a joint power control and resource allocation algorithm for...
TRANSCRIPT
A Joint Power Control and Resource AllocationAlgorithm for D2D Communications
M. Belleschi, G. Fodor, D. D. PendaEricsson Research, Sweden
Email: [email protected]
M. JohanssonRoyal Institute of Technology, Sweden
Email: [email protected]
A. AbrardoUniversity of Siena, Italy
Email: [email protected]
Abstract—We consider the problem of joint power control,signal-to-noise-and-interference-ratio (SINR) target setting, modeselection and resource allocation for cellular network assisteddevice-to-device (D2D) communications. This problem is impor-tant for fourth generation systems, such as the release under studyof the Long Term Evolution Advanced (LTE-A) system standard-ized by the Third Generation Partnership Project (3GPP). Whileprevious works on radio resource management (RRM) algorithmsfor D2D communications dealt with mode selection and powercontrol, the problem of resource allocation for the integratedcellular-D2D environment and in particular the joint problemof mode selection, resource allocation and power allocation hasnot been addressed. We propose a utility function maximizationapproach that allows to take into account the inherent tradeoff between maximizing spectrum efficiency and minimizing therequired sum transmit power. We implement the proposed RRMalgorithms in a realistic system simulator and report numericalresults that indicate large gains of D2D communications both interms of spectrum- and energy efficiency.
I. INTRODUCTION
Device-to-device (D2D) communications in cellular spec-trum supported by a cellular infrastructure has the potentialof increasing the spectrum and energy efficiency as well asallowing new peer-to-peer services by taking advantage ofthe so called proximity and reuse gains [1], [2], [3]. In fact,D2D communications in cellular spectrum is currently studiedby the 3rd generation partnership project (3GPP) to facilitateproximity aware internetworking services [4], [5].
However, D2D communications utilizing cellular spectrumposes new challenges, because relative to cellular communica-tion scenarios, the system needs to cope with new interferencesituations. 1 For example, in an orthogonal frequency divisionmultiplexing (OFDM) system in which user equipments (UE)are allowed to use D2D (also called direct mode) communica-tion, D2D communication links may reuse some of the OFDMtime-frequency physical resource blocks (RB).
Due to the reuse, intracell orthogonality is lost and intracellinterference can become severe due to the random positions ofthe D2D transmitters and receivers as well as of the cellularUEs communicating with their respective serving base stations
1It is advantageous to use uplink resources for the D2D link, becausein some countries regulatory requirements may not allow to use downlinkresources by user equipments in the future. Therefore, in this paper we onlydeal with the case when the D2D links use UL cellular resources, such asthe uplink OFDM resource blocks in a cellular Frequency Division Duplexingsystem or the uplink time slots in a Time Division Duplexing system [6], [7],[3].
[8], [9]. To realize the promises of D2D communicationsand to deal with intra- and intercell interference, the researchcommunity has proposed a number of important radio resourcemanagement (RRM) algorithms.
Although the objectives of such algorithms may be different(including enhancing the network capacity [10], improving thereliability [11], minimizing the sum transmission power [12],ensuring quality of service [13] or protecting the cellular layer(i.e. the cellular UEs) from harmful interference caused by theD2D layer [14]), there seems to be a consensus that the keyRRM techniques include:
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Cellular UE
D2D candidate
ULresource
D2D (Direct) Modewith Resource Reuse
Cellular Mode (through BS)
D2D (Direct) Mode with Dedicated Resources
D2D Communications andResource Allocation Mode
Figure 1. A D2D candidate consists of a D2D Transmitter and a D2DReceiver that are in the proximity of each other. The mode selection (MS)algorithm needs to decide on one of 3 possible communication modes and al-locate resources (resource blocks, RB) for the communication. In this paper weassume that the cellular uplink resources are used for D2D communications.We will use the indicator variable q to distinguish different modes.
1) Mode Selection (MS): MS algorithms determine whetherD2D candidates in the proximity of each other shouldcommunicate in direct mode using the D2D link or incellular mode (i.e. via the base station, BS) [15], [16],[17], see Figure 1;
2) Power Control (PC): PC algorithms taking into accountthe interference situation between the cellular and D2Dlayer play a key role to achieve various objectives [13],[18];
3) Resource Allocation (RA): Surprisingly, resource alloca-tion in the sense of selecting particular OFDM resourceblocks or frequency channels out of a set of availableones for each transmit-receive pair (cellular or D2D) is
seldom addressed in the literature ([8], [19], [20]);4) Pairing: In the D2D context, pairing refers to selecting
the D2D pair(s) and at most one cellular UE that share(reuse) the same OFDM resource block, similarly tomultiuser MIMO techniques. Pairing is a key techniqueto achieve high reuse gains [12];
5) Multiple Input Multiple Output (MIMO) Schemes: In-terference avoiding MIMO schemes have been proposedby [21]. Such schemes can be applied, for example forthe cellular transmissions to avoid generating interfer-ence to a D2D receiver.
In this paper we propose a framework that deals with powercontrol, mode selection and resource allocation in an iterativefashion. In our work we focus on single input single output(SISO) transceivers, although our methodology can be readilyapplied to the MIMO case as well.
The basic idea of our proposed scheme is to separate thetime scales for MS, RA and pairing, from the SINR targetsetting (rate control) and fast power control loops. The SINRtarget setting is exercised at the time scale of the tens ofmilliseconds by the outer-loop, where this outer loop assumesthat mode selection and resource allocation have been alreadydone by a heuristic mode selection and resource allocationalgorithm. A fast power control inner loop adjusts transmitpowers such that the SINR targets set by the outer loop aremet. The SINR target setting outer loop and the power controlinner loop work in concert and operate on each resourceblock in isolation maximizing a utility function that balancesbetween spectral and energy efficiency over feasible SINRtargets. Outer- and inner loops are executed iteratively untilconvergence is reached. We test this double loop approach in arealistic multicell system and find that the algorithm provideslarge gains in terms of spectral and energy efficiency, whencompared with traditional cellular communications.
We structure the paper as follows. The next section describesour system model. Section III formulates the SINR targetsetting and power control problem as an optimization task.Section IV develops a decomposition approach assuming thatthe specific resources (OFDM resource blocks) have beenallocated for the transmitter-receiver pairs in the system. Next,in Section V we formulate the MS problem and in Section VIwe address the joint MS and resource allocation problem byallocating the transmitter-receiver pairs to resource blocks suchthat the intracell interference due to resource reuse betweenD2D pairs and cellular-UEs is minimized. There are someintimate relationships between outer loop and inner loop thatare discussed in Section VII that describes the complete RRMprocedure. Numerical results are reported in Section VIII andthe conclusions are drawn by Section IX.
