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Coverage Performance in MIMO-ZFBF Dense HetNetswith Multiplexing and LOS/NLOS Path-Loss Attenuation

Mohammad G. Khoshkholgh, Memeber, IEEE, Keivan Navaie, Senior Memeber, IEEE,Kang G. Shin, Life Fellow, IEEE, Victor C. M. Leung, Fellow, IEEE

Abstract —The performance of multiple-input multiple-output (MIMO) multiplexing heterogenous cellular networks are often analyzedusing a single-exponent path-loss model. Thus, the effect of the expected line-of-sight (LOS) propagation in densified settings isunaccounted for, leading to inaccurate performance evaluation and/or inefficient system design. This is due to the complexity ofLOS/non-LOS models in the context of MIMO communications. We address this issue by developing an analytical framework based onstochastic geometry to evaluate the coverage performance. We focus on the zero-forcing beamforming where the maximumsignal-to-interference ratio is used for cell association. We analytically derive the coverage. We then investigate the cross-streaminterference correlation, and develop two approximations of the coverage: Alzer Approximation (A-A) and Gamma Approximation (G-A).The former is often used in the single antenna and single-stream MIMO. We extend A-A to a MIMO multiplexing system and evaluateits utility. We show that the inverse interference is well-fitted by a Gamma random variable, where its parameters are directly related tothe system parameters. The accuracy and robustness of G-A is higher than that of A-A. We observe that depending on the multiplexinggain, it is possible to attain the best coverage probability by proper densification.

Index Terms —Area spectral efficiency, coverage probability, densification, heterogenous cellular networks (HetNets), LOS/NLOSpath-loss model, multiple-input multiple-output (MIMO), multiplexing, numerical complexity, Poisson Point Process (PPP), stochasticgeometry, zero-forcing beamforming (ZFBF).

1 INTRODUCTION

D ENSIFICATION of heterogenous cellular networks (Het-Nets) as well as air interface technology based on

multiple-input multiple-output (MIMO) are viable ways toaddress the rapid and substantial growth of mobile datademand. In fact, these two technologies have been integralparts of the air interface technology in 5G and beyond [2],[3], thanks to a set of encouraging analytical results in [4],[5], [6], [7] demonstrating a nearly proportional growth ofthe area spectral efficiency (ASE) in HetNets by steadilyincreasing the number of BSs per unit area (densification)without deteriorating the coverage probability (scale invari-ance). These analytical results were developed by leveragingtools of stochastic geometry (e.g., [8], [9], [10], and validatedwith empirical studies, e.g., [11], [12].

• The paper received March 27, 2018, revised February 20, 2019 andMay 17, 2019. The paper is accepted June 5, 2019. An earlier versionof this work was presented in IEEE VTC-Fall, 2017, Toronto [1]. Thecorresponding author is V. C. M. Leung.

• M. G. Khoshkholgh (e-mail: m.g.khoshkholgh@gmail.com) is with theDepartment of Electrical and Computer Engineering, the University ofBritish Columbia, Vancouver, BC, Canada V6T 1Z4; V. C. M. Leungis with the College of Computer Science and Software Engineering,Shenzhen University, Shenzhen 518060, China, and also with the Depart-ment of Electrical and Computer Engineering, The University of BritishColumbia, Vancouver, BC V6T 1Z4, Canada (e-mail:,vleung@ieee.org); K.Navaie (e-mail: k.navaie@lancaster.ac.uk) is with the School of Computingand Communications, Lancaster University, Lancaster, United KingdomLA1 4WA; K. G. Shin (e-mail: kgshin@umich.edu) is with the Departmentof Electrical Engineering and Computer Science, University of Michigan,Ann Arbor, MI 48109-2121, U.S.A.

• This work was supported in part by the National Natural Science Foun-dation of China under Grant 61671088, in part by the Vanier CanadaGraduate Scholarship, in part by the National Engineering Laboratory forBig Data System Computing Technology at Shenzhen University, China,and in part by the Natural Sciences and Engineering Research Council ofCanada.

The scale invariance property of HetNets, however, de-pends heavily on the standard path-loss model (SPLM)L(‖x‖) = ‖x‖−α, where ‖x‖ is the Euclidean distancebetween the source and the destination, and 2 < α < 8is the path-loss exponent. Nevertheless, SPLM has intrin-sic disadvantages, such as singularity—where the receivedpower may increase significantly as ‖x‖ → 0, which re-sults from ultra densification [13]. It is shown in [13], [14]that in contrast to the analysis based on SPLM, the cover-age probability under a bounded path-loss function, e.g.,L(‖x‖) = ‖x + 1‖−α, is decreased by increasing the densityof base stations (BSs). A similar conclusion was drawn in[15], where the coverage probability in a double-slope path-loss environment was shown to decrease significantly dueto densification. Such conclusions are in contrast with thoseof made in [4], [5], [6], [7] based on SPLM. This is becauseSPLM fails to model the propagation in mobile systemssuch as small cells, where a combination of line-of-sight(LOS), and non-LOS (NLOS) links are involved with theinside/outside of buildings.

The need for an inclusive path-loss attenuation modelwhich is able to characterize propagation in the cellularnetworks in various environments is also recognized by the3GPP. In 3GPP TR36.814, Release 9 [16], a practical path-loss model is described as the one that can distinguishbetween Line of Sight (LOS), and non-LOS (NLOS) links.Such models will henceforth be referred to as LOS/NLOSmodels. Adopting a 3GPP LOS/NLOS path-loss model [16],scale invariancy is shown to be not preserved in ultra-densecellular networks [17], [18]. This is due to the fact that closerBSs show higher tendency to exhibit the LOS effect—witha smaller path-loss exponent—while farther BSs are mostlikely to demonstrate NLOS path-loss attenuation. There-

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fore, the inter-cell interference (ICI) eventually dominatesthe received signal power.

Hence, the analytical results obtained based on the SPLMare only reliable in cases where the network is, at most,moderately densified. In such a case, signals from mostof the BSs, both serving and interfering, close to the userequipment (UE) are propagating through a NLOS link.However, this might not be a valid assumption for heavilydensified HetNets [19], [20], where it is most likely for aUE to have LOS signals, both from the interferers and theserving BS. Therefore, there is an urgent need to re-visit theperformance evaluation of HetNets while considering theLOS/NLOS path-loss model .

Several models have been proposed to characterize thepath-loss attenuation in dense HetNets. A multi-ball path-loss model was proposed, and then the spectral efficiencyand coverage performance of a single-tier cellular networkwas investigated in [21]. The analysis was then validatedusing empirical data collected in various cities [21].

The effect of NLOS link propagation on the outageprobability was also studied in [22], where the authors con-structed a new analytical path-loss model, formulating theprobability of realizing the LOS propagation with distance,average size and density of the buildings per area. TheBoolean blockage model [22] was further utilized in [23] toincorporate the effect of the size and density of buildings aswell as the wall penetration on the performance evaluationof a two-tier HetNet. Their work distinguishes betweenindoor small cell BSs and outdoor Macro BSs, and also takesinto account the signal propagation characteristics of LOS,NLOS, and blocked modes.

Similarly, the area spectral efficiency (ASE) is closelyrelated to the path-loss model. The authors of [24] studiedthe fundamental limits of ultra-dense networks accordingto fading distribution, shadowing, and multi-slope path-loss attenuation. Adopting the tools of stochastic geometryalong with the extreme value theory, the authors of [24] ob-tained scaling laws governing the downlink SINR, coverageprobability, and spectral efficiency. It is also shown in [25]that the spectral efficiency may be reduced substantially bydensification if the LOS path-loss exponent becomes verysmall. The same behavior was reported for the uplink inan ultra-dense single-tier cellular network with multi-slopepath-loss attenuation [26].

The main focus in the above-mentioned studies, e.g.,[13], [14], [15], however, is on single-tier networks withSISO-based air interface. Despite its relevance and impor-tance, to the best of our knowledge, the coverage per-formance of MIMO multiplexing in a multi-tier HetNetwith LOS/NLOS path-loss attenuation has not been fullyinvestigated. An instance for practical applications of suchsystems is in sub-6 GHz spectrum [2], [16].

The importance of an accurate path-loss model is alsowitnessed in the emerging literature. For instance, the au-thors of [27] demonstrated the feasibility of MIMO com-munications in wireless energy harvesting applications, andhighlighted further the crucial role of LOS component inenhancing the rate of harvesting energy and decodingprobability. Moreover, [28] studied the coverage probabilityand spectral efficiency of multi-tier mmWave communi-cations for both noise- and interference-limited scenarios,

in the presence of practical beamforming alignment error,LOS/NLOS and blockage model. The results were then ex-tended in [29] to investigate the potential of cellular systemsfor simultaneous information and wireless power transfer.The antenna’s directionality was shown instrumental to(partially) cancel out the severe effect of LOS interferenceas a result of network densification. Understanding theimpact of LOS/NLOS on the coverage and ASE of MIMOmultiplexing systems, however, has not yet been considered.

We consider a link-level coverage performance analysisin which successful reception of all data steams is consid-ered as a successful transmission. This is different from theconventional approach, stream-level analysis, which definesa successful transmission as the successful reception of asingle data stream. In fact, our previous results [30], [31],[32], [33] indicate that the coverage performance of MIMOmultiplexing HetNets with SPLM is best represented bytheir link-level analysis. Thus, part of this paper could beconsidered as an extension of our previous results to sys-tems with LOS/NLOS path-loss attenuation.

The coverage probability as a function of different sys-tem parameters is often incorporated in adaptive HetNetresource allocations, as well as system design problems [34],[35], [36]. Therefore, a quick and accurate estimation ofthe coverage probability is important to the optimizationof the system operating parameters on-the-go. Adoptingthe LOS/NLOS propagation model makes it challengingto analyze coverage performance in such systems. In thispaper, we derive closed-form analytical results and thenpropose quick and accurate approximations.

We leverage stochastic geometry to obtain a closed-formexpression for the coverage probability. Calculating the cov-erage probability based on derived closed-form, however,requires substantial numerical calculations. The complexityof the problem has its root in the intrinsic correlation in theinter-cell interference (ICI) across streams. This is due to thepacked geometry of MIMO dense networks in which theinterferers to each data stream are not independent. To ad-dress this issue, we analytically investigate the cross-streamICI correlation in a MIMO HetNet setting with multiplexing.Our analysis indicates a very high correlation in ICI acrossstream. This justifies the construction of ’full-correlation‘(FC) approximation, where the ICI across all streams on agiven communication link is considered fully correlated.

