multi-channel resource allocation towards ergodic rate ...achieved d2d rate and the interference to...
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Multi-channel Resource Allocation towards ErgodicRate Maximization for Underlay Device-to-Device
CommunicationsRuhallah 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 toregular cellular users. Maximizing the performance of under-lay D2D communications requires joint consideration for theachieved D2D rate and the interference to cellular users. Inthis work, we consider the D2D power allocation optimizationover multiple resource blocks (RBs), aiming at maximizing theeither the ergodic D2D rate or the ergodic sum rate of D2D andcellular users, under the long-term sum-power constraint of theD2D users and per-RB probabilistic signal-to-interference-and-noise (SINR) requirements for all cellular users. We formulatestochastic optimization problems for D2D power allocation overtime. The proposed optimization framework is applicable to bothuplink and downlink cellular spectrum sharing. To solve theproposed stochastic optimization problems, we first convexifythe problems by introducing a family of convex constraints as areplacement for the non-convex probabilistic SINR constraints.We then present two dynamic power allocation algorithms: aLagrange dual based algorithm that is optimal but with ahigh computational complexity, and a low-complexity heuristicalgorithm based on dynamic time averaging. Through simulation,we show that the performance gap between the optimal andheuristic algorithms is small, and effective long-term stochasticD2D power optimization over the shared RBs can lead tosubstantial 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 localcommunications nature, D2D communication can be providedwith 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 theNatural Sciences and Engineering Research Council (NSERC) of CanadaCollaborative Research and Development Grant CRDPJ-466072-14, and inpart by NSERC Discovery Grants.
R. AliHemmati and B. Liang are with the Department of Electrical andComputer Engineering, University of Toronto, Toronto, Ontario M5S 3G4,Canada (e-mail: [email protected]; [email protected]).
M. Dong is with the Department of Electrical, Computer and SoftwareEngineering, University of Ontario Institute of Technology, Oshawa, OntarioL1H 7K4, Canada (e-mail: [email protected]).
G. Boudreau and S. H. Seyedmehdi are with Ericsson Canada,Ottawa, Ontario, Canada (e-mail: [email protected];[email protected]).
A preliminary version of this work [1] was presented at the 2016IEEE International Workshop on Signal Processing Advances in WirelessCommunications (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-basedpublic safety networks for its security and reliability [5]. Ad-ditionally, D2D communications is necessary for the scenarioswhere cellular transmission is not accessible [4].
To facilitate D2D communication, there are different chal-lenges which should be addressed carefully. A survey on thechallenges and proposed solutions for D2D communicationscan be found in [6]. In particular, sharing cellular recoursesbetween D2D and regular cellular users may cause intra-celland inter-cell interference. One possible option is to allocatedifferent resources for cellular and D2D communications, i.e.overlay D2D communications. However, to achieve the highestpossible spectral efficiency, underlay D2D communicationshas attracted more attention in the literature, where D2D andcellular users within a cell share the same spectrum resourceand hence interfere with each other. In this paper we mainlyfocus on underlay D2D communications.
Underlaying requires effective interference management andresource sharing among all users. Many methods have beenpresented in the literature to address these problems. Forexample, Graph-based [7], [8] and game theoretic frameworks[9]–[14] were considered. Power back-off approaches wereinvestigated in [15]–[17], and an interference cancelationmethod was proposed in [18]. These works do not directlyaddress the optimization of spectrum resource and powerallocation in D2D communications.
Closer to our interest, resource and power optimizationmethods have been proposed in [19]–[26] to maximize theD2D rate, D2D-cellular sum rate, or power-rate efficiency. Anoptimal power allocation solution for D2D users underlayingcellular users in downlink transmission was given in [19].The solutions in [19] were achieved without imposing anyconstraint on the D2D power. In [20], a solution to encom-pass mode selection, resource allocation, and power controlwithin a single framework was proposed. An energy efficientpower control design for resource sharing between cellularand 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 efficiencyunder signal-to-noise ratio (SNR) constraints were provided in[23]. The authors of [24] and [25] proposed sub-optimal powerallocation solutions for D2D users in uplink transmission,
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which divide the original problem into several easier sub-problems. In [26], an optimal power allocation method basedon maximizing application-dependent weighted cell utility wasproposed.
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 D2Dnode accesses only a single channel at a time. They cannot bedirectly applied to the multi-channel scenario that is prevalentin most practical systems, such as supporting multiple RBsin an LTE network. Second, [20]–[26] considers only short-term power constraints. Yet, D2D nodes are often powered bybatteries with limited energy storage capacity, which directlycorresponds to long-term D2D power constraints. Further-more, long-term D2D power allocation on individual RBs cangive probabilistic guarantees on the interference from D2Dtransmitters to cellular users over the shared RBs. These areimportant characteristics of D2D communications that requirefurther investigation beyond [19]–[26]. In [27] and [28], wesolved the D2D-cellular sum-rate maximization problem overmultiple RBs. However, the solution was short-term withregard 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 theergodic D2D-cellular sum rate by optimizing the power alloca-tion of the D2D users, under the long-term power constraint onthe D2D users and per-RB probabilistic SINR constraints forall cellular users. The combination of long-term power andSINR constraints with multi-channel communications leadsto a complicated non-convex stochastic optimization problem.Building on our preliminary results presented in [1], the maincontributions of this paper are as follows:
• We present a study on ergodic rate maximization withlong-term power constraints and per-RB probabilisticSINR constraints in D2D communications. To address thenon-convexity in our optimization problem, we proposea family of convex constraints that provides upper andlower bounds for the non-convex probabilistic SINRconstraints. In particular, using the Chernoff bound, wefurther propose a method to reduce the gap between theprobabilistic constraint and its convex replacement.
• Subsequently, to further convexify the D2D-cellular sum-rate maximization problem, we replace the objective by afunction which, depending on the values of parameters, iseither convex and decreasing, or concave and increasing.For the convex decreasing case, we show that optimalallocated power is zero, while for the concave decreasingcase, we obtain a convex optimization problem.
• To solve the resulting convex optimization problem, wepropose two dynamic algorithms for power allocationover time. The first algorithm is based on the Lagrangeduality which provides the optimal power levels overall RBs at each time slot. However, the computationalcomplexity of this algorithm can be prohibitive when thechannel state space is large. Therefore, we propose analternative heuristic algorithm based on dynamic timeaveraging, which drastically reduces the computational
Cu1
eNB1
Cu2
pDt,j|hj|2 Ij
p(k)r,j
pCr,j
pDt,j|h
I,(k)j |2
pDt,j|h
Ij|
2I0j
Dut Dur
eNBkI0,(k)j
Fig. 1: A cellular network with underlaying D2D users inuplink resource sharing. Dut and Dur: transmit and receivenodes of a D2D pair, respectively. Cu: cellular users. Solidand 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 toreformulate the original problems to derive an upperbound of the original problems for comparison.
• Finally, we show that the proposed algorithms are easilyscalable and can be applied to more general cases withmultiple cells and additional power constraints.
The rest of the paper is organized as follows. Section IIpresents the system model of the cellular network used inthis paper and the resource allocation problem is defined inthis section. The proposed methods for solving the D2D-rate maximization problem and the sum-rate maximizationproblem are presented in Section III and Section IV. InSection V, we discuss extensions of the proposed methodto multi-cell scenarios and to accommodate additional powerconstraints. Section VI presents the simulation results. SectionVII concludes the paper.
Notations: We use italic fonts and boldface small letters torepresent scalar variables and vectors respectively. The nota-tion a < 0 means all entries of vector a are nonnegative. Wedefine
[x]ba
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, σ2) to denote a Gaussian randomvariable with mean m and variance σ2.
