A Tractable Model for Device-to-Device Communication ... - IEEE Xplore

ICC'14 - W3: Workshop on Small Cell and 5G Networks

A Tractable Model for Device-to-Device Communication Underlaying Multi-Cell Cellular Networks Hao Feng∗ , Haibo Wang∗ , Xiaohui Xu∗ , Chengwen Xing† ∗

Research Institute of Broadband Wireless Mobile Communications, Beijing Jiaotong University Email: {12120064, hbwang, 12120167}@bjtu.edu.cn † School of Information and Electronics, Beijing Institute of Technology Email: [email protected]

Abstract—It is well-established that Device-to-Device (D2D) communication is a promising technology to improve spectral efficiency and reduce system power consumption simultaneously. In this paper, we build a tractable model for Device-to-Device communication underlaying multi-cell cellular networks to evaluate the coverage probability and area spectral efficiency by utilizing stochastic geometry. To mitigate the interference in this hybrid system, we adopt Exclusion Regions (ERs) around base stations to limit the locations of cellular users and active D2D users. Numerical results demonstrate that ERs could significantly improve the coverage probability and the system area spectral efficiency. Our results also show that the coverage probability target can still be achieved for the maximum allowed D2D communication distance or the maximum D2D pairs density by adjusting the ER radius.

I.

I NTRODUCTION

Device-to-Device (D2D) communications have recently been proposed as an underlaying method to cellular networks in order to increase the system spectral efficiency and reduce the power consumption of user equipments. In D2D communication systems, D2D user equipments (DUEs) transmit data signals to each other over a direct link by sharing the radio resource of cellular user equipments (CUEs). While D2D enables high speed local communications, it also brings potential interference between DUEs and CUEs. Many previous works have been done to get optimal coordination scheme to mitigate the interference to CUE, to achieve throughput enhancement of D2D system, or to select a best CUE for D2D link [1]-[5]. For example, Doppler [1] et al. proposed a mechanism to control the maximum transmit power of DUEs to alleviate D2D interference to CUE. Xu [2] designed an uplink spectrum-sharing scheme by tracking the near-far interference and identify the interfering cellular users to enhance system average throughput. Xiang [3] focused on a distance-dependent algorithm with power optimization to help BS determine the optimal mode to maximise the overall system capacity. Liu [4] studied mode selection for D2D with consideration of relay technology and found that relay node can increase both chance and area of D2D pair. Wang [5] proposed a distance-constrained resource-sharing criteria for BS to select a CUE for D2D link to keep the minimum distance between them. However, [1]-[5] only considered a single cell in their system models, while the cumulated interference from neighbor cells was not included.

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Andrews [6] et al. has proposed to employ Poisson point process (PPP) to model the locations of network nodes in a multi-cell system and to analyze the average interference. In [7], the author utilized stochastic geometry to optimize the sum-capacity and propose a linear searching algorithm to find the optimal parameters such as the density of DUEs and the optimal transmit power. Lin [8] proposed a tractable hybrid network model to study the optimal scheme to partition the spectrum between cellular and D2D transmissions and found the optimal threshold for mode selection. A joint optimal decision is also introduced in the paper. Nevertheless, [6]-[8] assume the distance between D2D link have been a priori condition. None of work mentioned above has investigated what the maximum allowed D2D communication distance should be, and most of them assumed CUEs sharing their Uplink (UL) radio resource with D2D pairs, the Downlink (DL) resource-sharing has not been fully discussed. In this work, we investigate D2D communication underlaying the downlink radio resources of a multi-cell OFDMA system, assuming that the locations of DUEs and BSs follow two independent PPPs. To mitigate the interference and improve the overall area spectral efficiency (ASE), Exclusion Regions (ERs) are employed which only allow the CUEs inside the ERs and DUEs outside the ERs to share the same spectrum. Moreover, we derive the close-form expressions of the coverage probability of both CUEs and DUEs, as well as ASE of the whole hybrid network. Through numerical performance evaluation, we prove that the ERs with carefully selected radius could improve the system performance significantly, and the coverage probability target can be achieved for the maximum allowed D2D communication distance or the maximum D2D pairs density by adjusting the ER radius. The rest of the paper is organized as follows. Section II describes the system model. In section III, four type interferences are investigated and the expressions of coverage probabilities are provided with closed-form solutions. The area spectral efficiency is given in section IV. Numerical results and discussion are addressed in section V followed by conclusions in Section VI. II.

