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Coverage and Rate Analysis forMillimeter Wave Cellular Networks

Tianyang Bai and Robert W. Heath, Jr.

Abstract—Millimeter wave (mmWave) holds promise as acarrier frequency for fifth generation cellular networks. BecausemmWave signals are sensitive to blockage, prior models forcellular networks operated in the ultra high frequency (UHF)band do not apply to analyze mmWave cellular networks di-rectly. Leveraging concepts from stochastic geometry, this paperproposes a general framework to evaluate the coverage and rateperformance in mmWave cellular networks. Using a distance-dependent line-of-site (LOS) probability function, the locationsof the LOS and non-LOS base stations are modeled as twoindependent non-homogeneous Poisson point processes, to whichdifferent path loss laws are applied. Based on the proposedframework, expressions for the signal-to-noise-and-interferenceratio (SINR) and rate coverage probability are derived. ThemmWave coverage and rate performance are examined as afunction of the antenna geometry and base station density.The case of dense networks is further analyzed by applying asimplified system model, in which the LOS region of a user isapproximated as a fixed LOS ball. The results show that densemmWave networks can achieve comparable coverage and muchhigher data rates than conventional UHF cellular systems, despitethe presence of blockages. The results suggest that the cellsizeto achieve the optimal SINR scales with the average size of thearea that is LOS to a user.

I. I NTRODUCTION

The large available bandwidth at millimeter wave(mmWave) frequencies makes them attractive for fifth gen-eration cellular networks [3]–[5]. The mmWave band rangingfrom 30 GHz to 300 GHz has already been considered in var-ious commercial wireless systems including IEEE 802.15.3cfor personal area networking [6], IEEE 802.11ad for local areanetworking [7], and IEEE 802.16.1 for fixed-point access links[8]. Recent field measurements reveal the promise of mmWavesignals for the access link (between the mobile station and basestation) in cellular systems [5], [9].

One differentiating feature of mmWave cellular commu-nication is the use of antenna arrays at the transmitter andreceiver to provide array gain. As the wavelength decreases,antenna sizes also decrease, reducing the antenna aperture. Forexample, from the Friis free-space equation [10], a mmWavesignal at 30 GHz will experience 20 dB larger path loss thana signal at 3 GHz. Thanks to the small wavelength, however,it is possible to pack multiple antenna elements into the

The authors are with The University of Texas at Austin, Austin, TX, USA.(email: tybai@utexas.edu, rheath@utexas.edu) This work is supported in partby the National Science Foundation under Grants No. 1218338and 1319556,and by a gift from Huawei Technologies, Inc.

Preliminary results related to this paper were presented atthe 1st IEEEGlobal Conference on Signal and Information Processing (GlobalSIP) [1] andthe 47th Annual Asilomar Conference on Signals, Systems, and Computers(Asilomar) [2].

limited space at mmWave transceivers [3]. With large antennaarrays, mmWave cellular systems can implement beamformingat the transmitter and receiver to provide array gain thatcompensates for the frequency-dependent path loss, overcomesadditional noise power, and as a bonus also reduces out-of-cellinterference [4].

Another distinguishing feature of mmWave cellular com-munication is the propagation environment. MmWave signalsare more sensitive to blockage effects than signals in lower-frequency bands, as certain materials like concrete wallsfound on building exteriors cause severe penetration loss [11].This indicates that indoor users are unlikely to be coveredby outdoor mmWave base stations. Channel measurementsusing directional antennas [5], [9], [12] have revealed anotherinteresting behavior at mmWave: blockages cause substantialdifferences in the line-of-sight (LOS) paths and non-line-of-sight (NLOS) path loss characteristics. Such differences havealso been observed in prior propagation studies at ultra highfrequency bands (UHF) from 300 MHz to 3 GHz, e.g. see[13]. The differences, however, become more significant formmWave since diffraction effects are negligible [4], and thereare only a few scattering clusters [14]. Measurements in [5],[9], [12] showed that mmWave signals propagate as in freespace with a path loss exponent of 2. The situation wasdifferent for NLOS paths where a log distance model wasfit with a higher path loss exponent and additional shadowing[5], [9]. The NLOS path loss laws tend to be more dependenton the scattering environment. For example, an exponent aslarge as 5.76 was found in downtown New York City [5],while only 3.86 was found on the UT Austin campus [9]. Thedistinguishing features of the propagation environment need tobe incorporated into the any comprehensive system analysisofmmWave networks.

The performance of mmWave cellular networks was simu-lated in prior work [14], [15] using insights from propagationchannel measurements [5]. In [15], using the NLOS path losslaw measured in the New York City, lower bounds of thesignal-to-noise-and-interference ratio (SINR) distribution andthe achievable rate were simulated in a 28 GHz pico-cellularsystem. In [14], a mmWave channel model that incorporatedblockage effects and angle spread was proposed and furtherapplied to simulate the mmWave network capacity. Bothresults in [14], [15] show that the achievable rate in mmWavenetworks can outperform conventional cellular networks intheultra high frequency (UHF) band by an order-of-magnitude.The simulation-based approach [14], [15] does not lead toelegant system analysis as in [16], which can be broadlyapplied to different deployment scenarios.

2

Stochastic geometry is a useful tool to analyze systemperformance in conventional cellular networks [16], [17].In[16], by modeling base station locations in a conventionalcellular network as a Poisson point process (PPP) on theplane, the aggregate coverage probability was derived in asimple form, e.g. a closed-form expression when the path lossexponent is 4. Moreover, the stochastic model was shownto provide a lower bound of the performance in a realcellular system [16]. There have been several extensions oftheresults in [16], such as analyzing a multi-tier network in [18]and predicting the site-specific performance in heterogeneousnetworks in [19]. It is not possible to directly apply resultsfrom conventional networks to mmWave networks due to thedifferent propagation characteristics and the use of directionalbeamforming. There has been limited application of stochasticgeometry to study mmWave cellular networks. The primaryrelated work was in [20], where directional beamforming wasincorporated for single and multiple user configurations, but asimplified path loss model was used that did not take mmWavepropagation features into account.

A systematic study of mmWave network performanceshould incorporate the impact of blockages such as buildingsin urban areas. One approach is to model the blockagesexplicitly in terms of their sizes, locations, and shapes usingdata from a geographic information system. This approach iswell suited for site-specific simulations [21] using electromag-netic simulation tools like ray tracing [22]. An alternativeis to employ a stochastic blockage model, e.g. [23], [24],where the blockage parameters are drawn randomly accordingto some distribution. The stochastic approach lends itselfbetter to system analysis and can be applied to study systemdeployments under a variety of blockage parameters such assize and density.

The main contribution of this paper is to propose a stochas-tic geometry framework for analyzing the coverage and ratein mmWave cellular networks. As a byproduct, the frameworkalso applies to analyze heterogenous networks in which thebase stations are distributed as certain non-homogeneous PPPs.We incorporate directional beamforming by modeling thebeamforming gains as marks of the base station PPPs. Fortractability of the analysis, the actual beamforming patterns arealso approximated by a sectored model, which characterizeskey features of an antenna pattern: directivity gain, half-powerbeamwidth, and front-back ratio. A similar model was alsoemployed in work on ad hoc networks [25]. To incorporateblockage effects, we model the probability that a communica-tion link is LOS as a function of the link length, and provide astochastic characterization of the region where a user doesnotexperience any blockage, which we define as theLOS region.Applying the distance-dependent LOS probability function, thebase stations are equivalently divided into two independentnon-homogenous point processes on the plane: the LOS andthe NLOS base station processes. Different path loss lawsand fading are applied separably to the LOS and NLOS case.Based on the system model, expressions for the SINR andrate coverage probability are derived in general mmWavenetworks. To simplify the analysis, we also propose a sys-tematic approach to approximate a complicated LOS function

as its equivalent step function. Our analysis indicates that thecoverage and rate are sensitive to the density of base stationsand the distribution of blockages in mmWave networks. It alsoshows that dense mmWave networks can generally achievegood coverage and significantly higher achievable rate thanconventional cellular networks.

A simplified system model is proposed to analyze densemmWave networks, where the infrastructure density is com-parable to the blockage density. For a general LOS function,the LOS region observed by a user has an irregular and randomshape. Coverage analysis requires integrating the SINR overthis region [1]. We propose to simplify the analysis by ap-proximating the actual LOS region as a fixed-sized ball calledthe equivalent LOS ball. The radius of the equivalent LOSball is chosen so that the ball has the same average number ofLOS base stations in the network. With the simplified networkmodel, we find that in a dense mmWave network, the cellradius should scale with the size of LOS region to maintainthe same coverage probability. We find that continuing toincrease base station density (leading to what we call ultra-dense networks) does not always improve SINR, and theoptimal base station density should be finite.