II. SYSTEM MODEL AND PROBLEM DEFINITION
A. System Model
We consider a cellular system consisting of multiple cellularusers and D2D users underlaying the cellular users. Weassume that an idle D2D pair arrives at the cell of interestrequesting access to spectrum for D2D communications. Dueto the localized and low-power transmission of D2D users,we assume the resource planning (e.g., spectrum allocationand power control) of existing cellular users in the networkis not modified. As a practical representation of cellularcommunications, e.g., LTE networks, we assume multiple RBs
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TABLE I: Notation Definition
N number of active cellular users in each cellC set of all available RBs in the cellCl set of allocated RBs to the lth D2D pairSj set of neighboring cellular users using RB jpDt,j D2D transmitted power over RB j
pCr,j cellular user received power over RB j
p(k)r,j neighboring cellular user received power over the RB j (for k ∈ Sj )
Ij D2D received interference power over RB jpDt,j |h
Ij |
2 cellular user received interference power over the RB j from the new D2D pair
pDt,j |h
I,(k)j |2 neighboring cellular user received interference power over the RB j (for k ∈ Sj ) from the new D2D pair
hj D2D channel coefficient over RB jI0j cellular user received interference over RB j before entering the new D2D user
I0,(k)j neighboring cellular user received interference over RB j (for k ∈ Sj ) before entering the new D2D user
PDmax maximum available power for a D2D pair
ζintraj,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 )
σ2 noise power over each RB
are allocated to each user in the network. Since the D2D de-vices use licensed cellular spectrum, we assume that resourceallocation is centrally controlled by the cellular operator. Inparticular, the RBs are allocated to the cellular and D2D usersby the Evolved Node B (eNB). Furthermore, we assume thatchanges to RB allocation occur at a time scale much largethan power allocation, so that when considering the powerallocation problem, the RB allocation is viewed as fixed. Thereis no intra-cell interference among cellular users in a cellbecause of orthogonal assignment of RBs to the cellular users.However, due to frequency reuse at neighboring cells, thesecellular users suffer from inter-cell interference. Fig. 1 showsthe interference scenarios for a cellular network with D2Dusers in uplink resource sharing. The proposed algorithms canbe similarly applied to the alternate case of downlink spectrumsharing.
We assume that there are N active cellular users in eachcell. A D2D pair attempts to reuse the assigned RBs of activecellular users in the cell and C is the set of all available RBswithin the cell. Let Cl indicate the set of allocated RBs to thelth D2D pair. For j ∈ Cl, let pD
t,j denote the transmit powerof the D2D pair over the jth RB and pC
r,j denote the receivedpower from the unique cellular user that is assigned to thejth RB. In addition, let Sj denote the set of all cellular usersin the neighboring cells that are using the jth RB. Let p
(k)r,j
denote the received power from the kth user in Sj over thejth RB.
The cellular users have both intra-cell interference from theD2D transmission and inter-cell interference from neighboringcells. For j ∈ Cl, let I0
j and I0,(k)j denote the received
interference power over the jth RB for the correspondingcellular user in the main cell and the kth neighboring user,respectively, excluding the interference from the D2D pairunder consideration. For the uplink sharing, let |hI
j |2 and
|hI,(k)j |2 denote the channel power gains over the jth RB
between the D2D transmitter and the eNB and between theD2D transmitter and the kth neighboring cellular user’s eNB,for k ∈ Sj , respectively (for the downlink case, the samenotation can be used, except that the eNBs are replaced by
the corresponding cellular users.). Furthermore, let Ij denotethe received interference power over the jth RB at the D2Dreceiver. And finally, let hj denote the D2D channel coefficientover the jth RB. Under the fading environment, all channelpower gains and interference power are random variables. Thenotation used throughout this paper is summarized in Table I.
B. Ergodic Rate Optimization Problem
For the uplink transmission, the received SINR of thecellular user over the jth RB at the eNB in the main cell,at the eNB of the kth neighboring cellular user in Sj , and atthe D2D receiver are respectively given by 1
SINRCj =
pCr,j
σ2 + I0j + pD
t,j |hIj |
2, (1)
SINRC,(k)j =
p(k)r,j
σ2 + I0,(k)j + pD
t,j |hI,(k)j |2
, k ∈ Sj , (2)
SINRDj =
|hj |2pDt,j
σ2 + Ij. (3)
In order to maintain the quality of service for the cellularusers at a specific level, it is important to control the inter-ference from the D2D transmitter to the cellular users in themain cell and also in the neighboring cells. Therefore, the D2Dpower over each RB must be confined. We first consider thefollowing constraints
Pr{
SINRCj ≤ ζ intra
j,min
}≤ ε, j ∈ Cl (4)
Pr{
SINRC,(k)j ≤ ζ
(k)j,min
}≤ ε, j ∈ Cl, k ∈ Sj (5)
where ζ intraj,min and ζ
(k)j,min are minimum SINR targets for the
cellular user in the main cell and the kth neighboring cellularuser in Sj , respectively. These constraints guarantee a specificlong-term QoS for the cellular users in the main cell and
1For the downlink, SINR is defined by replacing the eNB with the cellularuser.
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neighboring cells. We define
ηj , min
{pCr,j/ζ intra
j,min − (σ2 + I0j )
|hIj |
2,
{p(k)r,j /ζ
(k)j,min − (σ2 + I
0,(k)j )
|hI,(k)j |2
}
k∈Sj
. (6)
It is easy to show that (4) and (5) are equivalent to thefollowing constraint:
Pr{
pDt,j ≥ ηj
}≤ ε. (7)
Furthermore, in order to limit power usage for the D2D user,we additionally consider a long-term sum-power constraint forthe D2D pair as follows:
E{∑
j∈Cl
pDt,j
}≤ PD
max. (8)
The statistical constraints on the D2D transmission powerin (7) and (8) are more practical than the deterministic onescommonly assumed in the literature [19], [23]–[26]. Insteadof imposing instantaneous, strict SINR and power constraintsin each time slot, we allow their fluctuations over time.Constraint (7) models long-term QoS requirements, while theconstraint (8) corresponds to the need to conserve energyespecially for battery-powered D2D equipment. The resultantadditional degree of freedom in dynamic adjustment of theD2D transmission power, tailored to the time-varying channelconditions, can lead to substantial gains in the ergodic D2Drate and D2D-cellular sum rate. This will be numericallydemonstrated in Section III-E and IV-B, where we comparethe cases where the D2D transmission power is properlydesigned over time under statistical constraints, and where itis deterministically bounded in each time slot. Furthermore, inSection V-B, we will discuss how the proposed solution canbe easily extended to the case where there are both statisticaland deterministic constraints on the D2D transmission power.
Thus, in this paper, we study the following two stochasticpower allocation problems to find the optimal power in eachtime slot over each RB for the new D2D pair:
I) Ergodic D2D-Rate Maximization Problem
D1 : maxpDt <0
E{∑
j∈Cl
log(1 + SINRDj )}
subject to (7) and (8);
II) Ergodic Sum-Rate Maximization Problem
S1 : maxpDt <0
E{∑
j∈Cl
log(1 + SINRCj ) + log(1 + SINRD
j )}
subject to (7) and (8),
where we define pDt = [pD
t,1, ∙ ∙ ∙, pDt,|Cl|
]T . Note that, in theabove optimization problems, the transmission power pD
t isa mapping from the random channel state vector to a powerallocation vector.