II. SYSTEM MODEL
We consider a wireless network with a total of L communi-cating transmitter-receiver pairs. A transmitter-receiver pair canbe a cellular UE transmitting data to its serving base stationor a device-to-device (D2D) pair communicating in cellularuplink spectrum. D2D candidates are source-destination pairs
in the proximity of each other that may communicate in directmode, depending on the MS decision that is part of the RRMalgorithm developed in this paper. In this paper we assume anorthogonal frequency division multiplexing (OFDM) cellularnetwork, such as the 3GPP LTE-A system, in which the timeand frequency resources are organized in physical resourceblocks (RB) [7].
The network topology is represented by a directed graphwith connections labelled l = 1, . . . , L. All (i.e. cellular andD2D) transmitters are assumed to have data to send to theircorresponding receivers (saturated buffers) and sl denotes thetransmission rate of Transmitter-l. Associated with each linkl is a function ul(·), which describes the utility of commu-nicating at rate sl. The utility function ul is assumed to beincreasing and strictly concave, with ul → −∞ as sl → 0+.We let c = [cl] denote the vector of link capacities, whichdepend on the communication bandwidth W , the achievedSINR of the links (γl) as well as the specific modulation andcoding schemes used for the communication. Obviously, thetarget rate vector s (which is in one-to-one correspondencewith the SINR targets, γtgt
l ) must fulfill the following set ofconstraints:
s ≼ c(p), s ≽ 0.
In this formulation, it is convenient to think of the s vector asthe vector of the rate (translating to SINR) targets, while the cvector represents the actual capacity achieved by the particularpower vector p.
Let Gl,m denote the effective link gain between the trans-mitter of pair m and the receiver of pair l (including path-lossand shadowing) and let σl be the thermal noise power at thereceiver of link l, and Pl be the transmission power. The SINRof link l is
γl(p) =GllPl
σl +∑m ̸=l
GlmPm
(1)
where p = [P1, ..., PL] is the power allocation vector, and∑m ̸=l Gl,mPm is the interference experienced at the receiver
of link l.Equation (1) can also be written as
γl(PtotlRx
, Pl, Gll) =GllPl
σl + (P totlRx−GllPl)
(2)
where P totlRx
represents the total received power measured bythe receiver of link l. Hence, the SINR in (2) can be computedby Receiver-l without knowing either the power used by otherD2D pairs or cellular transmitters or any of the channel gains,except the one related to its corresponding Transmitter-l. Forease of notation, we will use the notation γl(p) for the SINRat Receiver-l. Each link is seen as a Gaussian channel withShannon-like capacity
cl(p) = W log2(1 +Kγl(p)
)(3)
which actually represents the maximum rate that can beachieved on link l. W is the system bandwidth and K modelsthe SINR-gap reflecting a specific modulation and coding
2
scheme. In the following we assume K = 1.
III. THE SINR TARGET SETTING AND POWER CONTROLPROBLEM
In this section we assume that mode has already beenselected for the D2D candidates and all (cellular and D2D)links have been assigned a frequency channel or an OFDMRB. From the concept of D2D communications reusing cellularspectrum and the system model of the previous section itfollows that the resource blocks may be used by multiplecellular and D2D transmitters. In this section we focus onhandling this interference by properly setting the SINR targetsand allocating transmit powers, while the MS and resourceassignment problems that determine the specific cellular andD2D transmitters that share a given resource block are ap-proached in Sections V and VI.
For the set of interfering links sharing the same resourceblock and thereby causing interference to one another, weformulate the problem of target rate setting and power controlas:
maximizep,s
∑l ul(sl)− ω
∑l Pl
subject to sl ≤ cl(p), ∀l,p, s ≽ 0
(4)
which aims at maximizing the utility while taking into accountthe transmit powers (through a predefined weight ω ∈ (0,+∞)[22] [23]), so as to both increase spectrum efficiency and re-duce the sum power consumption over all transmitters sharinga specific RB.
The constraints in Problem (4) formally ensure that the rate(SINR target) allocation of sources does not exceed the linkcapacities, which is a quantity that is optimized through thepower allocation. As it will become clear later, the operationof the so called outer loop is such that SINR targets are alwaysfeasible, and so the power allocation by the inner loop ensuresthat target rate (s) and capacity (c) vectors coincide at the endof the convergence of the outer and inner loop pair.
A. Convexifying the Problem of Equation (4)Unfortunately, Problem (4) is not convex, but exploiting the
results presented in [22] and [24], we can transform it into thefollowing equivalent form:
maximizes̃,p̃
∑l ul(e
s̃l)− ω∑
l eP̃l
subject to log(es̃l) ≤ log(cl(e
p̃))∀l,
(5)
where sl ← es̃l and Pl ← eP̃l . The transformed Problem (5) isproved to be convex (now in the s̃l-s and P̃l-s) since the utilityfunctions ul(·) are selected to be (log, x)-concave over theirdomains [22]. In this paper we use ul(x) , ln(x), ∀l. Underthis condition, we can solve Problem (5) to optimality bymeans of an iterative algorithm where the s̃l-s (or equivalentlythe SINR targets) are set by an outer loop. The transmitpowers P̃l-s that meet the particular SINR targets (set in eachouter loop cycle) are in turn set by a Zander type iterativeSINR target following inner loop [25]. This separation of thesetting of the SINR targets and corresponding power levels aredetailed in the next Section.
IV. DECOMPOSITION APPROACH
A. Formulating the Decomposed ProblemWe now reformulate Problem (5) as a problem in the user
rates s̃ (Problem-I), which, due to the convexification, canbe solved for a given (assumed known) power allocation (p̃).Note that the target rate vector s̃ can be uniquely mapped toa target SINR vector γtgt as it will be shown later. We defineProblem-I as:
maximizes̃
ν (̃s)
subject to s̃ ∈ S̃(6)
where S̃ = {s̃| log(es̃l) ≤ log(cl(ep̃)), ∀l} represents the set
of feasible rate vectors that, for a given power vector p̃, fulfillthe constraints of Problem (5).
Comparing (5) and (6), it follows that the objective func-tion in (6) is defined as ν (̃s) ,
∑l ul(e
s̃l) − φ(p̃), whereφ(p̃) , ω
∑l e
P̃l represents the cost in terms of the totaltransmit power for realizing a given target rate s̃. Accordingly,we denote with φ⋆(p̃) , ω
∑l e
P̃⋆l the cost of achieving the
optimum rates s̃⋆ that solve the utility maximization Problem(6).
Therefore, Problem-II, for a given s̃ vector, can be formu-lated as
minimizep̃
ω∑
l eP̃l
subject to log(es̃l) ≤ log(cl(e
p̃))∀l.
(7)
Solution approaches to Problem-I and Problem-II are proposedin the next subsection.
