optimization problem.
Building on our preliminary results presented in [1], the main
contributions of this paper are as follows:
•We present a study on ergodic rate maximization with
long-term power constraints and per-RB probabilistic
SINR constraints in D2D communications. To address the
non-convexity in our optimization problem, we propose
a family of convex constraints that provides upper and
lower bounds for the non-convex probabilistic SINR
constraints. In particular, using the Chernoff bound, we
further propose a method to reduce the gap between the
probabilistic constraint and its convex replacement.
•Subsequently, to further convexify the D2D-cellular sum-
rate maximization problem, we replace the objective by a
function which, depending on the values of parameters, is
either convex and decreasing, or concave and increasing.
For the convex decreasing case, we show that optimal
allocated power is zero, while for the concave decreasing
case, we obtain a convex optimization problem.
•To solve the resulting convex optimization problem, we
propose two dynamic algorithms for power allocation
over time. The first algorithm is based on the Lagrange
duality which provides the optimal power levels over
all RBs at each time slot. However, the computational
complexity of this algorithm can be prohibitive when the
channel state space is large. Therefore, we propose an
alternative heuristic algorithm based on dynamic time
averaging, which drastically reduces the computational
Cu1
eNB1
Cu2
pD
t,j|hj|2Ij
p(k)
r,j
pC
r,j
pD
t,j|hI,(k)
j|2
pD
t,j|hI
j|2I0
j
DutDur
eNBk
I0,(k)
j
Fig. 1: A cellular network with underlaying D2D users in
uplink resource sharing. Dutand Dur: transmit and receive
nodes of a D2D pair, respectively. Cu: cellular users. Solid
and dashed lines: desired and interfering signals, respectively.
complexity.
•To show the tightness of the power allocation solutions
by the proposed algorithms, we propose a method to
reformulate the original problems to derive an upper
bound of the original problems for comparison.
•Finally, we show that the proposed algorithms are easily
scalable and can be applied to more general cases with
multiple cells and additional power constraints.
The rest of the paper is organized as follows. Section II
presents the system model of the cellular network used in
this paper and the resource allocation problem is defined in
this section. The proposed methods for solving the D2D-
rate maximization problem and the sum-rate maximization
problem are presented in Section III and Section IV. In
Section V, we discuss extensions of the proposed method
to multi-cell scenarios and to accommodate additional power
constraints. Section VI presents the simulation results. Section
VII concludes the paper.
Notations: We use italic fonts and boldface small letters to
represent scalar variables and vectors respectively. The nota-
tion a<0means all entries of vector aare nonnegative. We
define xb
a
Δ
= max{a, min{x, b}} and x+
Δ
= max{x, 0}.
For a random process y,y[n]indicates its outcome at time-
slot n. We use x∼ N(m, σ 2)to denote a Gaussian random
variable with mean mand variance σ2.
II. SYSTEM MODEL AND PROBLEM DEFINITION
A. System Model
We consider a cellular system consisting of multiple cellular
users and D2D users underlaying the cellular users. We
assume that an idle D2D pair arrives at the cell of interest
requesting access to spectrum for D2D communications. Due
to the localized and low-power transmission of D2D users,
we assume the resource planning (e.g., spectrum allocation
and power control) of existing cellular users in the network
is not modified. As a practical representation of cellular
communications, e.g., LTE networks, we assume multiple RBs
3
TABLE I: Notation Definition
Nnumber of active cellular users in each cell
Cset of all available RBs in the cell
Clset of allocated RBs to the lth D2D pair
Sjset of neighboring cellular users using RB j
pD
t,j D2D transmitted power over RB j
pC
r,j cellular user received power over RB j
p(k)
r,j neighboring cellular user received power over the RB j(for k∈ Sj)
IjD2D received interference power over RB j
pD
t,j |hI
j|2cellular user received interference power over the RB jfrom the new D2D pair
pD
t,j |hI,(k)
j|2neighboring cellular user received interference power over the RB j(for k∈ Sj) from the new D2D pair
hjD2D channel coefficient over RB j
I0
jcellular user received interference over RB jbefore entering the new D2D user
I0,(k)
jneighboring cellular user received interference over RB j(for k∈ Sj) before entering the new D2D user
PD
max maximum available power for a D2D pair
ζintra
j,min cellular user minimum required SINR over RB j
ζ(k)
j,min neighboring cellular user minimum required SINR over RB j(for k∈ Sj)
σ2noise power over each RB
are allocated to each user in the network. Since the D2D de-
vices use licensed cellular spectrum, we assume that resource
allocation is centrally controlled by the cellular operator. In
particular, the RBs are allocated to the cellular and D2D users
by the Evolved Node B (eNB). Furthermore, we assume that
changes to RB allocation occur at a time scale much large
than power allocation, so that when considering the power
allocation problem, the RB allocation is viewed as fixed. There
is no intra-cell interference among cellular users in a cell
because of orthogonal assignment of RBs to the cellular users.
However, due to frequency reuse at neighboring cells, these
cellular users suffer from inter-cell interference. Fig. 1 shows
the interference scenarios for a cellular network with D2D
users in uplink resource sharing. The proposed algorithms can
be similarly applied to the alternate case of downlink spectrum
sharing.
We assume that there are Nactive cellular users in each
cell. A D2D pair attempts to reuse the assigned RBs of active
cellular users in the cell and Cis the set of all available RBs
within the cell. Let Clindicate the set of allocated RBs to the
lth D2D pair. For j∈ Cl, let pD
t,j denote the transmit power
of the D2D pair over the jth RB and pC
r,j denote the received
power from the unique cellular user that is assigned to the
jth RB. In addition, let Sjdenote the set of all cellular users
in the neighboring cells that are using the jth RB. Let p(k)
r,j
denote the received power from the kth user in Sjover the
jth RB.
The cellular users have both intra-cell interference from the
D2D transmission and inter-cell interference from neighboring
cells. For j∈ Cl, let I0
jand I0,(k)
jdenote the received
interference power over the jth RB for the corresponding
cellular user in the main cell and the kth neighboring user,
respectively, excluding the interference from the D2D pair
under consideration. For the uplink sharing, let |hI
j|2and
|hI,(k)
j|2denote the channel power gains over the jth RB
between the D2D transmitter and the eNB and between the
D2D transmitter and the kth neighboring cellular user’s eNB,
for k∈ Sj, respectively (for the downlink case, the same
notation can be used, except that the eNBs are replaced by
the corresponding cellular users.). Furthermore, let Ijdenote
the received interference power over the jth RB at the D2D
receiver. And finally, let hjdenote the D2D channel coefficient
over the jth RB. Under the fading environment, all channel
power gains and interference power are random variables. The
notation used throughout this paper is summarized in Table I.
