Optimized Opportunistic Multicast Scheduling (OMS) over Wireless Cellular Networks

Transcription

1 MITSUBISHI ELECTRIC RESEARCH LABORATORIES Optimized Opportuistic Multicast Schedulig OMS) over Wireless Cellular Networks Tze-Pig Low, Ma-O Pu, Y.-W. Peter Hog, C.-C. Jay Kuo TR March 010 Abstract Optimized opportuistic multicast schedulig OMS) is studied for cellular etworks, where the problem of efficietly trasmittig a commo set of foutai-ecoded data from a sigle base statio to multiple users over quasi-static fadig chaels is examied. The proposed OMS scheme better balaces the tradeoff betwee multiuser diversity ad multicast gai by trasmittig to a subset of users i each time slot usig the maximal data rate that esures successful decodig by these users. We first aalyze the system delay i homogeeous etworks by capitalizig o extreme value theory ad derive the optimal selectio ratio i.e., the portio of users that are selected i each time slot) that miimizes the delay. The, we exted results to heterogeeous etworks where users are subject to differet chael statistics. By partitioig users ito multiple approximately homogeeous rigs, we tur a heterogeeous etwork ito a composite of smaller homogeeous etworks ad drive the optimal selectio ratio for the heterogeeous etwork. Computer simulatios cofirm theoretical results ad illustrate that the proposed OMS ca achieve sigificat performace gais i both homogeeous ad heterogeeous etworks as compared with the covetioal uicast ad broadcast schedulig. IEEE Trasactios o Wireless Commuicatios This work may ot be copied or reproduced i whole or i part for ay commercial purpose. Permissio to copy i whole or i part without paymet of fee is grated for oprofit educatioal ad research purposes provided that all such whole or partial copies iclude the followig: a otice that such copyig is by permissio of Mitsubishi Electric Research Laboratories, Ic.; a ackowledgmet of the authors ad idividual cotributios to the work; ad all applicable portios of the copyright otice. Copyig, reproductio, or republishig for ay other purpose shall require a licese with paymet of fee to Mitsubishi Electric Research Laboratories, Ic. All rights reserved. Copyright c Mitsubishi Electric Research Laboratories, Ic., Broadway, Cambridge, Massachusetts 0139

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3 TW-MAR : OPTIMIZED OPPORTUNISTIC MULTICAST SCHEDULING OVER WIRELESS CELLULAR NETWORKS 1 Optimized Opportuistic Multicast Schedulig OMS) over Wireless Cellular Networks Tze-Pig Low, Studet Member, IEEE, Ma-O Pu, Member, IEEE, Y.-W. Peter Hog, Member, IEEE, ad C.-C. Jay Kuo, Fellow, IEEE Abstract Optimized opportuistic multicast schedulig OMS) is studied for cellular etworks, where the problem of efficietly trasmittig a commo set of foutai-ecoded data from a sigle base statio to multiple users over quasi-static fadig chaels is examied. The proposed OMS scheme better balaces the tradeoff betwee multiuser diversity ad multicast gai by trasmittig to a subset of users i each time slot usig the maximal data rate that esures successful decodig by these users. We first aalyze the system delay i homogeeous etworks by capitalizig o extreme value theory ad derive the optimal selectio ratio i.e., the portio of users that are selected i each time slot) that miimizes the delay. The, we exted results to heterogeeous etworks where users are subject to differet chael statistics. By partitioig users ito multiple approximately homogeeous rigs, we tur a heterogeeous etwork ito a composite of smaller homogeeous etworks ad derive the optimal selectio ratio for the heterogeeous etwork. Computer simulatios cofirm theoretical results ad illustrate that the proposed OMS ca achieve sigificat performace gais i both homogeeous ad heterogeeous etworks as compared with the covetioal uicast ad broadcast schedulig. I. INTRODUCTION The ever-icreasig demad for richer multimedia cotets has drive the developmet of highly efficiet cotet distributio techologies over wireless etworks. I particular, opportuistic schedulig has recetly emerged as oe of the most promisig techiques for cotet delivery. These methods effectively icrease throughput with reduced delay by exploitig multiuser diversity iheret i wireless etworks [1] [3]. Most previous research o opportuistic schedulig has focused o applicatios where the base statio BS) schedules differet data to each user. Here, we cocetrate o broadcast Mauscript received March 16, 009; revised August 7, 009; accepted October 5, 009. The associate editor coordiatig the review of this paper ad approvig it for publicatio was F. A. Cruz-Perez. This research was supported i part by the Croucher Foudatio uder a post-doctoral fellowship, ad i part by the Natioal Sciece Coucil, Taiwa, uder grat E This work was preseted i part at the IEEE Globecom 008, New Orleas, LA, U.S.A, November 008 ad i part at the IEEE Iteratioal Coferece o Acoustics, Speech ad Sigal Processig ICASSP) 009, Taiwa, April 009. Tze-Pig Low ad C.-C. Jay Kuo are with the Mig Hsieh Departmet of Electrical Egieerig, Uiversity of Souther Califoria, Los Ageles, CA 80089, USA s: tlow@usc.edu; cckuo@sipi.usc.edu). Ma-O Pu is with the Mitsubishi Electric Research Laboratories MERL), Cambridge, MA 0139, USA. This work was doe whe he was a Croucher post-doctoral research fellow at Priceto Uiversity, Priceto, NJ mpu@merl.com). Y.