Device-to-Device Communication in Wireless Mobile Social Networks Jemin Lee† and Tony Q. S. Quek†∗ †

Singapore University of Technology and Design, Singapore ∗ Institute for Infocomm Research, A*STAR

Abstract—Wireless mobile social networks have been introduced to consider characteristics of mobile user’s behaviors for network configuration. The device-to-device (D2D) communication is to increase the network spectral efficiency by allowing users to communicate directly with the corresponding destinations. This paper develops a framework for the design of reliable D2D communication provided wireless mobile social network (D2D-MSN), and introduces a D2D-MSN throughput as a new measurement of the spectral efficiency achieved by users communicating successfully. Our framework accounts for spatial user distributions and the link distance-based mode selection scheme that allows the D2D mode only for the users with shorter link distance than a threshold. We also account for the communication distance distribution defined in a social context to decrease the communication probability density according to the link distance with a power exponent. This research offers insights on the optimal configuration of the D2D-MSN, and also provide a new perspective on the effect of the link distance distribution on the D2D communication performance. Index Terms—device-to-device communication, mobile social network, transmission mode selection, stochastic geometry

I. I NTRODUCTION Wireless mobile social networks, which are the combination of the physical wireless mobile network and the virtual social network, have been introduced to enhance the network performance by exploiting the characteristics of the mobile user’s social relationship and behaviors [1], [2]. Recently, device-to-device (D2D) communications have been introduced to allow users communicate directly with each other without using a base station (BS) [3]. The D2D communication can offload the traffic handled by BSs and enhance the local service quality and the spectrum efficiency of the wireless network. For the design of a reliable D2D communication provided wireless mobile social network (D2D-MSN), algorithms for interference management and transmission mode selection are required by considering mobile user social behaviors. The throughput of D2D communications has been presented by taking into account the resource management and the D2D communication operation [4]–[10]. The resource management algorithms for D2D communications are presented to enhance the network throughput by the interference limited area control [4], the transmission power control [5], and the joint This work was partly supported by the Temasek Research Fellowship, the SRG ISTD 2012037, the SUTD-MIT International Design Centre under Grant IDSF1200106OH, and the A*STAR SERC Grant 1224104048.

frequency spectrum scheduling and power control [6]. For efficient operations of D2D communications, the transmission mode selection [7] and the receive mode selection [8] are also presented. The performance of D2D communication has recently been analyzed for various network settings including cognitive radio networks [9] and spectrum sharing networks [10]. However, most of these works have not taken into account for the spatial distribution of users and the distance distribution of communication links, which are paramount for the analysis of D2D communication throughput. In this paper, we provide a foundation for D2D-MSN, composed of spatially distributed users selectively transmitting their data in a cellular mode or a D2D mode. We consider the transmission mode selection algorithm that selects a D2D mode when a communication link distance is shorter than a threshold, otherwise it selects a cellular mode. In a mobile social network, it is shown that the probability of communication to a receiver is inversely proportional to the distance between the transmitter and the receiver [11]–[13]. Hence, we model the communication link distribution in a social context, for which probability density decreases according to the link distance with a power exponent. We also introduce the D2D-MSN throughput as a performance metric that measures the spectral efficiency of users communicating successfully in a cellular or a D2D mode. The key contributions of the paper can be summarized as follows: (i) introduction and definition of the D2D-MSN throughput in a D2D-MSN composed of D2D-mode and a cellular-mode users; (ii) development of a framework for the design and the analysis of D2D-MSN accounting for the spatial user distributions as well as the communication distance distribution; and (iii) quantification of the D2D-MSN throughput for characterizing the effects of D2D communication parameters such as the D2D mode selection threshold and the D2D resource portion on the D2D-MSN throughput. The remainder of this paper is organized as follows: Section II describes the D2D-MSN configuration including the distance distribution of communication link. Section III introduces and analyzes the D2D-MSN throughput. Section IV quantifies the effect of D2D parameters on the D2D-MSN throughput. Conclusion is given in Section V.

is the distance between transmitter and receiver; α is the pathloss exponent; N = N0 W ; N0 is the one-sided power spectral density (PSD); W is a channel bandwidth; ΠI is the distribution of interfering nodes following a PPP with the spatial density λI ; and interfering node transmission power is PI . The complementary cumulative distribution function (CCDF) of received SINR by a μ-mode user in Rayleigh fading channels is given by [14], [15] 2/α ηN Dα ηPI −cα λI D2 − (μ) (μ) ¯ P P Fζ (μ) (η) = ED e (2)

Fig. 1. An example of a D2D-MSN (blue circles are cellular-mode transmitting users, black circles are D2D-mode transmitting users, and white circles are receiving users).