The FC assumption is then used in our proposed AlzerApproximation (A-A) of the coverage probability. We fur-ther propose a new approximation based on a novel wayof modeling the inverse ICI using a fitted Gamma distribu-tion, i.e., Gamma Approximation (G-A). We then obtain theparameters of the fitted Gamma distribution as functions ofmain system and path-loss parameters. This approximationis presented for the first time.

Both A-A and G-A need significantly lower numericalcomputations than that of the originally obtained closed-form. Our extensive simulation and numerical results in-dicate that the G-A outperforms the A-A in terms ofaccuracy while its corresponding computational complex-ity is slightly higher. Our simulation results also revealthat G-A demonstrates a higher level of robustness to awide range of system parameters, including density ofBSs, multiplexing gains, and LOS path-loss exponent. As

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its practical importance, the proposed G-A, unlike A-A,pinpoints the optimal density for which the best coverageperformance is achieved. Our results suggest that G-A is amuch better choice than that of the A-A for MIMO commu-nications (including mmWave communications), and alsosingle-antenna system under Nakagami fading.

We further utilize the results in this paper to approxi-mate the coverage performance of diversity-only MIMO sys-tem in HetNet systems with homogeneous/nonhomogeneousSPLM. Our results show that compared to multiplexingsystems, diversity-only systems provide a higher coverageperformance without degrading ASE. Interestingly, in a 2-tier HetNet, when densification in Tier 2 improves thecoverage probability, it can counterproductively acts in anenvironment with LOS/NLOS path-loss attenuation.

Our numerical studies further provide quantitative in-sights into the impacts of densification, multiplexing, andthe propagation environment on the coverage probabilityand ASE. Our results also show that by careful selectionof BS density in each tier, one can exploit the existence ofthe LOS propagation to improve ASE. In such a setting,the ASE gain is higher for cases with smaller LOS path-lossexponents.

Note that the focus of the earlier conference version[1] of this paper was on a single-tier network, where onlya specific LOS/NLOS model was considered. Here theresults in [1] are extended to a K-tier HetNet with thegeneric LOS/NLOS, where we also develop A-A and G-A.Furthermore, this paper incorporates the cross-stream ICIcorrelation, which was absent in [1].

The rest of this paper is organized as follows. Section 2reviews the literature of performance evaluation of MIMOcommunications under the stochastic geometry. Section 3presents the system model which is followed by the defini-tion of the the coverage probability in multiplexing systemsin Section 4. Section 5 considers ICI correlation, introducesFC assumption, and develops the A-A and G-A methods.Section 6 utilizes our analytical results to evaluate the cov-erage performance in MIMO diversity only systems, ZFBFwith nonhomogenous and homogenous SPLM, and systemswith available CSI at the transmitter (CSIT). Numerical andsimulation results are then presented in Section 7, followedby conclusions in Section 8.

2 RELATED WORK

The main focus of this paper is on the MIMO multiplex-ing systems where stochastic geometry tools are used inour analysis. Using SPLM assumption, the authors of [37]investigate the coverage probability of several prominentMIMO techniques in ad hoc networks. Furthermore, in [38]the impact of inaccurate channel state information on thecoverage probability is investigated in a clustered MIMO adhoc network. It is shown in [38] that in interference-limitedscenarios using single-user MIMO communications can im-prove the coverage performance. Coverage performanceand ASE of multiple-user spatial-division multiple access(SDMA) in multiple-input single-output (MISO) HetNetsis also investigated in [7], [39] and [40] for cases wherecell association (CA) is based on range expansion, i.e., UEsare associated with BSs with the smallest path-loss scaled

by a range extension parameter. A novel technique wasdeveloped in [41] that can be used for the evaluation offunctionals of Poisson point processes and the SIR distribu-tion of wireless systems under Nakagami fading. Further,a moment-generating method for approximating the SIRdistribution of SISO systems under Nakagami fading wasdeveloped in [42]. The authors of [43] and [44] focused onmaximum ratio combining (MRC) and optimal combiningin downlink and uplink of cellular networks, respectively.The Gil-Pelaez inversion theorem [45] is found instrumentalin analyzing the symbol error probability (SEP) of MIMOmultiplexing systems [46].

An equivalent-in-distribution (EiD) technique was sug-gested in [47] to understand the SEP of MIMO communi-cations. A unified method for studying the SEP in MIMOcommunications was proposed in [48] by adopting theEiD method. Moreover, the impact of interference-drivencorrelation on receiver arrays in ad-hoc network as wellas the downlink of a single-tier cellular network was in-vestigated in [49] and [50]. The authors of [51], [52], [53]demonstrated the importance of theoretical results devel-oped in [7], [40], [42] for the optimization of MIMO ellularsystems. For instance, in [52], the coverage probability,spectral efficiency, and load balancing in MIMO systemswere considered. Further, [51] optimized ASE and energy-efficiency in uplink/downlink multi-user MIMO system.For managing inter-cell interference, [54] investigated thecoupled optimized offloading and coordinated MIMO com-munications. Energy-efficiency of MIMO downlink was alsothe subject of [55] the authors which attempted to highlightthe significance of beamforming schemes.

Although significant in their own rights, all of the aboveefforts rely on the restricted path-loss model of SPLM andoften consider single-stream MIMO communication. For amore practical path-loss model, researchers often adoptedAlzer’s Theorem to derive the coverage probability of aSISO system under Nakagami fading and single-streammulti-user MIMO systems [17], [23], [56], [57]. The coverageprobability of a heterogenous device-to-device mmWavesystems was studied in [58], confirming the utility andaccuracy of the Alzer method. The authors further intro-duced a mixed inverse-gamma log-normal distribution toapproximate the interference distribution under LOS/NLOSpath-loss model.

In this paper we extend the Alzer method to the caseof MIMO multiplexing systems to investigate its accuracyand use as a benchmark for comparison with G-A approx-imation. For MIMO-ZFBF and under LOS/NLOS path-lossmodel in [59], [60], the transmission delay of a single-tierwireless ad hoc network was investigated. The importanceof LOS/NLOS path-loss model for achievable optimiza-tion of the network was highlighted. Further, importanceof spatially-coded MIMO configuration, packet retransmis-sion, and advanced hybrid repeat request protocols weredemonstrated. However, their analyses are applicable onlyto single-tier ad hoc networks, and may not be extensible tothe multi-tier cellular systems under max-SIR CA rule.

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Fig. 1. A schematic of the system model for K = 1. A BS located closerto the origin has a higher chance of LOS propagation. Because of theNLOS path-loss, the signals received from farther BSs are substantiallyweaker. Nevertheless, the typical UE can still be associated with a BSin NLOS mode due to, for example, fast fading fluctuations.

3 SYSTEM MODEL

We investigate the downlink communication in a multi-tiercellular network. The network is comprised of K ≥ 1 tiersof randomly located base stations (BSs), where the BSs of tieri, i ∈ K = 1, . . . ,K, are spatially distributed according toa homogenous Poisson point process (PPP), Φi, with spatialdensity, λi (the number of BSs per unit area), λi ≥ 0. PPPs,Φi, Φj , ∀i, j ∈ K, i 6= j are mutually independent. A HetNetconsists of Φi, i ∈ K, i.e., Φi∀i, is referred to as Φ.

UEs are randomly positioned across the network andform a PPP, ΦU , independent of Φ, with given density, λU .According to Slivnayak’s Theorem [61], [62] and due to thestationarity of the point processes, the spatial performanceof the network can be obtained from the perspective of atypical UE positioned at the origin. We also assume thatUEs are equipped with Nr antennas.

Tier i is fully characterized by the corresponding spatialdensity of BSs, λi, their transmission power, Pi, the mini-mum required received SIR for the UEs in Tier i, βi ≥ 1,the number of the transmit antennas at the BSs, N t

i , andthe number of scheduled streams Si ≤ minN t

i , Nr also

referred to as multiplexing gain [30], [63], [64]. Fig. 1 shows aschematic model of the network for K = 1.

3.1 Generic LOS/NLOS Path-Loss Model

A block fading wireless channel is considered, where at thebeginning of each time slot, an independent realization ofthe fading is generated and stays fixed throughout that timeslot. The typical UE is associated with BS xi, transmitting Si

data streams. The received signal, yxi∈ CNr×1, is

yxi=

√Li(‖xi‖)Hxisxi+

∑

j∈K

∑

xj∈Φj\xi

√Lj(‖xj‖)Hxj sxj ,

(1)where ∀xi, i ∈ K, sxi = [sxi,1 . . . sxi,Si ]

T ∈ CSi×1,sxi,li ∼ CN (0, Pi/Si) is the transmitted signal correspond-ing to stream li in Tier i, Hxi ∈ CNr×Si is the fadingchannel matrix between BS xi and the typical UE, withentries independently drawn from CN (0, 1). Transmitted

signals across different BSs are also assumed to be mutuallyindependent, and also independent of the channel matrices.In (1), Li(‖xi‖) is a generic distance-dependent path-lossfunction, where ‖xi‖ is the Euclidean distance between BSxi and the origin, which is random.

As shown in Fig. 1, a BS experiences LOS or NLOSpropagation, depending on its relative distance to the UE,density of buildings, type of the clutter, etc. To modelLOS/NLOS pathloss, we adopt the path-loss model recom-mended in the 3rd Generation Partnership Project (3GPP)[16], [17], [18], where the path-loss attenuation in Tier i is

Li(‖xi‖) =

Li

L(‖xi‖) with probability of piL(‖xi‖),

LiN(‖xi‖) with probability of pi

N(‖xi‖).(2)

For ni ∈ L, N, function Lini

(‖xi‖) can adopt any fea-sible path-loss function, e.g., Li

ni(‖xi‖) = φi

ni‖xi‖−αi

ni ,Li

ni(‖xi‖) = φi

ni(1 + ‖xi‖)−αi

ni , or Lini

(‖xi‖) =φi

nimax1, ‖xi‖−αi

ni , where αiL (αi

N) is the path-loss ex-ponent associated with the LOS (NLOS) link, φi

L (φiN) is a

constant, characterizing the LOS (NLOS) wireless propaga-tion environment, and is related to various factors, e.g., theheight of transceivers, antenna’s beam-width, weather, etc.