-W. Peter Hog is with the Istitute of Commuicatios Egieerig Natioal Tsig Hua Uiversity, Hsichu 30013, Taiwa ywhog@ee.thu.edu.tw). Digital Object Idetifier xxxx applicatios i a sigle-cell system where the BS seds a commo set of data to multiple users udergoig idepedet chael fadig. I fact, with the icreased popularity of commercial services such as mobile TV, up-to-date iformatio distributio services e.g., ews, traffic, stock market etc), ad file distributio e.g., applicatios, software upgrades etc) [4], [5], the eed for such schedulig schemes has geerated strog iterests i recet years. I this work, our goal is to develop efficiet schedulig schemes for such applicatios by exploitig both multiuser diversity ad multicast gai. We cosider the problem of trasmittig a commo set of data to all users o the dowlik of a wireless cellular etwork. To achieve this task, two schedulig schemes have bee studied extesively i the literature. They are the opportuistic uicast schedulig scheme ad the broadcastig scheme. The former exploits multiuser diversity by limitig the BS to trasmit to the user with the best istataeous chael i each time slot at the user s highest supportable data rate. I this case, oly oe user is served at oe time ad the trasmissio must be repeated multiple times util all users receive the whole message. I cotrast, the latter exploits the broadcast ature of the wireless medium by trasmittig at a rate that is supported by all users i the etwork. I this case, all users are served at oce but the rate is costraied by the user with the poorest chael coditios i order to prevet chael outages. Beig motivated by recet advaces i erasure codes ad foutai codes [6], [7], opportuistic multicast schedulig OMS) schemes have bee proposed recetly to better balace the tradeoff betwee multiuser diversity ad the multicast gai [8] [11]. Assumig that each user experieces idepedet fadig over differet time slots, the OMS schemes select a differet group of users for service i each time slot at a rate supported by the worst user i the selected group. The adoptio of erasure ad foutai codes allows users to recover the full origial cotet of a message oce a miimum set of ecoded symbols is received, regardless of the specific received sequece of ecoded symbols. This is i sharp cotrast with covetioal schemes without erasure or foutai codes where the BS must keep track of the data that each user has received throughout the cotet delivery process. This is obviously a huge burde for etworks with a large umber of users. The work i [8], [9] adopted the media OMS scheme where the best half of the users are served i each time slot. Despite its good performace as compared to uicast ad broadcast, the optimality of fixig the selectio ratio at 50% has ot bee clearly addressed. The mai cotributio of this work is to derive the optimal

4 TW-MAR : OPTIMIZED OPPORTUNISTIC MULTICAST SCHEDULING OVER WIRELESS CELLULAR NETWORKS user selectio ratio that miimizes the total umber of time slots required for all users to successfully receive the commo set of messages. We first focus o homogeeous etworks where all users experiece idepedet ad idetically distributed i.i.d.) chael coditios. I this case, the optimal user selectio ratio ca be derived aalytically usig cetral order statistics ad extreme value theory [1]. Next, we exted results to heterogeeous etworks where users are assumed to be radomly distributed i a cellular etwork. Clearly, the system performace is costraied by the cell-edge users that have the worst average chael coditios. By exploitig this observatio, we devise OMS schemes with the absolute ad the ormalized istataeous SNR criteria by optimizig the performace of cell-edge users. Before proceedig further, some characteristics differetiatig our work from other existig studies o OMS [10], [11] ad adaptive trasmissio schemes [13] should be emphasized. Specifically, Ge et al. [10] proposed a threshold-based scheme that trasmits i the curret time slot oly if more tha G users where G is a pre-defied umber) ca support a predefied rate R. I cotrast, our scheme ivestigates the optimal user selectio ratio with a adaptive target data rate i each time slot. Aother closely related work was coducted by Kozat [11], where the performace of OMS is studied i terms of system throughput by cosiderig ifiite ecoded messages. The goal of our curret research is to miimize the time duratio required for trasmittig a fiite-legth message to all users. By capitalizig o extreme value theory, we are able to provide theoretical aalysis o the system performace. Fially, despite the fact that both our work ad [13] explore the tradeoff betwee multicast ad uicast schemes, our work develops a uified theoretical framework to compute the optimal percetage of users served i each trasmissio regardless of uicast or multicast desig. The rest of this paper is orgaized as follows. We first itroduce the system model i Sectio II. The, uder the assumptio that all users experiece i.i.d. fadig, OMS with optimized user selectio ratios is proposed to miimize the system delay for homogeeous etworks i Sectio III. By extedig results i Sectio III, we develop OMS for heterogeeous etworks i Sectio IV. Computer simulatio results are show i Sectio V. Fially, cocludig remarks are give i Sectio VI. The followig covetio is adopted i this paper: N µ, σ ) ad CN µ, σ ) beig the real- ad complex-valued Gaussia distributios with mea µ ad variace σ while log ) ad l ) deotig the logarithm operators of base ad e, respectively, ad beig the largest iteger o larger tha the eclosed quatity. Fially, E [ ] ad Var [ ] deote the expected value ad the variace of the eclosed radom variable, respectively. II. SYSTEM MODEL Cosider a sigle-atea dowlik cellular etwork where the BS is to trasmit a commo message to N users i the etwork. We examie a time-slotted system where the legth of each time-slot, T, is comparable with the chael Fig. 1. Illustratio of opportuistic multicast schedulig OMS). coherece time. As a result, the chael coefficiets are assumed to remai costat throughout each time slot but vary idepedetly from oe time slot to aother. Let h k) be the istataeous chael coefficiet betwee the BS ad the - th user i the k-th time slot. We assume that {h k),, k} are idepedet amog users ad i.i.d. over differet time slots. To distribute the commo message to N users, we adopt a multicast schedulig scheme where a subgroup of users is served simultaeously i each time slot with a rate that depeds o the istataeous chael coditio of the users. We assume that the message size S is