II. D2D-MSN C ONFIGURATION In this section, we describe a D2D-MSN and a transmission mode selection scheme. A. Network Configuration We consider a wireless mobile social network composed of users aiming to transmit their data via one of two transmission modes: a cellular mode that makes each user communicates to a BS, and a D2D mode that enables users to communicate directly to their corresponding destinations, which is determined in a social context. Users are distributed in the network according to a homogeneous Poisson point process (PPP) Π with spatial density λt . Channels are allocated to each cellular-mode users exclusively, and they are reused in other cells with a frequency reuse factor 1/η.1 When there are more cellular-mode users in a cell than the number of available channels for cellular mode, some cellular-mode users are blocked for accessing channels. On the other hand, allocated channels for the D2D mode are shared by all D2D-mode users in the network, and each D2Dmode user randomly accesses a channel. The β portion of the total frequency channels are allocated to D2D-mode users. Hence, the numbers of channels available for D2D-mode users and cellular-mode users in a cell are C (d) = ηβCt and C (c) = Ct − βCt , respectively,2 where x is the ceiling function of x and Ct is the total available channels in a cell. The instantaneous signal-to-interference and noise ratio (SINR) per symbol received by a node at y from a μ-mode user at x is given by (μ) ζx,y =

P (μ) Hx,y D−α x,y

−α Z∈ΠI PI HZ,y DZ,y

+N

(1)

where P (μ) is the transmission power of a user in μ mode; Hx,Y is the fading level of the link; Dx,Y = |x − Y| for cellular-mode users are reused in every adjacent η cells. total number of channels available in this network is ηCt since Ct channels are allocated to each cell and reused in every η cells. The β portion of total channels are allocated to D2D mode and C (d) = ηβCt .

where cα = (2π/α)Γ(2/α) Γ(1 − 2/α) and D is the random communication link distance. B. Transmission Mode Selection We assume that a source at xs ∈ Π communicates to the corresponding destination at Yd ∈ Πr ∩ Bxs (rd,u ) where Πr is a PPP of receiving nodes with spatial density λr ; rd,u is the maximum communication distance; and Bx (r) is the ball of radius r centered at x. The link-distance dependent mode selection scheme is considered to transmit in a D2D mode only if Dxs ,Yd is shorter than the mode selection threshold dth . Hence, the probability that a user selects the D2D mode is pm = P {Dxs ,Yd ≤ dth }. By the thinning property of a PPP [16], after the transmission mode selection, the distributions of cellular-mode users and D2D-mode users still follow PPPs (c) (d) (c) Π and Π with spatial densities λt = λt (1 − pm ) and (d) λt = λt pm , respectively. The distribution of communication distance Ds is determined according to a power-law communication probability, defined in a social context [11]–[13]. Specifically, a source communicates to one of uniformly distributed receiving nodes, and the communication probability is inversely proportional to the Euclidean distance between the source and the receiving node with a power exponent θ. Note that the power exponent for cellular-mode users is θ = 0 since users are uniformly distributed in a cell and communicate to a BS located at the origin, while the power exponent for D2D-mode users is determined by network social characteristics. Here, we assume 0 ≤ θ < 2 for analytic tractability. The probability density function (PDF) of Ds is defined as follows. Lemma 1: By the power-law communication probability with the exponent 0 ≤ θ < 2, the approximated PDF of the social communication distance Ds is given by fDs (d) =

(3)

Proof: By the power-law communication probability, the probability that a node at x communicates to a node at Yd ∈ Πr,x = Πr ∩ Bx (rd,u ) is given by

1 Channels 2 The

(2 − θ) d1−θ . 2−θ rd,u

pcx,Yd =

D−θ x,Yd Y∈Πr,x

D−θ x,Y

,

∀ Yd ∈ Πr,x .

(4)

The probability that the distance between a node at x and its destination is less than d is defined as P {Ds ≤ d} = EΠr,x{P {Ds ≤ d|Πr,x }} −θ (a) Y∈Πr,o ∩Bo (d) Do,Y = EΠr,o −θ Y∈Πr,o Do,Y

−θ EΠr,o Y∈Πr,o ∩Bo (d) Do,Y

≈ −θ EΠr,o Y∈Πr,o Do,Y

(5)

(6)

where o is the origin and (a) is from the stationarity of a PPP [16]. Due to the difficulty in the derivation of (5), we use the approximation as (6). By Campbell’s theorem [16], we have ⎧ ⎫ d ⎨ ⎬ EΠ D−θ r−θ+1 dr (7) = 2πλ r o,Y ⎩ ⎭ 0 Y∈Πr,o ∩Bo (d)

=

2πλr 2−θ d . 2−θ

Using (7) into (6), the cumulative distribution function (CDF) of Ds is given by P {Ds ≤ d} =

d2−θ 2−θ rd,u

(8)

and its first derivative by d results in the PDF of Ds , given by (3). By Lemma 1, the D2D mode selection probability, which is equal to the CDF of the link distance FDs (dth ), is given by pm =

d2−θ th . 2−θ rd,u

(9)

III. D2D-MSN T HROUGHPUT In this section, we define and and analyze the D2D-MSN throughput in closed form for Rayleigh fading channels. We now assume that the interference from other cell is negligible due to high frequency reuse factor, and cellular-mode users are in noise-limited environment. On the other hand, D2D mode users are in interference-limited environment as D2D-mode users access the same channel. We first define the random set of m-mode users T (μ) (∀ m ∈ {c, d}), who are successfully transmitting their data to corresponding receivers, as

(c) T (c) = x ∈ Rd : A(x) = 0, ζx,R(x) ≥ ζ (10)

(d) (11) T (d) = x ∈ Rd : ζx,R(x) ≥ ζ where ζ is the required SINR threshold; R(x) is the destination location of a node at x; and A(x) is an indicator of channel allocation to a node at x, i.e., A(x) = 0 when no channel is allocated. Here, ζ = 2R/W − 1 where R [bits/sec] is the required data rate. Using T (μ) , we define the D2D-MSN throughput as follows.

