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Spectrum Sharing for Device-to-DeviceCommunication in Cellular Networks

Xingqin Lin, Jeffrey G. Andrews and Amitabha Ghosh

Abstract—This paper addresses two fundamental and interre-lated issues in device-to-device (D2D) enhanced cellular networks.The first issue is how D2D users should access spectrum, and weconsider two choices: overlay (orthogonal spectrum between D2Dand cellular UEs) and underlay (non-orthogonal). The secondissue is how D2D users should choose between communicatingdirectly or via the base station, a choice that depends on distancebetween the potential D2D transmitter and receiver. We proposea tractable hybrid network model where the positions of mobilesare modeled by random spatial Poisson point process, with whichwe present a general analytical approach that allows a unifiedperformance evaluation for these questions. Then, we deriveanalytical rate expressions and apply them to optimize the twoD2D spectrum sharing scenarios under a weighted proportionalfair utility function. We find that as the proportion of poten tialD2D mobiles increases, the optimal spectrum partition in theoverlay is almost invariant (when D2D mode selection thresholdis large) while the optimal spectrum access factor in the underlaydecreases. Further, from a coverage perspective, we revealatradeoff between the spectrum access factor and the D2D modeselection threshold in the underlay: as more D2D links areallowed (due to a more relaxed mode selection threshold), thenetwork should actually make less spectrum available to themto limit their interference.

Index Terms—Device-to-device communication, spectrum shar-ing, mode selection, cellular networks, stochastic geometry.

I. I NTRODUCTION

Device-to-device (D2D) networking allows direct commu-nication between cellular mobiles, thus bypassing the basestations (BS). D2D opens up new opportunities for proximity-based commercial services, particularly social networking ap-plications [2], [3]. Other use cases include public safety,localdata transfer and data flooding [3]. Further, D2D may bringbenefits such as increased spectral efficiency, extended cellularcoverage, improved energy efficiency and reduced backhauldemand [4], [5].

A. Related Work and Motivation

The idea of incorporating D2D communication in cellularnetworks, or more generally, the concept of hybrid networkconsisting of both infrastructure-based and ad hoc networkshas long been a topic of considerable interest. In earlier studiesD2D was mainly proposed for relaying purposes [6], [7]. By

Xingqin Lin and Jeffrey G. Andrews are with Department of Electrical& Computer Engineering, The University of Texas at Austin, USA. (E-mail:xlin@utexas.edu, jandrews@ece.utexas.edu). Amitabha Ghosh is with Nokia.(E-mail: amitava.ghosh@nsn.com).

This research was supported by Nokia Siemens Networks and the NationalScience Foundation grant CIF-1016649. A part of this paper was presentedat IEEE Globecom 2013 in Atlanta, GA [1].

allowing radio signals to be relayed by mobiles in cellularnetworks, it was shown that the coverage and throughputperformance can be improved [6], [7]. More recently, D2Din cellular networks has been motivated by the trend ofproximity-based services and public safety needs [3], [5].Inparticular, Qualcomm publicized the necessity of developing atailored wireless technology to support D2D in cell phones andhas also built a D2D demonstration system known as Flash-LinQ [8], [9]. Now D2D is being studied and standardized bythe 3rd Generation Partnership Project (3GPP). In particular,potential D2D use cases have been identified in [3]; and anew 3GPP study item focusing on the radio access aspect wasagreed upon at the December 2012 RAN plenary meeting [10].Through the most recent 3GPP meetings, initial progress onD2D evaluation methodology and channel models has beenmade [11], and broadcast communication is the current focusof D2D study in 3GPP [12].

In parallel with the standardization effort in industry, basicresearch is being undertaken to address the many fundamen-tal problems in supporting D2D in cellular networks. Onefundamental issue is how to share the spectrum resourcesbetween cellular and D2D communications. Based on the typeof spectrum sharing, D2D can be classified into two types:in-band and out-of-band. In-band refers to D2D using thecellular spectrum, while out-of-band refers to D2D utilizingbands (e.g. 2.4GHz ISM band) other than the cellular band.In-band D2D can be further classified into two categories:overlay and underlay. Overlay means that cellular and D2Dtransmitters use orthogonal time/frequency resources, whileunderlay means that D2D transmitters opportunistically accessthe time/frequency resources occupied by cellular users.

Existing research relevant to this paper includes spectrumsharing in cognitive radio networks, where secondary cognitivetransmitters may access the primary spectrum if the primarytransmitters are not active or they do not cause unacceptableinterference [13]. For example, to protect the primary users,secondary transmissions in [14], [15] are regulated by sensingthe activities of primary transmissions, while [16] imposesstringent secondary access constraints on e.g. collision prob-ability. Multi-antenna techniques are used in [17]–[19] tominimize secondary interference to primary network. Auctionmechanisms are used in [20], [21] to control the spectrumaccess of secondary network. More recently, the economicaspects of spectrum sharing in cognitive radio networks havegained much interest. For example, [22] adopts a dynamicalgame approach to study the spectrum sharing among a primaryuser and multiple secondary users. Similarly, a three-stagedynamic game is formulated in [23] to study spectrum leasing

2

and pricing strategies, while [24] designs incentive schemesfor spectrum sharing with cooperative communication.

Unlike spectrum sharing in cognitive radio networks, D2Dspectrum sharing is controlled by the cellular network. HowD2D should access the spectrum is a largely open question,though some initial results exist (see e.g. [25]–[29]). D2Dspectrum sharing is further complicated by D2Dmode se-lection which means that a potential D2D pair can switchbetween direct and conventional cellular communications [4],[5]. Determining an optimum D2D mode selection threshold– which we define as the Tx-Rx distance under which D2Dcommunication should occur – is another objective of thispaper.

Note that D2D is different from ad hoc networks whoseanalysis and design are notoriously difficult (see e.g. [30]–[32]). A key difference is that D2D networking can be assistedby the cellular network infrastructure which is not availableto a typical ad hoc network [5], [33]. Nevertheless, supportingD2D communication requires a lot of new functionalities [3],[5], [10] and significantly complicates the cellular networkdesign. The main objectives of this paper are to provide atractable baseline model for D2D-enabled cellular networksand to develop a unified analytical framework for the analysisand design of D2D spectrum sharing. This goal is fairlyambitious since D2D spectrum sharing scenarios are quitediverse. The main tool used in this paper is stochastic ge-ometry, particularly the Poisson point processes (PPP). Notethat the PPP model for BS locations has been recently shownto be about as accurate in terms of both signal-to-interference-plus-noise ratio (SINR) distribution and handover rate as thehexagonal grid for a representative urban cellular network[34],[35].

B. Contributions and Outcomes

The main contributions and outcomes of this paper are asfollows.

1) A tractable hybrid network model:In Section II, weintroduce a hybrid network model, in which the random andunpredictable spatial positions of mobile users are modeled asa PPP. This model captures many important characteristics ofD2D-enabled cellular networks including D2D mode selection,transmit power control and orthogonal scheduling of cellularusers within a cell.

2) A unified performance analysis approach:In SectionIII, we present a general analytical framework. With thisapproach, a unified performance analysis is conducted for twoD2D spectrum sharing scenarios: overlay and underlay in-bandD2D. In particular, we derive analytical rate expressions andapply them to optimize spectrum sharing parameters.

3) Design insights:The following observations are madefrom the derived analytical and/or numerical results underthemodel studied in this paper and may be informative for systemdesign.

