Overlaid Device-to-Device Communication in Cellular Networks

Overlaid Device-to-Device Communication in Cellular Networks Geordie George, Ratheesh K. Mungara, and Angel Lozano Department of Information and Communication Technologies Universitat Pompeu Fabra (UPF) 08018 Barcelona, Spain Email:{geordie.george, ratheesh.mungara, angel.lozano}@upf.edu

Abstract—We present an analytical characterization of the spectral efficiency achievable by D2D communication links overlaid on a cellular network. The analysis relies on a stochastic geometry formulation with a novel approach to the modeling and spatial averaging of interference, which facilitates obtaining compact expressions. These expressions are then applied to gauge the potential of D2D communication, which is found to be very sizeable in situations of strong traffic locality.

I. I NTRODUCTION Device-to-device (D2D) communication, currently being touted as a potential ingredient of 5G cellular networks [1], allows users in close proximity to establish direct communication links, replacing two long hops via the base station (BS) with a single shorter hop. Provided there is sufficient spatial locality in the wireless traffic, this can bring about several benefits: reduced power consumption, lower end-toend latency, higher bit rates, and especially a much higher system spectral efficiency thanks to the denser spectral reuse. While the ad-hoc nature of D2D communication results in more complex and irregular topologies, in the context of a cellular network such approach can—unlike in traditional ad-hoc networks—benefit from infrastructure assistance to perform efficient user discovery and smart scheduling. Most initial attempts to quantify the benefits of D2D relied on simulations [2]–[5]. Recognizing that stochastic geometry tools allow for models that are more amenable to treatment, and arguably more representative of the heterogeneous structure of emerging wireless networks [6], some recent works modeled the user locations via PPP (Poisson point process) distributions and analytically tackled the system-level performance of D2D communication [7], [8] . In this paper, we continue down the path of [7], [8], but with a different approach to model interference and with a controllable degree of spatial averaging. This modified approach enables characterizing in simpler form—sometimes even in closed form—the spectral efficiencies achievable with D2D communication. Although the approach accommodates both overlay or underlay options, where respectively the D2D communication utilizes dedicated spectrum or reuses the cellular uplink, in this paper we concentrate on the former. The This work was supported in part by Intel’s University Research Program “5G: Transforming the Wireless User Experience”.

extension to underlay and the technical proofs of the results are relegated to a companion paper [9] II. S YSTEM M ODEL We consider an interference-limited cellular network where macro BSs are regularly placed on a hexagonal grid. In each cell, regular cellular transmissions are orthogonalized while, on separate spectrum, multiple D2D links coexist. Transmitters and receivers have a single antenna and each receiver knows the fading of only its own link, be it cellular or D2D. Our focus is on a given time-frequency signaling resource, where either one cellular uplink or multiple D2D links are active per cell. To facilitate the readability of the equations, we utilize distinct fonts for the cellular and D2D variables. A. User Locations The locations of the transmitters, both cellular and D2D, are modeled relative to that of a given receiver under consideration. For the cellular uplink, the receiver under consideration is a BS whereas, for the D2D link, it is a D2D receiver. In either case, and without loss of generality, we place such receiver at the origin and index the intended transmitter with zero. All other transmitters (interferers) are indexed in order of increasing distance. For the receiving BS, the intended transmitter is always the closest cellular transmitter while, for the D2D receiver, the intended transmitter need not be the closest D2D transmitter. 1) Cellular Uplink: To study this link, we place a receiving BS at the origin and locate an intended cellular transmitter uniformly within the corresponding cell, which (cf. Fig. 1) is circular with radius R and denoted by B(0, R). There is thus one and only one cellular transmitter within B(0, R), and its distance to the BS at the origin is denoted by r0 . The cellular interferers from other cells are all outside 1 B(0, R), modeled via a PPP Φ with density λ = πR 2 . With this density made to coincide with the number of BSs per unit area, this has been shown to be a fine model for a network with one cellular transmitter per cell [7], [10]. 2) D2D Link: To study this link, in turn, we place a D2D receiver at the origin and locate its intended D2D transmitter at a distance r0 .

