Uplink Interference Analysis for Two-tier Cellular Networks with Diverse Users under Random Spatial Patterns
aa r X i v : . [ c s . N I] O c t Uplink Interference Analysisfor Two-tier Cellular Networks with Diverse Usersunder Random Spatial Patterns
Wei Bao,
Student Member, IEEE and Ben Liang,
Senior Member, IEEE
Abstract —Multi-tier architecture improves the spatial reuse ofradio spectrum in cellular networks, but it introduces compli-cated heterogeneity in the spatial distribution of transmitters,which brings new challenges in interference analysis. In thiswork, we present a stochastic geometric model to evaluate theuplink interference in a two-tier network considering multi-type users and base stations. Each type of tier-1 users andtier-2 base stations are modeled as independent homogeneousPoisson point processes, and tier-2 users are modeled as locallynon-homogeneous clustered Poisson point processes centered attier-2 base stations. By applying a superposition-aggregation-superposition approach, we quantify the interference at bothtiers. Our model is also able to capture the impact of two typesof exclusion regions, where either tier-2 base stations or tier-2users are restricted in order to avoid cross-tier interference. Asan important application of this analytical model, an intensityplanning scenario is investigated, in which we aim to maximizethe total income of the network operator with respect to theintensities of tier-2 cells, under constraints on the outage prob-abilities of tier-1 and tier-2 users. The result of our interferenceanalysis suggests that this maximization can be converted to astandard convex optimization problem. Finally, numerical studiesfurther demonstrate the correctness of our analysis.
Index Terms —Two-tier cellular network, multi-type users,interference, outage probability, stochastic geometry
I. I
NTRODUCTION H Igher capacity, better service quality, lower power usage,and ubiquitous coverage are some of the most importantobjectives in the deployment of wireless cellular networks.To achieve these goals, one efficient approach is to installa second tier of small cells (termed tier-2 cells), such asfemtocells, overlapping the original tier-1 macro cells [2].Each tier-2 cell is centered at a base station with shorterrange and lower cost. By doing so, the tier-2 network couldprovide nearby user equipments (UEs) with higher-qualitycommunication links that require lower power usage.However, with such tier-2 facilities, interference manage-ment becomes more challenging.
First , the spatial patterns ofdifferent network components vary significantly. Tier-1 BSsare designed and deployed regularly by the network operator;tier-1 users are randomly distributed in the system; tier-2 BSsare deployed irregularly, sometimes in an “anywhere plug andplay” manner (e.g., femtocell BSs), implying a high level of
The authors are affiliated with the Department of Electrical and ComputerEngineering, University of Toronto, 10 King’s College Road, Toronto, Ontario,Canada (email: { wbao, liang } @comm.utoronto.ca). A preliminary version ofthis work has appeared in IEEE ICCC [1]. This work has been supported inpart by grants from Bell Canada and the Natural Sciences and EngineeringResearch Council (NSERC) of Canada. spatial randomness; the distribution of tier-2 users are evenmore complicated: they not only are randomly distributed,but also show spatial correlation, since they are likely toaggregate around tier-2 BSs. Because each network componentcontributes to the total interference differently, their overalleffect is difficult to characterize. Second , tier-2 cells may beclassified into different types according to their communicationrange (e.g., picocell and femtocell) and their local load (e.g.,intensity of local UEs), and UEs are different in terms oftheir transmission parameters (e.g., transmission power). Suchdiverse tier-2 cells and UEs introduce a more complicatedinterference environment.
Third , in order to alleviate cross-tier interference, a system operator may impose an exclusionregion around each tier-1 BS [3], [4], in which either tier-2BSs or UEs are restricted. This results in a unique patternof correlation between tiers, bringing additional challenges toaccurate interference analysis.Recent works have applied the theory of stochastic geometryto analyze interference in cellular networks [5]. Interferersare often modeled as a Poisson point process (PPP), sothat the interference created by them is the shot noise [6]–[8] in Euclidean space. The Laplace transform of the shotnoise can be derived directly from the Laplace functional[6], [7] or the generating functional [9] of the PPP. In thisway, the interference can be analyzed mathematically. Systemmetrics, such as outage probability and system throughputcan then be deduced from the Laplace transform. Employingthis approach, the downlink interference of multi-tier cellularnetworks was characterized in [10], [11], and the uplinkinterference of single-tier cellular network was studied in [12]–[16].However, to analyze the uplink interference in two-tiernetworks is more challenging, as we need to account for thespatial randomness and correlation of tier-2 UEs aggregatingaround tier-2 BSs. Innovative efforts have been made inprevious works, but as detailed in Section II, they only partiallyresolve the challenges. For example, [3] evaluated the uplinkperformance of two-tier networks based on several levels ofapproximations. [17] studied both uplink and downlink inter-ference of two-tier networks assuming that the UEs transmitat the same power without power control. Both [3] and [17]considered homogeneous UEs and tier-2 cells.In this work, we propose an accurate uplink interferencemodel of two-tier cellular networks, considering multiple typesof tier-1 UEs, tier-2 BSs, and tier-2 UEs. At tier-1, theinterference is studied as shot noise corresponding to PPPs. Attier-2, we develop a superposition-aggregation-superpositionSAS) approach to overcome the challenges in analysis. Inparticular, we show that the interference from all UEs ineach tier-2 cell can be equivalently aggregated as a single-point interference source. Through the SAS approach, weprecisely compute the interference of both tiers, avoiding anyapproximation.Furthermore, in order to alleviate cross-tier uplink interfer-ence, it is commonly proposed to impose an exclusion regionaround each tier-1 BS [3], [4], in which tier-2 BSs or tier-2 UEs are restricted. In this paper, we exam the effects oftwo types of exclusion regions: 1) no tier-2 BSs are allowedwithin the exclusion regions (BS exclusion); 2) no tier-2 UEsare allowed within the exclusion regions (UE exclusion). Ouranalytical and simulation observations demonstrate that usingexclusion regions only bring slightly improvement on theoutage performance at a tier-1 BS, but UE-exclusion regionsare more effective than BS-exclusion regions with the sameexclusion radius.Another important contribution of this paper is to providenew insights on system design. Through our SAS approach, weshow that the coverage probability at tier-1 and tier-2 BSs canbe expressed as a product of negative exponential functionsof the intensity of tier-2 cells. As an application example ofthis property, we present an intensity planning scenario, inwhich we aim to maximize the total income of the networkoperator with respect to the intensities of tier-2 cells, underconstraints on the required outage probabilities of tier-1 andtier-2 UEs. We demonstrate how our analysis can provide anefficient solution to this optimization problem.The rest of the paper is organized as follows. In section II,we discuss the relation between our work and prior works.In Section III, we present the system model. In Section IV,we analyze the interference at tier-1 cells. In Section V,we analyze the interference at tier-2 cells. In Section VI,we conduct case studies based on the interference analysis,and the intensity planning problem is presented. In SectionVII, we validate our analysis with simulation results. Finally,concluding remarks are given in Section VIII.II. R
ELATED W ORKS
For two-tier networks, the downlink interference was wellstudied through stochastic geometric approaches. For example,Dhillon et al. in [10] analyzed the downlink outage perfor-mance of a heterogeneous network with multiple tiers whenthe minimum required signal to interference plus noise ratio(SINR) threshold is greater than . Kim et al. in [11] studiedthe maximum tier-1 UE and tier-2 cell densities with downlinkoutage performance constraints. Singh et al. in [18] studied thedownlink interference with flexible user association.The analysis of uplink interference in two-tier networks ismore challenging compared with the downlink case, as weneed to account for the spatial randomness and correlation oftier-2 UEs aggregating around tier-2 BSs. Innovative effortshave been made in previous works. Kishore et al. in [19]studied the uplink performance of a single tier-1 cell and asingle tier-2 cell, while and the same authors in [20] extendedit to the case of multiple tier-1 cells and multiple tier-2 cells. However, their models were based on a fixed number of tier-1 and tier-2 cells, without considering the random spatialpatterns of UEs and BSs. Chandrasekhar and Andrews in [3]evaluated the uplink performance of two-tier networks withrandom UEs and tier-2 cells. However, several interferencecomponents were analyzed based on approximations: 1) theinter-interference of tier-1 cell was estimated as a truncatedGaussian random variable; 2) the radius of tier-2 cells wasregarded as zero when viewed from the outside; 3) tier-2UEs were assumed to transmit at the maximum power at theedge of tier-2 cells; and 4) the cross interference from tier-1UEs to tier-2 BSs only accounted for the interference froma reference tier-1 cell. Cheung et al. in [17] studied bothuplink and downlink interference of two-tier network basedon a Neyman-Scott Process [9], [21]. However [17] was alsolimited in two aspects: 1) all UEs were assumed to transmitat the same power; and 2) tier-2 UEs were assumed to beuniformly distributed in an infinitesimally thin ring around thetier-2 BS. In addition, neither [3] nor [17] considered multi-type UEs or tier-2 cells. In contrast, our work does not requireany of the above approximations, and we further considermultiple types of UEs and tier-2 BSs, and two types of tier-2exclusion regions. III. S YSTEM M ODEL
A. System Topology
An example of the system topology considered in thiswork is illustrated in Fig. 1. We use the terms “tier-1” and“tier-2” throughout this paper, which are synonymous with“macro tier” and “small-cell tier” respectively. First, followinga common convention in the literature, we assume that the tier-1 cells form an infinite hexagonal grid on the two-dimensionalEuclidean space R . Tier-1 BSs are located at the centers ofthe hexagons B = { ( aR c , √ aR c + √ bR c ) | a, b ∈ Z } ,where R c is the radius of the hexagon. Tier-1 UEs arerandomly distributed in the system, which are modeled asPPPs. We assume that there are M types of tier-1 UEs, definedby their different required received power levels. Each typeindependently forms a homogeneous PPP. Let Φ i denote thePPP corresponding to type- i tier-1 UEs. Its intensity is λ i .We consider N types of tier-2 BSs and K types of tier-2 UEs. Different tier-2 BS types are defined in terms oftheir communication ranges and their local UE densities;different tier-2 UE types are defined in terms of their requiredreceived power levels. Because tier-2 BSs generally have highspatial randomness, we assume each type of tier-2 BSs forma homogeneous PPP. Let Θ i denote the PPP corresponding totype- i tier-2 BS. Its intensity is µ i . Each tier-2 BS connectswith the Internet core via wired links, which has no influenceon the interference analysis.Each tier-2 BS communicates with different types of localtier-2 UEs surrounding it, composing a tier-2 cell. Let R i bethe communication radius of each type- i tier-2 BS, with itscorresponding tier-2 UEs located within R i from it. Given the location of a type- i tier-2 BS at x , we assume that eachtype of local tier-2 UEs are independently distributed as a non-homogenous PPP in the disk centered at x with radius R i . Let ype-1 tier-1 UEsType-2 tier-1 UEsType-1 tier-2 UEsType-2 tier-2 UEsTier-1 cellsType-1 tier-2 cellsType-2 tier-2 cells Fig. 1. System model. Ψ ij ( x ) denote the PPP of type- j tier-2 UEs around a type- i tier-2 BS at x . Its intensity at x is described by ν ij ( x − x ) ,a non-negative function of the vector x − x . Note that the UEintensity ν ij ( x − x ) = 0 if | x − x | > R i . We assume thetier-2 UEs in one tier-2 cell are also independent with tier-2 UEs in other tier-2 cells as well as tier-1 UEs. To betterunderstand the distribution of tier-2 BSs and tier-2 UEs, Θ i can be regarded as a parent point process on the plane, while Ψ ij ( · ) is a daughter process associated with a point in theparent point process. Note that the aggregating of tier-2 UEsaround tier-2 BSs implicitly defines the location correlationamong tier-2 UEs. In particular, we emphasize that due tothe randomness in the location of tier-2 cells, the locations ofdifferent types of tier-2 UEs are dependent and non-Poisson. In this paper, we focus on the closed access scenario. Localtier-2 UEs only connect to their serving tier-2 BSs, and tier-1 UEs only connect to tier-1 BSs. They are not transferredto the other tier if they are closer to a BS in that tier. It isalso possible that two tier-2 cells are overlapping with eachother. In this case, tier-2 UEs maintain connection to their ownserving tier-2 BSs.Our analytical model requires Poisson assumptions on tier-1UEs and tier-2 BSs. In Sections VII-H and VII-I, we presentsimulation data to study the impact of correlated tier-1 UEsand tier-2 BSs on analytical accuracy in general and on systemperformance in the intensity planning problem.