C. Feasibility Check
Consider the SINR constraints in (4)-(5). The feasible setis non-empty only if we have
Pr{ pC
r,j
σ2 + I0j
≤ ζ intraj,min
}≤ ε, j ∈ Cl, (9)
Pr{ p
(k)r,j
σ2 + I0,(k)j
≤ ζ(k)j,min
}≤ ε, j ∈ Cl, k ∈ Sj . (10)
For example, in the case that all signal and interface powersare exponentially distributed, the constraints in (4)-(5) areequivalent to
1 −e−λC
p,jζintraj,minσ2
1 +λC
p,jζintraj,min
λI,j
≤ ε, j ∈ Cl, (11)
1 −e−λ
(k)p,jζ
(k)j,minσ2
1 +λ
(k)p,jζ
(k)j,min
λ(k)I,j
≤ ε, j ∈ Cl, k ∈ Sj , (12)
where λCp,j , λ
(k)p,j , λI,j and λ
(k)I,j are the rates of exponentially
distributed random variables pCr,j , p
(k)r,j , I0
j and I0,(k)j , respec-
tively.
III. D2D RATE MAXIMIZATION
The stochastic optimization problem D1 can be reformu-lated as an equivalent deterministic optimization problem,in which all expectations in the optimization problem D1can be written as probability-weighted sums of functions ofrealizations of the random channel state vector over all RBs. Inthis reformulation, the decision variable is pD
t,j correspondingto every realization of the channel state vector. However, thecomplexity of directly solving such an optimization problemwould be prohibitive, due to the exponential size of the multi-dimensional channel state space. Instead, we propose to firstconvexify the optimization problem D1. We will then showthat using Lagrange multipliers, the problem of finding pD
t,j
can be solved separately over each observed realization ofchannel states.
A. Convexification of Problem D1
The probabilistic individual power constraint in (7) is notconvex. We consider instead stronger convex constraints usingthe following lemma.
Lemma 1. For any strictly increasing function f(∙) such thatf(0) = 1, we have
Pr{
pDt,j ≥ ηj
}≤ E
{f(pD
t,j − ηj)}
. (13)
Proof: Since f(∙) is a strictly increasing function, we have
Pr{
pDt,j ≥ ηj
}= Pr
{pDt,j − ηj ≥ 0
}
= Pr{
f(pDt,j − ηj) ≥ f(0)
}
≤ E{
f(pDt,j − ηj)
}, (14)
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where the last inequality is achieved by applying Markov’sinequality 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 willuse only convex increasing functions. We propose substitutingthe following constraint for the constraint in (7):
E{
fj(pDt,j − ηj)
}≤ ε, (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 D1 by using (15) andsolving the new convex optimization problem:
D2 : maxpDt,j<0
E{∑
j∈Cl
log
(
1 +|hj |2pD
t,j
σ2 + Ij
)}
subject to E{∑
j∈Cl
pDt,j
}≤ PD
max (16)
E{
fj(pDt,j − ηj)
}≤ ε, j ∈ Cl. (17)
B. Solution via the Lagrange Method
The Lagrange function of D2 can be written as
LD(pDt , λ, μ)
=∑
j∈Cl
E{
log
(
1 +|hj |2pD
t,j
σ2 + Ij
)
− μpDt,j − λjfj(p
Dt,j − ηj)
}
+ μPDmax +
∑
j∈Cl
λjε (18)
where λ = [λ1, ∙ ∙ ∙, λ|Cl|]T is a vector of Lagrange multipliers.
The corresponding Lagrange dual function is
gD(λ, μ) = maxpDt <0
LD(pDt , λ, μ). (19)
To find the optimal pDt,j , for fixed values of λj and μ, we
need to solve the following optimization problem for each jand each channel realization of the jth RB:
pD∗t,j (λj , μ) = arg max
pDt,j≥0
log
(
1 +|hj |2pD
t,j
σ2 + Ij
)
− μpDt,j − λjfj(p
Dt,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 befound efficiently.
The optimal values for λ and μ can be found through thedual optimization problem
minλ<0,μ≥0
gD(λ, μ) (21)
using the subgradient method [29]. To find subgradients, wenote that
gD(λ′, μ′)
= maxpDt <0
LD(pDt , λ′, μ′)
≥ LD(pD∗t (λ, μ), λ′, μ′)
= gD(λ, μ) +∑
j∈Cl
(λ′j − λj)(ε − E{fj(p
D∗t,j (λj , μ) − ηj)})
+ (μ′ − μ)(PDmax − E{
∑
j∈Cl
pD∗t,j (λj , μ)}). (22)
Hence, the following are subgradients of g(λ, μ):
∂μ = PDmax −
∑
j∈Cl
E{pD∗t,j (λj , μ)} (23)
∂λj = ε − E{fj(pD∗t,j (λj , μ) − ηj)} j ∈ Cl. (24)
Following the subgradient method, after a sufficient numberof iterations, we can find the value of optimal Lagrangemultipliers, i.e., μ∗ and λ∗
j for all j ∈ Cl. Then, for eachchannel realization, we need to solve the following optimiza-tion problem to find an optimal power allocation:
pD∗t,j (H) = arg max
pDt,j≥0
log
(
1 +|hj |2pD
t,j
σ2 + Ij
)
(25)
− μ∗pDt,j − λ∗
jfj(pDt,j − ηj(H))
where H is 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 assumethe channel for D2D users is stationary.
C. Special Case for Function fj(∙): Chernoff Bound
Using the Chernoff bound, (15) becomes
E{
eωj(pDt,j−ηj)
}≤ ε. (26)
If we use the Chernoff bound as a substitute of the probabilisticSINR constraint, i.e., (7), it is important to properly choosethe value of ωj to achieve the minimal gap between the twoconstraint. In fact, we have:
Pr{
pDt,j ≥ ηj
}≤ min
ωj
E{
eωj(pDt,j−ηj)
}≤ E
{eωj(p
Dt,j−ηj)
}.
(27)
To find an optimal ωj we have
∂
∂ωjE{
eωj(pDt,j−ηj)
}= E
{(pD
t,j − ηj)eωj(p
Dt,j−ηj)
}= 0. (28)
We define the random variable xj = pDt,j − ηj . Unfortunately,
the distribution of xj is not known before solving the optimiza-tion problem. However, we observe that xj is a complicatedmixture of multiple random quantities, and our numericalresults indicate it has roughly bell-shape distribution. Thus,we assume that xj ∼ N (mj , υj) and
E{
xjeωjxj
}
=∫ +∞
−∞xeωjx 1
υj
√2π
e−
(x−mj)2
2υ2j dx
= e
(ωjυ2j +mj)2−m2
j
2υ2j
∫ +∞
−∞y
1
υj
√2π
e−
(y−(ωjυ2j +mj))2
2υ2j dy. (29)
Note that the integral in (29) is the mean value for the randomvariable yj ∼ N (ωjυ
2j + mj , υj). Therefore, from (28) and
(29), a suitable value for ωj can be found as −mj
υ2j
. Since
6
mj and υj are not known, we use an iterative method where,starting from an initial point for ωj , in each step we updatethe value of ωj for the next step by estimating mj and 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(pDt,j−
ηj) = [1 + 1ηmax
j(pD
t,j − ηj)]m , where ηmaxj is the maximum
possible value for ηj , is convex and increasing with f(0) = 1.In this case, we have the following constraint:
E{
[1 + ωj(pDt,j − ηj)]
m}≤ ε, (30)
where ωj = 1ηmax
j.
