Coverage and Rate Analysis for Millimeter Wave Cellular Networks

Transcription

1 Coverage and Rate Anaysis for Miimeter Wave Ceuar Networks Tianyang Bai and Robert W. Heath, Jr. arxiv:42.643v3 cs.it 8 Oct 24 Abstract Miimeter wave mmwave) hods promise as a carrier frequency for fifth generation ceuar networks. Because mmwave signas are sensitive to bockage, prior modes for ceuar networks operated in the utra high frequency UHF) band do not appy to anayze mmwave ceuar networks directy. Leveraging concepts from stochastic geometry, this paper proposes a genera framework to evauate the coverage and rate performance in mmwave ceuar networks. Using a distancedependent ine-of-site LOS) probabiity function, the ocations of the LOS and non-los base stations are modeed as two independent non-homogeneous Poisson point processes, to which different path oss aws are appied. Based on the proposed framework, expressions for the signa-to-noise-and-interference ratio SINR) and rate coverage probabiity are derived. The mmwave coverage and rate performance are examined as a function of the antenna geometry and base station density. The case of dense networks is further anayzed by appying a simpified system mode, in which the LOS region of a user is approximated as a fixed LOS ba. The resuts show that dense mmwave networks can achieve comparabe coverage and much higher data rates than conventiona UHF ceuar systems, despite the presence of bockages. The resuts suggest that the ce size to achieve the optima SINR scaes with the average size of the area that is LOS to a user. I. INTRODUCTION The arge avaiabe bandwidth at miimeter wave mmwave) frequencies makes them attractive for fifth generation ceuar networks 3 5. The mmwave band ranging from 3 GHz to 3 GHz has aready been considered in various commercia wireess systems incuding IEEE c for persona area networking 6, IEEE 82.ad for oca area networking 7, and IEEE for fixed-point access inks 8. Recent fied measurements revea the promise of mmwave signas for the access ink between the mobie station and base station) in ceuar systems 5, 9. One differentiating feature of mmwave ceuar communication is the use of antenna arrays at the transmitter and receiver to provide array gain. As the waveength decreases, antenna sizes aso decrease, reducing the antenna aperture. For exampe, from the Friis free-space equation, a mmwave signa at 3 GHz wi experience 2 db arger path oss than a signa at 3 GHz. Thanks to the sma waveength, however, it is possibe to pack mutipe antenna eements into the The authors are with The University of Texas at Austin, Austin, TX, USA. emai: tybai@utexas.edu, rheath@utexas.edu) This work is supported in part by the Nationa Science Foundation under Grants No and 39556, and by a gift from Huawei Technoogies, Inc. Preiminary resuts reated to this paper were presented at the st IEEE Goba Conference on Signa and Information Processing GobaSIP) and the 47th Annua Asiomar Conference on Signas, Systems, and Computers Asiomar) 2. imited space at mmwave transceivers 3. With arge antenna arrays, mmwave ceuar systems can impement beamforming at the transmitter and receiver to provide array gain that compensates for the frequency-dependent path oss, overcomes additiona noise power, and as a bonus aso reduces out-of-ce interference 4. Another distinguishing feature of mmwave ceuar communication is the propagation environment. MmWave signas are more sensitive to bockage effects than signas in owerfrequency bands, as certain materias ike concrete was found on buiding exteriors cause severe penetration oss. This indicates that indoor users are unikey to be covered by outdoor mmwave base stations. Channe measurements using directiona antennas 5, 9, 2 have reveaed another interesting behavior at mmwave: bockages cause substantia differences in the ine-of-sight LOS) paths and non-ine-ofsight NLOS) path oss characteristics. Such differences have aso been observed in prior propagation studies at utra high frequency bands UHF) from 3 MHz to 3 GHz, e.g. see 3. The differences, however, become more significant for mmwave since diffraction effects are negigibe 4, and there are ony a few scattering custers 4. Measurements in 5, 9, 2 showed that mmwave signas propagate as in free space with a path oss exponent of 2. The situation was different for NLOS paths where a og distance mode was fit with a higher path oss exponent and additiona shadowing 5, 9. The NLOS path oss aws tend to be more dependent on the scattering environment. For exampe, an exponent as arge as 5.76 was found in downtown New York City 5, whie ony 3.86 was found on the UT Austin campus 9. The distinguishing features of the propagation environment need to be incorporated into the any comprehensive system anaysis of mmwave networks. The performance of mmwave ceuar networks was simuated in prior work 4, 5 using insights from propagation channe measurements 5. In 5, using the NLOS path oss aw measured in the New York City, ower bounds of the signa-to-noise-and-interference ratio SINR) distribution and the achievabe rate were simuated in a 28 GHz pico-ceuar system. In 4, a mmwave channe mode that incorporated bockage effects and ange spread was proposed and further appied to simuate the mmwave network capacity. Both resuts in 4, 5 show that the achievabe rate in mmwave networks can outperform conventiona ceuar networks in the utra high frequency UHF) band by an order-of-magnitude. The simuation-based approach 4, 5 does not ead to eegant system anaysis as in 6, which can be broady appied to different depoyment scenarios.

