decouple HetNet mmWave

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To analyze the performance of wireless networks efficiently, stochastic geometry has emerged as a unified mathematical paradigm due to its tractability and accuracy (I[8). Specially, it was first applied to analyze the mm Wave cellular networks in [191, where the locations of mm Wave BSs follow a two-dimensional homogeneous Poisson point process(PPP), and it is observed that mm Wave could provide comparable coverage and higher data rates than microwave systems Moreover, a comprehensive overview of mathematical models and analytical techniques for mmWave cellular systems was performed in[7, where the authors suggested that an mm Wave network should be overlaid on a sub-6GHz network to provide high data rate in hotspots Only limited work has been carried out in the hybrid sub-6GHz and mm Wave cellular net works. a HetNet consisting of sub-6GHz MCells and mm Wave cells was studied in where the locations of Bss are modeled as two independent Ppps, and it is observed that extremely high bias values are desirable for Cells. a general and tractable mm Wave cellular model was proposed to characterize the associated rate distribution of networks consisting of sub- 6GHz MCells and mm Wave SCells in [201, and the analysis indicated that spectral efficiency of mm Wave networks increases with the Bs density, particularly at the cell edge. However, sub-6GHz SCells are not taken into account in 8] In this paper, using the tools from stochastic geometry, we provide a tractable framework to characterize the hybrid sub-6GHz and mm Wave HetNets with decoupled access where the user equipments (UEs)select the downlink and uplink serving BSs separately. The main contributions of this paper are summarized as follows Besides the traditional mcells both sub-6Ghz and mm Wave cells are considered in our work. Moreover, the uplink power control is incorporated for uplink sub-6GHZ UES, and the power-limited mm Wave UEs transmit with constant power Area sum rate(ASr) is proposed to investigate the system performance of the hybrid frequency networks. The disparity of bandwidth in sub-6GHz and mm Wave is considered in ASR, and it is shown that mm Wave SCells are more efficiently in improving network ASR compared with sub-6GHz SCells with the same density The general expressions for the performance metrics, including sinr coverage probability user-perceived rate coverage probability, and ASR, are derived. Based on the analytical and numerical results, we investigate the impact of CElls densification, and give insights on the network design The rest of the paper is organized as follows. The system model is introduced in Section IIl In Section l the association probability with decoupled access is derived. The expressions of sinR coverage probability, user-perceived rate coverage probability, and asr are given in Section IV Numerical and simulation results are presented in Section V which are followed by the conclusions in Section VI II. SYSTEM MODEL We consider a two-tier sub-6GHz HetNets coexisting with mm Wave SCells, where the lo- cations of sub-6GHz MCells, sub-6GHZ SCells, and mm Wave cells are modeled as homo geneous PPP 1, 2, and p, with density A1, A2, and A3, respectively. The BSs of each tier are distinguished by their spatial densities, transmit powers, carrier frequencies as well as propagation characters. The UEs are spatially and independently distributed in R according to a homogeneous PPp qu with density Au. The analysis is performed, without loss of generality, for a typical UE yo located at the origin according to the Slivnyak-Mecke theorem [21 It is shown that the uplink transmit power in mm Wave networks is even smaller than that of sub-6GHZ System [22 and power control can be neglected for mm Wave networks [7].Therefore we assume that the mm wave ues transmit with constant power Pu and that the sub-6GHZ UES utilize fractional power control(FPC) in the uplink to partially compensate for the long-term channel variation [23]. Given a typical UE yo associated with a sub-6GHZ BS in the uplink, the transmit power with FPC can be formulated as Pusyo = Pure, where Syo is the FPC coefficient of the typical UE yo, 0<E< l is the power control fraction, a is the path loss exponent, and r is the distance from yo to its serving BS. Obviously, e is equal to O in the mm Wave SCells A. Directional Beamforming The sub-6GHz BSs are assumed to be equipped with omni-directional antennas, and the mm Wave Bss are assumed to be equipped with directional antenna arrays to compensate for the high path loss. For the sake of simplicity, the directional antenna arrays are approximated by a sectored antenna model 24 namely if|f|≤bb b(6) otherwise where Bb is the beamwidth of the main lobe and Gm and gm are the main-lobe and side-lobe gains, respectively. When the typical UE is associated with an mm Wave Bs, the mm