Power Adjustment and Scheduling in OFDMA Femtocell Networks

Transcription

1 Power Adjustment and Schedulng n OFDMA Femtocell Networks Mchael Ln CSE Department Pennsylvana State Unversty Emal:moln@cse.psu.edu Novella Bartoln Computer Scence Department Sapenza Unversty, Italy Emal:novella@d.unroma1.t Thomas La Porta CSE Department Pennsylvana State Unversty Emal:tlp@cse.psu.edu Abstract Densely-deployed femtocell networks are used to enhance wreless coverage n publc spaces lke offce buldngs, subways, and academc buldngs. These networks can ncrease throughput for users, but edge users can suffer from co-channel nterference, leadng to servce outages. Ths paper ntroduces a dstrbuted algorthm for network confguraton, called Radus Reducton and Schedulng (RRS), to mprove the performance and farness of the network. RRS determnes cell szes usng a Vorono-Laguerre framework, then schedules users usng a schedulng algorthm that ncludes vacancy requests to ncrease farness n dense femtocell networks. We prove that our algorthm always termnate n a fnte tme, producng a confguraton that guarantees user or area coverage. Smulaton results show a decrease n outage probablty of up to 50%, as well as an ncrease n Jan s farness ndex of almost 200%. I. INTRODUCTION Femtocells are small, locally-deployed wreless base statons that have become popular as a means of supportng or replacng exstng wreless macrocells. Ther key features are: a small coverage area, on the order of 100 square meters; automatc setup and the use of the customer s Internet connecton for backhaul, whch allow them to be deployed by customers rather than servce provders; customer-defned access control. Densely deployed femtocells can be used to establsh wreless networks that operate n a smlar manner to publc or enterprse WF networks, but serve users usng 3G or 4G wreless technology. These types of networks face dfferent challenges than both WF networks and ndependently deployed femtocells. The load on such networks s typcally much hgher than on solated femtocells, and co-channel nterference between femtocells can cause servce degradaton. Due to these ssues, recently dense femtocell networks have been an actve topc n the research communty. For nstance, Arslan, et al. [1] ntroduce FERMI, a system for nterference mtgaton n dense OFDMA femtocell networks, that uses concepts from fractonal frequency reuse to mtgate the negatve performance mpact of nterference. Wang, et al. [2] present a jont power-and-farness optmzaton algorthm for dense femtocell networks. Due to the network envronment, edge users n dense femtocell networks can be exposed to hgh nterference, leadng to reduced throughput and farness. Farness n wreless networks refers to the ablty of the network to share ts resources amongst all users, rather than usng ts resources to serve only the users wth the best sgnal qualty, whch maxmzes system throughput but can starve some users. Farness has typcally been approached as a schedulng or resource management [3], [4], [5], [6], [7] problem. See [8] for a comprehensve dscusson of farness n wreless networks. Ths paper takes a dfferent approach, and ntroduces a new algorthm, RADIUS REDUCTION AND SCHEDULING (RRS), that ncreases farness n densely deployed femtocell networks usng a combnaton of power management and resource block schedulng. Gven the locatons of users and femtocells, RRS determnes the network confguraton n two phases. In the frst phase, t uses a Vorono-Laguerre geometry-derved framework to reduce femtocell coverage overlaps and co-channel nterference. At the same tme, ths prelmnary phase seeks to maxmze the number of users that can be served wthn ther qualty of servce requrements. The frst phase of the algorthm provdes an teratve adjustment of each femtocell radus, on the bass of only local nformaton on the settng of neghbor cells. We prove the termnaton of ths phase n a fnte number of steps, and we show that t preserves ether area or user coverage whle reducng transmsson power and femtocell rado coverage overlaps and nterference. In the second phase, RRS provdes a resource block schedulng scheme that uses vacancy requests to mprove resource sharng and servce of users that are unable to meet ther performance targets. Smulatons of an LTE-based network are used to compare RRS to prevous approaches. We show that RRS can reduce the outage probablty, the percentage of users that cannot meet ther throughput requrements, by up to 100% over a baselne algorthm workng wth fxed cell rad and best-effort schedulng. Furthermore, our algorthm ncreases the Jan s ndex [9] of the network, a common measure of farness, by up to 190%. The contrbutons of ths paper can be summarzed as follows: We propose a new algorthm, called RRS, for reducng femtocell coverage rad and schedulng resource blocks among users. Ths algorthm can be formulated n several varants to consder dfferent performance objectves We prove that RRS termnates n a fnte tme and preserves ether area or user coverage, despte radus

