Downlink and Uplink Decoupling in Two-Tier Heterogeneous Networks with Multi-Antenna Base Stations
aa r X i v : . [ c s . I T ] F e b Downlink and Uplink Decoupling in Two-TierHeterogeneous Networks with Multi-Antenna BaseStations
Mudasar Bacha, Yueping Wu and Bruno Clerckx
Abstract —In order to improve the uplink performance offuture cellular networks, the idea to decouple the downlink (DL)and uplink (UL) association has recently been shown to providesignificant gain in terms of both coverage and rate performance.However, all the work is limited to SISO network. Therefore,to study the gain provided by the DL and UL decoupling inmulti-antenna base stations (BSs) setup, we study a two tierheterogeneous network consisting of multi-antenna BSs, andsingle antenna user equipments (UEs). We use maximal ratiocombining (MRC) as a linear receiver at the BSs and using toolsfrom stochastic geometry, we derive tractable expressions forboth signal to interference ratio (SIR) coverage probability andrate coverage probability. We observe that as the disparity inthe beamforming gain of both tiers increases, the gain in term ofSIR coverage probability provided by the decoupled associationover non-decoupled association decreases. We further observethat when there is asymmetry in the number of antennas of bothtier, then we need further biasing towards femto-tier on the topof decoupled association to balance the load and get optimal ratecoverage probability.
I. I
NTRODUCTION
The demand for high data rates is ever-growing and it isprojected that over the next decade a factor of a thousandtimes increase in wireless network capacity will be required[1]. In order to meet this challenge, a massive densificationof the current wireless networks characterized by the densedeployment of low power and low cost small cell is required,which will convert the existing single-tier homogeneous net-works into multi-tier heterogeneous networks (HetNets) [2].HetNets that consist of different types of base stations (Macro,Micro, Pico and Femto) can not be operated in the same wayas a single-tier homogeneous network (consisting of MacroBase stations only) and need some fundamental changes inthe design and deployment to meet the high data rate demand.Cellular networks have been designed mainly for downlink(DL) because initially the traffic was asymmetric (mostly in thedownlink direction). However, with the increase in real-timeapplications, online social-networking, and video-calling thetraffic in UL has greatly increased, which necessitates the needfor the uplink (UL) optimization. In current cellular networks,
M. Bacha and B. Clerckx is with the Communication and SignalProcessing Group, Department of Electrical and Electronic Engineering,Imperial College London, London SW7 2AZ, U.K. (e-mail: m.bacha13,[email protected]). This work has been partially supported by theEPSRC of UK, under grant EP/N015312/1.Y. Wu was with the Department of Electrical and Electronic Engineering,Imperial College London, London SW7 2AZ, U.K., and is now with the HongKong Applied Science and Technology Research Institute (ASTRI) (e-mail:[email protected]). cell association is based on downlink average received power,which is viable for homogeneous networks where the transmitpower of all the base stations (BSs) is the same. However, inheterogeneous networks there is a big disparity in the transmitpower of different BSs, and the association scheme based ondownlink received power is highly inefficient, therefore, theidea of downlink and uplink decoupling (DUDe) has beenproposed for 5G in [3]-[6].
A. Related Work
A simulation based study has been performed on two-tierlive network where the UL association is based on minimumpath-loss while the DL association is based on DL receivedpower [5]. This kind of association divides the users into threegroups: users attached to macro base station (MBS) both inthe DL and UL, users attached to femto base station (FBS)both in the DL and UL, and users attached to MBS in theDL and FBS in the UL. The authors in [5] showed that thegain in UL throughput is quite high when the UL associationis based on minimum path-loss. The gain comes from thoseusers which are connected to MBS in the DL and FBS inthe UL because they have better channel to the femto-celland they create less interference to the macro-cell. A networkconsisting of macro-tier and femto-tier is studied using toolsfrom stochastic geometry in [7], where the throughput gain dueto decoupling has been shown. In [8], the analytical resultsobtained from stochastic geometry-based model have beencompared with the results obtained from simulation in [5],and they found that both of them match with each other. Theyalso found that the association probability mainly depends onthe density of the deployment and not on the process used togenerate the deployment geometry. It has been shown in [11]that DUDe provides gain in term of system rate, spectrumefficiency, and energy efficiency. A joint study of DL and ULfor k tier SISO network has been performed in [10], whileconsidering a weighted path-loss association and UL powercontrol.Stochastic geometry has emerged as a powerful tool forthe analysis of cellular networks after the seminal work of[12]. It has been shown that stochastic geometry-based modelsare equally accurate as grid based models. In addition, theyprovide more tractability and their accuracy becomes better asthe heterogeneity of the network increases. Most of the work,which considered stochastic geometry-based model mainlystudied the DL performance of the HetNets. For instance, single input single output (SISO) HetNets have been studiedin [13], and [14], MIMO HetNets in [15], [16], [17], [18],and [19]. A complete survey can be found in [25] and thereferences therein. However, only limited work has beencarried out in UL because it is more involved due to UL powercontrol and correlation among interferers. The UL powercontrol is required because an interfering user may be closerto the BS than the scheduled user, which creates an additionalsource of randomness in the UL modeling. The correlationamong interfering users comes due to the orthogonal channelsassignment within a cell, which prohibits the use of the samechannel in a cell. In other words, there is only one UErandomly located within the coverage region of the BS, whichtransmits in a given resource block. Therefore, the interferencedoes not originate from the the PPP distributed UEs butinstead from Voronoi perturbed lattice process [10]. The exactinterference characterization for which is not available [10],and thus makes the UL analysis even more involved.An uplink model for the single tier network has been derivedin [20], which uses fractional power control (FPC) in the UL.A multi-tier UL performance has been studied in [21] and[10], where each tier differs only in terms of density, cutoffthreshold, and transmit power. In [21] a truncated channelpower inversion is used due to which mobile users suffer fromtruncation outage in addition to SINR outage. The performancegain of DUDe is only studied for SISO network and there isno work which studies the decoupled association in the MIMOnetwork . Therefore, in this work we consider multi-antennaBSs and we also consider UL biasing with the DUDe. B. Contributions and Outcomes
The main challenge in modeling the UL multi-antennasHetNets, in addition to the generic challenges discussed above,is to select an analytically tractable technique from the numberof possible multi-antenna techniques. We consider maximalratio combining (MRC) at the BSs and assume that the channelis perfectly known at the receiver. A receiver has knowledgeabout the channel between the transmitter and itself, but itdoes not have any knowledge about the interfering channel.Furthermore, we consider power control in the UL, whichpartially compensates for the path-loss [10], [20]. We considerRayleigh fading in addition to path-loss .We use a cell association technique with biasing, which canbe used in any MIMO HetNets. This association completelydecouples the DL and UL association, and is generic andsimple. Cell biasing in the UL can be used to balance theload across the tiers. This association scheme is motivated bythe technique used in [7] for SISO HetNets. Due to the DUDe,users are divided into three disjoint groups as shown in Fig. [22] studies the UL performance in multi antennas BSs network, whichwas not available online at the initial submission of this paper. However, ouranalysis approach is significantly different than [22]. We explicitly take intoaccount the beamforming gain in the cell association and use Fa ` a di Bruno’sformula [27] to find the high order derivative of the Laplace transform of theinterference, whereas [22] does not consider beamforming gain in the cellassociation and use Gil-Pelaez inversion theorem to avoid finding the higherorder derivative. For the sake of simplicity, we do not consider shadowing in this work.Shadowing in similar setup can be found in [10] and [22].