In the special case where (30) is linear, i.e., m = 1, wecan provide a closed-form solution to (20) as follows. Settingthe derivative of (20) equal to zero, for fj(pD
t,j − ηj) = 1 +ωj(pD
t,j − ηj), we have
|hj |2
|hj |2pDt,j + σ2 + Ij
− μ − λjωj = 0. (31)
Thus,
pD∗t,j (λj , μ) =
[ 1μ + λjωj
−σ2 + Ij
|hj |2
]
+. (32)
E. Time-Averaging Based Heuristic Solutions
In the optimal solutions above, the calculation of subgradi-ents has high complexity and needs full information aboutthe statistics of the channels between D2D users and allinterference channels. In this section, we present a new methodto directly find the Lagrange multipliers by approximating theprimal domain problem.
Considering the fact that μ is related to constraint (16), ineach step, we can find μ[n + 1] by associating it with theapproximated constraint En+1
{∑j∈Cl
pDt,j
}≤ PD
max, where
n is the time slot index and En{x} , 1n
∑nt=1 x[t]. It is easy
to show that we have En+1{x} = 1n+1 (x[n + 1] + nEn{x}).
Thus, the sum power constraint in (8) can be written as∑
j∈Cl
pDt,j [n + 1] ≤ (n + 1)PD
max − nEn
{∑
j∈Cl
pDt,j
}
, P′Dmax[n + 1]. (33)
Similarly, the individual power constraints in (17) can beapproximated as
pDt,j [n + 1]
≤ ηj [n + 1] + f−1j
((n + 1)ε − nEn{fj(p
Dt,j − ηj)}
)
, η′j [n + 1], (34)
where f−1j (∙) is the inverse function of fj(∙). Since, fj(∙) is
a strictly increasing function, f−1j (∙) is unique.
In time slot n, we may solve the following optimizationproblem
D3 : maxpDt <0
∑
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 primalproblem [29], we have two cases.Case 1)
∑j∈Cl
η′j [n] ≤ P
′Dmax: pD∗
t,j [n] = η′j [n], for all j ∈ Cl.
Case 2)∑
j∈Clη′
j [n] > P′Dmax[n]: for all j ∈ Cl we have
pD∗t,j [n] =
[1
μ[n]−
σ2 + Ij [n]|hj [n]|2
]η′j [n]
0
. (35)
Note that in Case 2, μ[n] must be found such that∑j∈Cl
pDt,j [n] = P
′Dmax[n], which can be achieved using the
bisection method.
Remark. It is worth mentioning that the sum-power constraintin (33) is equivalent to
En
{∑
j∈Cl
pDt,j
}≤ PD
max, (36)
and also the individual power constraint in (34) is equivalentto
En
{fj(p
Dt,j − ηj)
}≤ ε (37)
which are valid for all n. For sufficiently large n, by assumingergodicity for all channels in the network, satisfying the con-straints in (33) and (34) guarantees satisfying the constraintsin stochastic optimization problem in D2. For small values ofn, since there is no information on the future channel state,the feasible set of D3 is a subset of the feasible set of D2. Byincreasing the time-window size, i.e., for larger n, the feasibleset of D3 converges to the feasible set of D2. In other words,the per-time-slot optimization problem D3 provides a lowerbound for the stochastic optimization problem D2.
F. An Upper Bound for D1
For a performance benchmark, we propose an upper boundfor the optimization problem in D1 and consider the gapbetween the lower and upper bound. In this section we tryto reach a proper upper bound.
Theorem 1. For any ωj ≥ 0, Pr{
pDt,j ≥ ηj
}≥ 1 −
E{
e−ωj(pDt,j−ηj)
}.
Proof: We have Pr{
pDt,j ≥ ηj
}= Pr
{−ωj(pD
t,j −ηj) ≤
0}
= Pr{
e−ωj(pDt,j−ηj) ≤ 1
}. From the Markov inequality
we have
Pr{
e−ω(pDt,j−ηj) ≤ 1
}≥= 1 − E
{e−ωj(p
Dt,j−ηj)
}. (38)
The optimization problem for finding the upper bound canbe written as follows
D4 : maxpDt <0
E{∑
j∈Cl
log
(aj + bjp
Dt,j
aj + cjpDt,j
)}
(39)
7
subject to E{∑
j∈Cl
pDt,j
}≤ PD
max (40)
E{
1 − e−ωj(pDt,j−ηj)
}≤ ε, j ∈ Cl. (41)
Unfortunately, because the constraint in (41) is concave,D4 is not a convex optimization problem. However, the dualproblem is always convex and provides an upper bound forthe primal problem. Therefore, we can use the dual problemto find an upper bound for D1. The Lagrange function of (39)can be written as
L1(pDt , λ, μ)
,∑
j∈Cl
E{
log
(
1 +|hj |2pD
t,j
σ2 + Ij
)}
+ μ(PDmax −
∑
j∈Cl
E{
pDt,j
})
+∑
j∈Cl
λj(ε − 1 + E{
e−ωj(pDt,j−ηj)
})
=∑
j∈Cl
E{
log
(aj + bjp
Dt,j
aj + cjpDt,j
)
− μpDt,j + λje
−ωj(pDt,j−ηj)
}
+ μPDmax +
∑
j∈Cl
λj(ε − 1). (42)
The corresponding Lagrange dual function can be defined is
g1(λ, μ) = maxpDt <0
L1(pDt , λ, μ). (43)
To find an optimal pDt,j , for fixed values of λj and μ and for
j ∈ Cl, in (43) we need to solve the following optimizationproblem for each j
pD∗t,j (λj , μ) = arg max
pDt,j≥0
log
(
1 +|hj |2pD
t,j
σ2 + Ij
)
− μpDt,j + λje
−ωj(pDt,j−ηj). (44)
Note that (44) is not a convex optimization problem. Thuswe need to apply an extremum-search method to find theglobal optimum point. Then, to solve (43), the subgradientmethod can be used.
Improving the value of ωj: Similarly to Section III-C, toimprove ωj for the upper bound, we have
∂
∂ωj(1 − E
{e−ωj(p
Dt,j−ηj)
}) = E
{(pD
t,j − ηj)e−ωj(p
Dt,j−ηj)
}
= 0. (45)
Assuming that xj = pDt,j − ηj ∼ N (mj , υj), we have
E{
xjeωjxj
}
=∫ +∞
−∞xe−ωjx 1
υj
√2π
e−
(x−mj)2
2υ2j dx
= e
(mj−ωjυ2j )2−m2
j
2υ2j
∫ +∞
−∞y
1
υj
√2π
e−
(y−(mj−ωjυ2j ))2
2υ2j dy = 0.
From the above equation, a suitable value for ωj can be foundas mj
υ2j
. Hence, we can update the value of ωj using an iterativemethod as discussed in Section III-C.
IV. ERGODIC SUM-RATE MAXIMIZATION
Similarly to the previous section, we first convexify the non-convex optimization problem in S1 and then use the Lagrangeduality to find the D2D power allocation pD
t,j separately overeach observed outcome (or “realization”) of the channel statevector. However, the more complex non-convex form of theD2D-cellular sum-rate objective presents further challenges.In this section, we detail the additional procedures to solveS1. We use the same definition and special cases of fj(∙) asin the previous section.
Note that the sum-rate of cellular users over RBs in Cl, priorto the D2D pair entering the system, is given by
∑j∈Cl
log(1+pCr,j
σ2+ICj
). It is independent of D2D transmitter power allocation.Thus, the sum-rate maximization problem S1 is equivalent tothe problem of maximizing the ergodic sum-rate improvementdue to the addition of the new D2D pair, given by
S1′ : maxpDt <0
E{∑
j∈Cl
log(1 + SINRCj ) + log(1 + SINRD
j )
− log
(
1 +pCr,j
I0j + σ2
j
)}
subject to (7) and (8).