Overlay vs. underlay. We evaluate the rate performancein both overlay and underlay scenarios. We observe that D2Dmobiles can enjoy much higher data rate than regular cellularmobiles in both scenarios. As for cellular mobiles in the

overlay case, their rate performance also improves due tothe offloading capability of D2D communication. In contrast,the rate performance of cellular mobiles in the underlaycase does not improve or even slightly degrades with D2Dcommunication.1 This is because cellular mobiles suffer frominterference caused by the underlaid D2D transmissions, whichoffsets the D2D offloading gain. From an overall mean-rate(averaged across both cellular and D2D mobiles) perspective,both overlay and underlay provide performance improvement(vs. pure cellular).

D2D mode selection.We derive the optimal D2D modeselection threshold that minimizes the transmit power. Wefind that the optimal threshold is inversely proportional tothe square root of BS density and monotonically increaseswith the pathloss exponent. Moreover, it is invariant withthe distance distribution of potential D2D pairs. D2D modeselection and spectrum sharing may be jointly optimized frome.g. the rate perspective. From a coverage perspective, wereveal a tradeoff between the D2D spectrum access and modeselection threshold in the underlay case: as more D2D linksare allowed (due to a more relaxed mode selection threshold),the network should actually make less spectrum available tothem to limit their interference.

The rest of this paper is organized as follows. Section IIdescribes the system model and performance metrics. Prelim-inary analytical results are presented in Section III. Two D2Dspectrum sharing scenarios, overlay and underlay, are analyzedand optimized in Sections IV and V, respectively. Section VIpresents a case study of overlay vs. underlay, and is followedby our concluding remarks in Section VII.

II. SYSTEM MODEL AND PERFORMANCEMETRICS

A. Network Model

As shown in Fig. 1, we consider a hybrid network consistingof both cellular and D2D links and focus on the uplink.The BSs are regularly placed according to a hexagonal grid.Denoting by1/λb the area of a hexagonal cell,λb can beregarded as the average number of BSs per unit area. Thetransmit user equipments (UE) are randomly distributed andmodeled by an independentlymarkedPPP denoted as

Φ = {(Xi, δi, Li, Pi)}. (1)

Here{Xi} denote the spatial locations of the UEs. Denote byΦ ∈ R

2 the unmarkedPPP{Xi} with λ being its intensity.{δi} denote the types of the UEs and are assumed to be i.i.d.Bernoulli random variables withP(δi = 1) = q ∈ [0, 1]. Inparticular, UE i is called apotential D2D UE2 if δi = 1;otherwise, it is called acellular UE. So,q is a simple indicatorof the load of potential D2D traffic.{Li} denote the lengthsof radio links. For notational simplicity, denote byLc (resp.Ld) the generic random variable for the link length of a typicalcellular UE (resp. potential D2D UE).{Pi} denote the transmit

1Note that the underlay study in this paper assumes that D2D randomlyaccesses the cellular spectrum. With carefully designed dynamic schedulingin the underlay, the rate of cellular mobiles may also increase, and the rateof D2D mobiles may further increase.

2It is called potential D2D UE as a UE with D2D traffic can use eithercellular or D2D mode.

3

−2000 −1000 0 1000 2000

−2,000

−1,000

0

1,000

2,000

Fig. 1. A hybrid network consisting of both cellular and D2D links. Solidtriangles, solid squares and dots denote BSs, uplink cellular transmitters andD2D transmitters, respectively. For clarity we omit plotting D2D receivers,each of which is randomly located on the circle centered at the associatedD2D transmitter.

powers of UEs. In this paper we usechannel inversionforpower control, i.e.,Pi = Lα

i , whereα > 2 denotes the pathlossexponent; extension to distance-proportional fractionalpowercontrol (i.e.,Pi = Lαǫ

i where ǫ ∈ [0, 1]) is straightforward.Similarly, we usePc and Pd to denote the generic randomvariables for the transmit powers of cellular and potentialD2DUEs, respectively.

Remark on channel inversion.Note that channel inversionin this paper only compensates for the large-scale pathloss. Inparticular, it doesnot compensate for the small-scale fading.This channel inversion scheme has two advantages: 1) it doesnot lead to excessively large transmit power when the link ispoor (due to the small-scale fading), and 2) the transmitteronly needs a long-term statistic (i.e. pathloss) to decide itstransmit power, i.e., instantaneous channel state informationis not required to be available at the transmitter as small-scalefading is not compensated for. Note that for ease of exposition,we assume in the analysis that the average received power is1 due to channel inversion, i.e.,Pi = Lα

i . In other words,Pi

should be considered asvirtual transmit power, and should bescaled appropriately to map to theactual transmit powerPi,say, Pi = ρPi, whereρ is the coefficient of proportionality.Normally,ρ ≪ 1 since the practical transmit power of wirelessdevices is far less than the pathloss.

Next, let us introduce the notation SNRm to denote theaveragereceivedsignal power normalized by noise power, i.e.,

SNRm =PL−α

N0Bw

=ρPL−α

N0Bw

=1

ρ−1N0Bw

, (2)

whereN0 denotes the one-sided power spectral density of theadditive white Gaussian noise, andBw denotes the channelbandwidth. In the rest of this paper, if the average receivedpower is normalized to1, we useN0 to denote theequivalentnoise powerρ−1N0Bw. By choosing the operating regimeSNRm (or equivalently, the coefficientρ) appropriately, wecan make sure that the UE power constraints are satisfied and

thus there is no need to truncate UE transmit power to meetthe peak power constraint. We will give more detailed resultsin Section III-B to illustrate the above argument.

The potential of D2D will largely depend on the amountof local traffic that may be routed through local direct paths,instead of infrastructure paths. One possible approach to model“data localization” would be based on current user trafficstatistics. However, it appears very challenging to acquiresuch traffic data, which is typically owned by operators andcontains sensitive and proprietary information. Even if thecurrent traffic data could be obtained from the operators, itmight not be too useful, since presumably D2D’s availabilitycould change future traffic patterns. For example, once usersrealize high D2D speeds are possible, more local sharingis likely to occur. So far, no commonly agreed upon D2Ddistance distribution has appeared in the literature. In theabsence of such an accepted model, we assume that eachpotential D2D receiver is randomly and independently placedaround its associated potential D2D transmitter with isotropicdirection and Rayleigh distributed distanceD with probabilitydensity function (PDF) given by

fD(x) = 2πξxe−ξπx2

, x ≥ 0. (3)

In other words, the potential D2D receiver is randomly placedaround its associated potential D2D transmitter accordingtoa two-dimensional Gaussian distribution, which results in(3).A similar Gaussian assumption has also been used in [36]to analyze the performance of FlashLinQ. The analysis andcalculations in this paper can be used to study other D2Ddistance distributions as well.

In this paper, we considerdistance-based D2D mode se-lection: cellular mode is used ifD ≥ µ; otherwise, D2Dmode is selected. If we assume that the received signal power(averaged over fast fading) is only a function of distanceand pathloss exponent, distance-based D2D mode selectionis equivalent to the average received-signal-power or SNR-based mode selection, to which the results in this paper canbe directly applied.

B. Spectrum Sharing

The spectrum sharing models studied in this paper aredescribed as follows.

1) Overlay in-band D2D:The uplink spectrum is dividedinto two orthogonal portions. A fractionη is assigned to D2Dcommunication while the other1 − η is used for cellularcommunication. We termη spectrum partition factorin theoverlay.

2) Underlay in-band D2D: We assume that each D2Dtransmitter uses frequency hopping to randomize its inter-ference to other links. Specifically, we divide the channelinto B subchannels. Each D2D transmitter may randomly andindependently accessβB of them, where the factorβ ∈ [0, 1]measures the aggressiveness of D2D spectrum access. We termβ spectrum access factorin the underlay.