large or that the slot duratio T is small) so that the distributio of the commo message to all N users must be completed over multiple time slots. By employig a rateless ecodig scheme, such as the foutai code [7], the BS ca geerate a cotiuous stream of source bits from the iteded message so that ay collectio of S bits from the data stream ca guaratee reliable recovery of the origial message. I the proposed trasmissio scheme, a differet portio of the data stream is trasmitted i each time slot ad a user is assumed to be able to recover the origial message wheever it collects a total of S bits over a series of time slots, regardless of which portios of the bit stream were received. It is assumed that the BS has kowledge of the istataeous sigal-to-oise ratio SNR) of each user at the begiig of every time slot. Based o the istataeous SNR, the BS will select a group of target users, whose idex set is deoted by Ik) for the k-th slot, ad trasmit at a rate costraied by the lowest-snr user amog all selected users. The SNR of the -th user i the k-th time slot takes the followig form: γ k) = P h k) N 0 L p D ), 1) where P is the trasmissio power, N 0 is the oise variace, ad L p D ) is the path loss over the distace betwee the

5 TW-MAR : OPTIMIZED OPPORTUNISTIC MULTICAST SCHEDULING OVER WIRELESS CELLULAR NETWORKS 3 BS ad the -th user i.e., D ). Thus, the maximal rate supportable by the -th user i the k-th time slot is give by r k) = log 1 + γ k) ). ) Subsequetly, the maximal data rate that the BS ca trasmit without icurrig chael outage to the selected users i I k is give by Rk) = mi r k). 3) Ik) Therefore, each user i Ik) will successfully receive Rk) T source bits i the k-th slot. Let K be the total umber of time slots required for all N users to successfully recover the message i.e., the system delay). Clearly, there exists a desig tradeoff betwee the sequece of multicast groups {Ik), k 1} ad the system delay K. O the oe had, selectig more users will improve the multicast gai at the cost of a lower data rate Rk). O the other had, selectig fewer users eables higher data rates by better exploitig the multiuser diversity gai but limits the umber of users that are served simultaeously i each time slot. I the followig, optimized OMS is devised by carefully takig ito accout the tradeoff betwee multicast gai ad multiuser diversity i both homogeeous ad heterogeeous etworks. III. OMS IN HOMOGENEOUS NETWORKS We begi with homogeeous etworks where all users experiece i.i.d. fadig with the same average SNR, ρ 0. Mathematically, ρ 0 ca be defied as show i 1) with L p D ) = 1 for all : [ P h k) ] ρ 0 = E [γ k)] = E. 4) Without loss of geerality, let F γ0 z) be the cumulative distributio fuctio CDF) of the istataeous SNR of each user. From ), the CDF of each user s supportable data rate ca be expressed as N 0 F r y) = Pr {r = log1 + γ) y} = F γ0 y 1) 5) For example, uder the Rayleigh fadig sceario, the chael coefficiet h k) is assumed to be circularly symmetric complex Gaussia with zero mea ad uit variace, i.e. h k) CN 0, 1), ad the CDF of the istataeous SNR ad the supportable data rate are give by ) F γ0 z) = 1 exp zρ0, 6) ad ) F r y) = 1 exp y 1, 7) ρ 0 respectively. It is worthwhile to remark that o specific fadig distributios are assumed i this work ad, thus, the results are geerally applicable to arbitrary fadig scearios. To facilitate the followig discussio, we rak users accordig to their istataeous SNR i the k-th time slot ad defie a order-mappig fuctio as π k i) for i = 1,,, N with ordered istataeous SNR give by γ πk 1)k) γ πk )k)... γ πk N)k), 8) or equivaletly ordered data rates r πk 1)k) r πk )k) r πk N)k). Note that the positio of the -th idexed user i the k-th time slot ca be easily determied by ivertig the order-mappig fuctio, i.e., π k ). I the proposed OMS scheme, the BS trasmits to a fixed umber of users, deoted by M, i each time slot regardless of how may bits they have previously received. Specifically, i the k-th time slot, the BS chooses the multicast set Ik) = {π k 1), π k ),..., π k M)} ad trasmits at rate Rk) = r πm) so that oly the users i the selected set ca reliably decode the trasmitted data. It is assumed that the BS keeps a updated record of the istataeous SNRs of all users ad that it is able to determie the amout of iformatio each user has decoded i all previous time slots. The process completes whe all users are able to accumulate S ecoded bits. Hece, the system delay of the OMS scheme ca be computed as a fuctio of M, i.e., { KM) = mi k : mi [ k j=1 ] 1 {π j ) M} r π j M)j) T S where the idicator fuctio 1 {π j ) M} is defied as 1 {π { 1 if π j ) M} = j ) M 0 otherwise, }, 9) 10) which idicates whether or ot user is icluded i the selected group of the j-th time slot. It should be emphasized that 9) provides a geeralized form of may existig schedulig schemes, such as those give i the followig. Covetioal broadcast The broadcast scheme that schedules trasmissio to all users i each time slot is a special case of 9) with M = N, ad the system delay is give by { k } K BC KN) = mi k : r πjn)j) T S. j=1 11) Opportuistic uicast The opportuistic uicast schedulig method is a special case of 9) with M = 1 ad the correspodig system delay is K UC K1) = { { k } } mi k : mi 1 {πj1)=}r j) T S. 1) j=1 Media OMS The media user OMS scheme proposed i [8], [9] is the case with M = N/, assumig homogeeous users. The