2R

received from the D2D transmitters in Φ. For these user-touser links, we consider a different pathloss exponent η > 2.

Base station Cell Intended transmitter Cellular Interferer D2D Interferer

III. I NTERFERENCE M ODELING

R

r0

R

0

-R

-2R -2R

-R

0

R

2R

Fig. 1. System model. Located at the origin is a receiving BS and shown with a square marker in the surrounding circle is its intended cellular transmitter, while shown with square markers outside the circle are other cellular transmitters. On separate spectrum, and shown with diamond markers, are D2D transmitters.

A key differentiating feature of our framework is the interference modeling, expounded in this section and validated later in the paper. We depart from the approach in [7] where z and z are distributed as per (2) and (4), with all the products of signal and fading Gaussian variates therein explicitly retained, and also from the approach in [11] where a Gamma distribution fit is applied. Rather, recognizing that both z and z consist of a large number of independent terms whose fading is unknown by the receiver of interest, we model their short-term distributions as zero-mean complex Gaussian random variables with matched conditional covariances σ 2 = E |z|2 |{rk }  2 2 and σ = E |z| |{rj } , respectively, where the expectations are over the data and fading distributions. The conditional covariance σ 2 , which represents the power of z for given interferer locations, is easily found to equal σ2 =

∞ X

P rk−η

(5)

k=1

The D2D interferers conform to another PPP Φ with density λ = Kλ, reflecting the fact that there are on average K active D2D links per cell. B. Received Signal We denote by P and P the (fixed) powers transmitted by cellular and D2D users, respectively. 1) Cellular Uplink: The BS at the origin observes q y = P r0−η H0 b0 + z (1) where the first term is the signal from the intended cellular user while the second term represents the interference ∞ q X z= P rk−η Hk bk (2) k=1

made up from contributions from all other cellular transmitters in Φ\B(0, R), with η > 2 the pathloss exponent for cellular links, rk the distance between the kth cellular transmitter and the BS at the origin, Hk the corresponding fading, and bk the symbol transmitted by the kth cellular user. The fading coefficients {Hk } are independent identically distributed (IID) complex Gaussian random variables with zero mean and unit variance, i.e., drawn from NC (0, 1). Likewise, bk ∼ NC (0, 1). 2) D2D Link: To analyze this link, we shift the origin to the D2D receiver of interest, which observes q y = P r0−η H0 b0 + z (3) where the first term is the signal from the intended D2D transmitter while the second term is the interference ∞ q X z= P rj−η Hj bj (4) j=1

while its D2D counterpart σ2 equals σ2 =

∞ X

P rj−η .

(6)

j=1

Both (5) and (6) contain an infinite number of terms, of which a handful largely dominate the total interference. In recognition of this, we condition on the interferer locations within a circle surrounding the receiver of interest and replace the aggregate interference emanating from outside that circle with its expected (over the interference locations) value. As we shall see, this expected value is representative of most instances of the interference outside the circle—by virtue of the law of large numbers—and, thanks to the potency of stochastic geometry, this expected value can be computed explicitly. The introduction of the averaging circle allows reducing the number of variables retained in the formulation without the significant loss of information brought about by a complete averaging, a frequent recourse in stochastic geometry analyses. Specifically, the circle allows establishing the performance for specific user locations within the circle, and not only the average performance over all such locations. The radius of the averaging circle then becomes a modeling parameter that should be chosen to balance simplicity (the smaller the circle, the few interferers that are explicitly retained) and accuracy (the smaller the circle, the less fidelity in representing interference instances by their average). As a rather natural choice, we make the size of the circle coincide with that of a cell, B(0, R), and later validate this choice. With the foregoing choice for the averaging circle, the interference power in (5) is completely replaced by its average

(as all cellular interferers lie outside the circle) and we thus reset "∞ # X −η 2 σ =E P rk (7) Z

k=1 ∞

2πλP r1−η dr

=

(8)