Though we examine D2D spectrum sharing from the fre-quency domain perspective, it is straightforward to interpretthe derived results in this paper from a two-dimensional

4

DiskA

Cellular UE Typical cellular signal link

Potential D2D UE in cellular mode Cellular interfering link

Fig. 2. An approximate uplink interference analysis. The typical cellulartransmitter is uniformly distributed inA, while cellular interferers form aPPP with densityλb outside the diskA.

time-frequency perspective. Take the overlay for example:theequivalent interpretation is that a proportionη of the OFDMA(orthogonal frequency-division multiplexing access) resourceblocks is assigned to D2D while the remaining resource blocksare used by cellular.

C. Transmission Scheduling

Cellular transmitters including cellular UEs and potentialD2D UEs in cellular mode form a PPPΦc with intensityλc = (1−q)λ+qλP(D ≥ µ). We assume an orthogonal multi-ple access technique and that uplink transmitters are scheduledin a round-robin fashion. It follows that only one uplinktransmitter in each macrocell can be active at a time. Generallyspeaking, scheduling cellular transmitters in an orthogonalmanner leads todependentthinning of PPPΦc. This makesthe analysis intractable and some simplified assumptions areneeded (see e.g. [37]). In this paper, denoting byA thecoverage region of a hexagonal macrocell, we approximateA by a disk that has the same area as the hexagonal cell, i.e.,A = B(0, R) whereB(x, r) denotes the ball centered atx

with radiusr andR =√

1πλb

. To avoid triviality, we assumeλc ≥ λb, which is reasonable as the uplink transmitter densityis usually larger than the BS density. Further, we assume thatthe typical active cellular transmitter is uniformly distributedin the coverage regionA, and that the locations of cellularinterferers form a PPPΦc,a with intensityλb. For the typicaluplink transmission, cellular interferers are located outsidethe regionA. Fig. 2 illustrates the proposed approximateinterference analysis for a typical uplink transmission. Dueto the use of this approximation, the analytical results aboutcellular performance derived in this paper are approximations;for notation simplicity, we will present the results as equalitiesinstead of the more cumbersome approximations in the sequel.The approximation will be numerically validated later.

As for potential D2D UEs in D2D mode, they form a PPPΦd with intensity λd = qλP(D < µ). For D2D mediumaccess control, we consider a simple spatial Aloha accessscheme: in each time slot each potential D2D UE in D2D modetransmits with probabilityκ and is silent with probability1−κ;

the activity decisions are independently made over differenttime slots and different transmitters. The study of this simplebaseline medium access scheme can serve as a benchmark forfuture work on more sophisticated D2D scheduling schemes,e.g., carrier sense multiple access (CSMA) or centralizedscheduling.

D. Performance Metrics

We will analyze the average rates of cellular and potentialD2D UEs,Tc andTd. Recall that potential D2D UEs can useeither cellular or D2D mode. Denote byTd the average rateof potential D2D UEs in D2D mode. Conditioning on usingcellular mode, the rateTd of potential D2D UEs equals therate Tc of cellular UEs; conditioning on using D2D mode,the rateTd of potential D2D UEs equalsTd by definition.Under the assumed distance-based D2D mode selection, atypical potential D2D UE uses cellular mode with probabilityP(D ≥ µ) and D2D mode with probabilityP(D < µ). Tosum up, the average rate of potential D2D UEs can be writtenas

Td = P(D ≥ µ) · Tc + P(D < µ) · Td. (4)

III. PRELIMINARY ANALYSIS

In this section we present preliminary analytical results,which lay the foundation for the study of overlay and underlayin-band D2D in the next two sections.

A. A Unified Analytical Approach

Consider a typical transmitter and receiver pair interfered byK types of heterogeneous interferers. The set of thek-th typeof interferers is denoted asMk. In this paper, we focus onfrequency-flat narrowband channels; the results can be readilyextended to OFDM-based frequency-selective wideband chan-nels, each of which can conceptually be regarded as a set ofparallel frequency-flat narrowband channels.

The baseband received signalY0[n] at the typical receiverlocated at the origin can be written as

Y0[n] =√

P0L−α0 G0S0[n]

+K∑

k=1

∑

i∈Mk

√

Pi‖Xi‖−αGiSi[n] + Z[n], (5)

whereP0, L0, G0 andS0[n] are associated with the typical linkand denote the typical link’s transmit power, length, channelfading and unit-variance signal, respectively;Pi, ‖Xi‖, Gi andSi[n] are associated with the interfering link from transmitterito the typical receiver and denote the interfering link’s transmitpower, length, channel fading and unit-variance signal, respec-tively; Z[n] is additive white Gaussian noise with constantPSDN0 Watts/Hz. It follows that the received SINR is givenby SINR = W

I+N0Bw, where signal powerW = P0L

−α0 G0,

and interference powerI =∑K

k=1

∑

i∈MkPi‖Xi‖−αGi.

In this paper, recall we assume channel inversion, i.e.,P0L

−α0 = 1, and thusW = G0. For simplicity we consider

Rayleigh fading, i.e.,G0 ∼ Exp(1), and assume independent

5

fading over space. Then the following corollary will be par-ticularly useful.

Corollary 1. Suppose SINR= WI+N0

, whereW ∼ Exp(1)andI respectively denote the (random) signal and interferencepowers, andN0 denotes theequivalentnoise power. IfW andI are independent,

E [log(1 + SINR)] =∫ ∞

0

e−N0x

1 + xLI(x) dx, (6)

whereLI(s) = E[e−sI ] denotes the Laplace transform ofI.

Corollary 1 follows from a more general result given in[38] and its proof may also be directly found in [39]. Notethat in this paper, interference is not Gaussian but may beconsidered asconditionally Gaussian. Specifically, assumingall the transmitters use Gaussian signaling, the interferenceis Gaussian conditioned on the fading and node locationsin the network. Then treating the interference as noise, wemay invoke Shannon’s formula to determine the maximumachievable spectral efficiency of a typical link. If the randomfading and node locations are furthered averaged out, we arriveat the expression (6). Though not optimal in an information-theoretical sense, (6) serves as a good performance metric andhas been widely adopted in literature [39].

Next we define the ergodic link spectral efficiencyR, whichcombines modulation and coding schemes in the physical layerand multiple access protocols in the medium access controllayer, as follows.

R = E [∆ · log(1 + SINR)] , (7)

where∆ denotes the time and/or frequency resources accessedby the typical link. For example, in the overlay with spectrumpartition factor η, a typical D2D link with random Alohaaccess probabilityκ can effectively accessκη time-frequencyresources. We will analyze ergodic link spectral efficiencyRin detail in Sections IV and V.

B. Transmit Power Analysis

In this subsection, we analyze the transmit power distri-butions, particularlyE[Pc] and E[Pd], the average transmitpowers of cellular and potential D2D UEs. The derived resultsare not only interesting in its own right but also are extensivelyused for the analysis of rate performance later. To this end,denote byPd the generic random variable for the transmitpower of potential D2D UEs in D2D mode.

Lemma 1.The average transmit powers of a typical cellularUE, a potential D2D UE and, a D2D-mode potential D2D UEare respectively given by

E[Pc] =1

(1 + α2 )π

α2 λ

α2

b

(8)

E[Pd] = e−ξπµ2

E[Pc] + (ξπ)−α2 γ(

α

2+ 1, ξπµ2) (9)

E[Pd] =(ξπ)−

α2

1− e−ξπµ2γ(

α

2+ 1, ξπµ2), (10)

where γ(s, x) =∫ x

0 zs−1e−z dz is the lower incompleteGamma function.

