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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 15, NO. 8, AUGUST 2016 5695

Performance Analysis of Network-Assisted D2DDiscovery in Random Spatial Networks

Dionysis Xenakis, Marios Kountouris, Senior Member, IEEE, Lazaros Merakos,Nikos Passas, and Christos Verikoukis, Senior Member, IEEE

Abstract— Device-to-device (D2D) discovery is the inextricableprelude for the direct exchange of local traffic between cellularusers in proximity. The D2D discovery process can be basedon either autonomous actions taken by D2D-enabled devices,also known as network-assisted D2D discovery or core networkfunctionalities to estimate proximity, also known as network-assisted D2D discovery. A key advantage of network-assistedD2D discovery is its potential to reduce the energy, signaling,and interference required for D2D discovery, by exploitingknowledge of the network layout. We analyze the performance ofnetwork-assisted D2D discovery in random spatial networks andderive useful guidelines for its design. We derive approximateexpressions for the distance distribution between two D2D peersconditioned on the core network’s knowledge of the cellularnetwork layout, assuming that the base stations are distributedaccording to the Poisson point process. The expressions are usedto assess the interplay between the D2D discovery probability andkey system parameters, such as network intensity and transmitpower, as well as to identify conditions to maximize the D2Ddiscovery probability. Numerical results validate the accuracy ofour findings and provide insights on the performance tradeoffsof network-assisted D2D discovery.

Index Terms— Device-to-device discovery, network-assisteddiscovery, location information, evolved packet core.

I. INTRODUCTION

DEVICE-TO-DEVICE (D2D) communication has drawnsignificant attention due to the demand for direct

exchange of local traffic between nearby devices. AlthoughD2D communication has been primarily motivated by appli-cations such as social networking and public safety com-munications [1], it can also be used to offload the cellularaccess network, or even to establish bidirectional links forcarrying out local measurement/control data [2]. Two functionsare instrumental for direct proximity-based traffic exchange:D2D discovery and D2D communication [1]. D2D discoveryinvolves the detection of D2D-enabled devices in proximity,

Manuscript received May 29, 2014; revised February 4, 2015,September 23, 2015, and April 12, 2016; accepted April 28, 2016. Date ofpublication May 12, 2016; date of current version August 10, 2016. This workwas supported by the SMART-NRG Project FP7-PEOPLE-2013-IAPP underGrant 612294. The associate editor coordinating the review of this paper andapproving it for publication was A. Sprintson.

D. Xenakis, L. Merakos, and N. Passas are with the Department ofInformatics and Telecommunications, University of Athens, Athens 15784,Greece (e-mail: [email protected]; [email protected]; [email protected]).

M. Kountouris is with the Mathematical and Algorithmic SciencesLaboratory, France Research Center, Huawei Technologies Company, Ltd.,Paris 25536, France (e-mail: [email protected]).

C. Verikoukis is with the Centre Tecnologic de Telecomunicacions deCatalunya, Barcelona 08860, Spain (e-mail: [email protected]).

Color versions of one or more of the figures in this paper are availableonline at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TWC.2016.2568172

while D2D communication includes the establishment, andutilization of a physical link between D2D-enabled users todirectly exchange data without routing packets through theaccess network.

D2D discovery is within the scope of the 3rd GenerationPartnership Project (3GPP) for the Long Term Evolution (LTE)Release 12 system [3], a.k.a. LTE-Advanced (LTE-A). 3GPPfocuses on two types of D2D discovery: direct and network-assisted discovery. The direct discovery is based on the D2D-enabled devices to autonomously discover, or indicate theirpresence to, other devices. On the contrary, network-assisteddiscovery, often referred to as Evolved Packet Core (EPC)-level discovery, relies on the core network to determinethe proximity of D2D-enabled devices. Both methods havetheir own advantages and unique challenges. Direct D2Ddiscovery can be deployed even when outside network cov-erage, network-assisted discovery enables network operatorsto reduce the required energy, signaling, and interference.

Network-assisted discovery can provide more accurate esti-mation of the proximity between D2D-enabled devices byexploiting existing knowledge of the cellular network layout.Besides, the EPC is at least aware of the Base Station (BS)with which the cellular devices associate with, also coined asthe associated BS [4], [5]. This knowledge combined withadditional information on the spatial relation between therespective associated BSs, such as the distance or the relativeposition of the D2D peers with respect to their associated BSs,e.g. distance and angle of arrival (AoA), can be the cornerstonefor more accurate D2D discovery at the EPC. The LTE-Asystem already supports a suite of user and BS measurementsthat can be readily utilized towards this direction [6].

In this paper, we focus on the performance of network-assisted D2D discovery, where the EPC utilizes existingknowledge of the network layout to trigger (or not) theD2D discovery phase between two D2D-enabled devices ata given time. Under this viewpoint, we develop an analyticalframework that enables the EPC to evaluate the probabilitythat two tagged devices are in proximity, a.k.a. D2D discoveryprobability, conditioned on the available knowledge of thenetwork layout. Compared to D2D discovery with no knowl-edge of the network layout, network-assisted D2D discoveryenables network operators to better infer on ‘the probabilitythat two D2D peers are in proximity’. As highlighted in [7],the performance of D2D discovery is tightly coupled withthe notion of proximity. In the sequel, we assume that twoD2D-enabled devices are in proximity whenever the long-termaverage received signal power from the D2D source is greaterthan or equal to the receiver sensitivity at the D2D target.

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5696 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 15, NO. 8, AUGUST 2016

We choose to follow this definition for two main reasons.First, the D2D discovery process is most likely to be basedon the long-term average and not the instantaneous receivedpower at the D2D target, i.e. small-scale fading is averagedout. Second, this notion of proximity is closer to the one usedduring the cell search phase [8]. Assuming that the pathloss isinversely proportional to the distance between the D2D peersand governed by a pathloss exponent a, the D2D discoveryprobability is defined as follows:

AJ � P[Pt Z−a ≥ Pr

∣∣J], (1)

where J denotes the vector of location information parametersavailable for the two D2D peers at the EPC, Pt the transmitpower at the D2D source, Pr the receiver sensitivity at the D2Dtarget, and Z the distance between the D2D peers. The receiversensitivity Pr is typically fixed and depends on the systemparameters that specify the reference measurement channel [9],e.g. duplexing mode and bandwidth. In the sequel, we furtherassume that the transmit power Pt is fixed and known at theEPC for mathematical tractability.

By rearranging (1), it can be easily shown that the D2Ddiscovery probability equals the value of the cumulative

density function (cdf) of the distance Z at point(

PtPr

) 1a,

conditioned on the available knowledge given in J. In lightof the above remark, we derive the conditional probabilitydistribution of the distance Z between two D2D peers. Theanalysis is focused on the scenarios where the EPC is at leastaware of either the distance between the associated BS ofthe D2D source and the associated BS of the D2D target,denoted by Dk , or their neighboring degree, denoted by k.The neighboring degree between the associated BSs of the twoD2D peers is equal to k, if the associated BS of the D2D targetis the k-th nearest BS of the associated BS of the D2D source.We consider these two broad scenarios of practical interest,since they both allow the EPC to directly relate the locationsof the D2D source and the D2D target without requiring addi-tional information on their exact locations. Another importantreason for focusing on these two classes is the fact that thedistance Dk and the neighboring degree k typically remainfixed over time and do not depend on the D2D peers.

Note that analyzing the performance of D2D discoveryunder the impact of the interference caused by other D2Dusers or BSs, i.e. based on the Signal to Interference plus NoiseRatio (SINR), requires the use of Palm theory [10]. In moredetail, if we account for the interference caused by othersources, all subsequent derivations would involve probabilitygenerating functionals conditioned on the available knowledgeof the network layout. Such analysis goes beyond the scopeof this paper and is left as future work. However, note thatthe D2D discovery probability in (1) can also be viewed asan upper bound on the performance of the D2D discoveryprobability under the impact of interference from other cellularsources (SINR-based D2D discovery).