In (2), for a BS located at position xi the probability inLOS mode is pi

ni(‖xi‖), where

∑ni∈L,N

pini

(‖xi‖) = 1. For

instance, ITU-R UMi model is [16], [17]

piL(‖xi‖) = min

Di

0

‖xi‖ , 1 (

1− e− ‖xi‖

Di1

)+ e

− ‖xi‖Di

1 , (3)

where, parameters Di0, and Di

1 characteriize the near-field(LOS), and far-field (NLOS) critical distances, respectively.Therefore, if ‖xi‖ ≤ Di

0, BS xi is in LOS mode. For‖xi‖ > Di

0, the probability of LOS mode declines exponen-tially with the distance, and for ‖xi‖ > Di

1, it decreasesquickly to 0.

A similar approach was also adopted in [22], [23], [57]to characterize pi

L(‖xi‖). Note that the model in (3) andsimilar approaches in [21], [22], [57] all have a certain levelof adjustability to the communication environment (urban,dense urban, or suburban) or the clutter city (flat, scattered,hill-sided, etc.). The critical parameters of these mathemat-ical models, such as Di

0 and Di1 in (3), are often obtained

using experimental measurements complemented by dataanalysis techniques, see, e.g., [57]. Therefore, some of thehidden aspects of channel modeling, such as the correlationin LOS mode—caused by large obstacles/buildings in anarea which similarly affect the transmitted signals of adja-cent BSs—are eventually represented in the path-loss model.(It is straightforward to confirm that the SPLM abides by(2).)

For the simulation and numerical studies we considerthe path-loss attenuation function, Li

ni(‖xi‖) = φi

ni(1 +

‖xi‖)−αini , along with the LOS probability, (3), unless oth-

erwise stated. Although our main focus is on the genericpath-loss model, (2), with an arbitrary LOS probability, it isstraightforward to extend our analysis to other models, suchas multi-slope path-loss [15], and multi-ball path-loss [21].

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3.2 SIR of Data Streams

In the analysis we assume the availability of perfect chan-nel state information at the receiver (CSIR), while CSI atthe transmitter (CSIT) is not available. Each BS xi turnson Si transmit antennas and equally divides its transmitpower, Pi, among them. This transmission scheme is oftenreferred to as open-loop pre-coding, see, e.g., [63], [64]. Atthe receiver, the system employs zero-forcing beamforming(ZFBF) [30], [63]. To decode the li-th stream, in ZFBF, atypical UE uses the available CSIR, Hxi , to mitigate theinter-stream interference. The typical UE also obtains matrix(H†

xiHxi)

−1H†xi

, where (.)† is the conjugate transposeoperator, and then multiplies the conjugate of the li-th col-umn by the received signal in (1). In an interference-limitedregime, i.e., ignoring noise, the post-processing SINR asso-ciated with the li-th stream is

SIRxi,li =Pi

SiLi(‖xi‖)HZF

xi,li

I li, (4)

where

I li ∆=∑

j∈K

∑

xj∈Φj\xi

Pj

SjLj(‖xj‖)GZF

xj ,li , (5)

is the inter-cell interference (ICI) stream li experienced.As shown in [63], [64], the intended channel power gains

associated with the li-th data stream, HZFxi,li

, and the ICIcaused by xj 6= xi on data stream li, GZF

xj ,li, are chi-squared

random variables with 2(Nr − Si + 1), and 2Sj , degrees offreedom (DoF), respectively. For each li, HZF

xi,liand GZF

xj ,li

are independent random variables. For for l 6= li, HZFxi,li

(GZFxj ,li

) and HZFxi,l

(GZFxj ,l) are independent and identically

distributed (i.i.d.). In (4), for a given communication link,SIRZF

xi,li , are identical, but not independent across streams,see Section 5.1.

4 COVERAGE PROBABILITY IN MULTI-STREAM

MIMO HETNETS

The coverage probability is defined as the probability thatthe SIR stays above a given SIR threshold. The coverageprobability in a cellular network is often related to thecomplementary cumulative distribution function (CCDF)of the received SIR [9], [10], [62]. The same definition isalso used in SISO and single-stream MIMO communicationsystems, e.g., diversity systems and space division multipleaccess [6], [7]. In multiple stream MIMO, we consider thecoverage probability as the probability that all of a typicalUE’s streams are successfully decoded at the receiver. Sucha notion of coverage probability is also referred to as all-coverage probability in isolated scenarios, e.g., [30], [31], [32],[33], [65], [66].

4.1 Cell Association

To evaluate the coverage probability, we first need to charac-terize the mechanism used to associate UEs with BSs. Thismechanism is often referred to as cell association (CA). De-pending on the communication scenario, application contextand signaling structure, two main approaches have beenproposed in the literature, including max-SIR [1], [4], [6],

[32], [33], [34], [67], [68], and range expansion (a.k.a. closest-BS) [5], [7], [9].

In the closest-BS (max-CIR) CA, the BS located closest(thus providing the maximum CSI) to the user is consideredas the serving BS. Our previous work [14], [34] showedthat the coverage performance is substantially improved byadopting max-SIR CA. SIR-based CA is also integrated intovarious resource-allocation mechanisms by incorporatingthe physical-layer specifications, transmission policies, andscheduling and coordination across tiers, see, e.g., [36], [69],[70], [71], [72].

4.2 Coverage Probability

In a system with max-SIR, a BS is selected as the servingBS for a typical UE, if all SIRs across the streams are largerthan the SIR threshold, βi. Therefore, for a typical UE thecoverage probability of ZFBF is

cZF = PAZF 6= ∅, (6)

where

AZF =∃i ∈ K : max

xi∈Φi

minli=1,...,Si

SIRZFxi,li ≥ βi

. (7)

The NLOS signals are expected to be, on average, weakerthan that of LOS. In some cases however, the fading fluctu-ation and the impact of array processing may cause the CAto select a NLOS BSs.

Evaluating cZF, for LOS/NLOS, is challenging due tothe following issues. First, for each data stream, the fadingfluctuation in the intended signal is chi-squared, whichoften results in a less tractable analysis than that of Rayleighfading. Second, the unconventional LOS/NLOS path-lossmodel exacerbates the complexity of analysis. Third, the ICIcorrelation across data streams in a communication link fur-ther interrelates the stream and link coverage probabilities.The cross-stream ICI correlation is created by the existenceof the same interferers across data streams, see Section 5.1.

Therefore, conditioned on GZFxj ,li

, ∀li, the interferenceoriginated from BS xj , depends on Lj(‖xj‖), which is arandom variable (r.v.) independent of data stream li. In[30], [33], we have already developed analytical tools thatenable us to deal with the first and the third issues above incases with SPLM. In what follows, we use this to obtain thecoverage probability in cases with the LOS/NLOS path-lossmodel, addressing the above three issues.

Proposition 1: The coverage probability of a multi-stream MIMO-ZFBF cellular network with LOS/NLOSpath-loss, (2), is

cZF =∑i∈K

2πλi

Nr−Si∑m1=0

. . .

Nr−Si∑mSi

=0

(−1)m1+...+mSi

m1! . . . mSi !

∞∫

0

ridri

∂m1 . . . ∂mSi

(∑n∈L,N pi

n(ri)Ψin(ri)

)

∂tl1m1 . . . ∂tlSi

mlSi

∣∣∣tli

=1,∀li

,

where for each n ∈ L,N,

Ψi

n(ri) = exp

−2π

K∑

j=1

λj

∞∫

0

yj

(1−Ψi

n(ri, yj))dyj

,

6

Ψin(ri, yj) =

∑

n′∈L,Npj

n′(yj)Si∏

li=1

(1 +βiSiPjL

jn′(yj)

SjPiLin(ri)

tli)−Si

Proof: See Appendix A. ¥The coverage probability in Proposition 1 is a function

of tiers’ BS densities, their SIR thresholds, transmissionpowers, and multiplexing gains. The impact of of LOS,and NLOS path-loss for the intended link are captured byfunctions Ψ

i

N(ri), and Ψi

N(ri), respectively. Furthermore,Ψ

i

N(ri) and Ψi

N(ri), are functions of the LOS/NLOS modesof the interfering links, respectively, through Ψi

L(ri, yj), andΨi

N(ri, yj).As a key performance parameter, the coverage probabil-

ity is often required to be calculated many times in variousresource-allocation schemes to find the best combination ofthe network operational parameters. Calculating the cov-erage probability in Proposition 1 is, however, challengingdue to the requirement of extensive numerical calculations.Thus, we propose several approximations of the coverageprobability with acceptable accuracy and reasonable com-putational complexity.

5 COVERAGE PROBABILITY APPROXIMATION

The link-level coverage probability, as defined in Section3, is directly related to the SIR of every single stream inthat link. The SIR values for streams are, however, stronglycorrelated due to cross-stream ICI correlation. Therefore, toevaluate the coverage probability, we first need to quantifythe cross-stream ICI correlation.

5.1 Cross-Stream ICI Correlation

We use Pearson correlation coefficients to characterize theICI correlation between streams li 6= l′i:

ρli,l′i =E[I liI l′i ]− E[I li ]E[I l′i ]√

Var(I li)Var(I l′i), (8)

where Var(x) is the variance of random variable x. Wefurther note that the ICI is an identical shot-noise processacross streams, so ρli,l′i is1

ρli,l′i =E[I liI l′i ]− (E[I li ])2

Var(I li). (9)

Proposition 2: For a MIMO-ZFBF multiplexing systemwith the generic LOS/NLOS pathloss,

ρli,l′i =

∑j∈K

P 2j λj

∑nj∈L,N

∞∫0

rjpjnj

(rj)(Ljnj

(rj))2drj

∑j∈K

P 2j

Sj(Sj+1)−1

S2j

λj

∑nj∈L,N

∞∫0

rjpjnj (rj)(L

jnj (rj))2drj

.

(10)Proof: See Appendix B. ¥

1. Interference correlation is studied in [50], [62], [73], focusing onquantifying the impact of interference correlation on the time andreceive-array diversities. Here we are interested in quantifying theimpact of multiplexing gains and NLOS/NLOS path-loss model onthe interference correlation. We also use this analysis to corroborate thevalidity of the full-correlation (FC) assumption for approximating thecoverage probability.

For a single tier network, K = 1, Proposition 2 results inthe following corollary.