B. Ergodic Rate Optimization Problem
For the uplink transmission, the received SINR of the
cellular user over the jth RB at the eNB in the main cell,
at the eNB of the kth neighboring cellular user in Sj, and at
the D2D receiver are respectively given by 1
SINRC
j=pC
r,j
σ2+I0
j+pD
t,j |hI
j|2,(1)
SINRC,(k)
j=p(k)
r,j
σ2+I0,(k)
j+pD
t,j |hI,(k)
j|2, k ∈ Sj,(2)
SINRD
j=|hj|2pD
t,j
σ2+Ij
.(3)
In order to maintain the quality of service for the cellular
users at a specific level, it is important to control the inter-
ference from the D2D transmitter to the cellular users in the
main cell and also in the neighboring cells. Therefore, the D2D
power over each RB must be confined. We first consider the
following constraints
PrnSINRC
j≤ζintra
j,mino≤, j ∈ Cl(4)
PrnSINRC,(k)
j≤ζ(k)
j,mino≤, j ∈ Cl, k ∈ Sj(5)
where ζintra
j,min and ζ(k)
j,min are minimum SINR targets for the
cellular user in the main cell and the kth neighboring cellular
user in Sj, respectively. These constraints guarantee a specific
long-term QoS for the cellular users in the main cell and
1For the downlink, SINR is defined by replacing the eNB with the cellular
user.
4
neighboring cells. We define
ηj,min (pC
r,j /ζintra
j,min −(σ2+I0
j)
|hI
j|2,
(p(k)
r,j /ζ(k)
j,min −(σ2+I0,(k)
j)
|hI,(k)
j|2)k∈Sj
.(6)
It is easy to show that (4) and (5) are equivalent to the
following constraint:
PrnpD
t,j ≥ηjo≤. (7)
Furthermore, in order to limit power usage for the D2D user,
we additionally consider a long-term sum-power constraint for
the D2D pair as follows:
EnX
j∈Cl
pD
t,j o≤PD
max.(8)
The statistical constraints on the D2D transmission power
in (7) and (8) are more practical than the deterministic ones
commonly assumed in the literature [19], [23]–[26]. Instead
of imposing instantaneous, strict SINR and power constraints
in each time slot, we allow their fluctuations over time.
Constraint (7) models long-term QoS requirements, while the
constraint (8) corresponds to the need to conserve energy
especially for battery-powered D2D equipment. The resultant
additional degree of freedom in dynamic adjustment of the
D2D transmission power, tailored to the time-varying channel
conditions, can lead to substantial gains in the ergodic D2D
rate and D2D-cellular sum rate. This will be numerically
demonstrated in Section III-E and IV-B, where we compare
the cases where the D2D transmission power is properly
designed over time under statistical constraints, and where it
is deterministically bounded in each time slot. Furthermore, in
Section V-B, we will discuss how the proposed solution can
be easily extended to the case where there are both statistical
and deterministic constraints on the D2D transmission power.
Thus, in this paper, we study the following two stochastic
power allocation problems to find the optimal power in each
time slot over each RB for the new D2D pair:
I) Ergodic D2D-Rate Maximization Problem
D1 : max
pD
t<0
EnX
j∈Cl
log(1 + SINRD
j)o
subject to (7) and (8);
II) Ergodic Sum-Rate Maximization Problem
S1 : max
pD
t<0
EnX
j∈Cl
log(1 + SINRC
j) + log(1 + SINRD
j)o
subject to (7) and (8),
where we define pD
t= [pD
t,1,∙∙∙, pD
t,|Cl|]T. Note that, in the
above optimization problems, the transmission power pD
tis
a mapping from the random channel state vector to a power
allocation vector.
C. Feasibility Check
Consider the SINR constraints in (4)-(5). The feasible set
is non-empty only if we have
PrnpC
r,j
σ2+I0
j≤ζintra
j,mino≤, j ∈ Cl,(9)
Prnp(k)
r,j
σ2+I0,(k)
j≤ζ(k)
j,mino≤, j ∈ Cl, k ∈ Sj.(10)
For example, in the case that all signal and interface powers
are exponentially distributed, the constraints in (4)-(5) are
equivalent to
1−e−λC
p,j ζintra
j,minσ2
1 + λC
p,j ζintra
j,min
λI,j
≤, j ∈ Cl,(11)
1−e−λ(k)
p,j ζ(k)
j,minσ2
1 + λ(k)
p,j ζ(k)
j,min
λ(k)
I,j
≤, j ∈ Cl, k ∈ Sj,(12)
where λC
p,j ,λ(k)
p,j ,λI,j and λ(k)
I,j are the rates of exponentially
distributed random variables pC
r,j ,p(k)
r,j ,I0
jand I0,(k)
j, respec-
tively.
III. D2D RATE MAXIMIZATION
The stochastic optimization problem D1can be reformu-
lated as an equivalent deterministic optimization problem,
in which all expectations in the optimization problem D1
can be written as probability-weighted sums of functions of
realizations of the random channel state vector over all RBs. In
this reformulation, the decision variable is pD
t,j corresponding
to every realization of the channel state vector. However, the
complexity of directly solving such an optimization problem
would be prohibitive, due to the exponential size of the multi-
dimensional channel state space. Instead, we propose to first
convexify the optimization problem D1. We will then show
that using Lagrange multipliers, the problem of finding pD
t,j
can be solved separately over each observed realization of
channel states.
A. Convexification of Problem D1
The probabilistic individual power constraint in (7) is not
convex. We consider instead stronger convex constraints using
the following lemma.
Lemma 1. For any strictly increasing function f(∙)such that
f(0) = 1, we have
PrnpD
t,j ≥ηjo≤Enf(pD
t,j −ηj)o.(13)
Proof: Since f(∙)is a strictly increasing function, we have
PrnpD
t,j ≥ηjo= PrnpD
t,j −ηj≥0o
= Prnf(pD
t,j −ηj)≥f(0)o
≤Enf(pD
t,j −ηj)o,(14)
5
where the last inequality is achieved by applying Markov’s
inequality and the assumption that f(0) = 1.
Note that in Lemma 1, f(∙)does not need to be convex.
However, to obtain a convex optimization problem, we will
use only convex increasing functions. We propose substituting
the following constraint for the constraint in (7):
Enfj(pD
t,j −ηj)o≤, (15)
where fj(∙)’s are convex increasing functions for all j∈ Cl.
By satisfying (15), the constraint in (7) will be guaranteed.
Thus, we can find a lower-bound for D1by using (15) and
solving the new convex optimization problem:
D2 : max
pD
t,j <0
EnX
j∈Cl
log 1 + |hj|2pD
t,j
σ2+Ij!o
subject to EnX
j∈Cl
pD
t,j o≤PD
max (16)
Enfj(pD
t,j −ηj)o≤, j ∈ Cl.(17)
B. Solution via the Lagrange Method
The Lagrange function of D2can be written as
LD(pD
t,λ, μ)
=X
j∈Cl
Enlog 1 + |hj|2pD
t,j
σ2+Ij!−μpD
t,j −λjfj(pD
t,j −ηj)o
+μP D
max +X
j∈Cl
λj(18)
where λ= [λ1,∙∙∙, λ|Cl|]Tis a vector of Lagrange multipliers.
The corresponding Lagrange dual function is
gD(λ, μ) = max
pD
t<0LD(pD
t,λ, μ).(19)
To find the optimal pD
t,j , for fixed values of λjand μ, we
need to solve the following optimization problem for each j
and each channel realization of the jth RB:
pD∗
t,j (λj, μ) = arg max
pD
t,j ≥0log 1 + |hj|2pD
t,j
σ2+Ij!