Compared with our prior work in [1], this paper provides ageneralized mathematical framework and includes the detailedmathematical derivations. The system model applies for ageneral LOS probability function and includes the impact ofgeneral small-scale fading. We also provide a new approachto compute coverage probability, which avoids inverting theFourier transform numerically and is more efficient than priorexpressions in [1]. Compared with our prior work in [2], wealso remove the constraint that the LOS path loss exponent is2, and extend the results in [2] to general path loss exponents,in addition to providing derivations for all results, and newsimulation results.

This paper is organized as follows. We introduce the systemmodel in Section II. We derive expressions for the SINR andrate coverage in a general mmWave network in Section III. Asystematic approach is also proposed to approximate generalLOS probability functions as a step function to further simplifyanalysis. In Section IV, we apply the simplified system modelto analyze performance and examine asymptotic trends indense mmWave networks, where outdoor users observe morethan one LOS base stations with high probability. Finally,conclusions and suggestions for future work are provided inSection VI.

II. SYSTEM MODEL

In this section, we introduce our system model for evalu-ating the performance of a mmWave network. We focus ondownlink coverage and rate experienced by an outdoor user,as illustrated in Fig. 1(a). We make the following assumptionsin our mathematical formulation.

Assumption 1 (Blockage process):The blockages, typicallybuildings in urban areas, form a process of random shapes,e.g. a Boolean scheme of rectangles [24], on the plane. Weassume the distribution of the blockage process to be stationaryand isotropic - in other words - invariant to the motions oftranslation and rotation [26, Chapter 10].

3

(a) System model for mmWave cellular networks

Mmθ

(b) Sectored model to approximate beamforming patterns.

Fig. 1. In (a), we illustrate the proposed system model for mmWave cellularnetworks. Blockages are modeled as a random process of rectangles, whilebase stations are assumed to be distributed as a Poisson point process on theplane. An outdoor typical user is fixed at the origin. The basestations arecategorized into three groups: indoor base stations, outdoor base stations thatare LOS to the typical user, and outdoor base stations that are NLOS to theuser. Directional beamforming is performed at both base stations and mobilestations to exploit directivity gains. In (b), we illustrate the sectored antennamodelGM,m,θ(φ), which is used to approximate the beamforming patterns.

Assumption 2 (PPP BS):The base stations form a homo-geneous PPPΦ with densityλ on the plane. Note that a basestation can be located either inside a blockage or outside ablockage. In this paper, however, we will focus on the SINRand rate provided by the outdoor base stations as the blockagesare assumed to be impenetrable. LetΦ = {Xℓ} be the pointprocess of outdoor base stations,Xℓ the ℓ-th outdoor basestation, andRℓ = |OXℓ| denote the distance fromℓ-th basestation to the originO. Define τ as the average fraction ofthe land covered by blockages, i.e., the average fraction ofindoor area in the network. Further, we assume the base stationprocessΦ is independent of the blockage process. Therefore,each base station has an i.i.d. probability1 − τ to be locatedoutdoor. By the thinning theorem of PPP [26], the outdoorbase station processΦ is a PPP of densityλ = (1 − τ)λ onthe plane. In addition, all base stations are assumed to haveaconstant transmit powerPt.

Assumption 3 (Outdoor user):The users are distributed asa stationary point process independent of the base stationsand blockages on the plane. A typical user is assumed tobe located at the originO, which is a standard approachin the analysis using stochastic geometry [16], [26]. Bythe stationarity and independence of the user process, the

downlink SINR and rate experienced by the typical user havethe same distributions as the aggregate ones in the network.The typical user is assumed to be outdoors. The indoor-to-outdoor penetration loss is assumed to be high enough suchthat an outdoor user can not receive any signal or interferencefrom an indoor base station. Therefore, the focus in this paperis on investigating the conditional SINR and rate distributionof the outdoor typical user served by outdoor infrastructure.Indoor users can be served by either indoor base stationsor by outdoor base stations operated at UHF frequencies,which have smaller indoor-to-outdoor penetration losses inmany common building materials. We defer the extension toincorporate indoor users to future work.

We say that a base station atX is LOS to the typical userat the originO if and only if there is no blockage intersectingthe linkOX . Due to the presence of blockages, only a subsetof the outdoor base stationsΦ are LOS to the typical user.

Assumption 4 (LOS and NLOS BS):An outdoor base sta-tion can be either LOS or NLOS to the typical user. LetΦL

be the point process of LOS base stations, andΦN = Φ/ΦL bethe process of NLOS base stations. Define theLOS probabilityfunctionp(R) as the probability that a link of lengthR is LOS.Noting the fact that the distribution of the blockage process isstationary and isotropic, the LOS probability function dependsonly on the length of the linkR. Also,p(R) is a non-increasingfunction ofR; as the longer the link, the more likely it will beintersected by one or more blockages. The NLOS probabilityof a link is 1− p(R).

The LOS probability function in a network can be derivedfrom field measurements [14] or stochastic blockage models[23], [24], where the blockage parameters are characterized bysome random distributions. For instance, when the blockagesare modeled as a rectangle boolean scheme in [24], it followsthat p(R) = e−βR, whereβ is a parameter determined by thedensity and the average size of the blockages, and1/β is whatwe called the average LOS range of the network in [24].

For the tractability of analysis, we further make the follow-ing independent assumption on the LOS probability; takingaccount of the correlations in blockage effects generally makesthe exact analysis difficult.

Assumption 5 (Independent LOS probability):The LOSprobabilities are assumed to be independent between differentlinks, i.e., we ignore potential correlations of blockage effectsbetween links.

Note that the LOS probabilities for different links arenot independent in reality. For instance, neighboring basestations might be blocked by a large building simultaneously.Numerical results in [24], however, indicated that ignoringsuch correlations cause a minor loss of accuracy in the SINRevaluation. Assumption 5 also indicates that the LOS basestation processΦL and the NLOS processΦN form twoindependent non-homogeneous PPP with the density functionsp(R)λ and (1 − p(R))λ, respectively, whereR is the radiusin polar coordinates.

Assumption 6 (Path loss model):Different path loss lawsare applied to LOS and NLOS links. Given a link has length

4

TABLE IPROBABILITY MASS FUNCTION OFDℓ AND Dℓ

k 1 2 3 4ak MrMt Mrmt mrMt mrmt

bk crct cr(1− ct) (1− cr)ct (1− cr)(1− ct)ek Mr Mr/ξt mr mr/ξt

R, its path loss gainL(R) is computed as

L(R) = I(p(R))CLR−αL + (1− I(p(R))CNR

−αN , (1)

whereI(x) is a Bernoulli random variable with parameterx,αL, αN are the LOS and NLOS path loss exponents, andCL, CN are the intercepts of the LOS and NLOS path lossformulas. Typical values of mmWave path loss exponents andintercept constants are available in prior work, see e.g. [5],[9]. The model could be further enhanced by including log-normal shadowing, but this is deferred in our paper to simplifythe analysis.

Assumption 7 (Directional beamforming):Antenna arraysare deployed at both base stations and mobile stations toperform directional beamforming. For tractability of the anal-ysis, the actual array patterns are approximated by a sectoredantenna model, which was used in prior ad hoc networkanalysis [25]. LetGM,m,θ(φ) denote the sectored antennapattern in Fig. 1(b), whereM is the main lobe directivity gain,m is the back lobe gain,θ is the beamwidth of the main lobe,andφ is the angle off the boresight direction. In the sectoredantenna model, the array gains are assumed to be constantM for all angles in the main lobe, and another constantmin the side lobe in the sectored model. We letMt, mt, andθt be the main lobe gain, side lobe gain, and half powerbeamwidth of the base station antenna, andMr, mr, andθr thecorresponding parameters for the mobile station. Without lossof generality, we denote the boresight direction of the antennasas 0◦. Further, letDℓ = GMt,mt,θt(φ

ℓt)GMr,mr,θr(φ

ℓr) be the

total directivity gain in the link from theℓ-th base station tothe typical user, whereφℓr andφℓt are the angle of arrival andthe angle of departure of the signal.

Assumption 8 (User association):The typical user is asso-ciated with the base station, either LOS or NLOS, that has thesmallest path lossL(Rℓ). The serving base station is denotedasX0. Both the mobile station and its serving base stationwill estimate channels including angles of arrivals and fading,and then adjust their antenna steering orientations accordinglyto exploit the maximum directivity gain. Errors in channelestimation are neglected, and so are errors in time and carrierfrequency synchronizations in our work. Thus, the directivitygain for the desired signal link isD0 = MrMt. For theℓ-th interfering link, the anglesφℓr and φℓt are assumed tobe independently and uniformly distributed in(0, 2π], whichgives a random directivity gainDℓ.

By Assumption 7 and Assumption 8, the directivity gainin an interference linkDℓ is a discrete random variable withthe probability distribution asDℓ = ak with probability bk(k ∈ {1, 2, 3, 4}), whereak and bk are constants defined inTable I, cr = θr

2π , andct = θt2π .

