Tier-Aware Resource Allocation in OFDMA Macrocell-Small Cell Networks

Transcription

1 1 Tier-Aware Resource Allocatio i OFDMA Macrocell-Small Cell Networks Amr Abdelasser, Ekram Hossai, ad Dog I Kim arxiv:145.2v1 [cs.ni] 8 May 214 Abstract We preset a joit sub-chael ad power allocatio framework for dowlik trasmissio a orthogoal frequecydivisio multiple access (OFDMA)-based cellular etwork composed of a macrocell overlaid by small cells. I this framework, the resource allocatio (RA) problems for both the macrocell ad small cells are formulated as optimizatio problems. For the macrocell, we formulate a RA problem that is aware of the existece of the small cell tier. I this problem, the macrocell performs RA to satisfy the data rate requiremets of macro user equipmets (MUEs) while imizig the tolerable iterferece from the small cell tier o its allocated sub-chaels. Although the RA problem for the macrocell is show to be a mixed iteger oliear problem (MINLP), we prove that the macrocell ca solve aother alterate optimizatio problem that will yield the optimal solutio with reduced complexity. For the small cells, followig the same idea of tier-awareess, we formulate a optimizatio problem that accouts for both RA ad admissio cotrol (AC) ad aims at imizig the umber of admitted users while simultaeously miimizig the cosumed badwidth. Similar to the macrocell optimizatio problem, the small cell problem is show to be a MINLP. We obtai a upper boud o the optimal solutio with reduced complexity through covex relaxatio. I additio, we employ the dual decompositio techique to have a distributed solutio for the small cell tier. Numerical results cofirm the performace gais of our proposed RA formulatio for the macrocell over the traditioal resource allocatio based o miimizig the trasmissio power. Besides, it is show that the formulatio based o covex relaxatio yields a similar behavior to the MINLP formulatio. Also, the distributed solutio coverges to the same solutio obtaied by solvig the correspodig covex optimizatio problem i a cetralized fashio. Keywords:- Base statio desificatio, small cells, OFDMA, dowlik resource allocatio, sub-chael ad power allocatio, admissio cotrol, covex optimizatio, dual decompositio. I. INTRODUCTION As more ad more customers subscribe to mobile broadbad services, there is a tremedous growth i the demad for mobile broadbad commuicatios, together with the icreased requiremets for higher data rates, lower latecies ad ehaced quality-of-service (QoS). Fueled by the popularity of smartphoes ad tablets with powerful multimedia capabilities, services ad applicatios, it is aticipated that by 22, the existig wireless systems will ot be able to accommodate the expected 1-fold icrease i total mobile broadbad data [2]. Therefore, 5G cellular techologies are beig sought. Of A. Abdelasser ( asra@cc.umaitoba.ca) ad E. Hossai ( Ekram.Hossai@umaitoba.ca) are with the Departmet of Electrical ad Computer Egieerig at the Uiversity of Maitoba, Caada. D. I. Kim ( dikim@skku.ac.kr) is with the School of Iformatio ad Commuicatio Egieerig at the Sugkyukwa Uiversity (SKKU), Korea. A prelimiary versio of the paper [1] has bee submitted to IEEE Globecom 14. the several eablig techologies for 5G to hadle the expected traffic demad, base statio (BS) desificatio is cosidered as oe of the most promisig solutios [3]. BS desificatio ivolves the deploymet of a large umber of low power BSs. This decreases the load per BS ad leads to a better lik betwee a user equipmet (UE) ad its servig BS owig to a smaller distace betwee them [4]. This BS desificatio also creates a multi-tier etwork of odes with differet trasmit powers, coverage areas ad loads. Two issues arise i such a dese multi-tier etwork. The first issue is that the resource allocatio (RA) i oe tier caot be doe i isolatio of the resource allocatio i aother tier. I other words, oe tier should take ito cosideratio the cosequeces of its RA decisios o the other tiers. The secod issue is that cetralized RA solutios may ot be feasible. Hece, there is a eed for decetralized solutios for RA i differet etwork tiers. I this paper, we formulate the RA problem for a twotier orthogoal frequecy divisio multiple access (OFDMA) wireless etwork composed of a macrocell overlaid by small cells. The objective of the macrocell is to allocate resources to its macro UEs (MUEs) to satisfy their data rate requiremets. I additio, kowig about the existece of small cells, the macrocell allocates the radio resources (i.e., sub-chael ad power) to its MUEs i a way that ca sustai the highest iterferece level from the small cells. For this reaso, we formulate a optimizatio problem for the macrocell with a objective that is differet from those i the traditioal RA problems. Now, sice small cells create dead zoes aroud them i the dowlik directio, the MUEs should be protected agaist trasmissio from the small cells [3], [5]. Hece, kowig about the imum allowable iterferece levels for MUEs, the small cells perform RA by solvig a optimizatio problem whose objective fuctio combies both the admissio cotrol (AC) ad the cosumed badwidth (i.e., umber of allocated sub-chaels). The objective of the small cell tier is to admit as may small cell UEs (SUEs) as possible at their target data rates ad cosume the miimum amout of badwidth. Agai, this follows the same otio of tier-awareess by leavig as much badwidth as possible for other etwork tiers (e.g., for device-to-device (D2D) commuicatio). For this, a optimizatio problem is formulated for the small cell tier with the aforemetioed objective, give the QoS requiremets of SUEs ad the iterferece costraits for the MUEs. Dual decompositio is used to have a decetralized RA ad AC problem by decomposig the optimizatio problem ito subproblems for each small cell to solve. For this, oly local chael gai iformatio is used alog with some coordiatio