−μpD
t,j −λjfj(pD
t,j −ηj)(20)
where pD∗
t,j (λj, μ)is the optimal power allocation. Note that
the problem in (20) is convex and the optimal value can be
found efficiently.
The optimal values for λand μcan be found through the
dual optimization problem
min
λ<0,μ≥0gD(λ, μ)(21)
using the subgradient method [29]. To find subgradients, we
note that
gD(λ0, μ0)
= max
pD
t<0LD(pD
t,λ0, μ0)
≥ LD(pD∗
t(λ, μ),λ0, μ0)
=gD(λ, μ) + X
j∈Cl
(λ0
j−λj)(−E{fj(pD∗
t,j (λj, μ)−ηj)})
+ (μ0−μ)(PD
max −E{X
j∈Cl
pD∗
t,j (λj, μ)}).(22)
Hence, the following are subgradients of g(λ, μ):
∂μ =PD
max −X
j∈Cl
E{pD∗
t,j (λj, μ)}(23)
∂λj=−E{fj(pD∗
t,j (λj, μ)−ηj)}j∈Cl.(24)
Following the subgradient method, after a sufficient number
of iterations, we can find the value of optimal Lagrange
multipliers, i.e., μ∗and λ∗
jfor all j∈ Cl. Then, for each
channel realization, we need to solve the following optimiza-
tion problem to find an optimal power allocation:
pD∗
t,j (H) = arg max
pD
t,j ≥0log 1 + |hj|2pD
t,j
σ2+Ij!(25)
−μ∗pD
t,j −λ∗
jfj(pD
t,j −ηj(H))
where His the vector of channel state.
Note that considering (23) and (24), the above formulation
can only be applied over a time interval where we can assume
the channel for D2D users is stationary.
C. Special Case for Function fj(∙): Chernoff Bound
Using the Chernoff bound, (15) becomes
Eneωj(pD
t,j −ηj)o≤. (26)
If we use the Chernoff bound as a substitute of the probabilistic
SINR constraint, i.e., (7), it is important to properly choose
the value of ωjto achieve the minimal gap between the two
constraint. In fact, we have:
PrnpD
t,j ≥ηjo≤min
ωj
Eneωj(pD
t,j −ηj)o≤Eneωj(pD
t,j −ηj)o.
(27)
To find an optimal ωjwe have
∂
∂ωj
Eneωj(pD
t,j −ηj)o=En(pD
t,j −ηj)eωj(pD
t,j −ηj)o= 0.(28)
We define the random variable xj=pD
t,j −ηj. Unfortunately,
the distribution of xjis not known before solving the optimiza-
tion problem. However, we observe that xjis a complicated
mixture of multiple random quantities, and our numerical
results indicate it has roughly bell-shape distribution. Thus,
we assume that xj∼ N(mj, υj)and
Enxjeωjxjo
=Z+∞
−∞
xeωjx1
υj√2πe−(x−mj)2
2υ2
jdx
=e
(ωjυ2
j+mj)2−m2
j
2υ2
jZ+∞
−∞
y1
υj√2πe−(y−(ωjυ2
j+mj))2
2υ2
jdy. (29)
Note that the integral in (29) is the mean value for the random
variable yj∼ N(ωjυ2
j+mj, υj). Therefore, from (28) and
(29), a suitable value for ωjcan be found as −mj
υ2
j
. Since
6
mjand υjare not known, we use an iterative method where,
starting from an initial point for ωj, in each step we update
the value of ωjfor the next step by estimating mjand υj
using the available information to compute the statistics of xj
in the current step.
D. Special Case for Function fj(∙): Polynomial Functions
It is easy to show that, for any m∈N, the function f(pD
t,j −
ηj) = [1 + 1
ηmax
j(pD
t,j −ηj)]m, where ηmax
jis the maximum
possible value for ηj, is convex and increasing with f(0) = 1.
In this case, we have the following constraint:
En[1 + ωj(pD
t,j −ηj)]mo≤, (30)
where ωj=1
ηmax
j
.
In the special case where (30) is linear, i.e., m= 1, we
can provide a closed-form solution to (20) as follows. Setting
the derivative of (20) equal to zero, for fj(pD
t,j −ηj) = 1 +
ωj(pD
t,j −ηj), we have
|hj|2
|hj|2pD
t,j +σ2+Ij−μ−λjωj= 0.(31)
Thus,
pD∗
t,j (λj, μ) = h1
μ+λjωj−σ2+Ij
|hj|2i+.(32)
E. Time-Averaging Based Heuristic Solutions
In the optimal solutions above, the calculation of subgradi-
ents has high complexity and needs full information about
the statistics of the channels between D2D users and all
interference channels. In this section, we present a new method
to directly find the Lagrange multipliers by approximating the
primal domain problem.
Considering the fact that μis related to constraint (16), in
each step, we can find μ[n+ 1] by associating it with the
approximated constraint En+1nPj∈ClpD
t,j o≤PD
max, where
nis the time slot index and En{x},1
nPn
t=1 x[t]. It is easy
to show that we have En+1{x}=1
n+1 (x[n+ 1] + nEn{x}).
Thus, the sum power constraint in (8) can be written as
X
j∈Cl
pD
t,j [n+ 1] ≤(n+ 1)PD
max −nEnnX
j∈Cl
pD
t,j o
,P0D
max[n+ 1].(33)
Similarly, the individual power constraints in (17) can be
approximated as
pD
t,j [n+ 1]
≤ηj[n+ 1] + f−1
j(n+ 1)−nEn{fj(pD
t,j −ηj)}
,η0
j[n+ 1],(34)
where f−1
j(∙)is the inverse function of fj(∙). Since, fj(∙)is
a strictly increasing function, f−1
j(∙)is unique.
In time slot n, we may solve the following optimization
problem
D3 : max
pD
t<0X
j∈Cl
log 1 + |hj[n]|2pD
t,j
σ2+Ij[n]!
subject to (33) and (34).
After applying the KKT optimality condition to the primal
problem [29], we have two cases.
Case 1) Pj∈Clη0
j[n]≤P0D
max:pD∗
t,j [n] = η0
j[n], for all j∈ Cl.
Case 2) Pj∈Clη0
j[n]> P 0D
max[n]: for all j∈ Clwe have
pD∗
t,j [n] = 1
μ[n]−σ2+Ij[n]
|hj[n]|2η0
j[n]
0
.(35)
Note that in Case 2, μ[n]must be found such that
Pj∈ClpD
t,j [n] = P0D
max[n], which can be achieved using the
bisection method.
Remark. It is worth mentioning that the sum-power constraint
in (33) is equivalent to
EnnX
j∈Cl
pD
t,j o≤PD
max,(36)
and also the individual power constraint in (34) is equivalent
to
Ennfj(pD
t,j −ηj)o≤(37)
which are valid for all n. For sufficiently large n, by assuming
ergodicity for all channels in the network, satisfying the con-
straints in (33) and (34) guarantees satisfying the constraints
in stochastic optimization problem in D2. For small values of
n, since there is no information on the future channel state,
the feasible set of D3is a subset of the feasible set of D2. By
increasing the time-window size, i.e., for larger n, the feasible
set of D3converges to the feasible set of D2. In other words,
the per-time-slot optimization problem D3provides a lower
bound for the stochastic optimization problem D2.