B. Pathloss Model and Power Control
Let P t ( x ) denote the transmission power at x and P r ( y ) denote the received power at y . We assume that P r ( y ) = P t ( x ) g x g x , y h x , y A | x − y | γ , where A | x − y | γ is the propagation lossfunction with predetermined constants A and γ , g x g x , y is theshadowing term, which is composed of the near field factor g x and far field factor g x , y [15], [22], and h x , y is the fastfading term. Here, g x and g x , y are independently log-normally distributed with given parameters, and h x , y is independentlyexponentially distributed with unit mean (Rayleigh fading withpower normalization).We follow the conventional assumption that uplink powercontrol adjusts for propagation losses and shadowing [3], [12],[15], [16], [22], [23]. The targeted power level for type- i tier-1 UEs is P i , and the targeted power for type- j tier-2UEs in type- i tier-2 cells is Q ij . Given the targeted receivedpower P (i.e., P = P i or P = Q ij ) at y and transmitter at x , the transmission power is P A | x − y | γ g x g x , y . Then, the resultantcontribution to interference at y ′ = y is P | x − y | γ g x , y ′ h x , y ′ | x − y ′ | γ g x , y .Note that g x , y ′ /g x , y is still log-normally distributed andis i.i.d. with respect to different x , and h x , y is i.i.d. withrespect to different x and y . Let g ( · ) be the probability densityfunction (pdf) of g x , y ′ /g x , y (log-normal).In addition, we assume the system is interference limited,such that noise is negligible. C. Tier-2 Exclusion Regions
To reduce the interference from tier-2 UEs to tier-1 BSs,exclusion regions of tier-2 cells were proposed in [4]. In thispaper, we also consider two types of exclusion regions. For
BSexclusion , we assume that no tier-2 BSs are allowed to locatewithin R e, distance from a tier-1 BS, as shown in Fig. 2(a).For UE exclusion , we assume that no tier-2 UEs are allowedto locate within R e, distance from a tier-1 BS, as shownin Fig. 2(b). Let B ( x , R ) denote the disk region centered at x with radius R . The collection of BS-exclusion and UE-exclusion regions are denoted by F = S x ∈ B B ( x , R e, ) and F = S x ∈ B B ( x , R e, ) , respectively. Thus, with the BS-exclusion regions, the intensity of Θ i becomes in F ; withthe UE-exclusion regions, the intensity of Ψ ij becomes in F .We assume that tier-2 BSs are not protected by exclusionregions, since tier-1 macrocell BSs and UEs are the entrenchedequipment whose behavior is difficult to change. Open accesshas been recognized as an efficient approach to reduce thecross-tier interference from tier-1 UEs to tier-2 BSs. However,it will introduce additional practical concerns (e.g., signalingoverhead and network security) as well as substantial chal-lenges in analytical modeling, which is beyond the scopeof this paper. Interested readers are referred to [24] for adiscussion comparing the open and closed modes. D. Uplink Multiple Access
In this paper, we assume that all uplink communicationsoccur on the same channel. This analysis can be extended tonon-orthogonal multiple access schemes, such as CDMA. ForCDMA, a spreading code is applied to transmit a signal, sothat at the receiver, the SIR is equivalent to being multipliedby the spreading factor [3], [7].For systems with orthogonal multiple access schemes (e.g.,OFDMA), the frequency and time resources are partitioned By doing so, we capture the fact that the targeted received powers ofdifferent UEs may be different in reality. This provides a more general analysismodel than previous works, such as [3], [12], [15], [16], where the targetedreceived power level is assumed to be a constant. ier-2 BSsTier-2 UEsTier-1 BSsTier-1 cellsExclusion regionsTier-2 cells R e, R e, (a) BS-exclusion regions(b) UE-exclusion regions Fig. 2. Two types of exclusion regions. into multiple orthogonal resource blocks. Different UEs inthe same cell use different resource blocks. In this case, amodified version of our model is employed to provide closeapproximations for UEs’ performance in Section VI-B andVII-F.Furthermore, since we focus on uplink interference analysis,we assume that the downlink and uplink of the system areoperated separately in different frequency or time. Hence, thedownlink interference has no influence on the interferenceanalysis in this paper.IV. I
NTERFERENCE TO T IER -1 BS S In this section, we analyze the uplink interference at tier-1BSs. Given a reference type- k tier-1 UE, termed the typicaltier-1 UE , communicating with its BS, termed the typical tier-1 BS , we compute the interference from all other tier-1 andtier-2 UEs to the typical BS. The tier-1 cell corresponding tothe typical tier-1 BS is denoted as the typical tier-1 cell . Dueto stationarity of point processes corresponding to tier-1 UEs,tier-2 BSs, and tier-2 UEs, throughout this section we will re-define the coordinates so that the typical tier-1 BS is locatedat . Let H ( x ) denote the hexagon region centered at x withradius R c . Correspondingly, the typical UE is located at some TABLE IS
ELECTED D EFINITION OF V ARIABLES
Name Definition Φ i The point process of type- i tier-1 UEs. Θ i The point process of type- i tier-2 BSs.The point process of type- j tier-2 Ψ ij ( x ) UEs associating with a type- i tier-2BS located at x . λ i The intensity of type- i tier-1 UEs. µ i The intensity of type- i tier-2 BSs.Given a type- i tier-2 BS, the intensity of type ν ij ( x ) j tier-2 UEs associating to it, where x is therelative coordinate with respect to the tier-2 BS. I ,in,i The interference from type- i tier-1 UEsinside the typical tier-1 cell to the typical tier-1 BS. I ,out,i The interference from type- i tier-1 UEsoutside the typical tier-1 cell to the typical tier-1 BS. I ,i The interference from type- i tier-1 UEsto the typical tier-1 BS. I The interference from tier-1 UEsto the typical tier-1 BS. b I ,i,j The interference from type- j tier-2 UEs insidea single type- i tier-2 cell to the typical tier-1 BS. b I ,i The interference from tier-2 UEs inside a singletype- i tier-2 cell to the typical tier-1 BS. I ,i The interference from tier-2 UEs inside alltype- i tier-2 cells to the typical tier-1 BS. I The interference from all tier-2 UEs to thetypical tier-1 BS. I ′ ,i The interference from type- i tier-1 UEsto the typical tier-2 BS. I ′ The interference from tier-1 UEsto the typical tier-2 BS. b I ′ ,i,j The interference from type- j tier-2 UEs insidea single type- i tier-2 cell to the typical tier-2 BS. b I ′ ,i The interference from tier-2 UEs inside a singletype- i tier-2 cell to the typical tier-2 BS. I ′ ,i The inter-cell interference from tier-2 UEs insidetype- i tier-2 cells to the typical tier-2 BS. I ′ The inter-cell interference from tier-2 UEsto the typical tier-2 BS. I ′ ,j The intra-cell interference from type- j tier-2 UEs to the typical tier-2 BS. I ′ The intra-cell interference from tier-2 UEsto the typical tier-2 BS. x U that is uniformly distributed in H ( ) . Let Φ ! k denote thepoint process of all other type- k tier-1 UEs conditioned on thetypical UE (i.e., the reduced Palm point process with respectto Φ k ). Since the reduced Palm point process of a PPP has thesame distribution as the original PPP, Φ ! k is still a PPP withntensity λ k [6]. For presentation convenience, we define e Φ i ,such that e Φ i = Φ i if i = k and e Φ i = Φ ! i if i = k . Note thatthe above typicality definition and the coordination translationfollow standard stochastic geometric techniques.In the following, instead of directly computing the distri-bution of interference, we study its Laplace transform (i.e.,moment generating function), which fully characterizes itsdistribution. A. Interference from Tier-1 UEs to Tier-1 BS
It is not difficult to compute the Laplace transform of theinterference produced by tier-1 UEs to the typical tier-1 BSlocated at . The following analysis uses a standard stochasticgeometric approach and is provided for completeness.Let I ,in,i denote the total interference from type- i tier-1UEs inside the typical cell H ( ) , and I ,out,i denote the totalinterference from type- i tier-1 UEs outside the typical cell. Wehave I ,in,i = X x ∈ e Φ i T H ( ) P i h x , , (1)and I ,out,i = X x ∈ B \{ } X x ∈ e Φ i T H ( x ) P i | x − x | γ g x , h x , | x | γ g x , x , (2)in which P x ∈ e Φ i T H ( x ) P i | x − x | γ g x , h x , | x | γ g x , x is the interfer-ence from type- i tier-1 UEs in the cell H ( x ) , and P x ∈ B \{ } P x ∈ e Φ i T H ( x ) P i | x − x | γ g x , h x , | x | γ g x , x is the overall in-terference from all type- i tier-1 UEs outside the typical cell. I ,in,i and I ,out,i can be regarded as shot noises cor-responding to e Φ i . Their Laplace transforms can be derivedthrough Laplace functionals corresponding to PPP: L I ,in,i ( s ) = exp (cid:18) − (3 √ / λ i (cid:18) sP i R c sP i + 1 (cid:19)(cid:19) , (3)and L I ,out,i ( s ) = exp − λ i X x ∈ B \{ } Z H ( x ) Z R + sP i | x − x | γ g g ( g ) sP i | x − x | γ g + | x | γ dgd x ! . (4)Due to the independence of I ,in,i and I ,out,i , as well asthe independence of interference among different tiers, theLaplace transform of the overall interference from all tier-1UEs to the typical tier-1 BS can be computed as L I ( s ) = M Y i =1 L I ,i ( s ) = M Y i =1 (cid:0) L I ,in,i ( s ) L I ,out,i ( s ) (cid:1) . (5) In the numerical computation of (4) and (20), we have truncated thesummation terms up to | x | ≤ R c . B. Interference from Tier-2 UEs to Tier-1 BS
It is much more challenging to characterize the interferencefrom tier-2 UEs to the typical tier-1 BS, which is part ofthe core contribution of this work. Because tier-2 UEs arecorrelated as they aggregate around tier-2 BSs, the interfer-ence cannot be analyzed by a traditional stochastic geometricapproach. Instead, we propose the following superposition-aggregation-superposition (SAS) method, which exactly cap-tures the interference from tier-2 cells.