Fig. 1: System Model1; (I) users attached to the MBS both in the DL and UL, (II)users attached to the FBS both in the DL and UL, and (III)users attached to the MBS in the DL and FBS in the UL.The gain in the UL performance comes from the last kind ofusers because they have strong connection to the FBS (lowpath-loss) and they create less interference to the MBS (dueto larger distance).In this paper, we study both the SIR and rate coverageprobability of a two tier network where the association is basedon DL and UL decoupling. The novel and insightful findingsof this paper are as follows: • The gain in term of SIR coverage probability provided bythe DUDe association over a no-DUDe association (asso-ciation based on DL maximum received power averagedover fading) decreases as the difference in the numberBS’s antennas in the femto and macro-tier increases.When the number of MBS antennas is larger than that ofFBS, the association region of a MBS is enlarged due tothe larger beamforming gain provided by the MBS. As aresult of which UEs closer to the FBSs become associatedwith MBSs. These boundary UEs, which are connectedto macro-tier, create strong interference at nearby FBSswhen they transmit to their serving MBSs. On the otherhand, when both tiers have the same beamforming gain,the coverage region of both tiers are the same and theinterference created by the boundary UEs is not thatstrong. Thus, the DUDe gain over No-DUDe is high whenboth tier have the same beamforming. • It has been shown in [5], [7], [10], [11] that DUDeassociation improves the load balance and provides fair-ness in the UL performance of different UEs. In [10]it is shown that in the UL the optimal rate coverage isprovided by the minimum path-loss association. However,we observe that in the SIMO network DUDe associationdoes not completely solve the load imbalance problemand the optimal rate coverage is not provided by theminimum path-loss association. In the SIMO network,this load imbalance problem comes from the differentbeamforming gain of the femto and macro-tier, therefore,we still need biasing towards femto-tier to balance theload. We show that when the beamforming gain of themacro-tier is high as compared to the femto-tier thenbiasing towards femto-tier improves the rate coverageprobability. Y K = q P X α K ( η − K h K s K + X i ∈ Φ ′ K \ u q P X α K ηK i D − α K K i h K i s K i | {z } interference from K th tier scheduled UEs + X q ∈ Φ ′ J q P X α J ηJ q D − α K J q h J q s J q | {z } interference from J th tier scheduled UEs + n (1) Z K = h HK Y K = q P X α K ( η − K k h K k s K + X i ∈ Φ ′ K \ u q P X α K ηK i D − α K K i h HK h K i s K i + X q ∈ Φ ′ J q P X α J ηJ q D − α K J q h HK h J q s J q + h HK n (2) γ K = P k h K k X α K ( η − K P i ∈ Φ ′ K \ u P (cid:12)(cid:12)(cid:12)(cid:12) h HK h Ki k h K k (cid:12)(cid:12)(cid:12)(cid:12) X α K ηK i D − α K K i + P q ∈ Φ ′ J P (cid:12)(cid:12)(cid:12)(cid:12) h HK h Jq k h K k (cid:12)(cid:12)(cid:12)(cid:12) X α J ηJ q D − α K J q + σ n (3)The rest of the paper is organized as follows, in Section II,we present our system model and assumptions. In SectionIII, we derive the association probabilities and the distancedistribution of a user to its serving BS. Section IV is the maintechnical section, where we study the SIR coverage and therate coverage of the network. Section V presents simulationsand numerical results, while Section VI concludes the paperand provides further research directions.The key notations used in this paper are given in Table 1.II. S YSTEM M ODEL
A. Network Model
We consider a heterogeneous network that consists of macrobase stations (MBSs), femto base stations (FBSs) and userequipments (UEs). The location of MBSs, FBSs and UEsare modeled as 2-D independent homogeneous Poisson PointProcesses (PPPs). Let Φ M , Φ F , and Φ U represent the PPPs forMBSs, FBSs and UEs respectively. Furthermore, let λ M , λ F ,and λ U be the density of Φ M , Φ F , and Φ U respectively. Thetransmit power of a MBS and FBS are represented by P M and P F respectively, where P M > P F . We consider thatMBSs have N M and FBSs have N F antennas and N M ≥ N F ,while UEs have single antenna. Throughout the system model,we only consider inter-cell interference i.e., a BS schedules asingle UE in a given resource block. The analysis is performedfor a typical user located at the origin and the BS serving thistypical user is referred to as the tagged BS [23].
B. Uplink Power Control
We consider a fractional power control in the uplink [9],which partially compensates for path-loss. Let X K be thedistance between a UE and its serving K th-tier BS. The UEtransmits with P U = P X ηα K K , where α K is the path-lossexponent of the K th-tier, P is the transmit power of theUE before applying the UL power control, and ≤ η ≤ is the power control fraction. If η = 1 , the path-loss iscompletely inverted by the power control, and if η = 0 , nochannel inversion is applied and all UEs transmit with thesame power. We do not consider maximum transmit powerconstraint for tractability of the analysis. However, the analysis can be extended to include the maximum power constraintsimilar to [21] and [22]. C. Signal Model
The received signal vector Y K at a tagged BS whena typical UE u is served by a K th tier BS having N K antennas is given by (1) (at the top of this page), where α K is the path-loss exponent of K th tier ( α K > ; h K i = h h K , h K , . . . h K NK i T is the complex channel gainand the magnitude of each h i follows Rayleigh distribution(we assume Rayleigh fading channel); X J q represents theEuclidean distance between the q th UE of the J th tier andits serving BS; D J q is the Euclidean distance between the q th interfering UE of the J th tier to the tagged BS; s J q isthe signal transmitted by the q th UE of the J th tier havingunit power; n = [ n , n , · · · , n N K ] T is the vector of complexadditive white Gaussian noise at the tagged BS; Φ ′ K and Φ ′ J represent the point processes formed by the thinned PPPof the scheduled UEs of the K th and J th tier respectively.Since, we assume multiple antennas’ BS, we apply a receivercombiner g to s K of a typical UE. By using maximal ratiocombining (MRC), g = h HM , (1) can be written as in (2) (atthe top of this page). Similarly, the SINR γ K at the taggedBS K can be written as in (3), available at the top of thispage, where k h K k ∼ Gamma ( N K , , whereas (cid:12)(cid:12)(cid:12)(cid:12) h HK h Ki k h K k (cid:12)(cid:12)(cid:12)(cid:12) and (cid:12)(cid:12)(cid:12)(cid:12) h HK h Jq k h K k (cid:12)(cid:12)(cid:12)(cid:12) both follow exponential distribution [24]. Weassume high density for UEs such that each BS has at leastone UE in its association region and UEs always have data totransmit in the UL (saturated queues). Throughout the paperthe K th tier will always be the serving tier of the typicalUE while J th tier will be the interfering tier. We will usethe terms UE and user, and typical user and random userinterchangeably. D. Cell Association
The long term average received power (accounting forbeamforming gain) at a typical UE when a K th tier BS TABLE I: List of Notations
Notation Description Φ K , Φ U PPP of tier K BSs, PPP of UEs λ K , λ U density of tier K BSs, density of UEs P K , P U transmit power of each BS of the K th tier , transmit power of a UE X K , α K distance between the typical UE and the tagged
BS, path-loss exponent of K th tier X K i , X J q distance between an interfering UE of K th and J th tier, and their serving BSs respectively D K i , D J q distance between an interfering UE of K th and J th tier, and the tagged BS respectively A K , N K association probability of a typical UE to K th tier, number of antennas at a K th tier BS B bias factor, B = B F B M , where B F and B M is biasing towards femto-tier and macro-tier respectively τ K , ρ K , η SIR threshold and rate threshold of K th tier, UL power control fraction C , C K SIR coverage probability of the network, SIR coverage probability of the K th tier R , R K rate coverage probability of the network, rate coverage probability of the K th tier Ω K , ¯Ω K , W load on a K th tier BS, average load on K th tier BS, bandwidth in Hz h K , h K i , h J q complex channel gain between the tagged BS and typical UE, an interfering UE of K th and J th tier respectively transmits is P K N K X − α K K . Similarly, in the UL, the longterm average received power at a typical K th tier BS is P N K X − α K K (before employing UL power control). In theDL, a UE is associated to a BS from which it receives themaximum average power, while in the uplink it is associatedto a BS that receives the maximum average power. In the UL,each UE has the same transmit power, so the association isactually related to the number of antennas and the path-loss.Due to the cell association criterion, there are three sets ofUEs: 1) UEs connected to the MBSs both in the DL and theUL, 2) UEs associated to the MBSs in the DL and FBSs in theUL, and 3) UEs connected to the FBSs both in the DL and theUL as shown in Fig. 1. In the DL, the load imbalance problemarises due to the high transmit power and beamforming gainof the MBS as compared to the FBS, whereas in the UL itis only due to the larger number of antennas at the MBS. Inorder to balance the load among the macro-tier and femto-tierin the UL, we use bias factor B = B F B M , where B F and B M arethe bias towards femto- and macro-tier respectively. A biasing B > offloads UEs from the macro-tier to the femto-tier, B < offloads UEs form the femto-tier to the macro-tier, and B = 1 means no biasing. The association criterion is basedon long-term average biased-received power and the UEs indifferent region can be written as: • Case1- UEs connected to MBS both in the UL and DL: (cid:0) P M N M X − α M M > P F N F X − α F F (cid:1)| {z } DL association rule \(cid:0) N M B M X − α M M > N F B F X − α F F (cid:1)| {z } UL association rule , • Case2- UEs connected to MBS in the DL and FBS in theUL: n(cid:0) P M N M X − α M M > P F N F X − α F F (cid:1) \(cid:0) N M B M X − α M M ≤ N F B F X − α F F (cid:1)(cid:9) , • Case3- UEs connected to FBS both in the UL and DL: n(cid:0) P M N M X − α M M ≤ P F N F X − α F F (cid:1) \(cid:0) N M B M X − α M M ≤ N F B F X − α F F (cid:1)(cid:9) . III. P
RELIMINARIES
In this section, we find the association probabilities of UEsand the distance distribution of a UE to its serving BS. Thesewill be required in the next section to find the SIR coverageand rate coverage of the network.