Assumption 9 (Small-scale fading):We assume indepen-

dent Nakagami fading for each link. Different parametersof Nakagami fadingNL andNN are assumed for LOS andNLOS links. Lethℓ be the small-scale fading term on theℓ-th link. Then |hℓ|

2 is a normalized Gamma random variable.Further, for simplicity, we assumeNL andNN are positiveintegers. We also ignore the frequency selectivity in fading,as measurements show that the delay spread is generallysmall [5], and the impact of frequency-selective fading canbeminimized by techniques like orthogonal frequency-divisionmultiplexing or frequency domain equalization [10].

Measurement results indicated that small-scale fading atmmWave is less severe than that in conventional systems whennarrow beam antennas are used [5]. Thus, we can use a largeNakagami parameterNL to approximate the small-variancefading as found in the LOS case. Letσ2 be the thermal noisepower normalized byPt. Based on the assumptions thus far,the SINR received by the typical user can be expressed as

SINR=|h0|

2MrMtL(R0)

σ2 +∑

ℓ>0:Xℓ∈Φ |hℓ|2DℓL(Rℓ)

. (2)

Note that the SINR in (2) is a random variable, due to therandomness in the base station locationsRℓ, small-scale fadinghℓ, and the directivity gainDℓ. Using the proposed systemmodel, we will evaluate the mmWave SINR and rate coveragein the following section.

III. C OVERAGE AND RATE ANALYSIS IN GENERAL

NETWORKS

In this section, we analyze the coverage and rate in theproposed model of a general mmWave network. First, weprovide some SINR ordering results regarding different param-eters of the antenna pattern. Then we derive expressions forthe SINR and rate coverage probability in mmWave networkswith general LOS probability functionp(R). To simplifysubsequent analysis, we then introduce a systematic approachto approximatep(R) by a moment matched equivalent stepfunction.

A. Stochastic Ordering of SINR With Different Antenna Ge-ometries

One differentiating feature of mmWave cellular networks isthe deployment of directional antenna arrays. Consequently,the performance of mmWave networks will depend on theadaptive array pattern through the beamwidth, the directivitygain, and the back lobe gain. In this section, we establish someresults on stochastic ordering of the SINRs in the systems withdifferent antenna geometries. While we will focus on the arraygeometry at the transmitter, the same results, however, alsoapply to the receiver array geometry. The concept of stochasticordering has been applied in analysis of wireless systems [27],[28]. Mathematically, the ordering of random variables canbedefined as follows [27], [28].

Definition 1: Let X and Y be two random variables.Xstochastically dominatesY , i.e., X has a better distributionthanY , if P(X > t) > P(Y > t) for all t ∈ R.

Next, define the front-to-back ratio (FBR) at the transmitterξt as the ratio between the main lobe directivity gainMt and

5

the back lobe gainmt, i.e., ξt = Mt/mt. We introduce thekey result on stochastic ordering of the SINR with respect tothe directivity gains as follows.

Proposition 1 (Stochastic ordering w.r.t. directivity gains):Given a fixed beamwidthθt and FBRξt at the transmitter,the mmWave network with the larger main lobe directivitygainMt has a better SINR distribution. Similarly, with fixedbeamwidth θt and main lobe gainMt, a larger FBR ξtprovides a better SINR distribution.

Proof: From Definition 1, we need to show that for eachrealization of base station locationsRℓ, small-scale fadinghℓ, and anglesφℓr and φℓt, the value of the SINR increaseswith Mt and ξt. Given Rℓ, hℓ, φℓr , and φℓr (ℓ ∈ N), wecan normalize both the numerator and denominator of (2) byMt, and then write SINR= |h0|

2MrL(R0)

σ2/Mt+∑

ℓ>0:Xℓ∈Φ Dℓ(ξt)|hℓ|2L(Rℓ)

,

where Dℓ(ξt) = ek with probability bk, and bk, ek areconstants defined in Table I. Note thatDℓ(ξt) is independentof Mt, and is a non-increasing function ofξt. Hence, whenξtis fixed, largerMt provides larger SINR; whenMt is fixed,largerξt provides larger SINR.

Next, we provide the stochastic ordering result regardingbeamwidth as follows.

Proposition 2 (Stochastic ordering w.r.t. beamwidth):Given a fixed main lobe gainMt and FBR ξt at thetransmitter, a smaller beamwidthθt provides a better SINRdistribution.

The proposition can be rigorously proved using couplingtechniques. We omit the proof here and instead provide anintuitive explanation as below. Intuitively, with narrower mainlobes, fewer base stations will transmit interference to thetypical user via their main lobes, which gives a smaller inter-ference power. The desired signal term in (2) is independentofthe beamwidth, as we ignore the channel estimation errors andpotential angle spread. Hence, based on our model assump-tions, smaller beamwidths provide a better SINR performance.

We note that the ordering result in Proposition 2 assumesthat there is no angle spread in the channel. With anglespread, a narrow-beam antenna may capture only the signalenergy arriving inside its main lobe, missing the energy spreadoutside, which causes a gain reduction in the signal power[29]. Consequently, the results in Proposition 2 should beinterpreted as applying to the case where beamwidths arelarger than the angle spread, e.g. if the beamwidth is morethan55◦ per the measurements in [12]. We defer more detailedtreatment of angle spread to future work.

B. SINR Coverage Analysis

The SINR coverage probabilityPc(T ) is defined as theprobability that the received SINR is larger than some thresh-old T > 0, i.e., Pc(T ) = P(SINR > T ). We present thefollowing lemmas before introducing the main results on SINRcoverage. By Assumption 4, the outdoor base station processΦcan be divided into two independent non-homogeneous PPPs:the LOS base station processΦL and NLOS processΦN. Wewill equivalently considerΦL andΦN as two independent tiersof base stations. As the user is assumed to connect to the basestation with the smallest path loss, the serving base station can

only be either the nearest base station inΦL or the nearest onein ΦN. The following lemma provides the distribution of thedistance to the nearest base station inΦL andΦN.

Lemma 1:Given the typical user observes at least one LOSbase station, the conditional probability density function of itsdistance to the nearest LOS base station is

fL(x) = 2πλxp(x)e−2πλ∫

x0rp(r)dr/BL, (3)

wherex > 0, BL = 1 − e−2πλ∫

∞

0rp(r)dr is the probability

that a user has at least one LOS base station, andp(r) isthe LOS probability function defined in Section II. Similarly,given the user observes at least one NLOS base station, theconditional probability density function of the distance to thenearest NLOS base station is

fN(x) = 2πλx(1 − p(x))e−2πλ∫ x0r(1−p(r))dr/BN, (4)

where x > 0, and BN = 1 − e−2πλ∫

∞

0r(1−p(r))dr is the

probability that a user has at least one NLOS base station.Proof: The proof follows [24, Theorem 10] and is omitted

here.Next, we compute the probability that the typical user is

associated with either a LOS or a NLOS base station.Lemma 2:The probability that the user is associated with

a LOS base station is

AL = BL

∫ ∞

0

e−2πλ∫ ψL(x)

0 (1−p(t))tdtfL(x)dx, (5)

whereψL(x) = (CN/CL)1/αN xαL/αN . The probability that

the user is associated with a NLOS base station isAN =1−AL.

Proof: See Appendix B.Further, conditioning on that the serving base station is LOS

(or NLOS), the distance from the user to its serving basestation follows the distribution given in the following lemma.

Lemma 3:Given that a user is associated with a LOS basestation, the probability density function of the distance to itsserving base station is

fL(x) =BLfL(x)

ALe−2πλ

∫ ψL(x)

0 (1−p(t))tdt, (6)

whenx > 0. Given the user is served by a NLOS base station,the probability density function of the distance to its servingbase station is

fN(x) =BNfN(x)

ANe−2πλ

∫ ψN(x)

0 p(t)tdt, (7)

wherex > 0, andψN(x) = (CL/CN)1/αL xαN/αL .

Proof: The proof follows a similar method as that ofLemma 2, and is omitted here.

Now, based on Lemma 2 and Lemma 3, we present themain theorem on the SINR coverage probability as follows

Theorem 1:The SINR coverage probabilityPc(T ) can becomputed as

Pc(T ) = ALPc,L(T ) +ANPc,N(T ), (8)

6

where for s ∈ {L,N}, Pc,s(T ) is the conditional coverageprobability given that the user is associated with a base stationin Φs. Further,Pc,s(T ) can be evaluated as

Pc,L(T ) ≈NL∑

n=1

(−1)n+1

(

NL

n

)

×

∫ ∞

0

e−nηLx

αLTσ2

CLMrMt−Qn(T,x)−Vn(T,x)fL(x)dx, (9)

and

Pc,N(T ) ≈NN∑

n=1

(−1)n+1

(

NN

n

)

×

∫ ∞

0

e−nηNx

αNTσ2

CNMrMt−Wn(T,x)−Zn(T,x)fN(x)dx. (10)

where

Qn(T, x) = 2πλ

4∑

k=1

bk

∫ ∞

x

F

(

NL,nηLakTx

αL

NLtαL

)

p(t)tdt,

(11)

Vn(T, x) =2πλ

4∑

k=1

bk

∫ ∞

ψL(x)

F

(

NN,nCNηLakTx

αL

CLNNtαN

)

(1− p(t))tdt, (12)

Wn(T, x) =2πλ

4∑

k=1

bk

∫ ∞

ψN(x)

F

(

NL,nCLηNakTx

αN

CNNLtαL

)

p(t)tdt, (13)

Zn(T, x) =2πλ

4∑

k=1

bk

∫ ∞

x

F

(

NN,nηNakTx

αN

NNtαN

)

(1− p(t))tdt, (14)

and F (N, x) = 1 − 1/(1 + x)N . For s ∈ {L,N}, ηs =

Ns(Ns!)− 1Ns , Ns are the parameters of the Nakagami small-

scale fading; fork ∈ {1, 2, 3, 4}, ak = akMtMr

, ak and bk areconstants defined in Table I.

