Analytical Evaluation of Fractional Frequency Reuse for Heterogeneous Cellular Networks

Analytical Ev aluation of Fractional Frequenc y Reuse for Heterogeneous Cellular Networks Thomas D. Novlan, Radha Krishna Ganti, Arunabha Ghosh, Jef frey G. Andre ws Abstract Interference management techniques are critical to the performance of heterogeneous cellular net- works, which will have dense and ov erlapping coverage areas, and experience high le vels of interference. Fractional frequency reuse (FFR) is an attracti ve interference management technique due to its low complexity and overhead, and significant cov erage improv ement for low-percentile (cell-edge) users. Instead of relying on system simulations based on deterministic access point locations, this paper instead proposes an analytical model for ev aluating Strict FFR and Soft Frequency Reuse (SFR) deployments based on the spatial Poisson point process. Our results both capture the non-uniformity of heterogeneous deployments and produce tractable expressions which can be used for system design with Strict FFR and SFR. W e observe that the use of Strict FFR bands reserved for the users of each tier with the lowest av erage SINR provides the highest gains in terms of coverage and rate, while the use of SFR allows for more efficient use of shared spectrum between the tiers, while still mitigating much of the interference. Additionally , in the context of multi-tier networks with closed access in some tiers, the proposed framew ork shows the impact of cross-tier interference on closed access FFR, and informs the selection of key FFR parameters in open access. I . I N T RO D U C T I O N Modern cellular network deployments are currently transitioning from largely homogeneous (one-tier) voice-centric deployments to highly heterogeneous data-centric networks comprised of dif ferent classes (tiers) of access points [1]. These include operator-deployed picocells and distributed antenna systems [2], [3], [4], and home user -deployed femtocells [5]. T . D. Novlan, R. K. Ganti, and J. G. Andre ws are with the W ireless Networking and Communications Group, the University of T exas at Austin. A. Ghosh is with A T&T Laboratories. The contact author is J. G. Andrews. Email: jandrews@ece.ute xas.edu. This research has been supported by A T&T Laboratories. Date revised: June 23, 2011 2 Performance analysis of these networks is much more in volved than for a single-tier network because of the need to account for inter -cell and cross-tier interference and the non-uniformity of the access point deployments arising from both topographic and economic reasons. A further complication in heterogeneous network analysis arises from dif ferent user association policies. As a result, there is a need for new and general models for analyzing the important metrics of cov erage and rate in the context of these multi-tier networks. While prior work has relied on simulations based on deterministic models of AP locations, these hav e not led to general or tractable solutions. In this paper , instead, we model the AP locations as a Poisson point process (PPP) [6], [7], [8]. This modeling approach has been recently applied to the analysis of cellular networks due to the ability to deri ve tractable expressions for cov erage and rate both for one-tier [9] and very recently , heterogeneous networks [10], [11], [12], [13]. A. F ractional F requency Reuse Faced with increased traf fic demands in interference-limited cellular networks, fractional frequency reuse (FFR) is an attracti ve strategy due to its low complexity of implementation and its significant gains for the bottom percentile of mobile users. Recently , FFR has been included in fourth generation (4G) wireless standards including W iMAX 2 (802.16m) and 3GPP-L TE since release 8 [14]. This work extends our nov el analytical model of FFR in the downlink of a cellular network with a single-tier of base stations using the PPP model dev eloped in [15], [16] to a general multi-tier network with closed and open access between the tiers. This allows the de velopment of tractable expressions for the SINR distributions to be deri ved as a function of the FFR parameters which can be utilized for the system design of these networks. W e will consider the two most common types of FFR: Strict FFR and Soft F r equency Reuse (SFR). Under Strict FFR, which extends the traditional frequency reuse used extensi vely in current cellular networks [17], [18], users in the interior of a cell are allocated a common sub- band of frequencies f c while at the cell-edge, users are allocated separate subbands partitioned across cells with a reuse factor of ∆ . The left sub-figure in Fig. 2(a) illustrates potential Strict FFR allocations with ∆ = 3 in which edge users are giv en frequency resources corresponding to subbands f 1 , f 2 , or f 3 . The primary adv antage of Strict FFR is the significant reduction in interference for edge users, although there is a loss in spectral efficienc y since each cell cannot fully utilize all ∆ + 1 subbands [19]. The right sub-figure in Fig. 2 illustrates the frequency and transmit power allocation for SFR. 3 Edge users are allocated bandwidth subbands with a reuse factor of ∆ , but the main difference vs. Strict FFR is that each cell utilizes all ∆ subbands since interior users are allowed to share sub-bands with edge users in other cells. Because cell-edge users share the bandwidth with neighboring cells, their downlinks are typically transmitted with higher power lev els in order to reduce the impact of the inter-cell interference [20], [21]. T o accomplish this, a transmit power control factor β ≥ 1 is introduced to create two dif ferent classes, P int = P and P edge = β P , where P int is the transmit power of the base station if user y is an interior user and P edge is the transmit power of the base station if user y is a cell-edge user . The increased interference for edge users under SFR is traded off for greater spectral utilization [22]. B. Related W ork Early work on frequenc y partitioning for two-tier networks is found in [23]. Their proposed strategy maximizes the spectral ef ficiency for a minimum QoS requirement and the number of users per tier . They assume that the femtocells are gi ven a separate frequency band from the macrocells, such that there is no cross-tier interference. The authors in [24] consider an adaptiv e FFR strategy for mitigating inter -femtocell in- terference while k eeping