A. Association Probability
In this subsection, we find the association probabilities ofthe UEs.
Lemma 1.
The probability that a typical UE is associatedwith the MBS both in the UL and the DL is given by P ( case πλ M Z ∞ X M e − π (cid:20) λ F Υ /αF (cid:16) X αM /αFM (cid:17) + λ M X M (cid:21) d X M , (4) where for B F B M ≥ P F P M , Υ = B F N F B M N M and for B F B M < P F P M , Υ = P F N F P M N M . The association probability is independent ofthe density of the UEs.Proof: See Appendix A.
Lemma 2.
The probability that a typical UE is associatedwith a MBS in the DL and a FBS in the UL is P ( case πλ F "Z ∞ X F e − π (cid:20) λ M Υ ′ /αM (cid:16) X αF /αMF (cid:17) + λ F X F (cid:21) d X F − Z ∞ X F e − π (cid:20) λ M Υ ′ /αM (cid:16) X αF /αMF (cid:17) + λ F X F (cid:21) d X F , (5) where when B F B M ≥ P F P M , then Υ ′ = B M N M B F N F and Υ ′ = P M N M P F N F and when B F B M < P F P M then Υ ′ = P M N M P F N F and Υ ′ = B M N M B F N F .Proof: The proof follows similar steps as Lemma 1.
Lemma 3.
The probability that a typical UE associates withthe FBS both in the DL and the UL can be written as P ( case πλ F Z ∞ X F e − π (cid:20) λ M Υ ′ /αM (cid:16) X αF /αMF (cid:17) + λ F X F (cid:21) d X F , (6) where when B F B M ≥ P F P M , then Υ ′ = P M N M P F N F and when B F B M < P F P M then Υ ′ = B M N M B F N F .Proof: It can be easily proved by following the same stepsas in Lemma 1.
From Lemma 1, 2 and 3, the tier-association probabilitiesin the UL can be easily obtained. Thus the probability that atypical UE is associated with K th-tier BS is given by A K = 2 πλ K Z ∞ X K e − π (cid:20) λ J Υ /αJ (cid:16) X αK/αJK (cid:17) + λ K X K (cid:21) d X K (7)where K, J ∈ {
M, F } and K = J and for B F B M ≥ P F P M , Υ = B J N J B K N K and for B F B M < P F P M , Υ = P J N J P K N K . It is important tomention that the condition B F B M < P F P M in Lemma 1, 2, 3 and (7)is very unlikely to be true because usually we need to offloadthe UEs towards femto-tier instead of macro-tier. However, wespecifically mentioned it so that the expression in Lemma 1,2, 3 and (7) holds for the entire range of the bias B .For α K = α J = α , (7) simplifies to A K = λ K λ K + Υ /α λ J . (8)The probability that a typical UE associates to the K th tierincreases with increasing the density of K th tier BS, or biasingtowards K th tier or placing more antennas at K th tier BSs.However, the increase due to biasing and beamforming gainis not the dominant factor due to the presence of the exponent /α where α > . B. Distance Distribution to the Serving BS
In this subsection, we find the distance distribution of thescheduled user to the serving BS.
Lemma 4.
The distribution of the distance X K between thetypical UE and the tagged BS is f X K ( X K ) = 2 πλ K A K X K × exp ( − π λ K X K + λ J (cid:18) B J N J B K N K (cid:19) /α J X α K /α J ) K !) , (9) where K, J ∈ {
M, F } , K = J , and A K is the tierassociation probability.Proof: We provide the proof in Appendix B.
Remark 1.
It is important to mention that the distancedistribution of an interfering UE to its serving BS is differentfrom the distribution of the typical UE and the tagged BSbecause the distance between an interfering UE and its servingBS is upper bounded by a function of the distance between aninterfering UE and the tagged BS. Specifically, let both thetypical UE u and an interfering UE u i belong to the K thtier and let the distance between u i and its serving BS be X K i , and D K i be the distance between u i and the tagged BSthen ≤ X K i ≤ D K i . Similarly, if u i belongs to the J th tier(interfering tier) and the distance between u i and its servingBS is X J i and the distance between u i and the tagged BS is D J i then ≤ X J i ≤ (cid:18) N J B J D αKJi N K B K (cid:19) /α J . Remark 2.