Density of macrocellsλb (π5002)−1 m−2

Density of UEsλ 10× (π5002)−1 m−2

D2D distance parameterξ 10× (π5002)−1 m−2

Potential D2D UEsq 0.2Pathloss exponentα 3.5SNRm 10 dBMode selection thresholdµ 200 mAloha access probabilityκ 1Spectrum partition factorη 0.2UE weights(wc, wd) (0.6, 0.4)Spectrum access factorβ 1Number of subchannelsB 1

TABLE ISIMULATION /NUMERICAL PARAMETERS

Proof: See Appendix A.Note that bothE[Pc] andE[Pd] increase as pathloss expo-

nentα increases and are inversely proportional to the squareroot of BS density. Next we examine how to choose D2Dmode selection thresholdµ to minimize the average transmitpowerE[Pd] of potential D2D UEs.

Proposition 1.For any distribution functionfD(x) of thenonnegative random distanceD, E[Pd] is minimized when

µ⋆ = (E[Pc])1

α =

(

1

1 + α2

)1

α√

1

πλb. (11)

Proof: See Appendix B.Prop. 1 shows thatµ⋆ is only a function of the average

transmit powerE[Pc] of cellular UEs and is independent ofthe distribution ofD; in particular, the Rayleigh assumptionmade in (3) is not needed here. Specifically,µ⋆ is inverselyproportional to the square root of BS densityλb, whichis intuitive: cellular mode becomes more favorable whenmore BSs are available. In addition,( 1

1+α2

)1

α monotonicallyincreases asα increases. This implies thatµ⋆ increases inα,agreeing with intuition: local transmission with D2D mode ismore favorable for saving transmit power when the pathlossexponent increases.

Before ending this section, we give a numerical example inFig. 3 showing that UE power constraint can be satisfied bychoosing the right operating regime SNRm. Throughout thispaper, the parameters used in plotting numerical or simulationresults are summarized in Table I unless otherwise specified.In Fig. 3, the cellular peak powerPc,max is defined asthe minimum transmit power used by a cell-edge cellulartransmitter to meet the target SNRm, i.e.,Pc,max is determined

by SNRm =Pc,maxR

−α

N0Bw. Similar, the D2D peak powerPd,max

is determined by SNRm =Pd,maxµ

−α

N0Bw. The average power of a

cellular (resp. D2D) transmitter can be obtained by multiplying(8) (resp. (10)) with the scaling factorN0Bw · SNRm. Asshown in Fig. 3, the typical UE power constraint23 dBm(i.e. 200 mW) is well respected even when SNRm=10 dB, arelatively high average received SNR in the uplink. Besides,Fig. 3 also shows that compared to cellular transmitters, D2D

6

−6 −4 −2 0 2 4 6 8 10−15

−10

−5

0

5

10

15

20

25

SNRm

(dB)

Tra

nsm

it P

ower

(dB

m)

Peak transmit powerAverage transmit power

D2D TX

Cellular TX

UE Power Constrain: 23 dBm

Fig. 3. UE transmit power versus network operating regime SNRm withN0 = −174 dBm/Hz andBw = 1 MHz.

transmitters can save about15 dB transmit power in achievingthe same SNRm target, demonstrating the energy efficiency ofD2D communication. In other words, with the same powerbudget, the D2D links can enjoy about15 dB higher SNRmthan the cellular links.

IV. A NALYSIS OF OVERLAY IN-BAND D2D

A. Link Spectral Efficiency

Let us consider a typical D2D link. As the underlyingPPP is stationary, without loss of generality we assume thatthe typical receiver is located at the origin. Note that theanalysis carried out for a typical link indicates the spatiallyaveraged performance of the network by Slivnyak’s theorem[40]. Henceforth, we focus on characterizing the performanceof a typical link, which may be either a D2D or cellular linkdepending on the context.

With overlay in-band D2D, the interferers are cochannelD2D transmitters. Due to the random Aloha access, the effec-tive interferers constitute a homogeneous PPP with densityκλd, a PPP thinned fromΦd with thinning probability κ.Denoting this thinned PPP byκΦd, the interference at thetypical D2D receiver is given by

Id =∑

Xi∈κΦd\{o}Pd,iGi‖Xi‖−α.

Proposition 2.With overlay in-band D2D, the complementarycumulative distribution function (CCDF) of the SINR of D2Dlinks is given by

P(SINR≥ x) = exp(

−N0x− cx2

α

)

, x ≥ 0, (12)

wherec is a non-negative constant given by

c =κq(λξ − (λξ + λπµ2)e−ξπµ2

)

sinc( 2α )

. (13)

Further, the spectral efficiencyRd of D2D links is given by

Rd = κ

∫ ∞

0

e−N0x

1 + xe−cx

2

α dx. (14)

Proof: This proposition follows by evaluating the Laplacetransform ofId and using (6). See Appendix C for details.

Note that in Prop. 2, asµ increases,c monotonicallyincreases, which in turn results in monotonically decreasingRd. This agrees with intuition: the spectral efficiency of D2Dlinks decreases when more potential D2D UEs choose D2Dmode (leading to increased interference). In particular,

Rd,min = limµ→∞

Rd = κ

∫ ∞

0

e−N0x

1 + xe−κq λ

ξ(sinc( 2

α))−1x

2

αdx.

The typical D2D link experiences the most severe interferencein this case and thus has the minimum spectral efficiency. Onthe contrary,

Rd,max = limµ→0

Rd = κ

∫ ∞

0

1

1 + xe−N0x dx = κeN0E1(N0),

whereE1(z) =∫∞z

e−x

x dx is the exponential integral. Thetypical D2D link is free of interference in this case and thushas the maximum spectral efficiency.

Now let us consider a typical uplink. With overlay in-bandD2D, the interferers are cellular transmitters in the othercells.The interference at the typical BS is given by

Ic =∑

Xi∈Φc,a∩Ac

Pc,iGi‖Xi‖−α.

Proposition 3.With overlay in-band D2D, the CCDF of theSINR of cellular links is given by

P(SINR≥ x) = exp

(

−N0x− 2πλb

∫ ∞

R(

1− 2F1(1,2

α; 1 +

2

α;− x

(R/r)α)

)

r dr

)

, x ≥ 0, (15)

where2F1(a, b; c;x) denotes the hypergeometric function, and

R =√

1πλb

. Further, the spectral efficiencyRc of cellular linksis given by

Rc =λb

λc(1− e

− λcλb )

∫ ∞

0

e−N0x

1 + xexp

(

− 2πλb

×∫ ∞

R

(

1− 2F1(1,2

α; 1 +

2

α;− x

(R/r)α)

)

r dr

)

dx, (16)

whereλc = (1 − q)λ+ qλe−ξπµ2

.

Proof: See Appendix D.Unlike the closed form expression (12) for the CCDF of the

SINR of D2D links, the expression (15) for the CCDF of theuplink SINR involves an integration. To have some insights,we consider sparse (i.e.λb → 0) and dense (i.e.λb → ∞)networks in the following corollary.

Corollary 2. In a sparse network withλb → 0, the CCDF ofthe SINR of cellular links is given by

P(SINR≥ x) = exp

(

−(

N0 +4

α2 − 4

)

x

)

, x ≥ 0, (17)

7

In a dense network withλb → ∞, the CCDF of the SINR ofcellular links is given by

P(SINR≥ x) = exp

(

−N0x− 1

2sinc( 2α )

x2

α

)

, x ≥ 0. (18)

Proof: See Appendix E.In a sparse network, Corollary 2 implies that interference

and noise have the same impact on the SINR coverage per-formance (in the order sense). From this perspective, we maysimply consider interference as an extra source of noise. Thus,the sparse network is noise-limited. In contrast, in a densenetwork, the impact of interference behaves differently for UEswith different SINR targets. For users with low SINR target(i.e. x → 0), interference has a more pronounced impact onthe SINR coverage performance than the noise. The converseis true for users with high SINR target (i.e.x → ∞).