B. Solving the Rate (SINR Target) Allocation ProblemWe are now concerned with setting the SINR targets by
solving Problem-I. Provided that the objective function ν (̃s) in(6) is concave and differentiable we can determine the optimals̃⋆ by means of projected gradient iterations, with a fixedpredefined step ϵ:
s̃(k+1)i = s̃
(k)i + ϵ∇iν (̃s
(k)) ∀i (8)
where
∇iν (̃s) = ∂∂s̃i
[∑l ul(e
s̃l)− φ⋆(p̃)]
= ui′(es̃i)es̃i − ∂
∂s̃i
[φ⋆(p̃)
].
(9)
To compute (9), we first need to find φ⋆(p̃) by solving theprimal Problem-II (7). Since it is convex in p̃, it can beconveniently solved by Lagrangian Decomposition as follows.Let λ be the Lagrange multipliers (dual variables) for theconstraints in (7) and form the Lagrangian function:
L(λ, p̃) = ω∑l
eP̃l +∑l
λl
[log(es̃l)− log
(cl(e
p̃)) ]
. (10)
The Lagrangian dual problem of Problem-II is given by:
maximizeλ
[L(λ) = minp̃L(λ, p̃)]
subject to λ ≽ 0.(11)
Since the original problem is convex, if regularity conditionshold then the solution of Problem (11) correspond to the
3
solution of Problem (7), i.e. L(λ⋆) = φ⋆(p̃). Assuming that(λ⋆, p̃⋆) represent the optimum solution of Problem-II (7), weare now in the position of calculating φ⋆(p̃) from (10):
φ⋆(p̃) =∑l
[ωeP̃
⋆l − λ⋆
l log(cl(e
p̃⋆
)) ]
+∑l
λ⋆l log(e
s̃l)
and∂
∂s̃i[φ⋆(p̃)] = λ⋆
i .
Recalling (9), we have:
∇iν (̃s) = ui′(es̃i)es̃i − λ⋆
i = es̃i [ui′(es̃i)− λ⋆
i
es̃i] =
= si[ui′(si)− λ⋆
i
si],
and so the final target rate update, for all i, is:
si(k+1) = es̃
(k+1)i = si
(k) exp(ϵ∇iν (̃s(k))).
Combining the above with (9) we can write the SINR targetsetting outer loop in the following form, for all i:
si(k+1) = si
(k) exp
(ϵ si
(k)[ui
′(si(k))− λ⋆
i (si(k))
(si(k))
])(12)
The updating rule of the outer loop given by (12) is useful,because it determines the (k+1)-th rate and SINR that shouldbe targeted by the inner power control. Note that as a naturalconsequence of the decomposition approach, (12) requires theknowledge of the Lagrange multipliers λ⋆
i associated withProblem-II, which can be found by solving the power controlproblem associated with the k-th outer loop. We consider thisproblem next.
C. Solving the Power Allocation Problem for a given SINRTarget
We now consider the power control problem (Problem-II) fora given SINR target as follows. Given s̃(k) ∈ S̃ , the constraintsin (7) correspond to requiring that the SINR-s of the linksexceed a target value, i.e.
log(es̃l)≤ log
(cl(e
p̃))⇔ γl(p) ≥ γl
tgt(̃s(k)) ∀l,
where γl(p) is defined in (1), and
γtgtl
(s̃(k)l
), 2
es̃lW − 1. (13)
Therefore, Problem (7) can be rewritten as:
minimizep̃
ω∑
l eP̃l
subject to γl(p) ≥ γtgtl (s̃l) ∀l
p̃ ≽ 0
(14)
and solved with an iterative SINR target following closed-looppower control (CLPC) scheme [25]:
Pl(t+1) =
γtgtl (s̃l)
γl(p(t)
)Pl(t). (15)
Thus, for a given γtgtl (s̃l), the (15) power control inner loop
provides an efficient means to set the transmit powers at eachtransmitter in loop (t+1), provided that the transmitter knowsthe SINR measured at the receiver in the preceding loop(γl(p(t)
)).
D. Determining the λ⋆i -s
We can now determine the λ⋆i -s for the outer loop update
(12) by exploiting the intimate relationship between the opti-mal p⋆ and the associated Lagrange multipliers λ⋆
i -s. To thisend, we rewrite the constraints in (14) as:
GllPl
σl +∑m ̸=l
GlmPm
− γtgtl ≥ 0 ⇒
Pl − γtgtl
∑m ̸=l
Glm
GllPm −
γtgtl σl
Gll≥ 0 ∀l. (16)
Furthermore, let H ∈ RLxL and η ∈ RL be defined as follows:
H = [hlm] ,{−1 if l = m
γtgtl
Glm
Gllif l ̸= m
η = [ηl] ,[γtgtl σl
Gll
].
Using this notation, we can reformulate Problem (14) as thefollowing Linear Programming (LP) problem:
minimizep
ω1Tp
subject to Hp ≼ −ηp ≽ 0,
(17)
with the corresponding Dual Problem
maximizeλ(LP)
ηTλ(LP)
subject to HTλ(LP) ≽ −ω1λ(LP) ≽ 0
(18)
which is necessary to compute the Lagrange multipliers inEquation (12) for the rate update.The inequality constraints in (18) can be rewritten explicitlyas:
λ(LP)l
ω−∑k ̸=l
Gkl
Gkkγtgtk
λ(LP)k
ω≤ 1, ∀l. (19)
Proof :
HTλ(LP) ≽ −ω1∑k
hklλ(LP)k ≥ −ω, ∀l;
−hllλ(LP)l +
∑k ̸=l
hklλ(LP)k ≥ −ω, ∀l
λ(LP)l
ω−∑k ̸=l
Gkl
Gkkγtgtk
λ(LP)k
ω≤ 1, ∀l. �
By defining
µl ,λ(LP)l
ω
γtgtl σl
Gll=
λ(LP)l
ωηl, (20)
4
inequality (19) can be interpreted as an SINR requirement, i.e.
γFCl (µ) , µlGll
σl +∑k ̸=l
Gklσl
σkµk
≤ γtgtl , ∀l. (21)
Proof :
λ(LP)l
ω −∑k ̸=l
Gkl
Gkkγtgtk
λ(LP)k
ω≤ 1
λ(LP)l
ω
γtgtl σl
Gll
Gll
γtgtl σl
−∑k ̸=l
Gkl
Gkkγtgtk
λ(LP)k
ω
σk
σk≤ 1
µlGll
γtgtl σl
≤∑k ̸=l
Gkl
σkµk + 1
γFCl (µ) , µlGll
σl+
∑k ̸=l
Gklσl
σkµk
≤ γtgtl , ∀l. �
Therefore, Problem (18) can be reformulated as:
maximizeµ
ω1Tµ
subject to λlFC ≤ γl
tgt, ∀lµ ≽ 0
(22)
where the solution µ can be found through the followingdistributed iterations
µ(t+1)l =
γtgtl
γFCl (µ(t))
µ(t)l ∀l. (23)
Note that (23) can be interpreted as a reverse link powercontrol problem, in which Receiver-l becomes a transmitter(transmitting with power µl) and Transmitter-l measures theexperienced SINR at its position. In this sense Equation (20)represents the SINR requirement of the "forward channel"(FC), that is the SINR requirement related to the transmissionfrom the receiver to the transmitter of link-l.Once the iterative procedure (23) converges to the optimumµ⋆, the optimal dual variables λ⋆(LP) can be retrieved fromEquation (20) as
λ⋆(LP)l = ωµ⋆
l η−1l , ∀l. (24)
The original nonlinear power control problem (7) and its (LP)formulation (17) are equivalent in the sense that there isthe following specific relation between their optimal solutions(p̃⋆,λ⋆) and (p⋆,λ⋆(LP)):
P ⋆l = eP̃
⋆l ∀l
λ⋆l = log(1 + γtgt
l )1+γtgt
l
γtgtl
P ⋆l log(2)λ⋆(LP) ∀l.