2 2 Stochastic geometry is a usefu too to anayze system performance in conventiona ceuar networks 6, 7. In 6, by modeing base station ocations in a conventiona ceuar network as a Poisson point process PPP) on the pane, the aggregate coverage probabiity was derived in a simpe form, e.g. a cosed-form expression when the path oss exponent is 4. Moreover, the stochastic mode was shown to provide a ower bound of the performance in a rea ceuar system 6. There have been severa extensions of the resuts in 6, such as anayzing a muti-tier network in 8 and predicting the site-specific performance in heterogeneous networks in 9. It is not possibe to directy appy resuts from conventiona networks to mmwave networks due to the different propagation characteristics and the use of directiona beamforming. There has been imited appication of stochastic geometry to study mmwave ceuar networks. The primary reated work was in 2, where directiona beamforming was incorporated for singe and mutipe user configurations, but a simpified path oss mode was used that did not take mmwave propagation features into account. A systematic study of mmwave network performance shoud incorporate the impact of bockages such as buidings in urban areas. One approach is to mode the bockages expicity in terms of their sizes, ocations, and shapes using data from a geographic information system. This approach is we suited for site-specific simuations 2 using eectromagnetic simuation toos ike ray tracing 22. An aternative is to empoy a stochastic bockage mode, e.g. 23, 24, where the bockage parameters are drawn randomy according to some distribution. The stochastic approach ends itsef better to system anaysis and can be appied to study system depoyments under a variety of bockage parameters such as size and density. The main contribution of this paper is to propose a stochastic geometry framework for anayzing the coverage and rate in mmwave ceuar networks. As a byproduct, the framework aso appies to anayze heterogenous networks in which the base stations are distributed as certain non-homogeneous PPPs. We incorporate directiona beamforming by modeing the beamforming gains as marks of the base station PPPs. For tractabiity of the anaysis, the actua beamforming patterns are aso approximated by a sectored mode, which characterizes key features of an antenna pattern: directivity gain, haf-power beamwidth, and front-back ratio. A simiar mode was aso empoyed in work on ad hoc networks 25. To incorporate bockage effects, we mode the probabiity that a communication ink is LOS as a function of the ink ength, and provide a stochastic characterization of the region where a user does not experience any bockage, which we define as the LOS region. Appying the distance-dependent LOS probabiity function, the base stations are equivaenty divided into two independent non-homogenous point processes on the pane: the LOS and the NLOS base station processes. Different path oss aws and fading are appied separaby to the LOS and NLOS case. Based on the system mode, expressions for the SINR and rate coverage probabiity are derived in genera mmwave networks. To simpify the anaysis, we aso propose a systematic approach to approximate a compicated LOS function as its equivaent step function. Our anaysis indicates that the coverage and rate are sensitive to the density of base stations and the distribution of bockages in mmwave networks. It aso shows that dense mmwave networks can generay achieve good coverage and significanty higher achievabe rate than conventiona ceuar networks. A simpified system mode is proposed to anayze dense mmwave networks, where the infrastructure density is comparabe to the bockage density. For a genera LOS function, the LOS region observed by a user has an irreguar and random shape. Coverage anaysis requires integrating the SINR over this region. We propose to simpify the anaysis by approximating the actua LOS region as a fixed-sized ba caed the equivaent LOS ba. The radius of the equivaent LOS ba is chosen so that the ba has the same average number of LOS base stations in the network. With the simpified network mode, we find that in a dense mmwave network, the ce radius shoud scae with the size of LOS region to maintain the same coverage probabiity. We find that continuing to increase base station density eading to what we ca utradense networks) does not aways improve SINR, and the optima base station density shoud be finite. Compared with our prior work in, this paper provides a generaized mathematica framework and incudes the detaied mathematica derivations. The system mode appies for a genera LOS probabiity function and incudes the impact of genera sma-scae fading. We aso provide a new approach to compute coverage probabiity, which avoids inverting the Fourier transform numericay and is more efficient than prior expressions in. Compared with our prior work in 2, we aso remove the constraint that the LOS path oss exponent is 2, and extend the resuts in 2 to genera path oss exponents, in addition to providing derivations for a resuts, and new simuation resuts. This paper is organized as foows. We introduce the system mode in Section II. We derive expressions for the SINR and rate coverage in a genera mmwave network in Section III. A systematic approach is aso proposed to approximate genera LOS probabiity functions as a step function to further simpify anaysis. In Section IV, we appy the simpified system mode to anayze performance and examine asymptotic trends in dense mmwave networks, where outdoor users observe more than one LOS base stations with high probabiity. Finay, concusions and suggestions for future work are provided in Section VI. II. SYSTEM MODEL In this section, we introduce our system mode for evauating the performance of a mmwave network. We focus on downink coverage and rate experienced by an outdoor user, as iustrated in Fig. a). We make the foowing assumptions in our mathematica formuation. Assumption Bockage process): The bockages, typicay buidings in urban areas, form a process of random shapes, e.g. a Booean scheme of rectanges 24, on the pane. We assume the distribution of the bockage process to be stationary and isotropic - in other words - invariant to the motions of transation and rotation 26, Chapter.