spectral ef ficiency as high as possible. They vary the size of FFR partitions and transmit power based on the amount of estimated interference. Howe ver they use a deterministic model for the femtocells inside of a single building and neglect macrocell or femtocell interference outside of the building. V ery recent work in [25] considers a deterministic model analysis of the spectral ef ficiency of femtocells as a function of the femtocell’ s location in a two-tier network with base stations modeled as a hexagonal grid and femtocells uniformly deployed in each cell. They fix the macrocell FFR sub-band allocations and then consider the spectral ef ficiency of a femtocell as a function of its distance from the cell center . Frequency partitioning between macrocells and femtocells is revisited in [26]. They propose a model where some sub-bands are reserved for only macrocell or femtocell users in addition to a common group of sub-bands, similar in concept to the proposed Strict FFR model. They also alternately consider partitioning in the time domain. They provide a large number of simulation results based on a deterministic model for the AP locations and motiv ate a dynamic partitioning based on measured interference lev els by users in either tier . The two primary user association policies for heterogeneous networks are closed access and open access . Under closed access, mobiles are restricted from connecting with certain tiers of 4 access points based on system performance metrics or economic or leg al factors in some cases [5]. Open access instead allows users to connect to APs of different tiers based on the association policy , which may be measured signal-to-interference-ratio ( SIR ) or traffic load and can be used as an interference management technique [27]. The authors in [28] consider performance tradeof fs for closed and open femtocell networks. Their analysis uses stochastic geometry tools from [9] in order to deriv e SINR distributions for different deployment scenarios at the cell edge or interior and for varying femtocell densities. Howe ver their analysis is constrained to the interior of a single macrocell and does not consider the ef fect of inter-cell interference or the use of FFR on the SINR distributions. C. Contributions In this paper we present the following contributions. First, we extend the framework of [15], [16] to ev aluate the SINR distributions for users in a do wnlink K -tier network utilizing Strict FFR and SFR. W e first consider closed access, which limits users to associate with APs in only one tier , with all the other tiers contrib uting interference. In addition, by considering a special case relev ant to interference-limited networks, the analytical expressions for the SINR distributions reduce to simple expressions which are a function of the key FFR design parameters, allo wing for clear , intuitiv e comparisons between the reuse strategies and insight into system design. Secondly , we propose a new frame work for analyzing coverage for the open access do wnlink under Strict FFR and SFR in which users may associate with APs in more than one tier . Finally , we pro vide implications of the analysis to system design for closed and open access networks. The models allo w for in vestigation of FFR parameter selection based on the densities, transmit powers, and resource allocation strategies of the tiers. In the next section, we provide a detailed description of the system model and our assumptions. I I . S Y S T E M M O D E L W e consider an OFDMA cellular downlink with K-tiers of access points (APs). The locations of the base stations and femtocells are modeled as independent spatial Poisson point processes (PPP) [29] of density λ k with independence between the tiers. In other words, for a giv en PPP , the number of points in a bounded area is a Poisson-distrib uted random v ariable and those points are uniformly-distributed within the area. A realization of a three-tier netw ork with Poisson 5 distributed APs and V oronoi cell coverage regions based on strongest recei ved po wer is gi ven in Fig. 1. W ithout loss of generality , we assume a typical mobile user at the origin and compute the SINR for this typical mobile. W e assume that the mobile user is served by only one tier at a time and by the closest AP of that tier , which is at a distance r k . Since the underlying APs are distributed as PPPs, it follo ws that r k is Rayleigh distrib uted [29]. W e assume that all the access points of the k th transmit with an equal po wer P k . The path loss exponent is giv en by α , and σ 2 is the noise po wer . W e assume that the small-scale fading between an y interfering AP and the typical mobile in consideration, denoted by G z , is i.i.d exponentially distributed with mean µ (corresponds to Rayleigh fading). The set of interfering APs in the k th tier is Z k , i.e. access points that use the same sub-band as the mobile user . W e denote the distance between the interfering AP and the mobile node in consideration by R z . The associated signal-to-interference-plus-noise-ratio ( SINR ) is giv en as SINR = P k g k r k − α σ 2 + K X k =1 P k I k , (1) where for an interfering set of k th tier APs Z k , I k = X z ∈Z k G z R z − α . (2) In the abov e e xpression, we hav e assumed that the nearest AP to the mobile in the k th tier is at a distance r k , which is a random v ariable. Also the fading between the nearest AP in consideration is denoted by g k . W ith FFR, a mobile user first determines its SINR to the nearest AP of the k th tier and checks if it is less than the tier’ s FFR threshold T k . If so, then the user is classified as an edge user and the AP transmits its downlink on the reserved FFR band, randomly picked from ∆ sub-bands av ailable. Otherwise we classify the mobile as an interior user . These classifications arise dif ferently than prior work utilizing the typical grid model assumption which defines an interior radius [19], since constant SINR contours can no longer be defined as concentric circles around the AP [30]. In fact the edge or interior user classifications does not necessarily have the same geographic interpretation for each cell. As noted in [7], this consequence of the spatial PPP more closely reflects non-regular deployments and typically corresponds to a lower performance bound compared to the upper bound provided by the grid model. 