2 Notaton Descrpton F set of femtocells f F -th femtocell U set of users u j U j-th user P possble power values of a femtocell π P power level of f B Max number of resource blocks for any femtocell ς j estmated SINR of user u j t j nstantaneous throughput of user u j τ j throughput requrement of user u j V (C ) Laguerre polygon of crcular range C V (f ) Laguerre polygon of femtocell f V (k) (f ) polygon V (f ) at teraton k Û (k) farthest user of V (k) (f ) ˆV (k) farthest vertex of V (k) (f ) r (k) radus of femtocell f at teraton k S (k) area covered at teraton k U (k) users covered at teraton k U (k) (f ) set of users nsde V (k) (f ) radus reducton rate of f at teraton k ɛ α mnmum reducton rate reducton TABLE I: Summary of notatons We provde smulaton results from a dynamc, nterferng OFDMA system smulator showng that our algorthm reduces outage probabltes and ncreases farness. II. NETWORK MODEL AND PROBLEM FORMULATION In ths secton we defne the femtocell network model and ntroduce our assumptons and notaton. Table I summarzes the notaton used throughout the paper. We consder an OFDMA network wth several femtocells f F, nterferng wth each other. Femtocell f may adjust ts power π wthn a range P of possble values. We consder a set of users U, and we assume that femtocells are deployed densely enough to cover all users of U when they work at maxmum power. We assume that users preferentally assocate wth femtocells. Nevertheless, n order to have a realstc model, we also consder the presence of a macrocell. Thanks to the jont presence of both the macrocell and the femtocells n our model, we are able to capture the nterference generated by the macrocell. A femtocell s avalable bandwdth s tme- and frequencydvded nto resource blocks (RB), wth B Max numbered resource blocks per femtocell, as n an LTE system. Resource blocks delver a varyng number of bts, dependng on the sgnal-to-nterference-and-nose rato (SINR) receved by a user on that resource block, and on the subsequent channel qualty ndcator (CQI) measured on the resource block tself. CQIs are a measurement of the channel qualty between a user and ts servng base staton. CQIs are determned by a step-wse functon defned on the SINR of an ndvdual resource block. The estmated SINR ς j of a user u j s measured usng reference sgnals. Notce that, snce OFDMA sgnals are spread across a wde spectrum range, ndvdual subchannel SINRs may dffer sgnfcantly from ς j. The nstantaneous throughput t j of user u j s the sum of the throughput acheved on each Fg. 1: Vorono and Vorono-Laguerre cell boundares resource block assgned to the user durng a transmsson frame. We tackle the problem of ncreasng the farness of a wreless network, wth the ancllary goal of decreasng power consumpton. We consder users havng heterogeneous requrements. We defne a hard throughput requrement τ j for each user u j, used to determne whether a user s served or not. Our goal s to serve the maxmum number of users under ths constrant, at the possble expense of reduced global throughput. We address ths problem by means of a new algorthm, RADIUS REDUCTION AND SCHEDULING (RRS) that determnes power and resource block schedulng, gven femtocells and users postons. The algorthm RRS runs n all femtocells across the network at network ntalzaton, and n local femtocells when users arrve or move between femtocells. It works n two phases. The frst phase of RRS determnes the femtocell transmsson power. Each femtocell confgures ts own transmsson power on the bass of a dstrbuted coordnaton protocol that allows neghborng femtocells (femtocells that are n rado proxmty wth each other) to exchange nformaton and adapt ther workng settng cooperatvely. By usng ths dstrbuted coordnaton protocol, neghborng femtocells determne ther respectve responsblty regons, namely the regons where each cell s responsble for provdng enough rado resources to serve users. In the second phase of RRS the algorthm provdes a resource block schedulng scheme, so that users workng on conflctng rado resources are able to share them and to perform non conflctng transmssons. In Sectons IV and V we descrbe the two phases of radus reducton and schedulng, respectvely. In the next Secton, we ntroduce the mathematcal background of Laguerre geometry to motvate the desgn of the radus reducton phase. III. BACKGROUND ON VORONOI-LAGUERRE DIAGRAMS In typcal wreless networks, users assocate wth the base staton that has the strongest reference sgnal. Under the assumpton of homogeneous transmsson power, the coverage cells resultng from ths type of user assocaton can be approxmated usng Vorono cells, where f s Vorono cell conssts of all ponts closer to f than to any other femtocell. The Vorono cell approxmaton has been extensvely studed and shown to be good under certan wreless network scenaros [10].

3 In a Vorono dagram, cell polygons are defned by the axs of the segments generated by two femtocells, that s the locus of the ponts that are equdstant from them and perpendcular to ther connectng segment (Vorono lne n Fgure 1). Gven any two femtocells, ths lne dvdes the plane nto two halves. If the femtocell transmsson power s homogeneous, the two femtocells would have the same coverage radus, and the Vorono lne would properly delmt ther responsblty regons. Nevertheless, when the cells have dfferent rado coverage capabltes, the Vorono lne may not determne the responsblty regon correctly, as shown n Fgure 1. In both the dagrams the Vorono cell assgns some ponts that are better served by the femtocell on the rght, located at P 2, to the femtocell on the left, at P 1. The desred partton of the plane nto responsblty regons s through the ntersecton of the crcles representng the rado coverage range of the two cells, labelled the Vorono-Laguerre lne n Fgure 1. Ths lne s perpendcular to the segment connectng the postons P 1 and P 2 of the two femtocells, and equdstant n the Laguerre geometry. Vorono-Laguerre cells are defned usng the Laguerre dstance, d 2 L, whch defnes a radus-dependent dstance between two crcles, or between a crcle and a pont. Consder the crcles C 1 and C 2, wth respectve rad r 1 and r 2, wth r 2 > r 1, n Fgure 1. The boundary between the Vorono cells of C 1 and