On the Performance of Successive Interference Cancellation in D2D-enabled Cellular Networks

Transcription

1 On the erformane of Suessive Interferene Canellation in D2D-enable Cellular Networks Chuan Ma 1, Weijie Wu 1, Ying Cui 1, Xinbing Wang 1,2 1. Shool of Eletroni Info. & Eletrial Eng., Shanghai Jiao Tong University, China 2. National Mobile Communiations Researh Laboratory, Southeast University, China {oknewkimi, weijiewu, uiying, Abstrat Devie-to-evie D2D ommuniation unerlaying ellular networks is a promising tehnology to improve network resoure utilization. In D2D-enable ellular networks, the interferene among spetrum-sharing links is more severer than that in traitional ellular networks, whih motivates the aoption of interferene anellation tehniques suh as suessive interferene anellation SIC at the reeivers. However, to ate, how SIC an affet the performane of D2D-enable ellular networks is still unknown. In this paper, we present an analytial framework for stuying the performane of SIC in large-sale D2D-enable ellular networks using the tools from stohasti geometry. To failitate the interferene analysis, we propose the approah of stohasti uivalene of the interferene, whih onverts the two-tier interferene interferene from both the ellular tier an D2D tier to an uivalent single-tier interferene. Base on the propose stohasti uivalene moels, we erive the general expressions for the suessful transmission probabilities of ellular uplinks an D2D links with infinite an finite SIC apabilities respetively. We emonstrate how SIC affets the performane of large-sale D2D-enable ellular networks by both analytial an numerial results. I. INTRODUCTION Reently, there has been a rapi inrease in the eman of loal area servies an proximity servies rose among the highly-apable user uipments UEs in ellular networks. In this ontext, a new tehnology alle evie-to-evie D2D ommuniation, whih enables iret ommuniation between UEs that are in proximity, has been propose an has strongly appeale to both aaemia 1], 2] an inustry 3], 4]. The integration of D2D ommuniation to ellular networks hols the promise of many types of avantages 2]: allowing for high-rate low-elay low-power transmission for proximity servies, inreasing frueny reuse fator an network apaity, failitating new types of peer-to-peer servies, et. However, the introution of D2D ommuniation also brings a number of tehnial hallenges, suh as evie isovery, moe seletion an interferene management. Interferene management is a major issue in D2D-enable ellular networks, sine D2D links share the same spetrum resoure with regular ellular links an the interferene among the spetrum-sharing links severely hampers the performane of the network. To guarantee reliable ommuniations in D2Denable ellular networks, extensive researh has been unertaken on the topi of interferene management. Most propose shemes an be lassifie into three ategories: 1 Interferene avoiane: orthogonal time-frueny resoure alloation shemes are aopte to avoi interferene between D2D an ellular links 5]; 2 Interferene oorination: intelligent power ontrol an link sheuling shemes are employe to mitigate the interferene between D2D an ellular links 6], 7]; an 3 Interferene anellation: avane signal proessing tehniques are applie at ellular an/or D2D links to anel the interfering signals 8], 9]. In this paper, we fous on the topi of interferene anellation in D2D-enable ellular networks. Interferene anellation IC is regare as a promising tehnique to reue interferene an improve network apaity. Using interferene anellation tehniques, the interfering signals an be regenerate an subsuently anele from the esire signal 1]. Best known interferene anellation tehniques inlue suessive interferene anellation SIC, parallel interferene anellation IC an iterative interferene anellation IIC 11]. The key avantage of SIC ompare to other interferene anellation tehniques is that the SIC reeiver is arhiteturally similar to traitional non-sic reeivers in terms of harware omplexity an ost 11], as it uses the same eoer to eoe the omposite signal at ifferent stages an neither ompliate eoers nor multiple antennas are ruire. Furthermore, it is known that SIC an ahieve the Shannon apaity region bounaries for both the broaast an multiple aess networks. As suh, SIC has been wiely stuie an reently implemente in ommerial wireless systems suh as IEEE However, to ate, most analytial results on SIC are for a ho an ellular networks. It is still unknown how SIC an improve the performane of large-sale D2D-enable ellular networks, in whih the interferene among spetrum-sharing links is more severer than that in traitional a ho an ellular networks. In this paper, we present an analytial framework to evaluate how SIC affets the performane of large-sale D2D-enable ellular networks. The main ontributions of this paper are summarize as follows. 1 We moel a large-sale D2D-enable ellular network without SIC apabilities via stohasti geometry, an erive the general expressions for the suessful transmission probabilities of ellular uplinks an D2D links. Furthermore, to simplify the interferene analysis in the network, we propose the approah of stohasti uivalene of the interferene. By this approah, the two-tier interferene interferene from the ellular tier an D2D tier an be represente by an