6 TW-MAR : OPTIMIZED OPPORTUNISTIC MULTICAST SCHEDULING OVER WIRELESS CELLULAR NETWORKS 4 resultig system delay is give by K OMS med K N ) = { { k mi k : mi j=1 1 {π j ) N }r π j N )j) } } T S. 13) I the followig, we aalyze, for ay give N, the optimal multicast group size M or, equivaletly, the optimal user selectio ratio, α = M N that miimizes the average system delay, i.e., E[KM)]. To obtai aalytically tractable results, we ivestigate the asymptotic performace i terms of the average trasmissio rate ad system delay for large etworks i.e., large N) by resortig to extreme value theory [1]. Lemma 1: Let F r y) ad f r y) be the CDF ad the probability desity fuctio PDF) of the data rate, which are assumed to be cotiuous. Give α 0, 1) with f r Fr 1 M α)) 0, it follows that, if lim = α, the N N [ N rπk M)k) Fr 1 α) ] N 0, σ α) ), 14) with σ α) = α1 α) [ fr F r 1 α) )]. 15) The proof follows directly from the asymptotic ormality of cetral order statistics stated i Theorem 7.5 i [1]. More specifically, it follows from 14) that Var [ r πk M)k) Fr 1 α) ] = [ rπk E M)k) Fr 1 α) 0 ) ] = 1 N σ α), 16) which goes to zero as N approaches ifiity. This implies that r πk M)k) coverges to Fr 1 α) i the mea square sese as N goes to ifiity. That is, r πk M)k) ca be approximated by Fr 1 α) for sufficietly large N. For example, uder the Rayleigh fadig, the asymptotic trasmissio rate is give by lim r π k M)k) = log 1 ρ 0 l M ), N N where α is the user selectio ratio to be optimized. Cosequetly, the umber of times that each user must be selected to successfully receive S source bits is approximated by S κ = Fr 1 α) T. 17) Next, we aalyze the asymptotic characteristic of the system delay. For otatioal coveiece, we use X,K to deote the umber of time slots that the -th user is chose over total K time slots. Mathematically, it ca be expressed as X,K = K k=1 1 {π k ) M}. 18) By ivokig the cetral limit theorem, we ca approximate X,K as a Gaussia distributed radom variable such that X,K N Kα, Kα1 α)). 19) Clearly, the system delay is bottleecked by the user that is chose the least umber of times over the K time slots ad ca be characterized by X mi,k = mi X,K. 0) To study the asymptotic characteristics of X mi,k, we eed the followig result from extreme value theory [1]. Lemma : Let X mi = mix where X, for = 1,, N, is a set of i.i.d. radom variables with CDF F X x). If the followig coditio holds, lim N N F X F X 1 N ) + x [F 1 X N e ) F 1 ) ]) X N = e x, 1) there exists ormalizig costats c N ad d N such that X mi c N )/d N coverges i distributio to the Gumbel distributio for miima. The ormalizig costats ca be determied by ) 1 c N = F X N d N = c N F X ) ) 1. 3) N e I this case, we say that F X x) lies i the Domai of Attractio of the Gumbel distributio for miima. Proof: The proof follows from Theorem 3.4 i [1]. As show i [1], the Gaussia CDF lies i the Domai of Attractio of the Gumbel distributio for miima. Thus, by the approximatio i 19) ad by Lemma, we kow that X mi,k c N )/d N coverges to the Gumbel distributio with ormalizig costats give as ) + µ x 4) 1 c N = σ x Φ N [ 1 d N = σ x Φ N ) Φ 1 N e )], 5) where Φ ) is the stadard Gaussia CDF while σ x ad µ x are the stadard deviatio ad mea of X, respectively. By exploitig the properties of the Gumbel distributio [1], [14], we have ad, E [X mi,k ] c N + ε)d N, 6) [ ) )] 1 1 = σ x 1 ε) Φ + ε Φ + µ x, 7) N N e Var [X mi,k ] = π 6 d N 8) where ε = is the Euler-Mascheroi costat, µ x = Kα, ad σ x = Kα1 α). It is straightforward to show from 8) that Var [X mi,k ] = O1), 9) for sufficietly large N, which implies that the system delay variace is isesitive to the umber of users i a homogeeous system. This observatio will be cofirmed by

7 TW-MAR : OPTIMIZED OPPORTUNISTIC MULTICAST SCHEDULING OVER WIRELESS CELLULAR NETWORKS 5 simulatio i Sec.V. By settig E[X mi,k ] = κ, where κ is defied as i 17), we ca compute the umber of time slots K required for all users to be selected at least κ times. This value ca the be used to approximate the average system delay as E[KαN)] K [ ξ α1 α) + ] ξ = α1 α) + 4ακ, α 30) where ξ = 1 ε) Φ ) 1 N + ε Φ 1 N e). The optimal user selectio ratio for homogeeous etworks ca the be obtaied by solvig the followig optimizatio problem: αhomo = arg mi E [KαN)], 31) α where K ) is give i 9). It is iterestig to ote that the optimizatio i 31) is oly over a sigle parameter, α 0, 1], ad ca be solved efficietly usig a simple lie search. I practice, the user selectio ratio αhomo = M N ca take oly N possible values sice the umber of users selected i each time slot is a iteger, i.e. M = 1,,..., N. Hece, the lie search has a complexity that is liear with N ad, thus, ca be doe efficietly i practice. Fially, the operatios of the proposed opportuistic schedulig scheme ca be summarized as follows. Whe multicast service is requested by the users, the BS first computes the optimal user selectio ratio αhomo by substitutig ito 31) the average SNR ρ 0, the total umber of users N, ad the time slot duratio T. The BS trasmits the message to all users over multiple time slots durig which the rates are chose adaptively accordig to the selected group of users i each time slot. Specifically, at the begiig of each time slot, the BS raks the istataeous SNR of all users ad chooses a data rate that is supported oly by the best αhomo N users. The istataeous SNR ca be derived by either exploitig chael reciprocity i time-divisio duplex TDD) systems or receivig feedback from each user i frequecy-divisio duplex FDD) systems. By assumig that the source data are ecoded by erasure or foutai codes, each user will be able to decode the etire message wheever it receives a total of S bits over the sequece of time slots, regardless of which specific time slots it was able to receive from. This process cotiues util all users are able to successfully decode the etire message. IV. OMS IN HETEROGENEOUS NETWORKS I this sectio, we exted results of the last sectio to heterogeeous etworks comprisig of N active users uiformly distributed aroud the BS i a circular cell of radius D max as illustrated i Fig.. The PDF of the distace D betwee the -th user ad the BS is give by [15] f D d) = d D max, 0 < d D max. 3) We model the effect of path loss betwee the BS ad the - th user as L p D ) = ɛd β, where ɛ ad β are the path loss Fig.. Homogeeous rigs of users i the heterogeeous cell. costat ad expoet, respectively. By takig ito accout the path loss, the average SNR experieced by the -th user ca be expressed as ρd ) = E [γ k)] = ρ 0 ɛd β. 33) Subsequetly, for ay give D, the CDF of the istataeous SNR of the -th user durig the k-th time slot, deoted by γ k), ca be expressed as F γ D z) = Pr {γ k) = P h k) } N 0 ɛd β z D ) = F γ0 z ɛd β, 34) where F γ0 z) was defied i the previous sectio as the CDF of γ k) with L p D ) = 1, ad the correspodig CDF of the data rate as F r D y) = F γ0 y 1) ɛd β ). 