Interference from One Tier-2 Cell (Superposition):
In thefirst step, we study the interference from a single type of tier-2 UEs in a single tier-2 cell. Let b I ,i ( x ) be the interferencefrom a single type- i tier-2 cell, whose BS is at x , to thetypical tier-1 BS. b I ,i ( x ) = P Kj =1 b I ,i,j ( x ) , where b I ,i,j isthe interference from all type- j tier-2 UEs in the single type- i tier-2 cell. We have b I ,i,j ( x ) = X x ∈ Ψ ij ( x ) Q ij | x − x | γ g x , h x , | x | γ g x , x . (6)Its Laplace transform can be derived through the Laplacefunctional corresponding to Ψ ij ( x ) , L b I ,i,j ( x ) ( s )= exp − Z B ( ,R i ) Z R + sQ ij | x | γ g g ( g ) sQ ij | x | γ g + | x + x | γ dgν ij ( x ) d x ! . (7)In this step, because the location of the tier-2 BS x is given , different types of tier-2 UEs associated with this tier-2BS are independent. Thus, the Laplace transform of b I ,i ( x ) can be computed as L b I ,i ( x ) ( s ) = K Y j =1 L b I ,i,j ( x ) ( s ) . (8)Note that the expressions for L b I ,i,j ( x ) ( s ) and L b I ,i ( x ) ( s ) in (7) and (8) are functions related to a unique coordinate x . This provides an important property for our subsequentanalysis, that the interfering signals from all UEs in one tier-2 cell can be equivalently regarded as emission from one aggregation point at x . As a consequence, we can use afunction of the aggregation point to represent the overallinterference from one tier-2 cell. Overall Interference from One Type of Tier-2 Cells(Aggregation):
Based on the above conclusion, we can studythe overall interference from a single type of tier-2 cells.Let I ,i denote the total interference from type- i tier-2 cellsto the typical tier-1 BS, I ,i = X x ∈ Θ i b I ,i ( x ) . (9) Note that for two tier-2 cells (no matter whether or not they overlap), theinterfering signals can be equivalently regarded as emission from two points.In this way, (7) and (8) accommodate the potential overlapping of two tier-2cells. hus, we can derive the Laplace transform of I ,i as follows L I ,i ( s ) = E Y x ∈ Θ i e − s b I ,i ( x ) ! = E E Y x ∈ Θ i e − s b I ,i ( x ) (cid:12)(cid:12)(cid:12)(cid:12) Θ i !! (10) = E Y x ∈ Θ i E (cid:16) e − s b I ,i ( x ) (cid:12)(cid:12) Θ i (cid:17)! (11) = E Y x ∈ Θ i L b I ,i ( x ) ( s ) ! (12) = exp (cid:18) − µ i Z R (1 − L b I ,i ( x ) ( s )) d x (cid:19) , (13)where (11) holds because given Θ i , the interference fromeach type- i tier-2 cell is independent with each other; (13)is obtained since (12) is in exactly the same form as thegenerating functional of the PPP Θ i [9]. Overall Interference (Superposition):
Let I denote thetotal interference from tier-2 UEs to the typical tier-1 BS.Because multiple types of tier-2 BSs can be regarded as independent superposition of each type of tier-2 BSs, theLaplace transform L I can be computed as L I ( s ) = N Y i =1 L I ,i ( s ) . (14) C. Overall Interference and Outage at Tier-1 Cell
Since tier-1 UEs and tier-2 UEs are independent, theLaplace transform of the total interference is L I ( s ) = L I ( s ) L I ( s ) . (15)Note that the uplink interference occurs at tier-1 BSs andhence its statistics is irrelevant to the type of the typical tier-1UEs communicating with the typical BS. Then, given an SIRthreshold T , the outage probability for any type- k tier-1 UEis given by P out,k = P ( P k h x U , < T I )= 1 − L I ( T /P k ) . (16) D. Exclusion Region
In this subsection, we discuss the effect of two types ofexclusion regions on the uplink interference analysis at tier-1BSs. Considering tier-2 UE-exclusion regions, (7) is affectedand becomes L b I ,i,j ( x ) ( s ) = exp − Z B ( ,R i ) Z R + sQ ij | x | γ g ( x + x / ∈ F ) g ( g ) sQ ij | x | γ g + | x + x | γ dgν ij ( x ) d x ! . (17) The coverage (resp. outage) probability is defined as the probability thatSIR is larger (resp. less) than T . P cover,k = 1 − P out,k . Considering tier-2 BS-exclusion regions, (13) is affected and L I ,i ( s ) becomes L I ,i ( s ) = exp − µ i Z R \F (1 − L b I ,i ( x ) ( s )) d x ! . (18)All the other steps in Section IV-B remain the same.V. I NTERFERENCE TO T IER -2 BS S In this section, we analyze the uplink interference at areference type- l tier-2 BS, termed the typical tier-2 BS , whenit is communicating with a reference type- k tier-2 UE, termedthe typical tier-2 UE . The typical tier-1 BS (located at some x B ) in this section is defined as the tier-1 BS nearest to thetypical tier-2 BS. Throughout this section we will re-definethe coordinates so that the typical tier-2 BS is located at . Accordingly, x B is uniformly distributed in H ( ) . Thecoordinates of all tier-1 BSs are re-defined as B ( x B ) = { ( aR c , √ aR c + √ bR c ) + x B | a, b ∈ Z } . Note that thecoordinates in Sections IV and V are labeled differently. A. Interference from Tier-1 UEs to Tier-2 BS
First, the interference from type- i tier-1 UEs to the typicaltier-2 BS is I ′ ,i ( x B ) = X x ∈ B ( x B ) X x ∈ Φ i T H ( x ) P i | x − x | γ g x , h x , | x | γ g x , x , (19)with Laplace transform L I ′ ,i ( x B ) ( s ) = exp − λ i X x ∈ B ( x B ) Z H ( x ) Z R + sP i | x − x | γ g g ( g ) sP i | x − x | γ g + | x | γ dgd x ! . (20)Due to the independence of different types of tier-1 UEs,the Laplace transform of the interference from all tier-1 UEsis L I ′ ( x B ) ( s ) = M Y i =1 L I ′ ,i ( x B ) ( s ) . (21) B. Inter-Cell Interference from Tier-2 UEs to Tier-2 BS
Similar to the approach in Section IV-B, we can apply theSAS method to analyze the inter-cell interference from tier-2UEs to the typical tier-2 BS (excluding co-cell UEs connectedwith the typical tier-2 BS).Conditioned on the typical tier-2 BS at , let Θ ! l denotethe reduced Palm point process of the other type- l tier-2 BSs.Since Θ l is a PPP, Θ ! l has the same distribution as Θ l [6]. Forpresentation convenience, we define e Θ i , such that e Θ i = Θ i if i = l and e Θ i = Θ ! i if i = l .Let b I ′ ,i ( x , x B ) be the interference from a single type- i tier-2 cell whose BS is at x , to the typical tier-2 BS. I ′ ,i ( x , x B ) = P Kj =1 b I ′ ,i,j ( x , x B ) , where b I ′ ,i,j is the in-terference from all type- j tier-2 UEs in the single type- i tier-2cell. We have b I ′ ,i,j ( x , x B ) = X x ∈ Ψ ij ( x ) Q ij | x − x | γ g x , h x , | x | γ g x , x . (22)Its Laplace transform is L b I ′ ,i,j ( x , x B ) ( s )= exp − Z B ( ,R i ) Z R + sQ ij | x | γ g g ( g ) sQ ij | x | γ g + | x + x | γ dgν ij ( x ) d x ! . (23)Due to the independence of different types of UEs in a tier-2cell, the Laplace transform of b I ′ ,i ( x , x B ) is L b I ′ ,i ( x , x B ) ( s ) = K Y j =1 L b I ′ ,i,j ( x , x B ) ( s ) . (24)Let I ′ ,i ( x B ) denote the overall inter-cell interference fromtype- i tier-2 cells, I ′ ,i ( x B ) = X x ∈ e Θ i b I ′ ,i ( x , x B ) . (25)Similar to the derivation of (13), we can derive its Laplacetransform as L I ′ ,i ( x B ) ( s )= E Y x ∈ e Θ i E (cid:16) e − s b I ′ ,i ( x , x B ) | e Θ i (cid:17) = E Y x ∈ e Θ i L b I ′ ,i ( x , x B ) ( s ) = exp (cid:18) − µ i Z R (1 − L b I ′ ,i ( x , x B ) ( s )) d x (cid:19) . (26)Let I ′ ( x B ) denote the total inter-cell interference from tier-2 UEs to the typical tier-2 BS. Because multiple types of tier-2BSs can be regarded as independent superposition of each typeof tier-2 BSs, the Laplace transform of I ′ ( x B ) is L I ′ ( x B ) ( s ) = N Y i =1 L I ′ ,i ( x B ) ( s ) . (27)Note that (22)-(27) do not depend on x B , thus x B can beomitted in these formulas. However, if we consider the twotypes of exclusion regions, (22)-(27) are affected by x B , and x B cannot be omitted. C. Intra-Cell Interference from Tier-2 UEs to Tier-2 BS
In this subsection, we consider the interference within thetypical tier-2 cell, given the typical type- k tier-2 UE. Let Ψ ! lk ( ) denote the reduced Palm point process of the othertype- k UEs in the typical tier-2 cell. Since Ψ lk ( ) is a PPP, Ψ ! lk ( ) has the same distribution as Ψ lk ( ) . For presentationconvenience, we define e Ψ lj ( ) , such that e Ψ lj ( ) = Ψ lj ( ) if k = j and e Ψ lj ( ) = Ψ ! lj ( ) if k = j . The intra-cell interference from type- j tier-2 UEs is I ′ ,j ( x B ) = X x ∈ e Ψ lj ( ) Q lj h x , , (28)with Laplace transform L I ′ ,j ( x B ) = exp − sQ lj sQ lj + 1 Z B ( ,R l ) ν lj ( x ) d x , (29)Thus, the Laplace transform of the overall interferenceinside the typical tier-2 cell is L I ′ ( x B ) ( s ) = K Y j =1 L I ′ ,j ( x B ) ( s ) . (30)Note that (29)-(30) do not depend on x B , x B can be omitted inthese formulas. However, if we consider two types of exclusionregions (discussed in Section V-E), (29)-(30) are affected by x B , and x B cannot be omitted. D. Overall Interference and Outage at Tier-2 Cell
Since I ′ ( x B ) , I ′ ( x B ) , and I ′ ( x B ) are independent, theLaplace transform of the total interference given x B is L I ′ ( x B ) ( s ) = L I ′ ( x B ) ( s ) L I ′ ( x B ) ( s ) L I ′ ( x B ) ( s ) . (31)Thus, the Laplace transform of the overall interferenceunconditioned on x B is L I ′ ( s ) = R H ( ) L I ′ ( x B ) ( s ) d x B (cid:0) √ / R c (cid:1) . (32)Because L I ′ ( x B ) ( s ) and L I ′ ( x B ) ( s ) do not depend on x B , L I ′ ( s ) = L I ′ ( s ) · L I ′ ( s ) · L I ′ ( s ) , (33)where L I ′ ( s ) = R H ( ) L I ′ x B ) ( s ) d x B ( √ / R c ) .Note that if we consider the two types of exclusion regions, L I ′ ( x B ) ( s ) and L I ′ ( x B ) ( s ) depend of x B , (31)-(32) rather than(33) should be employed.Finally, given the SIR threshold T , the outage probabilityof the typical tier-2 UE (type- k tier-2 UE in the typical type- l tier-2 cell) is given by P ′ out,lk = 1 − L I ′ ( T /Q lk ) . (34) E. Effect of Exclusion Regions
In this subsection, we discuss the effect of the two typesof exclusion regions defined in Section III-C on the uplinkinterference analysis at tier-2 BSs. Considering tier-2 UE-exclusion regions, (23) and (29) are affected and become L b I ′ ,i,j ( x , x B ) ( s ) = exp − Z B ( ,R i ) Z R + sQ ij | x | γ g ( x + x / ∈ F ( x B )) g ( g ) sQ ij | x | γ g + | x + x | γ dgν ij ( x ) d x ! , (35) Note that the collection of BS-exclusion and UE-exclusion regionsare re-defined as F ( x B ) = S x ∈ B ( x B ) B ( x , R e, ) and F ( x B ) = S x ∈ B ( x B ) B ( x , R e, ) , respectively. nd L I ′ ,j ( x B ) = exp − sQ lj sQ lj + 1 Z B ( ,R l ) \F ( x B ) ν lj ( x ) d x , (36)Considering tier-2 BS-exclusion regions, (26) is affected andbecomes L I ′ ,i ( x B ) ( s ) = exp − µ i Z R \F ( x B ) (1 − L b I ′ ,i ( x , x B ) ( s )) d x ! . (37)All the other steps in Section V-B and V-C remain thesame. Note that by considering the exclusion regions, (22)-(30) depend on x B . In this case, we should employ (31)-(32),rather than (33) to compute the Laplace transform of I ′ .VI. C ASE S TUDIES
In this section, we present several important case studiesbased on our analysis in Section IV and V. First, we discussthe effects of several network parameters based on a single-type scenario. Second, we present a modified version of ourmodel, which provides close approximations for UEs’ perfor-mance in systems using orthogonal multiple access schemes.Third, we present a negative exponential property and theintensity planning problem.
A. Single-Type Scenario
In this subsection, we present a simple case with only onetype of tier-1 UEs, one type of tier-2 BSs, and one type oftier-2 UEs (i.e., M = N = K = 1 ). Exclusion regions are notconsidered.First, according to (1)-(16), the outage probability of tier-1UEs is P out = 1 − P ,in P ,out P , where P ,in = L I ,in, ( T /P ) = exp (cid:18) − λ (3 √ / (cid:18) T R c T + 1 (cid:19)(cid:19) , (38) P ,out = L I ,out, ( T /P ) = exp − λ X x ∈ B \{ } Z H ( x ) Z R + T | x − x | γ g g ( g ) T | x − x | γ g + | x | γ dgd x ! , (39) P = L I , ( T /P ) = exp − µ Z R (cid:18) − exp (cid:18) − Z B ( ,R ) Z R + Q TP | x | γ gν ( x ) g ( g ) Q TP | x | γ g + | x + x | γ dgd x (cid:19)(cid:19) d x ! . (40)Then, according to (19)-(34), the outage probability of tier-2 UEs is P ′ out = 1 − P ′ P ′ P ′ , where P ′ = L I ′ ( T /Q ) = 13 √ / R c Z H ( ) exp − λ X x ∈ B ( x B ) Z H ( x ) Z R + P TQ | x − x | γ g g ( g ) P TQ | x − x | γ g + | x | γ dgd x ! d x B , (41) P ′ = L I ′ ( T /Q ) = exp − µ Z R (cid:18) − exp (cid:18) − Z B ( ,R ) Z R + T | x | γ gν ( x ) g ( g ) T | x | γ g + | x + x | γ dgd x (cid:19)(cid:19) d x ! , (42) P ′ = L I ′ ( T /Q ) = exp − TT + 1 Z B ( ,R ) ν ( x ) d x . (43)Through (38)-(43), we can observe important effects ofdifferent network parameters:
1) Effects of Targeted Received Power: P ,in = L I ,in, ( T /P ) corresponds to the coverage probability of tier-1 UEs if there is only co-tier intra-cell interference from tier-1UEs, which is irrelevant to P . Even if P is changed, both thereceived signal level and the co-tier interference level at tier-1BSs are scaled by the same factor, leading to a constant signalto interference ratio. Thus, P ,in does not change in this case.Similarly, P ,out corresponds to the coverage probability oftier-1 UEs if there is only co-tier inter-cell interference fromtier-1 UEs, which is also irrelevant to P .For the same reason, P ′ (resp. P ′ ) corresponds to thecoverage probability of tier-2 UEs if there is only co-tier inter-cell interference (resp. intra-cell interference) from tier-2 UEs,which is irrelevant to Q . P and P ′ are related to the cross-tier SIR. Increasing Q P leads to higher cross-tier SIR at tier-2 BSs, but lower cross-tierSIR at tier-1 BSs. Thus, P ′ is increased and P is decreased.As a example, we may design Q P to minimize the overallaverage outage probabilities of both tier-1 and tier-2 UEs,which can be computed as P out = P out λ + P ′ out µ R B ( ,R ) ν ( x ) d x λ + µ R B ( ,R ) ν ( x ) d x . (44)Numerical methods can be applied to search for the optimal Q P values, which will be further discussed in Section VII-D.Note that P and P ′ are the only parts to be recomputed underdifferent Q P , which reduces the complexity in the numericalsearch.