2 uivalent single-tier interferene that maintains the same stohasti harateristis as the two-tier interferene. 2 Base on the stohasti uivalene moels, we erive the general expressions for the suessful transmission probabilities of ellular uplinks an D2D links with infinite an finite SIC apabilities respetively. We emonstrate the effet of SIC on the performane of large-sale D2D-enable ellular networks by analytial an numerial results. The rest of this paper is organize as follows. Setion II presents the relate work. Setion III esribes the system moel. Setions IV an V analyze the suessful transmission probabilities in D2D-enable ellular networks without an with SIC respetively an present the numerial results. Setion VI onlues the paper. A summary of the notations use in this paper is given in Table I. II. RELATED WORK D2D ommuniation. There have been numerous stuies on interferene management for D2D ommuniations. Xu et al. 5] propose a ombinatorial aution approah to alloate orthogonal resoures between ellular an D2D users. Kaufman et al. 6] presente an opportunisti ommuniation sheme in whih the D2D network an ommuniate as a fully loae ellular network. ei an Liang 7] esigne a spetrum sharing protool that enables D2D users to ommuniate bi-iretionally while assisting the two-way ommuniations between the base station an ellular users. Min et al. 8] esigne an interferene anellation sheme that exploits a retransmission of the interferene from the base station. Ma et al. 9] propose two superposition oing-base ooperative relaying shemes to exploit the transmission opportunities for D2D users without eteriorating the performane of ellular users. Suessive interferene anellation. Very reently, there is a growing interest to exploit SIC at the physial layer to improve network performanes at upper layers. In 12], Gelal et al. propose a topology ontrol framework for exploiting the benefits of multi-paket reeption using SIC in multiuser MIMO networks. In 13], Jiang et al. ombines SIC an interferene avoiane to improve the throughput of a multihop network. In 14], Xu et al. evelope a eentralize power alloation sheme to ahieve the maximum throughput for ranom aess systems with SIC reeivers. In 15], Lv et al. propose two layere moels to haraterize the impat of SIC, an presente orresponing link sheuling shemes uner these moels. In 16], Mollanoori an Ghaeri stuie the uplink sheuling problems for networks supporting SIC, an erive the optimal eoing orer of the onurrent transmissions. Stohasti geometry for moeling wireless networks. As a mathematial tool to stuy ranom spatial patterns, stohasti geometry an be use to moel an analyze the interferene, onnetivity an overage in large-sale wireless networks 17]. Most of the literature in the area of moeling networks via stohasti geometry fous on a ho 18], 19] an ellular 2], 21] networks. Reently, stohasti geometry has Notation Table I: Notations use in the paper Desription Φ oisson point proess of ellular users ensity λ Φ oisson point proess of D2D transmitters ensity λ Φ Equivalent oisson point proess of intf the interferers for ellular links ensity λ intf Φ Equivalent oisson point proess of intf the interferene for D2D links ensity λ intf Transmission power of ellular users Transmission power of D2D users α ath loss exponent 2 α T SIR threshol for suessful transmission p Suessful transmission prob. of ellular links without SIC p Suessful transmission prob. of D2D links without SIC p SIC, p N SIC Suessful transmission prob. of ellular links with infinite an finite N-level SIC p SIC, Suessful transmission prob. of D2D links pm SIC with infinite an finite M-level SIC also been employe to moel D2D-enable ellular networks 22] 24]. In 22] 24], the ellular an D2D networks were moele by inepenent s, an their SINR istributions were erive without onsiering interferene anellation tehniques. The stohasti geometry-base analysis of SIC has been presente in literature 25] 29]. In 25], 26], simplifie SIC moels were given by assuming that the interferene greater than a threshol an be ompletely anele. Exat SIC moels were investigate for a ho network in 27], 28] an for ellular networks in 29]. Different from 27] 29], in this paper, we fous on the analysis of the effet of SIC for D2Denable ellular networks. Sine the stohasti harateristis of heterogeneous networks omprising a ho users an ellular users are quite ifferent from those of homogeneous networks omprising only a ho users or ellular users, the analysis of SIC in this paper is more hallenging than that in 27] 29]. III. SYSTEM MODEL In this setion, we elaborate on the network moel an esribe the SIC sheme. A. Network Moel We onsier a spetrum-sharing D2D-enable ellular network onsisting of both ellular users an D2D users over a large two-imensional spae, an fous on the uplink transmission for ellular users. The ellular users are assume to be spatially istribute as a homogeneous oisson point proess Φ with ensity λ, an an inepenent olletion of base stations BSs is assume to be loate aoring to some inepenent stationary point proess Φ b. We assume that eah ellular user is assoiate with its nearest base station, an eah base station has only one ative uplink ellular user sheule. Uner suh assumptions, eah base station an be onsiere to be uniformly istribute in the Voronoi ell of its assoiate ellular user, as shown in Fig.1. It is note that the orthogonal sheuling poliy leas to oupling between the loations of ellular users an those of base stations. Nevertheless, it has been shown that the