Based on the association rule in the previoussection, we define the interference boundary here. For a UEwho is associated to K th tier and the association distance is X K , the interference boundary I X J for the J th tier is givenby I X J = X J > (cid:16) N J B J N K B K (cid:17) /α J X α K /α J K . Thus, both Remark 1 and Remark 2 define the regions wherethe interfering UEs can be located and these regions come dueto the association rule defined in the previous section.IV. SIR
AND R ATE C OVERAGE P ROBABILITY
A. SIR Coverage Probability
The UL SIR coverage probability can be defined as theprobability that the instantaneous UL SIR at a randomlychosen BS is greater than some predefined threshold. The ULSIR coverage probability C of our system model can be writtenas C = C F A F + C M A M , (10)where C F , C M , A F , and A M are the coverage and associationprobability of femto- and macro-tier respectively. The K th-tiercoverage probability C K for a target SIR τ K can be definedas C K , E X K [ P [SIR X K > τ K ]] . (11)In the UL, the interfering UEs do not constitute a homoge-neous PPP due to the correlation among the interfering UEs.This correlation is due to the orthogonal channel assignmentwithin a cell and can be better modeled by a soft-core process[26]. However, soft core processes are generally analyticallynot tractable [25]. Therefore, in most of the UL analysisthey approximate it as a single homogeneous PPP (becausein the UL the transmit power of the UEs are the same andthe association regions of BSs form a Voronoi tessellation)[7], [8], [11], [21]. However, in our system model we cannot approximate it as a single homogeneous PPP , due tobiasing and different beamforming gain for femto and macro-tier (the association regions of BSs form a weighted Voronoitessellation). Therefore, we approximate it as two independentPPPs, i.e., femto-tier interfering UEs constitute one homoge-neous PPP while macro-tier interfering UEs constitute anotherhomogeneous PPP. However, we do not approximate theinterfering UEs as PPPs in the entire 2-D plane but the regionsdefined in Remark 1 and 2. The constraints of Remark 1 and2 are taken into consideration in the rest of the analysis.The channel h K follows Gamma ( N K , , therefore, weneed to find the higher order derivative of the Laplace trans-form of the interference, which is a common problem inMIMO transmission in the PPP network. In the literature, dif-ferent techniques have been used to simplify the n th derivativeof the Laplace transform. A Taylor expansion-based approx-imation is used in [31] while [32] uses special functions toapproximate n th derivative of the Laplace transform. However,both of these techniques are applicable to ad-hoc networksonly. For cellular network, a recursive-technique is used in[33], but their final expression is still complicated, therefore,we use Fa ` a di Bruno’s formula [27] to find the n th derivativeof the Laplace transform of the interference.We state the coverage probability of a random user associ-ated to a K th tier BS in the following theorem. L I ( s ) = exp (cid:18) − πsα K − (cid:20) λ K Z ∞ X − α K (1 − η ) K i F (cid:20) , − α K , − α K ; − sX − α K (1 − η ) K i (cid:21) × f X Ki ( X K i ) d X Ki + λ J ζ − /α K Z ∞ X α J /α K − α J (1 − η ) J q F (cid:20) , − α K , − α K ; − sζX − α J (1 − η ) J i (cid:21) f X Jq (cid:0) X J q (cid:1) d X Jq (cid:21)(cid:19) . (13) L I ( s ) = exp (cid:18) − πsα K − (cid:20) λ K Z ∞ X K i F (cid:20) , − α K , − α K ; − s (cid:21) f X Ki ( X K i ) d X Ki + λ J ζ − /α K Z ∞ X α J /α K J q F (cid:20) , − α K , − α K ; − sζ (cid:21) f X Jq (cid:0) X J q (cid:1) d X Jq (cid:21)(cid:19) , (15) Theorem 1.
The UL coverage probability C K of a typical userwhen the serving BS is a K th tier BS and the SIR thresholdis τ K for the system model in Section II is given by C K ( τ K ) = 2 πλ K A K Z ∞ X K exp n − π (cid:16) λ K X K + λ J ( ζ ) /α J × X α K /α J ) K (cid:17)o N K − X n =0 s n ( − n n ! L nI ( s ) d X K , (12) where s = τ K X α K (1 − η ) K , ζ = N J B J N K B K , L I ( s ) is the Laplacetransform of the interference given in (13) , available at thetop of this page. L nI ( s ) represents the n th derivative of the L I ( s ) and to find it we utilize Fa ` a di Bruno’s formula [27] L nI ( s ) = X n ! b ! b ! · · · b n ! L kI ( s ) (cid:18) f ′ ( s )1! (cid:19) b (cid:18) f ′′ ( s )2! (cid:19) b · · · (cid:18) f n ( s ) n ! (cid:19) b n , where f ( s ) is the term inside the exponential of (13) and thesummation is to be performed over all different solutions innon-negative integers b , · · · , b n of b + 2 b + · · · + nb n = n and k = b + · · · + b n . Proof:
See Appendix C.We see that as the number of antennas N K increases,the summation term becomes larger, and after taking the n th derivative, the expression becomes very lengthy. Hence,numerically computing the coverage probability is computa-tionally very expensive. B. Special Cases
The SIR coverage in Theorem 1 can be simplified for thefollowing plausible special cases.
Corollary 1.
The K th tier SIR coverage probability withoutUL power control ( η = 0) is given by (12) while the L I ( s ) simplifies to L I ( s )=exp (cid:18) − πτ K α K − (cid:20) λ K s /α K F (cid:20) , − α K , − α K ; − τ K (cid:21) + λ J ζ − αKαJ s αJ − αKαJ F (cid:20) , − α K , − α K ; − τ K s − α K /α J ζ α K /α J (cid:21)(cid:21)(cid:19) , (14) where s = X α K K and the rest of the variables have the usualmeaning. The coverage probability can be found by evaluating just asingle integral.
Corollary 2.
The C K with full channel inversion ( η = 1) isgiven by (12) while the L I ( s ) simplifies to (15) , availableat the top of this page, where s = τ K while the rest of theparameters remain the same. Corollary 3.
For B K N K = B J N J and α K = α J = α the C K is given by C K ( τ K )= 2 πλ K A K Z ∞ X K exp (cid:8) − πλX K (cid:9) N K − X n =0 s n ( − n n ! L nI ( s )d X K , (16) where λ = λ K + λ J and L I ( s ) is L I ( s ) = exp (cid:18) − πsλα − Z ∞ X − α (1 − η ) i × F (cid:20) , − α , − α ; − sX − α (1 − η ) i (cid:21) f X i ( X i ) d X i (cid:19) . (17)The coverage probability behaves as if the interference isfrom a single tier network with density λ = λ K + λ J . Corollary 4.
For N K = N J , B K = B J , α K = α J = α , τ K = τ J = τ and λ K = λ J = λ then the coverage probabilityis given by C = C K = C J = 2 πλ A Z ∞ X K exp (cid:8) − πλX K (cid:9) × N K − X n =0 s n ( − n n ! L nI ( s ) d X K , (18) where A = A K = A J and L I ( s ) is L I ( s ) = exp (cid:18) − πsλα − Z ∞ X − α (1 − η ) i × F (cid:20) , − α , − α ; − sX − α (1 − η ) i (cid:21) f X i ( X i ) d X i (cid:19) . (19)The network coverage probability C becomes equal to thetier coverage probability C K , C J . Corollary 5.
For η = 0 , B K N K = B J N J , α K = α J = α the C K is given by (16) while the L I ( s ) simplifies to L I ( s )=exp (cid:18) − πτ K s /α λα − F (cid:20) , − α , − α ; − τ K (cid:21)(cid:19) , (20) where s = X αK and λ = λ K + λ J . The coverage probability is in the form of single integraland the interference behaves as if it originates from a singletier network.
Corollary 6.
For η = 0 , N K = 1 , α K = α J = α the C K is C K ( τ K )= λ k A K h λ K + λ J ζ − /α + τ K α − G ( α, τ K , ζ, λ K , λ J ) i , (21) where G( α, τ K , ζ, λ K , λ J ) = λ K F (cid:2) , − α , − α ; − τ K (cid:3) + λ J ζ /α − F h , − α , − α ; − τ K ζ i , and ζ = B K N J B J . The coverage probability reduces to closed form.
Corollary 7.
For η = 0 , N K = N J = 1 B K = B J = 1 , α K = α J = α the the C K can further be simplified to C K ( τ K ) = 11 + τ K α − F (cid:2) , − α , − α ; − τ K (cid:3) . (22)The coverage probability becomes density invariant. C. Rate Coverage Probability
In this subsection, we find the rate coverage probability ofthe network, which is the probability that a randomly chosenuser can achieve a target rate or the average fraction of usersthat achieve the target rate. The rate coverage probability ofthe network can be written as R = A F R F + A M R M , (23)where R F and R M are the rate coverage probability, and A F and A M are the association probability of the femto- andmacro-tier respectively. The rate coverage R K of the K th tierwhen the rate threshold is ρ K can be written as R K , P (cid:20) W Ω K log (1 + SIR K ) > ρ K (cid:21) , (24)where W is the frequency resources and Ω K is the loadon a K th-tier BS. The rate distribution captures the effectof both SIR K and load Ω K , which in turn depends onthe corresponding association area. The distribution of theassociation area is complex and not known. However, by using the association area approximation in [30], the probabilitymass function of the load is given by P (Ω K = n ) = 3 . . ( n − n + 3 . . (cid:18) λ U A K λ K (cid:19) n − × (cid:18) . λ U A K λ K (cid:19) − ( n +3 . , n ≥ , (25)where Γ ( t ) = R ∞ x t − exp ( − x ) dx is a gamma function.We state the rate coverage probability R K in the followingTheorem. Theorem 2.