C 2 s ndcated by the dashed lne, whle the boundary between the Vorono-Laguerre cells s ndcated by the sold lne. The Laguerre dstance between two crcles C 1 and C 2 centered on ponts P 1 and P 2, wth rad r 1 and r 2, respectvely, s defned as: d 2 L (C 1, C 2 ) = p 1 p 2 2 (r 1 r 2 ) 2. Ths defnton can also be used to calculate the dstance between a crcle and a pont, by consderng the pont as a crcle wth null radus. It s straghtforward to see that under the Laguerre geometry, gven two crcles wth dstnct centers and possbly dfferent rad, the locus of the ponts equally dstant from them s a lne, hereby called the Vorono-Laguerre lne, that s perpendcular to the segment connectng the centers, wth the followng propertes: f the two crcles ntersect each other, ther Vorono-Laguerre lne crosses ther ntersecton ponts, as n the left sde of Fgure 1, whle f two crcles are dsjont, the Vorono-Laguerre lne les between them, as n the dagram on the rght of Fgure 1. Notce also that ths defnton mples that, dependng on the overlap between two crcles, n the Laguerre geometry the two centers may fall on the same sde of the Vorono Laguerre lne, whch would mply that the responsblty regon of one femtocell would be located on the opposte sde of the Vorono Laguerre lne wth respect to ts center [11]. The noton of Vorono dagrams can be extended to the Laguerre geometry, as follows: gven N crcles C wth centers C = (x, y ) and rad r, = 1,..., N, the Vorono-Laguerre polygons V (C ) of the crcles C are defned as V (C ) = {P R 2 d 2 L(C, P) d 2 L(C j, P), j }. Notce, that unlke wth tradtonal Vorono dagrams, ths extenson to the Laguerre geometry may lead to the case that some polygons are ether null or empty [11], reflectng a stuaton of hgh coverage redundancy that s not captured by tradtonal Vorono dagrams. Null and empty polygons are descrbed n more detal n Secton IV. Vorono-Laguerre dagrams are extraordnarly powerful n modelng the responsblty regons of heterogeneous femtocells. Indeed, a fundamental property of the Vorono dagrams n Laguerre geometry s the followng: Theorem III.1. ([12]) Let us consder N crcles C, wth centers C = (x, y ) and rad r, = 1,..., N, and let V (C ) be the Vorono-Laguerre polygon of the crcle C. For all k, j = 1, 2,..., N, V (C k ) C j C k. Less formally, f a pont P s n the coverage range of at least one femtocell, t s certanly covered also by the femtocell f that generates the Vorono-Laguerre polygon V (C ) that ncludes P. In the followng, we refer to V (f ) as to the Vorono- Laguerre polygon of femtocell f, or alternatvely, the responsblty regon of cell f. Lkewse, we denote wth V (k) (f ) the Vorono Laguerre polygon of femtocell f at teraton k. IV. RADIUS REDUCTION Ths secton descrbes the radus reducton phase of RSS. Frst, we prove that our coverage crtera preserve ether user or area coverage, then we descrbe ther use n the algorthm n detal. After that, we prove that the algorthm termnates, and dscuss the reducton rate parameter, α. The goal of ths phase s to obtan a network confguraton wth less overlap n the rado coverage of neghborng femtocells, so as to lmt co-channel nterference whch would reduce rado resource avalablty. Ths s done by elmnatng unncessary femtocells and reducng the transmsson power of the remanng femtocells. Ths phase s executed n a dstrbuted manner and governs the reducton of a femtocell s transmt power and consequently, ts rado coverage range. Snce the range reducton s performed n a dstrbuted manner, t s necessary to desgn an algorthm that allows neghborng cells to coordnate wth each other, so as to avod conflctng decsons that may lead to a loss of servce n some regons. Each femtocell calculates ts transmsson power teratvely. The algorthm begns wth all femtocells transmttng at maxmum power. At any teraton k, each femtocell f reduces ts transmsson power p (k) and ts transmsson radus r (k) r (p (k) ) correspondngly. It must be noted that snce every cell potentally reduces ts radus by some amount, the Vorono- Laguerre dagram determnng the responsblty regons of each cell s recalculated locally at any teraton k untl the algorthm converges. The amount of radus reducton at each teraton s determned accordng to one of two crtera: Radus reducton preservng user coverage (UCR) Each femtocell reduces ts radus ensurng that t does not leave uncovered any of the users resdng n ts responsblty regon at the current teraton. It does so by lmtng the

4 amount of reducton so as to preserve coverage of the farthest user located n ts polygon. Radus reducton preservng area coverage (ACR) Each femtocell reduces ts radus ensurng that t does not reduce the coverage of ts current teraton Vorono- Laguerre polygon. It does so by lmtng the amount of reducton so as to preserve rado coverage of the farthest pont of ts polygon. The ACR approach s nspred to the works of Gupta et al. [13] and Bartoln et al. [14] that are desgned for selectve actvaton and radus adaptaton of sensor networks to provde sensng coverage, and am at preservng coverage completeness of a contnuous area of nterest. By contrast, UCR s meant to ensure rado coverage where t s actually needed, that s n the dscrete ponts of the area where users are located. Note that UCR does not am to ensure completeness of area coverage. A. Preservaton of Coverage In ths subsecton, we establsh our crtera for elmnatng femtocells whle mantanng ether user or area coverage. Let us consder a femtocell f, located at P. Let Û (k) be the poston of the farthest covered user lyng n the Vorono- (k) Laguerre polygon of f, and let ˆV be the farthest covered pont of the same polygon at the current