Proof: See Appendix C.Though as an approximation of the SINR coverage prob-

ability, we find that the expressions in Theorem 1 comparefavorably with the simulations in Section V-A. In addition,the expressions in Theorem 1 compute much more efficientlythan prior results in [1], which required a numerical inverse ofa Fourier transform. Last, the LOS probability functionp(t)may itself have a very complicated form, e.g. the empiricalfunction for small cell simulations in [13], which will makethenumerical evaluation difficult. Hence, we propose simplifyingthe system model by using a step function to approximatep(t)in Section III-D. Before that, we introduce our rate analysisresults in the following section.

C. Rate Analysis

In this section, we analyze the distribution of the achievablerateΓ in mmWave networks. We use the following definitionfor the achievable rate

Γ =W log2(1 + min{SINR, Tmax}), (15)

whereW is the bandwidth assigned to the typical user, andTmax is a SINR threshold determined by the order of theconstellation and the limiting distortions from the RF circuit.The use of a distortion thresholdTmax is needed because ofthe potential for very high SINRs in mmWave that may notbe exploited due to other limiting factors like linearity intheradio frequency front-end.

The average achievable rateE[Γ] can be computed using thefollowing Lemma from the SINR coverage probabilityPc(T ).

Lemma 4:Given the SINR coverage probabilityPc(T ),the average achievable rate in the network isE [Γ] =Wln 2

∫ Tmax

0Pc(T )1+T dT.

Proof: See [16, Theorem 3] and [20, Section V].Lemma 4 provides a first order characterization of the rate

distribution. We can also derive the exact rate distributionusing the rate coverage probabilityPR(γ), which is theprobability that the achievable rate of the typical user islarger than some thresholdγ: PR(γ) = P[Γ > γ]. The ratecoverage probabilityPR(γ) can be evaluated through a changeof variables as in the following lemma.

Lemma 5:Given the SINR coverage probabilityPc(T ), forγ < W logN(1 + Tmax), the rate coverage probability can becomputed asPR(γ) = Pc(2

γ/W − 1).Proof: The proof is similar to that of [30, Theorem 1].

For γ < W logN(1 + Tmax), it directly follows thatPR(γ) =P[

SINR> 2γ/W − 1]

= Pc(2γ/W − 1).

Lemma 5 will allow comparisons to be made betweenmmWave and conventional systems that use different band-widths, as presented in Section V-A.

D. Simplification of LOS Probability Function

The expressions in Theorem 1 generally require numericalevaluation of multiple integrals, and may become difficult toanalyze. In this section, we propose to simplify the analysisby approximating a general LOS probability functionp(t) bya step function. We denote the step function asSRB(x), whereSRB(x) = 1 when0 < x < RB, andSRB(x) = 0 otherwise.Essentially, the LOS probability of the link is taken to be onewithin a certain fixed radiusRB and zero outside the radius.An interpretation of the simplification is that the irregulargeometry of the LOS region in Fig. 2 (a) is replaced with itsequivalent LOS ball in Fig. 2 (b). Such simplification not onlyprovides efficient expressions to compute SINR, but enablessimpler analysis of the network performance when the networkis dense.

We will propose two criterions to determine theRB givenLOS probability functionp(t). Before that, we first reviewsome useful facts.

Theorem 2:Given the LOS probability functionp(x), theaverage number of LOS base stations that a typical userobserves isρ = 2πλ

∫∞

0p(t)tdt.

7

Actual LOS region

(a) Irregular shape of an acutal LOS region.

RB

(b) Approximation using the equivalent LOS ball.

Fig. 2. Simplification of the random LOS region as a fixed equivalentLOS ball. In (a), we illustrate one realization of randomly located buildingscorresponding to a general LOS functionp(x). The LOS region observed bythe typical user has an irregular shape. In (b), we approximate p(x) by a stepfunction. Equivalently, the LOS region is also approximated by a fixed ball.Only base stations inside the ball are considered LOS to the user.

Proof: The average number of LOS base stations can becomputed as

ρ = E

[

∑

Xℓ∈Φ

I(Xℓ ∈ ΦL)

]

(a)= 2πλ

∫ ∞

0

p(t)tdt,

where (a) follows directly from Campbell’s formula of PPP[26].

A direct corollary of Theorem 2 follows as below.Corollary 2.1: When p(x) = SR(x), the average number

of LOS base stations isρ = πλR2.Note that Theorem 2 also indicates that a typical user

will observe a finite number of LOS base stations almostsurely when

∫∞

0 p(t)tdt < ∞. Hence, if p(x) satisfies∫∞

0 p(t)tdt < ∞, the parameterRB in SRB(x) can bedetermined by matching the average number of LOS basestations a user may observe.

Criterion 1 (Mean LOS BS Number):When∫∞

0 p(t)tdt <∞, the parameterRB of the equivalent step functionSRB(x)is determined to match the first moment ofρ. By Theorem 2,it follows thatRB =

(

2∫∞

0p(t)tdt

)0.5.

In the case where∫∞

0 p(t)tdt <∞ is not satisfied, anothercriterion to determineRB is needed. Note that even if the firstmoment is infinite, the probability that the user is associatedwith a LOS base station exists and is naturally finite for allp(t). Hence, we propose the second criterion regarding the

LOS association probability as follows.Criterion 2 (LOS Association Probability):Given a LOS

probability functionp(t), the parameterRB of its equivalentstep functionSRB(x) is determined such that the LOS asso-ciation probabilityAL is unchanged after approximation.From Lemma 2, the LOS association probability for a stepfunction SRB(x) equals1 − e−λπR

2B . Hence, by Criterion 2,

RB can be determined asRB =(

− ln(1−AL)λπ

)0.5

.Last, we explain the physical meaning of the step function

approximation as follows. As shown in Fig. 2(a), with a gen-eral LOS probability functionp(x), the buildings are randomlylocated, and thus the actual LOS region observed by the typicaluser may have an unusual shape. Although it is possible to in-corporate such randomness of the size and shape by integratingover p(t), the expressions with multiple integrals can makethe analysis and numerical evaluation difficult [1]. In Fig.2(b), by approximating the LOS probability function as a stepfunctionSRB(x), we equivalently approximate the LOS regionby a fixed ballB(0, RB), which we define as theequivalentLOS ball. As will be shown in Section IV, approximatingp(x) as a step function enables fast numerical computation,simplifies the analysis, and provides design insights for densenetwork. Besides, we will show in simulations in Section V-Athat the error due to such approximation is generally small indense mmWave networks, which also motivates us to use thisfirst-order approximation of the LOS probability function tosimplify the dense network analysis in the following section.

IV. A NALYSIS OF DENSE MMWAVE NETWORKS

In this section we specialize our results to dense networks.This approach is motivated by subsequent numerical resultsin Section V-A that show mmWave deployments will bedense if they are expected to achieve significant coverage. Wederive simplified expressions for the SINR and provide furtherinsights into system performance in this important asymptoticregime.

A. Dense Network Model

In this section, we build the dense network model bymodifying the system model in Section II with a few additionalassumptions. We say that a mmWave cellular network isdenseif the average number of LOS base stations observed by thetypical userρ is larger thanK, or if its LOS associationprobability AL is larger than1 − ǫ, where K and ǫ arepre-defined positive thresholds. In this paper, for illustrationpurpose, we will letK = 1 andǫ = 5%. Further, we say that anetwork isultra-densewhenρ > 10. Note thatρ also equalsthe relative base station density normalized by the averageLOS area, in this special case, as we will explain below.

Now we make some additional assumptions that will allowus to further simplify the network model.

Assumption 10 (LOS equivalent ball):The LOS region ofthe typical user is approximated by its equivalent LOS ballB(0, RB) as defined in Section III-D.