2 2 with the Home enb Gateway (HeNB-GW) [6]. The key cotributios of this paper ca be summarized as follows: We develop a complete framework for tier-aware resource allocatio i a OFDMA-based two-tier macrocell-small cell etwork with ew objectives, which are differet from the traditioal sum-power or sum-rate objectives. For the macrocell tier, we formulate a resource allocatio problem that is aware of the existece of the small cell tier ad show that it is a mixed iteger oliear program (MINLP). We prove that the macrocell ca solve aother alterate optimizatio problem that yields the optimal solutio for the MINLP with polyomial time complexity. We compare the proposed method for the macrocell RA problem to the traditioal miimize the total sumpower problem ad show that the proposed method outperforms the traditioal oe i terms of the average umber of admitted SUEs. For the small cell tier, we formulate a joit resource allocatio ad admissio cotrol problem that aims at imizig the umber of admitted SUEs ad miimizig their badwidth cosumptio to accommodate additioal tiers, ad show that it is a MINLP. We offer a upper boud solutio to the MINLP through covex relaxatio ad propose a solutio to the covex relaxatio that ca be implemeted i a distributed fashio usig dual decompositio. The rest of this paper is orgaized as follows. Sectio II reviews the related work. Sectio III presets the system model ad assumptios for this work. I Sectio IV, the optimizatio problems are formulated for both the macrocell tier ad the small cell tier, followed by the use of dual decompositio to have a decetralized operatio. Numerical results are discussed i Sectio V ad fially Sectio VI cocludes the work. A summary of the importat symbols ad otatios used i the paper is give i Table I. II. RELATED WORK The RA problem i OFDMA-based multi-tier cellular etworks has bee extesively studied i the literature. The authors i [7] studied the RA problem i a multi-tier cellular etwork to imize the sum-throughput subject to simple power budget ad sub-chael allocatio costraits. However, o QoS costraits were imposed. I [8], the RA problem i a femtocell etwork was modeled, with iterferece costraits for MUEs, i order to achieve fairess amog femtocells. No QoS costraits, however, were imposed for femtocell users. I [9], the RA problem i a two-tier macrocell-femtocell OFDMA etwork was modeled as a Stackelberg game, where the macrocell acts as the leader ad the femtocells act as the followers. However, o iterferece costraits for MUEs were cosidered. Also, o QoS costraits were imposed for femtocells. Referece [1] studied the RA problem i a twotier etwork composed of macrocells ad femtocells which aimed at imizig the sum-throughput of femtocells subject to total sum-rate costrait for the macrocell. Nevertheless, o QoS costraits were imposed for femtocells. The authors i [11] studied the RA problem with QoS ad iterferece costraits i a two-tier cellular etwork ad used clusterig as a techique to reduce the overall complexity. I the above works, either o QoS costraits were imposed or the RA problems with QoS costraits were assumed feasible. I other words, admissio cotrol [12], which is a techique to deal with ifeasibility whe it is ot possible to support all UEs with their target QoS requiremets, was ot studied. The authors i [13] proposed a distributed selforgaizig RA scheme for a femtocell oly etwork, with the aim of miimizig the total trasmit power subject to QoS costraits. It was show that miimizig the trasmit power (which results i reduced iterferece) may improve throughput. Several works i the literature have cosidered the admissio cotrol problem. For cellular cogitive radio etworks, [14] studied the problem of admissio ad power cotrol to admit the imum umber of secodary liks ad imize their sum-throughput subject to QoS requiremets ad iterferece costraits for primary liks. However, power cotrol was doe cetrally. The authors i [15] cosidered the problem of admissio ad power cotrol, where the primary users are guarateed a premium service rate ad the secodary users are admitted (as may as possible) so log as the primary users are ot affected. I [16], the authors proposed a joit rate ad power allocatio scheme with explicit iterferece protectio for primary users ad QoS costraits for secodary users, where admissio cotrol was performed cetrally. However, [14]-[16] oly cosidered sigle-chael systems. The authors i [17] studied the problem of joit rate ad power allocatio with admissio cotrol i a OFDMA-based cogitive radio etwork subject to QoS requiremets for secodary users ad iterferece costraits for primary users. However, resource allocatio ad admissio cotrol were performed cetrally. I additio, chaels were radomly allocated to secodary users. I relay etworks, [18] studied the problem of power allocatio i amplify ad forward wireless relay systems for differet objectives, where admissio cotrol was employed as a first step precedig power cotrol. However, oly oe chael was cosidered. I additio, power ad admissio cotrol were doe cetrally. I [19], a joit badwidth ad power allocatio for wireless multi-user etworks with admissio cotrol i relay etworks was proposed for differet system objectives. Uequal chuks of badwidths were allocated. However, the resource allocatio was performed cetrally. For a two-tier small cell etwork, [2] studied joit admissio ad power cotrol. Small cells are admitted ito the etwork so log as the QoS of macrocell users is ot compromised. Admissio ad power cotrol were performed i a distributed fashio. However, oly a sigle chael system was cosidered. Referece [21] proposed a distributed admissio cotrol mechaism for load balacig amog sub-carriers with multiple QoS classes. I additio, small cells mitigate co-tier ad cross-tier iterfereces usig slot allocatio of differet traffic streams amog differet sub-carriers. However, o power allocatio was performed. We otice that oe of the quoted works cosiders the

3 3 TABLE I SUMMARY OF THE IMPORTANT SYMBOLS AND NOTATIONS Symbol B d d F F f F s g i,j g g s I m I I th I th,l ad I th,h I th,m L M M m N N N a,c N m N o P i,j P i,j P B, P s, R B R m R f S S s y s,f α γ B,m γ s,f Γ i,j f δ η η ɛ Descriptio Idex of the macrocell Sub-gradiet Elemet of the sub-gradiet d Set of all SUEs Number of SUEs Idex of a SUE Set of all SUEs served by small cell s Chael gai of the lik betwee UE j served by BS i o sub-chael Lagrage dual fuctio Lagrage dual fuctio for small cell s Maximum tolerable iterferece level o sub-chael allocated to MUE m Upper limit o the imum tolerable iterferece level I m Equal value for the imum tolerable iterferece level I m, m M, N Lower ad upper limits o I th i the bisectio method Mea of I th,l ad I th,h Lagragia fuctio Set of all MUEs Number of MUEs Idex of a MUE Set of all available sub-chaels Number of sub-chaels Idex of a sub-chael Number of allocated sub-chaels Set of all sub-chaels allocated to MUE m Noise power Power allocated to the lik betwee UE j served by BS i o sub-chael Actual power allocated to the lik betwee UE j served by BS i o sub-chael Total macrocell power Total small cell power Coverage radius of macrocell B Data rate requiremet of MUE m Data rate requiremet of SUE f Set of all small cells