6 T o accommodate the difference between SFR and Strict FFR in terms of the use of power control, we introduce the design parameter β . T ypical ranges for β are 0-20 dB [22], [31]. Since this extra downlink po wer is only applied to 1 / ∆ of the base stations on the first tier the interference power is gi ven by η P 1 I 1 + P K k =2 P k I k , where η = (∆ − 1 + β ) / ∆ consolidates the edge and interior downlinks into a single effecti ve interference term. I I I . C OV E R A G E P RO B A B I L I T Y W I T H C L O S E D A C C E S S W e initially consider coverage probability the do wnlink of a multi-tier network with closed access between the tiers. For e xample, in the conte xt of a tw o-tier network with underlaid femtocells, a mobile user connected to the macrocell may be in range of a femtocell, but is unable to connect to that femtocell, potentially resulting in cross-tier interference. Cov erage probability is the probability that a user’ s SINR is greater than a threshold T , ¯ F ( T ) = P (SINR > T ) , (3) equi v alently the CCDF of the SINR for a particular reuse strategy , denoted as ¯ F ( T ) . A. Single-tier cover age with FFR Our prior results in [15], [16] take advantage of the frame work recently de veloped in [9] utilizing the Poisson point process (PPP) model for base station locations. The authors of [9] determine e xpressions for the exact distrib ution of the typical mobile’ s SINR , with traditional per- cell frequency reuse for a single-tier of base stations. As a result, under reasonable assumptions for modern cellular networks, the results in [15] reduce to tractable expressions which provide insight into system design guidelines and the relati ve merits of Strict FFR and SFR, compared to univ ersal reuse for a two-tier network with open access between tiers. Also in [15], [16], the shape and values of the distributions deri ved for Strict FFR and SFR are shown to be closer to results obtained using location data from an actual base station deployment than simulations utilizing the standard grid model. W e now provide the distribution of SINR for cell-edge users with Strict FFR and SFR under closed access. B. Multi-tier cover age with Strict FFR In the case of Strict FFR, we assume that inter-cell and cross-tier interference is present on the common sub-band allocated to all macrocells, while the FFR sub-band is reserved for macrocell 7 users and does not experience cross-tier interference, only inter -cell interference thinned with a reuse factor of ∆ . First tier edge users are those who hav e SINR less than the macrocell’ s FFR threshold T 1 on the common sub-band shared by all cells and are therefore selected by the reuse strategy to have a new sub-band allocated to them from the ∆ total av ailable sub-bands reserved for the edge users. Theor em 1 (Strict FFR, closed access, edge user): The cov erage probability of a first tier edge user in a strict FFR system, assigned a FFR sub-band is ¯ F FFR , c ( T ) = π λ 1 R ∞ 0 e − π λ 1 v ( 1+ ρ ( T ,α ) ∆ ) − µT σ 2 P 1 v α/ 2 − e − π λ 1 v ( 1+2 ξ ( T ,T 1 ,α, ∆)+2 P K k =2 κ k ψ ( γ k T 1 ,α ) ) − µ ( T + T 1 ) σ 2 P 1 v α/ 2 d v 1 − π λ 1 R ∞ 0 e − π λ 1 v ( 1+ ρ ( T 1 ,α )+2 P K k =2 κ k ψ ( γ k T 1 ,α ) ) e − µ ( T + T 1 ) σ 2 P 1 v α/ 2 d v , (4) where ρ ( z , α ) = z 2 /α Z ∞ z − 2 /α 1 1 + u α/ 2 d u, (5) ξ ( T , T 1 , α, ∆) = Z ∞ r 1  1 − 1 1 + T 1 r 1 α x − α  1 − 1 ∆  1 − 1 1 + T r 1 α x − α  x d x, (6) and ψ ( z , α ) = csc  2 π α  π z 2 /α α , γ k = P k P 1 , κ k = λ k λ 1 . (7) Pr oof: The proof is giv en in Appendix A. An immediate observation of this framework is that it leads to expressions which are only a function of the relev ant FFR design parameters. The intra-tier interference before and after FFR is applied are captured in the ξ ( T , T 1 , α, ∆) and ρ ( z , α ) terms respecti vely , while the cross tier interference terms for each tier are expressed by ψ ( z , α ) . C. Multi-tier cover age with SFR W e now consider the CCDF of the SINR for edge users with SFR. In this case all the subbands ov erlap with those of the other tiers since SFR makes use of the entire spectrum but allocates edge users with SINR belo w the FFR threshold a higher transmit power determined by the β parameter . Theor em 2 (SFR, closed access, edge user): The cov erage probability of an SFR edge user 8 whose initial SINR is less than T 1 is ¯ F SFR , c ( T ) = π λ 1 R ∞ 0 e − πλ 1 v ( 1+ ρ ( ηT β ,α )+2 P K k =2 κ k ψ ( γ k β T ,α ) ) e − µ ( T ) σ 2 β P 1 v α/ 2 d v 1 − π λ 1 R ∞ 0 e − πλ 1 v ( 1+ ρ ( ηT 1 ,α )+2 P K k =2 κ k ψ ( γ k T 1 ,α ) ) e − µ ( ηT 1 ) σ 2 P 1 v α/ 2 d v − π λ 1 R ∞ 0 e − πλ 1 v ( 1+2 ζ ( T ,T 1 ,α, ∆ ,β ,η )+2 P K k =2 κ k ( ψ ( γ k β T ,α )+ ψ ( γ k T 1 ,α ) )) e − µ ( T + ηT 1 ) σ 2 P 1 v α/ 2 d v 1 − π λ 1 R ∞ 0 e − πλ 1 v ( 1+ ρ ( ηT 1 ,α )+2 P k =2 κ k ψ ( γ k T 1 ,α ) ) e − µ ( ηT 1 ) σ 2 P 1 v α/ 2 d v . (8) where ζ ( T , T 1 , α, ∆ , β , η ) = Z ∞ r 1 " 1 − 1 1 + η T 1 r 1 α x − α 1 1 + η β T r 1 α x − α # x d x, ρ ( z , α ) is giv en by (5), and ψ ( z , α ) , κ k and γ k are gi ven by (7). Pr oof: The proof is giv en in Appendix B. The expressions differ from Strict FFR both due to the effecti ve SINR and FFR thresholds shaped by the power control factor β and effecti ve interference po wer η respecti vely . D. Model Evaluation While all our cov erage probability results hold for general pathloss exponents α and different noise powers σ 2 , in this section we present a special case where α = 4 and σ 2 = 0 . For this case the cov erage probability results reduce to simple closed-form expressions, allowing clear insight into the performance of cell-edge users, something not pre viously possible with the grid model. This choice of pathloss exponent is in the range of commonly used v alues in practice [32]. Furthermore, most urban cellular networks - where FFR is of the most interest - are interference-limited and noise is negligible compared to the background interference from the adjacent BSs. In the case of α = 4 and no noise, for Strict FFR , the CCDF is gi ven as, ¯ F FFR , e ( T ) = 1 + ρ ( T 1 ) + π 2 P K k =2 κ k √ γ k T ρ ( T 1 ) + π 2 P K k =2 κ k √ γ k T 1 1 + ρ ( T ) ∆ − 1 1 + 2 ξ ( T , T 1 , λ, ∆) + π 2 P K k =2 κ k √ γ k T ! , (9) where ξ ( T , T 1 , 4 , ∆) = T ρ ( T ) − ρ ( T 1 ) ( T 1 ∆ − T (1 + ∆)) 4∆( T 1 − T ) , and ρ ( x ) = √ x arctan  √ x  . (10) In the case of α = 4 and no noise, for SFR , the CCDF is gi ven as, ¯ F SFR , c ( T ) = 1 + ρ ( η T 1 ) + π 2 P K k =2 √ γ k T ρ ( η T 1 ) + π 2 P K k =2 √ γ k T ×    1 1 + ρ ( η β T ) ∆ + π 2 P K k =2 q