Opportunistic Beamforming for Finite Horizon Multicast

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1 Opportunstc Beamformng for Fnte Horzon Multcast Gek Hong Sm, Joerg Wdmer, and Balaj Rengarajan and Insttute IMDEA Networks, Madrd, Span Unversty Carlos III, Madrd, Span Accelera Moble Broadband Abstract Wreless multcastng suffers from the problem that the transmt rate s usually determned by the recever wth the worst channel. Composte or adaptve beamformng allows usng beamformng patterns that trade off antenna gans between recevers. A common soluton for wreless multcast wth beamformng s to select the pattern that maxmzes the mnmum rate among all recevers (for a gven transmt power). However, when usng opportunstc multcast to transmt a fnte number of packets to all recevers the fnte horzon problem ths s no longer optmal. Instead, the optmum beamformng pattern depends on nstantaneous channel condtons as well as the number of receved packets at each recever. We formulate the fnte horzon multcast beamformng problem as a dynamc programmng problem to obtan the optmal soluton. We further desgn a heurstc that has suffcently low complexty to be mplementable n practce and show through extensve smulatons that our algorthm sgnfcantly outperforms pror solutons. I. INTRODUCTION Wreless multcast s an effcent technque to dssemnate multmeda data to groups of users. Usng the broadcast nature of the wreless medum allows data to be served to multple users smultaneously, but at the same tme ths constrans the transmt rate to the rate that can be supported by the recever wth the worst channel condtons. To address ths ssue, a range of dfferent mechansms have been proposed. Wth opportunstc multcast schedulng (OMS), a base staton (BS) explots multuser dversty and may opportunstcally transmt to a subset of recevers that experence good channel condtons. Ths trades off multcast gan (acheved by transmttng to as many recevers as possble) versus multuser dversty gan (acheved by transmttng to a subset of recevers that currently experence good channel condtons) [] [3]. Multuser dversty gan can be exploted f recevers wth a bad channel are lkely to experence better channels n the future, and vce versa. It s partcularly sutable for homogeneous scenaros where recevers long-term average rates are smlar, but nstantaneous rates are hghly varable. If, however there are some recevers that are clearly worse than the rest, the overall rates are stll lmted by those recevers. In ths case explotng multuser dversty and multcast to all recevers are detrmental to performance. However, ths also means that the transmt rate s lmted by these worst recevers. To overcome ths problem, transmt beamformng can be used to adjust antenna gans to the dfferent recevers. Ths allows mprovng the sgnal-to-nose ratos (SNR) of recevers /4/$3. c 4 IEEE wth bad nstantaneous channels at the expense of worsenng those of recevers wth better nstantaneous channels. There are two man technques for mult-user beamformng: () composte beamformng [4] and () adaptve beamformng [5]. A composte beam s composed of multple pre-determned sngle-lobe beam patterns. In contrast, adaptve beamformng calculates antenna weghts drectly based on the measured channels to the dfferent recevers. Whle composte beamformng has lower complexty, adaptve beamformng may acheve better performance n mult-path rch envronments. Smlar to opportunstc multcast, the man challenge when desgnng mult-user beamformng mechansms s the tradeoff between hgh gan beamformng to few recevers versus lower gan beamformng to a larger recever set [4], [6], [7]. Most of the pror work n these areas ams at maxmzng the rates of the recevers, whch s optmal for the nfnte horzon problem. In contrast, we analyze the more realstc fnte horzon problem where the BS sends a block of a certan sze to all recevers. In case there are multple blocks to be sent, the BS starts transmttng packets for the next block only after all recevers have receved the frst data block. Ths s relevant for most practcal scenaros where relable delvery as well as delay constrants are a concern, as for example for multcast vdeo streamng. Smlar to pror work (e.g., []) we assume erasure codng of transmtted data, whch s hghly benefcal n wreless multcast scenaros and ensures that each packet receved by a recever s useful (wth hgh probablty). Ths paper focuses on the fnte horzon opportunstc multcast beamformng problem. In the nfnte horzon problem, t s possble to explot opportunstc gan very aggressvely, snce laggng recevers have an nfnte amount of tme to catch up. In contrast, n the fnte horzon case the optmum decsons on when to explot opportunstc gan and when to favor laggng users very much depend on the state of the recevers (.e., the amount of data receved thus far and therefore how close the recevers are to fnshng). Ths substantally changes the formulaton of the problem compared to the nfnte horzon problem. In addton, the hgher the number of users, the hgher the mult-user dversty and therefore the potental for opportunstc gan. Explotng opportunstc gan aggressvely leads to hgher average rates but at the same tme may lead to some users fnshng early, thus reducng the future total achevable throughput. We frst model the problem and obtan the optmum soluton