F. An Upper Bound for D1
For a performance benchmark, we propose an upper bound
for the optimization problem in D1and consider the gap
between the lower and upper bound. In this section we try
to reach a proper upper bound.
Theorem 1. For any ωj≥0,PrnpD
t,j ≥ηjo≥1−
Ene−ωj(pD
t,j −ηj)o.
Proof: We have PrnpD
t,j ≥ηjo= Prn−ωj(pD
t,j −ηj)≤
0o= Prne−ωj(pD
t,j −ηj)≤1o. From the Markov inequality
we have
Prne−ω(pD
t,j −ηj)≤1o≥= 1 −Ene−ωj(pD
t,j −ηj)o.(38)
The optimization problem for finding the upper bound can
be written as follows
D4 : max
pD
t<0
EnX
j∈Cl
log aj+bjpD
t,j
aj+cjpD
t,j !o(39)
7
subject to EnX
j∈Cl
pD
t,j o≤PD
max (40)
En1−e−ωj(pD
t,j −ηj)o≤, j ∈ Cl.(41)
Unfortunately, because the constraint in (41) is concave,
D4is not a convex optimization problem. However, the dual
problem is always convex and provides an upper bound for
the primal problem. Therefore, we can use the dual problem
to find an upper bound for D1. The Lagrange function of (39)
can be written as
L1(pD
t,λ, μ)
,X
j∈Cl
Enlog 1 + |hj|2pD
t,j
σ2+Ij!o+μ(PD
max −X
j∈Cl
EnpD
t,j o)
+X
j∈Cl
λj(−1 + Ene−ωj(pD
t,j −ηj)o)
=X
j∈Cl
Enlog aj+bjpD
t,j
aj+cjpD
t,j !−μpD
t,j +λje−ωj(pD
t,j −ηj)o
+μP D
max +X
j∈Cl
λj(−1).(42)
The corresponding Lagrange dual function can be defined is
g1(λ, μ) = max
pD
t<0L1(pD
t,λ, μ).(43)
To find an optimal pD
t,j , for fixed values of λjand μand for
j∈Cl, in (43) we need to solve the following optimization
problem for each j
pD∗
t,j (λj, μ) = arg max
pD
t,j ≥0log 1 + |hj|2pD
t,j
σ2+Ij!
−μpD
t,j +λje−ωj(pD
t,j −ηj).(44)
Note that (44) is not a convex optimization problem. Thus
we need to apply an extremum-search method to find the
global optimum point. Then, to solve (43), the subgradient
method can be used.
Improving the value of ωj:Similarly to Section III-C, to
improve ωjfor the upper bound, we have
∂
∂ωj
(1 −Ene−ωj(pD
t,j −ηj)o) = En(pD
t,j −ηj)e−ωj(pD
t,j −ηj)o
= 0.(45)
Assuming that xj=pD
t,j −ηj∼ N(mj, υj), we have
Enxjeωjxjo
=Z+∞
−∞
xe−ωjx1
υj√2πe−(x−mj)2
2υ2
jdx
=e
(mj−ωjυ2
j)2−m2
j
2υ2
jZ+∞
−∞
y1
υj√2πe−(y−(mj−ωjυ2
j))2
2υ2
jdy = 0.
From the above equation, a suitable value for ωjcan be found
as mj
υ2
j
. Hence, we can update the value of ωjusing an iterative
method as discussed in Section III-C.
IV. ERGODIC SUM-R ATE MAXIMIZATION
Similarly to the previous section, we first convexify the non-
convex optimization problem in S1and then use the Lagrange
duality to find the D2D power allocation pD
t,j separately over
each observed outcome (or “realization”) of the channel state
vector. However, the more complex non-convex form of the
D2D-cellular sum-rate objective presents further challenges.
In this section, we detail the additional procedures to solve
S1. We use the same definition and special cases of fj(∙)as
in the previous section.
Note that the sum-rate of cellular users over RBs in Cl, prior
to the D2D pair entering the system, is given by Pj∈Cllog(1+
pC
r,j
σ2+IC
j
). It is independent of D2D transmitter power allocation.
Thus, the sum-rate maximization problem S1is equivalent to
the problem of maximizing the ergodic sum-rate improvement
due to the addition of the new D2D pair, given by
S10: max
pD
t<0
EnX
j∈Cl
log(1 + SINRC
j) + log(1 + SINRD
j)
−log 1 + pC
r,j
I0
j+σ2
j!o
subject to (7) and (8).
A. Convexification of S1
Typically, only those RBs over which the cellular users
have a sufficiently high SINR condition are allocated to the
D2D user. After D2D reuse, the SINR of the cellular user
over such an RB is still relatively high. Therefore, we assume
the minimum SINR requirement ζintra
j,min 1, for all j∈ Cl.
With this assumption, we can use following approximation to
approximate the objective of S10as
X
j∈Cl
log(1 + SINRC
j)−log 1 + pC
r,j
I0
j+σ2
j!+ log(1 + SINRD
j)
≈X
j∈Cl"log pC
r,j
pD
t,j |hI
j|2+I0
j+σ2!−log pC
r,j
I0
j+σ2!
+ log 1 + |hj|2pD
t,j
Ij+σ2!#
=X
j∈Cl
log aj+bjpD
t,j
aj+cjpD
t,j !,(46)
where aj,(σ2+I0
j)(σ2+Ij),bj,(σ2+I0
j)|hj|2, and
cj,(σ2+Ij)|hI
j|2. Thus, we can approximate S1as
S2 : max
pD
t<0
EnX
j∈Cl
log( aj+bjpD
t,j
aj+cjpD
t,j
)o
subject to (7) and (8).
As discussed in section III, for the D2D-rate maximization
problem, (7) is not a convex constraint, so we substitute it
with (16). Therefore, we propose the following optimization
problem:
S3 : max
pD
t<0
EnX
j∈Cl
log( aj+bjpD
t,j
aj+cjpD
t,j
)o
8
subject to (16) and (17).
Furthermore, the objective function of S3, for pD
t<0, can be
upper bounded as follows:
EnX
j∈Cl
log( aj+bjpD
t,j
aj+cjpD
t,j
)o
=X
j∈Cl
Pr{bj≥cj}Enlog( aj+bjpD
t,j
aj+cjpD
t,j bj≥cj)
+ Pr{bj< cj}Enlog( aj+bjpD
t,j
aj+cjpD
t,j
)bj< cjo
≤X
j∈Cl
Pr{bj≥cj}Enlog( aj+bjpD
t,j
aj+cjpD
t,j
)bj≥cjo.(47)
The last inequality comes from the fact that the function
log( aj+bjpD
t,j
aj+cjpD
t,j
), for bj< cjand pD
t,j ≥0, is a decreasing (and
convex) function. The upper bound is achievable if and only
if for bj< cjwe have pD
t,j = 0. In other words, pD
t,j = 0 is
an optimal solution when we have bj< cjif and only if all
the constraint in S3are satisfied.
The sum-power constraint is satisfied if we have
EnX
j∈Cl
pD
t,j o=X
j∈Cl
(Pr{bj≥cj}EnpD
t,j bj≥cjo
+ Pr{bj< cj}EnpD
t,j bj< cjo)
=X
j∈Cl
Pr{bj≥cj}EnpD
t,j bj≥cjo(48)
≤PD
max.