By Assumption 10, the LOS probability functionp(t) isapproximated by its equivalent step functionSRB(x), and the

8

LOS base station processΦL is made up of the outdoor basestations that are located inside the LOS ballB(0, RB). Notingthat the outdoor base station processΦ is a homogeneous PPPwith densityλ, the average number of LOS base stations isρ = λπR2

B, which is the outdoor base station density timesthe area of the LOS region. For ease of illustration, we callρ the the relative densityof a mmWave network. The relativedensityρ is equivalently: (i) the average number of LOS basestations that a user will observe, (ii) the ratio of the averageLOS areaπR2

B to the size of a typical cell1/λ [26], and (iii)the normalized base station density by the size of the LOSball. We will show in the next section that the SINR coveragein dense networks is largely determined by the relative densityρ.

Assumption 11 (No NLOS and noise):Both NLOS basestations and thermal noise are ignored in the analysis sincein the dense regime, the performance is limited by other LOSinterferers.

We show later in the simulations that ignoring NLOS basestations and the thermal noise introduces a negligible error inthe performance evaluation.

Assumption 12 (No Small-scale fading):Small-scale fad-ing is ignored in the dense network analysis, as the signalpower from a nearby mmWave LOS transmitter is found to bealmost deterministic in measurements [5].

Based on the dense network model, the signal-to-interference ratio (SIR) can be expressed as

SIR=MtMrR

−αL0

∑

ℓ:Xℓ∈Φ∩B(0,RB)DℓR−αL

ℓ

. (16)

Now we compute the SIR distribution in the dense networkmodel.

B. Coverage Analysis in Dense Networks

Now we present an approximation of the SINR distributionin a mmWave dense network. Our main result is summarizedin the following theorem.

Theorem 3:The SINR coverage probability in a densenetwork can be approximated as

Pc(T ) ≈ ρe−ρN∑

ℓ=1

(−1)ℓ+1

(

N

ℓ

)∫ 1

0

4∏

k=1

exp

(

−2

αLbkρt

× (ℓηT ak)2αL Γ

(

−2

αL; ℓηT ak, ℓηT akt

αL2

))

dt,

(17)

whereΓ(s; a, b) =∫ b

axs−1e−xdx is the incomplete gamma

function, ak = ak/(MtMr), ak and bk are defined in TableI, η = N(N !)

1N , andN is the number of terms used in the

approximation.Proof: See Appendix D.

WhenαL = 2, the expression in Theorem 3 can be furthersimplified as follows.

Corollary 3.1: WhenαL = 2, the SINR coverage probabil-ity approximately equals

Pc(T ) ≈ ρe−ρN∑

ℓ=1

(−1)ℓ+1

(

N

ℓ

)∫ 1

0

4∏

k=1

exp (ρbk×

(

e−ℓηT akt − te−ℓηT ak))

(

1− e−ℓµηT akt

1− e−ℓµηT ak

)ℓηTbk akρt

dt,

(18)

whereµ = e0.577.The results in Theorem 3 generally provide a close approxi-

mation of the SINR distribution when enough terms are used,e.g. whenN ≥ 5, as will be shown in Section V-B. Moreimportantly, we note that the expressions in Theorem 4 arevery efficient to compute, as most numerical tools supportfast evaluation of the gamma function in (17), and (18) onlyrequires a simple integral over a finite interval. Besides, giventhe path loss exponentαL and the antenna geometryak,bk, Theorem 3 shows that the approximated SINR is only afunction of the relative densityρ, which indicates the SIRdistribution in a dense network is mostly determined on theaverage number of LOS base station to a user.

C. Asymptotic Analysis in Ultra-Dense Networks

To obtain further insights into coverage in dense networks,we provide results on the asymptotic SIR distribution whenthe relative densityρ becomes large. We use this distributionto answer the following questions: (i) What is the asymptoticSIR distribution when the network becomes extremely dense?(ii) Does increasing base station density always improve SIRin a mmWave network?

First, we present the main asymptotic results as follows.Theorem 4:In a dense network, when the LOS path loss

exponentαL ≤ 2, the SIR converges to zero in probability,as ρ → ∞. WhenαL > 2, the SIR converges to a nonzerorandom variable SIR0 in distribution, asρ→ ∞; Based on [31,Proposition 10], a lower bound of the coverage probability forthe asymptotic SIR0 is that forT > 1,

P(SIR0 > T ) ≥αLT

−2/αL

2π sin(2π/αL).

Proof: See Appendix E.Note that Theorem 4 indicates that increasing base station

density above some threshold will hurt the system perfor-mance, and that the SINR optimal base station density is finite.

Now we provide an intuitive explanation of the asymptoticresults as follows. When increasing the base station density, thedistances between the user and base stations become smaller,and the user becomes more likely to be associated with a LOSbase station. When the density is very high, however, a usersees several LOS base stations and thus experiences significantinterference.

We note that the asymptotic trends in Theorem 4 are validwhen base stations are all assumed to be active in the network.A simple way to avoid “over-densification” is to simply turnoff a fraction of the base stations. This is a simple kind ofinterference management; study of more advanced interferencemanagement concepts is an interesting topic for future work.

V. NUMERICAL SIMULATIONS

In this section, we first present some numerical results basedon our analyses in Section III and Section IV. We concludewith some simulations using real building distributions tovalidate our proposed mmWave network model.

9

A. General Network Simulations

In this section, we provide numerical simulations to validateour analytical results in Section III, and further discuss theirimplications on system design. We assume the mmWavenetwork is operated at 28 GHz, and the bandwidth assignedto each user isW = 100 MHz. The LOS and NLOS path lossexponents areαL = 2 and αN = 4. The parameters of theNakagami fading areNL = 3 andNN = 2. We assume theLOS probability function isp(x) = e−βx, where1/β = 141.4meters. For the ease of illustration, we define the notion ofthe average cell radiusof a network as follows. Note thatif the base station density isλ, the average cell size in thenetwork is 1/λ [26]. Therefore, the average cell radiusrcin a network is defined as the radius of a ball that has thesize of an average cell, i.e.,rc =

√

1/πλ. The average cellradius not only directly relates to the inter-site distancethat isused by industry in base station planning, but also equivalentlycharacterizes the base station density in a network; as a largeaverage cell size indicates a low base station density in thenetwork.

5 10 15 20 25 30 35 400.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SINR threshold in dB

SIN

R C

ov

era

ge

Pro

ba

bilit

y

(Mt ,mt ,θt )=(10 dB, −10 dB, 30°)

(Mt ,mt,θt )=(20 dB, −10 dB, 30°)

(Mt ,mt,θt )=(10 dB, −10dB, 45 °)

Fig. 3. SINR coverage probability with different antenna geometry. Theaverage cell radius isrc = 100 meters. The receiver beam pattern is fixed asG

10dB,−10dB,90◦ .

First, we compare the SINR coverage probabilities withdifferent transmit antenna parameters in Fig. 3 using MonteCarlos simulations. As shown in Fig. 3, when the side lobegain mt is fixed, better SINR performance is achieved byincreasing main lobe gainMt and by decreasing the mainlobe beamwidthθt, as indicated by the analysis in SectionIII-A.

Next, we compare the LOS association probabilitiesAL

with different average cell radii in Fig. 4. The results showthat the probability that a user is associated with a LOS basestation increases as the cell radius decreases. The resultsinFig. 4 also indicate that the received signal power will bemostly determined by the distribution of LOS base stations ina sufficiently dense network, e.g. when the average cell sizeis smaller than 100 meters in the simulation.

We also compare the SINR coverage probability with differ-ent cell radii in Fig. 5. The numerical results in Fig. 5 (a) show

50 100 150 200 250 3000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Avg. cell radius in meters

Ass

ocia

tion

Pro

babi

lity

LOS Association Prob.NLOS Association Prob.

Fig. 4. LOS association probability with different averagecell radii. Thelines are drawn from Monte Carlos simulations, and the marksare drawnbased on Lemma 2.

−5 0 5 10 15

0.4

0.5

0.6

0.7

0.8

0.9

1

SINR threshold in dB

SIN

R c

over

age

prob

abili

ty

Simu: rc=50 m

Simu: rc=100 m

Simu: rc=200 m

Simu: rc=300 m

Analy: rc=50 m

Analy: rc=100 m

Analy: rc=200 m

Analy: rc=300 m

(a) Analytical bounds using Theorem 1.

−5 0 5 10 150.65

0.7

0.75

0.8

0.85

0.9

0.95

1

SINR threshold in dB

SIN

R C

ov

era

ge

Pro

ba

bili

ty

SINR: rc=100 m

SIR: rc=100 m

SINR: rc=150 m

SIR: rc=150 m

SINR: rc=200 m

SIR: rc=200 m

(b) Comparison between SINR and SIR.

Fig. 5. SINR coverage probability with different average cell sizes. Thetransmit antenna pattern is assumed to beG

100dB,0dB,30◦. In (a), analytical

results from Theorem 1 are shown to provide a tight approximation. In (b),it shows that SIR converges to SINR when the base station density becomeshigh, which implies that mmWave networks can be interference-limited.