S YSTEM M ODEL AND P ROBLEM F ORMULATION

A. Network Model We consider an OFDMA-based system where multiple DUEs can share one downlink sub-channel of a CUE per


cell. Assume the Base stations (BSs) have full knowledge of the locations of the DUEs, i.e. given the DUEs report their locations to the BSs. The locations of the BSs follow a homogeneous Poisson point process (PPP) on the two-dimensional plane which is denoted by ΦB = {x1, x2..} ⊂ R2 with the density λB and the locations of potential D2D transmitters, which are CUEs with D2D communication capability, follow another independent homogeneous Poisson point process ΦD = {y1, y2..} ⊂ R2 of density λD . Assuming all the D2D transmitters use the same transmission power µD and all the D2D receivers are at the distance rD from the corresponding D2D transmitters in a random direction. It is also assumed that only one CUE per cell shares its downlink sub-channel with DUEs in this cell, and rB represents the distance between this CUE and its corresponding BS.

%6 %6 %6

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Besides, open loop fraction power control (OFPC) is employed for DUE transmitters. OFPC should be able to reduce the DUEs power consumption when the D2D distance is small, which can also reduce the interference to the hybrid system. In OFPC, the DUEs’ transmit power is selected in decibel as kα µD = µ0 + k × P L(rD ), where µ0 = µDmax /rDmax is an initial power, k is a path loss compensation factor, P L(rD ) is the path loss between the DUE pairs. We do not consider downlink power control for cellular transmissions, hence all the BSs are transmitting with constant power µB . C. Exclusion region In order to mitigate the interference between cellular and D2D links, ERs around the BSs are employed (see Fig. 1 for illustration). That is, only the CUEs locate inside the ERs will be selected to share their downlink sub-channels with D2D transmitters, whereas the potential D2D transmitters will be activated (meaning switched to D2D mode) only when they are outside ERs. Since every BS selects one CUE in an ER as the typical CUE per sub-channel, typical CUEs will follow a PPP with the same density as the BSs. However, it is worth noting that the distribution of active D2D transmitters will no longer be a PPP but a Poisson hole process [10]. For tractability, we have made the following assumptions: Assumption 1: The radius D of all ERs is same and there exists at least one CUE inside each ER.

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Assumption 2: The SINR threshold T depending on the Quality-of-Service (QoS) requirement of each link (no matter cellular or D2D link) is assumed to be the same.

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Fig. 1. The triangles are the BSs and the squares are the intended CUEs and the distance between them is rB . The filled circles are the active D2D transmitters and D2D receivers are represented by × at a distance rD from the corresponding transmitters. The big circles around every BS are exclusion regions with radius D. The hollow circles inside the exclusion regions are the inactive DUEs.

In the investigated system, the total interference to a typical CUE is from all the unintended BSs and all the active D2D transmitters. For a typical D2D receivers, the total interference from both BSs and all unintended active D2D transmitters (except the intended D2D transmitter). Hence there exist four types of interference: the interference from other BSs to a CUE IBC ; the interference from all DUEs to a CUE IDC ; the interference from all BSs to a DUE IBD and the interference from other D2D transmitters to a D2D receiver IDD . III.

B. Channel Model and Power Control To model the channel, the large-scale path loss is assumed to be inversely proportional to distance with the path loss exponent given by α. We assume that all the links experience Rayleigh fading, which has an exponential distribution with E[h] = 1. All transmissions experience additive white Gaussian noise (AWGN) and the noise power is assumed to be σ 2 . The cumulative interference at y from the points of the process Φx can be defined as X Ixy = µx hxy ℓ(y − x), (1) x∈Φx

where µx is the transmitter power of the point x, hxy stands for Rayleigh fading and ℓ(y − x) = ky − xk−α is the largescale path fading model.