γ k β T − 1 1 + 2 ζ ( T , T 1 , λ, ∆) + π 2 P K k =2 √ γ k T    , (11) 9 where ζ ( T , T 1 , β , η ) = η 3 / 2 T β 4 √ T 1 ( T − T 1 β ) − η β T 3  2 arctan  q β η T  + π  ( T − T 1 β ) + η T 3 / 2 T 1 3 / 2 β 5 / 2  2 arctan  1 √ η T 1  − π  ( T − T 1 β ) . (12) Fig. 3 sho ws the deriv ed distributions for Strict FFR and SFR edge users for a three-tier network with no noise and α = 4 compared with Monte-Carlo simulations. The accuracy of the mathematical model is highlighted by the exact match of the curves with the simulation results. W e also see the improved coverage afforded to cell-edge users with FFR compared to uni versal frequency reuse. For Strict FFR, much of the gain results in the remov al of both cross-tier interference and 1 / ∆ of the intra-tier interference. SFR provides a lo wer coverage gain, but this can be mitigated by the use of higher β , or taking into account that more spectrum is a v ailable than under Strict FFR since each cell fully utilizes all subbands. Using similar techniques we can deri ve the distributions for interior macro or femto users using this framew ork. Additionally , these results are also v alid for α 6 = 4 , but the e xpressions no longer hav e the same simple closed-form. Instead they are integrals that can be e valuated using numerical techniques. I V . C O V E R A G E P RO B A B I L I T Y W I T H O P E N A C C E S S In the following analysis of open access downlinks we make the follo wing two assumptions, (i) that there are only two-tiers of access points, and (ii) we only consider the SIR , as the access metric, neglecting noise. While our general framew ork can accommodate an unlimited number of tiers and noise, making those assumptions greatly reduces the complexity of the expressions for the SIR distributions. The follo wing SIR distributions for Strict FFR and SFR are a function of two open access thresholds, T 1 set by the macro tier and T 2 set by the second tier of APs. The open access thresholds determine whether a user is switched to a reuse- ∆ sub-band or served by a either the common band of the macrocell or the nearest second-tier AP . Let SIR 1 and SIR 2 denote the SIR at the typical mobile of the closest first and second tier AP respecti vely , SIR 1 = P 1 g 1 r 1 − α P 1 I 1 + P 2 I 2 + P 2 g 2 r 2 − α , SIR 2 = P 2 g 2 r 2 − α P 1 I 1 + P 2 I 2 + P 1 g 1 r 1 − α . (13) Here r 1 denotes the distance of the mobile at the origin to the nearest macro BS, and r 2 the distance to the nearest femtocell. The interference caused by the macro BSs is denoted by I 1 , 10 while I 2 is the interference caused by the femtocells, excluding the closest one. If for a mobile user , SIR 1 < T 1 and SIR 2 < T 2 , then the mobile user is allocated a ne w FFR sub-band δ y , where δ ∈ { 1 , ..., ∆ } with uniform probability 1 ∆ and a new SIR giv en by ˆ SIR which is different under Strict FFR or SFR. The CCDF of the edge user SIR under open access is giv en by ¯ F FFR , open , e ( T ) = P  ˆ SIR > T | SIR 1 < T 1 , SIR 2 < T 2  . (14) As we can see from (14) the analysis of the cov erage probability is more complicated relativ e to closed access due to the inter-dependence of the terms SIR 1 and SIR 2 . A. Strict FFR First we consider the distribution of (14) for Strict FFR. Since the mobile user is allocated a dif ferent sub-band, it experiences ne w fading power ˆ g 1 and out-of-cell interference P 1 ˆ I 1 , which does not hav e cross-tier interference. Theor em 3 (Strict FFR, open access, edge user): The cov erage probability of an edge user in a strict FFR system, assigned a FFR sub-band is ¯ F FFR , o ( T ) = p c ( T , λ 1 , α, ∆) − R ∞ 0 R ∞ 0  2 π λ 1 r 1 e − π λ 1 r 1 2   2 π λ 2 r 2 e − π λ 2 r 2 2  g n ( r 1 , r 2 )d r 1 d r 2 R ∞ 0 R ∞ 0 (2 π λ 1 r 1 e − π λ 1 r 1 2 ) (2 π λ 2 r 2 e − π λ 2 r 2 2 ) g d ( r 1 , r 2 )d r 1 d r 2 where g d ( r 1 , r 2 ) = 1 −  1 e ( − 2 π λ 1 ρ 1 , 1 ( T 1 ,α )) e ( − 2 π λ 2 ρ 1 , 2 ( γ T 1 ,α )) −  2 e ( − 2 π λ 1 ρ 2 , 1 ( T 2 /γ ,α )) e ( − 2 π λ 2 ρ 2 , 2 ( T 2 ,α )) , g n ( r 1 , r 2 ) =  1 e − 2 π ( λ 1 ξ 1 , 1 ( T ,T 1 ,α, ∆)+ λ 2 ρ 1 , 2 ( T 1 ,α )) +  2 e − 2 π ( λ 1 ξ 2 , 1 ( T ,T 2 /γ ,α, ∆)+ λ 2 ρ 2 , 2 ( T 2 ,α )) , ξ a,b ( T , z , α, ∆) = Z ∞ r b  1 − 1 1 + z r α a x − α  1 − 1 ∆  1 − 1 1 + T r α b x − α  x d x, (15) ρ a,b ( z , α ) = Z ∞ r b  1 − 1 1 + z r α a x − α  xdx, (16) and γ = P 2 P 1 ,  1 = 1 T 1 γ r 1 α r 2 α + 1 ! , and  2 =    1 T 2  γ r 1 α r 2 α  − 1 + 1    . (17) Pr oof: The proof is giv en in Appendix C. Compared to the closed access results, the deriv ations are not nearly as clean due to the dependence of the user’ s SIR on r 1 and r 2 . The deriv ations require ev aluating a double integral which does not hav e a closed form. In fact, the number of tiers under consideration determines 11 the number of integrals which must be ev aluated. Despite this, we can still obtain insight into the underlying nature of the distributions. Also, it is e xpected that most practical deployments would not hav e more than about three tiers e ven in dense en vironments, making this analysis practical through the use of numerical ev aluation of the inte grals. B. SFR As was the case for closed access, the SFR expressions differ from Strict FFR due to the po wer control factor and effecti ve interference power . Additionally the full ∆ -reuse of subbands with SFR results in cross-tier interference for the edge users as well as interior users. W e now gi ve the expression for cov erage probability with open access and SFR based on the SIR in (14). Theor em 4 (SFR, open access, edge user): The coverage probability of an SFR edge user whose initial SIR is less than T 1 and T 2 is ¯ F SFR , o ( T ) = π λ 1 R ∞ 0 e − π λ 1 v ( 1+ ρ ( η β T ,α )+2 κψ ( γ β T ,α ) ) d v R ∞ 0 R ∞ 0 (2 π λ 1 r 1 e − π λ 1 r 1 2 ) (2 π λ 2 r 2 e − π λ 2 r 2 2 ) f d ( r 1 , r 2 )d r 1 d r 2 − R ∞ 0 R ∞ 0  2 π λ 1 r 1 e − π λ 1 r 1 2   2 π λ 2 r 2 e − π λ 2 r 2 2  f n ( r 1 , r 2 )d r 1 d r 2 R ∞ 0 R ∞ 0 (2 π λ 1 r 1 e − π λ 1 r 1 2 ) (2 π λ 2 r 2 e − π λ 2 r 2 2 ) f d ( r 1 , r 2 )d r 1 d r 2 . where f n ( r 1 , r 2 ) =  1 e − 2 π λ 1 ( ζ 1 , 1 ( T ,T 1 ,α, ∆ ,β ,η )+ κψ ( γ β T ,α ) + κρ 1 , 2 ( γ T 1 ,α ) ) +  2 e − 2 π λ 1 ( ζ 2 , 1 ( T ,T 2 /γ ,α, ∆ ,β ,η )+ κψ ( γ β T ,α ) + κρ 2 , 2 ( T 2 ,α ) ) , f d ( r 1 , r 2 ) = 1 −  1 e ( − 2 π λ 1 ( ρ 1 , 1 ( η T 1 ,α )+ κρ 1 , 2 ( γ T 1 ,α ))) −  2 e ( − 2 π λ 1 ( ρ 2 , 1 ( η γ T 2 ,α ) + κρ 2 , 2 ( T 2 ,α ) )) , ζ a,b ( y , z , β , η ) = 1 2( y − z ) ( y ρ a,b ( y , α ) + z ρ a,b ( z , α )) , and ρ a,b ( z , α ) giv en by (16) . (18) Pr oof: The proof is giv en in Appendix D. The expressions hav e a similar form b ut differ from Strict FFR due to the effect of η and β on the SIR and FFR thresholds. As with Strict FFR, the deriv ations do not reduce as simply as closed access expressions due to the dependence of the user’ s SIR on r 1 and r 2 , but still can be computed with a single integral in the case of σ 2 = 0 and α = 4 . 