Number of small cells Idex of a small cell Admissio cotrol variable for SUE f served by small cell s Scale factor for small cells trasmissio powers o sub-chael Received SINR of a MUE m served by macrocell B o sub-chael Received SINR of a SUE f served by small cell s o sub-chael Sub-chael allocatio idicator for sub-chael allocated to UE j served by BS i Sub-chael badwidth Termiatio tolerace i bisectio method Lagrage multiplier associated with the cross-tier iterferece Elemet of the Lagrage multiplier η Weightig factor iteractio betwee the differet etwork tiers ad the cosequeces of RA decisios of oe tier o the other oe. I additio, it is desirable to have a RA ad AC scheme that is implemetable i a distributed fashio i a dese multi-tier OFDMA etwork. Table II summarizes the related work ad their differeces from the work preseted i this paper. III. SYSTEM MODEL, ASSUMPTIONS, AND RESOURCE ALLOCATION FRAMEWORK A. System Model ad Assumptios We cosider the dowlik of a two-tier cellular etwork, where a sigle macrocell, referred to by the idex B ad with coverage radius R B, is overlaid with S small cells. Deote by S the set of small cells, where S = S. A closed-access scheme is assumed for all small cells, where access to a small cell is restricted oly to the registered SUEs. All small cells are coected to the mobile core etwork. For example, femtocells ca coect to the core etwork by usig the DSL or CATV modems via a itermediate etity called the Femto Gateway (FGW) or HeNB-GW [6] which ca take part i the resource allocatio operatio for femtocells. We deote by M the set of MUEs served by the macrocell B with M = M. Each MUE m has a data rate requiremet of R m. I additio, deote by F the set of SUEs i the system with F = F. Each SUE f has a data rate requiremet of R f. We refer to the set of SUEs served by small cell s by F s. We assume that all UEs are already associated with their BSs ad that this associatio remais fixed durig the rutime of the resource allocatio process. We have S F s = F ad S F s = φ. All MUEs exist outdoor ad all SUEs exist idoor. We have a OFDMA system, where we deote by N the set of available sub-chaels with N = N ad f is the badwidth of a sub-chael. Uiversal frequecy reuse is assumed, where the macrocell ad all the small cells have access to the set of sub-chaels N. Γ i,j is the sub-chael allocatio idicator, i.e., Γ i,j = 1, if sub-chael is allocated to UE j served by BS i ad takes the value of otherwise. The UEs are capable of usig two modes of sub-chael allocatio, amely, the exclusive mode ad the time sharig mode. For the exclusive mode, i a give trasmissio frame, sub-chael is used by oe UE oly. I the time sharig mode, a sub-chael is allocated to a certai UE a portio of the time. I this way, multiple UEs ca time share a subchael i a give trasmissio frame [22]. Deote by Pi,j ad g i,j the allocated power to ad the

4 4 TABLE II SUMMARY OF RELATED WORK Previous Network type Objective fuctio Multi- AC QoS Distributed Impact of works chael costraits solutio oe tier o aother [7] Two-tier macrocell/ Maximize sum-rate Yes No No Yes No small cell etwork for two tiers [8] Two-tier macrocell/ Maximize sum-mi Yes No For Yes No small cell etwork rate for small cells MUEs [9] Two-tier macrocell/ Maximize sum-rate Yes No For Yes No small cell etwork for two tiers MUEs [1] Two-tier macrocell/ Maximize sum-rate Yes No For Yes No small cell etwork for small cells MUEs [11] Two-tier macrocell/ Maximize sum-rate Yes No For Semi- No small cell etwork for small cells SUEs distributed [13] Sigle-tier small Miimize sum-power Yes No Yes Yes No cell etwork [14] Cogitive radio Maximize umber of liks No Yes Yes No No etworks with sum-rate [15] Cogitive radio Maximize umber of users No Yes Yes Yes No etworks with mi sum-power [16] Cogitive radio Maximize mi-rate ad No Yes Yes No No etworks imize sum-log rate [17] Cogitive radio Maximize sum-rate Yes Yes Yes No No etworks [18] Relay Maximize mi-sinr, mi No Yes Yes No No etworks -power ad sum-rate [19] Relay Maximize sum-rate, Yes Yes Yes No No etworks mi-rate ad miimize sum-power [2] Two-tier macrocell/ Miimize sum-power with No Yes For MUEs Yes No small cell etwork umber of SUEs ad SUEs [21] Two-tier macrocell/ Maximize product of Yes Yes For SUEs Yes No small cell etwork miimum of (2 target rate - achieved rate) ad achieved rate Our Two-tier macrocell/ Maximize sum-tolerable Yes Yes For MUEs Yes Yes proposed small cell etwork iterferece for MUEs ad SUEs scheme ad imize admitted SUEs with miimum badwidth chael gai of the lik betwee BS i ad UE j o sub-chael. Chael gais are time varyig ad accout for path-loss, log-ormal shadowig, ad fast fadig. The chael gais are assumed to remai static durig the resource allocatio process. The received sigal to iterferece plus oise ratio (SINR) γb,m of a MUE m served by macrocell B o a subchael is defied as: γ B,m = P B,m g B,m I m + N o (1) where I m is the imum tolerable iterferece level at MUE m o sub-chael ad N o is the oise power. Accordig to (1), the followig costrait holds for small cell trasmissio powers o sub-chael : Γ B,m Γ s,f Ps,f g s,m Γ B,mIm (2) where the costrait is active oly if sub-chael is allocated to MUE m, i.e., Γ B,m = 1. Similarly, we ca defie the received SINR γs,f of a SUE f served by small cell s o a sub-chael as: γ s,f = Ps,f g s,f M m=1 Γ B,m P B,m g B,f + N. (3) o I (3), we cosider cross-tier iterferece from macrocell B. 1 O the other had, co-tier iterferece from other small cells is assumed to be a part of the oise power N o due to the wall peetratio loss ad their relatively low trasmissio powers [23]. B. Tier-Aware Resource Allocatio Framework Fig. 1 describes the RA framework proposed i this paper. Give the rate requiremets for the MUEs, the macrocell starts by allocatig resources to its MUEs ad specifies the imum tolerable iterferece levels o each allocated subchael. The macrocell the seds this RA iformatio to the HeNB-GW which broadcasts it to the small cells. The small cells the perform RA ad AC for its SUEs. For the resultig resource allocatio for small cells, the MUEs perform iterferece measuremets ad report them to the macrocell BS. 1 Note that it is straightforward to accout for iterferece from other macrocells i (1) ad (3). Nevertheless, sice we are focusig o the iteractio betwee RA decisios of the macrocell tier represeted by macrocell B ad the overlayig small cells tier, iterferece from other macrocells will appear as a costat term i (1) ad (3). Besides, I m will represet the imum tolerable iterferece from the small cell tier.