35) I the followig, we ivestigate two differet schemes i applyig OMS: the absolute-snr scheme ad the ormalized- SNR scheme. I the absolute-snr scheme, the actual istataeous SNR is employed to compute the user selectio ratio whereas the ormalized-snr scheme first ormalizes each user s istataeous SNR with respect to its average SNR before applyig OMS. Similar to the proportioal fair schedulig PFS) scheme i [] that employs the ormalized SNR to esure fairess amog differet users, the ormalized- SNR scheme is devised to icrease the chace of selectig the weaker users whose performace hiders the system delay. A. OMS with Absolute SNR It is ot surprisig to observe that the average system delay i heterogeeous etworks is typically limited by the users farthest away from the BS, i.e., cell-edge users. For sufficietly dese etworks, we ca assume that a rig of users at the

8 TW-MAR : OPTIMIZED OPPORTUNISTIC MULTICAST SCHEDULING OVER WIRELESS CELLULAR NETWORKS 6 cell edge, which is referred to as the edge group, form a homogeeous group with average SNR approximately equal to ρ edge = ρ 0 ɛdmax β. 36) Cocetratig o this cell-edge group, we use αedge to deote the optimal selectio ratio for the edge group accordig to 31). Sice the delay performace of the heterogeeous etwork is domiated by this edge group, we argue that users selected by the heterogeeous OMS scheme should iclude exactly αedge proportio of users i this group. Based o this observatio, we derive i the followig optimal user selectio ratio for heterogeeous etworks, αabs. From Lemma 1, we kow that, i a homogeeous group of N edge users, the rate of the U-th user, where U = α edge N edge, will coverge to a costat r πk U)k) F r D max 1 α edge ) rth 37) for large N edge. I other words, as log as the OMS scheme chooses a multicast group that results i the trasmissio rate r th, the optimal portio of users i the edge group, i.e. αedge, will be chose. Suppose that M = α absn users are chose i each time slot i the heterogeeous OMS scheme. Iterestigly, by assumig that the locatio of users are i.i.d., we ca also show by Lemma 1 that the rate trasmitted i each time slot coverges to costat Fr 1 αabs ), where the CDF of the rate r is ow averaged over the radom locatio of the user, i.e., F r y) = F r D =uy) f D u)du 38) = Dmax 0 F r D =uy) u D max du. 39) The user selectio ratio αabs must be chose to satisfy 1 αabs ) = r th. Thus, we have F r α abs = 1 F r r th ). 40) For example, i the case of Rayleigh fadig, this is give by αabs = Dmax l α ) edge Dmax u exp u β du, 41) 0 = 1 D max a) = 1 Dmax = 1 + m=1 D max 0 D max D β max l α edge exp 1 + D β max z β/ ) dz, 4) [ lα ) edge ) m z mβ ]) dz, 0 m=1 Dmax β m! 43) l αedge )m ), 44) m! mβ + 1 where a) follows from the Taylor series expasio of the expoetial fuctio about z = 0. The system delay of the absolute SNR scheme is where K is give i 9). K OMS abs = Kα absn), 45) B. OMS with Normalized SNR I the previous subsectio, the proposed OMS scheme selects the best M = αabs N users accordig to their absolute istataeous SNR values. This method is simple, ad it requires o kowledge of the average SNR of each user. However, it suffers from the disadvatage that users far away from the BS are ofte deprived of their opportuity to trasmit eve if they have reached their relative maximal SNR. To address this issue, we propose a alterative method where the OMS is derived based o the ormalized istataeous SNR values with respect to the average SNR. Let us defie the ormalized istataeous SNR of the -th user as ˆγ k) = γ k), 46) ρ where ρ = ρd ) is the average SNR of the -th user. 1) Two-User Example: We first use a two-user etwork to provide some isights. Cosider a etwork cosistig of two users: User is closer to the BS tha User 1, i.e. ρ > ρ 1. For such a simplified etwork, the uicast scheme selects either User 1 or User, while the broadcast scheme selects both users at the same time. I cotrast, the OMS scheme selects either oe or both users depedig o the achievable delay performace. Iterestigly, it is show i the Appedix that whe the OMS scheme selects oe user, the delay experieced by User 1 uder the ormalized-snr scheme is always smaller tha that uder the absolute-snr scheme, i.e., K1 abs K1 orm. Recall that delay is limited by User 1 ad that, whe the OMS scheme selects two users, the delay is the same for both absolute-snr ad ormalized-snr versios. As a result, the ormalized-snr OMS scheme will perform better or equally well as compared to the absolute-snr OMS scheme for the two-user case. Simulatio results will be provided i Sec. V-B1 to cofirm the aalysis reported i the Appedix. ) Multiple-User Case: Thus ispired, we establish the ormalized-snr scheme for geeral etworks with N active users. Followig the precedig aalysis, let us defie the ordered-mappig fuctio φ k i) for i = 1,,, N accordig to the ormalized SNR such that ˆγ φk 1)k) ˆγ φk )k)... ˆγ φk N)k). 47) The positio of the -th idexed user i the order is give by the iverse fuctio φ k ). I the ormalized-snr OMS scheme, the scheduler selects a group of M = αn users idexed by Îk) = { : φ ) M} for trasmissio, where α is the user selectio ratio. Sice the ormalized SNR s are i.i.d., all users are selected with equal probability α. Furthermore, because delay is limited by the edge group users, the OMS scheme should set the ormalized selectio ratio equal to the optimal αedge accordig to 31), i.e., α orm = αedge. To esure that all users i Îk) receive the trasmissio, the BS eeds to trasmit at a rate costraied by the worst user i Îk); amely, R orm k) = mi Îk) r k) = mi Îk) log1 + γ k)). 48) It is importat to ote that this rate is typically smaller tha the M-th best user accordig to the absolute istataeous