Corollary 1: For a single tier MIMO-ZFBF,

ρl1,l′1 =S2

1

S1(S1 + 1)− 1, (11)

which only depends on the multiplexing gain.Proposition 3: In a MIMO-ZFBF multiplexing system,

0.5 ≤ 11 + maxi

Si−1S2

i

≤ ρli,l′i ≤1

1 + miniSi−1S2

i

≤ 1. (12)

Proof: See Appendix C. ¥Proposition 3 shows that in the MIMO multiplexing

system, ρli,l′i is larger than 0.5, so ICI is highly correlatedacross data streams. In Fig. (2-a), we illustrate ρl1,l′1 vs. S1 fora simulated system. ICIs across data streams are shown to besignificantly correlated. Fig. (2-a) also confirms the validityof the bound in Proposition 3.

Also, Fig. (2-a) shows that for S1 > 2 (S1 < 2), ρl1,l′1 isan increasing (decreasing) function of S1, and its maximum(minimum) occurs at S1 = 2. Setting the derivative of (??)to zero, one can analytically obtain S1 = 2:

∂ρl1,l′1

∂S1=

∑j∈K

λjAj

(∑j∈K

λjAjSj(Sj+1)−1

S2j

)2

S1(S1 − 2)S4

1

= 0. (13)

In Fig. (2-b), we present ρl1,l′1 vs. α2L. ρl1,l′1 in (2-b) is

shown to vary within the bounds given in Proposition 3.Fig. (2-b) shows ρl1,l′1 to be also highly correlated acrossstreams, and an increasing function of α2

L. This is also statedin the following corollary.

Corollary 2: Increasing α2L results in a higher ρl1,l′1 for

any combination of multiplexing gains for which

∑

j∈KλjAj

Sj(Sj + 1)− 1S2

j

>S2(S2 + 1)− 1

S22

∑

j∈KλjAj . (14)

Proof: See Appendix D. ¥The system considered for the simulation in Fig. (2-b) is

a two-tier system, K = 2. For this system (14) is reduced to:

S1(S1 + 1)− 1S2

1

>S2(S2 + 1)− 1

S22

,

which holds if S1 > S2 > 2.2 That is, for any S1 > S2 > 2,∂ρl1,l′1

∂α2L

> 0.

5.2 Full Correlation Assumption

Our analysis in Section 5.1 indicates that the ICI is highlycorrelated across the streams, justifying the construction of’full-correlation‘ (FC) approximation, where the ICI acrossall streams in a given communication link is considered fullycorrelated, i.e., ρli,l′i = 1. Such an assumption has also beenused for analyzing other aspects of MIMO systems in [32],[33], [43].

2. This is because g(x) =x(x+1)−1

x2 is an increasing function of x forfor x > 2.

7

1 2 3 4 5 6 7 80.8

0.85

0.9

0.95

1

(a): S2=3, α

L2=2.5, λ

1=10−3, λ

2=10−1

S1

ρ 1,2

Upper Bound

Lower Bound

1 1.5 2 2.5 3 3.5 4 4.50.84

0.85

0.86

0.87

0.88

0.89

0.9

αL2

ρ 1,2

(b): S1=5, S

2=3, λ

1=10−3, λ

2=10−1

Upper Bound

Lower Bound

Fig. 2. Cross-stream ICI correlation coefficient vs. S1, α2L, the simulation results and bounds, in a system with K = 2, α1

N = 3.75, α2N = 4.75,

P1 = 25 W, P2 = 1 W, Nt1 = 16, Nt

2 = 8, Nr = 8, D10 = 36, D2

0 = 9, D11 = 48, D2

1 = 18 meter, and φiL = φi

N = 1.

Assuming FC, ICI for the typical UE associated with BSxi is

IFC =∑

j∈K

∑

xj∈Φj\xi

Pj

SjLj(‖xj‖)GZF

xj, (15)

where for li = 1, 2, . . . , Si we simply replace the interferingchannel power gain, GZF

xj ,li, with GZF

xj(both chi-squared r.v.s

with 2Sj DoFs). For data stream li, the post-processing SIRis then approximated as

SIRZF−FCxi,li

=

PiSi

Li(‖xi‖)HZFxi,li

IFC. (16)

Therefore, for the max-SIR CA under the FC assumption,the max-SIR CA rule 7 is rewritten as

AZF−FC =

∃i ∈ K : max

xi∈Φi

PiSi

Li(‖xi‖)HZFxi,min

IFC≥ βi

, (17)

HZFxi,min

∆= minli=1,...,Si

HZFxi,li

. This implies that the prob-

ability of coverage for the typical UE is cZF−FC =Pr

AZF−FC 6= ∅. Similarly to Appendix A, we then useLemma 1 in [4] to derive the coverage probability as

cZF−FC = P

max⋃

i∈Kxi∈Φi

PiSi

Li(‖xi‖)HZFxi,min

IFC≥ βi

=∑i∈K

E∑

xi∈Φi

1

(PiSi

Li(‖xi‖)HZFxi,min

IFC≥ βi

)

=∑i∈K

2πλi

∞∫

0

riP

PiSi

Li(ri)HZFxi,min

IFC≥ βi

dri

=∑i∈K

2πλi

∞∫

0

riELi(ri),Φ,Lj(‖xj‖)∀xj,jP

HZFxi,min ≥

βiIFC

PiSi

Li(ri)

∣∣Φ, Li(ri), Lj(‖xj‖)∀xj ,j

dri. (18)

Evaluating the coverage probability based on (18) is shownto be a complicated task. To address this difficulty, wepropose two approximations for (18) which enable the nu-merical evaluation of the coverage probability as a functionof the main system parameters.

5.2.1 Alzer ApproximationAlzer’s inequality [56], [57] (see Appendix-C Lemma 1)has been used to evaluate the coverage probability underNakagami-type fading and multi-user MIMO systems usingstochastic geometry. Here, for the first time we extendAlzer’s inequality to approximate the coverage probabilityof a multi-stream multi-tier MIMO-ZFBF system. We referto this method as A-A, where Alzer’s lemma is utilizedto approximate the effective power gain of the attendingchannel for each data stream as an exponential randomvariable. This is presented in the following Proposition.

Proposition 4: In a multi-stream MIMO-ZFBF cellu-lar system with LOS/NLOS pathloss model, and S′i =((Nr − Si + 1)!)−

1(Nr−Si+1) , the coverage probability is ap-

proximated as

cA−A =∑i∈K

2πλi

(1 +

Si∑

l′i=1

(Nr−Si+1)l′i∑

l′′i =0

(−1)l′i+l′′i ×

((Nr − Si + 1)l′i

l′′i

)(Si

l′i

) ∑ni∈L,N

∞∫

0

ripini

(ri)Ψini

(ri)dri

), (19)

where Ψini

(ri) = exp

(−2π

∑j

λj

∞∫0

yj(1− Ψini

(ri, yj))dyj

),

Ψini

(ri, yj) =

∑

nj∈L,N

pjnj

(yj)(1 + S′il

′′i

βiSiPjLjnj

(yj)

SjPiLini

(ri)

)Sj

.

Proof: See Appendix E. ¥The numerical complexity of obtaining cA−A in Propo-

sition 4 is significantly lower than that of Proposition 1 asthere are no concatenated higher-order differentiations. Theimpact of LOS/NLOS model parameters on the intendedand interfering links can be seen in Ψi

L(ri)/ ΨiN(ri), and

ΨiL(ri, yj)/Ψi

N(ri, yj).

5.2.2 Gamma ApproximationThe coverage probability, cZF−FC, is

cZF−FC =∑

i∈K2πλi

∑

ni∈L,N

∞∫

0

ripini

(ri)×

8

0 50 100 1500

0.2

0.4

0.6

0.8

1

x

CD

F

sim. αL2=1.5

Gamma fit αL2=1.5

sim. αL2=2.09

Gamma fit αL2=2.09

sim. αL2=4

Gamma fit αL2=4

0 5000 10000 150000

0.2

0.4

0.6

0.8

1

x

CD

F

sim. S1=4

Gamma fit S1=4

sim. S1=1

Gamma fit S1=1

Fig. 3. CDF of 1IFC , (a): λ1 = 10−4, λ2 = 10−3, S1 = 2, S2 = 2,

α1L = 2.09, α1

N = 3.75, α2N = 4.75, P1 = 50W, P2 = 10W, Nt

1 = 16,Nt

2 = 8, Nr = 8, and φiL = φi

N = 1; (b) λ1 = 10−4, λ2 = 10−3,S2 = 2, α1

L = 2.09, α1N = 3.75, α2

L = 1.5, α2N = 4.75, P1 = 50W,

P2 = 10W, Nt1 = 16, Nt

2 = 8, Nr = 8, and φiL = φi

N = 1.

EIFCP

HZF

xi,min

IFC≥ βi

Pi

SiLi

ni(ri)

∣∣Φ, IFC

dri. (20)

Instead of the intended fading gain, one may approximatethe statistical characteristics of IFC. Note, however, that inthe max-SIR CA, in some cases the interferers are even closerto the typical UE than the serving BS. This can happen, forinstance, in LOS mode with a very small LOS path-loss ex-ponent. For LOS path-loss functions, where Li

L(x) ∝ x−αiL ,

the mean and variance of IFC could be very high. Instead,we model 1

IFC .The CDF of 1

IFC is plotted in Fig. (3) for the simulatedsystem with the parameters given in the caption. Fig. (3-a) shows that for different values of the LOS path-lossexponent, the CDF closely follows the Gamma distribution.Fig. (3-b) further shows that the approximation based onGamma distribution remains valid for various multiplexinggains. This has also been confirmed for a variety of othersystem parameters which are not reported here due to spacelimitation.