3 1.9.8 Reeive omposite signal + - Deoe strongest interfering signal Regenerate strongest interfering signal Deoe k-th strongest interfering signal Regenerate k-th strongest interfering signal.2 Base station.1 Cellular user D2D user Figure 1: Network moel. Deoe esire signal Figure 2: Shemati iagram of SIC proess. Desire signal epenene introue by oupling has negligible effets on the performane analysis 21], 29], Therefore, for analytial tratability, we assume that the point proesses of ellular users an base stations are inepenent. The D2D transmitters in the network are assume to be istribute aoring to a homogeneous Φ with ensity λ. For a given D2D transmitter, its assoiate reeiver is assume to be loate at a istane l away with isotropi iretion, where l is Rayleigh istribute: f l l 2πλ le πλ l 2. 1 This Rayleigh istribution assumption is of pratial interest an is employe in many works 22] 24]. Other istributions of l an be easily inorporate into the framework. The transmission powers are assume to be at uplink ellular users an at D2D transmitters respetively. We aopt a unifie hannel moel that omprises stanar path loss an Rayleigh faing for both ellular an D2D links: given transmission power of the transmitter loate at x i, the reeive power at the reeiver loate at x j an be expresse as h x i x j, where h is the faing fator following an exponential istribution with unit mean, i.e. h exp 1, an α > 2 is the path loss exponent. In later parts of this paper we use to enote 2 α for brevity of expressions. In aition, as interferene ominates noise in most moern ellular networks, we onsier the network to be interferenelimite. B. SIC Tehnique SIC is a promising interferene anellation tehnique that has been wiely stuie for wireless networks. The basi onept of SIC is to regenerate the interfering signals an subsuently anel them from the reeive omposite signal, so as to improve the signal-to-interferene ratio SIR of the esire signal. In this tehnique, the SIC reeiver first eoes the strongest interfering signal by treating other signals as noise. Then it regenerates the analog signal from the eoe signal, an anels it from the reeive omposite signal. After this stage, the remaining signal is free from the interferene of the strongest interfering signal. Then, the SIC reeiver proees to eoe, regenerate an anel the seon strongest interfering signaling from the remaining signal an so forth, until the esire signal an be eoe. The shemati iagram of SIC proess is shown in Fig.2. In this paper, we assume that the SIC tehnique is employe at both ellular reeivers BSs an D2D reeivers. For ellular reeivers, we onsier the ases of infinite an finite SIC apabilities respetively, an stuy the suessful transmission probabilities of ellular links for eah ase. However, for D2D reeivers, ue to their limite omputational apabilities, we fous only on the ase of finite SIC apability. IV. NETWORK ERFORMANCE WITHOUT SIC In this setion, we onsier the senario that neither the ellular reeivers BSs nor the D2D reeivers have SIC apabilities, an erive the suessful transmission probabilities of both ellular an D2D links. We also analyze the stohasti uivalene of interferene in the network, whih is essential for the SIC analysis in later setions. A. Suessful Transmission robability of Cellular Links Without loss of generality, we onut the analysis on a typial ellular link that omprises a typial BS loate at the origin an its assoiate ellular user loate at a ranom istane r away. Uner the nearest-bs assoiation poliy, the ranom variable r an be shown to be Rayleigh istribute an its probability ensity funtion pf follows 21]: f r r 2πλ re πλr2. 2 Denote the faing fator of the typial ellular link by g, whih is i.i. exponential with g exp 1. Then, the reeive SIR of the typial ellular link, i.e., the reeive SIR at the typial BS, an be expresse as SIR g r I, 3