Definition 1: The D2D-MSN throughput is defined as ⎧ ⎪ ⎨ 1 R E 1T (c) (X) ρDM = (c) B (rc ) ⎪ ⎩W (c) X∈Π ∩Bo (rc ) ⎫ ⎪ ⎬ 1 (12) + 1T (d) (X) ⎪ W ⎭ (d) X∈Π ∩Bo (rc )

where rc is the macrocell radius; B (rc ) = |Bo (rc )| is the cardinality of Bo (rc ); W (μ) is the required bandwidth to communicate from a μ-mode user to its destination3 ; and 1, if X ∈ P 1P (X) 0, otherwise . The D2D-MSN throughput measures the spectral efficiency achieved by users who communicate successfully, and its unit is bits/Hz/sec/m2 . Lemma 2: The D2D-MSN throughput for Rayleigh fading channels is given by 2/α 2 ζ N rcα 2R (1 − pm ) P (c) (c) , γ ρDM =λt p¯b ζ N rcα α P (c) αW (c) 1− θ2 R(2 − θ) ηCt β + λt 2/α 2 2W cα ζ λt pm rd,u ⎞ ⎛ 4−θ 2/α 2 cα ζ λt pm2−θ rd,u θ ⎠ . (13) × γ ⎝1 − , 2 ηCt β (μ)

where p¯b (c)

p¯b

is the no blocking probability, given by

C (c) +1 (c) (c) B (rc ) λt e−B(rc )λt =1 − (14) 2 1 + C (c) Γ 1 + C (c) (c) . × 2 F2 2, 1 + C (c) ; 2 + C (c) , 2 + C (c) ; B (rc ) λt (c)

(d)

Proof: In (12), Π and Π are independent, and the (μ) events A(x) = 0 and ζx,R(x) ≥ ζ are also independent in T (μ) , so we can present ρDM as 1 (c) (c) (c) 1 (d) (d) λ p λ p p + ρDM =R (15) W t s W (c) t ¯b s since we have ⎧ ⎫ ⎪ ⎪ ⎨ ⎬ (c) (c) E 1T (c) (X) = λt B (rc ) p¯b p(c) s ⎪ ⎪ ⎩ ⎭ (c) X∈Π ∩Bo (rc ) ⎧ ⎫ ⎪ ⎪ ⎨ ⎬ (d) E 1T (d) (X) = λt B (rc ) p(d) s ⎪ ⎪ ⎩ ⎭ (d)

(16)

(17)

X∈Π ∩Bo (rc )

3 For the network with equal channel bandwidth for both uplink and downlink, W (c) will be 2W .

TABLE I PARAMETER VALUES IF NOT OTHERWISE SPECIFIED Values 4 103 2W 10−3 1.5 30 0.1

Parameters rc , rd,u [m] pm λt λe R [bits/sec] η P (c) , P (d)

Values 102 , 103 0.4 5 × 10−4 10−5 103 6 1

x 10

Ct = 100 Ct = 50 Ct = 10

1.6 1.4 1.2 1

ρDM

Parameters α W [Hz] W (c) [Hz] T [sec] θ Ct β

−3

1.8

0.8 0.6

by the Campbell’s theorem and the stationarity of a PPP (c) [16]. The p¯b is the probability that a cellular-mode user is supported by a cellular BS without blocking, given by

n (c) (c) ∞ e−λt B(rc ) λt B (rc ) n − C (c) (c) p¯b = 1 − n! n (c) n=C

(μ)

which results in (14). The ps is the probability that a user in μ mode transmits its data successfully to its destination, given by

¯ F =E (ζ ) . (18) p(μ) (μ) D s ζ |D

0.4 0.2 0

=

2 αrc2

P (c) ζ N

2/α 2 ζ N rcα , γ α P (c)

1

1.5

2

2.5

3

3.5

4

4.5 −3

x 10

Fig. 2. ρDM as a function of user spatial density λt for difference values of total available channels Ct . −4

x 10

θ θ θ θ

2

ρDM

= 1.5 = 1.0 = 0.5 = 0.0

1

(19)

where γ(· , ·) is the lower incomplete Gamma function [17]. In a D2D mode, users communicate in the interference-limited (d) (d) environment with PI = P (d) and λI = λt /C (d) , so ps is given using (2), (3), and (18) by

2/α (d) −(cα ζ λt /C (d) )D2 e p(d) = E D s (d) 2 − θ dth 1−θ −(cα ζ 2/α λ(d) )x2 t /C x e dx . (20) = 2−θ dth 0 (d)

0.5

λt

c

In a cellular mode, the interference from other cells is neg(c) ligible and θ(c) = 0, so ps is presented using (2) and (18) as ζN − (c) Dα = E p(c) D e P s

0

2−θ

By using λt = λt pm , pm = (dth /rd,u ) , and C (d) = ηCt β, (20) is represented by 1− θ2 2−θ ηCt β (d) ps = (21) 2−θ 2/α pm rd,u cα ζ pm λt 4−θ 2/α 2 λt θ cα ζ pm 2−θ rd,u . ×γ 1− , 2 ηCt β Substituting (19) and (20) and into (15) results in (13). IV. N UMERICAL R ESULTS In this section, we analyze the effect of network parameters on the D2D-MSN throughput ρDM in Rayleigh fading channels. Specifically, we present how to optimize the user spatial density λt and D2D communication parameters including the

0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

pm

Fig. 3. ρDM as a function of D2D mode selection probability pm for difference values of power exponent θ.