2) Effects of ν ( x ) : Let ν = R B ( ,R ) ν ( x ) d x indicatethe average number of tier-2 UEs in a tier-2 cell. We areinterested in studying the outage performance under the same ν , but different ν ( x ) .If ν ( x ) becomes more dense at locations with larger | x | (but less dense at locations with smaller | x | ), tier-2 UEs aremore likely to locate at cell edges where they transmit withhigher power levels. As a consequence, the interference fromtier-2 UEs is increased and the outage probabilities of bothier-1 and tier-2 UEs are increased. Further numerical studiesare given in Sections VII-E. B. Orthogonal Multiple Access
It is desirable to study systems using orthogonal multipleaccess schemes, such as OFDMA. In this case, the frequencyand time resources are partitioned into n orthogonal resourceblocks. Each BS randomly allocates one unused resource blockto one UE in the cell. However, this introduces complicatedcoupling between the point processes of BSs and UEs, whichare difficult to characterize based on standard stochastic geo-metric analysis [14].In this paper, we employ the following modified versionof our analysis to approximately characterize orthogonal mul-tiple access systems: (a) Intra-cell interference terms are notaccounted (i.e., I ,in,i and I ′ ,j are regarded as zero). (b) Inter-cell interfering UEs are equivalently viewed as independentlythinned point processes with probability n . The equivalentintensity of type- i tier-1 UEs is λ i n , and the intensity of type- j tier-2 UEs in a type- i tier-2 cell is characterized by ν ij ( x − x ) n .Note that in the orthogonal multiple access mode, the resultantinterfering UEs actually correspond to dependently thinnedpoint processes. We use the independently thinned point pro-cesses to approximate their corresponding dependently thinnedones. Simulation results in Section VII-F indicate that the abovemethod leads to closely approximated outage probabilities forboth tier-1 and tier-2 UEs.
C. Negative Exponential Property and Intensity Planning1) Negative Exponential Property:
Through our discussionin Section IV and V, we observe that L I ( s ) and L I ′ ( s ) canbe expressed in the form of L I ( s ) = Q Ni =1 exp( − µ i C i ( s )) and L I ′ ( s ) = Q Ni =1 exp( − µ i C ′ i ( s )) , where C i ( s ) = Z R (cid:18) − L b I ,i ( x ) ( s ) (cid:19) d x , (45) C ′ i ( s ) = Z R (cid:18) − L b I ′ ,i ( x ) ( s ) (cid:19) d x , (46)according to (13)-(14) and (26)-(27) respectively. The neatform expressions for L I ( s ) and L I ′ ( s ) are referred to as the negative exponential property of tier-2 cell intensities. Next,we will present the usefulness of this negative exponentialproperty in an intensity planning problem.
2) Intensity Planning:
Consider a network upgrading sce-nario. The original network is a one-tier network with multi-type UEs, which matches the system model of the tier-1network in this paper (discussed in Section III). All the pa-rameters of the original one-tier network are given, including R c , λ i , and P i . The operator aims to update the network toa two-tier network, with diverse tier-2 cells and tier-2 UEs.The tier-2 network also matches the same system model of If there are more than n UEs in one cell, we assume that the BS randomlyselects n UEs to allocate resource blocks. This approximation is applicable to the systems where the average numberof UEs per cell is less than the number of resource blocks n . the tier-2 network in this paper. The parameters of each typeof tier-2 cells are also given, including R i , Q ij , and ν ij ( x ) .Exclusion regions are not considered. After upgrading thenetwork, suppose the operator makes extra income U i ( µ i ) by type- i tier-2 cells, where U i is a non-decreasing concavefunction of the intensity µ i . Thus, the total income of updatingthe network is P Ni =1 U i ( µ i ) . To guarantee the uplink qualityof tier-1 and tier-2 UEs, the outage probability of the type- j tier-1 UEs cannot be greater than P target,j ; and the outageprobability of the type- k tier-2 UEs in type- l tier-2 cells cannotbe greater than P ′ target,lk . These outage probability constraintsrestrict the intensities of tier-2 cells. In summary, we canestablish a utility maximization problem:maximize µ i , ∀ i , N X i =1 U i ( µ i ) (47)subject to P out,j ( µ , . . . , µ N ) ≤ P target,j , ∀ j, (48) P ′ out,lk ( µ , . . . , µ N ) ≤ P ′ target,lk , ∀ l, k, (49) µ i ≥ , ∀ i, (50)where P out,j ( µ , µ , . . . , µ N ) and P ′ out,lk ( µ , . . . , µ N ) are theoutage probabilities of type- j tier-1 UEs and type- k tier-2 UEsin type- l tier-2 cells under µ , µ , . . . , µ N , computed as (16)and (34) respectively.By the negative exponential property, the outage probabilityconstraints (48) can be converted to inequalities: N Y i =1 exp( − A ij µ i ) ≥ − P target,j L I ( T /P j ) , (51)which is equivalent to N X i =1 A ij µ i ≤ B j , (52)where A ij = C i ( T /P j ) and B j = − log (cid:16) − P target,j L I ( T/P j ) (cid:17) .Similarity, the outage probability constraints (49) can beconverted to inequalities: N Y i =1 exp( − A ′ ilk µ i ) ≥ − P ′ target,lk L I ′ ( T /Q lk ) L I ′ ( T /Q lk ) , (53)which is equivalent to N X i =1 A ′ ilk µ i ≤ B ′ lk , (54)where A ′ ilk = C ′ i ( T /Q lk ) and B ′ lk = − log (cid:18) − P ′ target,lk L I ′ ( T/Q lk ) L I ′ ( T/Q lk ) (cid:19) .Note that A ij , B j , A ′ ilk , and B ′ lk can be computed withthe given network parameters. Thus, the constraints (52) and(54) represent linear intensity tradeoff . As a consequence,the original optimization problem (47)-(50) is converted toa convex optimization problem with simple linear constraints,which can be solved efficiently. Further numerical studies aregiven in Sections VII-G.This application example demonstrates that our analysiscan be used to convert the complicated outage probabilityconstraints into simple linear constraints, facilitating tractableproblem solutions.II. N UMERICAL S TUDY
In this section, we present a numerical study to validate theaccuracy and utility of our analysis model. Unless otherwisestated, R c = 1 km, γ = 4 , the shadowing term is log-normal with mean and standard deviation dB, fast fadingis Rayleigh with unit mean. Each simulation data point isaveraged over trials. Error bars in the figures showthe confidence intervals for simulation results. Some plotpoints are slightly shifted to avoid overlapping error bars foreasier inspection. The network parameters used in all figuresare listed in Table II. A. Model Comparison
First, we compare our system model with the model withapproximations in [3]. Since the authors of [3] did not considermulti-type UEs or BSs, our comparison is based on the case ofsingle-type tier-1 UEs, tier-2 cells, and tier-2 UEs (i.e., M = N = K = 1 ). We use all four approximating assumptionsstated in Section II in the way of [3].Figs. 3(a)-3(b) show the uplink outage probability of tier-1 cells under different λ and µ respectively. The figuresillustrate that our analytical results are accurate and offerimprovement over the model in [3]. In [3], because tier-2UEs are assumed to be located at the edge of tier-2 cells andtransmitting with maximum power, the interference from tier-2 UEs to tier-1 BSs (as well as tier-2 BSs) is overestimated.Also, when the tier-1 inter-cell interference is approximatedas truncated Gaussian, larger evaluation error occurs. Overall,the outage probabilities of tier-1 uplinks are overestimated bythe model in [3].Figs. 3(c)-3(d) show the uplink outage probability of tier-2 cells under different λ and µ respectively. The figuresagain illustrate that our analytical results are accurate andoffer improvement. In [3], because the interference from tier-1UEs outside the reference tier-1 cell is ignored, the cross-tierinterference from tier-1 UEs to tier-2 BSs is greatly underes-timated. Even though the co-tier interference from tier-2 UEsto tier-2 BSs is overestimated, that cannot compensate for theunderestimation of the cross-tier interference. Hence, overall,the outage probabilities of tier-2 uplinks are underestimatedby the model in [3].Notably, as we increase λ , more tier-1 interference isignored, causing larger approximation error; when we increase µ , the overestimation of the co-tier interference cancels moreof the underestimation of the cross-tier interference, leadingto an overall smaller estimation error.Fig. 3(e) shows the outage probability under different T (i.e., the CDF of SIR). This figure further confirms that ouranalytical results are more accurate than the alternatives. B. Outage Probability of Different Tiers and Types
In this subsection, we study the outage probabilities ofdifferent tiers and types. Fig. 3(f) shows the analytical andsimulation outage probabilities of different types and tiers.The simulation results agree with the analytical results. Attier-1, because P > P , the outage probability of type- tier-1 UEs is smaller. At tier-2, Q > Q leads to smalleroutage probability for type-2 tier-2 UEs; while Q = Q leads to the same outage probabilities. Given an arbitrarytypical UE, the Palm distribution of other UEs (i.e., interferers)remains the same as their original Poisson distribution. Thus,the distribution of interference remains the same. The effectof multiple types of UEs only manifests in the shape of thecommon distribution of interference. C. Exclusion Regions
In this subsection, we discuss the outage performance underthe influence of the two types of exclusion regions. Eachsimulation data point is averaged over trials.Under different radii of the exclusion regions, we study theoutage probabilities at both tiers derived by our model andsimulations, as shown in Figs. 3(g)-4(a). These figures showthat almost all value points computed by the model are withinthe confidence intervals, illustrating the correctness ofour modeling of two types of exclusion regions.At tier-1 cells, the results show that the outage probabilitydoes not decrease significantly when we increase the radius ofexclusion regions (e.g., the outage probability is lowered by . − . when we increase the radius from m to m). This is because, first, the interference at tier-1 BSs is notdominated by the interference from tier-2 UEs; and second,the exclusion regions are small, so that the probability thatthere are some tier-2 UEs in the exclusion regions causinglarge interference is small. As a result, the outage probabilityonly slightly decreases as we introduce the exclusion regions.Under the same radius, using UE-exclusion regions is moreefficient to decrease the outage probability at tier-1 cells. Thisis because tier-2 UEs are strictly forbidden within a distanceof R e, from a tier-1 BS when UE exclusion is applied,while they may be located as near as ( R e, − R ) + underBS exclusion.At tier-2 cells, the outage probability is influenced by theexclusion regions in more complicated ways: (a) Increasing R e, or R e, leads to less tier-2 UEs, which then deceases theoutage probability. (b) Increasing R e, or R e, leads to a higherprobability that a tier-2 cell is located at the edge of a tier-1cell, where tier-1 UEs transmit at higher power levels, which inturn increases the outage probability. (c) For the UE exclusion,increasing R e, leads to a higher probability that a tier-2 cellis overlapping with the exclusion regions, where tier-2 UEsare restricted, decreasing the intra-cell interference and outageprobability. Considering all of these factors, if BS exclusion isapplied, it is not obvious whether factor (a) or (b) dominatesthe other. Figs. 3(i)-4(a) indicate that the outage probabilityat tier-2 cells increases when T = 0 . and decreases when T = 1 , if we increase the radius of BS-exclusion regions. Onthe other hand, if UE exclusion is applied, factor (c) dominatesand thus the outage probability decreases when we increasethe radius of exclusion regions. D. Optimal Targeted Received Power
In this subsection, we present a numerical study on theoptimal targeted received power ratio, as discussed in Section
ABLE IIL
IST OF P ARAMETERS U SED IN N UMERICAL S TUDIES . Section Figure
M, N, K
Power (dBm) λ, µ (units/km ) Radius (km) ν ( · ) (units/km ) a T Special
VII-A 3(a)-3(d) , , P , Q ) = ( − , − Various R = 0 . ν ( x ) = 20 0 . or None
VII-A 3(e) , , P , Q ) = ( − , −
70) ( λ , µ ) = (0 . , . R = 0 . ν ( x ) = 20 Various None ( P , P ) = ( − , − . ν ( x ) = 10 VII-B 3(f) , , Q , Q ) = ( − , −
64) ( λ , λ ) = (0 . , . R = 0 . ν ( x ) = 15 ( Q , Q ) = ( − , −
67) ( µ , µ ) = (1 , R = 0 . ν ( x ) = 5 ν ( x ) = 20 VII-C 3(g)-4(a) , , P , Q ) = ( − , −
70) ( λ , µ ) = (0 . , . R = 0 . ν ( x ) = 20 0 . or Exclusion Regions
VII-D 4(b) , , Various ( λ , µ ) = (0 . , R = 0 . ν ( x ) = 20 0 . None
VII-E 4(c)-4(d) , , P , Q ) = ( − , − λ = 0 . R = 0 . None-homogeneous . None ( P , P ) = ( − , − . ν ( x ) = 16 VII-F 4(e) , , Q , Q ) = ( − , −
64) ( λ , λ ) = (1 . , . R = 0 . ν ( x ) = 48 OFDM, n = 16 ( Q , Q ) = ( − , −
67) ( µ , µ ) = (1 , R = 0 . ν ( x ) = 32 ν ( x ) = 32 VII-G 4(f) , , P , P ) = ( − , −
66) ( λ , λ ) = (0 . , . R = 0 . ν ( x ) = 10 Intensity Tradeoff ( Q , Q ) = ( − , − .