10

−5 0 5 10 15 20 250.75

0.8

0.85

0.9

0.95

1

SINR threshold in dB

SIN

R C

ov

era

ge

Pro

ba

bili

ty

rc=100 m, step function

rc=100 m, p(t)=e

−β t

rc=150 m, step function

rc=150 m, p(t)=e

−β t

Fig. 6. Comparison of the SINR coverage between usingp(x) and itsequivalent step functionSRB

(x). The transmit antenna pattern is assumedto beG

20dB,−10dB,30◦ . It shows that the step function tends to provide amore pessimistic SINR coverage probability, but the gap becomes smaller asthe network becomes more dense.

that our analytical results in Theorem 1 match the simulationswell with negligible errors. Unlike in a interference-limitedconventional cellular network, where SINR is almost invariantwith the base station density [16], the mmWave SINR coverageprobability is also shown to be sensitive to the base stationdensity in Fig. 5. The results in Fig. 5 (a) also showsthat mmWave networks generally require a small cell radius(equivalently a high base station density) to achieve acceptableSINR coverage. Moreover, the results in Fig. 5 (b) showthat when decreasing average cell radius (i.e., increasingbasestation density), mmWave networks will transit from power-limited regime into interference-limited regime; as the SIRcurves will converge to the SINR curve when densifying thenetwork.

Specifically, comparing the curves forrc = 200 meters andrc = 300 meters in Fig. 5 (a), we find that increasing basestation density generally improve the SINR in a sparse net-work; as increasing base station density will increase the LOSassociation probability and avoid the presence of coverageholes, i.e. the cases that a user observes no LOS base stations.A comparison of the curves forrc = 100 meters andrc = 50meters, however, also indicates that increasing base stationdensity need not improve SINR, especially when the networkis already sufficiently dense. Intuitively, increasing base stationdensity also increases the likelihood to be interfered by strongLOS interferers. In a sufficiently dense network, increasingbase station will harm the SINR by adding more stronginterferers.

Now we apply Theorem 3 to compare the SINR coveragewith different LOS probability functionsp(x). We approx-imate the negative exponential functionp(x) = e−βx byits equivalent step functionSRB(x). Applying either of thecriteria in Section III-D, the radius of the equivalent LOS ballRB equals 200 meters. As shown in Fig. 6, the step functionapproximation generally provides a lower bound of the actualSINR distribution, and the errors due to the approximation

50 100 150 200 250 300 350 400 4500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Achievable rate in Mbps

Rat

e co

vera

ge p

roba

bilit

y

mmWave: rc=50 m

mmWave: rc=100 m

mmWave: rc=200 m

mmWave: rc=300 m

Microwave: 20 MHzMicrowave: 100 MHz

Fig. 7. Rate coverage comparison between mmWave and conventionalcellular networks. The mmWave transmit antenna pattern is assumed to beG

10dB,−10dB,30◦. We assume the conventional system is operated at 2 GHz

with a cell radius of 500 m, and the transmit power of the conventional basestation is46 dBm.

become smaller when the base station density increases. Theapproximation of step function also enables faster evaluationsof the coverage probability, as it simplifies expressions for thenumerical integrals.

We provide rate results in Fig. 7, where the lines are drawnfrom Monte Carlos simulations, and the marks are drawn basedon Lemma 5. In the rate simulation, we assume that 64 QAMis the highest constellation supported in the networks, andthus the maximum spectrum efficiency per data stream is 6bps/Hz. In Fig. 7, we compare the rate coverage probabilitybetween the mmWave network and a conventional networkoperated at 2 GHz. The mmWave bandwidth is 100 MHz(which conceivably could be much larger, e.g. 500 MHz [4],[15]), while we assume the conventional system has a basicbandwidth of 20 MHz, which can be potentially extendedto 100 MHz by enabling carrier aggregation [33]. Rayleighfading is assumed in the UHF network simulations. We furtherassume that conventional base stations have perfect channelstate information, and apply spatial multiplexing (4×4 singleuser MIMO with zero-forcing precoder) to transmit multipledata streams. More comparison results with other techniquescan be found in [34]. Results in Fig. 7 shows that, due to thefavorable SINR distribution and larger available bandwidth atmmWave frequencies, the mmWave system with a sufficientlysmall average cell size outperforms the conventional systemin terms of providing high data rate coverage.

B. Dense Network Simulations

Now we show the simulation results based on the densenetwork analysis in Section IV. First, we illustrate the resultsin Theorem 3 with the simulations in Fig. 8. In the simulations,we include the NLOS base stations and thermal noise, whichwere ignored in the theoretical derivation. The expressionderived in Theorem 3 generally provides a lower bound ofthe coverage probability. The approximation becomes moreaccurate when more terms are used in the approximation,

11

especially whenN ≥ 5. We find that the error due to ignoringNLOS base stations and thermal noise is minor in terms of theSINR coverage probability, primarily impacting low SINRs.

−10 −5 0 5 10 15 20 250.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

SINR threshold in dB

SIN

R c

ov

era

ge

pro

ba

blity

Simu: LOS+ NLOS + Noise

Simu: LOS only

Analy: N=1

Analy: N=3

Analy: N=5

Analy: N=10

Fig. 8. Coverage probability in a dense mmWave network. The mmWavetransmit antenna pattern is assumed to beG

10dB,−10dB,30◦. We assume

RB = 200 m, and the relative base station densityρ = 4. N is the numberof terms we used to approximate the coverage probability in Theorem 3.

Next, we compare the SINR coverage probability withdifferent relative base station density whenT = 20 dB. Recallthat ρ = λπR2

B is the base station density normalized by thesize of the LOS region. In 9(a), the path loss exponent isassumed to beα = 2. We compute the coverage probabilityfrom ρ = −20 dB meters toρ = 20 dB with a step of1 dB. The analytical expressions in Theorem 3 are muchmore efficient than simulations: the plot takes seconds tofinish using the analytical expression, while it approximatelytakes an hour to simulate 10,000 realizations at each step. Asshown in Fig. 9 (a), although there is some gap between thesimulation and the analytical results in the ultra-dense networkregime, both curves achieve their maxima at approximatelyρ = 5, i.e., when the average cell radiusrc is approximately1/2 of the LOS rangeRB. Moreover, when the base stationdensity grows very large, the coverage probability begins todecrease, which matches the asymptotic results in Theorem4. The results also indicate that networks in the environmentswith dense blockages, e.g. the downtown areas of large citieswhere the LOS rangeRB is small, will benefit from networkdensification; as they are mostly operated in the region wherethe relative density is (much) smaller than the optimal valueρ ≈ 5, and thus increasingρ by densifying networks willimprove SINR coverage.

We also simulate with other LOS path loss exponents inFig. 9 (b). The results show that the optimal base stationdensity is generally insensitive to the change of the path lossexponent. When the LOS path loss exponent increases from1.5 to 2.5, the optimal cell size is almost the same. The resultsalso illustrate that the networks with larger path loss exponentαL have better SINR coverage in the ultra-dense regime whenρ > 10. Intuitively, signals attenuate faster with a larger pathloss exponent, and thus the inter-cell interference becomes

weaker, which motivates a denser deployment of base stationsin the network with higher path loss.

10−2

10−1

100

101

102

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Relative base station density ρ

SIN

R c

ov

era

ge

pro

ba

bilit

y a

t 2

0 d

B

Simulation: LOS + NLOS+ Noise

Analytical using N=5 terms

(a) Analytical results using Theorem 3.

100

101

102

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Relative base station density ρ

SIN

R c

ov

era

ge

pro

ba

bilit

y a

t 2

0 d

B

LOS path loss exponent: αL=2.5

LOS path loss exponent: αL=2

LOS path loss exponent: αL=1.5

(b) Optimal density with different path loss exponents.

Fig. 9. SINR coverage probability with different relative base station densitywhen the target SINR=20 dB. In the simulations, we include the NLOS basestations outside the LOS region and the thermal noise. We also fix the radiusof the LOS ball asRB =200 meters, and change the base station densityλat each step according to the value of the relative base station densityρ. In(a), it shows that ignoring NLOS base stations and the noise power causesminor errors in terms of the optimal cell radius. In (b), we search for theoptimal relative density with different LOS path loss exponents. It shows thatthe optimal cell radius is generally insensitive to the pathloss exponent.

Finally, we compare the spectral efficiency and averageachievable rates as a function of the relative densityρ in TableII. We find with a reasonable amount of density, e.g. when therelative densityρ is approximately 1, the mmWave system canprovide comparable spectrum efficiency as the conventionalsystem at UHF frequencies. With high density, rates that canbe achieved are an order of magnitude better than that in theconventional networks, due to the favourable SINR distributionand larger available bandwidth at mmWave frequencies.

C. Comparison with Real-scenario Simulations

Now we compare our proposed network models with thesimulations using real data. In the real-scenario simulations,

12

TABLE IIACHIEVABLE RATE WITH DIFFERENTBS DENSITIES

Carrier frequency 28 GHz 28 GHz 28 GHz 28 GHz 2 GHz 2 GHzBase station density Ultra dense Dense Intermediate Sparse - -Relative densityρ 16 4 1 0.45 - -

Spectrum efficiency (bps/Hz) 5.5 5.8 4.3 2.7 4.6 4.6Signal bandwith (MHz) 100 100 100 100 20 100Achievable rate (Mpbs) 550 580 430 270 92 459

(a) Snapshot of the simulated area from Google map.