(25)
Hence, when we achieve both P ⋆l and µ⋆
l , by means ofEquations (24) and (25), we are able to compute λ⋆ as
λ⋆l = log(1 + γtgt
l )1 + γtgt
l
γtgtl
Pl⋆ log(2)ωµ⋆
l
Gll
σlγtgtl
∀l,
(26)and use it to update the user rates of Equation (12).
E. Summary
This section developed a dual loop iterative solution ap-proach to the convex optimization problem (5). The basic ideahas been to decompose the problem to separate subproblems ins̃ (Problem-I) and p̃ (Problem-II). Problem-I can be solved bygradient iterations and using Lagrangian duality to obtain theSINR targets, while Problem-II can be solved by an iterativeSINR target following inner loop. We exploited the relationshipbetween Problem-I and Problem-II such that the necessaryLagrange multipliers in the iterations of Problem-I are providedby solving Problem-II. In a practical setting, the outer and innerloop can be started off by setting a low SINR target vectorand running the inner loop to determine the transmit powerlevels and the corresponding λ⋆
i -s. The updated power levelsand Lagrange multipliers are then used as the input values tothe outer loop update rule (12), see Figure 2.
�� ��
�����
CellularUE
BS
Figure 2. An example of a D2D pair sharing a resource block (RB) with acellular UE. The D2D Tx node has a target SINR of γtgt
l set by the outerloop and runs the inner loop to set the necessary transmit power Pl. The D2DRx node transmits on the backward channel also targeting an SINR target ofγtgtl and runs its inner loop to find the correct transmit power level of µl.
µ is then used to find the Lagrange multiplier λ⋆l that is used to update the
SINR targets. At the end of the outer loop convergence, the optimal SINRtargets and associated transmit power levels are reached at all transmitters.
Figure3 shows the implementation of the Inner- and Outer-Loop mechanism in the network, clarifying which informationmust be exchanged between the transmitter and receiver ofeach pair and which computations must be performed by bothnodes in order to achieve the optimal transmit power.
V. MODE SELECTION (MS)While cellular users (UEs) communicate with their re-
spective serving base stations, D2D users may communicateboth directly or in cellular mode. In the former case, D2Dtransmitters are allowed to reuse cellular resource blocks (D2Dreuse mode). Alternatively, when D2D candidates use the directmode, they can be allocated orthogonal, i.e. dedicated resourceblocks (D2D dedicated mode), in which case the reuse gain ofD2D communications is not harvested. However, even in thisdedicated resource assignment case, D2D communications canrealize the proximity and hop gains [3].
5
Tx Rx
(0)
(1)Eq.(1)E
(q.(15)
)P
γ
P
(0)P
(0)µ(0)( )γ P(0)
(1)
(1)
Eq.(21)Eq.(15)Eq.(2
(
3
)
)
FC
Pγ
µ
µ
(1)P(0)( )FCγ µ
(1)
(1)
(2)
Eq.(1)Eq.(23)Eq.
( )
(15) P
γµ
P
(1)µ(1)( )γ P
Convergence Inner Loop: (P*,µ*)
(0)
(0) * * *
(1)
(1)
*
(1)
Eq.(26)
Eq.(12) ( ,
Eq.(13)
( , , )
)
tgt
tgt
S S
S
Pγ µ λλ
γ
↔
⇒
⇒
(
(0) * *
0) (1)
(
*
(1)1
*
)
Eq.(26)
( , Eq.(12)
Eq.(13
( , , )
)
)
tgt
tgt
S S
P
S
γ µ λλ
γ
⇒
⇒
↔
( 0 ) *
(1)
( 0 ) *
tgt
P P
γµ µ
= =
( 0 )
( 0 )
( 0 )
tg t
P
γµ
(0)P(0)µ
(0)( )γ P
Inner Loop
Inner Loop
......
Figure 3. Implementation of the iterative Power Control mechanism in thenetwork. In order to update its transmit power P, the Tx node must know theSINR evaluated at the corresponding receiver. On the other hand, also the Rxnode is able to update its power µ on the basis of the SINRFC computedby the corresponding transmitter. Hence, if both Tx node and Rx node areaware of the initial values of the iterative procedure (showed in figure in thered boxes), at each iteration of the Inner-Loop only two messages need to beexchanged between the two users.
When a D2D candidate communicates in cellular mode, theallocated resource blocks must naturally be orthogonal to theresources allocated to cellular UEs. Therefore, three differenttransmission modes can be considered for D2D communica-tions [20]: D2D mode with dedicated resources, D2D modereusing cellular resources and cellular mode. Note that whenthe D2D candidate pairs communicate in cellular mode, theyneed to be allocated downlink (DL) resources for the basestation-D2D receiver link, but this DL resource usage is notmodeled in this paper.
We now consider a cellular system with N cellular UEs andM D2D candidates (transmitters) belonging to the sets N andM respectively, such that L = N +M is the total number ofusers. We denote with xi,j(q) the allocation variable of 0 or 1corresponding to transmitter-i being assigned to resource-j incommunication mode q, where q = 0 denotes cellular modeand q = 1 indicates direct D2D mode. By definition, cellularUEs always transmit in mode q = 0, hence we can drop theindex q for xn,j(q) for all n ∈ N . To create benchmarkingcases for the adaptive MS algorithm proposed later in thepaper, in forced D2D mode, all D2D candidates operate usingthe direct link q = 1, while in forced cellular mode, allD2D candidates communicate through the base station q = 0.With this notation and terminology, formulating the resourceconstraints in the following manner will become useful:
• Forced D2D mode:xm,j(q) = xm,j(q = 1), ∀m ∈M and∑
n∈N
xn,j ≤ 1, ∀j;
• Forced cellular mode:
xm,j(q) = xm,j(q = 0), ∀m ∈M and∑n∈N
xn,j +∑
m∈M
xm,j(q) ≤ 1, ∀j;
• Adaptive mode selection:∑n∈N
xn,j +∑
m∈Mxm,j(q = 0) ≤ 1, ∀j;
xm,j(q = 1) + xm,j(q = 0) ≤ 1 ∀j, ∀m ∈M,
where the last inequality expresses that a specific D2D pair mcan only be either in D2D or cellular mode over Resource-j.Note that formally a specific D2D pair m is allowed to use aresource in D2D mode and another resource in cellular mode.