The individual power constraints are satisfied if we have
Enfj(pD
t,j −ηj)o= Pr{bj≥cj}Enfj(pD
t,j −ηj)bj≥cjo
+ Pr{bj< cj}Enfj(pD
t,j −ηj)bj< cjo
= Pr{bj≥cj}Enfj(pD
t,j −ηj)bj≥cjo
+ Pr{bj< cj}Enfj(−ηj)bj< cjo
≤. (49)
Therefore, S3is equivalent to the following optimization
problem
S4 : max
pD
t<0X
j∈ClPjEnlog( aj+bjpD
t,j
aj+cjpD
t,j
)bj≥cjo
subject to X
j∈ClPjEnpD
t,j bj≥cjo≤PD
max (50)
PjEnfj(pD
t,j −ηj)bj≥cjo≤0
jj∈ Cl,(51)
where we define Pj
Δ
= Pr{bj≥cj}for all j∈ Cland 0
j
Δ
=
−(1−Pj)E{fj(−ηj)bj< cj}. We will later show that there
is no need to calculate 0
j.
It is easy to show that any solution for S4, by applying
pD
t,j = 0 when we have bj< cj, is feasible for the problem
S3. Also, again by applying pD
t,j = 0 when we have bj< cj,
the objective function of the two problem are the same. Thus,
the two problems S3and S4are equivalent.
Note that the function log( aj+bjpD
t,j
aj+cjpD
t,j
), for bj≥cjand pD
t,j ≥
0, is a concave (and increasing) function, thus we can use the
method of Lagrange multipliers to solve S4. The Lagrange
function of S4can be written as
LS(pD
t,λ, μ)
,X
j∈Cl
PjEnlog( aj+bjpD
t,j
aj+cjpD
t,j
)
bj≥cjo
+μ(PD
max −X
j∈Cl
PjEnpD
t,j
bj≥cjo)
+X
j∈Cl
λj(0− PjEnfj(pD
t,j −ηj)
bj≥cjo)
=X
j∈Cl
PjEnlog( aj+bjpD
t,j
aj+cjpD
t,j
)−μpD
t,j −λjfj(pD
t,j −ηj)
bj≥cjo
+μP D
max +X
j∈Cl
λj0
j.(52)
By applying the fact that for bj< cjwe have pD
t,j = 0, the
Lagrange function also can be written as follows:
LS(pD
t,λ, μ)
=X
j∈Cl
Enlog( aj+bjpD
t,j
aj+cjpD
t,j
)−μpD
t,j −λjfj(pD
t,j −ηj)o
+μP D
max +X
j∈Cl
λj. (53)
The Lagrange dual function is
gS(λ, μ) = max
pD
t<0LS(pD
t,λ, μ).(54)
To find optimal pD
t,j , for fixed values of λjand μand for j∈
Cl, in (54) we need to solve following optimization problem
for each j
pD∗
t,j (λj, μ) = arg max
pD
t,j ≥0log aj+bjpD
t,j
aj+cjpD
t,j !
−μpD
t,j −λjfj(pD
t,j −ηj).(55)
Note that (55) is only valid for bj≥cj, and for bj< cj
we have pD∗
t,j (λj, μ) = 0. Hence, it is a convex optimization
problem thus the optimal value can be found efficiently.
We next consider the following convex optimization prob-
lem to find the optimal λand μ:
min
λ<0,μ≥0gS(λ, μ).(56)
To solve (56), the subgradient method can be exploited. It is
easy to show that the subgradients are
∂λj=0
j− PjE{fj(pD∗
t,j (λj, μ)−ηj)bj≥cj}, j ∈Cl,
∂μ =PD
max −X
j∈ClPjE{pD∗
t,j (λj, μ)bj≥cj}.
Using (48) and (49), the subgradients can be re-written as
∂λj=−E{fj(pD∗
t,j (λj, μ)−ηj)}, j ∈Cl,(57)
9
∂μ =PD
max −X
j∈Cl
E{pD∗
t,j (λj, μ)}.(58)
We note that, (57) and (58) are exactly the same as (23) and
(24). After a sufficient number of iterations in the subgradient
method, the value of optimal lagrange multipliers, i.e., λ∗
j,
for all j∈Cl∗, and μ∗, are found. Then, for each channel
state, we solve the following optimization problem to find the
optimal power allocation:
pD∗
t,j (H) = arg max
pD
t,j ≥0log aj(H) + bj(H)pD
t,j
aj(H) + cj(H)pD
t,j !
−μ∗pD
t,j −λ∗
jeωj(pD
t,j −ηj(H)).(59)
Note that considering (57) and (58), the above formulation can
only be applied over a time-interval where we can assume the
channel for D2D users is stationary.
Remark. It is worth mentioning that since we have the same
constraints for both optimization problem S1and D1, the same
approach, as discussed in Section III-C, can be applied to sum-
rate maximization to improve the Chernoff bound.
Remark. Using the family of linear constraints in (30), we can
provide a closed-form solution for (55). Setting the derivative
of (55) equal to zero, for fj(pD
t,j −ηj) = ωj(pD
t,j −ηj)+1,
we have
ajbj−ajcj
(aj+bjpD
t,j )(aj+cjpD
t,j )−μ−λjωj= 0,(60)
i.e., we have (ωjλj+μ)bjcj(pD
t,j )2+ (ωjλj+μ)aj(bj+
cj)pD
t,j + ((ωjλj+μ)a2
j−ajbj+ajcj)=0or equivalently
κj(pD
t,j )2+βjpD
t,j + (1 −γj
(ωjλj+μ))=0where κj,bjcj
a2
j>0,
βj,bj+cj
aj>0and γj,bj−cj
aj>0. Summation of the
roots of this equation is −βj
κj, which is negative, so we have
at least one negative root. Hence, only the greater root can be
accepted or otherwise the solution for (55) is zero. Therefore,
we have
pD∗
t,j (λj, μ) = "−βj+qβ2
j−4κj(1 −γj
(μ+ωjλj))
2κj#+
.(61)
B. Time-Averaging Based Heuristic Solutions
Using the same idea of time-averaging instead of statistical
means, as we explained previously for D2, we propose the
following a heuristic solution for S3. In time slot n, we solve
the following optimization problem:
S5 : max
pD
t<0X
j∈Cl
log aj[n] + bj[n]pD
t,j
aj[n] + cj[n]pD
t,j !
subject to (33) and (34).
After applying the KKT optimality condition to the primal
problem [29], we have two cases:
Case 1) Pj∈Clη0
j[n]≤P0D
max:pD∗
t,j [n] = η0
j[n], for all j∈ Cl.
Case 2) Pj∈Clη0
j[n]> P 0D
max[n]: for all j∈ Clwe have
pD∗
t,j [n] = "−βj[n] + qβj[n]2−4κj[n](1 −γj[n]
μ[n])
2κj[n]#η0
j[n]
0
.(62)
In each time slot, μ[n]should be found such that
Pj∈ClpD
t,j [n]≤P0D
max[n]. This can be achieved using the
bisection method.
Since we have the same constraints for the optimization
problems D3and S5, all other discussions in Section III-E on
solving D3are valid for S5.