A. Related Works

Related literature deals with the analysis and optimiza-tion of D2D communications. The authors in [11] analyze

the performance of a distributed multi-hop spectrum accessprotocol for D2D communication, focusing on the impact ofD2D communication on the macrocell network and the powersavings due to the employment of single or multi-hop D2Droutes. A bio-inspired algorithm for direct D2D discovery andsynchronization is proposed in [12], based on the formationof acyclic graphs in the D2D network and the deployment ofthe firefly algorithm. In [13], we have developed an analyticalframework to assess and optimize the performance of proxim-ity estimation in heterogeneous wireless networks dominatedby clustering. Nonetheless, the proximity estimation processin [13] is employed between heterogeneous nodes that are notnecessarily capable of communicating.

Poisson point processes (PPPs), which have been exten-sively used for the analysis of cellular networks [8], [14]–[17],are increasingly used to analyze the performance of D2Dcommunications. In [18], the authors analyze the perfor-mance of two different spectrum sharing schemes and pro-vide design guidelines for D2D communication in the uplinkof cellular networks. The D2D proximity is based on thephysical distance between the D2D peers. The work in [19]considers D2D communication in Poisson networks withtime/frequency hopping to randomize interference and ana-lytical expressions for the SINR and throughput are derived.The authors in [20] investigate how mobility and networkassistance affect the performance of multicast D2D trans-missions in Poisson networks. Optimal network assistancestrategies are discussed towards minimizing the retransmissiontimes of multicast messages given certain constraints. Thechallenging issue of power control is addressed in [21], inwhich the authors focus on a single macrocell BS withcircular coverage where the locations of the D2D transmittersare PPP distributed. Assuming that the distance between aD2D pair is fixed, the authors propose a centralized powercontrol algorithm that maximizes the SINR of the standardcellular link while satisfying the individual target SINR ofthe D2D links. The performance of distributed power controlis also analyzed and optimal D2D transmission policies arediscussed.

B. Key Contributions

To our knowledge, this is the first work to assess theperformance of network-assisted D2D discovery in randomspatial networks. Our key contributions are as follows:

• We provide new ideas on how to integrate existingknowledge of the network layout into the analysis ofrandom spatial networks, where the BSs are assumed tobe placed at random in the Euclidean space, and showthat such an approach is analytically tractable.

• We derive closed-form expressions for the conditionalprobability density function (pdf) and complementarycdf (ccdf) of the distance Z between two D2D peers,given various combinations of location informationparameters.

• We analyze the performance of network-assisted D2Ddiscovery given different combinations of locationinformation and quantify how different levels ofknowledge affect the D2D discovery probability. The

XENAKIS et al.: PERFORMANCE ANALYSIS OF NETWORK-ASSISTED D2D DISCOVERY 5697

derived expressions can also serve as an upper bound onthe performance of SINR-based D2D discovery.

• We assess the impact of the BS density on theperformance of network-assisted D2D discovery andprovide useful insights on how to transform the today’scellular network, which is optimized for coverageand capacity, into a D2D-centric network where thediscovery and communication between the cellulardevices is orchestrated by the core network. Also,we employ novel analytical techniques to assess themonotonicity of expressions involving special functions.

• We provide useful design guidelines for network-assistedD2D discovery. We show that above a certain BS density,the D2D discovery probability AJ is primarily affectedby the distance Dk and that denser network layoutsmay reduce the D2D discovery probability. We alsoshow that the accuracy of AoA measurements can berelaxed without significantly affecting the performanceof network-assisted D2D discovery.

The remainder of this paper is organized as follows.In Section II, we present our system model, while in Section IIIwe derive the conditional pdf and ccdf of the distance betweentwo D2D peers given certain combinations of location infor-mation. In Section IV, we investigate how the BS densityaffects the D2D discovery probability and derive analyticalexpressions for the optimal BS density (when relevant). Theimpact of the key system parameters on the D2D discoveryperformance is assessed in Section V, where we additionallyprovide useful design guidelines for network-assisted D2Ddiscovery. Section VI contains our conclusions.

II. SYSTEM MODEL

We consider a D2D-enabled cellular network, where thelocations of all cellular BSs, including both macrocells andsmall cells, are distributed according to a homogeneous PPP�B with intensity λB in the Euclidean plane. Note that thesingle-tier PPP model is in line with multi-tier cellular networkmodels that use individual and independent PPPs to modeleach tier [8], [15], [16]. Nevertheless, we use a single-tierPPP to model the locations of the cellular BSs since, on theone hand, the superposition of independent PPPs is again aPPP of intensity equal to the sum of each tier intensity and,on the other hand, we do not focus on the performance ofnetwork-assisted D2D discovery in a specific cellular tier.Without assuming a specific distribution for the users, weconsider that a) the Point Process (PP) �U describing theuser locations is stationary and isotropic, and b) the x and ycoordinates of a given user are independent of the coordinatesof other users. We consider that all users associate with thenearest BS in �B [14] and focus on the network-assistedD2D discovery process between a tagged user, referred to asD2D source, and a (specific) target D2D-enabled user, referredto as D2D target. We further focus on the scenario where thenetwork is capable of identifying the associated BS of theD2D peers, and utilize UE and BS positioning measurementsto enhance the performance of D2D discovery.

The positioning measurements considered in this paper arelisted in Table I and are illustrated in Fig. 1. In Table I we high-light how these measurements can be derived in LTE/LTE-A.

TABLE I

CELLULAR-BASED LOCATION INFORMATION PARAMETERS

Fig. 1. System model parameters and related RVs.

Note that we do not assume that all of these measurementsare available to the EPC. Instead, we investigate how certaincombinations of these measurements (location information)can enhance network-assisted D2D discovery. Fig. 1 depictsall parameters and random variables (RVs) involved in ouranalysis. The D2D discovery probability can be computed bythe ccdf of the distance Z conditioned on the set of availablelocation information parameters, which we denoted by J, at

point(

PtPr

) 1a. By letting F̄Z |J(x) denote the corresponding

conditional ccdf of the distance Z at point x we define theD2D discovery probability as follows:

AJ = 1 − F̄Z |J

((Pt

Pr

) 1a)

. (2)

In the following, we consider that the transmit power Pt

at the D2D source and the receiver sensitivity Pr at the D2Dtarget are fixed and known to the EPC. The following lemmastates that the distance Dk between a random point in the


system and its k-th nearest (neighboring) BS in �B followsa generalized Gamma distribution. Note that although thesubscript k of the RV Dk can be omitted, in the sequel wechoose to use it as indication of the neighboring degree kbetween the associated BSs of the two D2D peers.

Lemma 1: The pdf fDk (d) of the distance Dk between arandom point in the system and the k-th neighboring BS inthe PPP �B is given by

fDk (d) = 2(πλB)k

�[k] d2k−1e−πλBd2, (3)

where �[k] is the Gamma function.Proof: The proof directly follows from [17, Th. 1]. �

Lemma 1 provides the pdf of the distance Dk between atagged BS and its k-th neighbor. For k = 1 it also provides thepdf of the distances Rs and Rt , i.e. Rayleigh distribution withparameter b2 = 1

2πλB. Note that the joint distribution of the

RVs Rs and Rt strongly depends on the location informationgiven in J, due to the assumption that all users associate withthe nearest BS in �B . However, as discussed in [22] andshown in Section V, this dependence is weak. Accordingly,we proceed with the following assumption.

Assumption 1. Given that the distance Dk is fixed andknown, the random variables Rs and Rt are independent andRayleigh-distributed with parameter b2 = 1

2πλB.

If the angles θs and θt are not assumed to be fixed andknown, we assume them to satisfy the following property.

Assumption 2. Given that the distance Dk is fixed andknown, the angle θs is uniformly distributed in (−π, π] andindependent of the distance Rs. Similarly, given that Dk isfixed and known, the angle θt is uniformly distributed in(−π, π] and independent of the distance Rt .

Assumption 2 states that, conditioned on a fixed dis-tance Dk , the D2D peers are scattered around their associatedBSs uniformly, having no bias on residing towards a specificdirection or adapting their distance to their BS based on theirangle to other BSs. Although assumptions 1 and 2 result inan approximate analysis, they also lead to valuable insights onhow to optimize the performance of network-assisted D2D dis-covery in practical networks. Besides, as shown in Section V,assumptions 1 and 2 improve the tractability of the systemmodel with minimal impact on the accuracy of the analysis.