VI. MODE SELECTION AND RESOURCE ALLOCATIONPROBLEM
A. Problem FormulationWith the notation introduced in the previous section, we are
now interested in formulating and solving the resource allo-cation problem that is concerned with selecting the mode (q)for D2D candidates and allocating resource blocks to cellularUEs and D2D candidates. We formulate the resource allocationtask as a cell-based optimization problem, in which we wishto maximize the overall spectral efficiency assuming fixedtransmit powers P for each user and thermal noise σ for eachlink. Recalling the definition of spectral efficiency for user-lon a given resource-j:
ηl,j = log2(1 +Gll,jP
σ + Il,j)
we notice that it actually depends on the path gain Gkl,j
between Transmitter-k and Receiver-l on the RB-j, and on theintracell interference Il,j =
∑k ̸=l P ·Gk,l,j , due to the possible
in-cell resource sharing between D2D pairs and cellular-UEs.Hence, our target to maximize the spectral efficiency canbe interpreted as our wish to both minimize the intracellinterference and, when there are enough orthogonal resources,to select for D2D candidates the transmission mode (q) thattakes advantage of their potential proximity (i.e. higher pathgain).Let L = N +M and J denote the number of users i.e. bothcellular UE and D2D transmitters and resource blocks (RB)respectively. To formulate the resource allocation problem weassume that each transmitter can only be assigned a single RB.Thus, the user assignment task becomes (Problem-III):
maximizex(q)
∑l
∑j log2(1 +
Gll,jP ·xl,j(q)
σ+Il,j)
subject to∑
l xl,j(q = 0) ≤ 1, ∀j (C1)∑j xl,j = 1, ∀l (C2)
xl,j(q = 1) + xl,j(q = 0) ≤ 1, ∀j,∀l (C3)xl,j(q) ∈ {0, 1}
(27)
The constraints (C1) express that each RB can be allocatedto at most one user in cellular mode due to the orthogonalityconstraint. Constraints (C2) express that each user must getallocated exactly one RB and constraints (C3) ensure that toeach user is assigned only one of the two possible modes.Obviously, cellular-UEs are always assigned to mode q = 0.
6
B. A Heuristic Algorithm to Solve the User Assignment Prob-lem
To solve Problem (27) we propose a new straightforwardprocedure based on the shadowed path loss measurements. Thisscheme, that we call MinInterf, exploits the proximity betweenD2D candidates for the Mode Selection, and performs ResourceAllocation by minimizing the intracell interference. It involvestwo steps. Firstly, orthogonal resources are allocated to cellularUEs. Since MinInterf disregards frequency selective fading, toeach UE it randomly picks and assigns an available RB. Next,for each D2D candidate in the cell, MinInterf considers twopossible cases:
• D2D transmission with dedicated resource. If there areorthogonal resources left, they can be assigned to theD2D candidate so that the D2D transmission does notaffect others within the same cell. In this case, the D2Dtransmitter can also choose which is the best mode tocommunicate with his corresponding receiver (i.e. Cellu-lar Mode or D2D Mode) on the basis of the channel gainsit experiences both towards the D2D receiver (Gd2dMode)and towards the Base Station (GCellularMode). Specif-ically, if GCellularMode ≤ Gd2dMode , then the directmode is preferred.
����Tx
����RxCellularUE1
BS
GCellularModeGD2DMode
CellularUE2
��� ��� ���
�����������
�����������
�������
��������� ����
����� ��
RB: Resource BlockBS: Base Station
Figure 4. An example of a D2D transmission with dedicated resource. TheD2D Tx node selects the transmission-mode (Cellular Mode or D2D Mode)according to the shadowed path loss measurements towards the D2D Rx nodeand towards the BS. If the channel gain between the D2D pair is higher thanthe one towards the BS, then the D2D Mode is preferred.
• D2D transmission with resource reuse. When there areno unused RBs in the cell, the D2D pair must com-municate in direct mode (D2D Mode) and reuse RBs.Sharing resources with other users within the same cellproduces intra-cell interference. To reduce this intracellinterference, for each resource-j we consider the sum
S(j) = [G2Tx_1Rx,j +G1Tx_2Rx,j ] (28)
as a measure of the potential interference, that assigningthe D2D-pair to resource-j causes. Here G2Tx_1Rx,j
represents the channel gain between the D2D transmitter
and the receiver of link(s) already allocated to resource-j, which may be the cellular base station and/or anotherD2D receiver. G2Tx_1Rx,j takes into account the inter-ference that the D2D pair produces transmitting on RB-j. G1Tx_2Rx,j , on the other hand, is the channel gainbetween the transmitter(s) already allocated to resource-j (which can be both a cellular-UE and/or other D2Dtransmitters) and the receiver of the new D2D pair to beallocated. G1Tx_2Rx,j is therefore related to the interfer-ence that the D2D pair might perceive due to the reuse.Once expression (28) is computed for each availableresource-j, the D2D pair is assigned to that resourcecorresponding to the minimum value.
����Tx
����Rx
CellularUE1
BS
CellularUE2
��� ���
�������������
���������������
�����������
G2Tx_1Rx, RB1G2Tx_1Rx, RB2
G1Tx_2Rx,RB1
G1Tx_2Rx,RB2
[G2Tx_1Rx, RB1+G1Tx_2Rx,RB1] ≤ [G2Tx_1Rx, RB2 + G1Tx_2Rx,RB2]
Figure 5. An example of a D2D transmission with resource reuse. D2D Txnode communicates directly to its D2D Rx node sharing a resource block (RB)with the cellular user UE. The shared RB is selected in such a way to minimizean estimate (Eq. (28)) of the intracell interference that D2D communicationmight perceive (related to the gain G1Tx_2Rx between the UE and the D2DRx node) and produce (related to the gain G2Tx_1Rx between the D2D Txnode and the BS).
It is worth noting that the final Resource Allocation achievedwith the presented MinInterf scheme represents a suboptimalsolution of Problem (27), nevertheless numerical results showthat its interplay with the iterative Power Control procedure,which takes into account also the intercell interference, allowsto attain good performance in terms of spectrum and energyefficiency. Algorithm 1 summarizes the main steps of theMinInterf scheme.