TABLE I S YSTEM PARAMETERS Parameter K= η, η K0 aj

Value

λ λ

10 3.5, 4.5 K Γ(0.5+j) √ KΓ(j)

r=R

=

2P (η − 2)Rη

(9)

where the expectation in (7) is over the PPP, (8) follows from Campbell’s theorem [12] and (9) is obtained by evaluating the 1 integral and substituting λ = πR 2. For the D2D link, considering a circle B(0, R) around the D2D receiver at the origin, the interference power in (6) can be rewritten as 0

2

σ =

K X

P rj−η

j=1

{z

|

σ2in

|

{z

σ2out

(10)

}

where σ2in corresponds to the K 0 D2D transmitters in Φ ∩ B(0, R) whereas σ2out corresponds to the transmitters in Φ\B(0, R). Noting that E[K 0 ] = K, and by virtue of Campbell’s theorem, the average value of σ2out can be computed allowing us to reset 0

2

σ =

K X

P rj−η +

j=1

% = PK 0 =

∞ X

+ P rj−η 0 j=K +1

}

where the expectation in the numerator is over b0 , conditioned on the fading, and

2KP . (η − 2)Rη

(11)

r0−η

(18)

−η 2K + (η−2)R η j=1 rj −η a0 PK 0 −η 2K a + η−2 j=1 j

(19)

is the local-average SIR at the D2D receiver of interest. The normalized D2D transmitter distances are a0 = rR0 and aj = rj R. IV. SIR D ISTRIBUTION For a specific geometry, given the normalized distance a0 , the value of ρ becomes determined. Since |H0 |2 is exponentially distributed with unit mean, it follows from (13) that SIR exhibits an exponential distribution with local-average ρ and thus its conditional CDF (cumulative distribution function) is

A. SIR of the Cellular Uplink Applying our interference model, and recalling the intended signal term from (1), the instantaneous SIR of the uplink is   P r0−η E |H0 b0 |2 | H0 (12) SIR = σ2 = ρ|H0 |2 (13) where the expectation in (12) is over b0 , conditioned on the fading H0 , while η − 2  r0 −η ρ= (14) 2 R is the local-average SIR at the BS. Defining a normalized distance a0 = rR0 , η − 2 −η (15) a0 2 evidencing that our interference-limited formulation is invariant to the absolute scale of the network. ρ=

B. SIR of the D2D Link Similarly, for the D2D link, the instantaneous SIR is   P r0−η E |H0 b0 |2 | H0 SIR = (16) σ2 2 = %|H0 | (17)

FSIR|ρ (γ) = 1 − e−γ/ρ .

(20) 0

Similarly, given the normalized distances a0 and {aj }K j=1 the value of % becomes determined and it follows from (17) that FSIR|% (γ) = 1 − e−γ/% .

(21)

Let us now validate the foregoing distributions, and with that the interference modeling approach that underpins them. Example 1. Consider a network with an average of K = 10 D2D links per cell and with typical values for the pathloss exponents (cf. Table I). To render the system as typical as possible, K 0 is set to its expected value and, for j = 1, . . . , K 0 , aj is set to the expected value of the normalized distance to the jth nearest neighboring point in a PPP with density λ [13]. The resulting a1 , . . . , aK 0 are given in Table I, where Γ(·) is the Gamma function. Shown in Fig. 2 are the distributions in (20)-(21) alongside the corresponding numerically computed distributions with z as in (2) and z as in (4). A very satisfactory agreement is observed for cellular links corresponding to both average (a0 = 0.5) and cell-edge (a0 = 0.95) user locations, and for D2D links with a0 = 0.05 and a0 = 0.1. Similarly good agreements are observed for other settings, supporting our interference modeling approach.

1

20

0.9

0.7

CDF

0.6

a0 = 0.5 SIR

a0 = 0.95

0.5

SIR

a0 = 0.1

0.4 0.3

SIR

a0 = 0.05

0.2 0.1 0 −20

Link spectral efficiency (bits/s/Hz)

0.8

18

SIR

0

10

20

SIR(dB)

30

40

14 12 10 8 6 4 2

Simulation Analytical −10

16

50

0 0

0.2

0.4

a0

0.6

0.8

1

Fig. 2. CDF of instantaneous SIR for cellular and D2D links with the parameters of Table I, for a0 = 0.5, 0.95 and a0 = 0.05, 0.1.