12 C. Model Evaluation Fig. 4 shows the deri ved distrib utions for Strict FFR and SFR edge users for a two-tier network with no noise and α = 4 compared with Monte-Carlo simulations. As with closed access, the curves match exactly . W e also note that there is an upwards shift in the coverage probability curves, due to the impact of off-loading of users onto the secondary tier . W ith closed access, users whose SINR falls belo w the first tier FFR threshold T 1 = 1 dB would be assigned a FFR band and may or may not be able to be cov ered due to interference or propagation challenges, ho we ver if their SINR to a second tier AP is greater than T 2 = 5 dB, they are guaranteed coverage and af fect the distribution of the users who utilize FFR. The selection of the FFR thresholds is further in vestigated in the follo wing section. V . S Y S T E M D E S I G N I M P L I C A T I O N S In this section we present sev eral applications of the Strict FFR and SFR SINR and SIR distributions deri ved for closed and open access in Sections III and IV, which illustrate ho w they can be used to provide additional tools and insight for the system design of heterogeneous networks utilizing FFR. A. A verage Edge User Rate In modern cellular networks, the important metric of a verage achiev able rate can be deri ved from the SINR statistics. In this section we illustrate how the cov erage results deriv ed in Section III and IV can be straightforwardly extended to de velop av erage edge user rate expressions under Strict FFR or SFR. The av erage data rate ¯ τ = E [ln (1 + SINR)] is achie ved by the users, assuming adaptive modulation and coding, and the expressions are gi ven in terms of nats/Hz, where 1 bit = log e (2) nats. The av erage rate of an edge user is determined by integrating ov er the SINR distribution and fading. Due to the two-stage nature of FFR the SINR e of the edge user on the new subband is conditioned on the previous SINR i on the common subband. Thus we have ¯ τ = E [ln (1 + SINR)] = Z r> 0 e − π λr 2 E [ln (1 + SINR e )] 2 π λr d r, = Z r> 0 e − π λr 2 Z t> 0 P  ln (1 + SINR e ) > t     SINR i < T FR  2 π λ d t r d r . 13 where we use the fact that since the rate τ = ln(1 + SINR) is a positi ve random variable, E [ τ ] = R t> 0 P ( τ > t )d t . From the abo ve expression we see that the deriv ation of these terms in volv es substituting e t − 1 in place of the SINR threshold T and computing an additional inte gral. B. Multi-tier interfer ence and closed access W e now consider a two-tier network with Strict FFR for the macro users and closed access and show the connection between the density ratio of the tiers κ and the SINR distribution. Fig. 5 plots the distribution for edge users as an increasing function of κ , ef fectively increasing the density of second-tier APs. As κ increases we see in Fig. 5 that the SINR increases for macrocell users. This is a consequence of the use of Strict FFR, since the FFR bands are reserved for only macrocell users, any user moving from the common band to the FFR band will see a reduction in interference. As the interference from the second tier increases with κ , more and more macro users hav e SINR below T 1 and since they cannot connect to the second tier due to the closed access constraint, they must be moved onto a FFR sub-band. The implication of this result is that the size of the partitions will need to be increased, which for Strict FFR, can cause the ov erall sum rate of the macrocells to decrease due to the reduction in ov erall spectrum usage. C. Open access FFR thr esholds In Fig. 6 the SFR edge user SIR CCDF is sho wn for dif ferent v alues of T 2 , the second-tier FFR threshold under open access. Decreasing T 2 increases the number of mobile users which can connect to that AP on the common sub-band. From Fig. 6 we see that this results in the ov erall increase of the SIR of the edge users. In other words, as T 2 increases, only the users with the worst SIR are gi ven FFR sub-bands and they also are the users who can hav e the greatest benefit from the FFR sub-bands. A related concept is called biasing, in which the access thresholds of the femtocells or other secondary APs are adjusted in order to increase the offload from the macrocell. The reasons for biasing may not be solely related to the ability of the macrocell to provide cov erage for a user , but rather to reduce traffic for especially overloaded macrocells. Our proposed frame work can implicitly capture this effect in the design of T 1 and T 2 . By raising T 1 and lo wering T 2 we can define a middle SIR range T B ias = T 1 − T 2 , wherein a desired percentage of macrocell users are of floaded. 14 V I . C O N C L U S I O N This w ork has presented a ne w tractable analytical frame work for ev aluating cov erage probabil- ity in heterogeneous networks utilizing Strict FFR and SFR which captures the non-uniformity of these deployments and giv es insight into the performance tradeoffs of those FFR strategies. The model presented in this work can be utilized as a foundation for future research in interference management and performance analysis of heterogeneous networks utilizing dynamic FFR strate- gies for addressing changing channel conditions and user traffic in the network [33], [34], [35]. Additionally , in the uplink, the constraints of power control, mobility of the interfering mobiles, and the important metric of power consumption at the mobile device impact the system design, make analysis very challenging using the traditional grid model [36]. T ractable analysis should assist system designers in ev aluating the performance of potential algorithms in non-uniform and multi-tier deployments. A P P E N D I X A P RO O F O F S T R I C T F F R , C L O S E D A C C E S S T H E O R E M A macrocell connected user y with SINR 1 < T 1 is giv en a FFR sub-band δ y , where δ ∈ { 1 , ..., ∆ } with uniform probability 1 ∆ , and experiences ne w fading power ˆ g 1 and out-of-cell interference P 1 ˆ I 1 , instead of g 1 and P 1 I 