4 where I x i Φ \{x } g i x i + y i Φ h i y i is the umulative interferene from all other ellular users exept the typial ellular user x that are loate at x i with faing fator g i an D2D transmitters that are loate at y i with faing fator h i 1. The suessful transmission probability of ellular links an be efine as p SIR > T ], 5 where T is the SIR threshol. The expression of p is given by the following theorem. Theorem 1. The suessful transmission probability of ellular links without SIC apability is where p λ 4 λ µ λ ν, 6 µ 1 T 2F 1 1, 1 ; 2 ; T, 7 ν T Γ 1 Γ 1 +, 8 an 2 F 1, Γ are respetively the Hypergeometri funtion an Gamma funtion. roof: Starting from the efinitions of p, we have p SIR > T ] E r,i g SIR > T ]] E r,i g g > T r α ]] I a E r,i exp T r α I ] b E r LI T r α] T r α f r r r. 9 a follows from the Rayleigh istribution assumption of hannel faing. In b, enotes the Laplae transform of I. Let I I + I, where I x i Φ \{x } g i x i an I y i Φ h i y i enote the interferene from ellular links an D2D links respetively. Then it is straightforwar to get s s s. 1 The Laplae transform of I is given by s E exp s g i x i x i Φ \{x } 1 To istinguish ifferent links, in this paper we use g exp 1, h exp 1 to represent the faing fators of links relate to ellular transmitters ellular users an D2D transmitters respetively. It is note that there is no essential istintion between these two symbols. E Φ x i Φ \{x } ˆ exp λ E g exp s g i x i ] Φ Bo,r 1 E g e sgi xi ] x i ˆ 1 exp λ 1 Φ Bo,r 1 + s x i x i v exp λ 2π r 1 + s v v α e exp λ π 1 s r 2 2F 1 1, 1 ; 2 ; s r α. 11 follows from the probability generating funtional GFL of 3]: E x Φ f x] exp λ R 1 f x x. 2 follows from the ouble integral in polar oorinates. e follows from the efinite integral 31, ]: b v 1+av v 1 α α b 2 a1 α 2 2 F 1 1, 1 2 α ; 2 2 α ; 1 ab. α Similarly, we have s E exp s h i y i y i Φ ˆ exp λ 1 E h exp R ˆ λ 2 exp exp λ 2π exp s h i y i ] y i 1 1 R s y i y i u 1 + s u uα λ π s Γ 1 Γ By plugging into 1 an letting s T r α, we get ν] r 2 T r α exp π λ µ + λ 13 1 T 2F 1 1, 1 ; 2 ; T, ν T where µ Γ 1 Γ 1 +. Then by plugging 2 13 into 9, we omplete the proof. B. Suessful Transmission robability of D2D Links We onut the analysis on a typial D2D link that omprises a typial D2D transmitter loate at some point in the network an a typial D2D reeiver loate at a ranom istane l away. Shift the oorinates suh that the typial D2D reeiver is loate at the origin 2, an enote the faing fator of the typial D2D link by h, h exp 1. Then, the reeive SIR of the typial D2D link an be expresse as SIR h l I, 14 2 It is note that the translations o not hange the istribution of 32].,

5 where I y i Φ \{y } h i y i + x i Φ g i x i 15 is the umulative interferene from all other D2D transmitters exept the typial D2D transmitter loate at y that are loate at y i with faing fator h i an ellular users that are loate at x i with faing fator g i. The suessful transmission probability of D2D links an be efine as p SIR > T ], 16 where T is the SIR threshol. Note that the same SIR threshol T is assume for ellular an D2D links. The expression of p is given by the following theorem. Theorem 2. The suessful transmission probability of D2D links without SIC apability is p where ν is given in 8. λ λ ν λ ν, 17 roof: Starting from the efinitions of p an SIR, p SIR > T ] E l,i h SIR > T ]] E l LI T lα] T lα f l l l. 18 Following approahes similar to those in previous proofs an Slivnyak s theorem 3]:!x, we have s E I exp si ] E exp s h i y i y i Φ \{y } E exp s ] g i x i x i Φ exp λ π s Γ 1 Γ 1 + exp λ π s Γ 1 Γ Therefore, ] T lα exp π λ + λ νl 2, 2 where ν is given in 8. Then by plugging 1 2 into 18, we omplete the proof. Remark 1. Via the expressions of p an p shown in Theorem 1 an 2, we an observe that µ, ν represent the effet of the interferene from the ellular links an D2D links respetively, an /, / an be regare as the onversion fators of powers. C. Stohasti Equivalene of Interferene By 4 15, the umulative interferene at eah link is generate by two-tier interferers, i.e., the ellular-tier an D2D-tier interferers. The analysis of suh two-tier interferene is trivial, as shown in the erivations of Theorem 1 an 2. Therefore, to simplify the analysis an failitate the performane evaluation of SIC in later setions, we propose an approah to uate the two-tier interferene by a single-tier interferene that has the same stohasti harateristis in terms of suessful transmission probability as the two-tier interferene. We first stuy the stohasti uivalene of the interferene for ellular links. By 4, the interferers for the typial ellular link onstitute Φ intf Φ \ {x } Φ. We represent Φ intf by an uivalent Φ intf \ {x } with ensity λ intf an transmission power. Then, the uivalent interferene at the typial ellular link an be expresse as I x i Φ intf \{x} g i x i whih has the same stohasti harateristis as I., 21 Lemma 1. The ensity of the uivalent interferers for ellular links is λ intf λ ν + λ µ, 22 where µ, ν are given in 7, 8 respetively. has the same stohasti hara- roof: Consiering I teristis as I, we have T r α The Laplae transform of I s E I E exp s exp λ Therefore, exp si ] x i Φ intf π T r α. 23 is obtaine as g i x i intf \{x} 1 s r 2 2F 1 1, 1 ; 2 ; s r α 24 T r α exp λ intf πµr2, 25 where µ is given in 7. Then by plugging into 23, we omplete the proof. It is note that p an be obtaine as L I T r α f r r r, an by 24, we have p λ λ intf µ + λ, 26 whih is onsistent with the result of Theorem 1. We next stuy the stohasti uivalene of the interferene for D2D links. By 15, the interferers for the typial D2D link onstitute Φ intf Φ \ {y } Φ. We represent.