VII. THE COMPLETE JOINT POWER CONTROL, MODESELECTION AND RESOURCE ALLOCATION ALGORITHM
With the solution to Problem-III in hand, we are now in theposition to propose the solution to the joint mode selection,resource allocation and power control problem (based on theboxed equations in the previous sections). We assume that alltransmitter-receiver pairs in the system have been assignedexactly one resource block. Note that multiple transmitter-receiver pairs may be assigned to a single RB in order to
7
Algorithm 1 MinInterfAllocate orthogonal resources to cellular-UEs (Randomly)for Each D2D candidate do
if there is an orthogonal resource-l left thenif GCellularMode ≤ Gd2dMode then
D2D candidate transmits in D2D-Mode on resource-lelse
D2D candidate transmits in Cellular-Mode onresource-l
end ifelse
for Each available resource-j doS(j) = [G2Tx_1Rx,j +G1Tx_2Rx,j ]
end forD2D candidate transmits in D2D-Mode on resource-jcorresponding to the minimum value of S
end ifend for
accommodate for cases in which resources must be overallo-cated due to high load. Once performed the resource alloca-tion/scheduling for all users, we let the outer (Eq. (12), (13),(26)) and inner loops (Eq. (1), (15), (21), (23)) determine theoptimal transmit powers that maximize the utility associatedwith each RB.
VIII. NUMERICAL RESULTS
A. Simulation Setup and Parameter Setting
In this section we consider the uplink (UL) of a 7-cellsystem, in which the number of UL physical resource blocks(RB) is 4 (per cell). We perform Monte Carlo experimentsto build some statistics over the performance measure ofinterests when employing the joint resource allocation andpower control (outer-inner loop) described in the previoussection. In each cell we drop two users ("cellular UEs") thatcommunicate with their respective serving base station (BS),that is transmit data to their serving BS. In addition, two orfour D2D candidate pairs are also dropped in the coveragearea of each cell. When two D2D pairs are dropped, the D2Dpairs must use orthogonal resources with the cellular users,but the system may select D2D (also called "direct") mode orcellular mode for them to communicate. When the D2D pairuses the cellular mode, the D2D transmitter transmits data tothe BS in the UL band, and the BS sends this data to the D2Dreceiver in the DL band. In our study, we do not model theDL transmission, essentially assuming that the DL resourcesare in abundance so that we can focus on the UL performance.When the D2D pair communicates in the direct mode, the D2Dtransmitter sends data to the D2D receiver using UL resources.This case is referred to "MS" to emphasize the role of the modeselection for the D2D candidates.
When four D2D pairs are dropped in addition to the twocellular users, two of them must use direct mode and overlap-ping resources with either other D2D direct mode users or withcellular users. This is because we assume only four resources
Table IPARAMETERS OF THE 7-CELL SYSTEM UNDER STUDY
Parameter ValueSystem Bandwidth 5MHzCarrier Frequency 2GHzGain at 1 meter distance -37dBThermal noise N0 (/MHz) -107 dBmPath Loss coefficient 3.5Lognormal shadow fading 6dBCell Radius 500mNumber of cells 7Max Tx Power 200mWMin Tx Power 5e-6WNumber of RB’s requested by users 1Max. Number of Outer-Loop iterations 200Max. Number of Inner-Loop iterations 10Number of MonteCarlo simulations 100Initial power 0.01 WInitial SINRtgt 0.2Initial µ 0.01ϵ 0.05ω 0.1Distance between cellular UE and the BS 100-400mDistance between D2D pairs 50-250m
per cell accommodating 6 transmitters and we assume thatcellular users and D2D candidates in cellular mode (thatis transmissions to the cellular BS) must remain orthogonalwithin a cell. We refer to this case as the "MS Reuse" tohighlight that there is a degree of mode selection freedomfor two D2D candidates but cellular resources now must bereused by multiple transmitters in each cell. Intuitively, weexpect some SINR degradation on the reused resources, butan increase in the total rate (and spectrum efficiency) due tomore transmissions per cell.
Distinguishing the two D2D pairs case and the four D2Dpairs case allows us to study the proximity gain (there is noresue in the first case) and the reuse gain (expected in thesecond case). The main simulation parameters are given inTableVIII-A.
B. Operation of the Outer and Inner Loops
Figures 6-10 illustrate the interplay of the outer and innerloops for the case of a single Monte Carlo drop, now assuming7x4 transmitters in the 7-cell system. Figure 6 shows the SINRtarget(s) set at each iteration of the outer-loop, while Figure 7illustrates the users’ SINR measured at each receiver duringthe inner loop (power control) iterations. The initial SINR isdetermined by the initial power levels and the geometry ofthe whole system, but the final SINR is set by the SINR targetevolution, together with the power control procedure. The outerloop adjusts the SINR target for each link from an initial lowvalue to a user specific optimal (utility maximizing) value,taking into account the current interference situation in thenetwork and trying to push the SINR target of each user as
8
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1
SINR D2D [dB]
CD
FR
UE 400 [m]
UE−Mode RD2D
50 [m]
MS RD2D
50 [m]
MS−Reuse RD2D
50 [m]
UE−Mode RD2D
100 [m]
MS RD2D
100 [m]
MS−Reuse RD2D
100 [m]
Figure 6. The evolution of the SINR target for the 7x4=28 users during the200 iterations of the outer loop. Initially, all SINR targets are set to a very lowvalue (around - 8dB), and the outer loop gradually adjusts all SINR targets tothe optimal (overall utility maximizing) value.
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1
Power UE [dBm]
CD
F
RUE
200 [m]
UE−Mode RD2D
50 [m]
MS RD2D
50 [m]
MS−Reuse RD2D
50 [m]
UE−Mode RD2D
100 [m]
MS RD2D
100 [m]
MS−Reuse RD2D
100 [m]
Figure 7. The evolution of the individual SINR values during the course ofthe outer loop. Initially, the SINR is determined by the low initial transmitpower and the geometry of the system. The outer loop successively adjuststhe individual SINR targets and these SINR targets are reached by each user.
high as beneficial for the utility. Hence, the final SINR targetsare link specific, which is the main difference as compared tofixed SINR target setting (naive) approaches.
Recall from Section III, the SINR targets correspond tothe rate vectors s̃ of the iterative procedure used to solveProblem (6). Since vectors s̃ always belong to a set of feasiblerate vectors, the beauty of our scheme is that due to the iterative
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Power UE [dBm]
CD
F
RUE
400 [m]
UE−Mode RD2D
50 [m]
MS RD2D
50 [m]
MS−Reuse RD2D
50 [m]
UE−Mode RD2D
100 [m]
MS RD2D
100 [m]
MS−Reuse RD2D
100 [m]
Figure 8. As the outer loop evolves, each user increases its rate (s) from theinitial low value until the individual rates that are overall optimal are reached.The operqtion of the outer and inner loops ensure that the individual SINRtargets remain feasible.