A. Convexification of S1
Typically, only those RBs over which the cellular usershave a sufficiently high SINR condition are allocated to theD2D user. After D2D reuse, the SINR of the cellular userover such an RB is still relatively high. Therefore, we assumethe minimum SINR requirement ζ intra
j,min � 1, for all j ∈ Cl.With this assumption, we can use following approximation toapproximate the objective of S1′ as
∑
j∈Cl
log(1 + SINRCj ) − log
(
1 +pCr,j
I0j + σ2
j
)
+ log(1 + SINRDj )
≈∑
j∈Cl
[
log
(pCr,j
pDt,j |h
Ij |
2 + I0j + σ2
)
− log
(pCr,j
I0j + σ2
)
+ log
(
1 +|hj |2pD
t,j
Ij + σ2
)]
=∑
j∈Cl
log
(aj + bjp
Dt,j
aj + cjpDt,j
)
, (46)
where aj , (σ2 + I0j )(σ2 + Ij), bj , (σ2 + I0
j )|hj |2, andcj , (σ2 + Ij)|hI
j |2. Thus, we can approximate S1 as
S2 : maxpDt <0
E{∑
j∈Cl
log(aj + bjp
Dt,j
aj + cjpDt,j
)}
subject to (7) and (8).
As discussed in section III, for the D2D-rate maximizationproblem, (7) is not a convex constraint, so we substitute itwith (16). Therefore, we propose the following optimizationproblem:
S3 : maxpDt <0
E{∑
j∈Cl
log(aj + bjp
Dt,j
aj + cjpDt,j
)}
8
subject to (16) and (17).
Furthermore, the objective function of S3, for pDt < 0, can be
upper bounded as follows:
E{∑
j∈Cl
log(aj + bjp
Dt,j
aj + cjpDt,j
)}
=∑
j∈Cl
Pr{bj ≥ cj}E{
log(aj + bjp
Dt,j
aj + cjpDt,j
∣∣∣bj ≥ cj)
+ Pr{bj < cj}E{
log(aj + bjp
Dt,j
aj + cjpDt,j
)∣∣∣bj < cj
}
≤∑
j∈Cl
Pr{bj ≥ cj}E{
log(aj + bjp
Dt,j
aj + cjpDt,j
)∣∣∣bj ≥ cj
}. (47)
The last inequality comes from the fact that the function
log(aj+bjpD
t,j
aj+cjpDt,j
), for bj < cj and pDt,j ≥ 0, is a decreasing (and
convex) function. The upper bound is achievable if and onlyif for bj < cj we have pD
t,j = 0. In other words, pDt,j = 0 is
an optimal solution when we have bj < cj if and only if allthe constraint in S3 are satisfied.
The sum-power constraint is satisfied if we have
E{∑
j∈Cl
pDt,j
}=∑
j∈Cl
(Pr{bj ≥ cj}E{
pDt,j
∣∣∣bj ≥ cj
}
+ Pr{bj < cj}E{
pDt,j
∣∣∣bj < cj
})
=∑
j∈Cl
Pr{bj ≥ cj}E{
pDt,j
∣∣∣bj ≥ cj
}(48)
≤ PDmax.
The individual power constraints are satisfied if we have
E{
fj(pDt,j − ηj)
}= Pr{bj ≥ cj}E
{fj(p
Dt,j − ηj)
∣∣∣bj ≥ cj
}
+ Pr{bj < cj}E{
fj(pDt,j − ηj)
∣∣∣bj < cj
}
= Pr{bj ≥ cj}E{
fj(pDt,j − ηj)
∣∣∣bj ≥ cj
}
+ Pr{bj < cj}E{
fj(−ηj)∣∣∣bj < cj
}
≤ ε. (49)
Therefore, S3 is equivalent to the following optimizationproblem
S4 : maxpDt <0
∑
j∈Cl
PjE{
log(aj + bjp
Dt,j
aj + cjpDt,j
)∣∣∣bj ≥ cj
}
subject to∑
j∈Cl
PjE{
pDt,j
∣∣∣bj ≥ cj
}≤ PD
max (50)
PjE{
fj(pDt,j − ηj)
∣∣∣bj ≥ cj
}≤ ε′j j ∈ Cl, (51)
where we define PjΔ= Pr{bj ≥ cj} for all j ∈ Cl and ε′j
Δ=
ε−(1−Pj)E{fj(−ηj)∣∣∣bj < cj}. We will later show that there
is no need to calculate ε′j .It is easy to show that any solution for S4, by applying
pDt,j = 0 when we have bj < cj , is feasible for the problem
S3. Also, again by applying pDt,j = 0 when we have bj < cj ,
the objective function of the two problem are the same. Thus,the two problems S3 and S4 are equivalent.
Note that the function log(aj+bjpD
t,j
aj+cjpDt,j
), for bj ≥ cj and pDt,j ≥
0, is a concave (and increasing) function, thus we can use themethod of Lagrange multipliers to solve S4. The Lagrangefunction of S4 can be written as
LS(pDt ,λ, μ)
,∑
j∈Cl
PjE{
log(aj + bjp
Dt,j
aj + cjpDt,j
)∣∣∣bj ≥ cj
}
+ μ(PDmax −
∑
j∈Cl
PjE{
pDt,j
∣∣∣bj ≥ cj
})
+∑
j∈Cl
λj(ε′ − PjE
{fj(p
Dt,j − ηj)
∣∣∣bj ≥ cj
})
=∑
j∈Cl
PjE{
log(aj + bjp
Dt,j
aj + cjpDt,j
) − μpDt,j − λjfj(p
Dt,j − ηj)
∣∣∣bj ≥ cj
}
+ μPDmax +
∑
j∈Cl
λjε′j . (52)
By applying the fact that for bj < cj we have pDt,j = 0, the
Lagrange function also can be written as follows:
LS(pDt , λ, μ)
=∑
j∈Cl
E{
log(aj + bjp
Dt,j
aj + cjpDt,j
) − μpDt,j − λjfj(p
Dt,j − ηj)
}
+ μPDmax +
∑
j∈Cl
λjε. (53)
The Lagrange dual function is
gS(λ, μ) = maxpDt <0
LS(pDt , λ, μ). (54)
To find optimal pDt,j , for fixed values of λj and μ and for j ∈
Cl, in (54) we need to solve following optimization problemfor each j
pD∗t,j (λj , μ) = arg max
pDt,j≥0
log
(aj + bjp
Dt,j
aj + cjpDt,j
)
− μpDt,j − λjfj(p
Dt,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 λ and μ:
minλ<0,μ≥0
gS(λ, μ). (56)
To solve (56), the subgradient method can be exploited. It iseasy to show that the subgradients are
∂λj = ε′j − PjE{fj(pD∗t,j (λj , μ) − ηj)
∣∣∣bj ≥ cj}, j ∈ Cl,
∂μ = PDmax −
∑
j∈Cl
PjE{pD∗t,j (λj , μ)
∣∣∣bj ≥ cj}.
Using (48) and (49), the subgradients can be re-written as
∂λj = ε − E{fj(pD∗t,j (λj , μ) − ηj)}, j ∈ Cl, (57)
9
∂μ = PDmax −
∑
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 subgradientmethod, the value of optimal lagrange multipliers, i.e., λ∗
j ,for all j ∈ Cl∗ , and μ∗, are found. Then, for each channelstate, we solve the following optimization problem to find theoptimal power allocation:
pD∗t,j (H) = arg max
pDt,j≥0
log
(aj(H) + bj(H)pD
t,j
aj(H) + cj(H)pDt,j
)
− μ∗pDt,j − λ∗
jeωj(p
Dt,j−ηj(H)). (59)
Note that considering (57) and (58), the above formulation canonly be applied over a time-interval where we can assume thechannel for D2D users is stationary.