588

I NTERFERENCE A NALYSIS

In this section, we derive the probability of coverage for DUE and CUE separately. The probability of coverage can be formally defined as the complementary cumulative distribution function (ccdf) of SINR as: P = P {SIN R > T }.

(2)

A. Interference to Cellular Users The SINR of a typical user at the distance rB (rB ≤ D) from its associated base station can be expressed as: SIN R =

−α µ B rB hBC . IBC + IDC + σ 2

(3)

Since the power fading is exponential and the BSs are distributed as a PPP, the Laplace transform of IBC can be


written as below according to Lemma 1 in [9]. LIBC (s) = exp

n

2 o α µδB sδ . − λB 2 sin(π · ) α π2 ·

(4)

With regard to IDC , the active D2D transmitters follow a Poisson hole process, which is hard to model directly [10]. However, the independent thinning of potential DUEs outside the ERs with probability exp(−λB πD2 ) yields a good approximation for IDC [11]. Since the D2D transmitters are at least at distance (D − rB ) to the cellular receiver, we can obtain the Laplace transform of the approximation IeDC by using Lemma 2 in [9]. h LIeDC (s) = exp − πλD exp(−λB πD2 )EhDC (sµD hDC )δ × γ(1 − δ, sµD hDC (D − rB )−α )− sµD (D − rB )2−α i , 1 + sµD (D − rB )−α

The interference from other D2D transmitters IDD can be approximated as IeDD by assuming the DUEs distribution is thinned with probability exp(−λB πD2 ). Following Lemma 1 in [9], the Laplace transform of IeDD is LIeDD (s) = exp

(5) where γ(a, x) = 0 ta−1 e−t dt is the lower incomplete gamma 2 . DUE’s transmit power is determined by function and δ = α the OFPC scheme, such that µD = µ0 + k × P L(rD ). Theorem 1: The coverage probability of a typical CUE can be lower-bounded as Z D 2 e−sσ LIBC (s)LIeDC (s)f (rB )drB . (6) PC >

α where s = T µ−1 D rD .

Proof: Similarly with the proof of Theorem 1, LIˆBD (s) and LIeDD (s) have been presented in equation (7) and (8).

PˆC =P (SIN R > T ) −α hBC µ B rB > T) =P ( IBC + IDC + σ 2 Z D h i 2 E e−s(IBC +IDC +σ ) f (rB )drB = 0 Z D 2 = e−sσ EIBC e−sIBC EIDC e−sIDC f (rB )drB 0 Z D 2 ≃ e−sσ LIBC (s)LIeDC (s)f (rB )drB ,

A REA S PECTRAL E FFICIENCY

In this section, we derive the Area Spectral Efficiency (ASE), which is the network-wide spatially averaged product of the density of successful transmissions and the corresponding spectral efficiency [12] [13]. Specifically we compute the average spectral efficiency in units of nats/Hz (1 bit = ln(2) nats ≃ 0.693 nats). Assuming adaptive modulation and coding are employed, the average rate per UE is calculated based upon the Shannon capacity expression [6]. Theorem 3: The average rate of the CUE can be calculated

0

Proof: Since LIBC (s) and LIeDC (s) are independent and using the fact that hBC is exponentially distributed, we can get that

2 o α µδ s δ . − λD exp(−λB πD2 ) D 2 sin(π · ) α (8) π2 ·

Theorem 2: The lower-bound for the coverage probability of DUE is 2 (9) PD > e−sσ LIˆBD (s)LIeDD (s),

IV.