C. An Upper Bound for S1
Similarly to Section III-F, the optimization problem for
finding the upper bound can be written as follows
S6 : max
pD
t<0
EnX
j∈Cl
log( aj+bjpD
t,j
aj+cjpD
t,j
)o
subject to (40) and (41).
As discussed before, S6is not a convex problem and an
upper bound cannot be achieved using straight-forward convex
optimization methods. We instead use the dual problem to find
an upper bound. The Lagrange function of S6can be written
as
L2(pD
t,λ, μ)
,X
j∈Cl
Enlog( aj+bjpD
t,j
aj+cjpD
t,j
)o+μ(PD
max −X
j∈Cl
EnpD
t,j o)
+X
j∈Cl
λj(−1 + Ene−ωj(pD
t,j −ηj)o)
=X
j∈Cl
Enlog( aj+bjpD
t,j
aj+cjpD
t,j
)−μpD
t,j +λje−ωj(pD
t,j −ηj)o
+μP D
max +X
j∈Cl
λj(−1).(63)
The corresponding Lagrange dual function can be defined is
g2(λ, μ) = max
pD
t<0L2(pD
t,λ, μ).(64)
To find an optimal pD
t,j , for fixed values of λjand μand for
j∈Cl, in (64) we need to solve the following optimization
problem for each j:
pD∗
t,j (λj, μ) = arg max
pD
t,j ≥0log aj+bjpD
t,j
aj+cjpD
t,j !
−μpD
t,j +λje−ωj(pD
t,j −ηj).(65)
Noting that (65) is not a convex optimization problem, we
apply an extremum-search method to find the global optimum
point. After finding the global maximum point, we can use the
subgradient method to find the optimal Lagrange multipliers.
V. DISCUSSION ON GENERALIZATIONS
A. Multi-D2D Multi-cell Scenario
The proposed sum-rate and D2D rate maximization frame-
work can be applied to a realistic multi-cell network with
multiple D2D pairs and cellular users in each cell. In general,
each D2D user has its own arrival time and duration of stay
in the network. Upon the arrival of any D2D user, a scheduler
within each cell (which can reside within the eNB) can decide
10
the resources to be allocated to the D2D user. To avoid large
overhead and to decrease the computational complexity, the
resource allocation decision for each D2D user can be made
separately by the scheduler, which is in harmony with the
fact that the nodes in a cellular system have different arrival
time and duration of stay in the network. Furthermore, by
implementing the same algorithms (i.e. sum-rate maximiza-
tion or D2D rate maximization) in all neighboring cells, we
guarantee all the constraints in (7) and (8), i.e., sum-power
constraint for all D2D users and a pre-defined SINR level
for all cellular users in the network, while optimizing the
throughput performance separately for each cell.
However, in the multi-cell scenario, applying the subgra-
dient method for sum-rate and D2D rate maximization may
cause some difficulties. As we notice before, the subgradient
method can be applied only under the assumption that the
signal and interference channels to the D2D user under con-
sideration is stationary. In the multi-cell scenario, the entry
of a new D2D user into the network changes the interference
distribution for all other D2D users in the neighboring cells
which use the same RBs as allocated to the new user. In that
case, the stationarity assumption is not valid anymore. Only
in a very crowded network, or in a large cell with D2D users
that are not very close to the edges of the cell, can we assume
that the interference to D2D users is approximately stationary.
Note that for the proposed time-averaging heuristic solution
does not require such a stationarity assumption. Therefore,
it can be applied to any multi-cell network efficiently while
guaranteeing all the constraints in (7) and (8) for all users in
the network.
B. More Generalized Optimization Problem
So far, we have considered the optimization problems in D1
and S1with long-term sum-power constraint (8) on the D2D
users. The constraint in (8) addresses the battery usage of the
D2D users. In reality, the instantaneous power may also be
confined due to physical design of the mobile device such as
power amplifier and antenna power limitations. We can add a
short-term power constraint as follows:
X
j∈Cl
pD
t,j ≤PD
max,s.(66)
The same procedure as described in this paper can be used
to solve the new optimization problems with this additional
constraint. In particular, for the D2D-rate optimization prob-
lem, to find the optimal power allocation we need to solve the
maximization problem in (20) subject to (66). To solve the
sum-rate optimization problem, the maximization problem in
(55) must be solved subject to constraint in (66). Since (66) is
a linear constraint, solving (20) and (55) under this constraint
still admits efficient convex optimization methods.
VI. SIMULATION EVALUATION
We consider a cellular network with 10 cellular users in
each cell and N= 10 distinct RBs are assigned to each
cellular user. Cellular and D2D users are uniformly randomly
TABLE II: Default values for simulation parameters
N= 10
Number of RBs per cellular user = 10
RB bandwidth = 12x15 KHz
Cell radius (Rc) = 100 m
D2D distance (d) = 20 m
Number of D2D pairs in each cell (Nd)=7
Ave. cellular SNR = 30 dB
Free space path-loss factor = 3.5
Standard deviation for shadowing = 6 dB
distributed over each cell. We apply a power control mecha-
nism which compensates for the free-space path loss effects
for all cellular users. For the link between any two nodes, we
use a simple path loss model K0D−α, where we set the path
loss constant Ko= 0.01, and the pass loss exponent α= 3.5.
We assume Rayleigh fading for all interference, cellular and
D2D links. The default values of different system parameters
are presented in Table II.
In each cell, when there are multiple active D2D pairs
requesting to share RBs, they are queued based on a first-
come-first-serve rule. We solve the proposed optimization
problems for the D2D pairs one by one based on their order in
the queue 2. To allocate RBs to each D2D pair, first we need
to find all available RBs that provide a non-empty feasible set
for each optimization problem. Let us assume that C∗
lis the
set of all available RBs such that we have (11)-(12) satisfied.
We need to find Cl⊆ C∗
lsuch that |Cl| ≤ Nl. To find the
optimal Clwe would need to solve N∗
l
Nloptimization problems
where N∗
l=|C∗
l|(assuming N∗
l≥Nl). Instead, we use the
following heuristic method for a more practical solution. Using
the Markov’s inequality instead of the probabilistic constraints
in (7) yields
EnpD
t,j −ηj+ 1o≤. (67)
Equivalently, we have
EnpD
t,j o≤Enηjo+−1.(68)
This means that the larger E{ηj}value, the larger the feasible
set for the optimization problems S1and D1. Therefore, by
sorting the values of Enηjofor all j∈ C∗
l, the lth D2D user
be opportunistically assigned the RBs with the highest E{ηj}
such that (11)-(12) are satisfied. It is worth mentioning that
other methods with higher complexity can be adapted for RB
allocation., e.g. the readers may consider [31]- [32].
For each data point, we average the results over a large
number of random positions of cellular and D2D users and
also random channel realizations. To avoid redundancy, we
only present the results for the uplink case, in Figs. 2-5.
A. Efficacy of Convexification
In this section we analyze the efficacy of applying the
convexification approach we proposed in Section III-A and
IV-A to solve the ergodic sum-rate and D2D rate optimization
problems. To focus on convexification and remove the effects
2For alternative approaches the readers may consider [30]
11
-40 -30 -20 -10 0 10 20
0
2
4
6
8
10
12
14
Avg. D2D Rate (bits/s/Hz)
P max
D (dBm)
Non-Ergodic
Time-Ave. Heuristic
Linear Cons. Subgradient
Subgradient (imperfect ch. est.)