Wave Bs estimates the channel accurately, and then adjusts its antenna steering orientation to the typical ue to maximize the directivity gain Gb(e). The beam directions of the interference links are assumed to be independently and uniformly distributed in-I, T. Therefore, the mm Wave Bs's antenna gain of an interference link is Gm with a probability of pM= Bb/(2T), and is Gm with a probability of pm=1- 0b/(2T) B. Blockage and Channel Models Blockage model is adopted in mm Wave transmission to characterize the high near-field path loss and poor penetration through solid materials. An mm Wave link can be either line-of- sight (LoS)or non-line-of-sight (NLoS), depending on whether the bs is visible to the UE or not. In this paper, we use PL(r) to denote the probability that an mm Wave link with length r is LoS. According to the generalized blockage ball model [201, we have P(r)=pL·1(r<BB), where 1()is the indicator function, RB is the maximum length of a LoS link, and Pr is the average fraction of the los area in the ball of radius RB For the typical UE, mm Wave bss can be categorized into LoS BS set L and NLoS BS set N with distance dependent density Pl(r)A3 ane nd(1-Pi (r))A3, respectively. It is notable that L and N are no longer homogeneous PPP under the generalized blockage ball model The signals on different frequencies experience path loss with different intercepts and expo nents. In mm Wave networks. measurement results show a distinction between los and nlos links, where the nlos signals usually exhibit higher path loss than that of Los signals. There fore, the path loss between a UE and the serving Bs in the kth tier can be formulated as ek(r)=Ckr-ak, where k E K=1, 2, L,N, r is the length of the link, and Ck and ak are the path loss intercept and the path loss exponent of the kth tier, respectively. Here, the indices of“1,“2,“L,and“N" denote the tiers of the sub-6 GHz CElls, the sub-6 GHz SCells,the mmWave Los SCells, and the mm Wave NLos CElls, respectively. The fast fading is assumed to be subject to independent and identically distributed (i.i. d. Rayleigh fading with unit mean i. e, h N exp(1) C. Association strategy The downlink and uplink ue associations are performed based on the corresponding bias average received power (BARP)independently. Considering the typical UE, its downlink and 6 uplink serving bss are DL arg max Bk PDLkGks(x) (3) XuL= arg max Bk Pull ak Gk.Ek(x) respectively, where Bk and Bk are the downlink and uplink bias values of the kth tier, respectively, PDL.k and Pul x Ek ak are the downlink transmit power of the serving BS in the kith tier and the uplink transmit power of the typical UE associated with bs x in the kith tier, respectively, Gk is the antenna gain of BSs in the kith tier, and x is the distance from BS x to the typical UE It is worth noting that the downlink and uplink serving BSs of the typical UE may be different 1. e, XDL * Xur. With decoupled access, the uplink interference can be decreased, and thus the uplink network performance is enhanced [16] Since orthogonal multiple access is employed within a cell, intra-cell interference is mitigated here. If the typical UE is associated with the kith tier, the received downlink/uplink sinr can be formulated as SINRDL DL, A GkhEk(xDld (5) +IDlE SINRUL Py0Gkhk(|xtL‖) 2+Iu where ok is the thermal noise power, and lDL k and luL. k are the downlink and uplink interference, respectively. Specifically, the downlink interference IDL k can be formulated as ∑∑ ((|x|) ork∈{1,2} i∈{1,2}x∈AxbL DL. h ∑∑PGb(Ox→yo)bx(lx),fo ∈{LN}xe重2x where 8x-yo denotes thc angle between thc intcrfcrencc link x, yo and the desired link x DI, yo. As for the uplink interference lUL k, the applying of FPC for sub-6GHz cells makes a little difference between h (1, 2 and kC L N, and the expression of lUL. k is given b fork∈{1,2 ∈{1,2y∈重dyo UL. k (8) ∑∑ PGb(y xt)bx(y-xL),frk∈{N i∈{LNy∈中ayo 7 where pu,i is the set of UEs associated with the ith tier, Sy is the power control coefficient of UE y, and Bx+x denotes the angle between the interference link y, xuL and the desired link y0→>XtL III. ASSOCIATION ANALYSIS In order to investigate the hybrid frequency networks, we first calculate the probability of the typical UE being associated with the kth tier, i. e, Au, k, IE DL, UL and k E K. The following lemma provides the distribution of minimum distance Rh, which will be applied in calculating Lemma 1: Denote Rk as the distance from the typical ue to its nearest bs in the k: th tier; the cumulative distribution function(CDF)of Rk is given by 1-exD(-A丌2) k∈{1,2} exp(-2rAs P(c) c dx,kEL, N] and the probability density function(PDF)of Rk is given by 27入 kEep(-k丌2), k∈{1,2} (10) 27 A3 P(r)r exp(-27A3/P( a)cdx),kEL, N where PN(r)=1-Pi (r) Proof: The proof can be found in [6 2 and is omitted here Denote KDL and Kul as the tier index of bss that the typical UE is associated with in the downlink and