Rx

n

as E[RC ] =

Z

D

f (rB ) 0

Z

∞

2

e−sσ LIBC (s)LIDC (s)dtdrB . T

(10)

Proof: According to Shannon Capacity Theorem, the average rate of the CUE can be obtained E[RC ] =E ln(1 + SIN R) Z ∞ Z D f (rB ) P ln(1 + SIN R) > t dtdrB = 0 T Z D Z ∞ −α µ B rB hBC = f (rB ) P ln(1 + 2 ) > t dtdrB I + I + σ BC DC 0 T Z D Z ∞ (et − 1)(IBC + IDC + σ 2 ) = f (rB ) P hBC > ]dtdrB . −α µ B rB 0 T (11)

0

2 · rB . α where s = T µ−1 B rB and f (rB ) = D2 B. Interference to D2D Users Since the active D2D transmitters are outside of the ERs and all the D2D receivers are at the distance rD from the corresponding D2D transmitters, the minimum distance between D2D receivers and nearest BS is (D − rD ). Then the Laplace transform of IBD can be obtained from Lemma 2 in [9]. h LIˆBD (s) = exp − πλB EhBD (sµB hBD )δ γ(1 − δ, sµB hBD sµB (D − rD )2−α i −α . × (D − rD ) ) − 1 + sµB (D − rD )−α (7)

589

t 1 with By adopting (4) and (5) in (11), and substituting e −−α µ B rB s, (10) is proved.

Theorem 4: The average spectral efficiency of DUE is Z ∞ 2 E[RD ] = (12) e−sσ LIBD (s)LIDD (s)dt, T

t 1. where s = e −−α µ D rD

Proof: The proof is similar to Theorem 3.


V.

P ERFORMANCE E VALUATION AND N UMERICAL R ESULTS

In this section, we present numerical simulations of the coverage probabilities and area spectral efficiency. We evaluate these metrics in two scenarios: with ER and without ER (w/o ER) in which each BS selects intended CUE in its cell randomly and all the DUEs are active. Besides, in order to verify the accuracy of our close-form expressions, we generated curves from both analytical expressions and Monte Carlo simulations. Key parameters applied in the section are listed in Table I. In particular, we assume that there are 40 sub-channels in a cell, so the transmitting power of BSs in one sub-channel is 46dBm/40 = 30dBm. TABLE I.

S IMULATION PARAMETERS AND VALUES

Parameter Density of BSs λB Density of DUEs λD Transmitting power of BS µB Maximum transmitting power of DUE µDmax Maximum distance between D2D link rDmax SINR threshold T Path loss compensation factor k Path loss exponent α

Value 1 × 10−6 m−2 1 × 10−5 m−2 30 dBm 23 dBm 100 m -10 dB 0.8 4

1 0.95

Coverage probability of DUE

The ASE of the investigated hybrid system should be calculated as: ASE = λC E[RC ] + λ′D E[RD ]. Where E[RC ] and E[RD ] represent the average rate per CUE and per DUE, respectively. λ′D = λD ·exp(−λB πD2 ) is the density of active DUEs.

0.9 0.85 0.8 0.75 0.7 0.65

D = 300m D = 217m D = 100m w/o ER

0.6 0.55 0.5 10

20

30

40

50

60

70

80

90

100

rD (m)

Fig. 3.

Coverage probability of DUE vs. D2D pair distance

CUE and its corresponding BS becomes larger. Fig. 3 shows the relationship between the coverage probability of DUE and rD : when rD is larger, the coverage rate of DUE becomes lower. The reason is that for large rD , the receiving signal power of DUE is relatively small under OFPC while the interference from other D2D links are stronger. It also reveals that DUEs’ coverage probability will be improved as D increasing from 100m to 300m. This is because a larger D will lead to a reduced number of active D2D transmitters and the interference level among different D2D pairs will be less. Considering the results of both Fig. 2 and Fig. 3, it can be found that small D provides the best coverage for CUEs while large D leads to better coverage for DUEs. TABLE II.

Coverage probability of CUE

1

Exclusion Region radius D 100 m 150 m 200 m 217 m 250 m 300 m

0.9

0.8

0.7

0.6

0.5

0.4 10

D = 100m (ana) D = 100m (sim) D = 217m (ana) D = 217m (sim) D = 300m (ana) D = 300m (sim) w/o ER(ana) 20

30

40

50

60

70

80

90

100

rD (m)

Fig. 2.