D2D mode selection probability pm and the D2D resource portion β. Unless otherwise specified, the values of network parameters presented in Table II are used for numerical results. Figure 2 shows ρDM as a function of the user spatial density λt for different values of Ct . It can be shown that the ρDM increases with λt , which means increasing λt increases ρDM although the successful reception probability of D2D-mode users becomes lower due to large interference. Note that as Ct decreases, blocking probability increases and will mainly degrade ρDM . Figure 3 shows ρDM as a function of pm for different values of θ. It is shown that the optimal pm in terms of ρDM increases as θ increases. This can be attributed to the fact that the throughput of D2D-mode users increases with θ and having more D2D-mode users achieves higher ρDM . This also shows the D2D mode cannot be helpful for mobile social networks

−4

x 10

1

2.5

0.9 0.8

2 0.7 0.6

ρDM

β

1.5 0.5 0.4

1

0.3 0.2

0.5

0.1 0

0

0.2

0.4

0.6

0.8

1

0

pm

Fig. 4. ρDM as a function of the D2D selection probability pm and the D2D resource portion β.

when users communicate to destinations located far away with high probability, i.e., when θ is low. We now present the joint optimal values of the D2D resource ratio β and the D2D mode selection probability pm in terms of ρDM in Fig. 4. It can be observed that, the optimal β is small for small pm , but it becomes large for large pm . This can be attributed to the fact that for a high pm , the successful reception probability in D2D mode decreases due to large interference and it mainly degrade ρDM . Hence, having more channels for the D2D mode can enhance ρDM . For a small pm , the blocking probability of cellular-mode users increases as many cellular-mode users access the limited channels, and it mainly degrade ρDM . Hence, ρDM can be enhanced by allocating more channels for the cellular mode. V. C ONCLUSION In this paper, we establish a foundation for the D2D-MSN accounting for the social-context based link distance distribution and the user spatial distributions. By quantifying the D2D-MSN throughput, we showed how D2D communication parameters including the D2D mode selection probability, the D2D resource portion, and the link distribution influence the network throughput. Specifically, as the power exponent of social communication distance distribution increases, the optimal D2D resource portion decreases while the optimal D2D mode selection probability increases. The outcomes of our work provide guidelines for design of the reliable D2D-MSN. R EFERENCES [1] D. Niyato, P. Wang, W. Saad, and A. Hjørungnes, “Controlled coalitional games for cooperative mobile social networks,” IEEE Trans. Veh. Technol., vol. 60, no. 4, pp. 1812–1824, May 2011. [2] J. Hu, L.-L. Yang, and L. Hanzo, “Mobile social networking aided content dissemination in heterogeneous networks,” China Communications, vol. 10, no. 6, pp. 1–13, Jun. 2013. [3] K. Doppler, M. Rinne, C. Wijting, C. Ribeiro, and K. Hugl, “Deviceto-device communication as an underlay to LTE-advanced networks,” IEEE Commun. Mag., vol. 47, no. 12, pp. 42–49, Dec. 2009.