Fig. 3. Cellular uplink spectral efficiency as function of the normalized link distance.

V. L INK S PECTRAL E FFICIENCY We now turn our attention to the ergodic spectral efficiency, arguably the most operationally relevant quantity in contemporary systems where codewords span many fading realizations across frequency (because of the wide bandwidths) and time (because of hybrid-ARQ) [14].

1) Cellular Uplink: Retaining the averaging circle equal to the cell size, the spectral efficiency is then a functional of the random locations of the intended cellular user (uniformly distributed within the cell). Thus, averaging C(ρ) over the distribution of a0 , a compact result involving only the MeijerG function ! a , ..., a , a , ..., a 1 n n+1 p Gm,n z (26) p,q b1 , ..., bm , bm+1 , ..., bq

A. Specific Network Geometry For a specific network geometry, i.e., for a given ρ, the spectral efficiency of the cellular uplink is C(ρ) = E [log2 (1 + SIR|ρ)] (22) Z ∞ = log2 (1 + γ) dFSIR|ρ (γ) (23) 0   1 = e1/ρ E1 log2 e (24) ρ R∞ where E1 (ζ) = 1 t−1 e−ζt dt is an exponential integral Likewise, for a given %, the spectral efficiency of the D2D link is   1 log2 e. (25) C(%) = e1/% E1 % Example 2. Shown in Fig. 3 is the spectral efficiency in (24) for the system parameters in Table I. B. Average Network Geometry The link spectral efficiencies in (24) and (25) can be further averaged over the local-average SIRs in (15) and (19), i.e., over the spatial distribution of the cellular and D2D transmitters, in order to characterize the average performance over all possible network geometries. Although, as mentioned earlier, the quantities thus obtained are less informative, they do help calibrate system-level benefits. To effect the spatial averaging, slightly different approaches are computationally more convenient in our stochastic geometry framework depending on the type of link (cellular/D2D).

and the pathloss η is obtained. Proposition 1. The cellular uplink average spectral efficiency over all network geometries equals ! 0, η−2 2 2 log e 2,2 2 η G2,3 . (27) C¯ = η η − 2 0, 0, −2 η Proof. See [9].



The versatility of our interference modeling approach is in full display here, facilitating a closed-form expression for a quantity that in previous analyses had only been obtained in integral form. 2) D2D link: For the D2D link, the average performance is characterized in two steps. First, we establish the average over all possible transmitter locations but for a fixed D2D link distance. To do so, we directly average the link spectral efficiency over the spatial locations of all cellular and D2D interferers in the network, without any prior replacement of interference by its expected value, i.e., without invoking any averaging circle. Proposition 2. For a given a0 , the D2D link spectral efficiency averaged over all network geometries equals Z ∞ 2/η 2 2 1 ¯ 0 ) = log (e) C(a e−γ a0 K Γ(1− η ) dγ (28) 2 0 γ+1

which, for η = 4, reduces to  √ √ ¯ 0 ) = 2 log (e) sin( πKa2 )si( πKa2 )− C(a 2 0 0  √ √ cos( πKa20 )ci( πKa20 )

0.3

(29)

0.25

25 % 0.2

a0

where the trigonometric integrals R ∞si(·) and ci(·) are respectively given by si(x) = x sin(t) t dt and ci(x) = R ∞ cos(t) − x t dt

0.15

80 %



Proof. See [9].

0.1

The result in Prop. 2, of interest in itself, can then be further averaged over the distribution of a0 . Since no such distribution based on actual user behavior has yet been reported, [7], [8] propound a Rayleigh distribution, which we next apply to our formulation. √ δ√ π , 2 K

Proposition 3. Let a0 be Rayleigh distributed with mean where δ > 0. The D2D link spectral efficiency averaged over all network geometries equals Z ∞ log2 e ¯= h  i dγ C (30) 0 (γ + 1) 1 + γ 2/η δ 2 Γ 1 − η2 which for η = 4, reduces to 2 3/2 4 ¯ = δ π log2 e − log2 (δ π) . C 1 + δ4 π

(31) 

Proof. See [9].