teraton k. Accordng to the UCR crteron, f can reduce ts radus for the (k + 1)-th teraton to a value r (k+1) r (k) such that t stll covers the poston of the farthest user of ts polygon: r (k+1) P Û (k). (1) Smlarly, accordng to the ACR crteron the next teraton value of r (k+1) must meet the area coverage constrant: r (k+1) P ˆV (k). (2) Note that n the descrpton of these two crtera for radus reducton we only consdered non-empty and non-null polygons. An empty polygon s a Vorono-Laguerre polygon that does not contan ts generatng pont, that s the polygon V (f ) for whch the poston P of the femtocell f s such that P / V (f ). A null polygon s a degenerate polygon, wth no ponts, that s V (f ) = [11]. The Corollares 1 and 2 follow from Theorem III.1 and characterze the redundancy of femtocell f n the aforementoned stuatons. Corollary 1 (User coverage redundancy). If femtocell f has an empty polygon V (f ) and t does not cover any of the users contaned n t, or t has a null polygon, then f s redundant, that s, for any user u wth poston P u covered by f, there s another femtocell f j that also covers P u. Proof. Ths corollary extends Corollary 3.1 of [14] to the case n whch a femtocell f may cover a part of ts polygon but does not cover any user. We recall that the Vorono-Laguerre dagram of a regon R, consttutes a partton of R. Let C(f ) be the rado range of f. Any user postoned n P u C(f ) does not belong to V (f ) because, by assumpton, f does not cover any user lyng n ts polygon. Hence the locaton of any user covered by f must belong to the Vorono cell of some other femtocell f j. By Theorem III.1 and snce P u s covered by assumpton, we can conclude that the user locaton P u s also covered by femtocell f j. As ths s true for any user n C(f ) we can conclude that f s redundant. In less formal words, user coverage redundancy captures the stuaton n whch femtocell f covers only users that are better served by other femtocells (cells that are closer accordng to the Laguerre dstance), or the stuaton n whch f s located such that t s too far away to cover the uncovered users located n ts polygon, even at maxmum transmsson power. Analogously, we can prove Corollary 2, whch drectly derves from Corollary 3.1 of [14]; we omt the proof for space. Corollary 2 (Area coverage redundancy). If femtocell f has an empty polygon V (f ) and t does not cover any pont of t, or t has a null polygon, then f s redundant, that s, for any pont Z covered by f, there s another femtocell f j that covers Z. Area coverage redundancy captures the stuaton n whch a femtocell f covers only ponts that are better covered by other femtocells (cells that are closer accordng to the Laguerre dstance), or s located such that t cannot cover the ponts of ts area of responsblty. Corollares 1 and 2 allow us to defne two stuatons of elmnable redundancy n whch a femtocell (the elmnable femtocell) can be mmedately dsabled. Note that these corollares defne only suffcent condtons for redundancy and some femtocells can be redundant wthout meetng the crtera n Corollares 1 and 2. In these cases a gradual teratve reducton of femtocell rado coverage range s needed to fnd redundant femtocells, a process whch addresses possble conflcts n the concurrent elmnaton of several potentally redundant femtocells. The followng theorems show that f all femtocells apply an teratve radus reducton under the lmts posed by the constrants n Equaton 1 for the UCR crteron and Equaton 2 for the ACR crteron, user coverage and area coverage are respectvely preserved and elmnable femtocells are turned off. We defne a dstrbuted executon of radus reducton accordng to the UCR or the ACR crteron as the followng: every non elmnable femtocell reduces ts radus under the constrants gven by Equaton 1 or Equaton 2, and every elmnable femtocell s turned off. Theorem IV.1 (User coverage preservaton under UCR). Let us consder a set F of femtocells, randomly spread over a regon R. Let us also consder a set U of users over the same regon. Let U (k) U be the subset of users that the femtocells of F are able to cover when each femtocell f works wth radus r (k). Let U (k+1) be the set of users covered by the same femtocells of F after a dstrbuted executon of a radus reducton accordng to the UCR crteron. Then

5 U (k) = U (k+1), so the radus reducton preserves coverage of the users wthn the regon R. Proof. For smplcty of notaton let us consder the set U as a fnte set of ponts n R, representng user postons. The Vorono-Laguerre dagram determned by the postons and rad of the femtocells of F at teraton k creates a partton of the set of U (k) as follows: U (k) = f F U (k) (f ), wth U (k) (f ) U (k) (f j ) =, for j, and where U (k) (f ) s the set of users nsde V (k) (f ). By alterng the rad of the femtocells the Vorono-Laguerre dagram s also altered, and consequently users that were n the polygon of a cell at teraton k may fnd themselves n another cell at teraton k + 1. Nevertheless, n order to prove the theorem we need to ensure that any user that was covered at teraton k wll stll be covered by at least one of the avalable femtocells even after the radus reducton performed at teraton k + 1. To ths purpose, t s suffcent to prove that such a radus reducton preserves coverage of all the covered user postons of each polygon accordng to the partton determned by the dagram at the k-th teraton. Thanks to Theorem III.1 we know that the covered users of each polygon V (k) (f ) are also covered by f tself, and therefore we can wrte U (k) n terms of the unon of the covered sets of user postons of each polygon as follows: U (k) = f F