[4] H. Min, J. Lee, S. Park, and D. Hong, “Capacity enhancement using an interference limited area for device-to-device uplink underlaying cellular networks,” IEEE Trans. Wireless Commun., vol. 10, no. 12, pp. 3995– 4000, Dec. 2011. [5] G. Fodor and N. Reider, “A distributed power control scheme for cellular network assisted D2D communications,” in Proc. IEEE Global Telecomm. Conf., Houston, TX, Dec. 2011, pp. 1–6. [6] P. Phunchongharn, E. Hossain, and D. I. Kim, “Resource allocation for device-to-device communications underlaying LTE-advanced networks,” IEEE Wireless Commun. Mag., vol. 20, no. 4, pp. 91–100, Sep. 2013. [7] S. Hakola, T. Chen, J. Lehtomaki, and T. Koskela, “Device-to-device (D2D) communication in cellular network-performance analysis of optimum and practical communication mode selection,” in Proc. IEEE Wireless Commun. and Networking Conf., Sydney, Australia, Apr. 2010, pp. 1–6. [8] H. Min, W. Seo, J. Lee, S. Park, and D. Hong, “Reliability improvement using receive mode selection in the device-to-device uplink period underlaying cellular networks,” IEEE Trans. Wireless Commun., vol. 10, no. 2, pp. 413–418, Feb. 2011. [9] P. Cheng, L. Deng, H. Yu, Y. Xu, and H. Wang, “Resource allocation for cognitive networks with d2d communication: An evolutionary approach,” in Proc. IEEE Wireless Commun. and Networking Conf. IEEE, 2012, pp. 2671–2676. [10] B. Kaufman, J. Lilleberg, and B. Aazhang, “Spectrum sharing scheme between cellular users and ad-hoc device-to-device users,” IEEE Trans. Wireless Commun., vol. 12, no. 3, pp. 1038–1049, 2013. [11] J. Kleinberg, “The small-world phenomenon: an algorithm perspective,” in Proc. ACM Symp. Theory of computing, Portland, OR, May 2000, pp. 163–170. [12] L. Backstrom, E. Sun, and C. Marlow, “Find me if you can: improving geographical prediction with social and spatial proximity,” in Proc. ACM Int. Conf. World wide web, Raleigh, NC, Apr. 2010, pp. 61–70. [13] B. Azimdoost, H. Sadjadpour, and J. Garcia-Luna-Aceves, “Capacity of wireless networks with social behavior,” IEEE Trans. Wireless Commun., vol. 12, no. 1, pp. 60–69, Jan. 2013. [14] J. Lee, J. G. Andrews, and D. Hong, “Spectrum-sharing transmission capacity,” IEEE Trans. Wireless Commun., vol. 10, no. 9, pp. 3053– 3063, Sep. 2011. [15] J. Lee, J. G. Andrews, and D. Hong, “Spectrum-sharing transmission capacity with interference cancelation,” IEEE Trans. Commun., vol. 61, no. 1, pp. 76–86, Jan. 2013. [16] J. F. Kingman, Poisson Processes. Oxford University Press, 1993. [17] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions. Dover Publications, 1970.

Singapore University of Technology and Design, Singapore ∗ Institute for Infocomm Research, A*STAR

Abstract—Wireless mobile social networks have been introduced to consider characteristics of mobile user’s behaviors for network configuration. The device-to-device (D2D) communication is to increase the network spectral efficiency by allowing users to communicate directly with the corresponding destinations. This paper develops a framework for the design of reliable D2D communication provided wireless mobile social network (D2D-MSN), and introduces a D2D-MSN throughput as a new measurement of the spectral efficiency achieved by users communicating successfully. Our framework accounts for spatial user distributions and the link distance-based mode selection scheme that allows the D2D mode only for the users with shorter link distance than a threshold. We also account for the communication distance distribution defined in a social context to decrease the communication probability density according to the link distance with a power exponent. This research offers insights on the optimal configuration of the D2D-MSN, and also provide a new perspective on the effect of the link distance distribution on the D2D communication performance. Index Terms—device-to-device communication, mobile social network, transmission mode selection, stochastic geometry

I. I NTRODUCTION Wireless mobile social networks, which are the combination of the physical wireless mobile network and the virtual social network, have been introduced to enhance the network performance by exploiting the characteristics of the mobile user’s social relationship and behaviors [1], [2]. Recently, device-to-device (D2D) communications have been introduced to allow users communicate directly with each other without using a base station (BS) [3]. The D2D communication can offload the traffic handled by BSs and enhance the local service quality and the spectrum efficiency of the wireless network. For the design of a reliable D2D communication provided wireless mobile social network (D2D-MSN), algorithms for interference management and transmission mode selection are required by considering mobile user social behaviors. The throughput of D2D communications has been presented by taking into account the resource management and the D2D communication operation [4]–[10]. The resource management algorithms for D2D communications are presented to enhance the network throughput by the interference limited area control [4], the transmission power control [5], and the joint This work was partly supported by the Temasek Research Fellowship, the SRG ISTD 2012037, the SUTD-MIT International Design Centre under Grant IDSF1200106OH, and the A*STAR SERC Grant 1224104048.

frequency spectrum scheduling and power control [6]. For efficient operations of D2D communications, the transmission mode selection [7] and the receive mode selection [8] are also presented. The performance of D2D communication has recently been analyzed for various network settings including cognitive radio networks [9] and spectrum sharing networks [10]. However, most of these works have not taken into account for the spatial distribution of users and the distance distribution of communication links, which are paramount for the analysis of D2D communication throughput. In this paper, we provide a foundation for D2D-MSN, composed of spatially distributed users selectively transmitting their data in a cellular mode or a D2D mode. We consider the transmission mode selection algorithm that selects a D2D mode when a communication link distance is shorter than a threshold, otherwise it selects a cellular mode. In a mobile social network, it is shown that the probability of communication to a receiver is inversely proportional to the distance between the transmitter and the receiver [11]–[13]. Hence, we model the communication link distribution in a social context, for which probability density decreases according to the link distance with a power exponent. We also introduce the D2D-MSN throughput as a performance metric that measures the spectral efficiency of users communicating successfully in a cellular or a D2D mode. The key contributions of the paper can be summarized as follows: (i) introduction and definition of the D2D-MSN throughput in a D2D-MSN composed of D2D-mode and a cellular-mode users; (ii) development of a framework for the design and the analysis of D2D-MSN accounting for the spatial user distributions as well as the communication distance distribution; and (iii) quantification of the D2D-MSN throughput for characterizing the effects of D2D communication parameters such as the D2D mode selection threshold and the D2D resource portion on the D2D-MSN throughput. The remainder of this paper is organized as follows: Section II describes the D2D-MSN configuration including the distance distribution of communication link. Section III introduces and analyzes the D2D-MSN throughput. Section IV quantifies the effect of D2D parameters on the D2D-MSN throughput. Conclusion is given in Section V.