5 5 Fig. 1. No Macrocell performs RA for MUEs to satisfy their miimum rate requiremets while imizig the tolerable iterferece from small cells Macrocell seds the RA iformatio of MUEs to HeNB-GW HeNB-GW broadcasts the RA iformatio of MUEs to small cells Small cells perform RA ad AC to imize the umber of admitted SUEs while satisfyig their miimum rate requiremets usig miimum amout of resources MUEs report iterferece measuremets to macrocell which updates the HeNB-GW MUEs iterferece costraits are satisfied ad covergece achieved? Stop Yes The RA framework for the macrocell ad the small cells. IV. PROBLEM FORMULATIONS FOR RESOURCE ALLOCATION A. Problem Formulatio for Macrocell The macrocell is resposible for providig the basic coverage for the MUEs [24]. Hece, the target of the macrocell is to allocate the resources to its MUEs to satisfy their data rate requiremets ad specify the imum tolerable iterferece level by its MUEs o the allocated sub-chaels. Differet methods (i.e., correspodig to optimizatio problems for resource allocatio with differet objective fuctios), as will be show later, ca be followed to accomplish this task. It is of iterest to study ad uderstad the effects of differet RA methods o other etwork tiers, which will be the small cell tier i our case. 1) Maximize the sum of tolerable iterferece levels: Oe way of performig RA i the macrocell is to allocate resources to the MUEs i a way that imizes the sum of the imum tolerable iterferece levels o the allocated subchaels. The motivatio behid this objective is to allow the imum possible freedom for the small cell tier i usig the sub-chaels. I this cotext, equal trasmit power is assumed o the allocated sub-chaels i the macrocell, i.e., PB,m = P B, N ac [25], where P B, is the total macrocell power ad N ac N is the umber of allocated sub-chaels. Deote by N m the set of sub-chaels allocated to MUE m. Hece, we ca defie the optimizatio problem i (4), where the objective is to imize the sum of the tolerable iterferece levels Im for all MUEs m o all sub-chaels. C1 is the data rate costrait for each MUE m. C2 ad C3 idicate that the sets of sub-chaels allocated to the MUEs are disjoit ad costitute the etire set of sub-chaels N. C4 is a costrait added for umerical purposes, where I, is a very large umber ad Im = I meas that sub-chael is ot actually allocated to MUE m. Fially, C5 idicates that Im should be positive. The macrocell BS the updates the HeNB-GW ad the cycle repeats util the iterferece thresholds for all the MUEs are ot violated ad the RA ad AC coverge for all small cells. This cycle repeats due to the distributed ature of RA ad AC i small cells. This repetitio, however, does ot take place if RA ad AC i small cells are performed by a cetral cotroller. Note that the resource allocatio i the macrocell from the first step remais fixed throughout the etire operatio of the resource allocatio process i the macrocell ad small cells. The awareess of the macrocell about the small cell tier is reflected i the way the radio resources are allocated i the macrocell. The macrocell allocates resources to its MUEs i a way that ca tolerate the imum iterferece possible from the samll cell tier. Note, however, that the miimum rate costraits of all MUEs must be satisfied i the sese that the rate requiremet for oe of the MUEs is compromised for admittig ew SUEs. O the other had, the awareess of the small cell tier about the existece of other tiers is reflected i the fact that the resource allocatio i the small cell tier satisfies the rate requiremets of the SUEs usig the miimum amout of badwidth resources. M I {Im } m m=1 N m subject to C1 : f log 2 N m C2 : N i Nj =, i, j M C3 : M m=1 N m {1, 2,..., N} C4 : I m I, m M, N ( 1 + P B,m ) g B,m Im R m, m M + N o C5 : I m, m M, N. (4) I geeral, (4) is a MINLP whose feasible set is ocovex due to C1 ad the combiatorial ature of subchael allocatio. Besides, PB,m is ukow as the umber of allocated sub-chaels N ac is ot kow yet. However, by carefully ispectig (4), some iterestig features ca be revealed which lead to the possibility of obtaiig the optimal

6 6 solutio of (4) with polyomial time complexity. We shall assume first that (4) is always feasible ad that i the extreme case, a MUE ca have its rate requiremet satisfied with oe sub-chael oly. The last assumptio is possible thaks to the fact that the macrocell i our model has a cotrol o the imum iterferece level o the allocated sub-chael. The followig Lemmas reveal some of the iterestig features of (4). Lemma 1. At optimality, all data rate costraits C1 hold with equality. Proof: Sice the objective fuctio i (4) is mootoically icreasig i I m ad C1 is mootoically decreasig i I m, C1 must hold with equality at optimality for all MUEs. Lemma 2. At optimality, each MUE m is assiged a sigle sub-chael i with I i m < I. Proof: To establish this result, we assume that for a MUE m at optimality, Im < I, N m with a objective fuctio value Obj m = N m Im for MUE m. However, accordig to Lemma 1, the objective fuctio is mootoically icreasig i Im, whereas the costrait C1 is mootoically decreasig i Im. Therefore, we ca decrease the value of Im i o a certai sub-chael i N m ad icrease the values of all other Im, j j N m, j i. I this way, we ed up with Im j = I, for j N m, j i. Meawhile, Im i reaches a value Im i such that the rate costrait for MUE m is met with equality resultig i a ew objective fuctio value Obj m = ( ( N m 1) I + Im) i which is clearly higher tha Obj m. Hece, the iitial assumptio of optimality is cotradicted. Recall that, a value of I for a certai Im meas that subchael is ot actually allocated to MUE m. This leads to the fact that a system with N sub-chaels ad M MUEs, where M N, will ed up with M sub-chaels oly allocated to the M MUEs which leads to a miimal use of the available system badwidth. Hece, P B,m = P B, N ac = P B, M. Lemma 3. The