Based on the above, we approximate the CDF of 1IFC

by a Gamma distribution, 1IFC ∝ Gamma(a, b), where,

adopting a moment-matching technique, parameters a andb are obtained as

a =(E 1

IFC )2

(E 1(IFC)2

− (E 1IFC )2)

, (21)

b =E 1

IFC

(E 1(IFC)2 − (E 1

IFC )2). (22)

For a Gamma(a, b) random variable with pdf fX(x) =ba

Γ(a)xa−1e−bx, X = a

b and Var(X) = ab2 . To obtain a and b,

we need to calculate E 1IFC and E 1

(IFC)2. The former is given

by

E1

IFC= E

∞∫

0

e−vIFCdv =

∞∫

0

Ψ(v)dv, (23)

where Ψ(v) is the Laplace transform of IFC:

Ψ(v) = Ee−v

∑j∈K

∑xj∈Φj\xi

PjSj

Lj(‖xj‖)GZFxj

=∏

j∈KEΦj

∏

xj∈Φj\xi

EGZFxj

,Lj(‖xj‖)e−v

PjSj

Lj(‖xj‖)GZFxj

=∏

j∈KEΦj

∏

xj∈Φj\xi

∑

nj∈L,N

pjnj

(‖xj‖)(1 + v

Pj

SjLj

nj (‖xj‖))Sj

= exp(−2π∑

j

λj

∞∫

0

yj(1− ΨZF(v, yj))dyj), (24)

where

ΨZF(v, yj) =∑

nj∈L,N

pjnj

(yj)(1 + Pj

SjLj

nj (yj))Sj

. (25)

Note that in ΨZF(v, yj , Sj) the first (second) term is asso-ciated with the LOS (NLOS) mode of the BS located at yj .Similarly, the second moment of 1

IFC is:

E1

(IFC)2= E

∞∫

0

∞∫

0

e−(v1+v2)IFC

dv1dv2 =

∞∫

0

∞∫

0

Ψ(v1 + v2)dv1dv2 =

∞∫

0

vΨ(v)dv. (26)

Using the above, in the following we approximate thecoverage probability using Gamma distribution, which willhenceforth be called G-A.

Proposition 5: We define

∑i,k

∆=∑

k0+k2+...+kNr−Si=Si−1

(Si − 1

k0, k2, . . . , kNr−Si

),

and Si(k) ∆= Nr − Si + 1 +Nr−Si∑

l=0

lkl. An approximation,

namely G-A, of the coverage probability in a multi-streamMIMO-ZFBF cellular network with an LOS/NLOS attenua-tion, (2), is

cG−A =∑

i∈K

2πλiSi

(Nr − Si)!Γ(a)

∑

ni∈L,N

∑i,k

×∞∫

0

∞∫

0

ripini

(ri)hSi(k)−1e−Sih

Nr−Si∏l=0

(l!)kl

γ

(a, b

βi

Pi

SiLi

ni(ri)h

)dridh

(27)where γ(a, bx) is the CDF of random variable Gamma(a, b).

Proof: See Appendix F. ¥Compared to Proposition 1, in Proposition 4 the numer-

ical complexity of obtaining an approximation of the cov-erage probability is substantially reduced. Nevertheless, thenumerical complexity of obtaining cG−A is higher than thatof cA−A, because in G-A, a and b should also be obtainedas in (21) and (22), respectively. G-A further requires thecalculation of double integration which is not required inA-A, see (46).

In G-A, a and b are tier-independent—once they areobtained for a given setting (density of BSs, transmissionpowers, multiplexing gains, and LOS/NLOS parameters),they are valid regardless of the tier that the typical UE is as-sociated with. This substantially reduces the computationalcost of evaluating (52).

9

The simulation results in Section 7 reveal that the extracomplexity of G-A brings a higher accuracy and robustnessover a wide range of system parameters. Compared to A-A, G-A also captures the actual behavior of the coverageprobability against densification more precisely. Further, asshown in Section 7, G-A enables accurate evaluation of BSs’density for which the maximum coverage performance isachieved.

6 SPECIAL CASES

We use the results derived for open-loop ZFBF MIMOmultiplexing system to evaluate the coverage performancein MIMO diversity only, ZFBF with nonhomogeneous andhomogenous SPLM, and systems with available CSIT (fullCSIT and quantized CSIT). The main objective is to demon-strate how one can derive the coverage probability of vari-ous MIMO system settings using the analytical frameworkdeveloped in this paper. The analytical results here aresupported further by the simulations and numerical resultsin Section 7.

6.1 Diversity Only Systems

In this type of systems, Si = 1, ∀i, i.e., single-stream MIMOor single-input multiple-output (SIMO) systems.

A-A Method: Proposition 4 is reduced to

cSIMO−A−A =∑

i∈K2πλi

(1 + Si

Nr∑

l′′i =0

(Nr

l′′i

)(−1)l′′i +1

∑

ni∈L,N

∞∫

0

ripini

(ri)Ψini

(ri)dri

), (28)

where Ψini

(ri) is defined as

Ψini

(ri, yj) =∑

nj∈L,N

pjnj

(yj)(1 + (Nr!)−

1Nr l′′i

βiPjLjnj

(yj)

PiLini

(ri)

)Sj.

G-A Method: Using Proposition 5, G-A approximationimplies that

cSIMO−G−A =∑

i∈K

2πλi

(Nr − 1)!Γ(a)

∑

ni∈L,N

×∞∫

0

∞∫

0

ripini

(ri)hNr−1e−hγ

(a, b

βi

PiLini

(ri)h

)dridh,

(29)where

a =(E 1

IFC )2

(E 1(IFC)2 − (E 1

IFC )2), (30)

b =E 1

IFC

(E 1(IFC)2 − (E 1

IFC )2), (31)

and

E1

IFC=

∞∫

0

exp(−2π∑

j

λj

∞∫

0

yj(1− ΨD(v, yj))dyj)dv,

(32)

E1

(IFC)2=

∞∫

0

v exp(−2π∑

j

λj

∞∫

0

yj(1− ΨD(v, yj))dyj)dv,

(33)where ΨD(v, yj) is defined similarly to (25), where Sj = 1.

6.2 ZFBF Multiplexing with Non-homogeneous SPLM

In non-homogeneous SPLM, αiL = αi

N = αi and φiL = φi

N ,∀i.

A-A Method: It is straightforward to show that the ap-proximation of the coverage probability based on A-A is

cA−A =∑i∈K

2πλi

(1 +

Si∑

l′i=1

(Nr−Si+1)l′i∑

l′′i =0

((Nr − Si + 1)l′i

l′′i

)

(−1)l′i+l′′i

(Si

l′i

) ∞∫

0

rie

−2π∑j

λjW (αj)r

2αiαj

i

αj(S′il′′iβiSiPj

Pi)2+ 2

αj

dri, (34)

where W (αj) =∞∫0

w−1− 2

αj (1− e−w)dw.

G-A Method: Using Proposition 5,

cG−A =∑

i∈K

2πλiSi

(Nr − Si)!Γ(a)

∑i,k

×∞∫

0

∞∫

0

rihSi(k)−1e−Sih

Nr−Si∏l=0

(l!)kl

γ

(a, b

βirαii

Pi

Sih

)dridh, (35)

where parameters a and b are defined, respectively, in(30) and (31). To derive these parameters, we should re-place ΨZF(v, yj) in (25) with ΨZF−Nonhomogeneous(v, yj) =

1(1+

PjSj

y−αjj

)Sj, along with (32) and (33).

6.3 ZFBF Multiplexing with Homogeneous StandardPath-Loss Model

In Homogeneous SPLM, αi = α ∀i.A-A Method: One may start with (34) and apply a

straightforward integration, to show that the coverage prob-ability is approximated as

cA−A =α

W (α)∑j

λjP αj

∑

i∈K

λi( Pi

βiSi)2+α

(S′i)2+α

(1+

Si∑

l′i=1

(Nr−Si+1)l′i∑

l′′i =0

(−1)l′i+l′′i

(Si

l′i

)((Nr − Si + 1)l′i

l′′i

)l′′2+α

i

).

This expression is similar to the upper-bound provided inProposition 1 in [30]:

cZF ≤ π

C(α)

∑

i∈K

λi

(Pi

S2i βi

)α(

Nr−Si∑ri=0

Γ( αSi

+ri)

Γ( αSi

)Γ(1+ri)

)Si

∑j∈K λj

(Pj

Sj

)α(

Γ( αSi

+Sj)

Γ(Sj)

)Si,

(36)where C(α) = πΓ(1 − α). It is interesting to observe someresemblance between (36) and cA−A.

10

G-A Method: Proposition 5 yields:

cG−A =∑

i∈K

2πλiSi

(Nr − Si)!Γ(a)

∑i,k

×∞∫

0

∞∫

0

rihSi(k)−1e−Sih

Nr−Si∏l=0

(l!)kl

γ

(a, b

βirαi

Pi

Sih

)dridh, (37)

where parameters a and b are defined in (30) and (31),respectively. To derive these parameters, we set ΨZF(v, yj)in (25) with ΨZF−Homogeneous(v, yj) = 1(

1+PjSj

y−αj

)Sj, along

with (32) and (33).

6.4 Known CSIT

In the above derivations, we simply assume that the CSIT isnot known to the BSs. However, there are practical scenariosthat CSIT is available to BSs. So, our analysis is shown tocover such cases as well.

6.4.1 Single-Input Single-Output (SISO) Systems

For a SISO system, Si = N ti = Nr = 1 and Proposition 1

yields:

cSISO =∑

i∈K2πλi

∑

n∈L,N

∞∫

0

ripin(ri) exp(−2π

K∑

j=1

λj

∞∫

0

yj

(1−Ψn(ri, yj)

)dyj)dri,

where

Ψn(ri, yj) =∑

n′∈L,Npj

n′(yj)(1 +βiPjL

jn′(yj)

PiLin(ri)

)−1.

Note that for this scenario, both A-A and Proposition 1yield the same coverage performance.

G-A Method: Using Proposition 5,

cSISO−G−A =∑

i∈K

2πλi

Γ(a)

∑

ni∈L,N

×∞∫

0

∞∫

0

ripini

(ri)e−hγ

(a, b

βi

PiLini

(ri)h

)dridh, (38)

where a, and b are defined in (30), and (31), respectively. Toderive these parameters, one should replace ΨZF(v, yj) in

(25) with ΨSISO(v, yj) =∑

nj∈L,N

pjnj

(yj)

(1+PjLjnj

(yj)) , along with

(32) and (33).

6.4.2 Multiple-Input Single-Output (MISO) SystemsIn a MISO system, Nr = 1, and Si = 1, ∀i. We assume thatavailable CSIT at the BSs is utilized for eigen-beamforming,i.e., maximum ratio transmission (MRT) [74]. In this system,the SIR at the typical UE served by BS xi is

SIRMRTxi

=PiLi(‖xi‖)HMRT

xi∑j∈K

∑xj∈Φj/xi

PjLj(‖xj‖)GMRTxj

, (39)

where HMRTxi

and GMRTxj

are chi-squared with 2N ti DoFs,

and exponential random variables, respectively.A-A Method: Using Proposition 4,

cMRT−A−A =∑

i∈K2πλi

(1 +

Nr∑

l′′i =0

(−1)l′′i +1

(Nr

l′′i

) ∑

ni∈L,N

∞∫

0

ripini

(ri)Ψini

(ri)dri

), (40)

where Ψini

(ri) = exp(−2π∑j

λj

∞∫0

yj(1 − Ψini

(ri, yj))dyj),

and,

ΨiL(ri, yj) =

∑

nj∈L,N

pjnj

(yj)(1 + (Nr!)−

1Nr l′′i

βiPjLjnj

(yj)

PiLini

(ri)

)Sj.