is the distance between transmitter and receiver; α is the pathloss exponent; N = N0 W ; N0 is the one-sided power spectral density (PSD); W is a channel bandwidth; ΠI is the distribution of interfering nodes following a PPP with the spatial density λI ; and interfering node transmission power is PI . The complementary cumulative distribution function (CCDF) of received SINR by a μ-mode user in Rayleigh fading channels is given by [14], [15] 2/α ηN Dα ηPI −cα λI D2 − (μ) (μ) ¯ P P Fζ (μ) (η) = ED e (2)

Fig. 1. An example of a D2D-MSN (blue circles are cellular-mode transmitting users, black circles are D2D-mode transmitting users, and white circles are receiving users).

II. D2D-MSN C ONFIGURATION In this section, we describe a D2D-MSN and a transmission mode selection scheme. A. Network Configuration We consider a wireless mobile social network composed of users aiming to transmit their data via one of two transmission modes: a cellular mode that makes each user communicates to a BS, and a D2D mode that enables users to communicate directly to their corresponding destinations, which is determined in a social context. Users are distributed in the network according to a homogeneous Poisson point process (PPP) Π with spatial density λt . Channels are allocated to each cellular-mode users exclusively, and they are reused in other cells with a frequency reuse factor 1/η.1 When there are more cellular-mode users in a cell than the number of available channels for cellular mode, some cellular-mode users are blocked for accessing channels. On the other hand, allocated channels for the D2D mode are shared by all D2D-mode users in the network, and each D2Dmode user randomly accesses a channel. The β portion of the total frequency channels are allocated to D2D-mode users. Hence, the numbers of channels available for D2D-mode users and cellular-mode users in a cell are C (d) = ηβCt and C (c) = Ct − βCt , respectively,2 where x is the ceiling function of x and Ct is the total available channels in a cell. The instantaneous signal-to-interference and noise ratio (SINR) per symbol received by a node at y from a μ-mode user at x is given by (μ) ζx,y =

P (μ) Hx,y D−α x,y

−α Z∈ΠI PI HZ,y DZ,y

+N

(1)

where P (μ) is the transmission power of a user in μ mode; Hx,Y is the fading level of the link; Dx,Y = |x − Y| for cellular-mode users are reused in every adjacent η cells. total number of channels available in this network is ηCt since Ct channels are allocated to each cell and reused in every η cells. The β portion of total channels are allocated to D2D mode and C (d) = ηβCt .

where cα = (2π/α)Γ(2/α) Γ(1 − 2/α) and D is the random communication link distance. B. Transmission Mode Selection We assume that a source at xs ∈ Π communicates to the corresponding destination at Yd ∈ Πr ∩ Bxs (rd,u ) where Πr is a PPP of receiving nodes with spatial density λr ; rd,u is the maximum communication distance; and Bx (r) is the ball of radius r centered at x. The link-distance dependent mode selection scheme is considered to transmit in a D2D mode only if Dxs ,Yd is shorter than the mode selection threshold dth . Hence, the probability that a user selects the D2D mode is pm = P {Dxs ,Yd ≤ dth }. By the thinning property of a PPP [16], after the transmission mode selection, the distributions of cellular-mode users and D2D-mode users still follow PPPs (c) (d) (c) Π and Π with spatial densities λt = λt (1 − pm ) and (d) λt = λt pm , respectively. The distribution of communication distance Ds is determined according to a power-law communication probability, defined in a social context [11]–[13]. Specifically, a source communicates to one of uniformly distributed receiving nodes, and the communication probability is inversely proportional to the Euclidean distance between the source and the receiving node with a power exponent θ. Note that the power exponent for cellular-mode users is θ = 0 since users are uniformly distributed in a cell and communicate to a BS located at the origin, while the power exponent for D2D-mode users is determined by network social characteristics. Here, we assume 0 ≤ θ < 2 for analytic tractability. The probability density function (PDF) of Ds is defined as follows. Lemma 1: By the power-law communication probability with the exponent 0 ≤ θ < 2, the approximated PDF of the social communication distance Ds is given by fDs (d) =

(3)

Proof: By the power-law communication probability, the probability that a node at x communicates to a node at Yd ∈ Πr,x = Πr ∩ Bx (rd,u ) is given by

1 Channels 2 The

(2 − θ) d1−θ . 2−θ rd,u

pcx,Yd =

D−θ x,Yd Y∈Πr,x

D−θ x,Y

,

∀ Yd ∈ Πr,x .