6 Φ intf by an uivalent Φ intf \ {y } with ensity λ intf an transmission power. Then, the uivalent interferene at the typial D2D link an be expresse as I h i y i. 27 y i Φ intf \{y} Lemma 2. The ensity of the uivalent interferers for D2D links is λ intf λ + λ. 28 roof: Starting from the Laplae transform of I, s E I exp si ] E exp s h i y i y i Φ intf \{y} exp λ intf π s Γ 1 Γ Therefore, T lα exp λ where ν is given in 8. Then by letting T lα, we omplete the proof. It is note that p an be obtaine as f l l l, an by 3, we have p λ intf πνl2, 3 T lα L I T lα λ intf ν + λ, 31 whih is onsistent with the result of Theorem 2. V. NETWORK ERFORMANCE WITH SIC In this setion, we stuy how SIC affets the suessful transmission probabilities in D2D-enable ellular networks. The analysis is base on the stohasti uivalene moels propose in setion IV. A. Suessful Transmission robability of Cellular Links We onut the analysis on a typial ellular link that omprises a typial SIC-apable BS loate at the origin an its assoiate ellular user loate at x, where x r. The uivalent interferers for the typial ellular link are orere by their reeive power at the typial BS suh that g i x i > g j x j, < i < j. Before eriving the suessful transmission probability of the typial ellular link, we first present two lemmas. Given that n strongest uivalent interferers have been anele, the reeive SIR of the typial ellular link, i.e., the reeive SIR at the typial BS, an be expresse as where I n SIR n g r, 32 I n x i Φ intf \{x,x1,...xn} g i x i 33 is the umulative interferene for the typial ellular link. Then, the suessful transmission probability of the typial ellular link given that n strongest uivalent interferers have been anele an be efine as p n The expression of p n SIR n > T ]. 34 is given by the following lemma. Lemma 3. Given that n strongest uivalent interferers have been anele, the suessful transmission probability of the typial ellular link is p n n 2 λ intf π2 n n Γ n e λ intf πξr,n f r r n r, 35 where f r r is given in 2 an ξ r, n 1 T rα 2 n 2F 1 1, 1 ; 2 ; T rα α + 2 n. n 36 roof: Starting from the efinitions of p n ] p n SIR n > T E r L n I n an SIR n, T r α] T r α f r r r. 37 The alulation of L n I ruires the istribution of the sum of the orer statistis of interfering powers, whih is iffiult to obtain in the SIC senario. However, it has been shown that the orer statistis of reeive powers in moern networks are ominate by the istane 29]. Therefore, we an alulate L n I by relaxing the orering of interfering powers to that of interfering istanes. Denote the istane from n-th uivalent interferer to the origin by n, then we have ] L n I s E exp si n E n,φ E n intf,g e sgi xi x i Φ intf \{x,x1,...xn} e λ intf Φ intf Bo,n 1 E g e sg i x i ] x i ] e λ intf Φ 1 intf Bo,n 1 1+s x i x i f n n n e λ intf 2π v n 1+s e λ intf π 1 s2 n 2F 1 1,1 ;2 ; s α n v α v f n n n f n n n. 38 From 3], the probability ensity funtion of n is given by n 2 λ f n n e λ intf π2 intf π2 n n. 39 n Γ n

7 By plugging 39 into 38 an letting s T r α, we get T r α n 2 e λ intf πξr,n n λ intf π2 n n, 4 n Γ n n 2F 1 1, 1 ; 2 ; T rα + where ξ r, n 1 T rα 2 α n 2 n. Then by plugging 2 4 into 37, we omplete the proof. A possible approah for simplifying the expression of p n is to approximate n by a fixe value n, whih uals the expetation of n, i.e., Then, p n n E n ] an be approximate by p n where f r r is given in 2 an ξ r, n 1 T rα 2 n 2F 1 Γ n πλ intf Γ n e λ intf π ξr, n fr r r, 42. 1, 1 ; 2 ; T rα α n 43 Next, we stuy the probability of aneling n-th strongest uivalent interferer. Given that all n 1 stronger uivalent interferers have been anele, the reeive SIR of n-th strongest interferer at the typial BS an be expresse as where I n intf SIR n intf g n x n, 44 I n intf x i Φ intf \{xn,x1,...xn} g i x i 45 is the umulative interferene for n-th strongest uivalent interferer. Then, the probability of aneling eoing n- th strongest uivalent interferer given that all n 1 stronger uivalent interferers have been anele an be efine as p n intf The expression of p n intf SIR n intf > T ]. 