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Power D2D [dBm]
CD
F
RUE
200 [m]
UE−Mode RD2D
50 [m]
MS RD2D
50 [m]
MS−Reuse RD2D
50 [m]
UE−Mode RD2D
100 [m]
MS RD2D
100 [m]
MS−Reuse RD2D
100 [m]
Figure 9. The Shannon capacity of each link (transmit-receive pair) as dictatedby the outer loop. This Shannon capacity is then realized by the operation ofthe inner loop power control for both the cellular and D2D links.
interplay between the outer and inner loops, the final SINRtargets not only utility maximizing, but also remain feasible.This is the second (major) advantage over fix SINR targetschemes. As the initial SINR target is set to the same valuefor all users, also their initial rate is the same (Figure 8).Clearly the maximum rate achievable by each user dependson its specific link capacity, which is a quantity that theouter- inner loop procedure aims to maximize. As the outer
9
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Power D2D [dBm]
CD
FR
UE 400 [m]
UE−Mode RD2D
50 [m]
MS RD2D
50 [m]
MS−Reuse RD2D
50 [m]
UE−Mode RD2D
100 [m]
MS RD2D
100 [m]
MS−Reuse RD2D
100 [m]
Figure 10. Evolution of the individual transmit powers of the 7x4=28transmitters in the system during the outer loop and inner loop (zoomedsection). With a closer inspection, it is actually visible how the inner loopadjusts the individual power levels for each new SINR target as set by theouter loop, and how it typically reaches the convergence within a very fewiterations (2-3).
loop evolves, in fact, the Shannon capacity of each link isoptimized (Figure 9) and, at the same time, each users rateincreases until convergence to its maximum achievable value,represented precisely by the link capacity. Figures 8-9 showhow the final values of rates and capacities coincide at theend of the convergence of the outer and inner loop. Finally,Figure 10 illustrates the evolution of the individual transmitpowers of the transmitters in the system. At each iteration ofthe outer loop, power levels are set in such a way to fulfill therequired values of SINR (i.e. to achieve the rate defined by theouter loop), and those new power levels are obtained troughthe inner loop iterations (see zoomed section in Figure10).The merit of the inner loop is that it works in a distributedfashion and typically reaches the convergence within a veryfew iterations (2-3).
C. Resource Allocation and Mode Selection (MS)
Figures 11-18 examine the performance of the proposedscheme when D2D pairs are forced to use cellular mode (inthe Figures: "UE-Mode") or may choose between cellular anddirect D2D modes without/with resource reuse ("MS" and"MS-Reuse" respectively). In each of these figures there are6 curves showing the performance for the 3 modes (UE-Mode, MS, MS-Reuse) and 2 predetermined D2D distances(50m and 100m). The cellular users are dropped around theirserving base stations at a predetermined distance (either 200mor 400m). The MC experiments are executed such that whenthe D2D distance and the cellular UE-base station distance areset to some nominal value, there is an allowed ± 10m zonewithin which the actual drop may vary.
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SINR UE [dB]
CD
F
RUE
200 [m]
UE−Mode RD2D
50 [m]
MS RD2D
50 [m]
MS−Reuse RD2D
50 [m]
UE−Mode RD2D
100 [m]
MS RD2D
100 [m]
MS−Reuse RD2D
100 [m]
Figure 11. The distribution of the SINR of the cellular users (that is theusers transmitting to their respective serving base stations), when the cellularusers are approximately 200m from their respective serving base stations. Thefigure shows this SINR distribution in 6 different cases, depending on the D2Ddistance (50 m or 100m) and the D2D mode (D2D pair in cellular mode, D2Dpair in D2D (direct) mode using dedicated resources and D2D pair in D2Dmode reusing cellular resources). There is a ca. 1 dB loss in the SINR in thereuse mode when the D2D distance is large.
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SINR UE [dB]
CD
F
RUE
400 [m]
UE−Mode RD2D
50 [m]
MS RD2D
50 [m]
MS−Reuse RD2D
50 [m]
UE−Mode RD2D
100 [m]
MS RD2D
100 [m]
MS−Reuse RD2D
100 [m]
Figure 12. This figure is similar to Figure 11, but now the cellular users are400m far from their serving base stations. Also in this case, the impact of theD2D traffic is hardly visible - thanks to the operation of the outer-inner loopcombo.
Figures 11 and 12 compare the SINR distribution of thecellular users when they are kept at a predetermined distance(200 ± 10 and 400 ± 10 meters respectively) from theirrespective serving base stations. The D2D candidate pairs aredropped according to a surface uniform distribution, but the
10
D2D distances are constrained to 50 and 100 meters. We cansee that the impact of D2D communication on the cellular layeris basically negligible, since only a small SINR degradationcan be observed when the D2D distance is high and reusemode is enforced.
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SINR D2D [dB]
CD
F
RUE
200 [m]
UE−Mode RD2D
50 [m]
MS RD2D
50 [m]
MS−Reuse RD2D
50 [m]
UE−Mode RD2D
100 [m]
MS RD2D
100 [m]
MS−Reuse RD2D
100 [m]
Figure 13. The distribution of the SINR of the D2D users in the same 6 casesas in Figure 11. The mode selection (MS) with dedicated resources (dashedlines) show a 10-25 dB proximity gain. MS with resource reuse (2 extra D2Dtransmitters accommodated in each cell !) yields a bit lower SINR (dottedlines), but due to the large proximity gain, these SINR values are still muchhigher than when using the cellular mode for the D2D pairs (solid line).
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SINR D2D [dB]
CD
F
RUE
400 [m]
UE−Mode RD2D
50 [m]
MS RD2D
50 [m]
MS−Reuse RD2D
50 [m]
UE−Mode RD2D
100 [m]
MS RD2D
100 [m]
MS−Reuse RD2D
100 [m]
Figure 14. This figure is similar to Figure 13, but now the cellular usersare 400m far from their respective serving base stations. Also in this case,the proximity gain on the D2D SINR is large, thanks to the operation of theresource allocation and the outer and inner loop operation.
The D2D SINR distributions are shown in Figures 13 and14 (for cellular user distances 200m and 400m respectively). Itis obvious that the D2D SINR:s are radically improved whenmode selection is allowed as compared to the forced cellularcommunication mode: this improvement can be as high as20-25 dB. What is noteworthy is that even though in reusemode the D2D SINR drops a bit as compared to using cellularmode (remember that in reuse mode the system resources areoverloaded to allow for the reuse gain), this SINR degradationas compared with using orthogonal resources is small ascompared with the large proximity gain that is achieved bymode selection (MS). It is, in fact the large proximity gainthat helps to harvest the reuse gain by moving the D2DSINR into a high (+20 dB) regime and allowing D2D pairsreuse cellular resources. Our smart resource allocation ensuresminimal interference between transmitters using the sameresources and so the proposed scheme realizes the expectedgains of D2D communications.