If we denote byθc the SINR threshold for successful uplinktransmissions and consider the outage probabilityP(SINR <θc), Corollary 2 implies that, asθc → 0, the outage proba-bility of a sparse network scales asΘ(θc). In this case, theoutage performance is noise-limited. In contrast, the outageprobability of a dense network scales asΘ(θ

2

αc ). In this case,

the outage performance is interference-limited.Before ending this section, we validate the analytical results,

particularly the CCDF of the uplink SINR (since we adopt anapproximate approach for the analysis on the uplink SINR).As all the major analytical results presented in this paper arefunctions of SINR, it suffices to validate the analytical SINRdistributions by simulation rather than repetitively validatingeach analytical expression which in turn is a function of SINR.In Fig. 4(a), we compare the analytical uplink SINR CCDF(given in Prop. 3) to the corresponding empirical distributionobtained from simulation using the hexagonal model. Thesimulation steps are described as follows.

1) Place the BSs according to a hexagonal grid in a largeareaC; the area of a hexagon equals1/λb.

2) Generate a random Poisson numberN with parameterλc|C|.

3) GenerateN points that are uniformly distributed inC;theseN points represent the cellular transmitters.

4) For each BS, it randomly schedules a cellular transmitterif there is at least one in its coverage region.

5) Determine the transmit power of each scheduled trans-mitter.

6) Generate independently the fading gains from eachscheduled cellular transmitter to each BS.

7) Collect the SINR statistics of the cellular links locatedin the central hexagonal cells (to avoid boundary effect).

8) Repeat the above steps10, 000 times.Fig. 4(a) shows that the analytical results match the empiri-

cal results fairly well; the small gaps arise as we approximatethe hexagonal model using a PPP model with a guard radius.Recall that in Fig. 4(a),λb = (π5002)−1 m−2, and thusthe network is sparse. In the sparse regime, as established inCorollary 2, SNRm (or equivalently,N0) has a considerableimpact on the uplink SINR CCDF; Fig. 4(a) confirms theanalytical result.

We next compare the SINR distribution of a typical D2Dlink (given in Prop. 2) to the corresponding empirical distri-bution obtained from Monte Carlo simulation. The results areshown in Fig. 4(b), from which we can see that the analyticalresults closely match the empirical results as in this case noapproximation is made in the analysis.

−15 −10 −5 0 5 10 150

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SINR (dB)

SIN

R C

CD

F

Hexagonal: Monte CarloPPP: Analytical Evaluation

SNRm

−> ∞

SNRm

= 10dB

SNRm

= −3dB

SNRm

= 3dB

SNRm

= 0dB

(a)

−20 −15 −10 −5 0 5 10 150

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SINR (dB)

SIN

R C

CD

F

PPP: Monte CarloPPP: Analytical Evaluation

SNRm

= −3dB

µ = 375m

SNRm

= −3dB

µ = 150m

SNRm

= 10dB

µ = 375mSNR

m −> ∞

µ = 375m

SNRm

−> ∞

µ = 150m

SNRm

= 10dB

µ = 150m

(b)

Fig. 4. Validation of the analytical SINR CCDF of cellular and D2D links.Fig. 4(a) shows the SINR CCDF of cellular links, while Fig. 4(b) shows theSINR CCDF of D2D links.

B. Optimizing Spectrum Partition

In this section we study how to choose the optimal spectrumpartition factorη⋆ such that

η⋆ = arg maxη∈[0,1]

u(Tc, Td), (19)

whereu(Tc, Td) is a utility function that can take differentforms under different design metrics. In this paper we usethe popular weighted proportional fair function:wc logTc +wd logTd, wherewc, wd > 0 are weight factors such thatwc+wd = 1.

8

To optimize the spectrum partition, we first need the rateexpressions forTc andTd. For a given spectrum partitionη,the (normalized) rate (bit/s/Hz)Tc of cellular UEs equalsRc

multiplied by the available spectrum resource1−η. In contrast,the rateTd of potential D2D UEs equals(1− η)Rc if cellularmode is used; otherwise, i.e., D2D mode is used, it equalsηRd. To summarize,

Tc = (1− η)Rc

Td = (1− η)e−ξπµ2

Rc + η(1− e−ξπµ2

)Rd. (20)

Fig. 5(a) shows the average rates of cellular and potentialD2D UEs as a function of D2D mode selection thresholdµ.3

As expected, the average rate of cellular UEs increases asµincreases. This is because with increasingµ, less potentialD2D UEs choose cellular mode and correspondingly cellularUEs can be scheduled more often. In contrast, the averagerate of potential D2D UEs first increases and then decreasesas µ increases. This is because the average rate of potentialD2D UEs is co-determined by its cellular-mode rate andD2D-mode rate: cellular-mode rate increases withµ whileD2D-mode rate decreases withµ (due to the increased intra-tier interference). Fig. 5(a) also shows that with appropriatechoice ofµ, potentials D2D UEs can enjoy much higher ratethan cellular UEs. Meanwhile, cellular UEs also benefit fromoffloading the traffic by D2D communication.

In [1], we have derived the optimal weighted proportionalfair spectrum partition, which reads as follows.

Lemma 2.The optimal weighted proportional fair spectrumpartitionη⋆ is given by

η⋆ = 1− wc

wc + wd· 1

1− (eξπµ2 − 1)−1 RcRd

(21)

if Rd > wc+wd

wd

1eξπµ2−1

Rc; otherwise,η⋆ = 0. In particular,limµ→∞ η⋆ = wd.

From Lemma 2, we can see that ifµ → ∞, i.e., potentialD2D UEs are restricted to use D2D mode, the optimal partitionη⋆ converges towd. So the optimal partitionη⋆ simply equalsthe weight we assign to the potential D2D UEs. In Fig. 5(b),we plot the utility value vs.η under different values ofq,the proportion of potential D2D UEs. It can be seen that theoptimal η⋆ = 0.4 = wd, which is independent ofq. This plotvalidates Lemma 2.

Note that the optimal partitionη⋆ in Lemma 2 is givenfor a fixed D2D mode selection thresholdµ. With η⋆(µ)we can further optimize the objective functionu(Tc, Td) bychoosing the optimalµ⋆. With the derivedη⋆(µ) the objectivefunctionu(Tc, Td) is only a function of the scalar variableµ.Thus, the optimalµ⋆ can be numerically found efficiently, andthe computed(µ⋆, η⋆(µ⋆)) gives the optimal system designchoice.

3Note that for fair comparison, we normalize the transmit powers of cellularand D2D transmitters by taking into account their transmission bandwidthswhen plotting numerical results in this paper. For example,in Fig. 5(a) withη = 0.2, when SNRm = 10dB, the average received SNR of cellular linksand of D2D links are given by SNRm+10 log10

1

1−η= 10.97 dB, SNRm+

10 log101

η= 16.99 dB, respectively.

0 100 200 300 400 5000

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

D2D Mode Selection Threshold (m)

Nor

mal

ized

Ave

rage

Rat

e (b

it/s/

Hz)

Cellular UEPotential D2D UE

SNRm

=10dB

SNRm

=5dB

SNRm

=0dB

(a)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9−2.6

−2.4

−2.2

−2

−1.8

−1.6

−1.4

−1.2

D2D Spectrum Partition Factor

Obj

ectiv

e V

alue

q=0.1, 0.3, 0.6, 0.7, 0.8, 0.9

(b)

Fig. 5. Overlay in-band D2D: Fig. 5(a) shows the average rates of cellularand potential D2D UEs vs. D2D mode selection threshold. Fig.5(b) showsthe utility value vs. D2D spectrum partition factorη under different values ofq, the proportion of potential D2D UEs.