(4)

The probability that the distance between a node at x and its destination is less than d is defined as P {Ds ≤ d} = EΠr,x{P {Ds ≤ d|Πr,x }} −θ (a) Y∈Πr,o ∩Bo (d) Do,Y = EΠr,o −θ Y∈Πr,o Do,Y

−θ EΠr,o Y∈Πr,o ∩Bo (d) Do,Y

≈ −θ EΠr,o Y∈Πr,o Do,Y

(5)

(6)

where o is the origin and (a) is from the stationarity of a PPP [16]. Due to the difficulty in the derivation of (5), we use the approximation as (6). By Campbell’s theorem [16], we have ⎧ ⎫ d ⎨ ⎬ EΠ D−θ r−θ+1 dr (7) = 2πλ r o,Y ⎩ ⎭ 0 Y∈Πr,o ∩Bo (d)

=

2πλr 2−θ d . 2−θ

Using (7) into (6), the cumulative distribution function (CDF) of Ds is given by P {Ds ≤ d} =

d2−θ 2−θ rd,u

(8)

and its first derivative by d results in the PDF of Ds , given by (3). By Lemma 1, the D2D mode selection probability, which is equal to the CDF of the link distance FDs (dth ), is given by pm =

d2−θ th . 2−θ rd,u

(9)

III. D2D-MSN T HROUGHPUT In this section, we define and and analyze the D2D-MSN throughput in closed form for Rayleigh fading channels. We now assume that the interference from other cell is negligible due to high frequency reuse factor, and cellular-mode users are in noise-limited environment. On the other hand, D2D mode users are in interference-limited environment as D2D-mode users access the same channel. We first define the random set of m-mode users T (μ) (∀ m ∈ {c, d}), who are successfully transmitting their data to corresponding receivers, as

(c) T (c) = x ∈ Rd : A(x) = 0, ζx,R(x) ≥ ζ (10)

(d) (11) T (d) = x ∈ Rd : ζx,R(x) ≥ ζ where ζ is the required SINR threshold; R(x) is the destination location of a node at x; and A(x) is an indicator of channel allocation to a node at x, i.e., A(x) = 0 when no channel is allocated. Here, ζ = 2R/W − 1 where R [bits/sec] is the required data rate. Using T (μ) , we define the D2D-MSN throughput as follows.

Definition 1: The D2D-MSN throughput is defined as ⎧ ⎪ ⎨ 1 R E 1T (c) (X) ρDM = (c) B (rc ) ⎪ ⎩W (c) X∈Π ∩Bo (rc ) ⎫ ⎪ ⎬ 1 (12) + 1T (d) (X) ⎪ W ⎭ (d) X∈Π ∩Bo (rc )

where rc is the macrocell radius; B (rc ) = |Bo (rc )| is the cardinality of Bo (rc ); W (μ) is the required bandwidth to communicate from a μ-mode user to its destination3 ; and 1, if X ∈ P 1P (X) 0, otherwise . The D2D-MSN throughput measures the spectral efficiency achieved by users who communicate successfully, and its unit is bits/Hz/sec/m2 . Lemma 2: The D2D-MSN throughput for Rayleigh fading channels is given by 2/α 2 ζ N rcα 2R (1 − pm ) P (c) (c) , γ ρDM =λt p¯b ζ N rcα α P (c) αW (c) 1− θ2 R(2 − θ) ηCt β + λt 2/α 2 2W cα ζ λt pm rd,u ⎞ ⎛ 4−θ 2/α 2 cα ζ λt pm2−θ rd,u θ ⎠ . (13) × γ ⎝1 − , 2 ηCt β (μ)

where p¯b (c)

p¯b

is the no blocking probability, given by

C (c) +1 (c) (c) B (rc ) λt e−B(rc )λt =1 − (14) 2 1 + C (c) Γ 1 + C (c) (c) . × 2 F2 2, 1 + C (c) ; 2 + C (c) , 2 + C (c) ; B (rc ) λt (c)

(d)

Proof: In (12), Π and Π are independent, and the (μ) events A(x) = 0 and ζx,R(x) ≥ ζ are also independent in T (μ) , so we can present ρDM as 1 (c) (c) (c) 1 (d) (d) λ p λ p p + ρDM =R (15) W t s W (c) t ¯b s since we have ⎧ ⎫ ⎪ ⎪ ⎨ ⎬ (c) (c) E 1T (c) (X) = λt B (rc ) p¯b p(c) s ⎪ ⎪ ⎩ ⎭ (c) X∈Π ∩Bo (rc ) ⎧ ⎫ ⎪ ⎪ ⎨ ⎬ (d) E 1T (d) (X) = λt B (rc ) p(d) s ⎪ ⎪ ⎩ ⎭ (d)

(16)

(17)

X∈Π ∩Bo (rc )

3 For the network with equal channel bandwidth for both uplink and downlink, W (c) will be 2W .

TABLE I PARAMETER VALUES IF NOT OTHERWISE SPECIFIED Values 4 103 2W 10−3 1.5 30 0.1

Parameters rc , rd,u [m] pm λt λe R [bits/sec] η P (c) , P (d)

Values 102 , 103 0.4 5 × 10−4 10−5 103 6 1

x 10

Ct = 100 Ct = 50 Ct = 10

1.6 1.4 1.2 1

ρDM

Parameters α W [Hz] W (c) [Hz] T [sec] θ Ct β

−3

1.8

0.8 0.6

by the Campbell’s theorem and the stationarity of a PPP (c) [16]. The p¯b is the probability that a cellular-mode user is supported by a cellular BS without blocking, given by

n (c) (c) ∞ e−λt B(rc ) λt B (rc ) n − C (c) (c) p¯b = 1 − n! n (c) n=C

(μ)

which results in (14). The ps is the probability that a user in μ mode transmits its data successfully to its destination, given by