C(k) (f ) U (k) (f ). Each non elmnable cell performs a radus reducton that, accordng to constrant (1), preserves coverage of the users of ts polygon. Therefore, U (k) (f ) C (k+1) (f ) = U (k) (f ) C (k) (f ). The same s trvally true also for elmnable femtocells as they do not have any covered user n ther polygons at teraton k. Ths concludes the proof, as U (k+1) = f F C(k+1) (f ) U (k) (f ) = = f F C(k) (f ) U (k) (f ) = U (k). The followng Theorem also holds for the crteron ACR and can be proved usng a smlar argument. Theorem IV.2 (Area coverage preservaton under ACR). Let us consder a set F of femtocells, randomly spread over a regon R. Let S (k) R be the porton of R that the femtocells of F are able to cover when each femtocell f works wth radus r (k). Let S (k+1) be the area covered by the same femtocells of F after a dstrbuted executon of a radus reducton accordng to the ACR crteron. Then S (k) = S (k+1), so the radus reducton preserves coverage of the regon R. We underscore that even though the radus reducton preserves area coverage and user coverage, accordng to crtera ACR and UCR, respectvely, the modfcaton n the femtocell rad sgnfcantly alter the shape of the Vorono-Laguerre polygons at any teraton. Consequently, users can be logcally reassgned at any gven teraton. We wll show later n ths secton that the teratve radus reducton rapdly converges to the fnal settng of the femtocell rad. Thanks to Theorems IV.1 and IV.2, we are able to guarantee that even f every femtocell performs a radus reducton to the mnmum value provded by Equatons 1 and 2, user and area coverage s preserved. Nevertheless the range of possble values for each femtocell radus can be exploted to prortze the radus reducton of some femtocells over the others accordng to a gven performance objectve. B. Radus Reducton We propose that the radus reducton be performed gradually, at every teraton k, wth only a partal reducton [0, 1] for every femtocell f, at each step, as we descrbe n Algorthm 1. We call the parameter the radus reducton rate of femtocell f at teraton k. The formulaton of ths parameter s descrbed n detal n Secton IV-D. It requres neghborng femtocells to exchange addtonal nformaton regardng ther current teraton settng. Under ths approach, femtocell f starts workng at maxmum power p (0) = p Max at teraton k = 0. Let p (k) mn () UCR ACR be the mnmum value of power that ensures that f covers ether the farthest covered user Û (k) (under the UCR crteron) (k) or the farthest covered pont ˆV (under the ACR crteron) of V (k) (f ). Ths value can be expressed as follows: under the UCR crteron t s p (k) mn () UCR = mn{π : r(π) P Û (k), π P}), whle under the ACR crteron, p (k) mn () ACR = mn{π : (k) r(π) P ˆV, π P}). At any gven teraton k, the transmsson power of f s reduced to p k+1 p k, wth: p (k+1) = p (k) α k (p (k) p (k) mn () UCR ACR). (3) Accordng to Equaton 3 the maxmum reducton of power s obtaned for α k = 1. We consder a postve lower bound ɛ α to α k, such that ɛ α > 0 and ɛ α << 1 to ensure that all the femtocells that can reduce ther radus are actually able to do so, regardless of the behavor of ther neghborng cells. A prelmnary exchange of nformaton among neghborng femtocells s needed to let each cell know the poston and current radus of ts neghbor femtocells to calculate the current teraton Vorono-Laguerre polygon, and the current value of the radus reducton rate, as ndcated n lne 4 of Algorthm 1. In most cases femtocells only allow tunng of the transmsson power wthn a dscrete set of values. In such cases, the algorthm should take the mnmum of these dscrete values that exceeds the calculated value of p (k+1). For the sake of clarty and wthout loss of generalty we neglect ths aspect n the followng. When femtocells have overlappng coverage ranges, the radus reducton algorthm can lead some femtocells to reduce ther transmsson powers to zero. When ths occurs, femtocells wth zero transmsson power serve no users, and do not transmt at all durng the next frame, untl a network reconfguraton, whch may occur as a consequence of user movement, arrval, or departure. The dstrbuted radus reducton procedure s descrbed n detal n Algorthm 1. Ths procedure guarantees that f s transmsson power, p (k), s reduced to the mnmum value of

6 Algorthm 1: Radus Reducton Algorthm Result: Power confguraton p for femtocell f 1 k = 0; 2 p 0 = p Max ; 3 whle!termnaton condton do 4 exchange nfo wth neghbor cells; 5 local constructon of V (k) (f ) ; 6 f f s elmnable (Corollares 1, 2) then 7 termnaton condton := true ; 8 go to sleep ; 9 else 10 calculate p (k) mn () UCR ACR ; 11 f p (k) mn () UCR ACR = p (k) then 12 termnaton condton := true ; 13 p := p (k) ; 14 else 15 calculate 16 p (k+1) ; := p (k) 17 k := k + 1 ; 18 end 19 end 20 end (p (k) p (k) mn ()); power that ensures ether area or user coverage accordng to crteron ACR or UCR, respectvely. The speed of f s radus reducton s determned by the parameter. Secton IV-D s devoted to a dscusson of possble ways to set the parameter. C. Termnaton of RRS The radus reducton algorthm provably termnates provded that the radus reducton at each step s a fnte amount, whch s always the case when radus reducton s lmted to dscrete steps. Theorem IV.3 (Convergence of UCR). Gven a set of F femtocells wth tunable rad, executng the radus confguraton phase of the algorthm RRS under the UCR crteron, each femtocell converges [n a fnte tme] to a fnal radus confguraton. Proof. If V (k) (f ) s not covered, f s elmnable and goes to sleep. If V (k) (f ) s at least partally covered, then f can reduce ts radus up to an extent that preserves