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Power UE [dBm]
CD
F
RUE
200 [m]
UE−Mode RD2D
50 [m]
MS RD2D
50 [m]
MS−Reuse RD2D
50 [m]
UE−Mode RD2D
100 [m]
MS RD2D
100 [m]
MS−Reuse RD2D
100 [m]
Figure 15. The distribution of the transmit power levels used by the cellularusers that are 200m away from their serving base stations. When the D2Dpairs reuse cellular resources (dotted lines), the UE power levels are higherto ensure high utilities, especially for cel edge users. However, when D2Dpairs are in D2D mode using orthogonal resources the UE power levels arelower (dashed line) than when the D2D pairs are in cellular mode (solid line),because the intercell interference is somewhat lower.
The distribution of the power levels used by the cellularusers are shown in Figure 15 and 16. Again, we note thatthe impact of the D2D users on the cellular layer is basicallynegligible, except when the D2D distance is relatively large(100m) and the system is overloaded by D2D users. In thiscase, the cellular users tend to raise their transmit powers tokeep the utility at the optimal level, but still using feasiblepower levels. As we have seen in the SINR graphs, this powercontrol ensures an essentially unaffected SINR level for thecellular users.
Finally, the distribution of the D2D transmit powers are
11
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UE 400 [m]
UE−Mode RD2D
50 [m]
MS RD2D
50 [m]
MS−Reuse RD2D
50 [m]
UE−Mode RD2D
100 [m]
MS RD2D
100 [m]
MS−Reuse RD2D
100 [m]
Figure 16. This figure is similar to Figure 15, but here the distance betweenthe cellular users and the serving base station is 400m. When D2D pairs usededicated resources and D2D direct mode, they disturb the (basically cell edge)cellular users more than when they are in cellular mode (compare the solidline with the dashed lines). D2D pairs in reuse mode cause the cellular usersto use even higher powers (dotted lines).
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Power D2D [dBm]
CD
F
RUE
200 [m]
UE−Mode RD2D
50 [m]
MS RD2D
50 [m]
MS−Reuse RD2D
50 [m]
UE−Mode RD2D
100 [m]
MS RD2D
100 [m]
MS−Reuse RD2D
100 [m]
Figure 17. The distribution of the D2D pair transmit power levels when thecellular users are 200m far from their serving base stations. When the D2Ddistance is 100m (dashed and dotted lines to the right from the solid lines)the transmit power levels are higher than in cellular mode (solid lines). Incontrast, when the D2D distance is 50m, the D2D transmit power levels arelower than when the D2D pairs are in cellular mode.
shown in Figures 17 and 18. The transmit power of the D2Dtransmitters can actually increase somewhat when communi-cating in direct mode when the D2D distance is large (100 m)and the cellular user distance is low. This is because when the
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400 [m]
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50 [m]
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50 [m]
MS−Reuse RD2D
50 [m]
UE−Mode RD2D
100 [m]
MS RD2D
100 [m]
MS−Reuse RD2D
100 [m]
Figure 18. This figure is similar to Figure 17, but now the cellular users are400m far from their serving base stations.
cellular users are close to the BS, the D2D transmit powerin cellular mode is quite low (thanks to the low intercellinterference) and when the D2D pairs use D2D mode (at largeD2D distance), they need to increase their power from thisquite low level to maintain a high utility.
D. The Impact of ω
Recall from the problem formulation of 4 that the parameterω sets the weight of the overall power consumption in theutility. In this section we set ω to 0.1 (rendering the powerconsumption less important than the achieved sum rate) andto 10 (stimulating low sum power operation of the system)and we are interested in the overall (average) achieved ratesand power levels as the cellular user-base station and D2Ddistances are varied over the coverage area of a cell.
Figures 19-20 compare the average rate when ω is set to 0.1and 10 respectively and we again emphasize that the shownaverage rates are the averages of the optimum sum rates thatcan be achieved in a given Monte Carlo drop.
As expected, the utility maximizing (optimum) average rateis much (with around 50%) higher when ω is small. However,this rate increase comes at the price of a large power increase,especially when the transmitter-receiver distances (both forcellular and D2D pairs) increase, see Figures 21-22.
IX. CONCLUSIONS
In this work, we addressed the problem of selecting commu-nication mode, allocating resources and setting SINR targetsand transmission power in an integrated multicell cellular andD2D communication system. Our solution approach hingesupon a utility function that can easily takes into account theinherent trade off between spectrum and energy efficiencyand an iterative algorithm that solves the mode selection
12
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2
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6
8
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RD2D
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RUE
[m]
Ave
rage
Rat
e [b
/s]
Figure 19. The average rate (i.e. over all 7x4 links and over all Monte Carloexperiments) for ω = 0.1 as the function of the D2D and user-base stationdistances.
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e [b
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Figure 20. This figure is similar to Figure 19, but for ω = 10. The averagerate in the system is now much lower, since the utility discourages high powerlevels.
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[m]RUE
[m]
Ave
rage
Use
d P
ower
[mW
]
Figure 21. Average power used by (cellular and D2D) transmitters for ω =0.1 as the function of the D2D and the user - base station distances. Theoptimal (utility maximizing) power levels increase with the distances, sincethe utility is dominated by the achieved rate.
50100
150200
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100
200
300
4000
2
4
6
RD2D
[m]RUE
[m]
Ave
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ower
[mW
]
Figure 22. This figure is similar to Figure 21, but for ω = 10. The powerlevels are now significantly lower, but at this low level they are sensitive tothe distances, basically as expected.
(MS), resource allocation (RA) and power control problemat different time scales. The MS and RA problem is solvedat a coarse time scale, which is indeed in line with recentproposals on cellular network assisted D2D communications[4], while the SINR target adjustment and power control loopsare executed on a finer time scale. The strength of this approachis not only that it finds the utility maximizing joint allocationbut also that it operates in a distributed fashion and remainsfeasible. This is very important for D2D communications,where the SINR dynamics of the system is necessarily largeand in practice would render fix SINR target setting approachesinappropriate.
The numerical results provide a number of important andinteresting insights. Due to the utility optimal allocation, theimpact of the D2D layer on the cellular layer is small, which isa key requirement in any system that allows licensed spectrumresources to be used by D2D traffic. Also, the proximitygain of D2D communications is so large when the D2Dlie in the proximity of one another that there is a potentialfor overloading (reusing) cellular resources by multiple D2Dpairs. Finally, our proposed scheme can readily accommodate asingle parameter (ω) that tunes the spectral efficiency - energyefficiency trade off by setting the system operational pointinto, e.g. a lower average rate regime and drastically reducingthe overall energy consumption. A future work, triggered bythe insight of the large SINR dynamics (and potential fairnessproblems) among the population within the coverage area isto study the setting of the ω parameter on an individual basis,rather than using a single common ω.
ACKNOWLEDGMENTS
The Authors thank Dr. Claes Tidestav and Dr. Gunnar Barkwhose detailed comments helped to improve the presentationand the contents of the paper.
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