¯ F =E (ζ ) . (18) p(μ) (μ) D s ζ |D

0.4 0.2 0

=

2 αrc2

P (c) ζ N

2/α 2 ζ N rcα , γ α P (c)

1

1.5

2

2.5

3

3.5

4

4.5 −3

x 10

Fig. 2. ρDM as a function of user spatial density λt for difference values of total available channels Ct . −4

x 10

θ θ θ θ

2

ρDM

= 1.5 = 1.0 = 0.5 = 0.0

1

(19)

where γ(· , ·) is the lower incomplete Gamma function [17]. In a D2D mode, users communicate in the interference-limited (d) (d) environment with PI = P (d) and λI = λt /C (d) , so ps is given using (2), (3), and (18) by

2/α (d) −(cα ζ λt /C (d) )D2 e p(d) = E D s (d) 2 − θ dth 1−θ −(cα ζ 2/α λ(d) )x2 t /C x e dx . (20) = 2−θ dth 0 (d)

0.5

λt

c

In a cellular mode, the interference from other cells is neg(c) ligible and θ(c) = 0, so ps is presented using (2) and (18) as ζN − (c) Dα = E p(c) D e P s

0

2−θ

By using λt = λt pm , pm = (dth /rd,u ) , and C (d) = ηCt β, (20) is represented by 1− θ2 2−θ ηCt β (d) ps = (21) 2−θ 2/α pm rd,u cα ζ pm λt 4−θ 2/α 2 λt θ cα ζ pm 2−θ rd,u . ×γ 1− , 2 ηCt β Substituting (19) and (20) and into (15) results in (13). IV. N UMERICAL R ESULTS In this section, we analyze the effect of network parameters on the D2D-MSN throughput ρDM in Rayleigh fading channels. Specifically, we present how to optimize the user spatial density λt and D2D communication parameters including the

0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

pm

Fig. 3. ρDM as a function of D2D mode selection probability pm for difference values of power exponent θ.

D2D mode selection probability pm and the D2D resource portion β. Unless otherwise specified, the values of network parameters presented in Table II are used for numerical results. Figure 2 shows ρDM as a function of the user spatial density λt for different values of Ct . It can be shown that the ρDM increases with λt , which means increasing λt increases ρDM although the successful reception probability of D2D-mode users becomes lower due to large interference. Note that as Ct decreases, blocking probability increases and will mainly degrade ρDM . Figure 3 shows ρDM as a function of pm for different values of θ. It is shown that the optimal pm in terms of ρDM increases as θ increases. This can be attributed to the fact that the throughput of D2D-mode users increases with θ and having more D2D-mode users achieves higher ρDM . This also shows the D2D mode cannot be helpful for mobile social networks

−4

x 10

1

2.5

0.9 0.8

2 0.7 0.6

ρDM

β

1.5 0.5 0.4

1

0.3 0.2

0.5

0.1 0

0

0.2

0.4

0.6

0.8

1

0

pm

Fig. 4. ρDM as a function of the D2D selection probability pm and the D2D resource portion β.

when users communicate to destinations located far away with high probability, i.e., when θ is low. We now present the joint optimal values of the D2D resource ratio β and the D2D mode selection probability pm in terms of ρDM in Fig. 4. It can be observed that, the optimal β is small for small pm , but it becomes large for large pm . This can be attributed to the fact that for a high pm , the successful reception probability in D2D mode decreases due to large interference and it mainly degrade ρDM . Hence, having more channels for the D2D mode can enhance ρDM . For a small pm , the blocking probability of cellular-mode users increases as many cellular-mode users access the limited channels, and it mainly degrade ρDM . Hence, ρDM can be enhanced by allocating more channels for the cellular mode. V. C ONCLUSION In this paper, we establish a foundation for the D2D-MSN accounting for the social-context based link distance distribution and the user spatial distributions. By quantifying the D2D-MSN throughput, we showed how D2D communication parameters including the D2D mode selection probability, the D2D resource portion, and the link distribution influence the network throughput. Specifically, as the power exponent of social communication distance distribution increases, the optimal D2D resource portion decreases while the optimal D2D mode selection probability increases. The outcomes of our work provide guidelines for design of the reliable D2D-MSN. R EFERENCES [1] D. Niyato, P. Wang, W. Saad, and A. Hjørungnes, “Controlled coalitional games for cooperative mobile social networks,” IEEE Trans. Veh. Technol., vol. 60, no. 4, pp. 1812–1824, May 2011. [2] J. Hu, L.-L. Yang, and L. Hanzo, “Mobile social networking aided content dissemination in heterogeneous networks,” China Communications, vol. 10, no. 6, pp. 1–13, Jun. 2013. [3] K. Doppler, M. Rinne, C. Wijting, C. Ribeiro, and K. Hugl, “Deviceto-device communication as an underlay to LTE-advanced networks,” IEEE Commun. Mag., vol. 47, no. 12, pp. 42–49, Dec. 2009.

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