III. DISTANCE DISTRIBUTIONS IN D2D-ENABLED

NETWORKS WITH LOCATION-ASSISTANCE

A. Distance Distributions Given the Distance Dk

In this section, we derive closed-form expressions for thepdf and the ccdf of the distance Z , given four distinct combina-tions of location information parameters: a) knowledge of onlythe distance Dk , b) knowledge of Dk and the relative position

of the D2D target with respect to its associated BS in polarcoordinates, i.e. [Rt , θt ], c) knowledge of Dk and the relativeposition of the D2D source with respect to its associated BSin polar coordinates, i.e. [Rs, θs ], and d) knowledge of allthe aforementioned parameters. The last combination allows adirect computation of Z . Given knowledge on the parametersin the set J, we denote the corresponding conditional pdf andccdf of the distance Z by fZ |J and F̄Z |J, respectively.

Theorem 1: The conditional pdf fZ |{Dk}(z) of the distance Zbetween two D2D peers, given that the associated BS of theD2D source and the associated BS of the D2D target areseparated by a distance Dk = D, is given by

fZ |{Dk}(z) = πλB ze− πλB2

(z2+D2)

I0 [πλB z D] , (4)

where I0[x] is the modified Bessel function of the first kindand order zero. The respective ccdf F̄Z |{Dk}(z) is given by

F̄Z |{Dk}(z) = Q1

[√πλB D,

√πλB z

], (5)

where QM [a, b] = ∫ ∞b x

( xa

)M−1e− x2+a2

2 IM−1 [ax] dx is theMarcum-Q function.

Proof: See Appendix A. �Theorem 1 provides an analytical tool for handling the

uncertainty on the distance between two D2D peers, givenlocation information that typically remains fixed over time:the distance Dk . In Theorem 2, we derive the conditional pdfand ccdf of the distance Z given additional knowledge on therelative position of the D2D target.

Theorem 2: The pdf fZ |{Dk,Rt ,θt }(z) of the distance Zbetween two D2D peers, given that a) the associated BS ofthe D2D source and the associated BS of the D2D target areseparated by a distance Dk = D, and b) the relative positionof the D2D target with respect to its associated BS equals to[Rt = T, θt = �] in polar coordinates, is given by (6), asshown at the bottom of this page. The ccdf F̄Z |{Dk ,Rt ,θt }(z) isgiven by (7), as shown at the bottom of this page.

Proof: See Appendix B. �Theorem 2 further reduces the uncertainty on the distance

between the D2D peers by incorporating additional knowledgeon the relative position of the D2D target with respect toits associated BS. However, in contrast with the acquisitionand caching of the Dk parameter, the relative position of theD2D target is expected to vary over time, requiring monitoringmeasurements by the associated BS. Corollary 1 includes theconditional pdf and ccdf of the distance Z when the EPC isaware of the relative position of the D2D source.

Corollary 1: The pdf fZ |{Dk,Rs ,θs}(z) and ccdfF̄Z |{Dk ,Rs ,θs}(z) of the distance Z between two D2D peers,given that a) the associated BS of the D2D source and theassociated BS of the D2D target are separated by a distance

fZ |{Dk ,Rt ,θt }(z) = 2πλB ze−πλB(z2+D2+T 2−2DT cos(π−�)) I0

[2πλB z

√D2 + T 2 − 2DT cos(π − �)

](6)

F̄Z |{Dk ,Rt ,θt }(z) = Q1

[√2πλB

(D2 + T 2 − 2DT cos(π − �)

), z

√2πλB

](7)


Dk = D, and b) the relative position of the D2D source withrespect to its associated BS equals to [Rs = S, θs = ϕ] inpolar coordinates, are given by (6) and (7), respectively, forT = S and � = π − ϕ.

Proof: Corollary 1 is proved by working in the Cartesianplane x ′′y ′′ centered at the position of the associated BS ofthe D2D source with positive x-axis the direction from theassociated BS of the D2D source to the associated BS of theD2D target. The distance Z is given by

Z =√

(X ′t + Sk+1)2 + Y ′

t2, (8)

where X ′t and Y ′

t are independent normal RVs with zeromean and equal variance b2 = 1

2πλB. By the law of cosines

Sk+1 = √D2 + S2 − 2DS cos ϕ (fixed parameter). The rest of

the proof is similar to that of Theorem 2. �Proposition 1 provides the distance Z when the EPC is

aware of all parameters in {Dk, Rt , θt , Rs , θs}.Proposition 1: The distance Z between two D2D peers,

given that a) the distance between the associated BSs ofthe two D2D peers is Dk = D, b) the relative position ofthe D2D target with respect to its associated BS equals to[Rt = T, θt = �], and c) the relative position of the D2Dsource with respect to its associated BS equals to [Rs = S,θs = ϕ], is given by

Z =√

(D + T cos � − S cos ϕ)2 + (T sin � − S sin ϕ)2. (9)

Proof: The proof is derived by substituting Dk = D,Xt = T cos(�), Yt = T sin(�), Xs = S cos ϕ and Ys = S sin ϕin (21) (see Appendix A). �

Note that the scenario where the D2D peers associate withthe same BS applies for Dk = 0.

B. Distance Distributions Given the Neighboring Degree k

Let us now focus on the scenarios where, instead of thedistance Dk , the EPC is aware of the neighboring degree kbetween the associated BSs of the D2D peers. Such a sce-nario is of practical relevance when the distance betweenthe BSs is not known a priori, e.g. unplanned deployment,or when the BSs face difficulties in accurately estimatingtheir separation distance, e.g. indoor deployment. Consider forexample a small-sized BS inside a building that measures thereceived signal strength from all nearby BSs. Even thoughthe BS can face difficulties in translatting these measurementsto the exact separation distance for each neighboring BS,e.g. due to non-line-of-sight conditions, it can readily identifytheir neighboring degree by sorting the derived measurementsin descending order. Besides, the estimation of k is lessvulnerable to the effects of the wireless medium. Under thisviewpoint, in this section we relax the requirement of havingperfect knowledge of the distance Dk and extend our analysisto the scenario where the associated BSs can only identifytheir neighboring degree k (more loose information).

Theorem 3: The conditional pdf fZ |{k}(z) of the distance Zbetween two D2D peers, given that the associated BS of theD2D source is the n-th neighbor of the associated BS of the

D2D target, i.e. k = n, is given by

fZ |{k}(z) = πλB

(2

3

)n

ze− πλB z2

3 Ln−1

[−πλB z2

6

], (10)

where Ln[x] is the Laguerre polynomial. The respective ccdfF̄Z |{k}(z) is given by

F̄Z |{k}(z) = 1

3e− πλB z2

3

n−1∑

m=0

εm

(2

3

)m

Lm

[−πλBz2

6

], (11)

where εm = 1 ∀m < n − 1 and εn−1 = 3.Proof: See Appendix C. �

Remark 1: Theorem 3 extends the closed-form expressionsof Theorem 1 when the EPC is aware of only the neighboringdegree k. This is achived since the arguments of the specialfunctions in Theorem 1 are in a simple form. However,Theorem 2 and Corollary 1 involve arguments with squareroots and powers of Dk , only allowing numerical evaluations.

Corollary 2: The conditional pdf fZ |{k,Rt ,θt }(z) of the dis-tance Z between two D2D peers, given that a) the associatedBS of the D2D source is the n-th neighbor of the associatedBS of the D2D target, i.e. k = n, and b) the relative positionof the D2D target with respect to its associated BS equals to[Rt = T, θt = �] in polar coordinates, is given by

fZ |{k,Rt ,θt }(z) = 2(πλB)n

�[n]∫ ∞

0fZ |x,Rt ,θt (z)x2n−1e−πλB x2

dx,

(12)

where the conditional pdf fZ |{x,Rt ,θt }(z) is given in (6) forDk = x. The respective ccdf F̄Z |{k,Rt ,θt }(z)(z) is given by

F̄Z |{k,Rt ,θt }(z)(z)

= 2(πλB)n

�[n]∫ ∞

0F̄Z |{x,Rt ,θt }(z)x2n−1e−πλB x2

dx, (13)

where F̄Z |{x,Rt ,θt }(z) is given in (7) for Dk = x.Proof: The proof follows by using the law of total

probability, Theorem 2 and (3). �Corollary 3: The conditional pdf fZ |{k,Rt ,θt }(z) and ccdf

F̄Z |{k,Rs ,θs }(z) of the distance Z between two D2D peers,given that a) the associated BS of the D2D source is then-th neighbor of the associated BS of the D2D target,i.e. k = n, and b) the relative position of the D2D sourcewith respect to its associated BS equals to [Rs = S, θs = ϕ]in polar coordinates, are given by (12) and (13), respectively,for T = S and � = π − ϕ.