Remark. It is worth mentioning that since we have the sameconstraints for both optimization problem S1 and D1, the sameapproach, 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 canprovide a closed-form solution for (55). Setting the derivativeof (55) equal to zero, for fj(pD
t,j − ηj) = ωj(pDt,j − ηj) + 1,
we haveajbj − ajcj
(aj + bjpDt,j)(aj + cjpD
t,j)− μ − λjωj = 0, (60)
i.e., we have (ωjλj + μ)bjcj(pDt,j)
2 + (ωjλj + μ)aj(bj +cj)pD
t,j + ((ωjλj + μ)a2j − ajbj + ajcj) = 0 or equivalently
κj(pDt,j)
2 +βjpDt,j +(1− γj
(ωjλj+μ) ) = 0 where κj ,bjcj
a2j
> 0,
βj ,bj+cj
aj> 0 and 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 beaccepted or otherwise the solution for (55) is zero. Therefore,we have
pD∗t,j (λj , μ) =
[−βj +
√β2
j − 4κj(1 − γj
(μ+ωjλj))
2κj
]
+
. (61)
B. Time-Averaging Based Heuristic Solutions
Using the same idea of time-averaging instead of statisticalmeans, as we explained previously for D2, we propose thefollowing a heuristic solution for S3. In time slot n, we solvethe following optimization problem:
S5 : maxpDt <0
∑
j∈Cl
log
(aj [n] + bj [n]pD
t,j
aj [n] + cj [n]pDt,j
)
subject to (33) and (34).
After applying the KKT optimality condition to the primalproblem [29], we have two cases:Case 1)
∑j∈Cl
η′j [n] ≤ P
′Dmax: pD∗
t,j [n] = η′j [n], for all j ∈ Cl.
Case 2)∑
j∈Clη′
j [n] > P′Dmax[n]: for all j ∈ Cl we have
pD∗t,j [n] =
[−βj [n] +
√βj [n]2 − 4κj [n](1 − γj [n]
μ[n] )
2κj [n]
]η′j [n]
0
. (62)
In each time slot, μ[n] should be found such that∑j∈Cl
pDt,j [n] ≤ P
′Dmax[n]. This can be achieved using the
bisection method.Since we have the same constraints for the optimization
problems D3 and S5, all other discussions in Section III-E onsolving D3 are valid for S5.
C. An Upper Bound for S1
Similarly to Section III-F, the optimization problem forfinding the upper bound can be written as follows
S6 : maxpDt <0
E{∑
j∈Cl
log(aj + bjp
Dt,j
aj + cjpDt,j
)}
subject to (40) and (41).
As discussed before, S6 is not a convex problem and anupper bound cannot be achieved using straight-forward convexoptimization methods. We instead use the dual problem to findan upper bound. The Lagrange function of S6 can be writtenas
L2(pDt , λ, μ)
,∑
j∈Cl
E{
log(aj + bjp
Dt,j
aj + cjpDt,j
)}
+ μ(PDmax −
∑
j∈Cl
E{
pDt,j
})
+∑
j∈Cl
λj(ε − 1 + E{
e−ωj(pDt,j−ηj)
})
=∑
j∈Cl
E{
log(aj + bjp
Dt,j
aj + cjpDt,j
) − μpDt,j + λje
−ωj(pDt,j−ηj)
}
+ μPDmax +
∑
j∈Cl
λj(ε − 1). (63)
The corresponding Lagrange dual function can be defined is
g2(λ, μ) = maxpDt <0
L2(pDt , λ, μ). (64)
To find an optimal pDt,j , for fixed values of λj and μ and for
j ∈ Cl, in (64) we need to solve the following optimizationproblem for each j:
pD∗t,j (λj , μ) = arg max
pDt,j≥0
log
(aj + bjp
Dt,j
aj + cjpDt,j
)
− μpDt,j + λje
−ωj(pDt,j−ηj). (65)
Noting that (65) is not a convex optimization problem, weapply an extremum-search method to find the global optimumpoint. After finding the global maximum point, we can use thesubgradient 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 withmultiple D2D pairs and cellular users in each cell. In general,each D2D user has its own arrival time and duration of stayin the network. Upon the arrival of any D2D user, a schedulerwithin each cell (which can reside within the eNB) can decide
10
the resources to be allocated to the D2D user. To avoid largeoverhead and to decrease the computational complexity, theresource allocation decision for each D2D user can be madeseparately by the scheduler, which is in harmony with thefact that the nodes in a cellular system have different arrivaltime and duration of stay in the network. Furthermore, byimplementing the same algorithms (i.e. sum-rate maximiza-tion or D2D rate maximization) in all neighboring cells, weguarantee all the constraints in (7) and (8), i.e., sum-powerconstraint for all D2D users and a pre-defined SINR levelfor all cellular users in the network, while optimizing thethroughput performance separately for each cell.
However, in the multi-cell scenario, applying the subgra-dient method for sum-rate and D2D rate maximization maycause some difficulties. As we notice before, the subgradientmethod can be applied only under the assumption that thesignal and interference channels to the D2D user under con-sideration is stationary. In the multi-cell scenario, the entryof a new D2D user into the network changes the interferencedistribution for all other D2D users in the neighboring cellswhich use the same RBs as allocated to the new user. In thatcase, the stationarity assumption is not valid anymore. Onlyin a very crowded network, or in a large cell with D2D usersthat are not very close to the edges of the cell, can we assumethat the interference to D2D users is approximately stationary.
Note that for the proposed time-averaging heuristic solutiondoes not require such a stationarity assumption. Therefore,it can be applied to any multi-cell network efficiently whileguaranteeing all the constraints in (7) and (8) for all users inthe network.
B. More Generalized Optimization Problem
So far, we have considered the optimization problems in D1and S1 with long-term sum-power constraint (8) on the D2Dusers. The constraint in (8) addresses the battery usage of theD2D users. In reality, the instantaneous power may also beconfined due to physical design of the mobile device such aspower amplifier and antenna power limitations. We can add ashort-term power constraint as follows:
∑
j∈Cl
pDt,j ≤ PD
max,s. (66)
The same procedure as described in this paper can be usedto solve the new optimization problems with this additionalconstraint. In particular, for the D2D-rate optimization prob-lem, to find the optimal power allocation we need to solve themaximization problem in (20) subject to (66). To solve thesum-rate optimization problem, the maximization problem in(55) must be solved subject to constraint in (66). Since (66) isa linear constraint, solving (20) and (55) under this constraintstill admits efficient convex optimization methods.
VI. SIMULATION EVALUATION
We consider a cellular network with 10 cellular users ineach cell and N = 10 distinct RBs are assigned to eachcellular user. Cellular and D2D users are uniformly randomly
TABLE II: Default values for simulation parameters
N = 10Number of RBs per cellular user = 10RB bandwidth = 12x15 KHzCell radius (Rc) = 100 mD2D distance (d) = 20 mNumber of D2D pairs in each cell (Nd) = 7Ave. cellular SNR = 30 dBFree space path-loss factor = 3.5Standard deviation for shadowing = 6 dB
distributed over each cell. We apply a power control mecha-nism which compensates for the free-space path loss effectsfor all cellular users. For the link between any two nodes, weuse a simple path loss model K0D
−α, where we set the pathloss constant Ko = 0.01, and the pass loss exponent α = 3.5.We assume Rayleigh fading for all interference, cellular andD2D links. The default values of different system parametersare presented in Table II.