2) ( µ , µ ) various R = 0 . ν ( x ) = 8( P , P ) = ( − , − . ν ( x ) = 10 VII-H 4(g)-4(h) , , Q , Q ) = ( − , −
64) ( λ , λ ) = (0 . , . R = 0 . ν ( x ) = 15 With correlation ( Q , Q ) = ( − , −
67) ( µ , µ ) = (1 , R = 0 . ν ( x ) = 5 ν ( x ) = 20( P , P ) = ( − , − ν ( x ) = 10 VII-I 4(i) , , Q , Q ) = ( − , −
67) ( λ , λ ) = (0 . , . R = 0 . ν ( x ) = 5 With correlation ( Q , Q ) = ( − , −
64) ( µ , µ ) various R = 0 . ν ( x ) = 5 ν ( x ) = 5 a Corresponding to the intensities inside the cell range.
VI-A1. Fig. 4(b) shows the results. Increasing Q P leads tohigher outage probability for tier-1 UEs but lower outageprobability for tier-2 UEs. The overall outage probabilitydecreases and then increases, which is minimized around Q P = 15 dB. E. Effects of Tier-2 UE Intensity Function
In this subsection, we discuss the effect of the tier-2 UEintensity function. We focus on three intensity functions oftier-2 UEs, all in units/km : (a) ν ( x ) = 20 , where UEs arehomogeneously distributed; (b) ν +11 ( x ) = 30 | x | R , where UEsare likely to locate at cell edges; and (c) ν − ( x ) = 60 R −| x | R ,where UEs are likely to locate near cell centers. Note that thetier-2 UE intensities are when | x | > R . In addition, theaverage numbers of tier-2 UEs in one tier-2 cell are the samein all of the three cases.As expected in the discussion in Section VI-A2, Figs. 4(c)-4(d) show that compared with ν ( x ) , ν +11 ( x ) introduceshigher interference from tier-2 UEs, causing higher outageprobabilities at both tiers, while ν − ( x ) brings lower inter-ference from tier-2 UEs, causing lower outage probabilities atboth tiers. F. Orthogonal Multiple Access
In this subsection, we present a numerical study on the sys-tems using orthogonal multiple access schemes. We simulatedependent thinning of UEs, instead of the independent thinningmodel used in our analysis.Fig. 4(e) shows the comparison of simulated outage prob-abilities and analytical ones (derived through our modifiedmodel discussed in Section VI-B). The number of resourceblocks at each BS is n = 16 . The figure shows that theanalytical outage probabilities derived in Section VI-B are only slightly smaller than simulated ones, suggesting that themodified version of our model provides useful approximationsfor systems using orthogonal multiple access schemes. G. Linear Intensity Tradeoff of Tier-2 cells
In this subsection, we investigate the linear intensity tradeoffdiscussed in Section VI-C. The outage probability constraintsare P target, = P target, = P ′ target, = P ′ target, = 0 . . Allthe other parameters are shown in Table II. Computed by (52)and (54), the tradeoff between µ and µ follows the linearinequality, . µ + 0 . µ ≤ . . (55)Note that (55) corresponds to the outage constraint of type-1tier-1 UEs, which dominates all the other outage constraints.We also use simulation to search for the maximum µ under different µ values. Fig. 4(f) shows the analytical andsimulation results of the intensity tradeoff between µ and µ . The results show that the simulation results agree with theanalytically obtained linear tradeoff in Section VI-C. H. Impact of Correlated Tier-1 UE or Tier-2 BS Locations
In this subsection, we study the performance under corre-lated tier-1 UE or tier-2 BS locations via simulation, in orderto show that our model remains useful as an approximationwhen the spatial patterns are not strictly Poisson.The correlated locations are generated as follows: In eachtrail of simulation, let X = ( x , x , . . . x n ) T denote originalpoints corresponding to one type of tier-1 UEs or tier-2 BSsgenerated as PPP. We set X ′ = K T X as the new coordinatesby introducing correlations among the original coordinates.Then, L = K T K is the covariance matrix of the newcoordinates. We further set the i th row, j th column of L , (cid:17)(cid:20) (cid:19)(cid:17)(cid:21) (cid:19)(cid:17)(cid:22) (cid:19)(cid:17)(cid:23) (cid:19)(cid:17)(cid:24) (cid:19)(cid:17)(cid:25) (cid:19)(cid:17)(cid:26) (cid:19)(cid:17)(cid:27) (cid:19)(cid:17)(cid:28) (cid:20)(cid:19)(cid:19)(cid:17)(cid:20)(cid:19)(cid:17)(cid:21)(cid:19)(cid:17)(cid:22)(cid:19)(cid:17)(cid:23)(cid:19)(cid:17)(cid:24)(cid:19)(cid:17)(cid:25)(cid:19)(cid:17)(cid:26)(cid:19)(cid:17)(cid:27)(cid:19)(cid:17)(cid:28)(cid:20) λ units/km O u t ag e p r o b a b ili t y µ = 0 . (cid:54)(cid:76)(cid:80)(cid:88)(cid:79)(cid:68)(cid:87)(cid:76)(cid:82)(cid:81)(cid:15)(cid:3)(cid:55)(cid:32)(cid:20)(cid:36)(cid:81)(cid:68)(cid:79)(cid:92)(cid:87)(cid:76)(cid:70)(cid:68)(cid:79)(cid:15)(cid:3)(cid:55)(cid:32)(cid:20)(cid:36)(cid:81)(cid:68)(cid:79)(cid:92)(cid:87)(cid:76)(cid:70)(cid:68)(cid:79)(cid:3)(cid:82)(cid:73)(cid:3)(cid:62)(cid:22)(cid:64)(cid:15)(cid:3)(cid:55)(cid:32)(cid:20)(cid:54)(cid:76)(cid:80)(cid:88)(cid:79)(cid:68)(cid:87)(cid:76)(cid:82)(cid:81)(cid:15)(cid:3)(cid:55)(cid:32)(cid:19)(cid:17)(cid:20)(cid:36)(cid:81)(cid:68)(cid:79)(cid:92)(cid:87)(cid:76)(cid:70)(cid:68)(cid:79)(cid:15)(cid:3)(cid:55)(cid:32)(cid:19)(cid:17)(cid:20)(cid:36)(cid:81)(cid:68)(cid:79)(cid:92)(cid:87)(cid:76)(cid:70)(cid:68)(cid:79)(cid:3)(cid:82)(cid:73)(cid:3)(cid:62)(cid:22)(cid:64)(cid:15)(cid:3)(cid:55)(cid:32)(cid:19)(cid:17)(cid:20) (a) Tier-1 outage probability under different λ . (cid:19)(cid:17)(cid:20) (cid:19)(cid:17)(cid:21) (cid:19)(cid:17)(cid:22) (cid:19)(cid:17)(cid:23) (cid:19)(cid:17)(cid:24) (cid:19)(cid:17)(cid:25) (cid:19)(cid:17)(cid:26) (cid:19)(cid:17)(cid:27) (cid:19)(cid:17)(cid:28) (cid:20)(cid:19)(cid:19)(cid:17)(cid:20)(cid:19)(cid:17)(cid:21)(cid:19)(cid:17)(cid:22)(cid:19)(cid:17)(cid:23)(cid:19)(cid:17)(cid:24)(cid:19)(cid:17)(cid:25)(cid:19)(cid:17)(cid:26)(cid:19)(cid:17)(cid:27)(cid:19)(cid:17)(cid:28)(cid:20) µ units/km O u t ag e p r o b a b ili t y λ = 0 . (cid:54)(cid:76)(cid:80)(cid:88)(cid:79)(cid:68)(cid:87)(cid:76)(cid:82)(cid:81)(cid:15)(cid:3)(cid:55)(cid:32)(cid:20)(cid:36)(cid:81)(cid:68)(cid:79)(cid:92)(cid:87)(cid:76)(cid:70)(cid:68)(cid:79)(cid:15)(cid:3)(cid:55)(cid:32)(cid:20)(cid:36)(cid:81)(cid:68)(cid:79)(cid:92)(cid:87)(cid:76)(cid:70)(cid:68)(cid:79)(cid:3)(cid:82)(cid:73)(cid:3)(cid:62)(cid:22)(cid:64)(cid:15)(cid:3)(cid:55)(cid:32)(cid:20)(cid:54)(cid:76)(cid:80)(cid:88)(cid:79)(cid:68)(cid:87)(cid:76)(cid:82)(cid:81)(cid:15)(cid:3)(cid:55)(cid:32)(cid:19)(cid:17)(cid:20)(cid:36)(cid:81)(cid:68)(cid:79)(cid:92)(cid:87)(cid:76)(cid:70)(cid:68)(cid:79)(cid:15)(cid:3)(cid:55)(cid:32)(cid:19)(cid:17)(cid:20)(cid:36)(cid:81)(cid:68)(cid:79)(cid:92)(cid:87)(cid:76)(cid:70)(cid:68)(cid:79)(cid:3)(cid:82)(cid:73)(cid:3)(cid:62)(cid:22)(cid:64)(cid:15)(cid:3)(cid:55)(cid:32)(cid:19)(cid:17)(cid:20) (b) Tier-1 outage probability under different µ . (cid:19)(cid:17)(cid:20) (cid:19)(cid:17)(cid:21) (cid:19)(cid:17)(cid:22) (cid:19)(cid:17)(cid:23) (cid:19)(cid:17)(cid:24) (cid:19)(cid:17)(cid:25) (cid:19)(cid:17)(cid:26) (cid:19)(cid:17)(cid:27) (cid:19)(cid:17)(cid:28) (cid:20)(cid:19)(cid:19)(cid:17)(cid:20)(cid:19)(cid:17)(cid:21)(cid:19)(cid:17)(cid:22)(cid:19)(cid:17)(cid:23)(cid:19)(cid:17)(cid:24)(cid:19)(cid:17)(cid:25)(cid:19)(cid:17)(cid:26)(cid:19)(cid:17)(cid:27)(cid:19)(cid:17)(cid:28)(cid:20) λ units/km O u t ag e p r o b a b ili t y µ = 0 . (cid:54)(cid:76)(cid:80)(cid:88)(cid:79)(cid:68)(cid:87)(cid:76)(cid:82)(cid:81)(cid:15)(cid:3)(cid:55)(cid:32)(cid:20)(cid:36)(cid:81)(cid:68)(cid:79)(cid:92)(cid:87)(cid:76)(cid:70)(cid:68)(cid:79)(cid:15)(cid:3)(cid:55)(cid:32)(cid:20)(cid:36)(cid:81)(cid:68)(cid:79)(cid:92)(cid:87)(cid:76)(cid:70)(cid:68)(cid:79)(cid:3)(cid:82)(cid:73)(cid:3)(cid:62)(cid:22)(cid:64)(cid:15)(cid:3)(cid:55)(cid:32)(cid:20)(cid:54)(cid:76)(cid:80)(cid:88)(cid:79)(cid:68)(cid:87)(cid:76)(cid:82)(cid:81)(cid:15)(cid:3)(cid:55)(cid:32)(cid:19)(cid:17)(cid:20)(cid:36)(cid:81)(cid:68)(cid:79)(cid:92)(cid:87)(cid:76)(cid:70)(cid:68)(cid:79)(cid:15)(cid:3)(cid:55)(cid:32)(cid:19)(cid:17)(cid:20)(cid:36)(cid:81)(cid:68)(cid:79)(cid:92)(cid:87)(cid:76)(cid:70)(cid:68)(cid:79)(cid:3)(cid:82)(cid:73)(cid:3)(cid:62)(cid:22)(cid:64)(cid:15)(cid:3)(cid:55)(cid:32)(cid:19)(cid:17)(cid:20) (c) Tier-2 