9 TW-MAR : OPTIMIZED OPPORTUNISTIC MULTICAST SCHEDULING OVER WIRELESS CELLULAR NETWORKS 7 SNR. As a result, a total of ˆMk) users are able to receive the trasmissio i the k-th time slot with ˆMk) = max =1,...,M π k φ k )). 49) Thus, the system delay for the ormalized SNR scheme is give by { mi Korm OM = k : mi [ k j=1 1 {π j ) ˆMj)} R ormj) ] } T S. 50) For etworks with a large umber of users, we expect that the umber of users actually selected i the ormalized-snr scheme will also coverge to a costat ad the differece betwee the two schemes will be egligible. V. SIMULATION RESULTS I this sectio, we study the performace of the proposed OMS schemes via computer simulatio. Uless otherwise specified, we set N = 100 so that a selectio ratio of α = M% effectively selects M users i the followig experimets. The Rayleigh fadig model is assumed throughout the experimets. A. Homogeeous etworks We first verify the accuracy of the optimal selectio ratios of a homogeeous etwork predicted by 31). Fig. 3 shows the simulated optimal selectio ratios i.e., the user selectio ratios obtaied by miimizig the simulated average delay) ad the aalytically optimized ratios usig 31) with respect to the average SNR. We ca see that the media OMS that fixes the selectio ratio at 50% may ot be optimal for all SNR values. It is iterestig to ote that, i practice as well as i the simulatios, the umber of time slots eeded for each user to receive the etire message must be a iteger value, causig a quatizatio effect that is ot well predicted by our asymptotic aalysis. This quatizatio effect is less critical at low SNR where the umber of time slots eeded for each user to receive the etire message is large, but it is more evidet at moderate to high SNR where the etire message may be received withi oly a few time slots. Iterestigly, this effect may also yield more tha oe locally optimal values of αhomo. I fact, we observed i our experimets two locally optimal solutios of αhomo for certai SNR values. The two locally optimal user selectio ratios yield small differece i delay ad ca be used equally effective i the proposed OMS scheme. Both of these solutios are plotted i Fig. 3. The aalytical approach preseted i 31) is able to predict oe of these solutios with good accuracy. Next, we compare the delay performace of the proposed OMS scheme agaist three other existig schedulig schemes metioed i Sectio III i.e., the covetioal broadcast, uicast, ad the media OMS schemes) i Fig. 4. Ispectio of Fig. 4 cofirms the covetioal wisdom: the uicast scheme outperforms the broadcast scheme at the low SNR regio by exploitig the multiuser diversity gai while the broadcast scheme has more advatages whe the multicast gai becomes Selectio Ratio, α * OMS Homogeeous Aalysis) OMS Homogeeous Sim. lower mi) OMS Homogeeous Sim. upper mi) Average SNR, db Fig. 3. Compariso of theoretical ad simulated optimal selectio ratios i homogeeous etworks. Delay Uicast Broadcast Media OMS Cell Edge Average SNR, db Fig. 4. Performace compariso of four multicast schemes i homogeeous etworks. more domiat with the icrease i SNR. The media OMS scheme outperforms both uicast ad broadcast schemes for SNR below 0 db. For very large SNR values, the OMS scheme chooses to serve all users i every time slot, which results i the delay performace covergig to that of the broadcast scheme. Fially, i Fig. 5, we plot the average OMS delay ad its variace with respect to the umber of users at SNR of 10 db. It is easy to show from 30) that the average delay is O ξ ) = O l N), which is cofirmed i Fig. 5. Furthermore, ispectio of Fig. 5 also reveals that the delay variace is isesitive to the umber of users, which agrees

10 TW-MAR : OPTIMIZED OPPORTUNISTIC MULTICAST SCHEDULING OVER WIRELESS CELLULAR NETWORKS SNR = 10 db Delay mea Delay Broadcast Uicast absolute SNR) Uicast ormalized SNR) OMS absolute SNR) OMS ormalized SNR) Delay variace Number of users, N Fig. 5. Delay performace as a fuctio of the umber of users with SNR of 10dB SNR Separatio ρ ρ ) db 1 Fig. 6. Delay performace as a fuctio of SNR separatio for the two-user example with ρ 1 = 0dB. well with the discussio i Sec. III. B. Heterogeeous etworks Here, we simulate the heterogeeous sceario ad provide further isight ito the performace of the proposed OMS scheme whe users chael statistics are ot i.i.d.. 1) Two-User Example: We first perform simulatio to cofirm the aalysis o two-user etworks reported i the Appedix. I Fig. 6, we simulate the delay performace of the two-user example where User 1 is at a fixed distace from the BS ad has a average SNR of 0dB. The average system delay is depicted for a rage of SNR separatio betwee the two users. Fig. 6 idicates that the performace gai by ormalizig the SNR icreases as the separatio icreases ad attais its maximum at a SNR separatio of about db. For a SNR separatio larger tha 4.5 db, the broadcast scheme is optimal amog all schemes uder cosideratio. Similar results ca also be observed i Fig. 7, which is plotted by icreasig the edge user s average SNR to 0 db e.g. a smaller cell). The system delay is reduced for all schemes due to the higher average SNR values. However, the improvemet achieved by the ormalized-snr scheme remais o-egligible for a SNR separatio uder db. ) Networks with more tha two users: Next, we simulate geeral heterogeeous etworks with more tha two users. The users are distributed uiformly i a circular cell area of radius D max = ρo ɛρ edge ) 1 β, with path-loss costat ɛ = ad path-loss expoet β = 3.5 [16]. We first examie the impact o the whole system delay due to the worst users i the etwork. I Fig. 8, we fix the cell area ad the average cell-edge SNR at 0 db ad 0 db. The, we icrease the umber of users i the cell from 10 to 1000 ad uiformly distribute them over the cell. Note that the system delay performace of the etwork is limited by the two outermost Delay Broadcast Uicast absolute