V. A NALYSIS OF UNDERLAY IN-BAND D2D

A. Link Spectral Efficiency

Considering the random access of D2D in both frequencyand time domains, the effective D2D interferers constituteahomogeneous PPP with densityκβλd, a PPP thinned fromΦd with thinning probabilityκβ. Denote this thinned PPPby κβΦd. Considering further the interference from cellulartransmittersΦc,a with densityλb, the interference at the typicalD2D receiver is given by

Id =∑

Xi∈κβΦd\{o}Pd,iGi‖Xi‖−α +

∑

Xi∈Φc,a

Pc,iGi‖Xi‖−α.

Proposition 4.With underlay in-band D2D, the CCDF of the

9

SINR of D2D links is given by

P(SINR≥ x) =

exp

(

−N0x− cβx2

α − 1

2sinc( 2α )

(βx)2

α

)

, x ≥ 0. (22)

Further, the spectral efficiencyRd of D2D links is given by

Rd = κ

∫ ∞

0

e−N0x

1 + xexp

(

−cβx2

α − (βx)2

α

2sinc( 2α )

)

dx. (23)

Proof: See Appendix F.Now let us consider a typical uplink. With underlay in-

band D2D, the interferers are out-of-cell cellular transmittersΦc,a∩ Ac and D2D transmitters inκβΦd. The interference atthe typical BS is given by

Ic =∑

Xi∈Φc,a∩Ac

Pc,iGi‖Xi‖−α +∑

Xi∈κβΦd

Pd,iGi‖Xi‖−α.

Proposition 5.With underlay in-band D2D, the CCDF of theSINR of cellular links is given by

P(SINR≥ x) = exp

(

−N0x− cβ1− 2

αx2

α − 2πλb

×∫ ∞

R

(

1− 2F1(1,2

α; 1 +

2

α;− x

(R/r)α)

)

rdr

)

, x ≥ 0. (24)

Further, the spectral efficiencyRc of cellular links is given by

Rc =λb

λc(1− e

− λcλb )

∫ ∞

0

e−N0x

1 + xexp

(

− cβ1− 2

αx2

α − 2πλb

×∫ ∞

R

(

1− 2F1(1,2

α; 1 +

2

α;− x

(R/r)α)

)

r dr

)

dx. (25)

Proof: See Appendix G.Prop. 4 (resp. Prop. 5) implies that the spectral efficiency

Rd of D2D links (resp.Rc of cellular links) decreases asβincreases. In other words, with largerβ, the increased D2Dinterferer density in each subchannel has a more significantimpact than the decreased transmit power per subchannel ofD2D interferers. To sum up, from the perspective of maximiz-ing the spectral efficiency of either D2D or cellular links, thedesign insight here is that underlay in-band D2D should accesssmall bandwidth with high power density rather than spreadingthe power over large bandwidth. However, smallβ limits thespectrum resource available to the D2D transmissions, whichin turn limits the D2D throughput or rate.

B. Optimizing Spectrum Access

As in the case of overlay, we choose an optimal spectrumaccess factorβ⋆ in underlay case such that

β⋆ = arg maxβ∈[0,1]

wc logTc + wd logTd. (26)

To this end, we first need the rate expressions forTc andTd.By definition, it is easy to see that

Tc = Rc, Td = e−ξπµ2

Rc + β(1− e−ξπµ2

)Rd, (27)

whereRc andRd are given in (25) and (23), respectively.

Fig. 6(a) shows the average rates of cellular and potentialD2D UEs as a function ofµ in the underlay scenario. Recall inthe overlay case, the rate of cellular UEs increases withµ dueto D2D offloading gain. In contrast, here the rate of cellularUEs stays almost constant or even slightly decreases withµ.This is because cellular UEs now suffer from the interferencecaused by the underlaid D2D transmissions; this offsets theoffloading gain. Fig. 6(a) also shows that largerβ leads tohigher rate of potential D2D UEs but lower rate of cellularUEs, which is pretty intuitive.

Next we optimize the spectrum access. From (27), we cansee that the spectrum access factorβ in the underlay scenariohas a much more complicated impact onTc andTd thanη doesin the overlay scenario. As a result, a closed-form solutionforβ⋆ is hard to obtain. Nevertheless, the optimization problemis of single variable and can be numerically solved. In Fig.6(b), we plot the utility value vs.β under different values ofq, the proportion of potential D2D UEs. It can be seen thatthe optimalη⋆ decreases asq increases.

0 100 200 300 400 5000

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

D2D Mode Selection Threshold (m)

Nor

mal

ized

Ave

rage

Rat

e (b

it/s/

Hz)

Cellular UEPotential D2D UE

SNRm

= 0dB

β = 1β = 0.1SNRm

= 10dB

(a)

0.2 0.4 0.6 0.8 1−2.1

−2

−1.9

−1.8

−1.7

−1.6

−1.5

−1.4

−1.3

D2D Underlay Spectrum Access Factor

Obj

ectiv

e V

alue

q=0.1, 0.3, 0.6, 0.7, 0.8, 0.9

(b)

Fig. 6. Underlay in-band D2D: Fig. 6(a) shows the average rates of cellularand potential D2D UEs vs. D2D mode selection threshold. Fig.6(b) showsthe utility value vs. D2D spectrum access factorβ under different values ofq, the proportion of potential D2D UEs.

10

Thus far, spectrum sharing is optimized from the rateperspective. In practice, D2D spectrum sharing may have totake into account other factors. Take the underlay scenariofor example. In order to protect the cellular transmissions,we may have to limit the proportion of the spectrum thatcan be accessed by D2D. Specifically, assume that D2Dtransmissions have a target outage probabilityǫd. Then D2Dcoverage probability must satisfy

P(W

Id +N0≥ θd) = E[e−βBθd(Id+N0)] ≥ 1− ǫd ⇒

(

N0Bθd + θ2

α

d c(µ))

β +θ

2

α

d

2sinc( 2α )

β2

α ≤ log(1

1 − ǫd), (28)

whereθd is the SINR threshold for successful D2D transmis-sions, andc(µ) = c (given in Prop. 2) monotonically increasesasµ increases. Inequality (28) reveals the tradeoff betweenβandµ. In particular, if each D2D transmission has access tomore spectrum, i.e. largerβ, the signal power is spread overwider channel bandwidth and thus the effective SINR gets“thinner” in each subchannel. This in turn implies that givenlink reliability requirement onθd andǫd, less cochannel D2Dtransmissions can be supported, i.e.,µ has to be decreased tomake more potential D2D UEs use cellular mode rather thanD2D mode.

Similarly, if cellular transmissions have a target outageprobability ǫc, the cellular coverage probability must satisfy

P(W

Ic +N0≥ θc) = E[e−θc(Ic+N0)] ≥ 1− ǫc ⇒

θ2

αc c(µ)β1− 2

α ≤ log(1

1− ǫc)−N0Bθc − 2πλb

×∫ ∞

R

(

1− 2F1(1,2

α; 1 +

2

α;− θc

(R/r)α)

)

r dr. (29)

As in (28), a joint constraint onβ and µ is imposed by(29); a tradeoff betweenβ and µ exists. Incorporating theconstraints (28) and (29) into the D2D underlay spectrumaccess optimization problem is an interesting topic for futurework.

VI. OVERLAY VS . UNDERLAY: A CASE STUDY

In this section we provide a case study to compare the rateperformance of overlay with that of underlay. The results areshown in Fig. 7, in which the label “Overall” indicates the rateperformance averaged across both cellular UEs and potentialD2D UEs. In Fig. 7, the percentage of D2D links equalsq(1−e−ξπµ2

) = 0.2(1 − e−10

π5002π2002) = 16%. Even with only

16% of the links being D2D links, the overall rate increasesremarkably in both overlay and underlay due to the high ratesof D2D links. Fig. 7 further shows that a smallη (say,0.3) inoverlay leads to as good rate performance of potential D2DUEs as its counterpart in underlay with a largeβ (say,0.9)because D2D UEs in overlay are free of cellular interference.