3 3 m a) System mode for mmwave ceuar networks θ M b) Sectored mode to approximate beamforming patterns. Fig.. In a), we iustrate the proposed system mode for mmwave ceuar networks. Bockages are modeed as a random process of rectanges, whie base stations are assumed to be distributed as a Poisson point process on the pane. An outdoor typica user is fixed at the origin. The base stations are categorized into three groups: indoor base stations, outdoor base stations that are LOS to the typica user, and outdoor base stations that are NLOS to the user. Directiona beamforming is performed at both base stations and mobie stations to expoit directivity gains. In b), we iustrate the sectored antenna mode G M,m,θ φ), which is used to approximate the beamforming patterns. Assumption 2 PPP BS): The base stations form a homogeneous PPP Φ with density λ on the pane. Note that a base station can be ocated either inside a bockage or outside a bockage. In this paper, however, we wi focus on the SINR and rate provided by the outdoor base stations as the bockages are assumed to be impenetrabe. Let Φ = {X } be the point process of outdoor base stations, X the -th outdoor base station, and R = OX denote the distance from -th base station to the origin O. Define τ as the average fraction of the and covered by bockages, i.e., the average fraction of indoor area in the network. Further, we assume the base station process Φ is independent of the bockage process. Therefore, each base station has an i.i.d. probabiity τ to be ocated outdoor. By the thinning theorem of PPP 26, the outdoor base station process Φ is a PPP of density λ = τ) λ on the pane. In addition, a base stations are assumed to have a constant transmit power P t. Assumption 3 Outdoor user): The users are distributed as a stationary point process independent of the base stations and bockages on the pane. A typica user is assumed to be ocated at the origin O, which is a standard approach in the anaysis using stochastic geometry 6, 26. By the stationarity and independence of the user process, the downink SINR and rate experienced by the typica user have the same distributions as the aggregate ones in the network. The typica user is assumed to be outdoors. The indoor-tooutdoor penetration oss is assumed to be high enough such that an outdoor user can not receive any signa or interference from an indoor base station. Therefore, the focus in this paper is on investigating the conditiona SINR and rate distribution of the outdoor typica user served by outdoor infrastructure. Indoor users can be served by either indoor base stations or by outdoor base stations operated at UHF frequencies, which have smaer indoor-to-outdoor penetration osses in many common buiding materias. We defer the extension to incorporate indoor users to future work. We say that a base station at X is LOS to the typica user at the origin O if and ony if there is no bockage intersecting the ink OX. Due to the presence of bockages, ony a subset of the outdoor base stations Φ are LOS to the typica user. Assumption 4 LOS and NLOS BS): An outdoor base station can be either LOS or NLOS to the typica user. Let Φ L be the point process of LOS base stations, and Φ N = Φ/Φ L be the process of NLOS base stations. Define the LOS probabiity functionpr) as the probabiity that a ink of engthr is LOS. Noting the fact that the distribution of the bockage process is stationary and isotropic, the LOS probabiity function depends ony on the ength of the ink R. Aso, pr) is a non-increasing function of R; as the onger the ink, the more ikey it wi be intersected by one or more bockages. The NLOS probabiity of a ink is pr). The LOS probabiity function in a network can be derived from fied measurements 4 or stochastic bockage modes 23, 24, where the bockage parameters are characterized by some random distributions. For instance, when the bockages are modeed as a rectange booean scheme in 24, it foows that pr) = e βr, where β is a parameter determined by the density and the average size of the bockages, and /β is what we caed the average LOS range of the network in 24. For the tractabiity of anaysis, we further make the foowing independent assumption on the LOS probabiity; taking account of the correations in bockage effects generay makes the exact anaysis difficut. Assumption 5 Independent LOS probabiity): The LOS probabiities are assumed to be independent between different inks, i.e., we ignore potentia correations of bockage effects between inks. Note that the LOS probabiities for different inks are not independent in reaity. For instance, neighboring base stations might be bocked by a arge buiding simutaneousy. Numerica resuts in 24, however, indicated that ignoring such correations cause a minor oss of accuracy in the SINR evauation. Assumption 5 aso indicates that the LOS base station process Φ L and the NLOS process Φ N form two independent non-homogeneous PPP with the density functions pr)λ and pr))λ, respectivey, where R is the radius in poar coordinates. Assumption 6 Path oss mode): Different path oss aws are appied to LOS and NLOS inks. Given a ink has ength

4 4 TABLE I PROBABILITY MASS FUNCTION OF D AND D k a k M rm t M rm t m rm t m rm t b k c rc t c r c t) c r)c t c r) c t) e k M r M r/ξ t m r m r/ξ t R, its path oss gain LR) is computed as LR) = IpR))C L R αl + IpR))C N R αn, ) where Ix) is a Bernoui random variabe with parameter x, α L, α N are the LOS and NLOS path oss exponents, and C L, C N are the intercepts of the LOS and NLOS path oss formuas. Typica vaues of mmwave path oss exponents and intercept constants are avaiabe in prior work, see e.g. 5, 9. The mode coud be further enhanced by incuding ognorma shadowing, but this is deferred in our paper to simpify the anaysis. Assumption 7 Directiona beamforming): Antenna arrays are depoyed at both base stations and mobie stations to perform directiona beamforming. For tractabiity of the anaysis, the actua array patterns are approximated by a sectored antenna mode, which was used in prior ad hoc network anaysis 25. Let G M,m,θ φ) denote the sectored antenna pattern in Fig. b), where M is the main obe directivity gain, m is the back obe gain, θ is the beamwidth of the main obe, and φ is the ange off the boresight direction. In the sectored antenna mode, the array gains are assumed to be constant M for a anges in the main obe, and another constant m in the side obe in the sectored mode. We et M t, m t, and θ t be the main obe gain, side obe gain, and haf power beamwidth of the base station antenna, and M r, m r, and θ r the corresponding parameters for the mobie station. Without oss of generaity, we denote the boresight direction of the antennas as. Further, et D = G Mt,m t,θ t φ t )G M r,m r,θ r φ r ) be the tota directivity gain in the ink from the -th base station to the typica user, where φ r and φ t are the ange of arriva and the ange of departure of the signa. Assumption 8 User association): The typica user is associated with the base station, either LOS or NLOS, that has the smaest path oss LR ). The serving base station is denoted as X. Both the mobie station and its serving base station wi estimate channes incuding anges of arrivas and fading, and then adjust their antenna steering orientations accordingy to expoit the maximum directivity gain. Errors in channe estimation are negected, and so are errors in time and carrier frequency synchronizations in our work. Thus, the directivity gain for the desired signa ink is D = M r M t. For the -th interfering ink, the anges φ r and φ t are assumed to be independenty and uniformy distributed in, 2π, which gives a random directivity gain D. By Assumption 7 and Assumption 8, the directivity gain in an interference ink D is a discrete random variabe with the probabiity distribution as D = a k with probabiity b k k {,2,3,4}), where a k and b k are constants defined in Tabe I, c r = θr 2π, and c t = θt 2π. Assumption 9 Sma-scae fading): We assume independent Nakagami fading for each ink. Different parameters of Nakagami fading N L and N N are assumed for LOS and NLOS inks. Let h be the sma-scae fading term on the - th ink. Then h 2 is a normaized Gamma random variabe. Further, for simpicity, we assume N L and N N are positive integers. We aso ignore the frequency seectivity in fading, as measurements show that the deay spread is generay sma 5, and the impact of frequency-seective fading can be minimized by techniques ike orthogona frequency-division mutipexing or frequency domain equaization. Measurement resuts indicated that sma-scae fading at mmwave is ess severe than that in conventiona systems when narrow beam antennas are used 5. Thus, we can use a arge Nakagami parameter N L to approximate the sma-variance fading as found in the LOS case. Let σ 2 be the therma noise power normaized by P t. Based on the assumptions thus far, the SINR received by the typica user can be expressed as SINR = h 2 M r M t LR ) σ 2 + >:X Φ h 2 D LR ). 