46 is given by the following lemma. Lemma 4. Given that all n stronger uivalent interferers have been anele, the probability of aneling n-th strongest uivalent interferer for the typial ellular link is p n intf 1 µ + 1 n, 47 where µ is given in 7. roof: Following the relaxation approah for the orer statistis of interfering powers in the proof of Lemma 3, we an rewrite SIR n intf as intf g n n. 48 SIR n I n intf Then, we have ] p n intf SIR n intf > T E n,i exp n T α ni n intf E n L n I T α ] n intf n intf The Laplae transform of I n intf L n I s e λ intf Hene, n intf intf ] T α n fn n n. 49 is obtaine as intf π 1 s2 n 2F 1 1,1 ;2 ; s α n. 5 T α n exp λ intf πµ2 n, 51 where µ is given in 7. Then by plugging into 49, we omplete the proof. Base on Lemma 3 an 4, we an erive the suessful transmission probability of the typial ellular link. Theorem 3. The suessful transmission probability of ellular links with infinite SIC apability is n n p SIC p + 1 p i p n, 52 n1 are given in 6, 35, 47 respe- where p, p n tively. i1, p i intf p i intf i roof: For the typial ellular link, we efine the event of its suessful transmission without SIC as E : SIR > T, 53 an the event of its suessful transmission with n-level n 1 SIC as n n E n : SIR i intf > T SIR i < T SIR n > T. i1 54 Using the assumption that the interferene to eah user is inepenent, we get { p p, n, E n ] n 1 p i n i1 pi intf i p n, n Therefore, the suessful transmission probability of ellular links an be obtaine as p SIC i n E n]. Corollary 1. The suessful transmission probability of ellular links with finite N-level N 1 SIC apability is N n n p N SIC p + 1 p i p n, where p, p n tively. n1, p i intf i1 p i intf i 56 are given in 6, 35, 47 respe-

8 Suessful transmission probability Cellular Simulation Cellular Analysis Cellular Analysis, approx n D2D Simulation D2D Analysis D2D Analysis, approx k n SIC level Suessful transmission probability A B C Cellular, N Cellular, N1 Cellular, N2 Cellular, N Figure 3: Suessful transmission probabilities of ellular an D2D links with SIC. Figure 4: Suessful transmission probability of ellular links with SIC vs. ensity of D2D links. B. Suessful Transmission robability of D2D Links We onsier a typial D2D link with M-level SIC reeivers. The assumption of finite-level SIC reeiver is motivate by the limite omputational apabilities of D2D users. The erivation of the suessful transmission probability of D2D links is quite similar to that of ellular links, an hene we iretly present the results an omit their proofs. Lemma 5. Given that n strongest uivalent interferers have been anele, the suessful transmission probability of the typial D2D link is p n n 2 λ intf πk2 n k n Γ n e λ intf πκl,kn f l l k n l, 57 where f l l is given in 1 an κ l, k n 1 T lα kn 2 2F 1 1, 1 ; 2 ; T lα kn α +kn Lemma 6. Given that all n stronger uivalent interferers have been anele, the probability of aneling n-th strongest uivalent interferer for the typial D2D link is where µ is given in 7. p n intf 1 µ + 1 n, 59 Theorem 4. The suessful transmission probability of D2D links with finite M-level M 1 SIC apability is M n n p M SIC p + p i intf 1 p i p n, n1 i1 i 6 are given in 17, 57, 59 respe- where p, p n tively., pn intf C. Disussions an Numerial Results Now that we have evelope expressions for the suessful transmission probabilities of ellular an D2D links with SIC apabilities, base on the stohasti uivalene moels. It is note that the erive analytial expressions are not the exat results of orresponing suessful transmission probabilities, sine approximate moels are use in the erivation. We provie some numerial results to ompare the analytial results with the atual simulation results. The system parameters are set as α 4, 1, λ λ.1, T 1. Fig.3 shows the suessful transmission probabilities with SIC apabilities. As an be observe, there exist gaps between the analytial results of p N SIC, p M SIC an orresponing