G-A Method: Using Proposition 5,

cSIMO−G−A =∑

i∈K

2πλi

(Nr − 1)!Γ(a)

∑

ni∈L,N

×∞∫

0

∞∫

0

ripini

(ri)hNr−1e−hγ

(a, b

βi

PiLini

(ri)h

)dridh,

(41)where a and b are obtained by replacing ΨZF(v, yj) in (25)

with ΨSIMO(v, yj) =∑

nj∈L,N

pjnj

(yj)

(1+PjLjnj

(yj)) , along with (32)

and (33).

6.4.3 MISO-SDMA Systems

Another instance is when the BSs have access to CSIT inan MISO-SDMA system. Here Nr = 1, and Si = 1, ∀i.We further assume that each cell of tier i serves Ui ≤ N t

i

UEs using ZFBF at the transmitter, see, e.g., [6], [7] for moredetails. Assuming a fixed transmit power, the SIR of thetypical UE associated with BS xi is

SIRSDMAxi

=Pi

UiLi(‖xi‖)HSDMA

xi∑j∈K

∑xj∈Φj/xi

Pj

UjLj(‖xj‖)GSDMA

xj

, (42)

where HSDMAxi

and GSDMAxj

are both chi-squared randomvariables with 2(N t

i − Ui + 1) and 2Uj DoFs, respectively[6], [14].

A-A Method: Using Proposition 4,

cSDMA−A−A =∑

i∈K2πλi

(1 +

Nrl′i∑

l′′i =0

(−1)l′′i +1

11

(Nrl′il′′i

) ∑

ni∈L,N

∞∫

0

ripini

(ri)Ψini

(ri)dri

), (43)

where Ψini

(ri) = exp(−2π∑j

λj

∞∫0

yj(1 − Ψini

(ri, yj))dyj), in

which,

Ψini

(ri, yj) =∑

nj∈L,N

pjnj

(yj)(1 + U ′

i l′′i

βiUiPjLjnj

(yj)

UjPiLini

(ri)

)Uj

.

G-A Method: Using Proposition 5, the G-A approximationis

cSIMO−G−A =∑

i∈K

2πλi

(Nr − Ui)!Γ(a)

∑

ni∈L,N

×∞∫

0

∞∫

0

ripini

(ri)hNr−Uie−hγ

(a, b

βi

Pi

UiLi

ni(ri)h

)dridh,

(44)where parameters a and b are obtained by re-placing ΨZF(v, yj) in (25) with ΨSDMA(v, yj) =

∑nj∈L,N

pjnj

(yj)(

1+PjUj

Ljnj

(yj)

)Ujalong with (32) and (33).

6.5 Imperfect CSIT

In practice, instead of a perfect CIST, often a delayed and/orquantized version of channel directional information (CDI)is available at the BSs. Consider the MISO-SDMA systemdiscussed above, and assume UEs report CDI using Bi

feedback bits associated with tier i. Following the sameline of argument as in [75], [76], [77], we can obtain thereceived channel power and interference statistics. Never-theless, based on imperfect CDI, zero-forcing beamformingis unable to completely eliminate the inter-user interference.Therefore, assuming quantization cell approximation (QCA)[75], [77], [78], the SIR of the typical UE associated with BSxi is :

SIRQSDMAxi

≈ (45)

Pi

UiLi(‖xi‖)HSDMA

xi(1− φi)

Pi

UiLi(‖xi‖)GQSDMA

xi φi +∑j∈K

∑xj∈Φj/xi

Pj

UjLj(‖xj‖)GSDMA

xj

,

where φi = 2− Bi

Nti−1 and GQSDMA

xiis an exponentially

distributed random variable with unit mean and indepen-dent of HSDMA

xiand GSDMA

xi[77], [78]. Note that the SIR

expression in (45) is in fact an approximation derived basedon QCA. Therefore, assuming exponential distribution forrandom variable, GQSDMA

xi, highly relies on QCA and may

become inaccurate if this assumption does not hold. Never-theless, as our simulations also suggest, the QCA assump-tion provides a high level of accuracy, and therefore (45) israther a close approximation.

For a system with imperfect CDIR, we obtain the outageprobability using A-A and G-A methods.

A-A Method:

Using Proposition 4,

cQSDMA−A−A =∑

i∈K2πλi

(1 +

Nrl′i∑

l′′i =0

(−1)l′′i +1

(Nrl′il′′i

) ∑

ni∈L,N

∞∫

0

ripini

(ri)Ψini

(ri)1 + U ′

i l′′i

Pi

UiLi

ni(ri)φi

dri

), (46)

where Ψini

(ri) = exp(−2π∑j

λj

∞∫0

yj(1 − Ψini

(ri, yj))dyj), in

which,

Ψini

(ri, yj) =∑

nj∈L,N

pjnj

(yj)(1 + U ′

i l′′i

βi

1−φi

UiPjLjnj

(yj)

UjPiLini

(ri)

)Uj

.

G-A Method:

Using Proposition 5, the G-A approximation is

cQSDMA−G−A =∑

i∈K

2πλi

(Nr − Ui)!Γ(a)

∑

ni∈L,N

×∞∫

0

∞∫

0

ripini

(ri)hNr−Uie−h

1 + Pi

UiLi

ni(ri)φi

γ

(a, b

βi

(1−φi)

Pi

UiLi

ni(ri)h

)dridh,

(47)where a and b are obtained by replacing ΨZF(v, yj) in (25)

with ΨSDMA(v, yj) =∑

nj∈L,N

pjnj

(yj)(

1+PjUj

Ljnj

(yj)

)Ujalong with

(32) and (33).From the above analysis, one can see that an imper-

fect CSIT increases the SIR threshold from βi to βi

1−φi.

Imperfect CSIT also creates extra interference—inter-userinterference— but the level of this extra interference isreduced by increasing Bi.

7 SIMULATIONS AND NUMERICAL ANALYSIS

We first evaluate the accuracy of the proposed approxima-tions of the coverage probability. We then study the effectsof various system parameters on the coverage probability aswell as area spectral efficiency (ASE) to gain insight on theeffects of densification, multiplexing gains, and propagationenvironment.

The simulation results are based on the Monte Carlosimulation, where 40,000 snap-shots are independently sim-ulated and averaged. In each snap-shot, we randomly createBSs based on the given densities in a disk with radiusof 10,000 units. The fading matrices are then randomlygenerated for each snap-shot based on a Rayleigh fadingdistribution. The LOS/NLOS path-loss for each BS is alsospecified based on the probabilistic model in (3). Systemparameters are set as: P1 = 25 W, P2 = 1 W, N t

1 = 16,N t

2 = 8, Nr = 8, β1 = 5, β2 = 2.5, D10 = 36, D2

0 = 9,D1

1 = 48, D21 = 18 meters, and φi

L = φiN = 1.

12

1 1.5 2 2.5 3 3.5 4 4.50

0.2

0.4

0.6

0.8

1

αL2

Cov

erag

e P

roba

bilit

y

sim. S1=1

G−A S1=1

A−A S1=1

sim. S1=4

G−A S1=4

A−A S1=4

Fig. 4. Coverage probability vs. α2L, where

λ1 = 10−4, λ2 = 10−3, S2 = 2, α1L = 2.09,

α1N = 3.75, and α2

N = 4.75.

1 2 3 4 5 6 7 80.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

S1

Cov

erag

e P

roba

bilit

y

sim. S2=2

G−A S2=2

A−A S2=2

sim. S2=6

G−A S2=6

A−A S2=6

Fig. 5. Coverage probability vs. S1, where λ1 =10−4, λ2 = 10−3, α1

L = 2.09, α1N = 3.75,

α1L = 1.5, and α2

N = 4.75.

10−6

10−5

10−4

10−3

10−2

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

λ2

Cov

erag

e P

roba

bilit

y

sim. λ1=2.5× 10−3

G−A λ1=2.5× 10−3

A−A λ1=2.5× 10−3

sim. λ1=10−5

G−A λ1=10−5

A−A λ1=10−5

Fig. 6. Coverage probability vs. λ2, where S1 =4, S2 = 2, α1

L = 2.09, α1N = 3.75, α1

L = 1.5,and α2

N = 4.75.

7.1 Accuracy of A-A and G-A

Fig. 4 shows the coverage probability vs. α2L. The level of

accuracy achieved by cG−A is shown to be higher than thatof cA−A. The inaccuracy gap induced by cA−A is almosttwice as much of cG−A.3 G-A is also shown to act as anupper-bound on the actual coverage performance with analmost fixed inaccuracy gap. A-A, however, results in avarying inaccuracy gap depending on the value of αL.

Fig. 5 shows the coverage probability vs. S1 for S2 = 2.For S1 ≥ 4, both cA−A and cG−A are shown close to theactual value. The larger the S2, the higher the accuracyof cA−A. However, for S1 < 4, the A-A becomes lessaccurate and fails to follow the actual coverage probability.In contrast, cG−A preserves a reasonable accuracy for allranges of multiplexing gains, and is able to follow thevariations in the actual coverage probability. Furthermore,G-A is shown to always act as an upper-bound on thecoverage probability, while A-A alternates between beingupper- and lower-bounded.

Fig. 6 shows the coverage probability vs. λ2, where(S1 = 4, S2 = 2). The density is measured as the number ofnodes per square meter. We consider two choices of sparse(λ1 = 10−5) and moderately dense (λ1 = 2.5 × 10−3). Forλ1 = 2.5× 10−3, both cA−A and cG−A are shown very closeto the actual coverage probability. For λ1 = 10−5, whileacting as an upper-bound, cG−A closely follows the cover-age probability and accurately predicts the best achievedcoverage probability. For A-A, like previous cases, cA−A

alternates between being upper- and lower-bounded, andis also unable to predict the density for which the highestcoverage probability is achieved. The above results showthat cG−A provides a better approximation than that ofcA−A.