2) Note that the SINR in 2) is a random variabe, due to the randomness in the base station ocationsr, sma-scae fading h, and the directivity gain D. Using the proposed system mode, we wi evauate the mmwave SINR and rate coverage in the foowing section. III. COVERAGE AND RATE ANALYSIS IN GENERAL NETWORKS In this section, we anayze the coverage and rate in the proposed mode of a genera mmwave network. First, we provide some SINR ordering resuts regarding different parameters of the antenna pattern. Then we derive expressions for the SINR and rate coverage probabiity in mmwave networks with genera LOS probabiity function pr). To simpify subsequent anaysis, we then introduce a systematic approach to approximate pr) by a moment matched equivaent step function. A. Stochastic Ordering of SINR With Different Antenna Geometries One differentiating feature of mmwave ceuar networks is the depoyment of directiona antenna arrays. Consequenty, the performance of mmwave networks wi depend on the adaptive array pattern through the beamwidth, the directivity gain, and the back obe gain. In this section, we estabish some resuts on stochastic ordering of the SINRs in the systems with different antenna geometries. Whie we wi focus on the array geometry at the transmitter, the same resuts, however, aso appy to the receiver array geometry. The concept of stochastic ordering has been appied in anaysis of wireess systems 27, 28. Mathematicay, the ordering of random variabes can be defined as foows 27, 28. Definition : Let X and Y be two random variabes. X stochasticay dominates Y, i.e., X has a better distribution than Y, if PX > t) > PY > t) for a t R. Next, define the front-to-back ratio FBR) at the transmitter ξ t as the ratio between the main obe directivity gain M t and

5 5 the back obe gain m t, i.e., ξ t = M t /m t. We introduce the key resut on stochastic ordering of the SINR with respect to the directivity gains as foows. Proposition Stochastic ordering w.r.t. directivity gains): Given a fixed beamwidth θ t and FBR ξ t at the transmitter, the mmwave network with the arger main obe directivity gain M t has a better SINR distribution. Simiary, with fixed beamwidth θ t and main obe gain M t, a arger FBR ξ t provides a better SINR distribution. Proof: From Definition, we need to show that for each reaization of base station ocations R, sma-scae fading h, and anges φ r and φ t, the vaue of the SINR increases with M t and ξ t. Given R, h, φ r, and φ r N), we can normaize both the numerator and denominator of 2) by h M t, and then write SINR = 2 M rlr ) σ 2 /M t+ D, >:X Φ ξ t) h 2 LR ) where D ξ t ) = e k with probabiity b k, and b k, e k are constants defined in Tabe I. Note that D ξ t ) is independent of M t, and is a non-increasing function of ξ t. Hence, when ξ t is fixed, arger M t provides arger SINR; when M t is fixed, arger ξ t provides arger SINR. Next, we provide the stochastic ordering resut regarding beamwidth as foows. Proposition 2 Stochastic ordering w.r.t. beamwidth): Given a fixed main obe gain M t and FBR ξ t at the transmitter, a smaer beamwidth θ t provides a better SINR distribution. The proposition can be rigorousy proved using couping techniques. We omit the proof here and instead provide an intuitive expanation as beow. Intuitivey, with narrower main obes, fewer base stations wi transmit interference to the typica user via their main obes, which gives a smaer interference power. The desired signa term in 2) is independent of the beamwidth, as we ignore the channe estimation errors and potentia ange spread. Hence, based on our mode assumptions, smaer beamwidths provide a better SINR performance. We note that the ordering resut in Proposition 2 assumes that there is no ange spread in the channe. With ange spread, a narrow-beam antenna may capture ony the signa energy arriving inside its main obe, missing the energy spread outside, which causes a gain reduction in the signa power 29. Consequenty, the resuts in Proposition 2 shoud be interpreted as appying to the case where beamwidths are arger than the ange spread, e.g. if the beamwidth is more than55 per the measurements in 2. We defer more detaied treatment of ange spread to future work. B. SINR Coverage Anaysis The SINR coverage probabiity P c T) is defined as the probabiity that the received SINR is arger than some threshod T >, i.e., P c T) = PSINR > T). We present the foowing emmas before introducing the main resuts on SINR coverage. By Assumption 4, the outdoor base station process Φ can be divided into two independent non-homogeneous PPPs: the LOS base station process Φ L and NLOS process Φ N. We wi equivaenty considerφ L andφ N as two independent tiers of base stations. As the user is assumed to connect to the base station with the smaest path oss, the serving base station can ony be either the nearest base station in Φ L or the nearest one in Φ N. The foowing emma provides the distribution of the distance to the nearest base station in Φ L and Φ N. Lemma : Given the typica user observes at east one LOS base station, the conditiona probabiity density function of its distance to the nearest LOS base station is f L x) = 2πλxpx)e 2πλ x rpr)dr /B L, 3) where x >, B L = e 2πλ rpr)dr is the probabiity that a user has at east one LOS base station, and pr) is the LOS probabiity function defined in Section II. Simiary, given the user observes at east one NLOS base station, the conditiona probabiity density function of the distance to the nearest NLOS base station is f N x) = 2πλx px))e 2πλ x r pr))dr /B N, 4) where x >, and B N = e 2πλ r pr))dr is the probabiity that a user has at east one NLOS base station. Proof: The proof foows 24, Theorem and is omitted here. Next, we compute the probabiity that the typica user is associated with either a LOS or a NLOS base station. Lemma 2: The probabiity that the user is associated with a LOS base station is A L = B L e 2πλ ψ L x) pt))tdt f L x)dx, 5) where ψ L x) = C N /C L ) /αn x αl/αn. The probabiity that the user is associated with a NLOS base station is A N = A L. Proof: See Appendix B. Further, conditioning on that the serving base station is LOS or NLOS), the distance from the user to its serving base station foows the distribution given in the foowing emma. Lemma 3: Given that a user is associated with a LOS base station, the probabiity density function of the distance to its serving base station is ˆf L x) = B Lf L x) e 2πλ ψl x) pt))tdt, 6) A L when x >. Given the user is served by a NLOS base station, the probabiity density function of the distance to its serving base station is ˆf N x) = B Nf N x) e 2πλ ψn x) pt)tdt, 7) A N where x >, and ψ N x) = C L /C N ) /αl x αn/αl. Proof: The proof foows a simiar method as that of Lemma 2, and is omitted here. Now, based on Lemma 2 and Lemma 3, we present the main theorem on the SINR coverage probabiity as foows Theorem : The SINR coverage probabiity P c T) can be computed as P c T) = A L P c,l T)+A N P c,n T), 8)