1 + K X k =2 P k I k . The CCDF of the edge user ¯ F FFR , c ( T ) is no w conditioned on its previous SINR . Using Bayes’ rule we hav e, P P 1 ˆ g 1 r 1 − α σ 2 + P 1 ˆ I 1 > T     P 1 g 1 r 1 − α σ 2 + P 1 I 1 + P K k =2 P k I k < T 1 ! = P  P 1 ˆ g 1 r 1 − α σ 2 + P 1 ˆ I 1 > T , P 1 g 1 r 1 − α σ 2 + P 1 I 1 + P K k =2 P k I k < T 1  P  P 1 g 1 r 1 − α σ 2 + P 1 I 1 + P K k =2 P k I k < T 1  . (19) Conditioning on r 1 , the distance to the nearest BS, which is Rayleigh distributed and focusing on the numerator of (19), since ˆ g 1 and g 1 are i.i.d. exponentially distributed with mean µ , giv es E  e  − µ T P 1 r 1 α ( σ 2 + P 1 ˆ I 1 )   − E  e  − µ T P 1 r 1 α ( σ 2 + P 1 ˆ I 1 )  e  − µ T 1 P 1 r 1 α ( σ 2 + P 1 I 1 + P K k =2 P k I k )   , Factoring out terms dependent on the independent noise power σ 2 we observe that the expec- tation of the second term with respect to ˆ I 1 , I 1 , I 2 , ... , and I K is the joint Laplace transform 15 L ( ˆ s 1 , s 1 , s 2 , ..., s K ) of ˆ I 1 , I 1 , I 2 , ..., and I K gi ven by = E " exp − ˆ s 1 ˆ I 1 − s 1 I 1 − K X k =2 s k I k !# = E " exp − ˆ s 1 X z ∈ Z 1 ˆ G z R z − α 1 ( δ z = δ y ) − s 1 X z ∈ Z 1 G z R z − α − K X k =2 s k X z ∈ Z k G z R z − α !!# = E " Y z ∈ Z 1  1 − E [ 1 ( δ z = δ y )] (1 − e ( − ˆ s 1 ˆ G z R z − α ) )  e ( − s 1 G z R z − α ) # K Y k =2 E " Y z ∈ Z k e ( − s k G z R z − α ) # , where 1 ( δ y = δ z ) is an indicator function that takes the value 1 if base station z is transmitting to an edge user on the same sub-band δ as user y , and the third step arises from the independence of I 1 and ˆ I 1 with respect to I 2 ,..., I K . Since ˆ G z and G z are also exponential random variables with mean µ , we can ev aluate the above expression as E " Y z ∈ Z 1  1 − 1 ∆  1 − µ µ + ˆ s 1 R z − α  µ µ + s 1 R z − α # K Y k =2 E " Y z ∈ Z k µ µ + s k R z − α # . By using the probability generating functional (PGFL) of the PPP [29] we obtain L ( ˆ s 1 , s 1 , s 2 , ..., s K ) = e  − 2 π λ 1 R ∞ r 1 h 1 − µ µ + s 1 x − α  1 − 1 ∆  1 − µ µ + ˆ s 1 x − α i x d x  K Y k =2 e  − 2 π λ k ( s k µ ) 2 /α π csc( 2 π α ) α  . Substituting for the integration variables s and de-conditioning on r 1 , we hav e 2 π r 1 λ 1 Z ∞ 0 e − π λ 1 r 1 2 ( 1+2 ξ ( T ,T 1 ,α, ∆)+2 P K k =2 κ k ψ ( γ k T 1 ,α ) ) − µ ( T + T 1 ) σ 2 P 1 r 1 α d r 1 , (20) where ξ ( T , T 1 , α, ∆) = Z ∞ r 1  1 − 1 1 + T 1 r 1 α x − α  1 − 1 ∆  1 − 1 1 + T r 1 α x − α  x d x, and ψ ( z , α ) = csc  2 π α  π z 2 /α α , γ k = P k P 1 , κ k = λ k λ 1 . No w we focus on the denominator of (19), using the independence of I 1 and I 2 ,..., I K we ha ve 1 − E " exp − µ T 1 P 1 r 1 α ( σ 2 + P 1 I 1 + K X k =2 P k I k ) !# = 1 − E  e  − µ T 1 P 1 r 1 α ( σ 2 + P 1 I 1 )   K Y k =2 E  e  − µ T 1 P 1 r 1 α ( P k I k )   = 1 − 2 π r 1 λ 1 Z ∞ 0 e − π λ 1 r 1 2 ( 1+ ρ ( T 1 ,α )+2 P K k =2 κ k ψ ( γ k T 1 ,α ) ) e − µ ( T + T 1 ) σ 2 P 1 r 1 α d r 1 . (21) The first term of the numerator represents the SINR on the newly allocated subband we hav e π λ 1 Z ∞ 0 e − π λ 1 v ( 1+ ρ ( T ,α ) ∆ ) − µT σ 2 P 1 v α/ 2 , (22) 16 since the receiv ed interference is only from the first tier APs due to the closed access frequency allocation for edge users and is originally giv en in [9]. Thus plugging (20) and (21) back into (19), and substituting (22) for the first term of the numerator and substituting r 1 2 = v we have (4). A P P E N D I X B P RO O F O F S F R , C L O S E D A C C E S S T H E O R E M A macrocell connected user y with SINR < T 1 is assigned a FFR sub-band δ y , where δ ∈ { 1 , ..., ∆ } with uniform probability 1 ∆ , and experiences ne w fading power ˆ g 1 , transmit po wer β P 1 , and out-of-cell interference. The CCDF of the edge user ¯ F SFR , c ( T ) is now conditioned on its pre vious SINR , ¯ F SFR , e ( T ) = P β P 1 ˆ g 1 r 1 − α σ 2 + η P 1 ˆ I 1 + P K k =2 P k ˆ I k > T     P 1 g 1 r 1 − α σ 2 + η P 1 I 1 + P K k =2 P k I k < T 1 ! . (23) Using Bayes’ rule as in Theorem 1 and focusing on the resulting numerator , since ˆ g 1 and g 1 are i.i.d. exponentially distributed with mean µ , this giv es E  e  − µ T β P 1 r 1 α ( σ 2 + η P 1 ˆ I 1 + P K k =2 P k ˆ I k )   − E  e  − µ T β P 1 r 1 α ( σ 2 + η P 1 ˆ I 1 + P K k =2 P k ˆ I k )  e  − µ T 1 P 1 r 1 α ( σ 2 + η P 1 I 1 + P K k =2 P k I k )   , No w concentrating on the second term, factoring out terms corresponding to the independent noise power σ 2 , and conditioning on r 1 , we obtain the joint Laplace transform of ˆ I 1 , ˆ I 2 , ..., ˆ I K , and I 1 , I 2 , ..., I K gi ven by E " Y z ∈ Z 1 µ µ + ˆ s 1 R z − α µ µ + s 1 R z − α # K Y k =2 E   Y z ∈ ˆ Z k µ µ + ˆ s k R z − α   E " Y z ∈ Z k µ µ + s k R z − α # . (24) Using the same method as Theorem 1 and de-conditioning on r 1 we obtain 2 π r 1 λ 1 Z ∞ 0 e − π λ 1 r 1 2 ( 1+2 ζ ( T ,T 1 ,α, ∆ ,β ,η )+2 P K k =2 κ k ( ψ ( γ k β T ,α )+ ψ ( γ k T 1 ,α ) )) − µ ( T β + T 1 ) σ 2 P 1 r 1 α d r 1 , where ζ ( T , T 1 , α, ∆ , β , η ) = Z ∞ r 1 " 1 − 1 1 + η T 1 r 1 α x − α 1 1 + η β T r 1 α x − α # x d x, Using the same argument and analysis for the resulting denominator of (23) after Bayes’ rule is applied we hav e 1 − 2 π r 1 λ 1 Z ∞ 0 e − π λ 1 r 1 2 ( 1+ ρ ( η T 1 ,α )+2 P K k =2 κ k ψ ( T 1 ,α ) ) e − µ ( T + η T 1 ) σ 2 P 1 r 1 α d r 1 , (25) 17 Finally , the first term of the numerator is gi ven as 2 π r 1 λ 1 Z ∞ 0 e − π λ 1 r 1 2 ( 1+ ρ ( η β T ,α )+2 P K k =2 κ k ψ ( γ k β T ,α ) ) e − µ ( T ) σ 2 β P 1 r 1 α d r 1 . (26) Thus plugging (25), (25), and (26) back into (23) and substituting r 1 2 = v we have (8). A P P E N D I X C P RO O F O F S T R I C T F F R , O P E N A C C E S S T H E O R E M A user y with SIR 1 < T 1 when connected to the closes macrocell and SIR 2 < T 2 when connected to the closest microcell is gi ven a FFR sub-band δ y , where δ ∈ { 1 , ..., ∆ } with uniform probability 1 ∆ , and experiences new fading power ˆ g 1 and out-of-cell interference P 1 ˆ I 1 . The CCDF of the edge user ¯ F FFR , o ( T ) is now conditioned on its pre vious SIR and r 1 and r 2 , the distance to the nearest tier 1 and tier 2 AP respecti vely , gi ven by P  P 1 ˆ g 1 r 1 − α P 1 ˆ I 1 > T     P 1 g 1 r 1 − α P 1 I 1 + P 2 I 2 + P 2 g 2 r 2 − α < T 1 , P 2 g 2 r 2 − α P 1 I 1 + P 2 I 2 + P 1 g 1 r 1 − α < T 2  . (27) Using Bayes’ rule and initially focusing on the denominator , the conditional term in (27), and conditioning on g 2 gi ves P  r 1 α P 1  P 2 T 2 g 2 r 2 − α − ( P 1 I 1 + P 2 I 2 )  < g 1 < T 1 r 1 α P 1  P 1 I 1 + P 2 I 2 + P 2 g 2 r 2 − α      g 2  P ( g 2 ) . Since g 1 and g 2 are i.i.d. exponentially distributed with mean µ , and setting ¯ I = P 1 I 1 + P 2 I 2 , this gi ves E g 2 " Z T 1 r 1 α P 1 ( ¯ I + P 2 g 2 r 2 − α ) r 1 α P 1  P 2 T 2 g 2 r 2 − α − ¯ I  + µe − µx dx # = E g 2  e − µ r 1 α P 1  P 2 T 2 g 2 r 2 − α − ¯ I  + − e − µT 1 r 1 α P 1 ( ¯ I + P 2 g 2 r 2 − α )  , where ( x ) + = ( x : x > 0 0 : x < 0 Ev aluating the expectation, collecting terms and simplifying gi ves, = 1 −  1 e − ¯ I µT 1 r 1 α P 1 −  2 e − ¯ I µT 2 r 2 α P 2 , where (28) γ = P 2 P 1 ,  1 = 1 T 1 γ r 1 α r 2 α + 1 ! , and  2 =    1 T 2  γ r 1 α r 2 α  − 1 + 1    . 