Proof: Similar to that of Corollary 2. �Let us now focus on the scenario where the EPC is aware

of the neighboring degree k and the relative positions of theD2D peers with respect to their associated BSs. Different fromProposition 1, the knowledge of k leaves uncertainty on thedistance Dk between the associated BSs of the D2D peers.Interestingly, the respective conditional pdf and ccdf of thedistance Z can be derived in closed-form.

Theorem 4: The conditional pdf fZ |{k,Rt ,θt ,Rs ,θs}(z) of thedistance Z between two D2D peers, given that a) the asso-ciated BS of the D2D source is the n-th neighbor of theassociated BS of the D2D target, i.e. k = n, b) the relativeposition of the D2D source with respect to its associated BS


fZ |{k,Rt ,θt ,Rs ,θs}(z) =

⎧⎪⎪⎨

⎪⎪⎩

0, z ≤ |Qy|2(πλB)n

�[n]z√

z2−Q2y

∑2m=1

(−Qx +qm

√z2−Q2

y

)2n−1U

[−Qx +qm

√z2−Q2

y

]

eπλB

(−Qx +qm

√z2−Q2

y

)2 , z > |Qy| (14)

F̄Z |{k,Rt ,θt ,Rs ,θs}(z) =

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

0, f or z ≤ |Qy |(�[n] − �

[n, πλB d2

1

])U [d1] + �

[n, πλBd2

2

]

�[n] , f or z > |Qy|, d2 > 0

1, f or z > |Qy|, d2 ≤ 0

(15)

equals to [Rs = S, θs = ϕ], and c) the relative position ofthe D2D target with respect to its associated BS equals to[Rt = T, θt = �], is given by (14), as shown at the topof this page, where U [x] is the unit step function, Qx =−T cos � − S cos ϕ, Qy = T sin � − S sin ϕ, qm = 1 form = 1 and qm = −1 for m = 2. The respective ccdfF̄Z |{k,Rt ,θt ,Rs ,θs}(z) is given by (15), as shown at the topof this page, where �[n, x] = ∫ ∞

x tn−1e−t dt is the upper

incomplete Gamma Function, d1 = −Qx −√

z2 − Q2y , and

d2 = −Qx +√

z2 − Q2y .

Proof: See Appendix D. �

IV. OPTIMAL NETWORK DEPLOYMENT FOR

NETWORK-ASSISTED D2D DISCOVERY

In this section, we provide guidelines for optimal networkdeployment as a means to optimize the probability of success-ful network-assisted D2D discovery. This kind of analysis is ofhigh practical interest as it provides useful insights on how totransform the today’s cellular network, which is optimized forcoverage and capacity, into a D2D-centric network where thediscovery and communication between the cellular devices isorchestrated by the EPC. The presented analysis also providesuseful guidelines for the installation of additional BSs soas to maximize the capability of the EPC to infer on theoutcome of D2D discovery. Besides, maximizing the capabilityof the EPC to estimate proximity between two devices, e.g.an anchor point and a target device with unknown location, isof paramount importance in public emergency networks andprivate network installations for automated navigation/control,e.g. industrial installations and underground facilities. To thisend, we examine the monotonicity of the D2D discoveryprobability AJ with respect to the BS density λB and provideexpressions for the optimal BS density. Since AJ is given by

the cdf of the distance Z at(

PtPr

) 1a, AJ is a) proportional to the

transmit power Pt and b) inversely proportional to the receiversensitivity Pr and the pathloss exponent a.

Theorem 5: Let q = D(

PtPr

)− 1a

, where D is the value ofthe distance Dk between the associated BSs of the two D2Dpeers. Given the distance D, the D2D discovery probabilityADk increases with λB for q < 1. However, for q > 1 thereexists a BS density λ∗

B that maximizes the D2D discoveryprobability and satisfies the following property:

I0

[πλ∗

B D2

q

]

− q I1

[πλ∗

B D2

q

]

= 0. (16)

The optimal BS density can be analytically approximated as

λ∗B ≈ q(1 + 3q + √

39q2 − 6q − 17)

16π D2(q − 1). (17)

Proof: See Appendix E. �The parameter

(PtPr

) 1a

corresponds to the maximum dis-

tance for successful D2D discovery between the D2D peers.Also, the distance between a user and the associated BSis inversely proportional to λB since, by definition, it isRayleigh distributed with parameter 1

2πλB. Therefore, as the

BS density increases, the distance Z between the D2Dpeers tends to reach the distance Dk between their asso-ciated BSs, i.e. a higher λB reduces uncertainty on theuser position around the associated BS. Theorem 5 canbe interpreted as follows: as λB increases, the distance Zbetween the D2D peers tends to be statistically closer tothe distance Dk , which for q < 1 is by definition lowerthan the maximum range for successful D2D discovery,

i.e. Dk <(

PtPr

) 1a. However, for q > 1, the distance Dk is

greater than the maximum D2D discovery range and, abovea certain BS density, the distance Z tends to be statisticallygreater than the D2D discovery range.

Interestingly, Theorem 5 can be extended to the scenariowhere, apart from the distance Dk , the EPC is additionallyaware of the relative positions of the D2D pairs with respectto their associated BS. This can be shown by noticing thatthe ccdf results in Theorem 2 are in a similar form with theones in Theorem 1. In more detail, if the EPC is aware of therelative position of the D2D source, Theorem 5 applies for

q = √D2 + S2 − 2DS cos ϕ

(PtPr

)− 1a. If the EPC is aware of

the relative position of the D2D target, Theorem 5 applies for

q = √D2 + T 2 − 2DT cos(π − �)

(PtPr

)− 1a. We now focus

on the scenarios where, instead of the distance Dk , the EPCis aware of k.

Theorem 6: When the EPC is aware of the neighboringdegree k = n, the probability Ak increases with λB.

Proof: By using Theorem 3 and (1), Ak is given by

Ak = 1 − 1

3e− πλB z2

3

n−1∑

m=0

εm

(2

3

)m

Lm

[−πλB z2

6

], (18)

where εm = 1,∀m < n − 1 and εn−1 = 3. By differentiating


with respect to λB we get:

∂Ak

∂λB= πz2e− πλB z2

3

·n−1∑

m=0

εm2m−1

3m+2

(2Lm

[−πλB z2

6

]−L1

m−1

[−πλB z2

6

]),

where L1m[x] = (m + 1)

∑mv=0

(−m)k zk

(k+1)!k! is the generalized

Laguerre polynomial of the first order. Now, since all parame-ters in the last expression are positive real and by definitionL1

m−1[x] < Lm [x] for x < 0, it follows that ∂Ak∂λB

> 0. �Different from Theorem 5, Theorem 6 shows that a higher

BS density will always improve the performance of network-assisted D2D discovery if the EPC is aware of the neigh-boring degree k. This mainly follows from the fact that thedistance Dk is not a fixed parameter as in Theorem 5, yet, itis inversely proportional to the BS density λB (3).