simulation results. The analytial results an be regare as lower bouns on the suessful transmission probabilities. The analytial results base on approximate n see an k n are also plotte in the figure, through whih we an fin that the results of the approximate analytial expressions losely math those of the exat analytial expressions. Therefore, the approximate analytial expressions an be employe to simplify the alulation of the suessful transmission probabilities. In aition, as shown in the figure, 2-level SIC an provie almost 5% performane improvement for the network; however, when the SIC level is larger than 2, SIC annot further improve the network performane. Consiering the harware ost of multi-level SIC, 1- an 2-level SIC an be aopte in pratial networks. Fig.4 shows the suessful transmission probability of ellular links with SIC vs. the ensity of D2D links. As expete, inreasing the ensity of D2D links leas to a erease in the suessful transmission probability of ellular links. However, from this figure we an observe that SIC an ompensate part of the performane loss of ellular links. For example, 1-level

9 SIC at the ellular reeiver an ompensate the performane loss generate by D2D interferers of ensity λ.35 see point A, an 2-level SIC orrespons to λ.45 see point B. VI. CONCLUSION In this paper, we stuy the performane of SIC in largesale D2D-enable ellular networks using the tools from stohasti geometry. We erive the suessful transmission probabilities of the network without SIC as the baseline results. To simplify the interferene analysis, we propose the approah of stohasti uivalene of the interferene, by whih the twotier interferene an be represente by an uivalent single-tier interferene. Base on the stohasti uivalene moels, we erive the suessful transmission probabilities of ellular an D2D links with infinite an finite SIC apabilities. The SIC gains are valiate by analytial an numerial results. ACKNOWLEDGMENT This work is supporte by NSF China No , , , , , , ; China Ministry of Euation Dotor rogram No ; National Mobile Communiations Researh Laboratory, Southeast University No.212D13, 214D7; Shanghai Basi Researh Key rojet 12JC1452, 11JC1451; Shanghai International Cooperation rojet No ; National Basi Researh 973 rogram of China No.215CB35243; Jiangsu Future Network Researh rojet BY REFERENCES 1] K. Doppler, M. Rinne, C. Wijting, C. Ribeiro, an K. Hugl, Devie-toevie ommuniation as an unerlay to LTE-avane networks, IEEE Communiations Magazine, vol. 47, no. 12, pp , De ] G. Foor, E. Dahlman, G. Milh, S. arkvall, N. Reier, G. Miklos, an Z. Turanyi, Design aspets of network assiste evie-to-evie ommuniations, IEEE Communiations Magazine, vol. 5, no. 3, pp , Mar ] Wireless Worl Initiative New Raio - WINNER+ D2.1, reliminary WINNER+ system onept, 29. 4] 3G TS 23.33, Tehnial speifiation group servies an system aspets; roximity-base servies rose, Rel-12, ] C. Xu, L. Song, Z. Han, Q. Zhao, X. Wang, X. Cheng, an B. Jiao, Effiieny resoure alloation for evie-to-evie unerlay ommuniation systems: A reverse iterative ombinatorial aution base approah, IEEE Journal on Selete Areas in Communiations, vol. 31, no. 9, pp , Sep ] B. Kaufman, J. Lilleberg, an B. Aazhang, Spetrum sharing sheme between ellular users an a-ho evie-to-evie users, IEEE Transations on Wireless Communiations, vol. 12, no. 3, pp , Mar ] Y. ei an Y.-C. Liang, Resoure alloation for evie-to-evie ommuniations overlaying two-way ellular networks, IEEE Transations on Wireless Communiations, vol. 12, no. 7, pp , July ] H. Min, W. Seo, J. Lee, S. ark, an D. Hong, Reliability improvement using reeive moe seletion in the evie-to-evie uplink perio unerlaying ellular networks, IEEE Transations on Wireless Communiations, vol. 1, no. 2, pp , Feb ] C. Ma, G. Sun, X. Tian, K. Ying, H. Yu, an X. Wang, Cooperative relaying shemes for evie-to-evie ommuniation unerlaying ellular networks, in roeeings of 213 IEEE Global Communiations Conferene GLOBECOM, De. 213, pp ] G. Boureau, J. aniker, N. Guo, R. Chang, N. Wang, an S. Vrzi, Interferene oorination an anellation for 4G networks, IEEE Communiations Magazine, vol. 47, no. 4, pp , Apr ] J. Anrews, Interferene anellation for ellular systems: a ontemporary overview, IEEE Wireless Communiations,, vol. 12, no. 2, pp , Apr ] E. Gelal, K. elehrinis, T.