18 W e observ e that the e xpectation of (28) with respect to I 1 and I 2 is the joint Laplace transform of I 1 and I 2 e v aluated at ( µT 1 r 1 α P 1 , µT 2 r 2 α P 2 ) . The joint Laplace transform denoted by g d ( r 1 , r 2 ) is g d ( r 1 , r 2 ) = E I 1 ,I 2 h 1 −  1 e − s 1 ¯ I −  2 e − s 2 ¯ I i = 1 −  1 e ( − 2 π λ 1 ρ 1 , 1 ( T 1 ,α )) e ( − 2 π λ 2 ρ 1 , 2 ( γ T 1 ,α )) −  2 e ( − 2 π λ 1 ρ 2 , 1 ( T 2 /γ ,α )) e ( − 2 π λ 2 ρ 2 , 2 ( T 2 ,α )) , where ρ a,b ( z , α ) is giv en by (16). De-conditioning on r 1 and r 2 , we hav e Z ∞ r 2 =0 Z ∞ r 1 =0  2 π λ 1 r 1 e − π λ 1 r 1 2   2 π λ 2 r 2 e − π λ 2 r 2 2  g d ( r 1 , r 2 )d r 1 d r 2 . (29) No w we turn our attention to the numerator which equals, E h e ( − µ ˆ I 1 T r 1 α ) i − E  e ( − ˆ I 1 µT r 1 α )   1 e  − ¯ I µT 1 r 1 α P 1  +  2 e  − ¯ I µT 2 r 2 α P 2   . Concentrating on the second term we observe that the expectation with respect to ˆ I 1 , I 1 , and I 2 is the joint Laplace transform of ˆ I 1 , I 1 , and I 2 e v aluated at ( µT r 1 α , µT 1 r 1 α P 1 , µT 2 r 2 α P 2 ) . The joint Laplace transform g n ( r 1 , r 2 ) := L num  µT r 1 α , µT 1 r 1 α P 1 , µT 2 r 2 α P 2  is E ˆ I 1 ,I 1 ,I 2 h exp  − s 1 ˆ I 1  (  1 exp ( − s 2 ( P 1 I 1 + P 2 I 2 )) +  2 exp ( − s 3 ( P 1 I 1 + P 2 I 2 ))) i . Expanding the terms and applying a similar approach as before we ha ve g n ( r 1 , r 2 ) =  1 e − 2 π ( λ 1 ξ 1 , 1 ( T ,T 1 ,α, ∆)+ λ 2 ρ 1 , 2 ( T 1 ,α )) +  2 e − 2 π ( λ 1 ξ 2 , 1 ( T ,T 2 /γ ,α, ∆)+ λ 2 ρ 2 , 2 ( T 2 ,α )) , where ξ a,b ( T , z , α, ∆) is giv en by (15). De-conditioning on r 1 and r 2 , Z ∞ r 2 =0 Z ∞ r 1 =0  2 π λ 1 r 1 e − π λ 1 r 1 2   2 π λ 2 r 2 e − π λ 2 r 2 2  g n ( r 1 , r 2 )d r 1 d r 2 . (30) Finally , plugging (20) and (29) into (27), and substituting (22) for the first term of the numerator by definition and r 1 2 = v we have (15). A P P E N D I X D P RO O F O F S F R , O P E N A C C E S S T H E O R E M A user y with SIR 1 < T 1 and SIR 2 < T 2 is given a FFR sub-band with uniform probability 1 ∆ , and experiences new fading power ˆ g 1 , transmit power β P 1 , and out-of-cell interference ¯ I = η P 1 I 1 + P 2 I 2 . The CCDF of the edge user ¯ F SFR , o ( T ) is now gi ven by P  β P 1 ˆ g 1 r 1 − α η P 1 ˆ I 1 + P 2 ˆ I 2 > T     P 1 g 1 r 1 − α ¯ I + P 2 g 2 r 2 − α < T 1 , P 2 g 2 r 2 − α ¯ I + P 1 g 1 r 1 − α < T 2  (31) 19 Using the method of Theorem 3, applying Bayes’ rule we have the joint Laplace transform of I 1 and I 2 gi ven r 1 and r 2 , f d ( r 1 , r 2 ) = 1 −  1 e ( − 2 π λ 1 ( ρ 1 , 1 ( η T 1 ,α )+ κρ 1 , 2 ( γ T 1 ,α ))) −  2 e ( − 2 π λ 1 ( ρ 2 , 1 ( η γ T 2 ,α ) + κρ 2 , 2 ( T 2 ,α ) )) , where γ = P 2 P 1 ,  1 = 1 T 1 γ r 1 α r 2 α + 1 ! ,  2 =    1 T 2  γ r 1 α r 2 α  − 1 + 1    , and ρ a,b ( z , α ) gi ven by (16) . De-conditioning on r 1 and r 2 , we hav e Z ∞ r 2 =0 Z ∞ r 1 =0  2 π λ 1 r 1 e − π λ 1 r 1 2   2 π λ 2 r 2 e − π λ 2 r 2 2  f d ( r 1 , r 2 )d r 1 d r 2 . (32) Again, follo wing the method of Theorem 3, we observe that the numerator of (31) is given by π λ 1 Z ∞ 0 e − π λ 1 v ( 1+ ρ ( η β T ,α )+2 κψ ( γ β T ,α ) ) d v − Z ∞ r 2 =0 Z ∞ r 1 =0  2 π λ 1 r 1 e − π λ 1 r 1 2   2 π λ 2 r 2 e − π λ 2 r 2 2  f n ( r 1 , r 2 )d r 1 d r 2 , (33) where f n ( r 1 , r 2 ) =  1 e − 2 π λ 1 ( ζ 1 , 1 ( T ,T 1 ,α, ∆ ,β ,η )+ κψ ( γ β T ,α ) + κρ 1 , 2 ( γ T 1 ,α ) ) +  2 e − 2 π λ 1 ( ζ 2 , 1 ( T ,T 2 /γ ,α, ∆ ,β ,η )+ κψ ( γ β T ,α ) + κρ 2 , 2 ( T 2 ,α ) ) . Thus plugging (33) and (32) back into (31) and substituting r 1 2 = v we have (18). R E F E R E N C E S [1] Qualcomm, “L TE advanced: heterogeneous networks, ” white paper , Jan. 2011. [Online]. A vailable: http://qualcomm.com/ documents/files/lte- advanced- heterogeneous- networks.pdf [2] Picochip, “The case for home base stations, ” white paper , Apr . 2007. [Online]. A vailable: http://www .femtoforum.org/ femto/Files/File/picoChipFemtocellWhitePaper1.1.pdf [3] A. Saleh, A. Rustako, and R. Roman, “Distributed antennas for indoor radio communications, ” IEEE T ransactions on Communications , vol. 35, no. 12, pp. 1245 – 1251, Dec. 1987. [4] J. Zhang and J. G. Andrews, “Distributed antenna systems with randomness, ” IEEE T ransactions on W ir eless Communi- cations , vol. 7, no. 9, pp. 3636 –3646, Sep. 2008. [5] V . Chandrasekhar , J. G. Andrews, and A. Gatherer, “Femtocell networks: a survey , ” IEEE Communications Magazine , vol. 46, no. 9, pp. 59 –67, Sep. 2008. [6] F . Baccelli, M. Klein, M. Lebourges, and S. Zuyev , “Stochastic geometry and architecture of communication networks, ” J. T elecommunication Systems , vol. 7, no. 1, pp. 209–227, 1997. [7] T . Bro wn, “Cellular performance bounds via shotgun cellular systems, ” IEEE Journal on Sel. Ar eas in Communications , vol. 18, no. 11, pp. 2443–2455, Nov . 2000. [8] M. Haenggi, J. Andrews, F . Baccelli, O. Dousse, and M. Franceschetti, “Stochastic geometry and random graphs for the analysis and design of wireless networks, ” IEEE Journal on Sel. Areas in Communications , vol. 27, no. 7, pp. 1029–1046, Sep. 2009. [9] J. G. Andre ws, F . Baccelli, and R. K. Ganti, “ A new tractable model for cellular coverage, ” in Proc. Allerton Conf. on Communication, Contr ol, and Computing , Monticello, Illinois, Oct. 2010, pp. 1204–1211. [10] H. S. Dhillon, R. K. Ganti, F . Baccelli, and J. G. Andrews, “Modeling and analysis of K-tier downlink heterogenous cellular networks, ” IEEE J ournal on Selected Areas in Communications , to appear . [Online]. A vailable: 20 [11] H. Dhillon, R. Ganti, and J. Andrews, “ A tractable framework for cov erage and outage in heterogeneous cellular networks, ” in Pr oc., Information Theory and Applications W orkshop (IT A) , Feb . 