coverage of the farthest covered user. As ɛ (α) > 0, the algorthm performs a reducton at any teraton, untl the radus becomes null or equals the dstance to the farthest user covered exclusvely. Termnaton follows by settng a fnte and postve mnmum value of radus reducton ɛ r that can be performed n a sngle teraton. Theorem IV.4 (Convergence of ACR). Gven a set of F femtocells wth tunable rad, executng the radus confguraton phase of the algorthm RRS under the ACR crteron, each femtocell converges [n a fnte tme] to a fnal radus confguraton. Proof. The proof follows smlarly to the proof of Theorem IV.3, wth the only excepton that f V (k) (f ) s only partally covered, then f cannot reduce ts radus as t needs to preserve coverage of the farthest covered pont. So the algorthm mmedately termnates for those femtocells coverng ther polygons only partally. D. Radus Reducton Rate α The radus reducton rate α controls the prorty wth whch f reduces ts transmsson power wth respect to ts neghbor femtocells. In the followng we ntroduce three dfferent prortzaton crtera, that correspond to dfferent ways to calculate. In all the approaches, only the femtocells wth reducton rate = 1 are allowed to perform the maxmum allowed radus reducton. 1) Prortzaton based on throughput ncrease (TI): Let t (k) (f ) be the estmated ncrease n total throughput that would be acheved by femtocell f f t decreases ts transmsson power to the mnmum value. We denote wth N s the set of femtocells n rado proxmty to f, and defne t (k) mn (N ) and t (k) Max(N ) the mnmum and maxmum value of the same metrc n the neghbor cells of f. Namely, t (k) mn (N ) mn fj N t (k) (f j ). Smlarly, t (k) Max(N ) max fj N t (k) (f j ). The value of under the TI crteron of prortzaton s therefore the followng: = t(k) (f ) t (k) mn (N ) t (k) Max(N ) t (k) mn (N ). (4) 2) Prortzaton based on number of users meetng throughput requrements (UTR): When the metrc of nterest n the prortzaton s the number of users that acheve ther qualty requrements, we can denote wth u (k) (f ) the number of users that would be able to meet ther throughput requrement f the radus of cell f were reduced at a mnmum. Smlarly to what we dd for the throughput ncrease crteron, we denote wth u (k) mn (N ) and u (k) Max(N ), the mnmum and the maxmum of the same metrc over the neghbors of f. Under the UTR crteron of prortzaton we defne the value of as follows: = u(k) (f ) u (k) mn (N ) u (k) Max(N ) u (k) mn (N ). (5) 3) Prortzaton based on load (Load): Accordng to ths last crteron we am at prortzng the femtocells whch currently have the hghest load n ther neghborhood. Hence we denote wth l (k) (f ) the number of users that are attached to f when the radus of the cell s the one provded at teraton k. We denote wth l (k) mn (N ) and l (k) Max(N ), the mnmum and the maxmum of the same metrc over the neghbors of f. Under the Load crteron of prortzaton we defne the value of as follows: = l(k) (f ) l (k) mn (N ) l (k) Max(N ) l (k) mn (N ). (6)

7 Algorthm 2: Vacancy Schedulng Algorthm Data: Users U(f ) assgned to femtocell f Result: Schedule S of resource blocks on femtocell f 1 R ; 2 B ; 3 φ B Max for u j U(f ) do 4 f ς uj > ψ then 5 R R {u j}; 6 else 7 B B {u j}; 8 end 9 end 10 sort R and B by number of RBs requred; 11 for u j R do 12 f b(ς uj ) φ then 13 assgn b(ς uj ) odd-numbered RBs n S to u j; 14 φ φ b(ς uj ); 15 else 16 reject u j; 17 end 18 end 19 for u j B do 20 f b(ς uj ) φ then 21 assgn b(ς uj ) even-numbered RBs n S to u j; 22 φ φ b(ς uj ); 23 else 24 reject u j; 25 end 26 end 27 f φ > 0 then 28 assgn remanng RBs n S to users n R B; 29 end 30 for f n N do 31 request vacances for u B; 32 end V. RESOURCE BLOCK SCHEDULING In the second phase of RRS, users are scheduled onto femtocells usng an nterference- and throughput-aware algorthm. The algorthm dvdes users nto two classes: regular and borderlne users. Regular users are users whose measured and estmated SINRs are suffcent to meet ther throughput targets. Borderlne users have SINRs that are close to, but below ther targets. Regular users are scheduled frst, n ascendng order of the estmated number of resource blocks they requre to meet ther throughput targets. If a user s unable to be scheduled enough resource blocks to meet ts throughput target, t s rejected. Regular users are assgned frst to even-numbered resource blocks; f needed, they are scheduled on odd-numbered blocks as well. Under all algorthm varants, borderlne users are defned as users whose SINRs are wthn 3 db below the lowest SINR threshold for servce. They are scheduled after all regular users have been scheduled. Borderlne users are scheduled frst on odd-numbered resource blocks; they are only scheduled on even-numbered resource blocks f there s excess capacty. Borderlne and regular users are scheduled n ths fashon to ncrease the probablty that borderlne user vacancy requests Varable Value Descrpton L 20 log 10 (f) + log 10 (d) 28 db Path loss [15] f 1900 Mhz Carrer frequency p tx M 40 dbm Macrocell tx power p tx F 24 dbm Femtocell tx power µ s 0 db Shadow fadng mean σ s 10 db Shadow fadng std dev n r 9 db Recever nose fgure T k -174 dbm/hz Thermal nose densty B Max 100 Max RBs per base staton TABLE II: Smulaton detals wll be satsfed. If free resource blocks reman after all regular and borderlne users have been scheduled, both regular and borderlne users are scheduled on them n a round-robn fashon. A. Vacancy Requests Borderlne users are unable to meet ther throughput targets as-s. Therefore, we ntroduce vacancy requests, whch allow a femtocell to ask ts neghbors to release scheduled resource