The R K when the rate threshold is ρ K for thesystem model under consideration is given by R K ( ρ K ) = X n ≥ . . ( n − n + 3 . . (cid:18) λ U A K λ K (cid:19) n − × (cid:18) . λ U A K λ K (cid:19) − ( n +3 . C K (cid:16) ρ K n/W − (cid:17) , (26) where C K is given by (12) .Proof: The rate coverage probability of the K th tier forthreshold ρ K can be written as R K ( ρ K ) = P (cid:20) W Ω K log (1 + SIR K ) > ρ K (cid:21) = P h SIR K > ρ K Ω K /W − i . (27)By the definition of the SIR coverage probability the aboveexpression becomes R K ( ρ K ) = E Ω K h C K (cid:16) ρ K Ω K /W − (cid:17)i = X n ≥ P (Ω K = n ) C K (cid:16) ρ K n/W − (cid:17) . (28)By putting (25) in the above expression, we obtain (26).The rate coverage probability expression in (26) can befurther simplified by using the mean load approximation usedin [30]. The mean load is given by ¯Ω K = E [Ω K ] = 1 + 1 . λ U A K λ K , (29)where K ∈ { M, F } . By using the mean load ¯Ω K thesummation over n is removed from (26).V. R ESULTS AND D ISCUSSION
First, we discuss the accuracy of our analysis and systemmodel. MBSs, FBSs and UEs are deployed according to thesystem model, and we fix P M = 43 dBm, P F = 20 dBm, P = − dBm/Hz, and W = 10 MHz. All the densities λ M , λ F and λ U are per square kilometers / Km . We con-sider the same SIR thresholds ( τ = τ M = τ F ) , rate thresholds ( ρ = ρ M = ρ F ) and path-loss exponents ( α = α M = α F ) forboth tiers.Fig. 2 shows the association probabilities of UEs to differentcases (mentioned in Section II) versus ratio of λ F and λ M , ( λ F /λ M ) , for the given parameters. The solid lines showanalytical results, derived using (4), (5), and (6) while marked λ F / λ M A ss o c i a t i on P r obab ili t y case1 (MBS both DL & UL), Analysiscase1 (MBS both DL & UL), Simulationcase2 (MBS DL, FBS UL), Analysiscase2 (MBS DL, FBS UL), Simulationcase3 (FBS both DL & UL), Analysiscase3 (FBS both DL & UL), Simulation Fig. 2: UL Association probabilities vs. λ F /λ M , ( α = 4 , N M = 5 , N F = 1 , B = 1) . A ss o c i a t i on P r obab ili t y λ F / λ M case1 (MBS both DL & UL), B=1case1 (MBS both DL & UL), B=5case2 (MBS DL, FBS UL), B=1case2 (MBS DL, FBS UL), B=5case3 (FBS both DL & UL), B=1case3 (FBS both DL & UL), B=5 Fig. 3: Effect of biasing on the UL association probabilities, ( α = 4 , N M = 5 , N F = 1 , B = 5) .points are obtained using Monte Carlo simulations. It can benoticed that as the density of the FBS, λ F , increases, thenumber of UEs in case and case also increases, whereas thenumber of UEs in case decreases. It can further be noticedthat initially the association probability of case increases veryrapidly and reaches a maximum value, ( λ F /λ M = 7) , andthen starts decreasing because a larger number of UEs becomeattached to FBSs both in the DL and UL. The figure providesan estimate of the load in different tiers for design engineers.We can observe that at λ F /λ M = 5 , 30 % of the UEs isattached to macro-tier ( case ) while 70 % of UEs is attachedto femto-tier ( case case ), but if we increase N M = 25 andkeep the rest of the parameters the same then 50 % of the UEswill be attached to macro-tier and 50 % to femto-tier (using(7)). This shows that even using DUDe and higher density forthe femto-tier, we still need to balance the load between thetiers. Therefore, we use biasing to balance the load and thenext figure shows the effect of biasing on different UEs’ type.Fig. 3 depicts the effect of biasing on association proba-bilities. It can easily be noticed that by using B = 5 theassociation probability of case increases while the associa- tion probability of case decreases. When B > it offloadsthe boundary UEs of the macro-tier and these UEs becomeattached to femto-tier. Similarly, when B < the boundaryUEs of the femto-tier are offloaded to the macro-tier, whereas B = 1 means no biasing. By changing B we can balance theload among two tiers for optimal performance.Fig. 4 compares the SIR coverage probability obtainedthrough simulations and analysis for various network parame-ters. It can be noticed that the analysis and simulations curvesare close to each other, which shows that the independenthomogeneous PPPs approximation of the interfering UEs isreasonably accurate. The gap between the simulation and thenumerical curve is due to the homogeneous PPP approxima-tion of the interfering UEs. There is some correlation amongthe interfering UEs as discussed in Section IV. However, it isquite challenging to model this correlation. Therefore, in mostof the UL analysis this correlation is ignored [7], [20], [21] and[22]. In [10] and [34] the interfering UEs are approximated asnon-homogeneous PPP in a SISO network model. However,due to multi-antenna BSs in our system model, we need to findthe higher order derivative of the Laplace transform of theinterference, and approximating the interfering UEs as non-homogeneous PPP makes the analysis even more involved.Fig. 5 shows the effect of η on SIR coverage probabilitywhen the cell association is based on maximum downlinkreceived power and when it is based on DUDe. It can beobserved that power control affects the cell-centered (corre-sponds to large SIR threshold) and cell-edged (correspondsto small SIR threshold) UEs differently, i.e., the centeredUEs coverage decreases with power control, whereas thecell-edged UEs coverage increases with the middle value of η = 0 . and with full channel inversion ( η = 1) it decreases.With η = 1 the interference power become significant andhence decreases the overall coverage, therefore, η should beoptimized accordingly. Furthermore, comparing Fig. 5a andFig. 5b reveals that the effect of power control is moreprominent when the association scheme is No-DUDe. Thisis due to the large cell size of the MBSs in the No-DUDeassociation as compared to the cell size of the MBSs in theDUDe association.Fig. 6 shows how the gain provided by the DUDe asso-ciation over No-DUDe association in term of SIR coverageprobability changes with the beamforming gain of both tiers.It is important to mention that the UL coverage probabilityof the network when the association is based on maximumDL received power averaged over fading can be derived bysimilar tools and methods used in this paper. It is clear fromthe figure that the gain of DUDe association over No-DUDeis maximum when both tiers have the same beamforminggain and decreases otherwise. When N M is large comparedto N F , the beamforming gain provided by a MBS increases,which enlarges the association region of a MBS. As a resultof which UEs closer to the FBSs become associated withMBSs. These boundary UEs, which are connected to