Chernoff-Bound Subgradient
Upper-Bound
(a) Throughput of D2D users under D2D rate maximization,
-40 -30 -20 -10 0 10 20
4
6
8
10
12
14
16
Avg. Sum Rate (bits/Hz)
P max
D (dBm)
Non-Ergodic
Time-Ave. Heuristic
Linear Cons. Subgradient
Subgradient (imperfect ch. est.)
Chernoff-Bound Subgradient
Upper-Bound
No D2D
(b) Throughput of the main cell under sum-rate maximization.
Fig. 2: The effect of convexification on throughput in a single cell scenario.
of inter-cell dynamics, in this subsection, we only consider
the single-cell scenario. We compare the performance of the
proposed solution methods with the upper-bounds developed
in Sections III-F and IV-C, and also with a naive non-ergodic
huristic method. For the non-ergodic approach, similar to [27]
and [28], instead of long-term constraints in (7) and (8), we
use short-term constraints, i.e., we set = 0 and we use the
constraint Pj∈ClpD
t,j ≤PD
max as the sum-power constraint.
We set ζintra
j,min = 3dB, for all j. Also, for ergodic methods we
set = 0.05.
Fig. 2 investigates the effects of changing PD
max on the D2D
and cell throughput. Fig. 2(a) shows the average D2D rate per
RB for the cell under the D2D-rate maximization objective,
and Fig. 2(b) shows the average D2D-cellular sum rate per
RB for the cell under the sum-rate maximization objective.
It can be seen that through applying the convexification
method, we reach a small gap between the upper-bound and
the Chernoff-bound and linear-constraint approximations. It is
worth mentioning that by increasing PD
max, the percentage of
this gap, for specially D2D rate, is decreasing.
The Chernoff-bound method provides slightly higher
throughput than the linear-constraint method. This is because
we can improve the Chernoff-bound through the iterative
method proposed in Section III-C. On the other hand, the
linear-constraint method offers lower computational complex-
ity, due to its semi-closed-form solution. For the heuristic time-
averaging method, the Chernoff-bound constraints and the lin-
ear constraints lead to solutions with the same computational
complexity and negligible performance gap, so only one curve
is shown. We observe that the performance gap between the
time-averaging heuristic and the subgradient methods is less
than 17%, with drastically reduced computational complexity.
Finally, Fig. 2 shows that for D2D power more than −10dBm,
which is almost always true in practice, the performance
gap between ergodic and non-ergodic power allocation is
considerable, indicating the need for the proposed stochastic
optimization solutions.
Besides the performance under perfect CSI, in Fig. 2, we
also consider the performance of the convexification method
TABLE III: MATLAB Run-Time for Different Algorithms
Non-Ergodic 5.91 s
Time-Ave. Heuristic 5.92 s
Linear Cons. Subgradient 1302.89 s
Chernoff-Bound Subgradient 1772.59 s
with the Chernoff bound using imperfect CSI where channel
estimation errors present. We model the channel estimation
error as a complex Gaussian noise with zero mean and variance
being 5% of the corresponding true channel variance. As can
be seen from Figs. 2(a) and (b), the rate loss by using imperfect
CSI for power allocation is less than 10%.
In Table III, we compare the computational complexity of
the methods discussed in this paper based on the MATLAB
run time of simulating 900 LTE frames (equivalent to 4.5
seconds) under all algorithms. It can be seen from Table III
that the proposed time-averaging heuristic is nearly identical
to the naive non-ergodic method in run time and it is around
300 times faster than the standard subgradient methods with
Chernoff-bound or linear constraints.
Note that all these methods have the same communication
complexity. In general, beside a common overhead in LTE
that is necessary to estimate CSI and the interference level,
to calculate ηj, channel and interference feedback is required
over each time slot. To avoid a large amount of information
exchange, we can interpret power constraint (7) as per-RB
power constraints set by the eNB in the main cell. In other
words, ηj, for all j∈Cl, is set by the eNB such that we
have SINR constraints (4)-(5) satisfied. In this case, each D2D
transmitter directly receives the value of ηj, for j∈Cl, from
the eNB of its own cell. Furthermore, after the initial setting of
ηjby the eNB, any change in the value of ηjcan be reported
using limited feedback, e.g., using differential coding.
B. Multi-cell Performance Comparison
Under the multi-cell scenario, we compare the proposed
heuristic solution with the non-ergodic approach. As default
values, we set PD
max =−8.5dBm,ζintra
j,min =ζ(k)
j,min =−3dB,
12
-30 -20 -10 0 10 20
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Avg. D2D Rate (bits/s/Hz)
P max
D (dBm)
Ergodic D2D-Rate Max. Method
Ergodic Sum-Rate Max. Method
Non Ergodic D2D-Rate Max. Method
Non Ergodic Sum-Rate Max. Method
(a) Achieved throughput for D2D users,
-30 -20 -10 0 10 20
2
2.5
3
3.5
4
Avg. Sum Rate (bits/s/Hz)
P max
D (dBm)
Ergodic D2D-Rate Max. Method
Ergodic Sum-Rate Max. Method
Non Ergodic D2D-Rate Max. Method
Non Ergodic Sum-Rate Max. Method
No D2D
(b) Achieved throughput in the main cell.
Fig. 3: The achieved throughput vs. PD
max.
-2 0 2 4 6 8
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Avg D2D Rate(bits/s/Hz)
ζ min (dB)
Ergodic D2D-Rate Max. Method
Ergodic Sum-Rate Max. Method
Non Ergodic D2D-Rate Max. Method
Non Ergodic Sum-Rate Max. Method
(a) Achieved throughput for D2D users,
-2 0 2 4 6 8
1.9
2
2.1
2.2
2.3
2.4
2.5
2.6
Avg Sum Rate(bits/s/Hz)
ζ min (dB)
Ergodic D2D-Rate Max. Method
Ergodic Sum-Rate Max. Method
Non Ergodic D2D-Rate Max. Method
Non Ergodic Sum-Rate Max. Method
No D2D
(b) Achieved throughput for the main cell.
Fig. 4: The achieved throughput vs. minimum required SINR.
100 150 200 250 300
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
Avg D2D Rate(bits/s/Hz)
Rc
Ergodic D2D-Rate Max. Method
Ergodic Sum-Rate Max. Method
Non Ergodic D2D-Rate Max. Method
Non Ergodic Sum-Rate Max. Method
(a) Achieved throughput for D2D users,
100 150 200 250 300
1.8
2
2.2
2.4
2.6
2.8
3
3.2
3.4
3.6
Avg Sum Rate(bits/s/Hz)
Rc
Ergodic D2D-Rate Max. Method
Ergodic Sum-Rate Max. Method
Non Ergodic D2D-Rate Max. Method
Non Ergodic Sum-Rate Max. Method
No D2D
(b) Achieved throughput for the cell.
Fig. 5: The achieved throughput for different cell radius.
13
for all jand for all k∈ Sj, unless otherwise specified. For
ergodic methods we set = 0.1.