Coverage probability of CUE vs. D2D pair distance

In Fig. 2, dotted lines show the Monte Carlo simulation results with ER. When comparing them with analytical results (solid lines) employing the same D, it can be observed that the analytical results provide lower-bounds to the simulation results. Fig. 2 illustrates that ER can help improving the coverage rate of CUE significantly. It also reveals that the coverage rate of CUE is decreasing as rD increases, since higher transmit power of DUE transmitters will be allocated for bigger rD under OFPC leading to stronger interference to CUE. Meanwhile, with the increase of D, CUE’s coverage rate is decreasing because the average distance between the

590

Allowed rD 70 m 78 m 81 m 85 m 69 m 50 m

More importantly, Fig. 2 and Fig. 3 implies that D and rD can be jointly optimized to guarantee the coverage rate of both CUEs and DUEs. Thus we run extensive simulations to find different (D, rD ) combinations to ensure the coverage rate of CUE and DUE are no less than 90%. As shown in Table II, it is found that the maximum allowed D2D separation of 85m could be achieved when D is set to 217m under the 90% coverage rate constraint. Fig. 4 illustrates the ASE versus D2D pair distance for different D settings. Fig. 4 reveals that: a) Small D always brings significant ASE gain within the whole range of rD , because the selected CUEs are close to the BS and their average spectral efficiency is much greater than the case without ER, where the CUEs randomly distributed in the cells; b) whether large D achieves any ASE gain would rather depends on the value of rD , e.g., for rD ≤ 40m, without ER is the better choice when D = 300m since interference from DUE is quite small and a large ER seriously reduces


the number of active DUEs.

to only allow the CUEs inside the ERs and DUEs outside the ERs to share the same spectrum. Numerical results verified that a small size of ER can significantly improves both the coverage probability and the system ASE. We also studied how to jointly design the maximum allowed D2D communication distance and the ER radius, as well as the maximum D2D density under a certain coverage constraint. The findings in this paper may help network operators set the maximum allowed communication distance between D2D pairs based on given DUE’s density or configure the maximum number of D2D pairs which can share the downlink resource of one CUE per cell.

60

D = 100m

55

D = 217m D = 300m

2

ASE (nats/Hz/(km ))

50

w/o ER 45 40 35 30 25 20

ACKNOWLEDGMENT

15 10 10

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80

90

This research has been supported partly through the National Natural Science Foundation of China [No.61001071], and in part through the project [2012JBM018] of Beijing Jiaotong University.

100

rD (m)

Fig. 4.

ASE vs. D2D pair distance

R EFERENCES 1

Coveragerate Probability

0.95

0.9

0.85

D = 100 m,CUE D = 100 m,DUE D = 220 m,CUE D = 220 m,DUE D = 300 m,CUE D = 300 m,DUE

0.8

0.75 1

2

3

4

5

6

7

8

9

10

n (λ’ /λ ) D

B

Fig. 5. Coverage probability vs. Active DUE density, with rD = rDmax = 100 m and λ′D = λD · exp(−λB πD 2 ) is the density of active DUEs

Fig. 5 depicts the coverage probability of both CUE and DUE versus the relative density λ′D /λB for different D settings by assuming rD = rDmax = 100m, from which the maximum allowed active D2D pair density under a certain coverage probability constraint can be found with the corresponding D setting. E.g., under the coverage constraint (PC ≥ 90%, PD ≥ 90%), the maximum allowed active D2D density λ′D /λC = 6 can be achieved when D is set to be 220m. That is, there could be at most 6 active D2D pairs sharing the downlink sub-channel of a CUE per cell, corresponding to the D2D communication distance is 100m. In this case, the allowed potential DUEs’ density can be approximately set as 7λB . This example tells us that there exists an optimal DUE’s density for each given D2D communication distance in our model.

VI.

C ONCLUSIONS

In this paper we have investigated the coverage probability and area spectral efficiency of a hybrid multi-cell system where D2D links reuse the downlink cellular spectrum, and derived the closed-form expressions. Exclusion regions are employed

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