In each cell, when there are multiple active D2D pairsrequesting to share RBs, they are queued based on a first-come-first-serve rule. We solve the proposed optimizationproblems for the D2D pairs one by one based on their order inthe queue 2. To allocate RBs to each D2D pair, first we needto find all available RBs that provide a non-empty feasible setfor each optimization problem. Let us assume that C∗
l is theset of all available RBs such that we have (11)-(12) satisfied.We need to find Cl ⊆ C∗
l such that |Cl| ≤ Nl. To find theoptimal Cl we would need to solve
(N∗
lNl
)optimization problems
where N∗l = |C∗
l | (assuming N∗l ≥ Nl). Instead, we use the
following heuristic method for a more practical solution. Usingthe Markov’s inequality instead of the probabilistic constraintsin (7) yields
E{
pDt,j − ηj + 1
}≤ ε. (67)
Equivalently, we have
E{
pDt,j
}≤ E
{ηj
}+ ε − 1. (68)
This means that the larger E{ηj} value, the larger the feasibleset for the optimization problems S1 and D1. Therefore, bysorting the values of E
{ηj
}for all j ∈ C∗
l , the lth D2D userbe opportunistically assigned the RBs with the highest E{ηj}such that (11)-(12) are satisfied. It is worth mentioning thatother methods with higher complexity can be adapted for RBallocation., e.g. the readers may consider [31]- [32].
For each data point, we average the results over a largenumber of random positions of cellular and D2D users andalso random channel realizations. To avoid redundancy, weonly present the results for the uplink case, in Figs. 2-5.
A. Efficacy of Convexification
In this section we analyze the efficacy of applying theconvexification approach we proposed in Section III-A andIV-A to solve the ergodic sum-rate and D2D rate optimizationproblems. To focus on convexification and remove the effects
2For alternative approaches the readers may consider [30]
11
-40 -30 -20 -10 0 10 200
2
4
6
8
10
12
14
Avg
. D2D
Rat
e (b
its/s
/Hz)
P max D (dB
m)
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 10 204
6
8
10
12
14
16
Avg
. Sum
Rat
e (b
its/H
z)
P max D (dB
m)
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 considerthe single-cell scenario. We compare the performance of theproposed solution methods with the upper-bounds developedin Sections III-F and IV-C, and also with a naive non-ergodichuristic method. For the non-ergodic approach, similar to [27]and [28], instead of long-term constraints in (7) and (8), weuse short-term constraints, i.e., we set ε = 0 and we use theconstraint
∑j∈Cl
pDt,j ≤ PD
max as the sum-power constraint.We set ζ intra
j,min = 3dB, for all j. Also, for ergodic methods weset ε = 0.05.
Fig. 2 investigates the effects of changing PDmax on the D2D
and cell throughput. Fig. 2(a) shows the average D2D rate perRB for the cell under the D2D-rate maximization objective,and Fig. 2(b) shows the average D2D-cellular sum rate perRB for the cell under the sum-rate maximization objective.It can be seen that through applying the convexificationmethod, we reach a small gap between the upper-bound andthe Chernoff-bound and linear-constraint approximations. It isworth mentioning that by increasing PD
max, the percentage ofthis gap, for specially D2D rate, is decreasing.
The Chernoff-bound method provides slightly higherthroughput than the linear-constraint method. This is becausewe can improve the Chernoff-bound through the iterativemethod proposed in Section III-C. On the other hand, thelinear-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 computationalcomplexity and negligible performance gap, so only one curveis shown. We observe that the performance gap between thetime-averaging heuristic and the subgradient methods is lessthan 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 performancegap between ergodic and non-ergodic power allocation isconsiderable, indicating the need for the proposed stochasticoptimization solutions.
Besides the performance under perfect CSI, in Fig. 2, wealso consider the performance of the convexification method
TABLE III: MATLAB Run-Time for Different Algorithms
Non-Ergodic 5.91 sTime-Ave. Heuristic 5.92 sLinear Cons. Subgradient 1302.89 sChernoff-Bound Subgradient 1772.59 s
with the Chernoff bound using imperfect CSI where channelestimation errors present. We model the channel estimationerror as a complex Gaussian noise with zero mean and variancebeing 5% of the corresponding true channel variance. As canbe seen from Figs. 2(a) and (b), the rate loss by using imperfectCSI for power allocation is less than 10%.
In Table III, we compare the computational complexity ofthe methods discussed in this paper based on the MATLABrun time of simulating 900 LTE frames (equivalent to 4.5seconds) under all algorithms. It can be seen from Table IIIthat the proposed time-averaging heuristic is nearly identicalto the naive non-ergodic method in run time and it is around300 times faster than the standard subgradient methods withChernoff-bound or linear constraints.
Note that all these methods have the same communicationcomplexity. In general, beside a common overhead in LTEthat is necessary to estimate CSI and the interference level,to calculate ηj , channel and interference feedback is requiredover each time slot. To avoid a large amount of informationexchange, we can interpret power constraint (7) as per-RBpower constraints set by the eNB in the main cell. In otherwords, ηj , for all j ∈ Cl, is set by the eNB such that wehave SINR constraints (4)-(5) satisfied. In this case, each D2Dtransmitter directly receives the value of ηj , for j ∈ Cl, fromthe eNB of its own cell. Furthermore, after the initial setting ofηj by the eNB, any change in the value of ηj can be reportedusing limited feedback, e.g., using differential coding.
B. Multi-cell Performance Comparison
Under the multi-cell scenario, we compare the proposedheuristic solution with the non-ergodic approach. As defaultvalues, we set PD
max = −8.5dBm, ζ intraj,min = ζ
(k)j,min = −3dB,
12
-30 -20 -10 0 10 200
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2A
vg. D
2D R
ate
(bits
/s/H
z)
P max D (dB
m)
Ergodic D2D-Rate Max. MethodErgodic Sum-Rate Max. MethodNon Ergodic D2D-Rate Max. MethodNon 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
Rat
e (b
its/s
/Hz)
P max D (dB
m)
Ergodic D2D-Rate Max. MethodErgodic Sum-Rate Max. MethodNon Ergodic D2D-Rate Max. MethodNon Ergodic Sum-Rate Max. MethodNo D2D
(b) Achieved throughput in the main cell.
Fig. 3: The achieved throughput vs. PDmax.
-2 0 2 4 6 80
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Avg
D2D
Rat
e(bi
ts/s
/Hz)
ζ min
(dB)
Ergodic D2D-Rate Max. MethodErgodic Sum-Rate Max. MethodNon Ergodic D2D-Rate Max. MethodNon Ergodic Sum-Rate Max. Method
(a) Achieved throughput for D2D users,
-2 0 2 4 6 81.9
2
2.1
2.2
2.3
2.4
2.5
2.6
Avg
Sum
Rat
e(bi
ts/s
/Hz)
ζ min
(dB)
Ergodic D2D-Rate Max. MethodErgodic Sum-Rate Max. MethodNon Ergodic D2D-Rate Max. MethodNon Ergodic Sum-Rate Max. MethodNo D2D
(b) Achieved throughput for the main cell.
Fig. 4: The achieved throughput vs. minimum required SINR.