allocated sub-chael i for MUE m is the oe with the highest chael gai g i B,m, i N m. ( Proof: Accordig to ) Lemma 2, at optimality, Obj m = ( Nm 1) I + Im i for MUE m, where I i m is selected such that the achieved data rate o sub-chael i is equal to ( R m. Hece, from the rate costrait formula, Im i = P i ) B,m gi B,m 2 Rm/ f 1 N o. It is clear that Im i is directly proportioal to gb,m i. Therefore, to imize Obj m, we eed to imize Im. i Hece, MUE m should be allocated sub-chael i such that i = arg gb,m. N m Based o the give Lemmas, we have the followig Theorem. Theorem 1. To imize the sum of tolerable iterferece levels, the macrocell ca solve the followig alterate optimizatio problem: M Γ B,mgB,m {Γ B,m} m=1 =1 subject to C1 : Γ B,m = 1, m M C2 : =1 M Γ B,m 1, N m=1 C3 : Γ B,m {, 1}, m M, N where the objective i (5) is to imize the sum of the allocated sub-chael gais. C1 restricts the umber of allocated sub-chaels to ay MUE m to oe sub-chael oly, whereas C2 restricts sub-chael to be allocated to at most oe MUE. The for each MUE m with allocated subchael, the imum tolerable ) iterferece level is give by: I m = ( P B,m g B,m 2 Rm/ f 1 N o (5). Hece, (4) is solved optimally. Proof: Accordig to Lemmas 2 ad 3, at optimality, each MUE will have oly oe sub-chael which is the oe with the highest gai. Hece, we ca defie the optimizatio problem i (5) which is the well-kow assigmet problem that ca be efficietly solved i polyomial time usig the Hugaria method [26]. I this way, the macrocell allocates sub-chaels to its MUEs i a way that satisfies their data rate requiremets ad that ca tolerate the imum possible iterferece from the small cell tier. 2) Miimize the total sum-power: As a baselie, we cosider aother way of performig RA i the macrocell by miimizig the total sum-power of the macrocell give the data rate requiremets of the MUEs. This problem has bee studied extesively i the literature [27]. However, the formulatio developed i [27] does ot accout for the imum tolerable iterferece level I m. Hece, we iclude it here with the required modificatio to determie the imum tolerable iterferece level I m. We have, thus, the followig optimizatio problem: mi M PB,m {PB,m} m=1 N m subject to C1 : f log 2 N m C2 : N i Nj =, i, j M C3 : M m=1 N m {1, 2,..., N} C4 : P B,m, m M, N. ( 1 + P B,m ) g B,m Im R m, m M + N o (6)

7 7 I (6), give the imum tolerable iterferece level o each allocated sub-chael I m, the macrocell seeks a power ad sub-chael allocatio solutio that miimizes the sumpower. Although (6) is a MINLP whose feasible set is ocovex, it has bee solved efficietly i [27] i the dual domai usig dual decompositio relyig o the fact that the duality gap becomes virtually zero whe the umber of sub-chaels i the system is sufficietly large. Remark 1. The reaso for choosig the resource allocatio solutio based o the formulatio i (6) as the baselie is the followig. With this solutio, at optimality, all MUEs have their rate requiremets satisfied with equality. This is also the case for the solutio obtaied from the formulatio i (4). I other words, with both the solutios, the MUEs achieve the same performace. The differece however lies i the way the resources are allocated, which subsequetly impacts the performace of the small cell tier. I (6), the same value of Im is assumed m M, N, i.e., Im = I th. For a further fair compariso betwee (4) ad (6), the macrocell adjusts the imum tolerable iterferece N m P B,m = P B,. This ca level I th such that M m=1 be accomplished by usig the bisectio method accordig to Algorithm 1 as give below, where I th,h > I th,l. Algorithm 1 Bisectio method to fid optimal I th 1: Macrocell iitializes I th,l, I th,h, ad δ 2: while M m=1 N m PB,m P B, > δ do 3: I th,m = (I th,l + I th,h ) /2 4: Macrocell solves the optimizatio problem i (6) 5: if M m=1 N m PB,m > P B, the 6: I th,h = I th,m 7: else if M m=1 N m PB,m < P B, the 8: I th,l = I th,m 9: ed if 1: ed while After Algorithm 1 termiates, I th,m gives the optimal value of I th. The optimizatio problem i (6) eds up with the power ad sub-chael allocatio to the MUEs with a uiform imum tolerable iterferece level I th o all allocated subchaels. I geeral, as will be show i the umerical results, (6) leads to a higher umber of allocated sub-chaels to the MUEs tha that (4) does. It is of iterest to study the effect of the two differet RA methods o the small cell tier. B. Problem Formulatio for Small Cells Due to the small distace ad the good chael coditios betwee small cells ad SUEs, small cells are capable of servig registered SUEs with higher data rates tha the macrocell. However, this should ot be at the cost of QoS degradatio at MUEs as they are served by the macrocell ad provided with basic coverage at possibly lower rates [1]. Hece, give the imum tolerable iterferece levels o each allocated sub-chael for the MUEs, each small cell ow tries to admit as may SUEs as possible at their target data rate by usig the miimum possible badwidth. Agai, the idea here is to leave as much badwidth as possible for the other etwork tiers (e.g., for device-to-device (D2D) commuicatio). 1) Cetralized operatio: To accomplish the aforemetioed requiremets, we defie the optimizatio problem i (7), where the objective fuctio accouts for both admissio cotrol ad sub-chael allocatio. We have the admissio cotrol variable y s,f which takes the value of 1 if SUE f is admitted i small cell s ad otherwise. By cotrollig the weightig factor ɛ [, 1], admissio cotrol ca be give higher priority over the umber of used sub-chaels. (1 ɛ) {Γ s,f,p s,f,y s,f} =1 f F s y s,f ɛ =1 subject to ) Ps,f C1 : f log 2 (1 + g s,f M m=1 Γ B,m P B,m g B,f + N o C2 : f F s =1 Γ s,f y s,f R f, s S, f F s Ps,f P s,, s S C3 : Γ B,m Ps,f g s,m Γ B,mIm, N C4 : P s,f Γ s,f P s, s S, f F s, N C5 : f F s Γ s,f 1, s S, N C6 : P s,f, s S, f F s, N C7 : Γ s,f {, 1}, s S, f F s, N C8 : y s,f {, 1}, s S, f F s, N. (7) 1 Propositio 1. By choosig ɛ < 1+SN, (7) admits the imum umber of SUEs while cosumig the miimum umber of sub-chaels. Proof: This Propositio ( ca be ) proved i a way similar to that i [28]. Let Γ s,f, P s,f, y s,f, s S, f F s, ( ) N deote a optimal solutio of (7). Let Γ ˆ s,f, Pˆ s,f, y s,f ˆ, s S, f F s, N be a feasible solutio that admits oe more SUE tha the optimal solutio, i.e., S y s,f ˆ = S ys,f + 1. The objective of the feasible solutio ca be writte as: (1 ɛ) y s,f ˆ ɛ ˆ (1 ɛ) =1 Γ s,f f F s y s,f + (1 ɛ) ɛsn (1 ɛ) ys,f (3) (2) (1)