It can be observed from Fig. 7 that the rate of potentialD2D UEs in overlay increases almost linearly asη increases.In contrast, asβ increases, the rate of potential D2D UEs inunderlay increases in a diminishing way. This is because asβ increases, the interference from cellular transmissions and

−4 −2 0 2 4 6 8 100

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

SNRm

(dB)

Nor

mal

ized

Ave

rage

Rat

e (b

it/s/

Hz)

η = 0.1η = 0.2η = 0.3

Potential D2D UE

Overall

Cellular UE

(a)

−4 −2 0 2 4 6 8 100

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

SNRm

(dB)

Nor

mal

ized

Ave

rage

Rat

e (b

it/s/

Hz)

β = 0.1β = 0.5β = 0.9

Cellular UE

Overall

Potential D2D UE

(b)

Fig. 7. Overlay vs. underlay: Fig. 7(a) and Fig. 7(b) show theaverage rateperformance of overlay and underlay, respectively. The benchmark curve forNo D2D is almost indistinguishable from the cellular curve withη = 0.1 inFig. 7(a) (resp. the cellular curve withβ = 0.1 in Fig. 7(b)). For clarity, wedo not plot the benchmark curves.

the mutual interference of D2D links increase and thus thereceived SINR degrades. Meanwhile, asη (resp.β) increases,the rate of cellular UEs decreases due to the less spectrumresource (resp. more D2D interference). Further, the rateperformance of cellular UEs is relatively sensitive toη inoverlay but is robust toβ in the underlay.

Recall that the higher the SNRm, the higher the transmitpowers. As SNRm increases from−4 dB to 10 dB, the rateof cellular UEs in both overlay and underlay and the rate of po-tential D2D UEs in overlay increase linearly, implying thatthetransmit powers are not high enough to make the performanceinterference-limited. In contrast, the rate of potential D2DUEs in underlay increases in a diminishing way, especiallywhen SNRm exceeds4 dB, implying that the performance isgradually limited by the interference caused by the increasedtransmit powers.

We summarize the main lessons drawn from the above

11

discussion as follows. In underlay, the rate of potential D2DUEs is more resource-limited; a linear increase in the spectrumresource can generally lead to a linear rate increase. Incontrast, the rate of potential D2D UEs is more interference-limited, mainly due to the cochannel cellular interference. Asfor the rate of cellular UEs, it is sensitive to the reductionofspectrum in overlay but is more robust to the cochannel D2Dinterference in underlay.

VII. C ONCLUSIONS ANDFUTURE WORK

In this paper we have jointly studied D2D spectrum sharingand mode selection using a hybrid network model and a unifiedanalytical approach. Two scenarios, overlay and underlay,havebeen investigated. Though spectrum sharing has been mainlystudied from a rate perspective, we also show in the underlaycase how to apply the derived results to study spectrum sharingfrom a coverage perspective, leading to the discovery of thetradeoff between the underlay D2D spectrum access and modeselection.

The ground cellular network studied in this paper is tradi-tional cellular architecture consisting of only tower-mountedmacro BSs. It is interesting to extend this work to (possiblymulti-band) heterogeneous networks [41], [42] consistingofdifferent types of lower power nodes besides macro BSs.This work can be further extended in a number of ways.At the PHY layer, it would be of interest to study theimpact of multiple antenna techniques and/or extend the point-to-point D2D communication to group communication andbroadcasting as they are important D2D use cases. At the MAClayer, it would be useful to explore efficient D2D schedulingmechanisms. At the network layer, our work can be used asa stepstone for a wide range of interesting topics includingmulti-hop and cooperative D2D communications.

ACKNOWLEDGMENT

The authors thank Editor Nikos Sagias and the anonymousreviewers for their valuable comments and suggestions.

APPENDIX

A. Proof of Lemma 1

Recall that we adopt an approximate approach on the uplinkanalysis. In particular, we approximate the coverage region of

a hexagonal macrocell as a ball with radiusR =√

1πλb

andassume that the typical active cellular transmitter is uniformlydistributed inA. Thus,P(Lc ≤ x) = x2/R2, 0 ≤ x ≤ R.

Taking the derivative and usingR =√

1πλb

, we obtain that

the PDF of the length of a typical cellular link isfLc(x) =2πλbx · Ix∈[0, 1√

πλb]. Then the average transmit power of a

cellular transmitter equals

E[Pc] = E[Lαc ] =

∫ 1/√πλb

0

2πλbxα+1 dx

=1

(1 + α2 )π

α2 λ

α2

b

. (30)

The PDF of the length of a typical D2D link is

fLd|D<µ(x) =

fD(x)

P(D < µ)=

2πξxe−ξπx2

1− e−ξπµ2if x ∈ [0, µ);

0 otherwise.

Correspondingly, itsα-th moment can be computed as follows:

E[Lαd |D < µ] =

1

P(D < µ)

∫ µ

0

2πξxα+1e−ξπx2

dx

=(ξπ)−

α2

1− e−ξπµ2γ(

α

2+ 1, ξπµ2), (31)

where we have usedP(D < µ) = 1 − e−ξπµ2

in the lastequality. By definition,

E[Pd] = P(D ≥ µ)E[Lαc ] + (1− P(D ≥ µ))E[Lα

d |D < µ].

SubstitutingE[Lαc ] andE[Lα

d |D < µ] into the above equationcompletes the proof.

B. Proof of Proposition 1

As in the proof of Lemma 1, the average transmit powerof a potential D2D UE equalsE[Pc] conditioned on cellularmode is used; otherwise, i.e., conditioned on D2D mode isused, it equalsE[Pd]. Conditioning on D2D mode, the D2Dlink lengthLd is distributed as 1

P(D<µ)fD(x), 0 ≤ Ld < µ. It

follows thatE[Pd] = E[Lαd |D < µ] =

∫ µ

0 xα fD(x)P(D<µ) dx and

thus

E[Pd] = (1 −∫ µ

0

fD(x) dx) · E[Pc]

+

∫ µ

0

fD(x) dx ·∫ µ

0

xα fD(x)

P(D < µ)dx

= (1 −∫ µ

0

fD(x) dx) · E[Pc] +

∫ µ

0

xαfD(x) dx. (32)

Taking the derivative ofE[Pd] with respect toµ, ddµE[Pd] =

fD(µ)(µα − E[Pc]). Setting the derivative to zero, we obtainthe stationary point(E[Pc])

1/α. It is easy to see thatE[Pd] isdecreasing whenµ ∈ [0, (E[Pc])

1/α) and is increasing whenµ ∈ [(E[Pc])

1/α,∞). Hence,E[Pd] is minimized atµ⋆ =(E[Pc])

1/α. The proof is complete by plugging the explicitexpression forE[Pc] (given in (8)).

C. Proof of Proposition 2

Consider the conditional Laplace transform

LId(s) = E[e−s

∑

Xi∈Φd\{o}Pd,iGi‖Xi‖−α

|o ∈ Φd]

= E!o[e

−s∑

Xi∈ΦdPd,iGi‖Xi‖−α

]

= E[e−s

∑

Xi∈ΦdPd,iGi‖Xi‖−α

]

= exp

(

−2πκλd

∫ ∞

0

(1 − E[exp(−sPdGr−α)])r dr

)

= exp

(

− πκλd

sinc( 2α )

E[P2

α

d ]s2

α

)

, (33)

where E!o[·] denotes the expectation with respect to the

reduced Palm distribution, the third equality follows fromSlivnyak’s theorem [40], the fourth equality is due to the

12

probability generating functional of PPP [39], and we haveused G ∼ Exp(1) in the last equality. Note thatλd =qλ(1 − e−ξπµ2

), and from the proof of Lemma 1,

E[P2

α

d ] = E[L2d] =

1

ξπ− µ2e−ξπµ2

1− e−ξπµ2. (34)

Pluggingλd andE[P2

α

d ] into LId(s) yields LId(s) = e−cs2

α ,wherec is given in Prop. 2. Invoking (6) yields the spectralefficiencyRd = κE[log(1 + SINR)] of D2D links.