The above result is independent of K because a0 is made to shrink as K increases, in an attempt to model a natural reduction in transmission range as the user density intensifies.

Average(link(spectral(efficiency(Dbits/s/Hz)

Cellular(uplink D2D(link

6 5 4 3 2 1 0 3

3.2

3.4

3.6

98.8 % 0 0

0.2

0.4

a0

0.6

0.8

1

Fig. 5. Contour plot for the relationship in (32). Within the unshaded region, C(%) > C(ρ).

VI. B ENEFITS OF D2D To conclude, we provide some examples comparing the uplink and D2D spectral efficiencies so as to gauge the benefits of the latter and to demonstrate the usefulness of the tools developed in earlier sections. Unless otherwise specified, the system parameters in Table I are applied. Example 4. Equating the link spectral efficiencies C(ρ) and C(%) for the parameters in Table I we obtain the relationship 4.5

a0 = 0.512 a03.5

(32)

a contour plot of which is presented in Fig. 5. Within the unshaded region, the D2D link has a better spectral efficiency than a corresponding uplink transmission from the same user would have, and thus D2D becomes advantageous. The share of geometries for which C(%) > C(ρ) for a given value of a0 is easily obtained as P[a0 > x] = 1 − x2 where x is the corresponding x-axis value of the contour. Some such shares are displayed, e.g., for a0 = 0.15 D2D is preferable in 80% of situations.

8 7

0.05

3.8

4

4.2

4.4

4.6

Pathloss(exponent

Fig. 4. Average spectral efficiencies of uplink and D2D link (with Rayleighdistributed distance having δ = 0.3) for varying η and η.

Example 3. Shown in Fig. 4 are the average spectral efficiencies in (27) and (30). For the D2D link, the distance is Rayleigh-distributed with δ = 0.3.

The above example shows how, for a rather typical network geometry, from a link vantage D2D is very often a better alternative than cellular communication via the BS even if only the uplink is considered, with the downlink bandwidth consumption neglected. With the costs of both uplink and downlink considered, the appeal of D2D would increase even further. Example 5. Considering a fixed link distance a0 = 0.08 for all D2D pairs, CDFs of C(ρ) and C(%) over the values of a0 0 and {aj }K j=1 are plotted in Fig. 6. Even for very high K, when the system brims with D2D interference, thanks to their short range many D2D links enjoy higher spectral efficiencies than the corresponding cellular uplink.

1 0.9

The D2D advantage grows with a shrinking a0 , approaching two orders of magnitude for a reasonable value of a0 = 0.09. Altogether, D2D shows great promise in situations where traffic locality is present, even if no attempt is made to optimize the scheduling of transmissions. With a smart such scheduling [15]–[17], this promise could only be reinforced. •

D2D link Cellular uplink

0.8 0.7

K = 80

CDF

0.6

K = 40 K = 20

0.5

R EFERENCES

K = 10

0.4

K= 5

0.3 0.2

K= 1

0.1 0 0

2

4

6

8

Link spectral efficiency (bits/s/Hz)

10

Average system spectral efficiency (bits/s/Hz/cell)

Fig. 6. CDFs of cellular and D2D link spectral efficiencies with a0 = 0.08. 90

D2D link Cellular uplink

80

a0 =0.09

70

60 50 40

a0 =0.12

30 20

a0 =0.15

10 0 0

50

K

100

150

Fig. 7. Average system spectral efficiency with a fixed D2D pair distance.

The foregoing example indicates that the number of D2D links that can coexist on a given signaling resource is large, and to better appreciate the benefits of such dense spectral reuse we next turn our attention to the system spectral efficiency (bits/s/Hz/cell), which reflects the benefits of this reuse. Example 6. Since there are K active D2D links per cell on average, the average system spectral efficiency of the D2D ¯ 0 ) whereas, for the cellular uplink, the average traffic is K C(a system spectral efficiency is C¯ as there is only one active cellular user per cell. Shown in Fig. 7 is the comparison of these quantities as function of K, for various a0 . Example 6 prompts three final observations: • For each value of a0 , there is an optimum “load” K. • For a very wide range of K, and especially around its optimum value, the D2D system spectral efficiency is much higher than its cellular counterpart.