outage probability under different λ . (cid:19)(cid:17)(cid:20) (cid:19)(cid:17)(cid:21) (cid:19)(cid:17)(cid:22) (cid:19)(cid:17)(cid:23) (cid:19)(cid:17)(cid:24) (cid:19)(cid:17)(cid:25) (cid:19)(cid:17)(cid:26) (cid:19)(cid:17)(cid:27) (cid:19)(cid:17)(cid:28) (cid:20)(cid:19)(cid:19)(cid:17)(cid:20)(cid:19)(cid:17)(cid:21)(cid:19)(cid:17)(cid:22)(cid:19)(cid:17)(cid:23)(cid:19)(cid:17)(cid:24)(cid:19)(cid:17)(cid:25)(cid:19)(cid:17)(cid:26)(cid:19)(cid:17)(cid:27)(cid:19)(cid:17)(cid:28)(cid:20) µ units/km O u t ag e p r o b a b ili t y λ = 0 . (cid:54)(cid:76)(cid:80)(cid:88)(cid:79)(cid:68)(cid:87)(cid:76)(cid:82)(cid:81)(cid:15)(cid:3)(cid:55)(cid:32)(cid:20)(cid:36)(cid:81)(cid:68)(cid:79)(cid:92)(cid:87)(cid:76)(cid:70)(cid:68)(cid:79)(cid:15)(cid:3)(cid:55)(cid:32)(cid:20)(cid:36)(cid:81)(cid:68)(cid:79)(cid:92)(cid:87)(cid:76)(cid:70)(cid:68)(cid:79)(cid:3)(cid:82)(cid:73)(cid:3)(cid:62)(cid:22)(cid:64)(cid:15)(cid:3)(cid:55)(cid:32)(cid:20)(cid:54)(cid:76)(cid:80)(cid:88)(cid:79)(cid:68)(cid:87)(cid:76)(cid:82)(cid:81)(cid:15)(cid:3)(cid:55)(cid:32)(cid:19)(cid:17)(cid:20)(cid:36)(cid:81)(cid:68)(cid:79)(cid:92)(cid:87)(cid:76)(cid:70)(cid:68)(cid:79)(cid:15)(cid:3)(cid:55)(cid:32)(cid:19)(cid:17)(cid:20)(cid:36)(cid:81)(cid:68)(cid:79)(cid:92)(cid:87)(cid:76)(cid:70)(cid:68)(cid:79)(cid:3)(cid:82)(cid:73)(cid:3)(cid:62)(cid:22)(cid:64)(cid:15)(cid:3)(cid:55)(cid:32)(cid:19)(cid:17)(cid:20) (d) Tier-2 outage probability under different µ . (cid:20)(cid:19) (cid:237)(cid:22) (cid:20)(cid:19) (cid:237)(cid:21) (cid:20)(cid:19) (cid:237)(cid:20) (cid:20)(cid:19) (cid:19) (cid:20)(cid:19) (cid:20) (cid:20)(cid:19) (cid:21) (cid:19)(cid:19)(cid:17)(cid:20)(cid:19)(cid:17)(cid:21)(cid:19)(cid:17)(cid:22)(cid:19)(cid:17)(cid:23)(cid:19)(cid:17)(cid:24)(cid:19)(cid:17)(cid:25)(cid:19)(cid:17)(cid:26)(cid:19)(cid:17)(cid:27)(cid:19)(cid:17)(cid:28)(cid:20) T O u t ag e p r o b a b ili t y λ = 0 . , µ = 0 . (cid:54)(cid:76)(cid:80)(cid:88)(cid:79)(cid:68)(cid:87)(cid:76)(cid:82)(cid:81)(cid:15)(cid:3)(cid:87)(cid:76)(cid:72)(cid:85)(cid:237)(cid:20)(cid:36)(cid:81)(cid:68)(cid:79)(cid:92)(cid:87)(cid:76)(cid:70)(cid:68)(cid:79)(cid:15)(cid:3)(cid:87)(cid:76)(cid:72)(cid:85)(cid:237)(cid:20)(cid:36)(cid:81)(cid:68)(cid:79)(cid:92)(cid:87)(cid:76)(cid:70)(cid:68)(cid:79)(cid:3)(cid:82)(cid:73)(cid:3)(cid:62)(cid:22)(cid:64)(cid:15)(cid:3)(cid:87)(cid:76)(cid:72)(cid:85)(cid:237)(cid:20)(cid:54)(cid:76)(cid:80)(cid:88)(cid:79)(cid:68)(cid:87)(cid:76)(cid:82)(cid:81)(cid:15)(cid:3)(cid:87)(cid:76)(cid:72)(cid:85)(cid:237)(cid:21)(cid:36)(cid:81)(cid:68)(cid:79)(cid:92)(cid:87)(cid:76)(cid:70)(cid:68)(cid:79)(cid:15)(cid:3)(cid:87)(cid:76)(cid:72)(cid:85)(cid:237)(cid:21)(cid:36)(cid:81)(cid:68)(cid:79)(cid:92)(cid:87)(cid:76)(cid:70)(cid:68)(cid:79)(cid:3)(cid:82)(cid:73)(cid:3)(cid:62)(cid:22)(cid:64)(cid:15)(cid:3)(cid:87)(cid:76)(cid:72)(cid:85)(cid:237)(cid:21) (e) Outage probability under different T (CDFof SIR). (cid:20)(cid:15)(cid:3)(cid:20) (cid:20)(cid:15)(cid:3)(cid:21) (cid:21)(cid:15)(cid:3)(cid:20)(cid:15)(cid:3)(cid:20) (cid:21)(cid:15)(cid:3)(cid:20)(cid:15)(cid:3)(cid:21) (cid:21)(cid:15)(cid:3)(cid:21)(cid:15)(cid:3)(cid:20) (cid:21)(cid:15)(cid:3)(cid:21)(cid:15)(cid:3)(cid:21)(cid:19)(cid:19)(cid:17)(cid:20)(cid:19)(cid:17)(cid:21)(cid:19)(cid:17)(cid:22)(cid:19)(cid:17)(cid:23)(cid:19)(cid:17)(cid:24)(cid:19)(cid:17)(cid:25)(cid:19)(cid:17)(cid:26)(cid:19)(cid:17)(cid:27) (cid:55)(cid:76)(cid:72)(cid:85)(cid:3)(cid:68)(cid:81)(cid:71)(cid:3)(cid:87)(cid:92)(cid:83)(cid:72) (cid:50) (cid:88) (cid:87) (cid:68) (cid:74) (cid:72) (cid:3) (cid:83) (cid:85) (cid:82)(cid:69) (cid:68) (cid:69) (cid:76)(cid:79)(cid:76)(cid:87) (cid:92) (cid:36)(cid:81)(cid:68)(cid:79)(cid:92)(cid:87)(cid:76)(cid:70)(cid:68)(cid:79)(cid:54)(cid:76)(cid:80)(cid:88)(cid:79)(cid:68)(cid:87)(cid:76)(cid:82)(cid:81)(cid:40)(cid:85)(cid:85)(cid:82)(cid:85)(cid:69)(cid:68)(cid:85) (f) Outage probability at different tiers. , i indi-cates type- i tier-1 UEs; , i, j indicates type- j tier-2 UEs in type- i tier-2 cells. (cid:19) (cid:19)(cid:17)(cid:20) (cid:19)(cid:17)(cid:21) (cid:19)(cid:17)(cid:22) (cid:19)(cid:17)(cid:23) (cid:19)(cid:17)(cid:24)(cid:19)(cid:17)(cid:26)(cid:19)(cid:17)(cid:26)(cid:19)(cid:24)(cid:19)(cid:17)(cid:26)(cid:20)(cid:19)(cid:17)(cid:26)(cid:20)(cid:24)(cid:19)(cid:17)(cid:26)(cid:21)(cid:19)(cid:17)(cid:26)(cid:21)(cid:24)(cid:19)(cid:17)(cid:26)(cid:22)(cid:19)(cid:17)(cid:26)(cid:22)(cid:24)(cid:19)(cid:17)(cid:26)(cid:23)(cid:19)(cid:17)(cid:26)(cid:23)(cid:24) (cid:53)(cid:68)(cid:71)(cid:76)(cid:88)(cid:86) (cid:50) (cid:88) (cid:87) (cid:68) (cid:74) (cid:72) (cid:3) (cid:83) (cid:85) (cid:82)(cid:69) (cid:68) (cid:69) (cid:76)(cid:79)(cid:76)(cid:87) (cid:92) (cid:37)(cid:54)(cid:3)(cid:72)(cid:91)(cid:70)(cid:79)(cid:88)(cid:86)(cid:76)(cid:82)(cid:81)(cid:15)(cid:3)(cid:86)(cid:76)(cid:80)(cid:88)(cid:79)(cid:68)(cid:87)(cid:76)(cid:82)(cid:81)(cid:37)(cid:54)(cid:3)(cid:72)(cid:91)(cid:70)(cid:79)(cid:88)(cid:86)(cid:76)(cid:82)(cid:81)(cid:15)(cid:3)(cid:68)(cid:81)(cid:68)(cid:79)(cid:92)(cid:87)(cid:76)(cid:70)(cid:68)(cid:79)(cid:56)(cid:40)(cid:3)(cid:72)(cid:91)(cid:70)(cid:79)(cid:88)(cid:86)(cid:76)(cid:82)(cid:81)(cid:15)(cid:3)(cid:86)(cid:76)(cid:80)(cid:88)(cid:79)(cid:68)(cid:87)(cid:76)(cid:82)(cid:81)(cid:56)(cid:40)(cid:3)(cid:72)(cid:91)(cid:70)(cid:79)(cid:88)(cid:86)(cid:76)(cid:82)(cid:81)(cid:15)(cid:3)(cid:68)(cid:81)(cid:68)(cid:79)(cid:92)(cid:87)(cid:76)(cid:70)(cid:68)(cid:79) (g) Tier-1 outage probability under different dif-ferent radius of exclusion regions, T = 1 . (cid:19) (cid:19)(cid:17)(cid:20) (cid:19)(cid:17)(cid:21) (cid:19)(cid:17)(cid:22) (cid:19)(cid:17)(cid:23) (cid:19)(cid:17)(cid:24)(cid:19)(cid:17)(cid:20)(cid:26)(cid:24)(cid:19)(cid:17)(cid:20)(cid:27)(cid:19)(cid:17)(cid:20)(cid:27)(cid:24)(cid:19)(cid:17)(cid:20)(cid:28)(cid:19)(cid:17)(cid:20)(cid:28)(cid:24)(cid:19)(cid:17)(cid:21)(cid:19)(cid:17)(cid:21)(cid:19)(cid:24)(cid:19)(cid:17)(cid:21)(cid:20)(cid:19)(cid:17)(cid:21)(cid:20)(cid:24)(cid:19)(cid:17)(cid:21)(cid:21)(cid:19)(cid:17)(cid:21)(cid:21)(cid:24) (cid:53)(cid:68)(cid:71)(cid:76)(cid:88)(cid:86) (cid:50) (cid:88) (cid:87) (cid:68) (cid:74) (cid:72) (cid:3) (cid:83) (cid:85) (cid:82)(cid:69) (cid:68) (cid:69) (cid:76)(cid:79)(cid:76)(cid:87) (cid:92) (cid:37)(cid:54)(cid:3)(cid:72)(cid:91)(cid:70)(cid:79)(cid:88)(cid:86)(cid:76)(cid:82)(cid:81)(cid:15)(cid:3)(cid:86)(cid:76)(cid:80)(cid:88)(cid:79)(cid:68)(cid:87)(cid:76)(cid:82)(cid:81)(cid:37)(cid:54)(cid:3)(cid:72)(cid:91)(cid:70)(cid:79)(cid:88)(cid:86)(cid:76)(cid:82)(cid:81)(cid:15)(cid:3)(cid:68)(cid:81)(cid:68)(cid:79)(cid:92)(cid:87)(cid:76)(cid:70)(cid:68)(cid:79)(cid:56)(cid:40)(cid:3)(cid:72)(cid:91)(cid:70)(cid:79)(cid:88)(cid:86)(cid:76)(cid:82)(cid:81)(cid:15)(cid:3)(cid:86)(cid:76)(cid:80)(cid:88)(cid:79)(cid:68)(cid:87)(cid:76)(cid:82)(cid:81)(cid:56)(cid:40)(cid:3)(cid:72)(cid:91)(cid:70)(cid:79)(cid:88)(cid:86)(cid:76)(cid:82)(cid:81)(cid:15)(cid:3)(cid:68)(cid:81)(cid:68)(cid:79)(cid:92)(cid:87)(cid:76)(cid:70)(cid:68)(cid:79) (h) Tier-1 outage probability under different dif-ferent radius of exclusion regions, T = 0 . . (cid:19) (cid:19)(cid:17)(cid:20) (cid:19)(cid:17)(cid:21) (cid:19)(cid:17)(cid:22) (cid:19)(cid:17)(cid:23) (cid:19)(cid:17)(cid:24)(cid:19)(cid:17)(cid:28)(cid:19)(cid:17)(cid:28)(cid:19)(cid:24)(cid:19)(cid:17)(cid:28)(cid:20)(cid:19)(cid:17)(cid:28)(cid:20)(cid:24)(cid:19)(cid:17)(cid:28)(cid:21)(cid:19)(cid:17)(cid:28)(cid:21)(cid:24)(cid:19)(cid:17)(cid:28)(cid:22) (cid:53)(cid:68)(cid:71)(cid:76)(cid:88)(cid:86) (cid:50) (cid:88) (cid:87) (cid:68) (cid:74) (cid:72) (cid:3) (cid:83) (cid:85) (cid:82)(cid:69) (cid:68) (cid:69) (cid:76)(cid:79)(cid:76)(cid:87) (cid:92) (cid:37)(cid:54)(cid:3)(cid:72)(cid:91)(cid:70)(cid:79)(cid:88)(cid:86)(cid:76)(cid:82)(cid:81)(cid:15)(cid:3)(cid:86)(cid:76)(cid:80)(cid:88)(cid:79)(cid:68)(cid:87)(cid:76)(cid:82)(cid:81)(cid:37)(cid:54)(cid:3)(cid:72)(cid:91)(cid:70)(cid:79)(cid:88)(cid:86)(cid:76)(cid:82)(cid:81)(cid:15)(cid:3)(cid:68)(cid:81)(cid:68)(cid:79)(cid:92)(cid:87)(cid:76)(cid:70)(cid:68)(cid:79)(cid:56)(cid:40)(cid:3)(cid:72)(cid:91)(cid:70)(cid:79)(cid:88)(cid:86)(cid:76)(cid:82)(cid:81)(cid:15)(cid:3)(cid:86)(cid:76)(cid:80)(cid:88)(cid:79)(cid:68)(cid:87)(cid:76)(cid:82)(cid:81)(cid:56)(cid:40)(cid:3)(cid:72)(cid:91)(cid:70)(cid:79)(cid:88)(cid:86)(cid:76)(cid:82)(cid:81)(cid:15)(cid:3)(cid:68)(cid:81)(cid:68)(cid:79)(cid:92)(cid:87)(cid:76)(cid:70)(cid:68)(cid:79) (i) Tier-2 outage probability under different dif-ferent radius of exclusion regions, T = 1 . Fig. 3. Numerical studies. L ij = exp( − α | i − j | ) , where α is referred to as the cor-relation parameter. Smaller α indicates stronger correlationsamong the points. Details to derive K from L can be found inSection 6.6 of [25]. Except the above location transformation,all the other parts of simulation remain the same.Figs. 4(g)-4(h) show the outage probability of tier-1 UEsand tier-2 UEs under different correlation parameters. Bothfigures show that the correlations of tier-1 UEs or tier-2 BSswill cause higher outage probabilities at both tiers. When thetier-1 UEs or tier-2 BSs are weakly correlated (e.g., α ≥ ),the gap between the correlated case and the non-correlatedcase is almost . Even when the tier-2 BSs are stronglycorrelated (e.g., α ≤ . ), our model is still useful inapproximating the outage probabilities. I. Impact of Correlated Tier-1 UE or Tier-2 BS Locations onthe Intensity Planning Problem
In this subsection, we study the intensity planning problem(presented in Section VI-C) under correlated tier-1 UE or tier-2 BS locations. The correlated locations of tier-1 UEs or tier-2BSs are generated by the method presented in Section VII-H.The network parameters are shown in Table II. We alsoset the outage constraints for tier-1 UEs are P target, = P target, = 0 . ; and there are no outage constraints for tier-2UEs. The utility functions are U ( µ ) = 1 . µ ) and U ( µ ) = ln(1 + 10 µ ) , U ( µ , µ ) = U ( µ ) + U ( µ ) . Eachsimulation data point is averaged over 30000 trials.First, without location correlations, we derive the optimal U ∗ , µ ∗ , and µ ∗ analytically based on the method in SectionVI-C. Second, given α , we obtain simulated outage probabili-ties of tier-1 UEs, P out, ( α ) and P out, ( α ) , under µ ∗ and µ ∗ . P out, ( α ) and P out, ( α ) are referred to as the relaxed outageprobabilities . Note that if the operator intends to attain U ∗ (cid:19)(cid:17)(cid:20) (cid:19)(cid:17)(cid:21) (cid:19)(cid:17)(cid:22) (cid:19)(cid:17)(cid:23) (cid:19)(cid:17)(cid:24)(cid:19)(cid:17)(cid:23)(cid:25)(cid:19)(cid:17)(cid:23)(cid:25)(cid:24)(cid:19)(cid:17)(cid:23)(cid:26)(cid:19)(cid:17)(cid:23)(cid:26)(cid:24)(cid:19)(cid:17)(cid:23)(cid:27)(cid:19)(cid:17)(cid:23)(cid:27)(cid:24) (cid:53)(cid:68)(cid:71)(cid:76)(cid:88)(cid:86) (cid:50) (cid:88) (cid:87) (cid:68) (cid:74) (cid:72) (cid:3) (cid:83) (cid:85) (cid:82)(cid:69) (cid:68) (cid:69) (cid:76)(cid:79)(cid:76)(cid:87) (cid:92) (cid:37)(cid:54)(cid:3)(cid:72)(cid:91)(cid:70)(cid:79)(cid:88)(cid:86)(cid:76)(cid:82)(cid:81)(cid:15)(cid:3)(cid:86)(cid:76)(cid:80)(cid:88)(cid:79)(cid:68)(cid:87)(cid:76)(cid:82)(cid:81)(cid:37)(cid:54)(cid:3)(cid:72)(cid:91)(cid:70)(cid:79)(cid:88)(cid:86)(cid:76)(cid:82)(cid:81)(cid:15)(cid:3)(cid:68)(cid:81)(cid:68)(cid:79)(cid:92)(cid:87)(cid:76)(cid:70)(cid:68)(cid:79)(cid:56)(cid:40)(cid:3)(cid:72)(cid:91)(cid:70)(cid:79)(cid:88)(cid:86)(cid:76)(cid:82)(cid:81)(cid:15)(cid:3)(cid:86)(cid:76)(cid:80)(cid:88)(cid:79)(cid:68)(cid:87)(cid:76)(cid:82)(cid:81)(cid:56)(cid:40)(cid:3)(cid:72)(cid:91)(cid:70)(cid:79)(cid:88)(cid:86)(cid:76)(cid:82)(cid:81)(cid:15)(cid:3)(cid:68)(cid:81)(cid:68)(cid:79)(cid:92)(cid:87)(cid:76)(cid:70)(cid:68)(cid:79) (a) Tier-2 outage probability under different dif-ferent radius of exclusion regions, T = 0 . . (cid:20)(cid:19) (cid:20)(cid:24) (cid:21)(cid:19) (cid:21)(cid:24) (cid:22)(cid:19)(cid:19)(cid:17)(cid:22)(cid:21)(cid:19)(cid:17)(cid:22)(cid:22)(cid:19)(cid:17)(cid:22)(cid:23)(cid:19)(cid:17)(cid:22)(cid:24)(cid:19)(cid:17)(cid:22)(cid:25)(cid:19)(cid:17)(cid:22)(cid:26)(cid:19)(cid:17)(cid:22)(cid:27) Q /P dB A v e r ag e o u t ag e p r o b a b ili t y (cid:36)(cid:81)(cid:68)(cid:79)(cid:92)(cid:87)(cid:76)(cid:70)(cid:68)(cid:79)(cid:54)(cid:76)(cid:80)(cid:88)(cid:79)(cid:68)(cid:87)(cid:76)(cid:82)(cid:81) (b) Average overall outage probability underdifferent Q /P . (cid:20) (cid:21) (cid:22) (cid:23) (cid:24)(cid:19)(cid:17)(cid:19)(cid:24)(cid:19)(cid:17)(cid:20)(cid:19)(cid:17)(cid:20)(cid:24)(cid:19)(cid:17)(cid:21)(cid:19)(cid:17)(cid:21)(cid:24)(cid:19)(cid:17)(cid:22)(cid:19)(cid:17)(cid:22)(cid:24)(cid:19)(cid:17)(cid:23) µ units/km O u t ag e p r o b a b ili t y ν ( · ), Simulation ν ( · ), Analytical ν +11 ( · ), Simulation ν +11 ( · ), Analytical ν − ( · ), Simulation ν − ( · ), Analytical (c) Tier-1 outage probability under different ν ( · ) . (cid:20) (cid:21) (cid:22) (cid:23) (cid:24)(cid:19)(cid:17)(cid:21)(cid:19)(cid:17)(cid:21)(cid:24)(cid:19)(cid:17)(cid:22)(cid:19)(cid:17)(cid:22)(cid:24)(cid:19)(cid:17)(cid:23)(cid:19)(cid:17)(cid:23)(cid:24)(cid:19)(cid:17)(cid:24)(cid:19)(cid:17)(cid:24)(cid:24) µ units/km O u t ag e p r o b a b ili t y ν ( · ), Simulation ν ( · ), Analytical ν +11 ( · ), Simulation ν +11 ( · ), Analytical ν − ( · ), Simulation ν − ( · ), Analytical (d) Tier-2 outage probability under different ν ( · ) . (cid:20)(cid:15)(cid:3)(cid:20) (cid:20)(cid:15)(cid:3)(cid:21) (cid:21)(cid:15)(cid:3)(cid:20)(cid:15)(cid:3)(cid:20) (cid:21)(cid:15)(cid:3)(cid:20)(cid:15)(cid:3)(cid:21) (cid:21)(cid:15)(cid:3)(cid:21)(cid:15)(cid:3)(cid:20) (cid:21)(cid:15)(cid:3)(cid:21)(cid:15)(cid:3)(cid:21)(cid:19)(cid:19)(cid:17)(cid:20)(cid:19)(cid:17)(cid:21)(cid:19)(cid:17)(cid:22)(cid:19)(cid:17)(cid:23)(cid:19)(cid:17)(cid:24)(cid:19)(cid:17)(cid:25)(cid:19)(cid:17)(cid:26) Tier and type O u t ag e p r o b a b ili t y (cid:36)(cid:81)(cid:68)(cid:79)(cid:92)(cid:87)(cid:76)(cid:70)(cid:68)(cid:79)(cid:54)(cid:76)(cid:80)(cid:88)(cid:79)(cid:68)(cid:87)(cid:76)(cid:82)(cid:81)(cid:40)(cid:85)(cid:85)(cid:82)(cid:85)(cid:69)(cid:68)(cid:85) (e) Outage probability at different tiers (orthogo-nal multiple access). (cid:19) (cid:19)(cid:17)(cid:24) (cid:20) (cid:20)(cid:17)(cid:24) (cid:21) (cid:21)(cid:17)(cid:24) (cid:22)(cid:19)(cid:19)(cid:17)(cid:24)(cid:20)(cid:20)(cid:17)(cid:24)(cid:21)(cid:21)(cid:17)(cid:24)(cid:22)(cid:22)(cid:17)(cid:24) µ µ (cid:36)(cid:81)(cid:68)(cid:79)(cid:92)(cid:87)(cid:76)(cid:70)(cid:68)(cid:79)(cid:3)(cid:87)(cid:85)(cid:68)(cid:71)(cid:72)(cid:82)(cid:73)(cid:73)(cid:54)(cid:76)(cid:80)(cid:88)(cid:79)(cid:68)(cid:87)(cid:76)(cid:82)(cid:81)(cid:3)(cid:87)(cid:85)(cid:68)(cid:71)(cid:72)(cid:82)(cid:73)(cid:73) (f) The tradeoff between µ and µ . (cid:20)(cid:19) (cid:19) (cid:20)(cid:19) (cid:20) (cid:20)(cid:19) (cid:21) (cid:20)(cid:19) (cid:22) (cid:20)(cid:19) (cid:23) (cid:19)(cid:17)(cid:22)(cid:24) (cid:19)(cid:17)(cid:23)(cid:19)(cid:17)(cid:23)(cid:24)(cid:19)(cid:17)(cid:24)(cid:19)(cid:17)(cid:24)(cid:24)(cid:19)(cid:17)(cid:25)(cid:19)(cid:17)(cid:25)(cid:24)(cid:19)(cid:17)(cid:26) (cid:38)(cid:82)(cid:85)(cid:85)(cid:72)(cid:79)(cid:68)(cid:87)(cid:76)(cid:82)(cid:81)(cid:3)(cid:83)(cid:68)(cid:85)(cid:68)(cid:80)(cid:72)(cid:87)(cid:72)(cid:85) (cid:50) (cid:88) (cid:87) (cid:68) (cid:74) (cid:72) (cid:3) (cid:51) (cid:85) (cid:82)(cid:69) (cid:68) (cid:69) (cid:76)(cid:79)(cid:76)(cid:87) (cid:92) (cid:38)(cid:82)(cid:85)(cid:85)(cid:72)(cid:79)(cid:68)(cid:87)(cid:76)(cid:82)(cid:81)(cid:3)(cid:82)(cid:73)(cid:3)(cid:87)(cid:76)(cid:72)(cid:85)(cid:237)(cid:20)(cid:3)(cid:56)(cid:40)(cid:86)(cid:15)(cid:3)(cid:87)(cid:92)(cid:83)(cid:72)(cid:237)(cid:20)(cid:3)(cid:56)(cid:40)(cid:38)(cid:82)(cid:85)(cid:85)(cid:72)(cid:79)(cid:68)(cid:87)(cid:76)(cid:82)(cid:81)(cid:3)(cid:82)(cid:73)(cid:3)(cid:87)(cid:76)(cid:72)(cid:85)(cid:237)(cid:21)(cid:3)(cid:37)(cid:54)(cid:86)(cid:15)(cid:3)(cid:87)(cid:92)(cid:83)(cid:72)(cid:237)(cid:20)(cid:3)(cid:56)(cid:40)(cid:49)(cid:82)(cid:81)(cid:237)(cid:70)(cid:82)(cid:85)(cid:85)(cid:72)(cid:79)(cid:68)(cid:87)(cid:76)(cid:82)(cid:81)(cid:15)(cid:3)(cid:87)(cid:92)(cid:83)(cid:72)(cid:237)(cid:20)(cid:3)(cid:56)(cid:40)(cid:38)(cid:82)(cid:85)(cid:85)(cid:72)(cid:79)(cid:68)(cid:87)(cid:76)(cid:82)(cid:81)(cid:3)(cid:82)(cid:73)(cid:3)(cid:87)(cid:76)(cid:72)(cid:85)(cid:237)(cid:20)(cid:3)(cid:56)(cid:40)(cid:86)(cid:15)(cid:3)(cid:87)(cid:92)(cid:83)(cid:72)(cid:237)(cid:21)(cid:3)(cid:56)(cid:40)(cid:38)(cid:82)(cid:85)(cid:85)(cid:72)(cid:79)(cid:68)(cid:87)(cid:76)(cid:82)(cid:81)(cid:3)(cid:82)(cid:73)(cid:3)(cid:87)(cid:76)(cid:72)(cid:85)(cid:237)(cid:21)(cid:3)(cid:37)(cid:54)(cid:86)(cid:15)(cid:3)(cid:87)(cid:92)(cid:83)(cid:72)(cid:237)(cid:21)(cid:3)(cid:56)(cid:40)(cid:49)(cid:82)(cid:81)(cid:237)(cid:70)(cid:82)(cid:85)(cid:85)(cid:72)(cid:79)(cid:68)(cid:87)(cid:76)(cid:82)(cid:81)(cid:15)(cid:3)(cid:87)(cid:92)(cid:83)(cid:72)(cid:3)(cid:21)(cid:3)(cid:56)(cid:40) (g) Tier-1 outage probability under different cor-relation parameters. (cid:20)(cid:19) (cid:19) (cid:20)(cid:19) (cid:20) (cid:20)(cid:19) (cid:21) (cid:20)(cid:19) (cid:22) (cid:20)(cid:19) (cid:23) (cid:19)(cid:17)(cid:25)(cid:24)(cid:19)(cid:17)(cid:26)(cid:19)(cid:17)(cid:26)(cid:24)(cid:19)(cid:17)(cid:27)(cid:19)(cid:17)(cid:27)(cid:24)(cid:19)(cid:17)(cid:28) (cid:38)(cid:82)(cid:85)(cid:85)(cid:72)(cid:79)(cid:68)(cid:87)(cid:76)(cid:82)(cid:81)(cid:3)(cid:83)(cid:68)(cid:85)(cid:68)(cid:80)(cid:72)(cid:87)(cid:72)(cid:85) (cid:50) (cid:88) (cid:87) (cid:68) (cid:74) (cid:72) (cid:3) (cid:51) (cid:85) (cid:82)(cid:69) (cid:68) (cid:69) (cid:76)(cid:79)(cid:76)(cid:87) (cid:92) (cid:38)(cid:82)(cid:85)(cid:85)(cid:72)(cid:79)(cid:68)(cid:87)(cid:76)(cid:82)(cid:81)(cid:3)(cid:82)(cid:73)(cid:3)(cid:87)(cid:76)(cid:72)(cid:85)(cid:237)(cid:20)(cid:3)(cid:56)(cid:40)(cid:86)(cid:15)(cid:3)(cid:87)(cid:92)(cid:83)(cid:72)(cid:237)(cid:20)(cid:3)(cid:37)(cid:54)(cid:15)(cid:3)(cid:87)(cid:92)(cid:83)(cid:72)(cid:237)(cid:20)(cid:3)(cid:56)(cid:40)(cid:38)(cid:82)(cid:85)(cid:85)(cid:72)(cid:79)(cid:68)(cid:87)(cid:76)(cid:82)(cid:81)(cid:3)(cid:82)(cid:73)(cid:3)(cid:87)(cid:76)(cid:72)(cid:85)(cid:237)(cid:21)(cid:3)(cid:37)(cid:54)(cid:86)(cid:15)(cid:3)(cid:87)(cid:92)(cid:83)(cid:72)(cid:237)(cid:20)(cid:3)(cid:37)(cid:54)(cid:15)(cid:3)(cid:87)(cid:92)(cid:83)(cid:72)(cid:237)(cid:20)(cid:3)(cid:56)(cid:40)(cid:49)(cid:82)(cid:81)(cid:237)(cid:70)(cid:82)(cid:85)(cid:85)(cid:72)(cid:79)(cid:68)(cid:87)(cid:76)(cid:82)(cid:81)(cid:15)(cid:3)(cid:87)(cid:92)(cid:83)(cid:72)(cid:237)(cid:20)(cid:3)(cid:37)(cid:54)(cid:15)(cid:3)(cid:87)(cid:92)(cid:83)(cid:72)(cid:237)(cid:20)(cid:3)(cid:56)(cid:40)(cid:38)(cid:82)(cid:85)(cid:85)(cid:72)(cid:79)(cid:68)(cid:87)(cid:76)(cid:82)(cid:81)(cid:3)(cid:82)(cid:73)(cid:3)(cid:87)(cid:76)(cid:72)(cid:85)(cid:237)(cid:20)(cid:3)(cid:56)(cid:40)(cid:86)(cid:15)(cid:3)(cid:87)(cid:92)(cid:83)(cid:72)(cid:237)(cid:21)(cid:3)(cid:37)(cid:54)(cid:15)(cid:3)(cid:87)(cid:92)(cid:83)(cid:72)(cid:237)(cid:21)(cid:3)(cid:56)(cid:40)(cid:38)(cid:82)(cid:85)(cid:85)(cid:72)(cid:79)(cid:68)(cid:87)(cid:76)(cid:82)(cid:81)(cid:3)(cid:82)(cid:73)(cid:3)(cid:87)(cid:76)(cid:72)(cid:85)(cid:237)(cid:21)(cid:3)(cid:37)(cid:54)(cid:86)(cid:15)(cid:3)(cid:87)(cid:92)(cid:83)(cid:72)(cid:237)(cid:21)(cid:3)(cid:37)(cid:54)(cid:15)(cid:3)(cid:87)(cid:92)(cid:83)(cid:72)(cid:237)(cid:21)(cid:3)(cid:56)(cid:40)(cid:49)(cid:82)(cid:81)(cid:237)(cid:70)(cid:82)(cid:85)(cid:85)(cid:72)(cid:79)(cid:68)(cid:87)(cid:76)(cid:82)(cid:81)(cid:15)(cid:3)(cid:87)(cid:92)(cid:83)(cid:72)(cid:237)(cid:21)(cid:3)(cid:37)(cid:54)(cid:15)(cid:3)(cid:87)(cid:92)(cid:83)(cid:72)(cid:237)(cid:21)(cid:3)(cid:56)(cid:40) (h) Tier-2 outage probability under different cor-relation parameters. (cid:20)(cid:19) (cid:19) (cid:20)(cid:19) (cid:20) (cid:20)(cid:19) (cid:21) (cid:20)(cid:19) (cid:22) (cid:20)(cid:19) (cid:23) (cid:237)(cid:19)(cid:17)(cid:19)(cid:24)(cid:19)(cid:19)(cid:17)(cid:19)(cid:24)(cid:19)(cid:17)(cid:20)(cid:19)(cid:17)(cid:20)(cid:24)(cid:19)(cid:17)(cid:21)(cid:19)(cid:17)(cid:21)(cid:24)(cid:19)(cid:17)(cid:22) (cid:38)(cid:82)(cid:85)(cid:85)(cid:72)(cid:79)(cid:68)(cid:87)(cid:76)(cid:82)(cid:81)(cid:3)(cid:83)(cid:68)(cid:85)(cid:68)(cid:80)(cid:72)(cid:87)(cid:72)(cid:85) η ( α ) (cid:38)(cid:82)(cid:85)(cid:85)(cid:72)(cid:79)(cid:68)(cid:87)(cid:76)(cid:82)(cid:81)(cid:3)(cid:82)(cid:73)(cid:3)(cid:87)(cid:76)(cid:72)(cid:85)(cid:237)(cid:20)(cid:3)(cid:56)(cid:40)(cid:86)(cid:38)(cid:82)(cid:85)(cid:85)(cid:72)(cid:79)(cid:68)(cid:87)(cid:76)(cid:82)(cid:81)(cid:3)(cid:82)(cid:73)(cid:3)(cid:87)(cid:76)(cid:72)(cid:85)(cid:237)(cid:21)(cid:3)(cid:37)(cid:54)(cid:86) (i) η ( α ) under different correlation parameters. Fig. 4. Numerical studies (cont’d). under the correlated case with parameter α , the original outageprobability constraints should be modified as the relaxedoutage probability constraints (i.e., P target, ( α ) = P out, ( α ) and P target, ( α ) = P out, ( α ) ). Third, under these relaxedoutage probability constraints, we can again derive the optimalutility U ( α ) analytically, without location correlations. U ( α ) is referred to as the ameliorated utility when the correlationsdiminish. Let η ( α ) = U ( α ) − U ∗ U ( α ) denote relative performancegap. We study η ( α ) against α to show the performance gapunder different α values.In the presented experiment, the optimal solution of theoriginal non-correlation case is µ ∗ = 0 . , µ ∗ = 0 . ,and U ∗ = 4 . . Fig. 4(i) shows η ( α ) against α with locationcorrelations of tier-1 UEs and tier-2 BSs, respectively. Theresults show that the performance gap is small when thelocations of tier-1 UEs or tier-2 BSs are weakly correlated( α ≥ ). VIII. C ONLUSIONS
In this paper, we propose a stochastic geometric model toaccurately quantify the uplink interference and outage perfor-mance of two-tier cellular networks with diverse users and tier-2 cells. By applying our SAS approach, we derive numericalexpressions for the Laplace transform of interference at bothtiers, avoiding the approximations required in prior works,leading to accurate numerical calculation of the outage proba-bility. Our model is also able to capture the impact of two typesof exclusion regions, in which either tier-2 base stations or tier-2 users are restricted in order to avoid cross-tier interference.As an application example, an intensity planning problemis investigated, in which the outage probability constraintsare converted to linear intensity tradeoff, facilitating efficientsolutions. Finally, numerical studies further demonstrate thecorrectness and usefulness of our analysis.R
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