Theorem 7: Let d1 and d2 denote the (fixed) parametersdefined in Theorem 4. Given that a) the associated BS ofthe D2D source is the n-th neighbor of the associated BSof the D2D target, i.e. k = n, b) the relative position ofthe D2D source with respect to its associated BS equals to[Rs = S, θs = ϕ] in polar coordinates, and c) the relativeposition of the D2D target with respect to its associatedBS equals to [Rt = T, θt = �] in polar coordinates, theD2D discovery probability Ak,Rt ,θt ,Rs ,θs satisfies the followingproperties with respect to λB:

1. For z ≤ |Qy|: Ak,Rt ,θt ,Rs ,θs = 1.2. For z > |Qy | and d2 ≤ 0: Ak,Rt ,θt ,Rs ,θs = 0.3. For z > |Qy |, d2 > 0, and d1 ≤ 0: Ak,Rt ,θt ,Rs ,θs increases

with λB.4. For z > |Qy |, d2 > 0, and d1 > 0: there exists an

optimal BS density λ∗B that maximizes Ak,Rt ,θt ,Rs ,θs

λ∗B =

n lnd2

1d2

2

π(d2

1 − d22

) . (19)

Proof: Properties 1 and 2 follow from (15). By combining(15) and (1) for z > |Qy |, d2 > 0, and d1 ≤ 0, we get

Ak,Rt ,θt ,Rs ,θs = 1 − �[n,πλBd2

2

]

�[n] . Now, by differentiating with

respect to λB , it follows that∂Ak,Rt ,θt ,Rs ,θs

∂λB> 0. When z > |Qy |,

d2 > 0, and d1 > 0, it follows that Ak,Rt ,θt ,Rs ,θs =�

[n,πλBd2

1

]−�[n,πλBd2

2

]

�[n] (Eqs. (1) (15)). By differentiating with

respect to λB we get:

∂Ak,Rt ,θt ,Rs ,θs

∂λB= (πλB)n d2n

2 e−πλBd22 − d2n

1 e−πλBd21

λB�[n] . (20)

Solving∂Ak,Rt ,θt ,Rs ,θs

∂λB= 0 with respect to λB yields (19). �

V. NUMERICAL RESULTS AND DESIGN GUIDELINES

In this section, we study the impact of the key systemparameters on the D2D discovery probability and derive usefulguidelines for the design of network-assisted D2D discovery.The receiver sensitivity is set to Pr = −93.5 dBm, which istypical for the LTE system with Frequency Division Duplex-ing (FDD) and bandwidth equal to (or greater than) 5 MHz.

Fig. 2. D2D Discovery Probability given Dk = 600m vs. BS density.

Fig. 3. D2D Discovery Probability given Dk = 900m vs. BS density.

A. Effect of BS Density

In this section, we study the impact of the BS densityλB on the D2D discovery probability given at least thedistance Dk . We focus on two distances Dk = 600 m andDk = 900 m which, in combination with the parameters underscope, result in D2D discovery success (ADk,Rt ,θt ,Rs ,θs = 1)and failure (ADk ,Rt ,θt ,Rs ,θs = 0), respectively. The D2D dis-covery probability for Dk = 600 m and Dk = 900 m isgiven in Figs. 2 and 3, respectively. To validate the impactof Assumptions 1 and 2 on the accuracy of our analysis,in the same figures we compare the analytical results withthe ones following from system-level simulations given thesame set of parameters. In the simulation model, we haveconsidered that the users are distributed according to a PPPwith intensity λU = 10λB . The network size was adapted from106 × 106 m2 to 104 × 104 m2 depending on the BS densityunder scope. Notably, as shown in Figs. 2 and 3, the results ofour analysis match very closely the simulated ones, indicatingthat the impact of Assumptions 1 and 2 on the accuracy ofthe presented analysis is weak. Given that the expressions inTheorem 1 constitute the basis of all subsequent analyticalderivations, in the remainder of section V we only include theresults of our analysis due to space limitations. As depicted inFig. 2, when the EPC is aware of only the distance Dk the D2Ddiscovery probability ADk increases with λB for Dk = 600 m,since q < 1 (as shown in Theorem 5). On the other hand, for


Fig. 4. D2D Discovery Probability given k vs. BS density.

Dk = 900 m (Fig. 3), which corresponds to q > 1, there existsan optimal BS density that maximizes the D2D discoveryprobability and is well approximated by (17) (highlighted withthe red circle). Similar properties are shown when the EPChas additional knowledge on the relative positions of the D2Dtarget or the D2D source, i.e. ADk ,Rt ,θt and ADk ,Rs ,θs in Fig. 3,respectively. The approximations on the optimal BS density forADk ,Rt ,θt and ADk ,Rs ,θs , indicated by the blue and green circles,respectively, are also shown to be close to the λB parameterthat maximizes the respective D2D discovery probabilities.Recall that the approximation accuracy can be increased byusing more terms from (41) and (42).

The results in Figs. 2 and 3 reveal that conditioned onknowledge of the relative position of the D2D source, orthe D2D target, the statistical behavior of the D2D discoveryprobability and the optimal λ∗

B significantly alters comparedto ADk , especially when q > 1 (the ratio of the Marcum-Qarguments in the ccdf results is higher than one). This followsfrom the fact that, the locations of the D2D peers are consid-ered to follow a symmetric normal distribution around theirassociated BS, i.e. Rayleigh-distributed distance combinedwith uniformly distributed angle. However, when a D2D peeris located in between the two associated BSs and its locationis known to the EPC, the probability of successful D2D dis-covery increases. This relation can be verified in Figs. 2 and 3,by comparing the results for ADk ,Rs ,θs and ADk and taking intoaccount that Rs = 200 m and θs = π/3 (Fig. 1).

In Fig. 4, we plot the impact of λB on the D2D discoveryprobability, given knowledge of the neighboring degree k,or the relative positions of the D2D peers. As provided byTheorem 6, the D2D discovery probability Ak always increaseswith λB , while given additional knowledge on the relativeposition of the D2D peers the corresponding D2D probabilityAk,Rt ,θt ,Rs ,θs is also increasing with respect to λB for Rs = 200m. However, for Rs = 400 m the probability Ak,Rt ,θt ,Rs ,θs

is maximized for a BS density that can be computed by(19) (highlighted with a star). These results are in line withTheorem 7 since the parameters Rs = 200 m and Rs = 400 m,correspond to d1 < 0 and d1 > 0, respectively.

We now explore how the neighboring degree k affects theD2D discovery probability. As expected, a higher k reducesthe D2D discovery probability given knowledge only on k,i.e. Ak=1 > Ak=3. The same applies when the relative

Fig. 5. D2D Discovery Probability given Dk vs. distance Dk .

position of the D2D peers is known to the EPC andRs = 200 m (magenta dashed), which corresponds to d1 < 0,i.e. Ak=1,Rt ,θt ,Rs ,θs > Ak=3,Rt ,θt ,Rs ,θs . However, this is not ineffect for Rs = 400 m (d1 > 0) in medium to very highBS densities (λB > 10−5), where the D2D probability fork = 3 (green continuous) is shown to be higher compared tothe one for the same parameters and k = 1 (red continuous).This follows since for a given k, a higher λB increases thestatistical distance Dk between the associated BSs of theD2D peers which, combined with the given positions of theD2D peers, shifts the peak of the D2D probability to higherBS densities.

B. Effect of Distance Dk

We now examine the impact of the distance Dk on theD2D discovery probability given knowledge of at least thedistance Dk (Fig. 5). First, we observe that given the sameset of location parameters, a higher BS density prolongs thetail of the D2D discovery probability, owing to the increaseduncertainty on the D2D source and/or D2D target positionsaround their associated BS. This is also expected if weconsider that for the given set of system parameters, themaximum range for successful D2D discovery is equal to(

PtPr

) 1a = 813.3 m, whereas the average distance between a

user and its associated BS is equal to 157m and 1570m forλB = 10−4 and λB = 10−6, respectively, i.e. expected valueof the Rayleigh distribution with parameter 1√

2πλB. Hence,

above a certain BS density, the performance of D2D discoveryis primarily affected by the distance Dk and not the relativepositions of the D2D peers, which can be approximated bythe position of their associated BSs. This approach can reducethe overhead required for user positioning while leaving theD2D discovery probability unaffected.

For λB = 10−4, we observe that the D2D discoveryprobability ADk is higher compared to the one given addi-tional knowledge on [Rt , θt ], i.e. ADk ,Rt ,θt . This can beexplained as follows: conditioned on [Rt = 200m, θt = π/3](Fig. 1), the distance Z between the D2D peers is statisticallyhigher compared to the scenario with no knowledge on [Rt , θt ](Fig. 1), where the position of the D2D target is considered


Fig. 6. D2D Discovery Probability given Dk vs. Transmit power Pt .

to follow a symmetrical normal distribution around its asso-ciated BS

(σ 2 = 1

2πλB

). This effect is more prominent for

λB = 10−4, where the uncertainty on position of the D2Dsource is significantly reduced compared to the one forλB = 10−6. Similar arguments can be used to compareADk and ADk ,Rs ,θs .