6 6 where for s {L,N}, P c,s T) is the conditiona coverage probabiity given that the user is associated with a base station in Φ s. Further, P c,s T) can be evauated as and N L ) P c,l T) ) n+ NL n n= e nη L x α LTσ 2 N N ) P c,n T) ) n+ NN n where Q n T,x) = 2πλ n= e nη N x α NTσ 2 V n T,x) =2πλ W n T,x) =2πλ C L MrM t Q nt,x) V nt,x) ˆfL x)dx, 9) C N MrM t W nt,x) Z nt,x) ˆfN x)dx. ) 4 b k F k= x 4 b k k= ψ Lx) N L, nη ) Lā k Tx αl pt)tdt, N L t αl F N N, nc ) Nη L ā k Tx αl C L N N t αn ) pt))tdt, 2) 4 b k k= Z n T,x) =2πλ ψ Nx) F N L, nc ) Lη N ā k Tx αn C N N L t αl pt)tdt, 3) 4 b k F k= x N N, nη ) Nā k Tx αn N N t αn pt))tdt, 4) and FN,x) = / + x) N. For s {L,N}, η s = N s N s!) Ns, N s are the parameters of the Nakagami smascae fading; for k {,2,3,4}, ā k = a k M tm r, a k and b k are constants defined in Tabe I. Proof: See Appendix C. Though as an approximation of the SINR coverage probabiity, we find that the expressions in Theorem compare favoraby with the simuations in Section V-A. In addition, the expressions in Theorem compute much more efficienty than prior resuts in, which required a numerica inverse of a Fourier transform. Last, the LOS probabiity function pt) may itsef have a very compicated form, e.g. the empirica function for sma ce simuations in 3, which wi make the numerica evauation difficut. Hence, we propose simpifying the system mode by using a step function to approximate pt) in Section III-D. Before that, we introduce our rate anaysis resuts in the foowing section. C. Rate Anaysis In this section, we anayze the distribution of the achievabe rate Γ in mmwave networks. We use the foowing definition for the achievabe rate Γ = W og 2 +min{sinr,t max }), 5) where W is the bandwidth assigned to the typica user, and T max is a SINR threshod determined by the order of the consteation and the imiting distortions from the RF circuit. The use of a distortion threshod T max is needed because of the potentia for very high SINRs in mmwave that may not be expoited due to other imiting factors ike inearity in the radio frequency front-end. The average achievabe rate EΓ can be computed using the foowing Lemma from the SINR coverage probabiity P c T). Lemma 4: Given the SINR coverage probabiity P c T), the average achievabe rate in the network is EΓ = W n2 Tmax P ct) +T dt. Proof: See 6, Theorem 3 and 2, Section V. Lemma 4 provides a first order characterization of the rate distribution. We can aso derive the exact rate distribution using the rate coverage probabiity P R γ), which is the probabiity that the achievabe rate of the typica user is arger than some threshod γ: P R γ) = PΓ > γ. The rate coverage probabiityp R γ) can be evauated through a change of variabes as in the foowing emma. Lemma 5: Given the SINR coverage probabiity P c T), for γ < W og N +T max ), the rate coverage probabiity can be computed as P R γ) = P c 2 γ/w ). Proof: The proof is simiar to that of 3, Theorem. For γ < W og N +T max ), it directy foows that P R γ) = P SINR > 2 γ/w = P c 2 γ/w ). Lemma 5 wi aow comparisons to be made between mmwave and conventiona systems that use different bandwidths, as presented in Section V-A. D. Simpification of LOS Probabiity Function The expressions in Theorem generay require numerica evauation of mutipe integras, and may become difficut to anayze. In this section, we propose to simpify the anaysis by approximating a genera LOS probabiity function pt) by a step function. We denote the step function as S RB x), where S RB x) = when < x < R B, and S RB x) = otherwise. Essentiay, the LOS probabiity of the ink is taken to be one within a certain fixed radius R B and zero outside the radius. An interpretation of the simpification is that the irreguar geometry of the LOS region in Fig. 2 a) is repaced with its equivaent LOS ba in Fig. 2 b). Such simpification not ony provides efficient expressions to compute SINR, but enabes simper anaysis of the network performance when the network is dense. We wi propose two criterions to determine the R B given LOS probabiity function pt). Before that, we first review some usefu facts. Theorem 2: Given the LOS probabiity function px), the average number of LOS base stations that a typica user observes is ρ = 2πλ pt)tdt.

7 7 Actua LOS region a) Irreguar shape of an acuta LOS region. RB b) Approximation using the equivaent LOS ba. Fig. 2. Simpification of the random LOS region as a fixed equivaent LOS ba. In a), we iustrate one reaization of randomy ocated buidings corresponding to a genera LOS function px). The LOS region observed by the typica user has an irreguar shape. In b), we approximate px) by a step function. Equivaenty, the LOS region is aso approximated by a fixed ba. Ony base stations inside the ba are considered LOS to the user. Proof: The average number of LOS base stations can be computed as a) ρ = E IX Φ L ) = 2πλ pt)tdt, X Φ where a) foows directy from Campbe s formua of PPP 26. A direct coroary of Theorem 2 foows as beow. Coroary 2.: When px) = S R x), the average number of LOS base stations is ρ = πλr 2. Note that Theorem 2 aso indicates that a typica user wi observe a finite number of LOS base stations amost surey when pt)tdt <. Hence, if px) satisfies pt)tdt <, the parameter R B in S RB x) can be determined by matching the average number of LOS base stations a user may observe. Criterion Mean LOS BS Number): When pt)tdt <, the parameter R B of the equivaent step function S RB x) is determined to match the first moment of ρ. By Theorem 2, it foows that R B = 2 pt)tdt ).5. In the case where pt)tdt < is not satisfied, another criterion to determine R B is needed. Note that even if the first moment is infinite, the probabiity that the user is associated with a LOS base station exists and is naturay finite for a pt). Hence, we propose the second criterion regarding the LOS association probabiity as foows. Criterion 2 LOS Association Probabiity): Given a LOS probabiity function pt), the parameter R B of its equivaent step function S RB x) is determined such that the LOS association probabiity A L is unchanged after approximation. From Lemma 2, the LOS association probabiity for a step function S RB x) equas e λπr2 B. Hence, by Criterion 2, n AL) λπ ).5. R B can be determined as R B = Last, we expain the physica meaning of the step function approximation as foows. As shown in Fig. 2a), with a genera LOS probabiity functionpx), the buidings are randomy ocated, and thus the actua LOS region observed by the typica user may have an unusua shape. Athough it