macro-tier, create strong interference at nearby FBSs when theytransmit to their serving MBSs. Whereas, when both tiers havethe same beamforming gain then the coverage region of bothtiers are the same and the interference created by the boundary −30 −20 −10 0 10 20 3000.10.20.30.40.50.60.70.80.91 SIR Threshold (dB) S I R C o v e r age P r obab ili t y Analytical, η =0Simulation, η =0Analytical, η =1Simulation, η =1 B=0dB (a) λ M = 3 , λ F = 10 , N M = 4 , N F = 2 , α = 3 −30 −20 −10 0 10 20 3000.10.20.30.40.50.60.70.80.91 SIR Threshold (dB) S I R C o v e r age P r obab ili t y Analytical, η =0Simulation, η =0Analytical, η =1Simulation, η =1 B=10dB (b) λ M = 3 , λ F = 10 , N M = 4 , N F = 2 , α = 3 −30 −20 −10 0 10 20 3000.10.20.30.40.50.60.70.80.91 SIR Threshold (dB) S I R C o v e r age P r obab ili t y Analytical, η =0Simulation, η =0Analytical, η =1Simulation, η =1 α =3 (c) λ M = 1 , λ F = 4 , N M = N F = 1 , B = 10 dB −30 −20 −10 0 10 20 3000.10.20.30.40.50.60.70.80.91 SIR Threshold (dB) S I R C o v e r age P r obab ili t y Analytical, η =0Simulation, η =0Analytical, η =1Simulation, η =1 α =4 (d) λ M = 1 , λ F = 4 , N M = N F = 1 , B = 10 dB Fig. 4: SIR coverage probability simulations vs analytical −30 −20 −10 0 10 20 3000.10.20.30.40.50.60.70.80.91 S I R C o v e r age P r obab ili t y SIR Threshold (dB) UL No−DUDe, η =1UL No−DUDe, η =0.5UL No−DUde, η =0 (a) No-DUDe −30 −20 −10 0 10 20 3000.10.20.30.40.50.60.70.80.91 SIR Threshold (dB) S I R C o v e r age P r obab ili t y UL DUDe, η =1UL DUDe, η =0.5UL DUDe, η =0 (b) DUDe Fig. 5: Effect of Power Control fraction η on the SIR coverage Probability, ( λ M = 2 , λ F = 12 , α = 3 , N M = 12 , N F = 4 , B = 1) . −30 −20 −10 0 10 20 3000.10.20.30.40.50.60.70.80.91 SIR Threshold (dB) S I R C o v e r age P r obab ili t y UL (DUDe)UL (No−DUDe) (a) N M = 12 , N F = 12 −30 −20 −10 0 10 20 3000.10.20.30.40.50.60.70.80.91 SIR Threshold (dB) S I R C o v e r age P r obab ili t y UL (DUDe)UL (No−DUDe) (b) N M = 1 , N F = 1 −30 −20 −10 0 10 20 3000.10.20.30.40.50.60.70.80.91 SIR Threshold (dB) S I R C o v e r age P r obab ili t y UL (DUDe)UL (No−DUDe) (c) N M = 12 , N F = 4 −30 −20 −10 0 10 20 3000.10.20.30.40.50.60.70.80.91 SIR Threshold (dB) S I R C o v e r age P r obab ili t y UL (DUDe)UL (No−DUDe) (d) N M = 12 , N F = 1 Fig. 6: Beamforming gain effect on the DUDe gain in term of SIR coverage probability with power control, ( η = 0 . , λ M = 2 , λ F = 12 , α = 3 , B = 1) .UEs is not that strong. Thus, the DUDe gain over No-DUDeis high when both tier have the same beamforming. In otherwords, we can say that as the difference in beamforming gainof both tiers increases, the gain provided by the DUDe overNo-DUDe decreases. Fig. 7 shows the same effect when ULpower control is not utilized.Fig. 8 shows the effect of the number of MBS’s antennasand biasing on rate coverage probability. For no biasing case B = 1 , increasing N M from to decreases the ratecoverage. To explain this effect, we know that the rate coveragedepends on the load on a BS (24). When N M is high, thecoverage region of macro-tier increases and most of the UEsbecome attached to MBSs due to which the macro-tier isoverloaded. Thus the overall rate coverage probability drops.Further, we can see from the figure that when N M = 1 , thenno-biasing gives us the maximum rate coverage, which is inaccordance with the result of [10]. However, for higher N M wesee that biasing improves the rate coverage. From the networkdesign perspective, we see that increasing N M can degradethe rate coverage, therefore, to benefit from a large number ofMBSs’ antennas we need a suitable biasing towards femto-tier.Fig. 9 illustrates the effect of FBSs’ density and path-lossexponent α on the rate coverage probability for the association scheme of DUDe and No-DUDe. It can be observed that bychanging α from to increases the rate coverage probabilityfor both DUDe and No-DUDe, which comes from the decreasein the interference power. It can be further observed that anincrease in λ F increases the rate coverage for the DUDe case.This improvement in the rate coverage comes from the inher-ent property of the DUDe to better handle interference. Onthe other hand, for No-DUDe association scheme, increasing λ F slightly improves the rate coverage for centered UEs (largerate threshold) while decreases the rate coverage of cell-edgedUEs (small rate threshold). When λ F increases then the loadon BS decreases due to which the rate coverage improves forthe cell-centered users. However, with the increase in λ F , thecell size of a BS decreases and by using channel inversion thecell-edged UEs transmit power also reduces, thus the coverageof cell-edge UEs reduces. A. Optimal bias and optimal power control fraction
Fig. 10 shows the effect of biasing on
SIR coverageprobability for η = 0 and η = 1 . For η = 0 the optimalcoverage probability is given by no biasing i.e., B = B F B M = 1 or B = 0 dB as shown by Fig. 10a. The SIR is independentof the load and depends on the density of BSs, path-loss, −30 −20 −10 0 10 20 3000.10.20.30.40.50.60.70.80.91 SIR Threshold (dB) S I R C o v e r age P r obab ili t y UL (DUDe)UL (No−DUDe) (a) N M = 12 , N F = 12 −30 −20 −10 0 10 20 3000.10.20.30.40.50.60.70.80.91 SIR Threshold (dB) S I R C o v e r age P r obab ili t y UL (DUDe)UL (No−DUDe) (b) N M = 1 , N F = 1 −30 −20 −10 0 10 20 3000.10.20.30.40.50.60.70.80.91 SIR Threshold (dB) S I R C o v e r age P r obab ili t y UL (DUDe)UL (No−DUDe) (c) N M = 12 , N F = 4 −30 −20 −10 0 10 20 3000.10.20.30.40.50.60.70.80.91 SIR Threshold (dB) S I R C o v e r age P r obab ili t y UL (DUDe)UL (No−DUDe) (d) N M = 12 , N F = 1 Fig. 7: Beamforming gain effect on the DUDe gain in term of SIR coverage probability without power control, ( η = 0 , λ M = 2 , λ F = 12 , α = 3 , B = 1) . R a t e C o v e r age P r obab ili t y N M =20, B=1N M =20, B=5N M =1, B=1N M =1, B=5 Fig. 8: Effect of number of MBS antennas and biasing on ratecoverage, ( λ M = 3 , λ F = 18 , λ u = 3000 , α = 3 , η = 1) beamforming gain of the BSs, and the SIR threshold τ , andwhen η = 0 then all UEs transmit with the same power. Byusing biasing we force a UE to associate to a BS to whichthe UE connection is not strong and thus the SIR coverageprobability reduces. However, from Fig. 10b we see that when η = 1 the optimal SIR is given by B = 5 dB. With power control the transmit power of a UE is proportional to itsdistance from the BS and the transmit power of the UEs atthe cell-edged is greater than the cell-centered UEs. Further,when the beamforming gain N M of the macro-tier is greaterthan the femto-tier then cell-edge UEs of macro cells transmitwith large