SNR) Uicast ormalized SNR) OMS absolute SNR) OMS ormalized SNR) SNR Separatio ρ ρ ) db 1 Fig. 7. Delay performace as a fuctio of SNR separatio for the two-user example with ρ 1 = 0dB. users havig the worst average SNR. Hece, we ivestigate the delay performace as the average SNR separatio betwee the two outermost users chages. Clearly, as the umber of users icreases, the user desity i the cell icreases ad the average SNR separatio betwee the two outermost users decreases. Ispectio of Fig. 8 shows that, as the average SNR separatio becomes large i.e. the etwork becomes less dese), the advatage of the ormalized-snr scheme is more apparet. Furthermore, Fig. 8 suggests that ormalizig the SNR i OMS results i less improvemet for the same SNR separatio for a smaller cell i.e. the average cell-edge SNR of 0dB). Next, we use Fig. 9 to cofirm the optimal selectio ratio for the OMS scheme usig the absolute-snr ad the ormalized- SNR selectio schemes as described i Sec. IV. For the

11 TW-MAR : OPTIMIZED OPPORTUNISTIC MULTICAST SCHEDULING OVER WIRELESS CELLULAR NETWORKS ρ 1 = 0dB Delay OMS absolute SNR) OMS ormalized SNR) ρ 1 = 0dB Selectio Raio, α * Average SNR separatio betwee two worst users, db Fig. 8. Compariso of OMS with the ormalized-snr scheme ad the absolute-snr scheme i heterogeeous etworks. 0.4 OMS Heterogeeous Aalysis) OMS Heterogeeous Sim. lower mi) OMS Heterogeeous Sim. upper mi) Cell Edge Average SNR, db Fig. 9. The optimal selectio ratio as a fuctio of the cell-edge average SNR i heterogeeous etworks. absolute-snr selectio, we compare the optimized selectio ratio aalytically derived from 44) with that obtaied by exhaustive search over the simulated average delay. As with the homogeeous case, we also observe, for certai SNR values, two locally optimal user selectio ratios that yield similar delay performaces. Ispectio of Fig. 9 reveals that the aalytical result closely matches the optimal selectio ratio obtaied by simulatios whe the miimum is uique ad it matches oe of the solutios whe more tha oe exists. Furthermore, we examie the ormalized-snr selectio by comparig the aalytical ad simulated ratios. Fig. 9 suggests that the aalytical ad simulated results agree well with each other i a similar maer. Compariso of selectio ratios obtaied from the absolute-snr ad the ormalized-snr selectio schemes shows that the ormalized-snr scheme teds to select a smaller selectio ratio. However, it should be bore i mid that the ormalized-snr scheme chooses the user selectio ratio based o the ormalized SNRs while the actual ratio of users whose supportable data rates are higher tha the selected rate is larger tha the selectio ratio. Fially, the advatage of the ormalized-snr scheme is exemplified i Fig. 10, where we compare the delay performace of the proposed OMS schemes agaist other existig schedulig schemes i the heterogeeous sceario. For the uicast ad media OMS schemes, ormalized SNRs are employed as the user selectio criteria to guaratee each user a fair chace to be served regardless of their average SNRs. Fig. 10 shows that the uicast scheme suffers from the worst delay performace i the SNR rage uder cosideratio. Furthermore, the broadcast scheme is more advatageous whe the cell-edge average SNR is sufficietly large whereas the media scheme caot fully exploit the multicast gai at large cell-edge average SNRs. I cotrast, the proposed OMS schemes with the ormalized- SNR scheme achieve robust performace throughout the SNR rage examied. Delay 10 1 Uicast > 30 OMS Norm. SNR, Simulated α) OMS Abs. SNR, Simulated α) OMS Norm. SNR, Aalytical α) OMS Abs. SNR, Aalytical α) Uicast Broadcast Media Cell Edge Average SNR, db Fig. 10. Performace compariso of differet schedulig schemes i heterogeeous etworks. VI. CONCLUSION I this work, we studied the opportuistic multicast schedulig OMS) scheme with optimal user selectio for both homogeeous ad heterogeeous cellular etworks. Capitalizig o extreme value theory, we provided theoretical aalysis o the optimal user selectio ratio that miimizes the system delay. Moreover, we demostrated that the proposed OMS scheme with the ormalized-snr scheme ca obtai further performace gai i heterogeeous etworks via aalysis o a two-user etwork example. Computer simulatio cofirmed that the proposed optimized OMS scheme achieves robust delay performace i both homogeeous ad heterogeeous

12 TW-MAR : OPTIMIZED OPPORTUNISTIC MULTICAST SCHEDULING OVER WIRELESS CELLULAR NETWORKS 10 etworks by exploitig the optimal tradeoff betwee multiuser diversity ad multicast gai. Throughout this work, it has bee assumed that the istataeous SNR of each user is available to the BS. This assumptio has sigificatly simplified the theoretical aalysis while providig isights ito the performace upper boud achieved by the proposed scheme. However, such a assumptio may be restrictive i practice, particularly for etworks with a large umber of users. Some modificatios of the proposed scheme are ecessary for its practical deploymet. For istace, limited-feedback techiques similar to that proposed i [17] merit future ivestigatio. APPENDIX A TWO-USER OMS WITH NORMALIZED SNR Suppose that the etwork cosists of two users. Oe is farther from the BS ad labeled as User 1, ad the other is closer to the BS ad labeled as User. Let γ k) be the istataeous SNR of the -th user i the k-th time slot. Deote by ρ the average SNR of the -th user such that ρ 1 < ρ. With a slight abuse of otatio, let us defie K as the delay required by the -th user to successfully receive S ecoded bits. The average system delay uder a particular schedulig scheme ca be expressed as E[K] = E [max{k 1, K }] max {E[K 1 ], E[K ]} = E[K 1 ]. 