3. Fig. 4 shows that Alzer approximation is neither an upper-boundnor a lower-bound on the coverage probability, while the Alzer methodbasically provides an upper-bound when it is used for SISO systemsunder Nakagami-type fading, MISO-SDMA and SIMO communicationsystems, and also in mmWave communication systems with directionalantennas [53], [56], [57]. By Proposition 4, the obtained coverage prob-ability is in fact a summation over indices of multiplexing gains, wheredepending on the multiplexing gains, some terms may adopt a negativesigns. Therefore, although the Alzer’s inequality provides an upper-bound for each data stream, summing up the upper-bounds results inan approximation based on the bounds provided by the Alzer method.

7.2 Coverage Probability

7.2.1 Impact of α2L

Fig. 4 shows that increasing α2L may increase/decrease the

coverage probability, depending on the value of multiplex-ing gain, S1. For (S1 = 1, S2 = 2), increasing α2

L improvesthe coverage probability. In this case, the LOS signals in Tier2 are weakened, while the ICI is increased. These make itdifficult to decode S2 = 2 transmitted data streams. Thus,the serving BS will most likely be selected from Tier 1.Similarly, for a large enough α2

L, successful decoding ofS1 = 1 data stream is easier than that of S2 = 2, and hencethe serving BS is most likely selected from Tier 1. In contrast,for (S1 = 4, S2 = 2), increasing α2

L degrades the coverageperformance, because the serving BS is most likely selectedfrom Tier 2, as successful decoding of S2 = 2 data streamsis much more likely than that of S1 = 4.

7.2.2 Impact of Multiplexing Gains

To study the impact of multiplexing gain, Fig. 5 presentsthe coverage probability vs. S1. Increasing S1 is shownto decrease the coverage probability, because the largerthe multiplexing gain, the less probable the typical UEcan simultaneously decode all data streams. Furthermore,increasing S1 also increases the power of ICI on each datastream. In general, system diversity is shown to provide ahigher coverage than that of system multiplexing.

This finding can be substantiated by an analysis. Weuse the following upper-bound (see Appendix G) on thecoverage probability

cZF−FC ≤( ∞∫

0

Ψ(v)dv

)2π

∑

i∈K

λiPiLi

βi

(Nr + 1Si

−1), (48)

where Li =∞∫0

riELi(ri)[Li(ri)]dri, and Ψ(v) is given in (24).

In (48),∞∫0

Ψ(v)dv represents the effect of interference that is

a decreasing function of multiplexing gain Si.4 Moreover,

4. To see this, one needs to show that function ΨZF(v, yj) de-fined in (25) is a decreasing function of Sj . Note that ΨZF(v, yj) ∝(1 +

Pj

SjLj

nj(yj)

)−Sj. Therefore, the differentiation of the logarithm

function in the right-hand-side with respect to Sj (assuming that Sj is

continues) is always negative, i.e., ∂∂Sj

log(1 +

Pj

SjLj

nj(yj)

)−Sj ≤ 0.

13

Nr+1Si

− 1 is also related to the effective DoF of each datastream, which is a decreasing function of Si. Therefore, theupper-bound (48) of the coverage probability declines byincreasing Si. The derived upper-bound is shown to provideinsights on the impact of multiplexing gain of the coverageperformance. However, as it is not the tightest upper-bound,the actual decline rate of the coverage probability due to theincrease of Si might be slightly different from that suggestedby (48). A more accurate upper-bound can be obtained viaChebyshev’s inequality. However, it requires the evaluationof second-order statistics of the accumulated data rate acrossdata streams.

7.2.3 Impact of Densification

To study the impact of densification, Fig. 6 plots the cover-age probability vs. λ2. For λ1 = 10−5, the highest coverageprobability is achieved when λ2 is around 10−3. Fig. 6 alsoshows that for λ2 < 10−3, the coverage probability can beimproved by increasing the density of BSs in Tier 2. In suchcases, although excessive ICI is created due to densification,many of the Tier-2 BSs are close enough to the typical UE tohave an LOS path-loss. Fig. 6 shows that the excessive ICI isapparently dominated by the existence of many strong LOSTier-2 BSs suitable for association. Furthermore, associationwith a BS in Tier 2 might be preferred to one in Tier 1assuccessful decoding of all S2 = 2 data streams is moreprobable than that of S1 = 4 data streams.

For λ2 > 10−3, increasing the density of Teir 2 mightlead to a significant decline of the coverage probability. Thisis attributed to the growth of ICI, especially caused by manyLOS interferer BSs in Tier 2. In this case, even associationwith a very close LOS BS cannot compensate the ICI growth.In this case, Tier 1 remains less qualified for association dueto its high multiplexing gain. It is further shown in Fig. 6that the coverage probability is substantially degraded byincreasing λ1 to 2.5×10−3. This is due to the increase of ICIas well as Tier 1’s high multiplexing gain.

The bell-shaped curve observed in Fig. 6 can also be ex-plained as follows. In Appendix H, we derive the followingupper-bound on the coverage probability:

cZF−FC ≤∑

i∈K2πλi

∞∫

0

rELi(r)exp− 2π

∑

j

λj

∞∫

0

ELj(y)

Ωi,j

( PiSjLi(r)PjSiLj(y)βi

)dy

dr, (49)

where

Ωi,j

( PiSjLi(ri)PjSiLj(y)βi

)= yF GZF

j

HZFi,min

( PiSjLi(ri)PjSiLj(y)βi

), (50)

and F GZFj

HZFi,min

(.) is the CCDF of random variableGZF

j

HZFi,min

, the

exact shape of which is not important for our purpose. Asshown in (49), by increasing λi, the coverage probability in-creases proportionally by a multiplicative scale, λi, while itis simultaneously decreased exponentially. This conflictingbehavior suggests that by densification, depending on thesystem parameters, one of the above phenomena becomesdominant and then the coverage probability can either

grow or decline. By differentiating (49) with respect to thedensities λi and letting the resultants equal to 0, a set of thefollowing K equations are obtained:

∞∫

0

rELi(r)

[e−2π

∑j λj

∞∫0ELj(y)Ωi,j

(PiSjLi(r)

PjSiLj(y)βi

)dy

×(

1− 2πλi

∞∫

0

ELj(y)Ωi,i

( Li(r)Lj(y)βi

))]dr = 0, ∀i ∈ K.

This suggests that there exist a set of densities maximizingthe coverage probability.

7.2.4 Impact of Interference Correlation

Our analysis in this paper is based on FC assumption. Tobetter understand the utility of this approximation, we com-pare the coverage probability under FC assumption with amanufactured scenario whereby for each data stream li, Kindependent sets of interferers denoted by Φli

j with givendensity λj are produced. To highlight this approach, wecall it No-Correlation (NC) assumption. Similarly to Propo-sitions 4 and 5, one can estimate the coverage probabilityvia A-A and A-G methods, respectively.

Corollary 3: Define S′i = ((Nr − Si + 1)!)−1

(Nr−Si+1) .Under NC assumption and based on the A-A method, thecoverage probability is approximated as

cA−A =∑i∈K

2πλi

∞∫

0

ri

∑ni∈L,N

pini

(ri)

(Nr−Si+1∑

l′′i =1

(Nr − Si + 1

l′′i

)

×(−1)l′′i +1Ψini

(ri)

)Si

dri, (51)

where Ψini

(ri) is as given in Proposition 4.Proof: See Appendix I. ¥Corollary 4: Under NC assumption, and based on the

G-A method, the coverage probability is approximated as

cG−A =∑

i∈K2πλi

∞∫

0

ri

∑

ni∈L,Npi

ni(ri)

×( ∞∫

0

hNr−Sie−h

(Nr − Si)!Γ(a)γ

(a, b

βi

Pi

SiLi

ni(ri)h

)dh

)Si

dri,

(52)where γ(a, bx) is the CCDF of random variableGamma(a, b).

Proof: See Appendix J. ¥Fig. 7 shows the coverage probability vs. β1. The FC as-

sumption is shown to provide a much higher accuracy thanNC assumption. One may assume that the intrinsic receiveddiversity gain in adopting NC assumption overestimatesthe coverage probability compared to the FC assumption.Surprisingly, the opposite is observed in Fig. 7, showingthat the coverage probability under NC is much smallerthan that of FC. This is due mainly to the existence of LOScomponent, as in the NC case for each data stream, thetypical UE is interfered with by a new set of K independentPPP interferers. Therefore, it is highly likely for the typicalUE to have a larger number of LOS-dominated interfering

14

2 4 6 8 100

0.05

0.1

0.15

0.2

0.25

0.3

0.35

β1

Cov

erag

e P

roba

bilit

y

NC assumption, A−AFC assumption, A−ANC assumption, G−AFC assumption, G−Asim.

Fig. 7. Coverage probability vs. β1, where S1 = 2, S2 = 6, β2 = 2.5,λ1 = 10−4, and λ2 = 10−3.

links from the BSs. In contrast, under the FC assumptionacross all data streams, the very same set of K independentPPP interferes are interfering with the typical UE. Com-pared to the FC assumption, this results in a significantlyhigher interference and therefore a much lower coverageprobability under the NC assumption. In other words, theimproved receive diversity due to no-correlated interferenceis compromised by a larger number of LOS interfering links.

7.2.5 Nonhomogeneous Path-Loss Model

We study the non-homogeneous path-loss scenario as de-scribed in Section 6. Fig. 8 shows the corresponding cover-age performance vs. λ2 with the same system parameterconsidered in Fig. 6. G-A is shown to provide a higherlevel of accuracy than that of A-A. For λ1 = 2.5 × 10−3

and 5 × 10−3 < λ2 < 10−2, the coverage probabilityobtained by A-A is not reliable. Nevertheless, G-A pre-serves a consistent accuracy and robustness. This justifiesG-A’s higher numerical complexity. Densification in Tier 2is also shown to improve the coverage probability. This isin contrast with what Fig. 6 shows. This can be explainedby noting that under the non-homogenous scenario, thepath-loss exponents in Tier 1 are increased, reducing theaggregated interference. Therefore, even receiving a weakersignal power per each data stream is enough to overcomethe impact of the interference and establish a communica-tion link with a BS in Tier 2. On the other hand, as in thecase of Fig. 6, densification in Tier 1 lowers the coverageperformance in a non-homogeneous path-loss environment.In this case, larger path-loss exponents do not improvethe coverage probability because setting S1 = 4 makes itincreasingly unlikely for a typical user to be associated witha BS in Tier 1.