C. Effect of Transmit Power

D2D discovery will be performed under unfavorable channelconditions, due to the lower height of the transmitter-receiverpair, the increased number of obstacles between the D2Dpeers, and the low transmit power required to avoid interfer-ence with other cellular connections. Under this viewpoint,in Fig. 6 we plot the impact of the transmit power Pt on theD2D discovery probability under high pathloss exponents andgiven information for at least the distance Dk . As expected,the D2D discovery probability increases with Pt under allcombinations of location information (Section IV). However,the (positive) impact of increasing Pt on the D2D discov-ery probability strongly depends on the pathloss exponentgoverning the D2D channel. For example, for a = 3.7, weobserve that Pt = 200 mW suffices to attain a D2D discoveryprobability higher than 90% for all combinations of locationinformation parameters. On the other hand, for a = 4.4, theD2D discovery probability can be greatly improved with aslight increase in the transmit power whereas, for a = 5, itremains roughly unaffected. For the given set of parameters,additional knowledge on [Rs , θs] and/or [Rt , θt ] significantlyalters the statistical behavior of the D2D discovery probability,especially for a = 4.4.

In Fig. 7, we depict the relation between Pt and the D2Ddiscovery probability given at least the neighboring degree k.As expected, a higher k reduces the D2D discovery probability,due to the statistical increase on the distance Dk . By compar-ing the impact of Pt on Ak,Rt ,θt ,Rs ,θs and Ak for k = 1, it canalso be seen that the knowledge of [Rs, θs ] and [Rt , θt ] altersthe statistical behavior of the D2D discovery probability withrespect to Pt . However, differently from the results in Fig. 6,this applies for all pathloss exponents. This property weakensfor higher neighboring degrees (k = 3).

To summarize, when the EPC is aware of the neighboringdegree k instead of the distance Dk , additional knowledge

Fig. 7. D2D Discovery Probability given k vs. Transmit Power Pt .

Fig. 8. D2D Discovery Probability vs. D2D target angle θt .

on the relative positions of the D2D peers may significantlyimprove the accuracy of network-assisted D2D discovery,especially for low k. On the other hand, the employmentof network-assisted D2D discovery can significantly reduceunnecessary transmissions of D2D discovery signals thatincrease the network interference and deplete the battery atthe mobile terminals. To this direction, the presented resultscan be used to assist the D2D source upon selecting anappropriate transmit power for a prescribed D2D discoveryprobability target, by exploiting fundamental location infor-mation at the EPC.

D. Effect of the Angle of the D2D target

In Fig. 8 we plot the D2D discovery probability withrespect to θt (Fig. 1), for all combinations that includeθt . Obviously, the statistical behavior of the remainderD2D discovery probabilities remains unchanged with respectto θt . Given full knowledge on the network layout, weobserve that the D2D discovery can be either success-ful or not. However, depending on the fixed parame-ters, there exists a θt interval within which the D2Ddiscovery is always successful, i.e. ADk ,Rt ,θt ,Rs ,θs = 1.Moreover, we observe that the probability ADk ,Rt ,θt ,Rs ,θs forθs = π/3 is a mirror function of ADk ,Rt ,θt ,Rs ,θs for θs = −π/3with respect to 180o, which corresponds to the directiontowards the associated BS of the D2D source. For the given


parameters, an increase to the distance Rs symmetricallyexpands the θt interval where ADk ,Rt ,θt ,Rs ,θs = 1 towards bothdirections. This result follows from the given θs under scope,since for θs = π/3 a higher distance Rs reduces the distanceZ between the D2D peers.

When the EPC is aware of only the parameters {Dk , Rt , θt },the D2D discovery probability is higher for all angles θt thatreside closer to the associated BS of the D2D source (greenline), i.e. 180o. On the other hand, an increase to the distanceRt enlarges the D2D discovery probability for θt towards thesame direction and reduces it for the ones residing towardsthe opposite one (green dashed line). Both these results areexpected if we consider that in the absence of knowledge of[Rs, θs ], the D2D source is considered to follow a symmetricnormal distribution around its associated BS.

Similar to ADk ,Rt ,θt ,Rs ,θs , the D2D discovery probabilityAk,Rt ,θt ,Rs ,θs for Rs = π/3 is mirrored for Rs = −π/3,with respect to the direction towards the associated BS of theD2D source (180o). However, in contrast with ADk ,Rt ,θt ,Rs ,θs ,an increase to the distance Rt ’stretches’ the probabil-ity Ak,Rt ,θt ,Rs ,θs towards the direction of the D2D sourcein a non-symmetric manner. This effect follows from theknowledge of the relative position of the D2D source andthe fact that the Rt is now higher (Rt = 400 m). Thecombination of these conditions creates bias on the D2Ddiscovery probability towards specific coordinates for theD2D target. It follows that the knowledge on the relativepositions of the D2D peers majorly impacts the D2D discoveryperformance, especially in sparse to medium networks wherethe uncertainty on the relative positions of the users (aroundtheir associated BSs) is high. The results in Fig. 8 alsoindicate that, under certain conditions, the estimation accuracyfor the angles θt and θs can be relaxed without affectingthe performance of network-assisted D2D discovery. Such anapproach can significantly reduce the overheads required forAoA measurements at the associated BSs.

VI. CONCLUSION

In this paper, we analyzed the statistical behavior of thedistance between two D2D peers conditioned on existingknowledge for the cellular network deployment. The ccdfexpressions were used to analyze the performance of network-assisted D2D discovery and provide useful insights on howdifferent levels of location awareness affect its performance.We also examined how unplanned cellular network densi-fication affects the performance of network-assisted D2Ddiscovery and provided analytical expressions for the optimalBS density that maximizes the D2D discovery probability.Accordingly, we investigated the key performance tradeoffsinherent to the network-assisted D2D discovery and provideduseful guidelines for its design in random spatial networks.Among others, the present results can be used to select thetransmit power at the D2D source for a given D2D discoveryprobability target, reduce unnecessary D2D discovery signals,identify the optimal BS density for network-assisted D2Ddiscovery, and relax the accuracy of user positioning whileleaving the D2D discovery probability unaffected.

APPENDIX

A. Proof of Theorem 1

Since the user and BS point processes are stationary andisotropic, we work in the Cartesian plane xy centered at theposition of the associated BS of the D2D source and having asa positive x-axis the direction from the associated BS of theD2D source to the associated BS of the D2D target (Fig. 1).Let Xs and Xt be the projections of the distances Rs andRt on the x-axis, respectively, and Ys and Yt the respectiveprojections in the y-axis. Note that [Xs, Ys ] are the Cartesiancoordinates of the D2D source in the xy plane, whereas therespective ones for the D2D target are [Dk + Xt , Yt ]. FromFig. 1, it follows that the distance Z between the D2D sourceand the D2D target is given by

Z =√

(Dk + Xt − Xs)2 + (Yt − Ys)2. (21)

We now define the auxiliary RVs: Qx = D + Xt − Xs andQy = Yt −Ys . The proof is based on the fact that the RVs Qx

and Qy are independent normal RVs with different mean yetequal variance. To prove this, we will show that a) the RVsXs and Ys are independent normal RVs with zero mean andequal variance b2 = 1

2πλB, b) the RVs Xt and Yt are indepen-

dent normal RVs with zero mean and equal variance b2, andc) the RVs Xs , Ys , Xt , and Yt , are mutually independent. Sincethe distance Rs between the D2D source and its associatedBS is Rayleigh distributed with parameter b2 (Assumption 1)and the angle θs is uniform in (−π, π] and independent ofthe distance Rs (Assumption 2), by using a similar approachwith[23, p. 146] it can be proved that the RVs Xs = Rs cos θs

and Ys = Rs sin θs are independent normal RVs with zeromean and equal variance: Xs, Ys ∼ N

(0, 1

2πλB

). The same

arguments can be used to show that the RVs Xt = Rt cos θt

and Yt = Rt sin θt are independent normal RVs with zero meanand equal variance b2 as well.

Since the coordinates of the D2D source do not dependon the ones of the D2D target (Section II), Qx equals tothe difference of two independent normal RVs plus a fixedvalue Dk = D. Thus, Qx is normally distributed with meanD and variance 2b2 = 1

πλB: Qx ∼ N (D, 1

πλB). By following a

similar approach, it can be shown that Qy is a normal RV withzero mean and variance 2b2: Qy ∼ N (0, 1

πλB). In addition,

Qy is independent of Qx , since the RV Xt , Xs , Yt , and Ys areshown to be mutually independent. Therefore,

fQx ,Q y (x, y) = λB

2e− πλB

2

((x−D)2+y2

). (22)

Now, let �Az denote the region of the plane such thatz <

√x2 + y2 < z + dz. �Az is a circular ring with inner

radius z and thickness dz. By working in polar coordinates,i.e. x = z cos ξ , y = z sin ξ , and dxdy = zdzdξ , we get

fZ |{Dk}(z)dz =∫

�Az

fQx ,Q y (x, y)dxdy

= λB

2

∫ 2π

0e− πλB

2

(z cos ξ−D)2+(z sin ξ)2

)zdzdξ.


Therefore,

fZ |{Dk}(z) = λB z

2e− πλB

2

(z2+D2

) ∫ 2π

0eπλB z D cos ξ dξ

= πλB ze− πλB2

(z2+D2

)I0 [πλB z D] ,

where the last step follows from the integral representation ofthe modified Bessel function of the first and zero-th order. Letus now derive the conditional ccdf F̄Z |{Dk}(z) as follows:

F̄Z |{Dk}(z) =∫ ∞

zπλB xe− πλB

2

(x2+D2

)I0 [πλB x D] dx

=∫ ∞

√πλB z

qe− q2+(√

πλB D)2

2 I0

[q

(√πλB D

)]dq

(23)

= Q1

[√πλB D,

√πλB z

], (24)

where QM [a, b] is the Marcum-Q function, IM−1[x] themodified Bessel function of the first kind and (M−1)-th order,(23) is derived by employing a change of variables q = x√

πλB,

and (24) by the definition of the Marcum-Q function.

B. Proof of Theorem 2

Since the user and BS PP are stationary and isotropic, wework in the Cartesian plane x ′y ′ centered at the position ofthe associated BS of the D2D source and having as a positivex-axis the direction from the associated BS of the D2D sourceto the D2D target (Fig. 1). Let (X ′

s , Y ′s) be the Cartesian

coordinates of the D2D source in the x ′y ′ plane. Given that thedistance Rs between the user and its associated BS is Rayleighdistributed (Assumption 1) and Assumption 2 holds, the RVsX ′

s and Y ′s are independent normal RVs with zero mean and

equal variance b2 = 12πλB

[23, p. 146]: X ′s , Y ′

s ∼ N(0, b2

).

The distance Z between the D2D source and the D2D target

is given by Z =√

(Tk+1 − X ′s)

2 + Y ′s

2, where Tk+1 is thedistance between the D2D target and the associated BS ofthe D2D source. Since the parameters Dk = D, Rt = T andθt = � are fixed, the distance Tk+1 is a fixed parameter thatcan be readily estimated by using the law of cosines in thetriangle |Rt ||Dk ||Tk+1|

Tk+1 =√

D2 + T 2 − 2DT cos(π − �). (25)

Let us now define the auxiliary RV Qx = Tk+1 − X ′s . Since

Tk+1 is fixed and X ′s ∼ N

(0, b2

)is normally distributed with

zero mean and variance b2, the RV Qx is normally distributedwith mean Tk+1 and variance b2. Given that X ′

s is independentof Y ′

s , it also follows that Qx is independent of Y ′s . Therefore,

fQx ,Y ′s(x, y) = 1

2πb2 e− (x−Tk+1)2+y2

2b2 . (26)

Provided that Z =√

Q2x + Y ′

s2, where Y ′

s ∼ N(0, b2

)and

Qx ∼ N(Tk+1, b2

)are independent, it can be proved that the

conditional pdf fZ |{Dk ,Rt ,θt }(z) is given by

fZ |{Dk ,Rt ,θt }(z) = ze− z2+T 2

k+12b2

b2 I0

[zTk+1

b2

]. (27)

The proof is similar to Theorem 1 (23) - (23). The condi-tional ccdf F̄Z |{Dk ,Rt ,θt }(z) is given as

F̄Z |{Dk ,Rt ,θt }(z) =∫ ∞

z

xe− x2+T 2

k+12b2

b2 I0

[xTk+1

b2

]dx

= Q1

[Tk+1

b,

z

b

], (28)

where (28) is derived by employing the change of variablesq = x

b and using the Marcum-Q function. The proof completesby substituting (25) and b = 1√

2πλB, in (27) and (28).

C. Proof of Theorem 3

The pdf fZ |{k}(z) and ccdf F̄Z |{k}(z) are derived by integrat-ing the results of Theorem 1 with respect to the distance Dk

given that the neighboring degree equals to k = n.

fZ |{k}(z)

=∫ ∞

0P[Z |Dk = x]P[Dk = x |k = n]dx (29)

= 2(πλB)n+1

�[n] ze− πλB z2

2

∫ ∞

0x2n−1e− 3πλB x2

2 I0 [πλB zx] dx

(30)

= πλB

(2

3

)n

ze− πλB z2

2 1 F1

[n, 1,

πλB z2

6

], (31)

where 1 F1[n, 1, x] is the confluent Hypergeometric Func-tion. (30) is derived by substituting (3) and (4) in (29),and (31) by solving the integral [24, p. 303]. By using thefunctional identities 1 F1[n, 1, x] = ex

1 F1[1 − n, 1,−x] andLn[x] =1 F1[−n, 1, x], where Ln[z] = ∑n

m=0

(nm

) (−1)m

m! xm

is the Laguerre polynomial, we reach to the pdf expressionin (10). The conditional ccdf F̄Z |{k}(z) is derived as

F̄Z |{k}(z) =∫ ∞

x=z

∫ ∞

y=0fZ |Dk=y,k=n(x) fDk |k=n(y)dydx

=∫ ∞

y=0fDk |k=n(y)

∫ ∞

x=zfZ |Dk=y(x)dxdy (32)

=∫ ∞

y=0

2(πλB)n

�[n] y2n−1e−πλB y2 · Q1

[ y

b,

z

b

]dy

(33)

= 1

3e− πλB z2

3

n−1∑

m=0

εm

(2

3

)m

Lm

[−πλBz2

6

], (34)

where (32) follows from the law of total probability, (33) bysubstituting (24) and (3), and (34) by solving the integral usingthe result in [25] for εm = 1,∀m < n − 1 and εn−1 = 3.

D. Proof of Theorem 4

By using Proposition 1, the distance Z is given by

Z = g(Dn) �√

(Dn + Qx )2 + Q2y, (35)

where Qx � Rt cos �− Rs cos ϕ and Qy � Rt sin �− Rs sin ϕare fixed. From (35), the distance Z is at least equal to |Qy|and thus, fZ |{k,Rt ,θt ,Rs ,θs} = 0 for Z < |Qy |. Now, since Z is a


function of the distance Dn with known pdf (3), for z ≥ |Qy |,fZ |{k,Rt ,θt ,Rs ,θs } is given by [23, p. 93]

fZ |{k,Rt ,θt ,Rs ,θs}(z) = fDn (d1)

|g′(d2)| + fDn (d2)

|g′(d2)| , (36)

where d1 = −Qx −√

Z2 − Q2y and d2 = −Qx +

√Z2 − Q2

y

are the real roots of (35) and g′(d) = d+Qx√(d+Qx )2+Q2

y

= d+Qxz

is the derivative of g(d) in ((35). Therefore,

fZ |{k,Rt ,θt ,Rs ,θs}(z) =2∑

m=1

2(πλB)nz

�[n]d2n−1

m e−πλBd2m U [dm]

|dm + Qx | ,

where the unit step function U [x] is used since Dk canonly take positive values. The pdf in 14 is derived by usingthe parameter qm , where q1 = 1 and q2 = −1. The ccdfF̄Z |k,Rt ,θt ,Rs ,θs (z) is defined only for z ≥ Qy and can beeasily derived by a) integrating the pdf 14 to ∞, b) solving therespective integrals and c) plugging d1 = −Qx −

√z2 − Q2

y

and d2 = −Qx +√

z2 − Q2y in the final expression.