1) Maximum Power for D2D Users: Fig. 3 shows the
effects of changing PD
max on the D2D and cell throughput. It
can be seen that, by increasing PD
max, at first the D2D-rate and
sum-rate increase, but after some point they start to degrade.
This happens because all neighboring cells use the same power
allocation algorithm, and by increasing the D2D power, we
increase the interference to and from the neighboring cells.
2) Minimum SINR Requirement for Cellular Users: Fig. 4
shows the effect of changing the the minimum required SINR
on the D2D and cell throughput. It can be seen that, by
increasing the minimum required SINR for the cellular users,
there is less room for controlling the power of D2D users
and this decreases the D2D throughput and the total cell
throughput.
3) Cell Size: Fig. 5 shows the effect of changing the cell
radius on the D2D and cell throughput. It can be seen from
that, by increasing the cell radius, we see an increase and
then a decrease in the throughput. In fact by increasing the
cell radius, we have two different effects. The cellular and
D2D users are spread over a larger area and thus the distance
between interfering users is decreased, but on the other hand
because of the uplink power control method for cellular users,
their power increases, leading to more interference to D2D
users.
VII. CONCLUSION
In this paper, we have considered optimal power allocation
by the D2D users in a cellular network for underlay D2D
communications in order to maximize the D2D rate and the
sum rate between D2D and cellular users. The proposed
optimization problems accommodates a long-term sum-power
constraint and probabilistic individual power constraints over
each accessible RB. This enables consideration for battery
energy limits at D2D transmitters and the interference created
by D2D communications. To solve the optimization problem,
several approximate convex constraints are introduced, as
replacement for the non-convex probabilistic individual power
constraints. After such convexification, optimal solutions to the
approximate D2D-rate and sum-rate maximization problems
are developed, which are shown to give throughput perfor-
mance that is close to an upper bound. We then propose a
heuristic method by using time-averaging to approximate for
long-term measures. The time-averaging heuristic method has
low computational complexity, and it can be easily applied to
the multi-cell scenario. Through simulation we observe that
the performance gap between the standard subgradient solution
and the proposed time-averaging heuristic method is less than
17%, with drastically reduced computational complexity for
the heuristic method.
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Ruhallah AliHemmati Ruhallah AliHemmati re-
ceived the B.Sc. degree in electrical engineering
from University of Tehran, Tehran, Iran, in 2002,
and the M.Sc. and Ph.D. degrees in telecommuni-
cation systems enngineering from Tarbiat Modares
University, Tehran, Iran, in 2004 and 2008, respec-
tively. From 2003 to 2005, he was with the Iranian
Telecommunication Research Center, and from 2005
to 2007, he was with the Advanced Communications
Research Institute, Tehran, Iran. From 2008 to 2009,
he was with the Advanced Multi-Dimensional Signal
Processing Laboratory, Queens University, Kingston, Canada, as a Visiting
Researcher. From 2012 to 2014, he was with University of Ontario Institute
of Technology (UOIT), Oshawa, Canada, as a Post-Doctoral Researcher. From
2014 to 2016 he was with University of Toronto, Toronto, Canada as a
Post-Doctoral Researcher. His main research interests are statistical signal
processing, detection and estimation theory, and relay networks.
Min Dong Min Dong (S’00-M’05-SM’09) received
the B.Eng. degree from Tsinghua University, Bei-
jing, China, in 1998, and the Ph.D. degree in electri-
cal and computer engineering with minor in applied
mathematics from Cornell University, Ithaca, NY, in
2004. From 2004 to 2008, she was with Corporate
Research and Development, Qualcomm Inc., San
Diego, CA. In 2008, she joined the Department
of Electrical, Computer and Software Engineering
at University of Ontario Institute of Technology,
Ontario, Canada, where she is currently an Associate
Professor. Her research interests are in the areas of statistical signal processing
for communication networks, cooperative communications and networking
techniques, and stochastic network optimization in dynamic networks and
systems.
Dr. Dong received the the 2004 IEEE Signal Processing Society Best
Paper Award, the Best Paper Award at IEEE ICCC in 2012, and the Early
Researcher Award from Ontario Ministry of Research and Innovation in 2012.
She is the co-author of the Best Student Paper Award of Signal Processing
for Communications and Networks in IEEE ICASSP’16. She served as an
Associate Editor for the IEEE TRANSACTIONS ON SIGNAL PROCESSING
(2010-2014), and as an Associate Editor for the IEEE SIGNAL PROCESSING
LETTERS (2009-2013). She was a symposium lead co-chair of the Commu-
nications and Networks to Enable the Smart Grid Symposium at the IEEE
International Conference on Smart Grid Communications (SmartGridComm)
in 2014. She has been an elected member of IEEE Signal Processing Society
Signal Processing for Communications and Networking (SP-COM) Technical
Committee since 2013.
Ben Liang Ben Liang received honors-simultaneous
B.Sc. (valedictorian) and M.Sc. degrees in Electrical
Engineering from Polytechnic University in Brook-
lyn, New York, in 1997 and the Ph.D. degree in
Electrical Engineering with a minor in Computer
Science from Cornell University in Ithaca, New
York, in 2001. In the 2001 - 2002 academic year,
he was a visiting lecturer and post-doctoral research
associate with Cornell University. He joined the
Department of Electrical and Computer Engineering
at the University of Toronto in 2002, where he is
now a Professor. His current research interests are in networked systems and
mobile communications. He has served on the editorial boards of the IEEE
Transactions on Mobile Computing since 2017 and the IEEE Transactions on
Communications since 2014, and he was an editor for the IEEE Transactions
on Wireless Communications from 2008 to 2013 and an associate editor
for Wiley Security and Communication Networks from 2007 to 2016. He
regularly serves on the organizational and technical committees of a number
of conferences. He is a senior member of IEEE and a member of ACM and
Tau Beta Pi.
Gary Boudreau GARY BOUDREAU [M’84-
SM’11] received a B.A.Sc. in Electrical Engineering
from the University of Ottawa in 1983, an M.A.Sc.
in Electrical Engineering from Queens University in
1984 and a Ph.D. in electrical engineering from Car-
leton University in 1989. From 1984 to 1989 he was
employed as a communications systems engineer
with Canadian Astronautics Limited and from 1990
to 1993 he worked as a satellite systems engineer
for MPR Teltech Ltd. For the period spanning 1993
to 2009 he was employed by Nortel Networks in
a variety of wireless systems and management roles within the CDMA
and LTE basestation product groups. In 2010 he joined Ericsson Canada
where he is currently employed in the LTE systems architecture group. His
interests include digital and wireless communications as well as digital signal
processing.
15
Hossein Seyedmehdi Hossein Seyedmehdi received
a Master’s degree from the National University of
Singapore in 2008 and a PhD degree from the
University of Toronto in 2014 both in Electrical
and Computer Engineering. He is currently affiliated
with Ericsson Canada where he is working on the
5thgeneration (5G) of wireless technologies includ-
ing the commercialization of massive MIMO. He
has numerous papers in the area of wireless systems
in addition to more than 10 patents. His research
interests include Information Theory for Wireless
Communications, Signal Processing for MIMO channels, Algorithms for
Radio Access Networks, and Future Radio Technologies.