blocks. Vacancy requests consst of a lst of resource block dentfers and duratons. Duratons are determned by the estmated number of transmsson frames each user wll need. Femtocells recevng vacancy requests wll only honor them on resource blocks that are assgned to users that exceed ther throughput targets. Ths ensures that users wll not fall below ther throughput targets due to vacancy requests. Vacancy requests ncrease farness at the expense of capacty by tradng resource blocks from users that exceed ther throughput targets to users that are close to meetng ther throughput targets. Snce borderlne users are preferentally scheduled to oddnumbered resource blocks, and regular users are preferentally scheduled to even-numbered resource blocks, f all regular users can be served at ther throughput targets wthout spllng over to the odd-numbered resource blocks, any vacancy requests from borderlne users on neghborng femtocells wll be satsfed. A. Expermental Detals VI. SIMULATIONS AND RESULTS The algorthms descrbed n Secton IV and V address two aspects of network operaton: transmt power, va radus reducton, and resource block schedulng. Recall that the radus reducton phase can be performed accordng two crtera: area coverage reducton (ACR) and user coverage reducton (UCR). Addtonally, recall that the radus reducton of a femtocell s performed wth dfferent prorty wth respect to ts neghbor femtocells, accordng to dfferent performance crtera, that are: estmated ncrease n total user throughput (TI), number of users meetng throughput requrements (UTR) and load of the femtocell (Load). Of the sx possble varants, n the followng experments, we consder the ACR-TI, the UCR-UTR, and the UCR-load varants. For each of these varants we consdered both the varants for the schedulng phase, namely wth or wthout vacancy requests. Our experments showed that the varants UCR-TI and ACR-UTR and ACR-load performed worse than UCR-UTR, ACR-TI, and UCR-load, respectvely. For the sake

8 Fg. 2: Outage probablty of brevty, we omt these results. We denote wth ACR-TI-V, UCR-UTR-V, UCR-load-V the varants of our algorthms n whch we schedule wth vacancy requests. We evaluated a baselne algorthm that does not perform any radus reducton, and uses a nave schedulng algorthm that schedules users n a best-effort manner wthout vacancy requests. In the fgures the baselne algorthm s referred to as Baselne. We also evaluated vacancy request schedulng on ts own, wth no radus reducton, ndcated n the graphs as Vacancy. To compare the performance of our algorthms to exstng work, we mplemented the Interferng Lnk Conflct algorthm, n the fgures referred to as Lnk Conflct, from [6]. Ths algorthm uses graph colorng to produce mutually exclusve resource block schedules, whle allowng unused resource blocks to be assgned f they meet certan nterference requrements. The orgnal paper descrbes a system wth bnary nterference, where nterference durng a transmsson results n zero throughput. By contrast, our smulated system s an SINR-based system, where nterference reduces sgnal qualty, but does not necessarly cause transmsson falure. Our results show that the colorng approach of the lnk conflct algorthm s unsuted for SINR-based systems. These algorthms were evaluated n a smulaton of an OFDMA network modeled usng LTE desgn prncples. Detals of the smulaton parameters are n Table II. Users and femtocells are unformly dstrbuted across a 50m 50m area of nterest. To quanttatvely measure farness, we use Jan s ndex [9], defned as f(x) = [ U j=1 xj]2 U U j=1 x2 j, where 0 f(x) 1, and t x j = j U. Jan s ndex s a commonly used measure of j=1 tj farness, wth f(x) = 1 when each user gets a 1 U share of total throughput, and f(x) = 1 U when one users gets all throughput. A hgher value for f(x) ndcates a more far throughput dstrbuton n the network. B. Results Fgure 2 shows the network outage probablty, defned as the percentage of users who are unable to reach ther throughput targets, as the number of concurrent users ncreases. We consdered a network composed of 30 femtocells, Fg. 3: Femtocell transmt power level where user throughput requrements are dstrbuted unformly random between 10 kbps and 50 kbps. The Baselne algorthm exhbts unform performance as the number of users ncreases, ndcatng that the network s underloaded. However, due to nterference, edge users are unable to meet ther throughput targets. UCR-UTR-V performs the best, closely followed by UCR-Load-V. The versons of the algorthms that do not use vacancy schedulng have hgher outage probabltes, but vacancy schedulng on ts own does not sgnfcantly reduce outage probabltes over the baselne. Vacancy schedulng benefts users at cell boundares the most, snce they experence the most nterference from other cells. However, when femtocell transmt powers are fxed, the reducton n nterference from vacancy schedulng s typcally not enough to ncrease an unserved user s throughput over ts throughput mnmum requrement. It s the combned reducton n nterference due to both reduced transmt powers and vacancy schedulng that ncreases throughput enough for more users to meet ther throughput requrements. The Lnk Conflct algorthm results n outage rates that are unacceptably hgh. Snce the algorthm elmnates conflctng transmssons between neghborng femtocells, t leaves a large number of resource blocks unassgned to prevent nterference. In a system where conflcts cause transmsson falure, ths leads to near-optmal performance, however, n systems that can tolerate co-channel nterference, the Lnk Conflct algorthm severely underutlzes avalable resources. Fgure 3 shows femtocell power levels as the number of users ncreases. UCR-UTR and UCR-Load consstently use less power than ACR-TI, resultng n smaller cells and lower nterference. The vacancy varants decrease outage probabltes wthout consumng more power, but gve up some resource block usage for reduced nterference. Ths tends to reduce the throughput of hgh-throughput users whle ncreasng the throughput of lower throughput users who are unable to reach ther mnmum throughput wthout assstance. The Baselne, Lnk Conflct, and Vacancy algorthms all use a constant amount of power as the number of users ncrease. Fgure 4 shows the total sum of throughput across all users. Intally, the Lnk Conflct and Baselne algorthms result n the hghest global throughput, although the Lnk Conflct algorthm s global throughput falls quckly as the number