is possibe to incorporate such randomness of the size and shape by integrating over pt), the expressions with mutipe integras can make the anaysis and numerica evauation difficut. In Fig. 2b), by approximating the LOS probabiity function as a step functions RB x), we equivaenty approximate the LOS region by a fixed ba B,R B ), which we define as the equivaent LOS ba. As wi be shown in Section IV, approximating px) as a step function enabes fast numerica computation, simpifies the anaysis, and provides design insights for dense network. Besides, we wi show in simuations in Section V-A that the error due to such approximation is generay sma in dense mmwave networks, which aso motivates us to use this first-order approximation of the LOS probabiity function to simpify the dense network anaysis in the foowing section. IV. ANALYSIS OF DENSE MMWAVE NETWORKS In this section we speciaize our resuts to dense networks. This approach is motivated by subsequent numerica resuts in Section V-A that show mmwave depoyments wi be dense if they are expected to achieve significant coverage. We derive simpified expressions for the SINR and provide further insights into system performance in this important asymptotic regime. A. Dense Network Mode In this section, we buid the dense network mode by modifying the system mode in Section II with a few additiona assumptions. We say that a mmwave ceuar network is dense if the average number of LOS base stations observed by the typica user ρ is arger than K, or if its LOS association probabiity A L is arger than ǫ, where K and ǫ are pre-defined positive threshods. In this paper, for iustration purpose, we wi et K = and ǫ = 5%. Further, we say that a network is utra-dense when ρ >. Note that ρ aso equas the reative base station density normaized by the average LOS area, in this specia case, as we wi expain beow. Now we make some additiona assumptions that wi aow us to further simpify the network mode. Assumption LOS equivaent ba): The LOS region of the typica user is approximated by its equivaent LOS ba B,R B ) as defined in Section III-D. By Assumption, the LOS probabiity function pt) is approximated by its equivaent step function S RB x), and the

8 8 LOS base station process Φ L is made up of the outdoor base stations that are ocated inside the LOS ba B,R B ). Noting that the outdoor base station process Φ is a homogeneous PPP with density λ, the average number of LOS base stations is ρ = λπrb 2, which is the outdoor base station density times the area of the LOS region. For ease of iustration, we ca ρ the the reative density of a mmwave network. The reative density ρ is equivaenty: i) the average number of LOS base stations that a user wi observe, ii) the ratio of the average LOS area πrb 2 to the size of a typica ce /λ 26, and iii) the normaized base station density by the size of the LOS ba. We wi show in the next section that the SINR coverage in dense networks is argey determined by the reative density ρ. Assumption No NLOS and noise): Both NLOS base stations and therma noise are ignored in the anaysis since in the dense regime, the performance is imited by other LOS interferers. We show ater in the simuations that ignoring NLOS base stations and the therma noise introduces a negigibe error in the performance evauation. Assumption 2 No Sma-scae fading): Sma-scae fading is ignored in the dense network anaysis, as the signa power from a nearby mmwave LOS transmitter is found to be amost deterministic in measurements 5. Based on the dense network mode, the signa-tointerference ratio SIR) can be expressed as M t M r R αl SIR = :X Φ B,R D. 6) B) R αl Now we compute the SIR distribution in the dense network mode. B. Coverage Anaysis in Dense Networks Now we present an approximation of the SINR distribution in a mmwave dense network. Our main resut is summarized in the foowing theorem. Theorem 3: The SINR coverage probabiity in a dense network can be approximated as N ) 4 P c T) ρe ρ exp 2 = ) + N k= ηtā k ) 2 α L Γ 2 ;ηtā k,ηtā k t αl 2 α L b k ρt α L )) dt, 7) where Γs;a,b) = b a xs e x dx is the incompete gamma function, ā k = a k /M t M r ), a k and b k are defined in Tabe I, η = NN!) N, and N is the number of terms used in the approximation. Proof: See Appendix D. When α L = 2, the expression in Theorem 3 can be further simpified as foows. Coroary 3.: When α L = 2, the SINR coverage probabiity approximatey equas N ) N 4 P c T) ρe ρ ) + expρb k = k= e ηtā k t te ηtā k)) e µηtā kt e µηtā k ) ηtbk ā k ρt dt, 8) where µ = e.577. The resuts in Theorem 3 generay provide a cose approximation of the SINR distribution when enough terms are used, e.g. when N 5, as wi be shown in Section V-B. More importanty, we note that the expressions in Theorem 4 are very efficient to compute, as most numerica toos support fast evauation of the gamma function in 7), and 8) ony requires a simpe integra over a finite interva. Besides, given the path oss exponent α L and the antenna geometry a k, b k, Theorem 3 shows that the approximated SINR is ony a function of the reative density ρ, which indicates the SIR distribution in a dense network is mosty determined on the average number of LOS base station to a user. C. Asymptotic Anaysis in Utra-Dense Networks To obtain further insights into coverage in dense networks, we provide resuts on the asymptotic SIR distribution when the reative density ρ becomes arge. We use this distribution to answer the foowing questions: i) What is the asymptotic SIR distribution when the network becomes extremey dense? ii) Does increasing base station density aways improve SIR in a mmwave network? First, we present the main asymptotic resuts as foows. Theorem 4: In a dense network, when the LOS path oss exponent α L 2, the SIR converges to zero in probabiity, as ρ. When α L > 2, the SIR converges to a nonzero random variabe SIR in distribution, asρ ; Based on 3, Proposition, a ower bound of the coverage probabiity for the asymptotic SIR is that for T >, PSIR > T) α LT 2/αL 2πsin2π/α L ). Proof: See Appendix E. Note that Theorem 4 indicates that increasing base station density above some threshod wi hurt the system performance, and that the SINR optima base station density is finite. Now we provide an intuitive expanation of the asymptotic resuts as foows. When increasing the base station density, the distances between the user and base stations become smaer, and the user becomes more ikey to be associated with a LOS base station. When the density is very high, however, a user sees severa LOS base stations and thus experiences significant interference. We note that the asymptotic trends in Theorem 4 are vaid when base stations are a assumed to be active in the network. A simpe way to avoid over-densification is to simpy turn off a fraction of the base stations. This is a simpe kind of interference management; study of more advanced interference management concepts is an interesting topic for future work. V. NUMERICAL SIMULATIONS In this section, we first present some numerica resuts based on our anayses in Section III and Section IV. We concude with some simuations using rea buiding distributions to vaidate our proposed mmwave network mode.