7.3 Impact of Imperfect CSIT

In Fig. 11, we investigate the impact of imperfect CSITon the coverage performance based on the derivations inSection 6.5 using QCA. As shown in Fig. 11, the QCA

10 12 14 16 18 20 220.1

0.15

0.2

0.25

0.3

0.35

B1

Cov

erag

e P

roba

bilit

y

QCA app.sim.QCA app.sim.QCA app.sim.

B2=10

B2=8B

2=6

Fig. 11. Coverage probability vs. B1, where λ1 = 10−3, λ2 = 10−2,U1 = 10, and U2 = 2. The analytical result is based on G-A approxima-tion.

approximation closely follows the actual coverage perfor-mance. Note that for clarity we only consider the G-Aapproximation in this simulation.

7.4 Area Spectral Efficiency (ASE)

Area spectral efficiency (ASE) of the network is often consid-ered as a crucial performance metric in cellular communica-tion systems. ASE is defined as ASEZF =

∑i

λiSicZFi log(1+

βi) nat/s/Hz/unit area [6], [7], [33], where cZFi is the

coverage probability from Tier i. Note that ASE is linearlyproportional to the multiplexing gain Si, but there is noguarantee that increasing the multiplexing gains improvesASE, as it may degrade the coverage probability.

Here we consider two systems: system 1 (SYS1) whereinboth LOS and NLOS components exist, and system 2 (SYS2)which has non-homogeneous path-loss model in which allBSs are subject to the NLOS component. The density of BSsin both SYS1 and SYS2 are the same.

7.4.1 Impact of α2L, and Multiplexing Gain

Fig. 9 shows the impact of near-field path-loss exponent inTier 2, α2

L, on the ASE. As expected, in SYS2 by increasingα2

L, the ASE does not change. This is because in the non-homogenous path-loss model, LOS/NLOS path-loss expo-nents are considered equal in each tier, and different acrossdifferent tiers. Increasing S1 = 1 to S1 = 4 does not changeASE in both SYS1 and SYS2. Thus, it may not be necessarilysuitable to increase the multiplexing gains, as it does notdirectly improve the ASE while it may degrade coverageperformance. Fig. 9 also shows that the ASE in SYS1 ishigher than that in SYS2 and a higher ASE gain is achievedfor smaller α2

L. Therefore, unlike the cases with all NLOSlinks, the LOS links can improve the ASE.

7.4.2 Impact of Densification and Multiplexing Gain

Fig. 10 plots ASE vs. S1. In general, G-A is shown to bemore accurate than A-A. Further, the figure shows that theASE is slightly reduced by increasing S1. Recall the coverageperformance from Fig. 5, showing that the coverage proba-bility is decreased by increasing multiplexing gains. Thus,

15

10−6

10−5

10−4

10−3

10−2

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

λ2

Cov

erag

e P

roba

bilit

y

sim. λ1=2.5× 10−3

G−A λ1=2.5× 10−3

A−A λ1=2.5× 10−3

sim. λ1=10−5

G−A λ1=10−5

A−A λ1=10−5

Fig. 8. Coverage probability vs. λ2, where S1 =4, S2 = 2, α1 = 2.09, and α2 = 4.75.

1 1.5 2 2.5 3 3.5 4 4.50

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6x 10

−3

αL2

AS

E

sim. sys2 S1=1

G−A sys2 S1=1

A−A sys2 S1=1

sim. sys1 S1=1

G−A sys1 S1=1

A−A sys1 S1=1

sim. sys2 S1=4

G−A sys2 S1=4

A−A sys2 S1=4

sim. sys1 S1=4

G−A sys1 S1=4

A−A sys1 S1=4

Fig. 9. ASE vs. α2L, where λ1 = 10−4, λ2 =

10−3, S2 = 2, α1L = 2.09, α1

N = 3.75, andα2

N = 4.75.

1 2 3 4 5 6 7 86

7

8

9

10

11

12x 10

−5

S2

AS

E

sim. S2=2

G−A S2=2

A−A S2=2

sim. S2=6

G−A S2=6

A−A S2=6

Fig. 10. ASE vs. S1, where λ1 = 10−4, λ2 =10−3, α1

L = 2.09, α1N = 3.75, α1

L = 1.5, andα2

N = 4.75.

10−6

10−5

10−4

10−3

10−2

10−5

10−4

10−3

10−2

10−1

λ2

AS

E

sim. sys2 λ

1=2.5× 10−3

G−A sys2 λ1=2.5× 10−3

A−A sys2 λ1=2.5× 10−3

sim. sys1 λ1=2.5× 10−3

G−A sys1 λ1=2.5× 10−3

A−A sys1 λ1=2.5× 10−3

sim. sys2 λ1=10−5

G−A sys2 λ1=10−5

A−A sys2 λ1=10−5

sim. sys1 λ1=10−5

G−A sys1 λ1=10−5

A−A sys1 λ1=10−5

Fig. 12. ASE vs. λ2, where S1 = 4, S2 = 2, α1L = 2.09, α1

N = 3.75,α1

L = 1.5, and α2N = 4.75.

we conclude that in many cases adopting a diversity onlysystem (see Section 6) is justifiable.

Fig. 12 plots ASE vs. λ2 for S1 = 4 and S2 = 2.Regardless of the multiplexing gains and the density of Tier1, Fig. 12 shows that increasing λ2 improves the ASE in bothsystems. One should also note that densification of Tier 1may not necessarily improve the ASE. In fact, in SYS2, ASEis degraded by increasing the density of Tier 1. Furthermore,in SYS1, for λ2 > 5 × 10−3, growing the density of Tier1 improves the ASE. In contrast, for λ2 < 5 × 10−3, theASE is not related to the density of Tier 1. For this setting(λ2 > 5× 10−3), the rate of ASE increase by increasing λ2 issmaller for higher Tier 1’s density. This is consistent with theobservation made in Fig. 6, where for S1 = 4, densificationin Tier 1 lowers coverage performance.

Note that for λ2 > 5 × 10−5, λ1 = 2.5 × 10−3, SYS1outperforms SYS2. This suggests that the existence of theLOS component has a positive effect on the ASE. In contrast,for λ2 < 5× 10−5, ASE in SYS1 is, in fact, smaller than thatin SYS2 due to the LOS component.

8 CONCLUSIONS

For open-loop multi-stream MIMO-ZFBF communicationsin random networks subject to LOS/NLOS propagation,we evaluated the coverage probability. Adopting the toolof stochastic geometry, we derived two easy-to-compute

approximations of the coverage probability, A-A and G-Amethods, as the function of densities of tiers, multiplex-ing gains, LOS/NLOS parameters, the number of receiverantennas, and the number of tiers. Our extensive simula-tion and numerical evaluations revealed that G-A is moreaccurate than A-A. Compared to A-A, G-A is also morerobust to various system parameters and can accuratelypredict the best density responsible for the peak of thecoverage probability. We therefore recommend its use if onewants to investigate other aspects of the system or carryout system design. Our results also showed that, undercertain scenarios, the existence of LOS mode can renderperceivable ASE performance boost over the case where allcommunication links are in NLOS mode. To achieve this,one must judiciously choose the density of BSs.

We also studied the cross-stream ICI correlation. Ouranalysis showed that in the MIMO multiplexing system, theICI is highly correlated across data streams. This findingcan substantially ease the performance evaluation of multi-stream systems, as shown in this paper.

The analytical results in this paper can also facilitateanalysis of mmWave multiplexing for which researcherscommonly focused on the stream-level performance eval-uation instead of the link-level performance [56], [57].

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Mohammad G. Khoshkholgh received hisB.Sc. degree in Electrical Engineering from Isfa-han University, Isfahan, Iran, in 2006, his M.Sc.degree in Electrical Engineering from the TarbiatModares University, Tehran, Iran, in 2008. He isnow with the University of British Columbia. Hisresearch interests are mainly in wireless com-munications and networks. He is the holder ofVanier Canada Graduate Scholarship.

Keivan Navaie is with the School of Comput-ing and Communications, Lancaster University,United Kingdom. His research interests lie in thefield of mobile computing, radio resource alloca-tion, cognitive radio networks, and cooperativecommunications. He is on the editorial boardof the IEEE Communications Surveys & Tutori-als, and IEEE Communications Letters. He hasserved in the technical program committee ofseveral IEEE conferences. He is a senior mem-ber of the IEEE, an IET Fellow, and a Chartered

Engineer.

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Kang G. Shin is the Kevin & Nancy O’ConnorProfessor of Computer Science in the Depart-ment of Electrical Engineering and ComputerScience, The University of Michigan, Ann Arbor.His current research focuses on QoS-sensitivecomputing and networking as well as on embed-ded real-time and cyber-physical systems.

He has supervised the completion of 82 PhDs,and authored/coauthored about 900 technicalarticles (more than 330 of these are in archivaljournals), a textbook and more than 40 patents

or invention disclosures, and received numerous best paper awards,including the Best Paper Awards from the 2011 ACM International Con-ference on Mobile Computing and Networking (MobiCom’11), the 2011IEEE International Conference on Autonomic Computing, the 2010 and2000 USENIX Annual Technical Conferences, as well as the 2003 IEEECommunications Society William R. Bennett Prize Paper Award and the1987 Outstanding IEEE Transactions of Automatic Control Paper Award.He has also received several institutional awards.

He was a co-founder of a couple of startups and also licensed someof his technologies to industry.

Victor C. M. Leung [S75, M89, SM97, F03]received the B.A.Sc. (Hons.) degree in electri-cal engineering from the University of BritishColumbia (UBC) in 1977, and was awardedthe APEBC Gold Medal as the head of thegraduating class in the Faculty of Applied Sci-ence. He attended graduate school at UBC ona Canadian Natural Sciences and EngineeringResearch Council Postgraduate Scholarship andreceived the Ph.D. degree in electrical engineer-ing in 1982. From 1981 to 1987, Dr. Leung was

a Senior Member of Technical Staff and satellite system specialist atMPR Teltech Ltd., Canada. He is also serving as the Overseas Dean ofthe School of Information and Electronic Engineering at Zhejiang Gong-shang University, China. Dr. Leung has co-authored more than 1000journal/conference papers, 37 book chapters, and co-edited 12 booktitles. Dr. Leung is a registered Professional Engineer in the Province ofBritish Columbia, Canada.