8 8 (1 ɛ) f F s y s,f ɛ =1 Γ s,f. The first iequality holds due to the fact that S N ˆ =1 Γ s,f is upper bouded by SN whe all sub-chaels i all small cells are allocated. The secod iequality holds by settig (1 ɛ) ɛsn >. Hece, 1 we have ɛ < 1+SN. The last iequality holds due to the o-egativity of S N f F s =1 Γ s,f. I this way, the value of the objective fuctio for the feasible solutio is higher ( tha the optimal ) oe, which cotradicts the optimality of Γ s,f, P s,f, y s,f. Thus, there is o other solutio that admits a higher umber of SUEs uder the costrait i (7). Give the optimum value for the admissio cotrol variable ys,f, (7) reduces to a feasible sub-chael ad power allocatio problem with respect to the variables Γ s,f ad P s,f that aims at miimizig the umber of used sub-chaels subject to the give costraits. I (7), C1 is a data rate costrait for a SUE f which is active oly if SUE f is admitted, i.e., y s,f = 1. C2 is the power budget costrait for each small cell s restrictig the total trasmissio power of small cell s to be less tha or equal to P s,. C3 is a costrait o the imum crosstier iterferece itroduced to MUE m usig sub-chael. C4 esures that if sub-chael is ot allocated to SUE f, its correspodig trasmit power Ps,f =. C5 costrais subchael to be allocated to at most oe SUE f i small cell s. C6 esures that the power Ps,f should be positive, ad fially, C7 ad C8 idicate that Γ s,f ad y s,f are biary variables. Claim 1. The optimizatio problem i (7) is always feasible. Proof: A trivial feasible solutio of (7) is Γ s,f =, Ps,f = ad y s,f =, s S, f F s, N. The problem i (7) is a MINLP whose feasible set is o-covex due to the combiatorial ature of sub-chael allocatio ad admissio cotrol. However, for small-sized problems, we use OPTI [29], which is a MATLAB toolbox to costruct ad solve liear, oliear, cotiuous ad discrete optimizatio problems, to obtai the optimal solutio. Obtaiig the optimal solutio, however, for larger problems is itractable. Aother approach that ca reder the problem i (7) more tractable is to have a covex reformulatio of (7) by relaxig the costraits C7 ad C8 ad allowig Γ s,f ad y s,f to take ay value i the rage [, 1]. Thus, Γ s,f is ow a time sharig factor that idicates the portio of time sub-chael is allocated to SUE f [3], [31], whereas y s,f idicates the ratio of the achieved data rate for SUE f. Hece, we defie the covex optimizatio problem i (8), where P s,f ca be related to Ps,f i (7) as P s,f = Γ s,f P s,f to deote the actual trasmit power [22]. Now, (8) is a covex optimizatio problem with a liear objective fuctio ad covex feasible set. It ca be solved efficietly by the iterior poit method [32]. Note that the system model assumed i (8) differs from the origial oe i (7) as it allows time sharig of sub-chaels ad fractioal satisfactio of the required data rates. Hece, the solutio of (8) gives a upper boud to the optimal solutio of (7). However, it helps by revealig some isights about the behavior of (7). Note that the solutio of (8) ecessitates the existece of a cetral cotroller which ca be, for example, the HeNB-GW. However, for dese small cell etworks, havig a decetralized solutio with some coordiatio with a cetral etity will be a more viable optio. (1 ɛ) y s,f ɛ Γ {Γ s,f, P s,f,y s,f s,f} =1 subject to ( ) P C1 : Γ s,f gs,f s,f f log 2 1 /Γ s,f + M m=1 Γ B,m P B,m g B,f + N o =1 C2 : f F s =1 y s,f R f, s S, f F s P s,f P s,, s S C3 : Γ B,m P s,f gs,m Γ B,mIm, N C4 : f F s Γ s,f 1, s S, N C5 : P s,f, s S, f F s, N C6 : Γ s,f (, 1], s S, f F s, N C7 : y s,f [, 1], s S, f F s, N. 2) Distributed operatio: To fulfill the requiremet of havig a decetralized solutio for (8), we use the dual decompositio method [33]. For this purpose, we defie the followig partial Lagragia fuctio of the primal problem i (8) formed by dualizig the costrait C3: ( L Γ s,f, P ) s,f, y s,f, η = (1 ɛ) y s,f ɛ Γ s,f =1 + η Γ B,mIm Γ B,m P s,f gs,m (9) =1 where η is the Lagrage multiplier vector (with elemets η ) associated with the cross-tier iterferece costrait C3. The the Lagrage dual fuctio is represeted as g(η) = subject to ( {Γ s,f, P s,f,y s,f} L Γ s,f, P ) s,f, y s,f, η (8) C1, C2, C4 C7. (1) From (9), the imizatio of L ca be decomposed ito S idepedet optimizatio problems for each small cell s as

9 9 follows: g s (η) = subject to (1 ɛ) y s,f ɛ {Γ s,f, P s,f,y s,f} f F s f F s η Γ P B,m s,f gs,m f F s =1 =1 Γ s,f C1, C2, C4 C7, s S. (11) Thus, the Lagrage dual fuctio is g(η) = S g s (η) + η Γ B,mIm. (12) =1 The, the dual problem is give by: mi g(η). (13) η I order to solve the dual problem, η ca be updated efficietly usig the ellipsoid method [34]. A sub-gradiet d of this problem required for the ellipsoid method is derived i the followig propositio. Propositio 2. For the optimizatio problem i (8) with a dual objective g(η) defied i (1), the followig choice of d is a sub-gradiet for g(η): d = Γ B,mIm Γ B,m P (14) s,f g s,m where d is a elemet of d ad Γ s,f, P s,f, ad y s,f optimize the imizatio problem i the defiitio of g(η). Proof: For ay ξ, s,f, P g(ξ) L(Γ s,f, ys,f, ξ) = g(η) + (ξ η ) Γ B,m =1 P s,f gs,m Γ B,mI m. Algorithm 2 gives a practical implemetatio of the distributed joit RA ad AC operatio for the small cells. After the macrocell has performed RA for its MUEs, it seds the sub-chael allocatio iformatio