9 9 A. Genera Network Simuations In this section, we provide numerica simuations to vaidate our anaytica resuts in Section III, and further discuss their impications on system design. We assume the mmwave network is operated at 28 GHz, and the bandwidth assigned to each user is W = MHz. The LOS and NLOS path oss exponents are α L = 2 and α N = 4. The parameters of the Nakagami fading are N L = 3 and N N = 2. We assume the LOS probabiity function is px) = e βx, where /β = 4.4 meters. For the ease of iustration, we define the notion of the average ce radius of a network as foows. Note that if the base station density is λ, the average ce size in the network is /λ 26. Therefore, the average ce radius r c in a network is defined as the radius of a ba that has the size of an average ce, i.e., r c = /πλ. The average ce radius not ony directy reates to the inter-site distance that is used by industry in base station panning, but aso equivaenty characterizes the base station density in a network; as a arge average ce size indicates a ow base station density in the network. Association Probabiity LOS Association Prob. NLOS Association Prob Avg. ce radius in meters Fig. 4. LOS association probabiity with different average ce radii. The ines are drawn from Monte Caros simuations, and the marks are drawn based on Lemma 2..9 SINR Coverage Probabiity Mt,mt,θt)= db, db, 3 ) Mt,mt,θt)=2 db, db, 3 ) Mt,mt,θt )= db, db, 45 ) SINR threshod in db SINR coverage probabiity Simu: r =5 m c Simu: r c = m Simu: r =2 m c Simu: r =3 m c Anay: r =5 m c Anay: r c = m Anay: r =2 m c Anay: r =3 m c SINR threshod in db a) Anaytica bounds using Theorem. Fig. 3. SINR coverage probabiity with different antenna geometry. The average ce radius is r c = meters. The receiver beam pattern is fixed as G db, db,9. First, we compare the SINR coverage probabiities with different transmit antenna parameters in Fig. 3 using Monte Caros simuations. As shown in Fig. 3, when the side obe gain m t is fixed, better SINR performance is achieved by increasing main obe gain M t and by decreasing the main obe beamwidth θ t, as indicated by the anaysis in Section III-A. Next, we compare the LOS association probabiities A L with different average ce radii in Fig. 4. The resuts show that the probabiity that a user is associated with a LOS base station increases as the ce radius decreases. The resuts in Fig. 4 aso indicate that the received signa power wi be mosty determined by the distribution of LOS base stations in a sufficienty dense network, e.g. when the average ce size is smaer than meters in the simuation. We aso compare the SINR coverage probabiity with different ce radii in Fig. 5. The numerica resuts in Fig. 5 a) show SINR Coverage Probabiity SINR: r c = m SIR: r c = m SINR: r c =5 m SIR: r c =5 m SINR: r c =2 m SIR: r c =2 m SINR threshod in db b) Comparison between SINR and SIR. Fig. 5. SINR coverage probabiity with different average ce sizes. The transmit antenna pattern is assumed to be G db,db,3. In a), anaytica resuts from Theorem are shown to provide a tight approximation. In b), it shows that SIR converges to SINR when the base station density becomes high, which impies that mmwave networks can be interference-imited.

10 SINR Coverage Probabiity r = m, step function c r = m, pt)=e β t c r =5 m, step function c r c =5 m, pt)=e β t Rate coverage probabiity mmwave: r c =5 m mmwave: r c = m mmwave: r c =2 m mmwave: r c =3 m Microwave: 2 MHz Microwave: MHz SINR threshod in db Achievabe rate in Mbps Fig. 6. Comparison of the SINR coverage between using px) and its equivaent step function S RB x). The transmit antenna pattern is assumed to be G 2dB, db,3. It shows that the step function tends to provide a more pessimistic SINR coverage probabiity, but the gap becomes smaer as the network becomes more dense. Fig. 7. Rate coverage comparison between mmwave and conventiona ceuar networks. The mmwave transmit antenna pattern is assumed to be G db, db,3. We assume the conventiona system is operated at 2 GHz with a ce radius of 5 m, and the transmit power of the conventiona base station is 46 dbm. that our anaytica resuts in Theorem match the simuations we with negigibe errors. Unike in a interference-imited conventiona ceuar network, where SINR is amost invariant with the base station density 6, the mmwave SINR coverage probabiity is aso shown to be sensitive to the base station density in Fig. 5. The resuts in Fig. 5 a) aso shows that mmwave networks generay require a sma ce radius equivaenty a high base station density) to achieve acceptabe SINR coverage. Moreover, the resuts in Fig. 5 b) show that when decreasing average ce radius i.e., increasing base station density), mmwave networks wi transit from powerimited regime into interference-imited regime; as the SIR curves wi converge to the SINR curve when densifying the network. Specificay, comparing the curves for r c = 2 meters and r c = 3 meters in Fig. 5 a), we find that increasing base station density generay improve the SINR in a sparse network; as increasing base station density wi increase the LOS association probabiity and avoid the presence of coverage hoes, i.e. the cases that a user observes no LOS base stations. A comparison of the curves for r c = meters and r c = 5 meters, however, aso indicates that increasing base station density need not improve SINR, especiay when the network is aready sufficienty dense. Intuitivey, increasing base station density aso increases the ikeihood to be interfered by strong LOS interferers. In a sufficienty dense network, increasing base station wi harm the SINR by adding more strong interferers. Now we appy Theorem 3 to compare the SINR coverage with different LOS probabiity functions px). We approximate the negative exponentia function px) = e βx by its equivaent step function S RB x). Appying either of the criteria in Section III-D, the radius of the equivaent LOS ba R B equas 2 meters. As shown in Fig. 6, the step function approximation generay provides a ower bound of the actua SINR distribution, and the errors due to the approximation become smaer when the base station density increases. The approximation of step function aso enabes faster evauations of the coverage probabiity, as it simpifies expressions for the numerica integras. We provide rate resuts in Fig. 7, where the ines are drawn from Monte Caros simuations, and the marks are drawn based on Lemma 5. In the rate simuation, we assume that 64 QAM is the highest consteation supported in the networks, and thus the maximum spectrum efficiency per data stream is 6 bps/hz. In Fig. 7, we compare the rate coverage probabiity between the mmwave network and a conventiona network operated at 2 GHz. The mmwave bandwidth is MHz which conceivaby coud be much arger, e.g. 5 MHz 4, 5), whie we assume the conventiona system has a basic bandwidth of 2 MHz, which can be potentiay extended to MHz by enabing carrier aggregation 33. Rayeigh fading is assumed in the UHF network simuations. We further assume that conventiona base stations have perfect channe state information, and appy spatia mutipexing 4 4 singe user MIMO with zero-forcing precoder) to transmit mutipe data streams. More comparison resuts with other techniques can be found in 34. Resuts in Fig. 7 shows that, due to the favorabe SINR distribution and arger avaiabe bandwidth at mmwave frequencies, the mmwave system with a sufficienty sma average ce size outperforms the conventiona system in terms of providing high data rate coverage. B. Dense Network Simuations Now we show the simuation resuts based on the dense network anaysis in Section IV. First, we iustrate the resuts in Theorem 3 with the simuations in Fig. 8. In the simuations, we incude the NLOS base stations and therma noise, which were ignored in the theoretica derivation. The expression derived in Theorem 3 generay provides a ower bound of the coverage probabiity. The approximation becomes more accurate when more terms are used in the approximation,