-S. Kim, I. Broustis, S. Krishnamurthy, an B. Rao, Topology ontrol for effetive interferene anellation in multi-user MIMO networks, in roeeings of 21 IEEE INFOCOM, Mar. 21, pp ] C. Jiang, Y. Shi, Y. Hou, W. Lou, S. Kompella, an S. Mikiff, Squeezing the most out of interferene: An optimization framework for joint interferene exploitation an avoiane, in roeeings of 212 IEEE INFOCOM, Mar. 212, pp ] C. Xu, L. ing,. Wang, S. Chan, an X. Lin, Deentralize power ontrol for ranom aess with suessive interferene anellation, IEEE Journal on Selete Areas in Communiations, vol. 31, no. 11, pp , Nov ] S. Lv, W. Zhuang, M. Xu, X. Wang, C. Liu, an X. Zhou, Unerstaning the sheuling performane in wireless networks with suessive interferene anellation, IEEE Transations on Mobile Computing, vol. 12, no. 8, pp , Aug ] M. Mollanoori an M. Ghaeri, Uplink sheuling in wireless networks with suessive interferene anellation, IEEE Transations on Mobile Computing, vol. 13, no. 5, pp , May ] M. Haenggi, J. Anrews, F. Baelli, O. Dousse, an M. Franeshetti, Stohasti geometry an ranom graphs for the analysis an esign of wireless networks, IEEE Journal on Selete Areas in Communiations, vol. 27, no. 7, pp , Sep ] R. Vaze an R. Heath, Transmission apaity of a-ho networks with multiple antennas using transmit stream aaptation an interferene anellation, IEEE Transations on Information Theory, vol. 58, no. 2, pp , Feb ] M. Haenggi, The loal elay in poisson networks, IEEE Transations on Information Theory, vol. 59, no. 3, pp , Mar ] J. Anrews, F. Baelli, an R. Ganti, A tratable approah to overage an rate in ellular networks, IEEE Transations on Communiations, vol. 59, no. 11, pp , Nov ] T. Novlan, H. Dhillon, an J. Anrews, Analytial moeling of uplink ellular networks, IEEE Transations on Wireless Communiations, vol. 12, no. 6, pp , Jun ] X. Lin an J. G. Anrews, Optimal spetrum partition an moe seletion in evie-to-evie overlai ellular networks, in roeeings of 213 IEEE Global Communiations Conferene GLOBECOM, De. 213, pp ] X. Lin, R. Ratasuk, A. Ghosh, an J. Anrews, Moeling, analysis an optimization of multiast evie-to-evie transmissions, IEEE Transations on Wireless Communiations, vol., no. 99, ] Q. Ye, M. Al-Shalash, C. Caramanis, an J. G. Anrews, Devieto-evie moeling an analysis with a moifie matern harore BS loation moel, in roeeings of 213 IEEE Global Communiations Conferene GLOBECOM, De. 213, pp ] S.. Weber, J. G. Anrews, X. Yang, an G. De Veiana, Transmission apaity of wireless a ho networks with suessive interferene anellation, IEEE Transations on Information Theory, vol. 53, no. 8, pp , Aug ] J. Lee, J. Anrews, an D. Hong, Spetrum-sharing transmission apaity with interferene anellation, IEEE Transations on Communiations, vol. 61, no. 1, pp , Jan ] J. Blomer an N. Jinal, Transmission apaity of wireless a ho networks: Suessive interferene anellation vs. joint etetion, in roeeings of 29 IEEE International Conferene on Communiations ICC, Jun. 29, pp ] X. Zhang an M. Haenggi, The aggregate throughput in ranom wireless networks with suessive interferene anellation, in roeeings of 213 IEEE International Symposium on Information Theory roeeings ISIT, July 213, pp ] M. Wilemeersh, T. Q. Quek, M. Kountouris, an C. H. Slump, Suessive interferene anellation in uplink ellular networks, in roeeings of 213 IEEE Workshop on Signal roessing Avanes in Wireless Communiations SAWC, Jun. 213, pp ] M. Haenggi, Stohasti geometry for wireless networks. Cambrige University ress, ] I. S. Grashteyn, I. M. Ryzhik, A. Jeffrey, an D. Zwillinger, Table of integrals, series, an prouts. Elsevier/Aaemi ress, ] F. Baelli an B. Blaszzyszyn, Stohasti Geometry an Wireless Networks. Now ublishers In, 29.