for its MUEs ad the iitialized multiplier η to the HeNB-GW. For a give η, all small cells solve their optimizatio problem i (11) simultaeously. For the give resource allocatio i the small cells, the MUEs estimate the resultig iterferece levels ad sed them to the macrocell which updates the multiplier values usig the ellipsoid method. The macrocell the iforms the updated multiplier values to the HeNB-GW, which broadcasts them to the small cells, ad the etire operatio repeats. Note that the small cells ca obtai the chael gais g s,m relyig o the techiques proposed i [7]. Fially, the remaiig issue is to obtai a feasible primal solutio to (8) based o the resultig solutio from the Lagragia dual i (13). It has bee reported i [33] ad [35] that the iteratios of the dual decompositio method are, i geeral, ifeasible with respect to (8). This ifeasibility, however, is ot severe as large costrait violatios usually get pealized. Hece, usig a simple procedure, oe ca recover a primal feasible solutio that serves as a lower boud for the optimal solutio of (8). Suppose that the reported iterferece level by a MUE m allocated a sub-chael was foud to be: f F s P s,f g s,m = α I m, α > 1. (15) A straightforward way to recover feasibility is for the HeNB-GW to istruct all small cells trasmittig o subchael to scale dow their trasmissio powers by the factor α. For the updated power values, the etire problem is solved to obtai the updated values of sub-chael allocatio ad admissio cotrol variables. The gap betwee the lower boud offered by this procedure ad the upper boud offered by (13), referred to as the duality gap, dimiishes with iteratios. Covergece to the optimal solutio is guarateed sice the primal optimizatio problem i (8) is covex. Algorithm 2 Distributed joit RA ad AC algorithm 1: Macrocell iitializes η, L, seds sub-chael allocatio iformatio Γ B,m ad η to HeNB-GW ad sets iteratio couter l = 1 2: HeNB-GW broadcasts Γ B,m ad η values to all small cells 3: repeat 4: for s = 1 : S do 5: All small cells solve (11) i parallel 6: ed for 7: All MUEs estimate iterferece levels o allocated subchaels ad report them to the macrocell 8: Macrocell evaluates the sub-gradiet (14) ad updates η usig the ellipsoid method 9: Macrocell seds updated η to HeNB-GW 1: HeNB-GW broadcasts updated η to all small cells 11: Macrocell sets l = l : util Covergece or l = L A. Parameters V. NUMERICAL RESULTS AND DISCUSSIONS We evaluate the system performace through extesive simulatios uder various topologies ad scearios. We have a macrocell located at the origi with radius 3 m. A hot spot of small cells exists at a distace of 1 m from the macrocell. The MUEs exist outdoor i this hot spot ad are served by the macrocell. Each small cell has 2 idoor SUEs located radomly o a circular disc aroud the small cell with a ier radius of 3 m ad a outer radius of 1 m [36]. The macrocell has a total power budget of P B, = 2 W. To model the propagatio eviromet, the chael models from [36] are used. The chael gais iclude path-loss,

10 1 log-ormal shadowig, ad multipath Rayleigh fadig. The path-loss betwee a small cell ad its served SUE, P L = log R ad the path-loss betwee a small cell ad the outdoor MUEs, P L = ( log R, log R) + L ow, where R is the distace betwee a small cell ad the UE ad L ow accouts for losses due to walls. For path-loss betwee the macrocell ad a SUE existig idoor, P L = log R + L ow ad for path-loss betwee the macrocell ad its MUE, P L = log R. We have the followig values for the stadard deviatio of logormal shadowig: 4 db for shadowig betwee SUE ad its small cell, 8 db for shadowig betwee MUE ad small cell ad 1 db for shadowig betwee macrocell ad SUE or MUE. The Rayleigh fadig gai is modeled as a expoetial radom variable with uit mea. We assume f = 18 KHz, ɛ =.9 1+SN, ad oise power, N o = 1 13 W. I is set to ay arbitrary large umber. All the rate requiremets i the umerical results are specified i terms of spectral efficiecy (bps/hz). I the umerical results, the followig performace metrics are used: Average percetage of admitted SUEs = S f Fs y s,f F 1. Average percetage of chael usage = S f Fs N =1 Γ s,f SN 1. B. Numerical Results 1) Compariso betwee the traditioal ad the proposed RA method for macrocell: I this sectio, we compare the two proposed schemes for RA i the macrocell, amely, the formulatio i (4), which we refer to as proposed ad the formulatio i (6), which we refer to as traditioal. Fig. 2 shows the chael gai realizatios for a sapshot of 3 MUEs with 1 sub-chaels. power PB,m by the macrocell ad the imum tolerable iterferece level Im o allocated sub-chael to MUE m. No values for power PB,m o the x-axis idicate uallocated sub-chael with the correspodig value for Im set to I, which meas that this sub-chael ca be used by the small cell tier ucoditioally. For further clarificatio, Table III shows the absolute values of PB,m ad I m. (dbm) P B,m I m (dbm) Sub chael, Sub chael, MUE 1 MUE 2 MUE 3 MUE 1 MUE 2 MUE 3 Fig. 3. Allocated power PB,m ad imum tolerable iterferece level Im for MUEs {1, 2, 3} usig the traditioal scheme. (dbm) P B,m MUE 1 MUE 2 MUE 3 g B,m g B,m 5 x 1 8 MUE Sub chael, 4 x 1 9 MUE 2 2 I m (dbm) Sub chael, MUE 1 MUE 2 MUE 3 g B,m.5 1 x Sub chael, Sub chael, Fig. 2. Chael gais gb,m for MUEs {1, 2, 3}. MUE 3 Figs. 3-4 compare the two RA results for the give sapshot with R m = 5 bps/hz. Each figure shows the allocated Sub chael, Fig. 4. Allocated power PB,m ad imum tolerable iterferece level Im for MUEs {1, 2, 3} usig the proposed scheme. It is clear from Fig. 3 that most of the sub-chaels are allocated to the MUEs (9 sub-chaels out of 1 are allocated to the 3 MUEs), whe usig the traditioal scheme for RA. We otice also that the macrocell favors good sub-chaels as they require less trasmit power to achieve the rate requiremets for the MUEs, leadig at the ed to miimum trasmit power requiremets.