100 150 200 250 3000
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
Avg
D2D
Rat
e(bi
ts/s
/Hz)
Rc
Ergodic D2D-Rate Max. MethodErgodic Sum-Rate Max. Method
Non Ergodic D2D-Rate Max. MethodNon Ergodic Sum-Rate Max. Method
(a) Achieved throughput for D2D users,
100 150 200 250 3001.8
2
2.2
2.4
2.6
2.8
3
3.2
3.4
3.6
Avg
Sum
Rat
e(bi
ts/s
/Hz)
Rc
Ergodic D2D-Rate Max. MethodErgodic Sum-Rate Max. Method
Non Ergodic D2D-Rate Max. MethodNon Ergodic Sum-Rate Max. MethodNo D2D
(b) Achieved throughput for the cell.
Fig. 5: The achieved throughput for different cell radius.
13
for all j and for all k ∈ Sj , unless otherwise specified. Forergodic methods we set ε = 0.1.
1) Maximum Power for D2D Users: Fig. 3 shows theeffects of changing PD
max on the D2D and cell throughput. Itcan be seen that, by increasing PD
max, at first the D2D-rate andsum-rate increase, but after some point they start to degrade.This happens because all neighboring cells use the same powerallocation algorithm, and by increasing the D2D power, weincrease the interference to and from the neighboring cells.
2) Minimum SINR Requirement for Cellular Users: Fig. 4shows the effect of changing the the minimum required SINRon the D2D and cell throughput. It can be seen that, byincreasing the minimum required SINR for the cellular users,there is less room for controlling the power of D2D usersand this decreases the D2D throughput and the total cellthroughput.
3) Cell Size: Fig. 5 shows the effect of changing the cellradius on the D2D and cell throughput. It can be seen fromthat, by increasing the cell radius, we see an increase andthen a decrease in the throughput. In fact by increasing thecell radius, we have two different effects. The cellular andD2D users are spread over a larger area and thus the distancebetween interfering users is decreased, but on the other handbecause of the uplink power control method for cellular users,their power increases, leading to more interference to D2Dusers.
VII. CONCLUSION
In this paper, we have considered optimal power allocationby the D2D users in a cellular network for underlay D2Dcommunications in order to maximize the D2D rate and thesum rate between D2D and cellular users. The proposedoptimization problems accommodates a long-term sum-powerconstraint and probabilistic individual power constraints overeach accessible RB. This enables consideration for batteryenergy limits at D2D transmitters and the interference createdby D2D communications. To solve the optimization problem,several approximate convex constraints are introduced, asreplacement for the non-convex probabilistic individual powerconstraints. After such convexification, optimal solutions to theapproximate D2D-rate and sum-rate maximization problemsare developed, which are shown to give throughput perfor-mance that is close to an upper bound. We then propose aheuristic method by using time-averaging to approximate forlong-term measures. The time-averaging heuristic method haslow computational complexity, and it can be easily applied tothe multi-cell scenario. Through simulation we observe thatthe performance gap between the standard subgradient solutionand the proposed time-averaging heuristic method is less than17%, with drastically reduced computational complexity forthe heuristic method.
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Ruhallah AliHemmati Ruhallah AliHemmati re-ceived the B.Sc. degree in electrical engineeringfrom University of Tehran, Tehran, Iran, in 2002,and the M.Sc. and Ph.D. degrees in telecommuni-cation systems enngineering from Tarbiat ModaresUniversity, Tehran, Iran, in 2004 and 2008, respec-tively. From 2003 to 2005, he was with the IranianTelecommunication Research Center, and from 2005to 2007, he was with the Advanced CommunicationsResearch Institute, Tehran, Iran. From 2008 to 2009,he was with the Advanced Multi-Dimensional Signal
Processing Laboratory, Queens University, Kingston, Canada, as a VisitingResearcher. From 2012 to 2014, he was with University of Ontario Instituteof Technology (UOIT), Oshawa, Canada, as a Post-Doctoral Researcher. From2014 to 2016 he was with University of Toronto, Toronto, Canada as aPost-Doctoral Researcher. His main research interests are statistical signalprocessing, detection and estimation theory, and relay networks.
Min Dong Min Dong (S’00-M’05-SM’09) receivedthe 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 appliedmathematics from Cornell University, Ithaca, NY, in2004. From 2004 to 2008, she was with CorporateResearch and Development, Qualcomm Inc., SanDiego, CA. In 2008, she joined the Departmentof Electrical, Computer and Software Engineeringat 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 processingfor communication networks, cooperative communications and networkingtechniques, and stochastic network optimization in dynamic networks andsystems.
Dr. Dong received the the 2004 IEEE Signal Processing Society BestPaper Award, the Best Paper Award at IEEE ICCC in 2012, and the EarlyResearcher Award from Ontario Ministry of Research and Innovation in 2012.She is the co-author of the Best Student Paper Award of Signal Processingfor Communications and Networks in IEEE ICASSP’16. She served as anAssociate Editor for the IEEE TRANSACTIONS ON SIGNAL PROCESSING(2010-2014), and as an Associate Editor for the IEEE SIGNAL PROCESSINGLETTERS (2009-2013). She was a symposium lead co-chair of the Commu-nications and Networks to Enable the Smart Grid Symposium at the IEEEInternational Conference on Smart Grid Communications (SmartGridComm)in 2014. She has been an elected member of IEEE Signal Processing SocietySignal Processing for Communications and Networking (SP-COM) TechnicalCommittee since 2013.
Ben Liang Ben Liang received honors-simultaneousB.Sc. (valedictorian) and M.Sc. degrees in ElectricalEngineering from Polytechnic University in Brook-lyn, New York, in 1997 and the Ph.D. degree inElectrical Engineering with a minor in ComputerScience from Cornell University in Ithaca, NewYork, in 2001. In the 2001 - 2002 academic year,he was a visiting lecturer and post-doctoral researchassociate with Cornell University. He joined theDepartment of Electrical and Computer Engineeringat the University of Toronto in 2002, where he is
now a Professor. His current research interests are in networked systems andmobile communications. He has served on the editorial boards of the IEEETransactions on Mobile Computing since 2017 and the IEEE Transactions onCommunications since 2014, and he was an editor for the IEEE Transactionson Wireless Communications from 2008 to 2013 and an associate editorfor Wiley Security and Communication Networks from 2007 to 2016. Heregularly serves on the organizational and technical committees of a numberof conferences. He is a senior member of IEEE and a member of ACM andTau Beta Pi.
Gary Boudreau GARY BOUDREAU [M’84-SM’11] received a B.A.Sc. in Electrical Engineeringfrom the University of Ottawa in 1983, an M.A.Sc.in Electrical Engineering from Queens University in1984 and a Ph.D. in electrical engineering from Car-leton University in 1989. From 1984 to 1989 he wasemployed as a communications systems engineerwith Canadian Astronautics Limited and from 1990to 1993 he worked as a satellite systems engineerfor MPR Teltech Ltd. For the period spanning 1993to 2009 he was employed by Nortel Networks in
a variety of wireless systems and management roles within the CDMAand LTE basestation product groups. In 2010 he joined Ericsson Canadawhere he is currently employed in the LTE systems architecture group. Hisinterests include digital and wireless communications as well as digital signalprocessing.
15
Hossein Seyedmehdi Hossein Seyedmehdi receiveda Master’s degree from the National University ofSingapore in 2008 and a PhD degree from theUniversity of Toronto in 2014 both in Electricaland Computer Engineering. He is currently affiliatedwith Ericsson Canada where he is working on the5thgeneration (5G) of wireless technologies includ-ing the commercialization of massive MIMO. Hehas numerous papers in the area of wireless systemsin addition to more than 10 patents. His researchinterests include Information Theory for Wireless
Communications, Signal Processing for MIMO channels, Algorithms forRadio Access Networks, and Future Radio Technologies.