11 especiay when N 5. We find that the error due to ignoring NLOS base stations and therma noise is minor in terms of the SINR coverage probabiity, primariy impacting ow SINRs. weaker, which motivates a denser depoyment of base stations in the network with higher path oss..9 SINR coverage probabity Simu: LOS+ NLOS + Noise Simu: LOS ony Anay: N= Anay: N=3 Anay: N=5 Anay: N= SINR coverage probabiity at 2 db Simuation: LOS + NLOS+ Noise Anaytica using N=5 terms SINR threshod in db Fig. 8. Coverage probabiity in a dense mmwave network. The mmwave transmit antenna pattern is assumed to be G db, db,3. We assume R B = 2 m, and the reative base station density ρ = 4. N is the number of terms we used to approximate the coverage probabiity in Theorem 3. Next, we compare the SINR coverage probabiity with different reative base station density when T = 2 db. Reca that ρ = λπrb 2 is the base station density normaized by the size of the LOS region. In 9a), the path oss exponent is assumed to be α = 2. We compute the coverage probabiity from ρ = 2 db meters to ρ = 2 db with a step of db. The anaytica expressions in Theorem 3 are much more efficient than simuations: the pot takes seconds to finish using the anaytica expression, whie it approximatey takes an hour to simuate, reaizations at each step. As shown in Fig. 9 a), athough there is some gap between the simuation and the anaytica resuts in the utra-dense network regime, both curves achieve their maxima at approximatey ρ = 5, i.e., when the average ce radius r c is approximatey /2 of the LOS range R B. Moreover, when the base station density grows very arge, the coverage probabiity begins to decrease, which matches the asymptotic resuts in Theorem 4. The resuts aso indicate that networks in the environments with dense bockages, e.g. the downtown areas of arge cities where the LOS range R B is sma, wi benefit from network densification; as they are mosty operated in the region where the reative density is much) smaer than the optima vaue ρ 5, and thus increasing ρ by densifying networks wi improve SINR coverage. We aso simuate with other LOS path oss exponents in Fig. 9 b). The resuts show that the optima base station density is generay insensitive to the change of the path oss exponent. When the LOS path oss exponent increases from.5 to 2.5, the optima ce size is amost the same. The resuts aso iustrate that the networks with arger path oss exponent α L have better SINR coverage in the utra-dense regime when ρ >. Intuitivey, signas attenuate faster with a arger path oss exponent, and thus the inter-ce interference becomes SINR coverage probabiity at 2 db 2 2 Reative base station density ρ a) Anaytica resuts using Theorem 3. LOS path oss exponent: α L =2.5 LOS path oss exponent: α L =2 LOS path oss exponent: α L =.5. 2 Reative base station density ρ b) Optima density with different path oss exponents. Fig. 9. SINR coverage probabiity with different reative base station density when the target SINR=2 db. In the simuations, we incude the NLOS base stations outside the LOS region and the therma noise. We aso fix the radius of the LOS ba as R B =2 meters, and change the base station density λ at each step according to the vaue of the reative base station density ρ. In a), it shows that ignoring NLOS base stations and the noise power causes minor errors in terms of the optima ce radius. In b), we search for the optima reative density with different LOS path oss exponents. It shows that the optima ce radius is generay insensitive to the path oss exponent. Finay, we compare the spectra efficiency and average achievabe rates as a function of the reative density ρ in Tabe II. We find with a reasonabe amount of density, e.g. when the reative density ρ is approximatey, the mmwave system can provide comparabe spectrum efficiency as the conventiona system at UHF frequencies. With high density, rates that can be achieved are an order of magnitude better than that in the conventiona networks, due to the favourabe SINR distribution and arger avaiabe bandwidth at mmwave frequencies. C. Comparison with Rea-scenario Simuations Now we compare our proposed network modes with the simuations using rea data. In the rea-scenario simuations,

12 2 TABLE II ACHIEVABLE RATE WITH DIFFERENT BS DENSITIES Carrier frequency 28 GHz 28 GHz 28 GHz 28 GHz 2 GHz 2 GHz Base station density Utra dense Dense Intermediate Sparse - - Reative density ρ Spectrum efficiency bps/hz) Signa bandwith MHz) 2 Achievabe rate Mpbs) SINR Coverage Probabiity a) Snapshot of the simuated area from Googe map. Rea buiding distribution Proposed genera network mode: pr)=e β r Equivaent LOS ba mode: R B =225 m SINR Threshod in db b) Comparison of SINR distribution Fig.. Comparison of SINR coverage resuts with rea-scenario simuations. The snapshot of The University of Texas at Austin campus is from Googe map. We use the actua buiding distribution of the area in the rea-scenario simuation. In the simuations of our proposed anaytica modes, we et β =.63 m in the LOS probabiity function pr) = e βr, and R B = 225 m in the simpified equivaent LOS ba mode, to match the buiding statistics in the area 24. we use the buiding distribution on the campus at The University of Texas at Austin. We aso appy a modified version of the base station antenna pattern in 35 with a smaer beam width of 3. The directivity gain at the base station is M t = 2 db. The mobie station is assumed to use uniform inear array with 4 antennas. When appying our anaytica modes, we fit the parameters of the LOS probabiity functions to match the buiding statistics 24, and use the sectored mode for beamforming pattern. We aso assume the mmwave base stations are distributed as a PPP with r c = 5 m. As shown in Fig., though some deviations in the high SINR regime, our anaytica modes generay show a cose characterization of the reaity. The deviation is expained as foows: the proposed anaytica mode computes the aggregated SINR coverage probabiity, averaging over a reaizations of buiding distributions over the infinite pane, whie the rea-scenario curve ony considers a specific reaization of buidings in a finite snapshot window. In this case, our mode overestimates the coverage probabiity in the ow SINR regime, and underestimates in the high SINR regime, as both signas and interference become more ikey to be bocked in the rea scenario simuation. We have found in other simuation exampes that the reverse can aso be true. Our mode shoud be viewed as a characterization of the average distribution and does not necessariy ower or upper bound the distribution for a given reaization. VI. CONCLUSIONS In this paper, we proposed a stochastic geometry framework to anayze coverage and rate in mmwave networks for outdoor users and outdoor infrastructure. Our mode took bockage effects into account by appying a distance-dependent LOS probabiity function, and modeing the base stations as independent inhomogeneous LOS and NLOS point processes. Based on the proposed framework, we derived expressions for the downink SINR and rate coverage probabiity in mmwave ceuar networks, which were shown to be efficient in computation and aso a good fit with the simuations. We further simpified the bockage mode by approximating the random LOS region as a fixed-size equivaent LOS ba. Appying the simpified framework, we anayzed the performance and asymptotic trends in dense networks. We used numerica resuts to draw severa important concusions about coverage and rate in mmwave networks. SINR coverage can be comparabe to conventiona networks at UHF frequency when the base station density is sufficienty high. Achievabe rates can be significanty higher than in conventiona networks, thanks to the arger avaiabe bandwidth. The SINR and rate performance is argey determined by the reative base station density, which is the ratio of the base station density to the bockage density. A transition from a power-imited regime to an interference-imited regime is aso observed in mmwave