9 Fg. 4: Global throughput of users ncreases. As the number of users ncreases, the throughput of the ACR-TI ncreases over the baselne. The ACR-TI algorthms always use less power than the baselne, ndcatng that the ACR-TI algorthms are able to reduce nterference and ncrease effcency. The UCR-UTR and UCR- Load algorthms decrease global throughput wth respect to the baselne, but do so whle ncreasng the number of users who meet ther throughput targets. The Jan s ndex of each algorthm, a measure of farness, s shown n Fgure 5. Recall that a Jan s ndex of 1 ndcates a perfectly unform throughput dstrbuton, wth each user 1 beng allocated U of the throughput, and a Jan s ndex 1 of U ndcates a completely unfar throughput dstrbuton, wth one user beng allocated all of the throughput. The UCR-Vacancy algorthms exhbt a U-shaped Jan s ndex curve, wth farness ncreasng agan as the number of users ncreases. The ncrease n farness at hgh load s due to the ncreased use of vacancy schedulng n the UCR varants as transmsson powers ncrease. Vacancy has a hgher Jan s ndex than Baselne, despte servng the same number of users, ndcatng a farer throughput dstrbuton. Due to ts overall poor performance, the Lnk Conflct algorthm has a very low farness ndex, as most users get no servce at all. Overall, the UCR-Load-V and UCR-UTR-V algorthms are able to trade off some total throughput to ncrease the number of users that are able to connect to the network. VII. CONCLUSION In ths paper we presented RRS, a dstrbuted algorthm for network management. RRS conssts of two parts: an algorthm for ncreasng farness n dense femtocell networks by managng femtocell transmsson power usng a Vorono-Laguerre geometry-based cell radus reducton, and a schedulng algorthm that allows femtocells to request vacances on resource blocks that are experencng heavy nterference. Smulatons show that RRS reduces outage probabltes by up to 50%, and ncreases Jan s ndex, a measure of farness, by up to 190%. ACKNOWLEDGMENTS Fg. 5: Jan s ndex The work of Novella Bartoln s partally supported by NATO under the SPS grant G4936 SONCS. REFERENCES [1] M. Arslan, J. Yoon, K. Sundaresan, S. Krshnamurthy, and S. Banerjee, A resource management system for nterference mtgaton n enterprse ofdma femtocells, Networkng, IEEE/ACM Transactons on, vol. 21, no. 5, pp , Oct [2] Y. Wang, J. Zhang, J. Wang, and B. Bensaou, Coordnated far resource sharng n dense ndoor wreless networks, n Networkng Conference, 2014 IFIP, June 2014, pp [3] T. Novlan, J. Andrews, I. Sohn, R. Gant, and A. Ghosh, Comparson of fractonal frequency reuse approaches n the ofdma cellular downlnk, n Global Telecommuncatons Conference (GLOBECOM 2010), 2010 IEEE, Dec 2010, pp [4] M. Pschella and J.-C. Belfore, Power control n dstrbuted cooperatve ofdma cellular networks, Wreless Communcatons, IEEE Transactons on, vol. 7, no. 5, pp , May [5] H. Fuj and H. Yoshno, Theoretcal capacty and outage rate of ofdma cellular system wth fractonal frequency reuse, n Vehcular Technology Conference, VTC Sprng IEEE, May 2008, pp [6] Z. Lu, T. Bansal, and P. Snha, Achevng user-level farness n openaccess femtocell-based archtecture, Moble Computng, IEEE Transactons on, vol. 12, no. 10, pp , Oct [7] J. Garca-Morales, G. Femenas, and F. Rera-Palou, Analytcal performance evaluaton of ofdma-based heterogeneous cellular networks usng ffr, n Vehcular Technology Conference (VTC Sprng), 2015 IEEE 81st, May 2015, pp [8] H. Sh, R. Prasad, E. Onur, and I. Nemegeers, Farness n wreless networks:ssues, measures and challenges, Communcatons Surveys Tutorals, IEEE, vol. 16, no. 1, pp. 5 24, Frst [9] R. Jan, D. Chu, and W. Hawe, A quanttatve measure of farness and dscrmnaton for resource allocaton n shared systems, Dgtal Equpment Corporaton, Tech. Rep. DEC-TR-301, [10] F. Baccell and B. Blaszczyszyn, On a coverage process rangng from the boolean model to the posson vorono tessellaton wth applcatons to wreless communcatons, INRIA, Tech. Rep. RR-4019, [11] H. Ima, M. Ir, and K. Murota., Vorono dagram n the Laguerre geometry and ts applcatons, SIAM J. Comput., vol. 14, no. 1, pp , [12] N. Bartoln, T. Calamoner, T. La Porta, and S. Slvestr, Autonomous deployment of heterogeneous moble sensors, IEEE Trans. on Moble Computng, vol. 10, no. 6, [13] Z. Zhou, S. Das, and H. Gupta, Varable rad connected sensor cover n sensor networks, ACM Trans. on Sensor Networks, vol. 5, no. 1, [14] N. Bartoln, T. Calamoner, T. La Porta, C. Petrol, and S. Slvestr, Sensor actvaton and radus adaptaton (sara) n heterogeneous sensor networks, ACM Trans. on Sensor Networks, vol. 8, no. 3, [15] ITU, Recommendaton tu-r p , Internatonal Telecommuncaton Unon, Recommendaton ITU-R P , The work of Mchael Ln s supported by the Wllam Leonhard Char at the Pennsylvana State Unversty, wth partal support from NSF grant CNS