[1] F. Boccardi, R. W. Heath Jr., A. Lozano, T. Marzetta, and P. Popovski, “Five disruptive technology directions for 5G,” IEEE Commun. Mag., vol. 52, no. 2, pp. 74–80, Feb. 2014. [2] B. Kaufman and B. Aazhang, “Cellular networks with an overlaid device to device network,” in Proc. Annual Asilomar Conf. Signals, Syst., Comp., Oct. 2008. [3] S. Hakola, T. Chen, J. Lehtom¨aki, and T. Koskela, “Device-to-Device communication in cellular network - performance analysis of optimum and practical communication mode selection,” in Proc. IEEE Wireless Commun. and Networking Conf., Apr. 2010. [4] C. Yu, K. Doppler, C. B. Ribeiro, and O. Tirkkonen, “Resource sharing optimization for device-to-device communication underlaying cellular networks,” IEEE Trans. Wireless Commun., vol. 10, no. 8, pp. 2752– 2763, Aug. 2011. [5] K. Doppler, C. Yu, C. B. Ribeiro, and P. Janis, “Mode selection for device-to-device communication underlaying an LTE-advanced network,” in Proc. IEEE Wireless Commun. and Networking Conf., Apr. 2010. [6] J. G. Andrews, F. Baccelli, and R. K. Ganti, “A tractable approach to coverage and rate in cellular networks,” IEEE Trans. Commun., vol. 59, no. 11, pp. 3122–3134, Nov. 2011. [7] X. Lin, J. G. Andrews, and A. Ghosh, “Spectrum sharing for deviceto-device communication in cellular networks,” Available online: http://arxiv.org/abs/1305.4219/, 2013. [8] Q. Ye, M. Al-Shalash, C. Caramanis, and J. G. Andrews, “Resource optimization in device-to-device cellular systems using time-frequency hopping,” Available online: http://arxiv.org/abs/1309.4062/, 2013. [9] G. George, R. K. Mungara, and A. Lozano, “An analytical framework for device-to-device communication in cellular networks,” Available online: http://arxiv.org/abs/1407.2201, 2014. [10] T. D. Novlan, H. S. Dhillon, and J. G. Andrews, “Analytical modeling of uplink cellular networks,” IEEE Trans. Commun., vol. 12, no. 6, pp. 2669–2679, June 2013. [11] R. W. Heath Jr., M. Kountouris, and T. Bai, “Modeling heterogeneous network interference using Poisson point processes,” IEEE Trans. Signal Processing, vol. 61, no. 16, pp. 4114–4126, Aug. 2013. [12] D. Stoyan, W. Kendall, and J. Mecke, Stochastic Geometry and Its Applications, John Wiley and Sons, 2nd edition, 1996. [13] M. Haenggi, “On distances in uniformly random networks,” IEEE Trans. Inform. Theory, vol. 51, no. 10, pp. 3584–3586, Oct. 2005. [14] A. Lozano and N. Jindal, “Are yesterday’s information-theoretic fading models and performance metrics adequate for the analysis of today’s wireless systems?,” IEEE Commun. Mag., vol. 50, no. 11, pp. 210–217, Nov. 2012. [15] X. Wu, S. Tavildar, S. Shakkottai, T. Richardson, J. Li, R. Laroia, and A. Jovicic, “FlashLinQ: A synchronous distributed scheduler for peerto-peer ad hoc networks,” IEEE/ACM Trans. Networking, vol. 21, no. 4, pp. 1215–1228, Aug. 2013. [16] N. Naderializadeh and A. S. Avestimehr, “ITLinQ: A new approach for spectrum sharing in device-to-device communication systems,” Available online: http://arxiv.org/abs/1311.5527/, 2013. [17] R. K. Mungara, X. Zhang, A. Lozano, and R. W. Heath Jr., “On the spatial spectral efficiency of ITLinQ,” in Proc. Annual Asilomar Conf. Signals, Syst., Comp., Nov. 2014.