The CCDF of the SINR of D2D links can be obtained asfollows:

P(SINR≥ x) = P (Go ≥ x(Id +N0))

= E[e−x(Id+N0)] = e−xN0LId(x). (35)

PluggingLId(x) into (35) completes the proof.

D. Proof of Proposition 3

The spectral efficiencyRc of cellular links is given by

Rc = Eo[

1

Nlog(1 + SINR)]

= Eo[

1

N] · E[log(1 + SINR)], (36)

whereN is the random number of potential uplink transmitterslocated in the cell. Due to the PPP assumption,N is a Poissonrandom variable with parameterλc/λb. Denoting byN thenumber ofother potential uplink transmitters located in thecell except the one under consideration, i.e.,N = 1 + N .Thus,

Eo[

1

N] =

∞∑

n=1

1

n· Po(N = n) =

∞∑

n=1

1

n· Po(N = n− 1)

=

∞∑

n=1

1

n·(λcλb)n−1

(n− 1)!e− λc

λb =λb

λc

∞∑

n=1

(λcλb)n

n!e−λc

λb

=λb

λc(1− e

− λcλb ), (37)

where the third equality is due to Slivnyak’s theorem [40]:conditioning on the transmitter under consideration, the otherpotential uplink transmitters are still PPP distributed and thusN ∼ Poisson(λc/λb) under the Palm probabilityPo.

Next we calculateE[log(1 + SINR)]. To this end, we firstcalculate the Laplace transform

LIc(s) = E[e−s

∑

Xi∈Φc,a∩Ac Pc,iGi‖Xi‖−α

]

= E[e−s

∑

Xi∈Φc,aPc,iGi‖Xi‖−α

I{‖Xi‖≥R} ]

= E[∏

Xi∈Φc,a

e−sPc,iGi‖Xi‖−αI{‖Xi‖≥R} ]

= exp

(

−∫ 2π

0

∫ ∞

0

1− E[e−sPcGr−αI{r≥R} ]λbr dr dθ

)

= exp

(

−2πλb

∫ ∞

R

(1 − E[e−sPcGr−α

])r dr

)

= exp

(

−2πλb

∫ ∞

R

(1 − E[e−sLαc Gr−α

])r dr

)

, (38)

where we have used the probability generating functional

of PPP in the fourth equality, andPc = Lαc in the last

equality. Using thatLc is distributed asfLc(x) = 2πλbxI(x ∈[0, 1/

√πλb]), andG ∼ Exp(1),

EG,Lc[e−xLα

c Gr−α

] = ELc

[

1

1 + xLαc r

−α

]

= 2πλb

∫

√

1

πλb

0

t

1 + xr−αtαdt

= 2F1(1,2

α; 1 +

2

α;− x

(r√πλb)α

).

PluggingLIc(s) into (6) yields

E[log(1 + SINR)] =∫ ∞

0

e−N0x

1 + xexp

(

− 2πλb

×∫ ∞

R

(1− 2F1(1,2

α; 1 +

2

α;− x

(r√πλb)α

))r dr

)

dx. (39)

Plugging (37) and (39) into (36) gives the spectral efficiencyRc of cellular links. The CCDF of the SINR of cellular linkscan be similarly obtained as in (35).

E. Proof of Corollary 2

For a sparse cellular network with smallλb, R is large andthus forr ∈ [R,∞),

1− E[e−xGLαc r−α

] ≈ xr−αE[GLα

c ] = xr−αE[G]E[Pc]

=xr−α

(α2 + 1)(πλb)α2

, (40)

where we have used the approximation1− e−y ≈ y for smallvalue y, the independence of fadingG and link lengthLc

andE[Pc] = E[Lαc ] in the first equality, and have plugged in

E[G] = 1 andE[Lαc ] = 1/(α2 + 1)(πλb)

α2 (c.f. Lemma 1) in

the last equality. Accordingly, the Laplace transform of theuplink interference is given by

LIc(s) = exp

(

−2πλb

∫ ∞

R

(1− E[e−sGLαc r−α

])r dr

)

≈ exp

(

−2πλbs

∫ ∞

R

r−α+1

(α2 + 1)(πλb)α2

dr

)

= exp

(

− 4

α2 − 4s

)

. (41)

For a dense network with largeλb, R is small and thus

LIc(s) ≈ exp

(

−2πλb

∫ ∞

0

(1− E[e−sGLαc r−α

])r dr

)

= exp

(

− πλb

sinc( 2α )

E[P2

αc ]s

2

α

)

= exp

(

− 1

2sinc( 2α )

s2

α

)

, (42)

where we have plugged inE[P2

αc ] = E[L2

c ] =1

2πλb(c.f. (30))

in the last equality. Combining the above asymptotic resultswith Prop. 3 completes the proof.

F. Proof of Proposition 4

Note that the average D2D transmit powerE[Pd] here is only1/βB of the one given in Lemma 1 as each D2D transmitter

13

accessesβB subchannels and needs to split its power accord-

ingly, i.e., Pd = 1βBLα

d . Hence,E[P2

α

d ] = ( 1βB )

2

αE[L2d]. The

corresponding spectral efficiency (normalized by bandwidthB) is given by

Rd =κE[log(1 + SINR)] = κ

∫ ∞

0

e−N0x

1 + xLId(βBx) dx

=κ

∫ ∞

0

e−N0x

1 + xexp

(

− πκλd

sinc( 2α )

E[P2

α

d ](βBx)2

α

)

× exp

(

− πλb

sinc( 2α )

E[P2

αc ](βBx)

2

α

)

dx. (43)

Hereλd = β ·qλ(1−e−ξπµ2

), the density of D2D transmittersthat is “seen” from each subchannel, and from the proof inAppendix A,

E[P2

α

d ] = (1

βB)

2

αE[L2d] = (

1

βB)

2

α

(

1

ξπ− µ2e−ξπµ2

1− e−ξπµ2

)

E[P2

αc ] = (

1

B)

2

αE[L2c ] = (

1

B)

2

α1

2πλb.

Plugging λd, E[P2

α

d ] and E[P2

αc ] into (43) establishes the

expression for the D2D link spectral efficiency. The CCDFof the SINR of D2D links can be similarly obtained as in(35).

G. Proof of Proposition 5

As in the proof of Prop. 4, the Laplace transform of theinterference can be calculated as follows.

LIc(s) = exp

(

− πκλd

sinc( 2α )

E[P2

α

d ]s2

α − 2πλb

×∫ ∞

R

(

1− 2F1(1,2

α; 1 +

2

α;− s/B

(r√πλb)α

)

)

r dr

)

. (44)

Then as in the proof of Prop. 3, the spectral efficiencyRc ofcellular links is given by

Rc = Eo[

1

Nlog(1 + SINR)]

=λb

λc(1 − e

−λcλb )

∫ ∞

0

e−N0x

1 + xLIc(Bx) dx. (45)

PluggingLIc(s) (44) into the above equation establishes theexpression for the cellular link spectral efficiency. The CCDFof the SINR of D2D links can be similarly obtained as in (35).

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