−10 −5 0 5 10 15 20 25 30

0.4

0.5

0.6

0.7

0.8

0.9

1

SINR Threshold in dB

SIN

R C

over

age

Pro

babi

lity

Real building distribution

Proposed general network model: p(r)=e −β r

Equivalent LOS ball model: RB

=225 m

(b) Comparison of SINR distribution

Fig. 10. Comparison of SINR coverage results with real-scenario simulations.The snapshot of The University of Texas at Austin campus is from Googlemap. We use the actual building distribution of the area in the real-scenariosimulation. In the simulations of our proposed analytical models, we letβ =0.0063 m−1 in the LOS probability functionp(r) = e−βr, andRB = 225m in the simplified equivalent LOS ball model, to match the building statisticsin the area [24].

we use the building distribution on the campus at The Univer-sity of Texas at Austin. We also apply a modified version of thebase station antenna pattern in [35] with a smaller beam widthof 30◦. The directivity gain at the base station isMt = 20dB. The mobile station is assumed to use uniform linear

array with 4 antennas. When applying our analytical models,we fit the parameters of the LOS probability functions tomatch the building statistics [24], and use the sectored modelfor beamforming pattern. We also assume the mmWave basestations are distributed as a PPP withrc = 50 m. As shown inFig. 10, though some deviations in the high SINR regime, ouranalytical models generally show a close characterizationofthe reality. The deviation is explained as follows: the proposedanalytical model computes the aggregated SINR coverageprobability, averaging over all realizations of building distribu-tions over the infinite plane, while the real-scenario curveonlyconsiders a specific realization of buildings in a finite snapshotwindow. In this case, our model overestimates the coverageprobability in the low SINR regime, and underestimates in thehigh SINR regime, as both signals and interference becomemore likely to be blocked in the real scenario simulation. Wehave found in other simulation examples that the reverse canalso be true. Our model should be viewed as a characterizationof the average distribution and does not necessarily lower orupper bound the distribution for a given realization.

VI. CONCLUSIONS

In this paper, we proposed a stochastic geometry frameworkto analyze coverage and rate in mmWave networks for outdoorusers and outdoor infrastructure. Our model took blockageeffects into account by applying a distance-dependent LOSprobability function, and modeling the base stations as in-dependent inhomogeneous LOS and NLOS point processes.Based on the proposed framework, we derived expressions forthe downlink SINR and rate coverage probability in mmWavecellular networks, which were shown to be efficient in com-putation and also a good fit with the simulations. We furthersimplified the blockage model by approximating the randomLOS region as a fixed-size equivalent LOS ball. Applyingthe simplified framework, we analyzed the performance andasymptotic trends in dense networks.

We used numerical results to draw several important con-clusions about coverage and rate in mmWave networks.

• SINR coverage can be comparable to conventional net-works at UHF frequency when the base station density issufficiently high.

• Achievable rates can be significantly higher than inconventional networks, thanks to the larger availablebandwidth.

• The SINR and rate performance is largely determined bythe relative base station density, which is the ratio of thebase station density to the blockage density.

• A transition from a power-limited regime to aninterference-limited regime is also observed in mmWave

13

networks, when increasing base station density.• The optimal SINR and rate coverage can be achieved

with a finite base station density; as increasing basestation density need not improve SINR in a (ultra) densemmWave network.

In future work, it would be interesting to analyze thenetworks with overlaid microwave macrocells, as mmWavesystems will co-exist with base stations operated in UHFbands. It would be another interesting topic to incorporatemmWave hardware constraints in the system analysis, andinvestigate the performance of mmWave networks applyinganalog/ hybrid beamforming [36] or using low-resolution A/Dconverters at the receivers [37].

ACKNOWLEDGEMENT

The authors would like to thank Dr. Xinchen Zhang for hisvaluable feedback on early drafts of this paper.

APPENDIX A

We provide two useful inequalities in the following lemmas.The first lemma approximates the tail probability of a gammarandom variable.

Lemma 6 (From [38]):Let g be a normalized gamma ran-dom variable with parameterN . For a constantγ > 0, theprobabilityP(g < γ) can be tightly upper bounded by

P(g < γ) <[

1− e−aγ]N

,

wherea = N(N !)−1N .

The following inequality will be used in the dense networkanalysis.

Lemma 7 (From [38]):For x > 0, it holds that

− log(1 − e−ax) ≤

∫ ∞

x

e−t

tdt ≤ − log(1− e−bx),

where a = e0.5772 and b = 1. Further, the lower boundgenerally provides a close approximation.

APPENDIX B

Proof of Lemma 2: For s = {L,N}, let ds be the distance fromthe typical user to its nearest base station inΦs. Note that itis possible that the user observes no base stations inΦs. Theuser is associated with a base station inΦL if and only if ithas a LOS base station, and its nearest base station inΦL hassmaller path loss than that of the nearest base station inΦN.Hence, it follows that

AL = BLP(

CLd−αL

L > CNd−αN

N

)

(a)= BL

∫ ∞

0

P (dN > ψL(x)) fL(x)dx, (19)

whereBL is the probability that the user has at least one LOSbase stations, (a) follows that by Lemma 1, andfL(x) is theprobability density function ofdL. Next, note that

P (dN > ψ(x)) = P (ΦN ∩ B(0, ψL(x)) = ∅)

= e−2πλ∫ ψL(x)

0 (1−p(t))tdt, (20)

whereB(0, x) denotes the ball centered at the origin of radiusx. Substituting (20) for (19) gives Lemma 2.

APPENDIX C

Proof of Theorem 1: Given that the user is associated with abase station inΦL, by Slivnyak’s Theorem [26], the condi-tional coverage probability can be computed as

Pc,L(T ) =

∫ ∞

0

P[

CLh0MrMtx−αL > T

(

σ2 + IL + IN)]

fL(x)dx, (21)

where IL =∑

ℓ:Xℓ∈ΦL∩B(0,x) CL |hℓ|2DℓR

−αL

ℓ and IN =∑

ℓ:Xℓ∈ΦN∩B(0,ψL(x))CN |hℓ|

2DℓR

−αN

ℓ are the interferencestrength from the tiers of LOS and NLOS base stations,respectively. Next, noting that|h0|

2 is a normalized gammarandom variable with parameterNL, we have the followingapproximation

P[

CLh0MrMtx−αL > T

(

σ2 + IL + IN)]

= P[

h0 > xαLT(

σ2 + IL + IN)

/(CLMrMt)]

(a)≈ 1− EΦ

(

1− e−ηLx

αLT(σ2+IL+IN)CLMrMt

)NL

(b)=

NL∑

n=1

(−1)n+1

(

NL

n

)

EΦ

[

e−nηLx

αLT(σ2+IL+IN)CLMrMt

]

(c)=

NL∑

n=1

(−1)n+1

(

NL

n

)

e−nηLx

αLTσ2

CLMrMt EΦL

[

e−nηLx

αLTILCLMrMt

]

EΦN

[

e−nηLx

αLTINCLMrMt

]

, (22)

where ηL = NL(NL!)− 1NL , (a) is from Lemma 6 [38] in

Appendix A, (b) follows from Binomial theorem and theassumption thatNL is an integer, and(c) follows from the factthatΦL andΦN are independent. Now we apply concepts fromstochastic geometry to compute the term for LOS interfering

links EΦL

[

e−nηLx

αLTILCLMrMt

]

in (22) as

EΦL

[

e−nηLx

αLTILCLMrMt

]

= E

[

e−nηLx

αLT∑

ℓ:Xℓ∈ΦL∩B(0,x)|hℓ|2DℓR

−αLℓ

MrMt

]

(c)= e

(

−2πλ∑4k=1 bk

∫

∞

x

(

1−Eg

[

e−nTηLgak(x/t)αL])

p(t)tdt)

(d)=

4∏

k=1

e−2πλbk∫

∞

x (1−1/(1+ηLaknT (x/t)αL/NL)NL)p(t)tdt

= e−Qn(T,x),

whereg in (c) is a normalized gamma random variable withparameterNL, ak = ak

MtMr, and for 1 ≤ k ≤ 4, ak and bk

are defined previously in Table I; (c) is from computing theLaplace functional of the PPPΦL [26]; (d) is by computingthe moment generating function of a gamma random variableg.

Similarly, for the NLOS interfering links, the small-scalefading term |hℓ|2 is a normalized gamma variable with pa-

14

rameterNN. Thus, we can computeEΦN

[

e−nηLx

αLTINCLMrMt

]

as

EΦN

[

e−nηLx

αLTINCLMrMt

]

= E

[

e−nηLx

αLTCN∑

ℓ:Xℓ∈ΦN∩B(0,ψ(x))|hℓ|2DℓR

−αNℓ

CLMrMt

]

=

4∏

k=1

e−2πλbk

∫

∞

ψL(x)

(

1−1/(

1+ηLaknTCNx

αL

CLtαNNN

)NN)

(1−p(t))tdt

= e−Vn(T,x).