uplink, respectively. From (3p and (4), the event of KDL=k and Kul - can be, respectively, described as (KdL=k)=1B:RROR>U 乃:T 1(A=)=1Br+1>∪BT3) (12) i∈K\k where Tk= PDL Gk CK, and Tk= PuGk Ck. Leveraging the distance distribution FR(r)in emma I we can derive the association probability for each tier, as shown in Theorem I Theorem 1: The probability of the typical ue being associated with the kth tier is v, k A I FR (Work (G)re (r)dr Ck where v∈{DL,UL},k∈K,F1(r)=1-FR:(r),amd B T 业DL,k(r)=ck(r) (14) BrTk l (1-∈k)a uL,2(r)=92k(7) Bi Ti (15) Pmo: Scc appendix囚 Based on the results of Theorem I we can derive the distribution of the conditional distance lu, k=Ru,k Ky=k. If the typical UE is associated with the hth tier in the downlink or uplink. the pdf of v k is given by the following corollary Corollary 1: The Pdf of the lu k is Rk (a)I 16 k ∈K\k Proof: Scc Appendix Bk Remark 1 Obviously, the UEs can be roughly categorized into two groups, i c, the uEs that associated to the same bs in both downlink and uplink, termed as coupled UEs, and the UEs that associated to different bss in the downlink and uplink, termed as decoupled uEs. The percentage of the decoupled uEs is given by =1-∑P(Ku=k,KDL=k) k∈K ∑P(B≤∪(B),B1<∪(B ∈C\k 21->Ⅱ(n>mx{k(R,(R)}) I2(mx{,9x()(d Here,(a)follows from the independence of different tiers and the property of k i(Pi. k(r)) Or, i(oi, k(r))=r, and(b)follows from the fact that Ey [IP(X>Y)= Fx(g)fr(y) dy for positive random variables X and y IV, PERFORMANCE ANALYSIS In this section, we will analyze the network performance in terms of the SInr coverage probability, rate coverage probability, and area sum rate A. INR Coverage analysis The SINR coverage probability Cv(T), VE DL, ULJ, is defined as the probability that the instantaneous received sinR is greater than a threshold T, and can be described as (7)=∑ (17 where Cv k(T), which is the SINR coverage probability conditioned on the typical UE being associated with the kth tier, can be expressed as C以k(7)=P(SINR,k>T|压D=h)=P Pik Gkhek(x K,= k (18) k千1m,k where Pik is the transmit power of the serving bs in the kth tier for v= DL and the transmit power of the typical ue for v= UL In our analysis, we assume that each bs has at least one ue in its coverage area and thus all BSs are active in both downlink and uplink. To facilitate the calculation of uplink interference, we use qu e qu to denote the set of the active ues in the uplink. we also assume that each bs has only one active UE at a given time, and the active ue is uniformly distributed in the uplink coverage area of its serving Bs. Therefore it is reasonable to conclude that the uplink active UE set u is approximately distributed in the plane with density 2k=l k. The exact distribution of u, however, is unknown due to the dependance induced by the Voronoi tessellation[25 , [26] Here, we assume that u approximately forms a homogeneous PPP with density 2k-l Xk,and more precisely, the UEs associated with BSs belonging to pk in the uplink form a homogeneous PPP pu, k with density Xk[17 ,[251,[27] Before deriving the sinr coverage probability, we first present the Laplace transforms of the interference terms IDL k and IUL k in following lemma Lemma 2: The laplace transforms of interference IDL k and IUL k, conditioned on the typical ue being associated with the kth tier in downlink and uplink, are given by exp(-2A1V(k,1(x),a,s1)-2xA2V(Ok2(x),a,s12),k∈{1,2} 1(S;U exp{-2丌入 Ge(Mm) P WL.(PK L(2), aL, SILG,) (r),CN, SING, kE[L, N (19) 10 exp(-2丌入10V(mx1{4k1(91(x2),},a,s1a)x(n)d -2T 2 So V(maxYk,2(k, 2(a)),u], C, sTluea)xuL ,(u)du), ke(1, 2) UL F exp3-2TA3CiC(M. m) P,WL(Dk,L(c),aL, sIlG +WN(K, N(a), aN, S\Gj )} ,k∈{L.N respectively, where Q+2 V(, c, B) 1+B Pi(r)da WN(, a, B) 1+B- PN(r)da with Gm=1, Gm= Gm/GM, and 2Fi denoting the hypergeometric function Proof: See Appendix Ck Based on Lemma 2 we now present the SINR coverage probability in the following theorem Theorem 2: The SINR coverage probability is given by (r)=∑4k TO k∈KC D 么(2) (21) where vE DL, UL), Suk(a)=Pu Gkek(a), fxm(r)is given by Lemmaz and iw,k(s: c)is given by Lemma 2 Proof: See Appendix DI Remark 2: As can be sccn from Thcorem 2, the distribution of the minimum distance Rk, the distribution of the conditional serving distance ay. k, and the interference Iuk play active roles determining the valuc of Cv (r), and their impacts on the nctwork performance will be shown in section ⅣMo moreover, it is noticed that a double integral is required for the calculation of CDL (T)and a triple integral is required for CuL(T) Remark 3: Theorem 2 gives the downlink and uplink SINR coverage probability with decou pled access. As a special case, the uplink sinr coverage probability with coupled access can be easily derived by replacing AUL, k and k (a)in(2I) with ADL k and Ok , 2(3), respectively, and is given b (r)=∑Am T exp fxp (a)d, (22) kCk 0

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