11 11 Fig. 4, o the other had, shows that the 3 MUEs require oly 3 sub-chaels to achieve their rate requiremets, as was proved before, usig the proposed scheme. Agai, the macrocell allocates the best sub-chaels to the MUEs. From Figs. 3-4 ad Table III, we ca otice that the etire power budget of macro BS, P B,, is used i both cases. It is worth metioig that whe we use the traditioal scheme for macrocell resource allocatio, it does ot ecessarily mea that it will cosume less power tha the proposed scheme, sice the imum tolerable iterferece level I m is adjusted accordig to Algorithm 1 by the macrocell to use the etire power budget. It rather meas that, give the imum tolerable iterferece levels, the resultig sub-chael ad power allocatio for the traditioal scheme will cosume the miimum power ad ay other allocatio will cosume a higher power. Now, for the imum tolerable iterferece levels I m, it is obvious from Figs. 3-4 ad Table III that the proposed scheme ca sustai higher iterferece levels from the small cell tier. For the traditioal scheme, the sum of the tolerable iterferece is give by: ( ) + I. O the other had, the sum of the tolerable iterferece levels for the proposed scheme ca be give by: ( ) + ( ) + ( ) + (7 I ). It is of iterest to compare the effect of the two differet RA schemes for the macrocell o the small cell tier. Fig. 5 compares the average percetage of admitted SUEs whe the macrocell performs RA i order to miimize the sum-power (as i (6)) ad imize the sum of tolerable iterferece levels (as i (4)) with two differet wall loss scearios. We have the followig sceario: 2 small cells located at ( 1, 1), (1, 1) i a square area hot spot of dimesios 2 2 m 2, 1 sub-chaels, P s, = 3 mw, R f = 5 bps/hz, ad R m = 5 bps/hz. Numerical results are obtaied ad averaged for 5 differet realizatios, where i each realizatio, the UE positios ad the chael gais are varied. The small cell problem is solved cetrally usig the covex formulatio i (8). It is clear from the figure that the proposed RA method for the macrocell outperforms the traditioal oe. Whe the macrocell performs RA accordig to the proposed method, it cosumes the miimum badwidth, ad therefore, frees as may sub-chaels as possible for the small cells. O the other had, the traditioal method cosumes more badwidth tha the proposed oe, hece, the small cells have more iterferece costraits to abide by. We also otice that as the wall losses icrease, the small cells ted to be more isolated ad the impact of resource allocatio i the macrocell o the small cell performace is low. 2) Compariso betwee the differet formulatios for the RA problem for small cells: Fig. 6 compares the values of the objective fuctio for the MINLP formulatio i (7), the cetralized covex formulatio i (8), ad the distributed formulatio i (9) for a sapshot of the followig sceario: 2 small cells located at ( 1, 1), (1, 1) i a square area hot spot of dimesios 2 2 m 2, 3 sub-chaels, 3 MUEs, P s, = 3 mw, L ow = 1 db, ad R f = 5 bps/hz. As was stated previously, the covex formulatio provides a upper boud for the solutio of the MINLP formulatio. Also, we otice that the cetralized ad distributed formulatios have Average percetage of admitted SUEs Traditioal, L ow =1dB Proposed, L ow =1dB Traditioal, L ow =1dB Proposed, L ow =1dB Number of MUEs, M Fig. 5. Average percetage of admitted SUEs vs. umber of MUEs M whe the macrocell employs both the proposed ad the traditioal methods for RA with differet wall loss scearios. the same solutio due to the covexity of the cetralized formulatio i (8). It is worth metioig that the covex formulatio exhibits a behavior similar to the MINLP formulatio. Hece, solvig the covex formulatios reveals isights ito the behavior of the solutio of the MINLP formulatio. We also otice that as R m icreases, the iterferece costraits for the MUEs become tighter. Hece, the average umber of admitted SUEs decreases. Sice the objective fuctio i our formulatio gives more priority to admissio cotrol, the value of objective fuctio decreases with icreasig R m. Objective fuctio value MINLP Cetralized Distributed R (bps/hz) m Fig. 6. The values of objective fuctio for differet formulatios vs. R m. Figs. 7-8 show the average percetage of admitted SUEs ad chael usage i a small cell vs. R m for the same sceario cosidered i Fig. 6. As was discussed i Fig. 6, as the rate requiremets for the MUEs icrease, they have tighter

12 12 TABLE III ABSOLUTE VALUES OF PB,m AND I m FOR THE TRADITIONAL AND PROPOSED MACROCELL RA SCHEMES Traditioal scheme Proposed scheme Sub-chael# P B,m (W) MUE MUE MUE MUE-1 I I I I I I I I m 1 1 MUE I I I I I I I P B,m (W) MUE MUE MUE MUE-1 I I I I I I I I I m 1 1 MUE I I I I I I I I I Average percetage of admitted SUEs Cetralized Distributed Average percetage of chael usage Cetralized Distributed R m (bps/hz) R m (bps/hz) Fig. 7. Average percetage of admitted SUEs vs. R m. Fig. 8. Average percetage of chael usage vs. R m. iterferece costraits. Hece, the percetage of admitted SUEs geerally decreases. We otice i Fig. 7 that, iitially, the average percetage of admitted SUEs is almost costat due to the icreased umber of used sub-chaels as show i Fig. 8. As the MUEs rate requiremets icrease further, the icrease i the umber of used sub-chaels is ot eough to accommodate the rate requiremets of the SUEs, hece, the average percetage of admitted SUEs decreases. 3) Covergece behavior: Usig the same sceario described for the previous figure, Fig. 9 shows the covergece behavior of Algorithm 2, where the upper boud refers to (13) ad the lower boud refers to the feasible objective obtaied by the procedure at the ed of Sectio IV-B2. I the figure, the best lower boud is obtaied by keepig track of the best primal feasible objective resultig through iteratios. It is clear that Algorithm 2 coverges to the optimal solutio of (8) withi a few iteratios. 4) Average percetage of admitted SUEs vs. R f : I this sceario, we have the followig setup: 5 small cells located at ( 2, 1), ( 2, 14), (2, 14), (2, 1), (, 12) i a square area hot spot of dimesios 4 4 m 2, 5 subchaels, 5 MUEs, P s, = 3 mw, L ow = 1 db ad R m = 4 bps/hz. Numerical results are obtaied ad averaged for 5 differet realizatios, where i each realizatio, the UE positios ad chael gais are varied. Fig. 1 shows the average percetage of admitted SUEs vs. R f. We otice that, geerally, as the rate requiremet icreases, more SUEs are i outage. We also otice that the distributed scheme coverges approximately to the same solutio as the cetralized solutio. 5) Average percetage of admitted SUEs vs. P s, : We have the same setup as the oe for the previous figure except for R f = 1 bps/hz. Fig. 11 shows the average percetage of admitted SUEs vs. P s,. We otice that as the imum trasmit power of the small cells icreases, the average umber of admitted SUEs icreases. This rate of icrease, however, is ot fixed as the system is limited by the iterferece costraits for MUEs. C. Summary of Major Observatios The major observatios from the umerical aalysis ca be summarized as follows: I a multi-tier etwork, it is critical to cosider the impact of RA decisios i oe tier o the other oe. For the macrocell etwork, as differet RA schemes are used to achieve the same rate requiremets for the MUEs, they affect the performace of the small cell tier differetly.