power and generate high interference. Therefore,offloading these cell-edged UEs to femto-tier improves theSIR coverage.The rate depends on the load and using appropriate valueof biasing can maximize the rate coverage. To find the closedform expression for the optimal bias is too challenging in oursystem model. However, the optimal value can be found by alinear search. Fig. 11 shows the rate coverage against biasingfor different rate threshold ρ . It is clear from the figure that themaximum rate coverage is given by offloading UEs towardsfemto-tier. However, this optimal bias value changes with ρ .When ρ is small (corresponds to cell-edged UEs) then weneed a small value of B whereas for large ρ (correspondsto cell-centered UEs) then we need more aggressive biasingas shown in Fig. 11a and Fig. 11b, respectively. One canobserve that for B < dB the rate coverage is very low. Whenthe beamforming gain of the macro-tier is high, the coverageregion is also large as compared to femto-tier and biasing R a t e C o v e r age P r obab ili t y UL (DUDe, λ F =9)UL (DUDe, λ F =12)UL (DUDe, λ F =18) α =3 (a) α = 3 R a t e C o v e r age P r obab ili t y UL (DUDe, λ F =9)UL (DUDe, λ F =12)UL (DUDe, λ F =18) α =4 (b) α = 4 R a t e C o v e r age P r obab ili t y UL (No−DUDe, λ F =9)UL (No−DUDe, λ F =12)UL (No−DUDe, λ F =18) α =3 (c) α = 3 R a t e C o v e r age P r obab ili t y UL (No−DUDe, λ F =9)UL (No−DUDe, λ F =12)UL (No−DUDe, λ F =18) α =4 (d) α = 4 Fig. 9: Effect of λ F and α on the rate coverage for DUDe and No-DUDe association ( η = 1 , λ M = 3 , N M = 6 , N F = 2 , B = 5 , λ U = 3000) . −15 −10 −5 0 5 10 150.30.40.50.60.70.80.9 Bias B (dB) S I R C o v e r age P r obab ili t y τ =−5dB τ =0dB τ =5dB (a) η = 0 −15 −10 −5 0 5 10 1500.10.20.30.40.50.60.70.80.9 Bias B (dB) S I R C o v e r age P r obab ili t y τ =−5dB τ =0dB τ =5dB (b) η = 1 Fig. 10: Optimal bias for SIR coverage ( N M = 20 , N F = 2 , λ M = 2 , λ F = 10 , λ U = 3000 , α = 3) −15 −10 −5 0 5 10 1500.10.20.30.40.50.60.70.80.91 Bias B (dB) R a t e C o v e r age P r obab ili t y ρ =5kbps ρ =20kbps ρ =40kbps (a) −15 −10 −5 0 5 10 1500.050.10.150.20.25 Bias B (dB) R a t e C o v e r age P r obab ili t y ρ =100kbps ρ =120kbps ρ =150kbps (b) Fig. 11: Optimal bias for rate coverage ( η = 1 , N M = 20 , N F = 2 , λ M = 2 , λ F = 10 , λ U = 3000 , α = 3) towards macro-tier further increases the coverage region ofMBSs (see Fig. 3). Due to this enlargement of the coverageregion, a large number of UEs becomes attached to the macro-tier and it becomes overloaded, which drops the rate coverageprobability. In [10] it is shown that for SISO network the ULrate coverage is maximized when the association is based onminimum path-loss. However, for MIMO setup this is not thecase. Comparing the UL offloading with the DL one can seethat in the DL we need more aggressive offloading of UEs tothe small cell, because there is a high disparity in both thetransmit powers and beamforming gains of macro and femtoBSs. Whereas, in the uplink the load imbalance is only dueto the difference in the beamforming gain of the macro andfemto BSs.Fig. 12 shows the rate coverage against η . The powercontrol fraction η affects the cell-edged and cell-centered UEsdifferently. For cell-edged UEs the optimal rate coverage isgiven by the median value of η is shown in Fig. 12a. Whereasfor cell-centered UEs the optimal rate coverage is given bywithout uplink power control η = 0 as shown in Fig. 12b.Therefore, based on the target rate threshold, the appropriatevalue of η can be chosen to optimize the rate coverage.VI. C ONCLUSION
Using tools from stochastic geometry, the UL performanceof a two-tier random network is studied, where the cell asso-ciation is based on DL and UL decoupling. Multiple antennasare considered at BSs, and single antennas are considered atUEs. The position of the MBSs, FBSs, and UEs are modeledusing a 2-D PPP. Maximal ratio combining has been usedat the MBS and tractable analytical expressions have beenderived for the rate and SIR coverage probability. It has beenshown that the gain (in term of SIR coverage probability) ofthe decoupled DL and UL association over the coupled DLand UL association is maximum when both tiers have thesame number of antennas (same beamforming gain). It has alsobeen observed that in order to leverage the benefits of multipleantennas in DUDe network, offloading of UEs to small cellis required. A future extension might consider to study theperformance of both DL and UL for MIMO network, and to find the optimal offloading strategy, which jointly optimizesboth the DL and the UL performance. To investigate thepotential gain offered by using multiple antennas BSs andusing interference cancellation would also be an interestingresearch direction. A
PPENDIX A Proof of Lemma 1:
The association criterion when a typicalUE connects to a MBS both in the UL and DL is given by P hn P M E n k h M k o X − α M M > P F E n k h F k o X − α F F o \n B M E n k h M k o X − α M M > B F E n k h F k o X − α F F oi where the expectation is over the channel fading. The E n k h M k o = N M , E n | h F | o = N F , where N M , and N F are the array gains and represent the number of antennasat a MBS and FBS respectively [28]. B F and B M are biasfactors toward femto-tier and macro-tier respectively. Theabove equation can be equivalently written as P h(cid:8) P M N M X − α M M > P F N F X − α F F (cid:9) \(cid:8) B M N M X − α M M > B F N F X − α F F (cid:9)(cid:3) . We know that P F < P M and when B F B M ≥ P F P M , it canbe easily observed that the common region in the aboveequation is N M X − α M M > B F B M N F X − α F F , or equivalently X F > (cid:16) B F N F B M N M (cid:17) (1 /α F ) X α M /α F M . Similarly, when B F B M < P F P M then the common region is X F > (cid:16) P F N F P M N M (cid:17) (1 /α F ) X α M /α F M and the probability is calculated as P ( case P ( X F > a )= Z ∞ (1 − F X F ( a )) f X M ( X M )d X M , where a = Υ /α F X α M /α F M , while for B F B M ≥ P F P M , Υ = B F N F B M N M and for B F B M < P F P M , Υ = P F N F P M N M . Using thenull probability of 2D PPP, F X F ( X M ) = 1 − e − πλ F X M , f X M ( X M ) = 2 πλ M X M e − πλ M X M and evaluating the inte-gral we obtain (4). η R a t e C o v e r age P r obab ili t y ρ =10kbps ρ =15kbps ρ =20kbps (a) η R a t e C o v e r age P r obab ili t y ρ =100kbps ρ =120kbps ρ =150kbps (b) Fig. 12: Optimal η for rate coverage ( N M = 4 , N F = 2 , λ M = 2 , λ F = 10 , λ U = 3000 , α = 3) A PPENDIX B Proof of Lemma 4:
The distance X K between a typicalUE and the tagged BS is a random variable (r.v). The event X K > x is equivalent to the event that X K > x given that atypical user is attached to the K th tier (proof follows similarmethod as in [13]) P [ X K > x ] = P [ X K > x | n = K ] = P [ X K > x, n = K ] A K , (30)where P [ n = K ] = A K is the tier association probabilitygiven (7). Let P r K and P r J be respectively the receivedpower from a typical UE at the nearest K th tier and J th tierBS then the joint probability P [ X K > x, n = K ] is P [ X K > x, n = K ] = P [ X K > x, P r K ( X K ) > P r j ]= Z ∞ x P (cid:2) B K N K X − α K K > B J N J X − α J J (cid:3) f X K ( X K ) d X K = Z ∞ x P " X J > (cid:18) B J N J B K N K (cid:19) /α J X α K /α J K f X K ( X K ) d X K . From the 2D null probability of PPP we obtain, P (cid:20) X J > (cid:16) B J N J B K N K (cid:17) /α J X α K /α J K (cid:21) =exp (cid:26) − πλ J (cid:16) B J N J B K N K (cid:17) /α J (cid:16) X α K /α J K (cid:17) (cid:27) , and f X K ( X K ) = 2 πλ K X K exp (cid:8) − πλ K X K (cid:9) , and pluggingin the above equation we get P [ X K > x, n = K ] = 2 πλ K Z ∞ x X K × exp ( − π λ K X K + λ J (cid:18) B J N J B K N K (cid:19) /α J ! (cid:16) X α K /α J K (cid:17) ) d X K . (31) By plugging (31) in (30) we get P [ X K > x ] = 2 πλ K A K Z ∞ x X K × exp ( − π λ K X K + λ J (cid:18) B J N J B K N K (cid:19) /α J !(cid:16) X α K /α J K (cid:17) ) d X K , (32)which is the complementary cumulative distribution function(CCDF) of X K , while it CDF is F X K ( x ) = 1 − P [ X K > x ] ,and probability density function (pdf) is f X K ( x ) = ddx F X K ( x ) , we obtain (9).A PPENDIX C Proof of Theorem 1:
We consider multiple antenna BSs anduse MRC combining, therefore, the signal channel follows
Gamma ( N K , , whereas the interfering channel still followsexponential distribution [24]. Let X K be the distance betweena typical UE and its serving K th tier BS then the coverageprobability C K for a given threshold can be written as C K ( τ K ) , E X K [ P [SIR X K > τ | X K ]]= Z ∞ P [SIR X K > τ K | X K ] f X K ( X K ) d X K = 2 πλ K A K Z ∞ P [SIR X K > τ K | X K ] X K × exp ( − π λ K X K + λ J (cid:18) B J N J B K N K (cid:19) /α J X α K /α J ) K !) d X K , (33)where the last expression follows by plugging f X K ( . ) from(9). For interference limited network the P [SIR X K > τ K | X K ] L I ( s ) = E I (cid:2) e − sI K (cid:3) a = E g i ,X Ki ,D Ki exp − s X i ∈ Φ ′ K \ u g i X α K ηK i D − α K K i E g q ,X Jq ,D Jq exp − s X q ∈ Φ ′ J g q X α J ηJ q D − α K J q b = E X Ki ,D Ki Y i ∈ Φ ′ K \ u E g i (cid:2) exp (cid:0) − sg i X α K ηK i D − α K K i (cid:1)(cid:3) E X Jq ,D Jq Y q ∈ Φ ′ J E g q h exp (cid:16) − sg q X α J ηJ q D − α K J q (cid:17)i c = E D Ki Y i ∈ Φ ′ K \ u E X Ki "
11 + sX α K ηK i D − α K K i E D Jq Y q ∈ Φ ′ J E X Jq "
11 + sX α J ηJ q D − α K J q d = exp (cid:18) − πλ K Z ∞ X K (cid:18) − E X Ki (cid:20)
11 + sX α K ηK i u − α K (cid:21)(cid:19) u du (cid:19) exp − πλ J Z ∞ (cid:18) NJBJXαKKNKBK (cid:19) /αJ − E X Jq "
11 + sX α J ηJ q v − α K v dv e = exp − πλ K Z ∞ X K Z u
11 + s − X − α K ηK i u α K f X Ki ( X K i ) d X Ki ! u du ! × exp − πλ J Z ∞ (cid:18) NJBJXαKKNKBK (cid:19) /αJ Z (cid:16) NJBJ vαKNKBK (cid:17) /αJ
11 + s − X − α J ηJ q v α K f X Jq (cid:0) X J q (cid:1) d X Jq v dv f = exp − πλ K Z X K s /α K X ηK i Z ∞ s − /αK X − η ) Ki
11 + Z α K / K d Z K ! f X Ki ( X K i ) d X Ki ! × exp − πλ J Z ζ αJ + αKα J X α K/α JK s /α K X α J η/α K J q Z ∞ ζ − /αK s − /αK X αJ (1 − η ) /αKJq
11 + Z α K / J d Z J ! f X Jq (cid:0) X J q (cid:1) d X Jq g = exp − πλ K sα K − Z X K X − α K (1 − η ) K i F (cid:20) , − α K , − α K ; − sX − α K (1 − η ) K i (cid:21) f X Ki ( X K i ) d X Ki ! × exp − πλ J ζ − /α K sα K − Z ζ αJ + αKα J X α K/α JK X α J /α K − α J (1 − η ) J q F (cid:20) , − α K , − α K ; − sζX − α J (1 − η ) J i (cid:21) f X Jq (cid:0) X J q (cid:1) d X Jq (35)can be written as P [SIR X K > τ K | X K ] = P k h K k X α K ( η − K P i ∈ Φ ′ K \ u (cid:12)(cid:12)(cid:12)(cid:12) h HK h Ki k h K k (cid:12)(cid:12)(cid:12)(cid:12) X α K ηK i D − α K K i + P q ∈ Φ ′ J (cid:12)(cid:12)(cid:12)(cid:12) h HK h Jq k h K k (cid:12)(cid:12)(cid:12)(cid:12) X α J ηJ q D − α K J q = P h k h K k > sI | X K i = E I " N K − X n =0 s n I n e − sI = N K − X n =0 s n ( − n n ! d n d s n L I ( s ) (34)where s = τ K X α K (1 − η ) K , I = P i ∈ Φ ′ K \ u g i X α K ηK i D − α K K i + P q ∈ Φ ′ J g q X α J ηJ q D − α K J q , g i = (cid:12)(cid:12)(cid:12)(cid:12) h HK h Ki k h K k (cid:12)(cid:12)(cid:12)(cid:12) , and g q = (cid:12)(cid:12)(cid:12)(cid:12) h HK h Jq k h K k (cid:12)(cid:12)(cid:12)(cid:12) . (1) follows due to the definition of SIR , (2) follows due to h K ∼ Gamma ( N K , , and (3) follows due to the Laplacetransform identity L (cid:8) I n e − sI (cid:9) = ( − n d n d s n L I ( s ) .Now, we find the Laplace transform L I ( s ) of the inter-ference, which can be written as in (35), available at thetop of this page, where ( a ) follows because the interfer-ence is from both the femto-tier and macro-tier’s scheduledusers, and also they are independent of each other, ( b ) isdue to the i.i.d assumption of g i and g q , and both g i and g q are further independent of point process Φ , ( c ) is dueto g i ∼ exp (1) and g q ∼ exp (1) , ( d ) follows due tothe probability generating functional (PGFL) of PPP, whichconvert an expectation over a point process to an integral E (cid:2)Q x ∈ Φ f ( x ) (cid:3) = exp (cid:0) − λ R R (1 − f ( x )) dx (cid:1) . It is impor-tant to mentioned that in step ( d ) the integration limits in bothof the integrals are not the same, i.e., the closest interferer ofthe serving tier can be at a distance X K from the typical BS,whereas the closest interferer of the non serving tier should beat a distance (cid:16) N J B J X αKK N K B K (cid:17) /α J , as mentioned in Remark 2. Instep ( e ) , we apply the inner expectations, which are required for the power control. Again, it is important to note that thedistance distribution of an interfering UE to its serving BS isdifferent from that of the typical UE to the tagged BS and fordifferent tiers the distance distribution of an interfering UE toits serving BS are also different, as mentioned in Remark 1.This difference can be seen by the limits of the inner integralin both exponential. ( f ) follows by changing the integrationorder, putting ζ = N J B J N K B K and some manipulations while ( g ) follows by writing the inner integrals as Gauss hypergeometricfunctions [29]. We combine the two exponential and pluggingit in (34) and then (34) into (33). Thus the proof is completed.A CKNOWLEDGMENT
The authors gratefully acknowledge the excellent feedbackprovided by the anonymous reviewers.R
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