2011, pp. 1–6. [12] S. Mukherjee, “UE coverage in L TE macro network with mixed CGS and open access femto overlay , ” in Proc., IEEE International Confer ence on Communications , Kyoto, Jun. 2011. [13] ——, “ Analysis of UE outage probability and macrocellular traffic offloading for WCDMA macro network with femto ov erlay under closed and open access, ” in Pr oc., IEEE International Confer ence on Communications , Kyoto, Jun. 2011. [14] N. Himayat, S. T alwar , A. Rao, and R. Soni, “Interference management for 4G cellular standards [WIMAX/L TE update], ” IEEE Communications Magazine , vol. 48, no. 8, pp. 86 –92, Aug. 2010. [15] T . D. Novlan, R. K. Ganti, A. Ghosh, and J. G. Andrews, “ Analytical e valuation of fractional frequency reuse for OFDMA cellular networks, ” Under Re vision, IEEE T ransactions on W ir eless Comm. [Online]. A vailable: [16] T . Novlan, R. Ganti, J. Andrews, and A. Ghosh, “ A ne w model for cov erage with fractional frequency reuse in OFDMA cellular networks, ” in Pr oc. IEEE Globecom , Houston, Dec. 2011, to appear . [17] K. Begain, G. Rozsa, A. Pfening, and M. T elek, “Performance analysis of GSM networks with intelligent underlay-ov erlay , ” in Pr oc. Intl. Symp. on Computers and Communications , T aormina, Italy , Jul. 2002, pp. 135–141. [18] M. Sternad, T . Ottosson, A. Ahlen, and A. Svensson, “ Attaining both coverage and high spectral efficiency with adapti ve OFDM do wnlinks, ” in Proc. IEEE V ehicular T echnology Conf. , vol. 4, Orlando, Florida, Oct. 2003, pp. 2486–2490. [19] T . Novlan, J. Andrews, I. Sohn, R. Ganti, and A. Ghosh, “Comparison of fractional frequency reuse approaches in the OFDMA cellular downlink, ” in Pr oc. IEEE Globecom , Miami, Florida, Dec. 2010, pp. 1–5. [20] J. Li, N. Shroff, and E. Chong, “ A reduced-power channel reuse scheme for wireless packet cellular networks, ” IEEE/ACM T rans. on Networking , vol. 7, no. 6, pp. 818–832, Dec. 1999. [21] Huawei, “R1-050507: Soft frequency reuse scheme for UTRAN L TE, ” 3GPP TSG RAN WG1 Meeting #41 , May 2005. [22] K. Doppler, C. W ijting, and K. V alkealahti, “Interference aware scheduling for soft frequency reuse, ” in Pr oc. IEEE V ehicular T echnology Conf. , Barcelona, Apr . 2009, pp. 1–5. [23] V . Chandrasekhar and J. G. Andrews, “Spectrum allocation in tiered cellular networks, ” IEEE T ransactions on Communi- cations , vol. 57, no. 10, pp. 3059 –3068, Oct. 2009. [24] H. Lee, D. Oh, and Y . H. Lee, “Mitigation of inter-femtocell interference with adaptive fractional frequency reuse, ” in Pr oc.,, IEEE International Confer ence on Communications , May 2010, pp. 1 –5. [25] J. Lee, S. Bae, Y . Kwon, and M. Chung, “Interference analysis for femtocell deployment in OFDMA systems based on fractional frequency reuse, ” IEEE Communications Letters , vol. PP , no. 99, pp. 1 –3, Apr . 2011. [26] M. Andrews, V . Capdevielle, A. Feki, and P . Gupta, “ Autonomous spectrum sharing for mixed L TE femto and macro cells deployments, ” in Pr oc., IEEE Conference on Computer Communications W orkshops , Mar . 2010, pp. 1 –5. [27] P . Xia, V . Chandrasekhar , and J. Andrews, “Open vs. closed access femtocells in the uplink, ” IEEE Tr ansactions on Wir eless Communications , vol. 9, no. 12, pp. 3798 –3809, Dec. 2010. [28] H. Jo, P . Xia, and J. G. Andrews, “Downlink femtocell networks: Open or closed?” in Pr oc., IEEE International Conference on Communications , Kyoto, Jun. 2011. [29] D. Stoyan, W . Kendall, and J. Mecke, Stochastic Geometry and Its Applications, 2nd Edition . John W iley and Sons, 1996. [30] A. Hernandez, I. Guio, and A. V aldovinos, “Interference management through resource allocation in multi-cell OFDMA networks, ” in Pr oc. IEEE V ehicular T echnology Conf. , Barcelona, Apr . 2009, pp. 1–5. [31] M. Al-Shalash, F . Khafizov , and C. Zhijun, “Interference constrained soft frequency reuse for uplink ICIC in L TE networks, ” in Pr oc. IEEE Intl. Symp. on P ersonal Indoor and Mobile Radio Communications , Istanbul, Turk ey , Sep. 2010, pp. 1882– 1887. [32] T . S. Rappaport, W ireless Communications Principles and Practice, 2nd Edition . Prentice Hall, 2002. [33] S. Ali and V . Leung, “Dynamic frequency allocation in fractional frequency reused OFDMA networks, ” IEEE T rans. on W ireless Communications , vol. 8, no. 8, pp. 4286–4295, Aug. 2009. [34] K. Son, S. Chong, and G. de V eciana, “Dynamic association for load balancing and interference avoidance in multi-cell networks, ” IEEE T ransactions on W ir eless Communications , vol. 8, no. 7, pp. 3566 –3576, Jul. 2009. [35] A. Stolyar and H. V iswanathan, “Self-organizing dynamic fractional frequency reuse for best-effort traffic through distributed inter-cell coordination, ” in Pr oc., IEEE Infocom , Apr . 2009, pp. 1287 –1295. [36] F . W amser , D. Mittelsta, and D. Staehle, “Soft frequency reuse in the uplink of an OFDMA network, ” in Pr oc. IEEE V ehicular T echnology Conf. , T aipei, T aiwan, May 2010, pp. 1–5. 21 Macro BSs Pico BSs Femto BSs Fig. 1. A realization of a Poisson distributed three-tier cellular network with coverage regions defined by the highest receiv ed power . f c f 3 P f 1 f 2 Strict FFR f 3 P f 1 f 2 SFR ɴ P f c f 3 P f 1 f 2 Allocation 3 T x power f 3 P f 1 f 2 ɴ P f c f 3 P f 1 f 2 subband f 3 P f 1 f 2 ɴ P subband Allocation 2 Allocation 1 T x power Inner User Edge User Fig. 2. Strict FFR (left) and SFR (right) subband and transmit power allocations with ∆ = 3 cell-edge reuse factor . 22 − 15 − 10 − 5 0 5 10 15 20 25 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 SINR Threshold (dB) Coverage Probability α =4; κ 2 = 4 γ 2 = .01; κ 3 = 9 γ 3 = .001; β = 5; T 1 = 3 dB No Reuse SFR Monte Carlo Strict FFR Monte Carlo SFR Analytical Strict FFR Analytical Fig. 3. Downlink edge user SINR distributions for closed access with three tiers of APs. − 15 − 10 − 5 0 5 10 15 20 25 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 κ = 5; γ = .01; β = 5; T 1 = 5 dB; T 2 = 1 dB SINR Threshold (dB) Coverage Probability No Reuse SFR Monte Carlo Strict FFR Monte Carlo SFR Analytical Strict FFR Analytical Fig. 4. Downlink edge user SINR distributions for open access with two tiers of APs. 23 − 5 0 5 10 15 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 γ = .01; T 1 = 5 dB; T 2 = 2 dB SINR Threshold Coverage Probability κ = 1 κ = 5 κ = 10 κ = 25 κ = 100 Fig. 5. Downlink edge user SINR distributions for Strict FFR and closed access as a function of the tier density ratio κ . Ŧ 5 0 5 10 15 0.1 0.2 0.3 0.4 0.5 0.6 0.7 κ = 5; γ = .01; β = 5; T 1 = 5 dB SINR Threshold Coverage Probability T 2 = Ŧ 12 dB T 2 = Ŧ 6 dB T 2 = 0 dB T 2 = 6 dB T 2 = 12 dB Fig. 6. Downlink edge user SIR distributions for SFR and open access as a function of T 2 .