51) Notice that the expectatio ca be closely approximated by its lower boud whe the differece of the average SNR of the two users are sufficietly large. I this case, the system delay is domiated by User 1, i.e., the user with the worst average SNR. To compare the performace betwee the absolute-snr ad the ormalized-snr schemes, we cosider the case with M = 1 i.e. oly oe user is selected i each time slot) sice the case with M = is idetical for both schemes i the twouser example. I this case, User 1 is selected as the target user ad, thus, able to successfully receive the packet) uder the absolute-snr scheme if ad oly if γ 1 γ. However, it also holds that { {γ 1 γ } = a) = γ 1 γ, γ 1 ρ 1 γ ρ { γ 1 γ, γ 1 ρ 1 γ ρ } } { γ 1 γ, γ 1 < γ } ρ 1 ρ { γ1 γ }, ρ 1 ρ where a) follows from the fact that γ 1 < γ ρ 1 ρ < γ sice ρ 1 < ρ. This shows that wheever User 1 is able to successfully receive a packet uder the absolute-snr scheme for a particular chael realizatio, it will also be able to successfully receive uder the ormalized-snr scheme. Hece, the delay of User 1 ca oly be smaller uder the ormalized-snr scheme, i.e. K1 abs K1 orm ad, hece, the ormalized-snr scheme will ot perform worse tha the absolute-snr scheme for the twouser case. REFERENCES [1] R. Kopp ad P. Humblet, Iformatio capacity ad power cotrol i sigle-cell multiuser commuicatios, i Proc. IEEE Iteratioal Coferece o Commuicatios ICC), Seattle, WA, Jue 005. [] P. Viswaath, D. N. C. Tse, ad R. Laroia, Opportuistic beamformig usig dumb ateas, IEEE Trasactios o Iformatio Theory, vol. 48, pp , Jue 00. [3] M. Sharif ad B. 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Thomas, Statistical aalysis of Extreme Values with Applicatios to Isurace, Fiace, Hydrology ad Other Fields, d ed. Bosto, MA : Birkhauser Verlag, 001. [15] S. Y. Baek, H. Y. Hwag, ad D. K. Sug, Performace aalysis of schedulig-based systems i Rayleigh fadig chaels, i Proc. IEEE 17th Iteratioal Symposium o Persoal, Idoor ad Mobile Radio Commuicatios, Helsiki, Filad, September 006. [16] IEEE, 80.0 Chael Models Documet for IEEE 80.0 MBWA System Simulatios, IEEE 80, IEEE 80.0-PD-08, Sep [Olie]. Available: http : // D ocs/ieee P D 08.doc [17] N. Ravidra ad N. Jidal, Multi-user diversity vs. accurate chael state iformatio i MIMO dowlik chaels, Submitted to IEEE Tras. Commuicatios, 009. Tze-Pig Low S 08) received his B.Eg. ad M.S. degrees i Electrical Egieerig from Natioal Uiversity of Sigapore i 1999 ad 00, respectively. From 000 to 008, he worked as a radio etwork egieer i the telecommuicatios idustry, desigig ad implemetig cdma000/evdo ad WCDMA/HSDPA radio etworks. He is curretly pursuig his Ph.D. at the Uiversity of Souther Califoria, Los Ageles. His curret research iterests are i sigal processig for commuicatios ad wireless multicast schedulig.

13 TW-MAR : OPTIMIZED OPPORTUNISTIC MULTICAST SCHEDULING OVER WIRELESS CELLULAR NETWORKS 11 Ma-O Pu M 06) received the BEg. Ho.) degree i electroic egieerig from the Chiese Uiversity of Hog Kog i 1996, the MEg. degree i computer scieces from Uiversity of Tsukuba, Japa i 1999 ad the Ph.D. degree i electrical egieerig from Uiversity of Souther Califoria, Los Ageles, i 006, respectively. He is a research scietist at Mitsubishi Electric Research Laboratories MERL), Cambridge, MA. He held research positios at Priceto Uiversity, Priceto, NJ from 006 to 008 ad Soy Corporatio i Tokyo, Japa, from 1999 to 001. Dr. Pu received the best paper award - ruer-up from the IEEE Coferece o Computer Commuicatios Ifocom), Rio de Jaeiro, Brazil i 009, the best paper awards from the IEEE Iteratioal Coferece o Commuicatios, Beijig, Chia i 008 ad the IEEE Vehicular Techology Fall Coferece, Motreal, Caada i 006. He is a recipiet of several scholarships icludig the Japaese Govermet Mobusho) Scholarship, the Sir Edward Youde Memorial fellowship for Overseas Studies ad the Croucher postdoctoral fellowship. Y.-W. Peter Hog M 05) received his B.S. degree i Electrical Egieerig from Natioal Taiwa Uiversity, Taipei, Taiwa, i 1999, ad his Ph.D. degree i Electrical Egieerig from Corell Uiversity, Ithaca, NY, i 005. He joied the Istitute of Commuicatios Egieerig ad the Departmet of Electrical Egieerig at Natioal Tsig Hua Uiversity, Hsichu, Taiwa, i Fall 005, where he is ow a Associate Professor. He was also a visitig scholar at the Uiversity of Souther Califoria durig Jue-August 008. His research iterests iclude cooperative commuicatios, distributed sigal processig for sesor etworks, ad PHY-MAC cross-layer desigs for ext geeratio wireless etworks. Dr. Hog received the best paper award for youg authors from the IEEE IT/COM Society Taipei/Taia chapter i 005, the best paper award amog uclassified papers i MILCOM 005, ad also the Juior Faculty Research Award from the College of EECS at Natioal Tsig Hua Uiversity i 009. He is a co-editor alog with A. Swami, Q. Zhao, ad L. Tog) of the book etitled Wireless Sesor Networks: Sigal Processig ad Commuicatios Perspectives published by Joh-Wiley & Sos i 007. C.-C. Jay Kuo S 83, M 86, SM 9, F 99) received the B.S. degree from the Natioal Taiwa Uiversity, Taipei, i 1980 ad the M.S. ad Ph.D. degrees from the Massachusetts Istitute of Techology, Cambridge, i 1985 ad 1987, respectively, all i Electrical Egieerig. Dr. Kuo was Computatioal ad Applied Mathematics CAM) Research Assistat Professor i the Departmet of Mathematics at the Uiversity of Califoria, Los Ageles, from October 1987 to December Sice Jauary 1989, he has bee with the Uiversity of Souther Califoria, where he is curretly Professor of Electrical Egieerig ad Computer Sciece ad Director of the Sigal ad Image Processig Istitute. His research iterests are i the areas of digital sigal ad image processig, multimedia compressio, commuicatio ad etworkig techologies. He is a co-author of about 160 joural papers, 790 coferece papers ad 9 books. Dr. Kuo is a Fellow of IEEE ad SPIE. He is co-editor-i-chief for the Joural of Visual Commuicatio ad Image Represetatio, ad Editor for the Joural of Iformatio Sciece ad Egieerig, LNCS Trasactios o Data Hidig ad Multimedia Security ad the EURASIP Joural of Applied Sigal Processig. Dr. Kuo received the Natioal Sciece Foudatio Youg Ivestigator Award NYI) ad Presidetial Faculty Fellow PFF) Award i 199 ad 1993, respectively.