E. Proof of Theorem 5

From Theorem 1 and (1) we get ADk = 1 −Q1

[√πλB D,

√πλB

(PtPr

) 1a]

. Let q = D(

PtPr

)− 1a

denote the

ratio of the arguments in the Marcum-Q function. By usingthe Marcum-Q f transform in [26, eq. (4.16)] for q < 1, wecan rewrite ADk as follows

ADk = 1 − 1

2π

∫ π

−π

(1 + q sin θ)e− πλ

2

(PtPr

) 2a(1+2q sin θ+q2)

1 + 2q sin θ + q2 dθ.

Now, by differentiating with respect to λB we get:

∂ADk

∂λB=

(PtPr

) 2a

4

∫ π

−π(1 + q sin θ)e

− πλ2

(PtPr

) 2a(1+2q sin θ+q2)

dθ

=I0

[πλB

(PtPr

) 2a

q

]− q I1

[πλB

(PtPr

) 2a

q

]

2

(π

(PtPr

) 2a)−1

eπλ2

(PtPr

) 2a(1+q2)

. (37)

Since all parameters in (37) are positive real and I0[x] >I1[x] ∀x > 0, for q < 1 the sign of (37) is always positive.Hence,

∂ADk∂λB

> 0, which implies that for q < 1 the D2Ddiscovery probability always increases with λB . Let us nowexamine the scenario where q > 1. By using the transform in[26, eq. (4.19)] for ζ = 1

q , we rewrite ADk as follows

ADk = − 1

2π

∫ π

−π

(ζ 2 + ζ sin θ)e− πλ2 D2(1+2ζ sin θ+ζ 2)

1 + 2ζ sin θ + ζ 2 dθ.

(38)

By differentiating with respect to λB we obtain

∂ADk

∂λB= D2

4

∫ π

−π(ζ 2 + ζ sin θ)e− πλ

2 D2(1+2ζ sin θ+ζ 2)dθ

= π D2

2eπλ2 D2(1+ζ 2)

(ζ I0

[πλB D2ζ

]− I1

[πλB D2ζ

]).

(39)

Since all parameters in (39) are positive real and the

existence of an optimal point requires∂ADk∂λB

= 0, (39) impliesthat the optimal BS density λ∗

B satisfies the condition

I0

[πλ∗

B D2

q

]

− q I1

[πλ∗

B D2

q

]

= 0. (40)

If the (common) argument in the two modified Besselfunctions in (40) is sufficiently large (typically higher than 2),then the Bessel functions can be approximated by

I0[x] = ex

√2πx

(1 + 1

8x

(1 + 9

2(8x)(1 + . . . )

)), (41)

I1[x] = ex

√2πx

(1 − 3

8x

(1 + 5

2(8x)(1 + + . . . )

)). (42)

Obviously, the use of more terms from (41) and (42) results

in more accurate approximation. Assuming thatπλ∗

B D2

q issufficiently large, we use the first three terms of (41) and (42)to approximate the Bessel functions in (40). Therefore, (40)can be converted to the following quadric equation

−128λ∗B

2 D4π2(q − 1) + 16λ∗B D2πq(1 + 3q)

+ 3q2(3 + 5q) = 0

which for q > 1 and D > 0 has the unique solution in (17).

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Dionysis Xenakis received the B.Sc. degree incomputer science in 2007, the M.Sc. degree incommunications systems and networks in 2009, andthe Ph.D. degree in 2014 from the Departmentof Informatics and Telecommunications, Universityof Athens, Greece. He has participated in variousEU-funded projects and served as a TPC memberin prestigious IEEE conferences. He is currentlya member of the Green Adaptive and IntelligentNetworking Group with the University of Athens.

Marios Kountouris (S’04–M’08–SM’15)received the Diploma degree in electrical andcomputer engineering from the National TechnicalUniversity of Athens, Greece, in 2002, and theM.S. and Ph.D. degrees in electrical engineeringfrom the Ecole Nationale Supérieure des Télécom-munications, France, in 2004 and 2008, respectively.His doctoral research was funded by Orange Labs,France. From 2008 to 2009, he was with the Depart-ment of ECE, The University of Texas at Austin,as a Research Associate, working under DARPA’s

IT-MANET program. From 2014 to 2015, he was an Adjunct Professor withthe School of EEE, Yonsei University, South Korea. Since 2015, he has beena Principal Researcher with the Mathematical and Algorithmic SciencesLaboratory, Huawei Technologies Company, Ltd., France. He is an Editorof the IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, the IEEETRANSACTIONS ON SIGNAL PROCESSING, the EURASIP Journal on WirelessCommunications and Networking, and the Journal of Communications andNetworks. He is currently an Associate Professor with the Department ofTelecommunications, Supélec. He received the 2013 IEEE ComSoc Outstand-ing Young Researcher Award for the EMEA Region, the 2014 EURASIP BestPaper Award for the EURASIP Journal on Advances in Signal Processing, the2012 IEEE SPS Signal Processing Magazine Award, and the Best Paper Awardin the Communication Theory Symposium at the IEEE Globecom 2009.

Lazaros Merakos received the Diploma degreein electrical and mechanical engineering from theNational Technical University of Athens, Greece, in1978, and the M.S. and Ph.D. degrees in electricalengineering from the State University of New York,Buffalo, NY, USA, in 1981 and 1984, respectively.He was a Faculty Member with the ElectricalEngineering and Computer Science Department,University of Connecticut (1983-1986) and theElectrical and Computer Engineering Department,Northeastern University, Boston, MA, USA (1986-

1994), and served as the Director of the Communications and Digital SignalProcessing Research Center with Northeastern University (1993-1994). He iscurrently a Professor with the Department of Informatics and Telecommuni-cations and the Director of the Communication Networks Laboratory with theUniversity of Athens. Since 1994, he has led several EU-funded projects thatshaped European research and development. His research interests are in theanalysis of communication networks and the design of mobile communicationsystems and services. He has authored more than 250 papers in the aboveareas. He is a member of the Board of the National Research Network ofGreece and the Chairman of the Board of the Greek Universities Network.

Nikos Passas received the Diploma (Hons.) degreefrom the Department of Computer Engineering,University of Patras, Greece, in 1992, and thePh.D. degree from the Department of Informaticsand Telecommunications, University of Athens,Greece, in 1997. Over the years, he has participatedand coordinated numerous national and Europeanresearch projects. He is currently a member of theLaboratory Teaching Staff with the Department ofInformatics and Telecommunications, Universityof Athens, and the Leader of the Green, Adaptive

and Intelligent Networking Research Group within the department. He hasauthored more than 120 papers in peer-reviewed journals and internationalconferences. His research interests are in the area of mobile networkarchitectures and protocols. He has also served as a Guest Editor andTechnical Program Committee Member in prestigious magazines andconferences, such as the IEEE WIRELESS COMMUNICATIONS, the IEEEVTC, the IEEE PIMRC, and the IEEE Globecom.

Christos Verikoukis (S’95–M’04–SM’07) receivedthe Ph.D. degree from the Technical University ofCatalonia, Barcelona, Spain, in 2000. He is cur-rently a Senior Researcher and the Head of theSMARTECH Department with the Centre Tecno-logic de Telecomunicacions de Catalunya, Spain,and an Adjunct Associate Professor with the Uni-versity of Barcelona. He has supervised 15 Ph.D.students and 5 post-doctoral researchers since 2004.He has participated in more than 30 competitiveprojects, while serving as the Principal Investigator

in national projects in Greece and Spain, and the Technical Manager forMarie Curie and Celtic projects. He is the Chair of the IEEE ComSocTechnical Committee on Communication Systems Integration and Modeling.He received the Best Paper Awards of the CQRM Symposium in the IEEE ICC2011, the IEEE Globecom 2015, the SAC Symposium of the IEEE Globecom2014, and the EURASIP 2013 Best Paper Award for the Journal on Advancesin Signal Processing.

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