Then, we obtain (9) from (22) by the linearity of integrals.Given the user is associated with a NLOS base station, we

can also derive the conditional coverage probabilityPc,N(T )following same approach as that ofPc,L(T ). Thus, we omitthe detailed proof of (10) here.

Finally, by the law of total probability, it follows thatPc(T ) = ALPc,L(T ) +ANPc,N(T ).

APPENDIX D

Proof Sketch of Theorem 3: For a generalαL, the coverageprobability can be computed as

Pc(T ) = ALPc,L(T ) = ALP(SIR> T )

= AL

∫ RB

0

P(CLMrMtr−αL > TIr)

2πλr

ALe−λπr

2

dr,

where Ir =∑

Xℓ∈Φ∩(B(0,RB)/B(0,RB))DℓCLR−αL

ℓ is theinterference power given that the distance to the user’sserving base station isR0 = r. Next, the probabilityP(CLMrMtr

−αL > TIr) can be approximated as

P(CLMrMtr−αL > TIr)

(a)≈ P(g > TrαLIr/(CLMrMt))

(b)≈ 1− EΦL

[

(

1− e−ηTrαLIr/(CLMrMt)

)N]

=

N∑

ℓ=1

(

N

ℓ

)

(−1)ℓEΦL

[

e−ℓηTrαLIr/(CLMrMt)

]

. (23)

In (a), the dummy variableg is a normalized gamma variablewith parameterN , and the approximation in(a) followsfrom the fact that a normalized Gamma distribution con-verges to identity when its parameter goes to infinity, i.e.,limn→∞

nnxn−1e−nx

Γ(n) = δ(x−1) [39], whereδ(x) is the Diracdelta function. In(b), it directly follows from Lemma 6 bytaking η = N(N !)1/N .

Next, we can computeEΦL

[

e−ℓηTrαLIr/(CLMrMt)

]

as

EΦL

[

e−ℓηTrαLIr/(CLMrMt)

]

(c)= exp

(

4∑

k=1

−2πλbk

∫ RB

r

1− e−ℓηakT (r/t)αLtdt

)

(d)=e−πλ(R

2B−r2)×

e∑4k=1

2αLπλr2bk(Tℓηak)

2/αL∫ ℓηT akℓηT ak(r/RB)αL

e−s

s1+2/αLds

(24)

=e−πλ(R2B−r2)×

e∑4k=1

2αLπλr2bk(Tℓηak)

2/αLΓ(

−2αL

;ℓηT ak(r/RB)αL ,ℓηT ak

)

,(25)

where (c) follows from computing the Laplace functional ofthe PPPΦL [26], and (d) follows from changing variableas s = ℓηakT (r/t)

αL . Hence, (17) directly follows fromsubstituting (25) for (23) and lettingρ = πλR2

B.When αL = 2, the steps above hold true till (24), which

can be further simplified as

EΦL

[

e−ℓηTrαLIr/(CLMrMt)

]

= e−πλ(R2B−r2)e

∑4k=1

(

πλr2bkTℓηak∫ ℓηT akℓηT ak(r/RB)2

e−s

s2ds

)

(e)= e−πλ(R

2B−r2)×

e

∑4k=1 πλr

2bk

(

e−(r/RB)2ℓTℓηak

(r/RB)2−e−ℓTℓηak+

∫ ℓTℓηakℓTℓηak(r/RB)2

e−s

s ds

)

(f)≈ e−πλ(R

2B−r2) exp

(

4∑

k=1

πλr2bk

(

e−(r/RB)2ℓTℓηak

(r/RB)2

−e−ℓTℓηak − log1− eµℓTℓηak(r/RB)2

1− eµℓTℓηak

))

, (26)

where(e) is from computing integration by part, (f) followsfrom Lemma 7 by lettingµ = e0.5772. Lastly, (18) followsfrom substituting (26) for (23) and lettingρ = πλR2

B.

APPENDIX E

Proof Sketch of Theorem 4: First, we show that the exact SIRdistribution in our dense network model is unchanged whenthe relative densityρ = λπR2

B is fixed. LetFℓ = RαL

ℓ be thepath loss gain in theℓ-th link. Then the SIR expression in (16)can be rewritten as SIR= MtMrF

−10 /

(∑

ℓ:Xℓ∈ΦLDℓF

−1ℓ

)

,whereDℓ is the directivity gain in theℓ-th link. Further, usingdisplacement theorem [26] and the method in [1, Lemma 7],we can show that{Fℓ}ℓ:Xℓ∈ΦL forms a PPP on the interval(0, RαL

B ) with density measure functionΛ(0, t) = λπt2/αL .Next, for c > 0, we define an class ofequivalent networksas the networks with base station densitycλ and LOS rangec−0.5RB. Note that all networks in the equivalent class havethe same relative densityρ = λπR2

B as the original network.Then we can show that a scaled version of path loss gain pro-cess

{

cαL/2Fℓ}

ℓin an equivalent network has the exact same

distribution, more specifically, the measure density function,as{Fℓ}ℓ in the original network. Thus, all equivalent networkshave the same SIR distribution as the original network; as thescaling constantcαL/2 cancels in the SIR expression. So far,we have shown that the SIR distribution in dense networks isunchanged when the relative densityρ is fixed.

Given the SIR invariant property with respect toρ, oneway to investigate the asymptotic SIR whenρ→ ∞ is to fixλ as a constantλ0 and examine the SIR performance whenRB → ∞. In other words, whenρ → ∞, the asymptoticSIR in the original network has the same distribution as theSIR in its asymptotic equivalent network, which has a basestation density ofλ0, and an infinitely largeRB. Let SIR0

be the SIR in the asymptotic equivalent network. Note that inthe asymptotic equivalent network, LOS base stations form ahomogeneous PPP with densityλ0 on the entire plane.

15

Next, for any realization of the path loss process{Fℓ}ℓ:Xℓ∈ΦL , the SIR in the asymptotic equivalent networkcan be lower and upper bounded by assuming all interferinglinks achieve the maximum directivity gainMrMt and theminimum directivity gainmrmt, respectively, as

F−10

∑

ℓ:Xℓ∈ΦLF−1ℓ

≤ SIR0 ≤MrMtF

−10

mrmt

∑

ℓ:Xℓ∈ΦLF−1ℓ

. (27)

ForαL > 2, an lower bound of the SIR coverage probabilityin the asymptotic equivalent network can be computed as forT > 1,

P(SIR0 > T ) ≥ P

(

F−10

∑

ℓ:Xℓ∈ΦLF−1ℓ

> T

)

(a)=

αLT−2/αL

2π sin(2π/αL),

where(a) follows from Proposition 10 in [31].Finally, for αL ≤ 2, we show that the upper bound of SIR0

in (27) converges to zero in probability. By [40, Proposition1, Fact 1], the distribution of the upper bound expressionin (27) is invariant with the small-scale fading distributionin the asymptotic equivalent network, and thus has the samedistribution as in theRayleigh-fading networkinvestigated in[18], where independent Rayleigh fading is assumed in eachlinks. Consequently, we compute the coverage probability forthe SIR0 upper bound in (27) as follows: for allT > 0,

P (SIR0 > T )(b)< P

[

MrMtF−10

mrmt

∑

ℓ>0:Xℓ∈ΦLF−1ℓ

> T

]

= P

[

⋃

ℓ:Xℓ∈ΦL

MrMtF−1ℓ

mrmt

∑

ℓ′ 6=ℓ:X′ℓ∈ΦL

F−1ℓ′

> T

]

= P

[

⋃

ℓ:Xℓ∈ΦL

F−1ℓ

mrmt

∑

ℓ′ 6=ℓ:X′ℓ∈ΦL

F−1ℓ′

>mrmtT

MrMt

]

(c)

≤∑

Xℓ∈ΦL

P

[

F−1ℓ

mrmt

∑

ℓ′ 6=ℓ:X′ℓ∈ΦL

F−1ℓ′

>mrmtT

MrMt

]

(d)= πλ

∫ ∞

0

exp

(

−πλT 2/αLt

∫ ∞

0

1

1 + rαL/2dr

)

dt

=T−2/αL

∫∞

01

1+rαL/2dr

(e)= 0,

whereT = mrmtTMrMt

, (b) is obtained from (27), (c) is from theunion bound, (d) follows from the fact that the upper boundin (27) has the same distribution as in the Rayleigh-fadingnetwork in [18], and the similar algebra in [18, Appendix B,Eqn (15)-(17)], and (e) is from the fact that

∫∞

01

1+rαL/2dr is

infinity whenαL ≤ 2. By now, we have shown that an upperbound of SIR in the asymptotic equivalent network, which alsoupper bounds the SIR in the original network whenρ → ∞,converges to zero in probability.

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