2 va dynamc programmng. Ths allows us to study the mpact of recever state, nstantaneous channel condtons, and average channels on the optmum recever set to transmt to, and hence the optmum beamformng pattern. Frst, we study these tradeoffs n toy scenaros wth two users. These nsghts allow us to desgn a low complexty heurstc algorthm that captures the man characterstcs of the optmum soluton and at the same tme can run n real-tme n the practcal wreless scenaros. We perform a range of smulatons for larger scenaros for homogeneous and heterogeneous recever sets wth realstc channels (Raylegh fadng). Whle the complexty of the dynamc programmng soluton prevents us from solvng those larger problem nstances optmally, we see that our proposed heurstc provdes sgnfcantly better performance than solutons based on broadcastng or greedly maxmzng rates. Note that both the broadcast and greedy mechansms use beamformng and take the nstantaneous channel condtons nto account. The greedy mechansm s thus optmal for the nfnte horzon case (or problem nstances wth very large block szes) n homogeneous scenaros as presented n [8]. The broadcast mechansm makes use of beamformng to maxmze the mnmum rate and hence does not suffer from recevers wth a bad channel severely lmtng the transmt rate as n OMS. It corresponds to the soluton n [7] that s optmal for scenaros wth fxed channels but may be too conservatve n case of varable channels. It s also optmal for scenaros wth varable channels where the recever set s very heterogeneous and one recever has a sgnfcantly worse average channel than the other recevers. The paper s structured as follows. A revew of state-of-theart for opportunstc multcast and multcast beamformng s gven n Secton II. In Secton III, we model the fnte horzon opportunstc multcast beamformng problem and provde an optmum soluton based on dynamc programmng. We desgn a low complexty heurstc FH-OMB n Secton IV and analyze ts performance n comparson to the optmum soluton and other pror schemes n Secton V. Secton VI concludes the paper and provdes an outlook on future work. II. RELATED WORK Opportunstc Multcastng: Opportunstc multcastng has been well studed for both the nfnte horzon problem [], [9] [] as well as the fnte horzon problem [], [3]. Among the frst deas to address the nfnte horzon problem for homogeneous scenaros was to splt the recevers nto two groups accordng to ther nstantaneous channels and serve the group wth the better channel qualty. As the composton of the group changes from slot to slot, all users have equal chances to be served [9]. Ths work was extended n [] by optmzng the selecton rato,.e., the sze of the recever set to transmt to. As a sngle pre-computed selecton rato s not always optmal, [] and [] propose a dynamc user selecton mechansm that depends on the nstantaneous channel at each transmsson. The authors of [] solve the user selecton problem for the fnte horzon case usng extreme value theory to mnmze completon tme. However, the user selecton s only based on the nstantaneous channel but not on the user state (.e., the amount of data receved by users). In wreless systems wth packet loss, ths s suboptmal snce users may have receved a dfferent number of packets. The problem s addressed n [3], where t s shown that the optmal soluton for the fnte horzon problem needs to take recever state nto account. The paper analyzes the trade off between multcastng gan and multuser dversty and provdes an optmal algorthm as well as a low complexty heurstc. The man challenge of opportunstc multcastng s to cope effcently wth recevers wth bad channel condtons. In ths context, transmt beamformng can be used to balance the users SNRs. Multcast Beamformng: Multcast beamformng provdes a trade off between multcast gan and beamformng gan. Beamformng to recevers wth poor channel condtons mproves the rate of these recevers (but at the same tme lowers SNR at other recevers). The basc algorthm proposed n [6] frst transmts omndrectonally to the recevers that have a hgh SNR and then beamforms sequentally to the remanng weak recevers. Better performance can be acheved by selectng the beamformng vector that maxmzes the mnmum SNR among all multcast recevers [3], [4]. In [4], recevers are parttoned nto groups that are scheduled sequentally, whch may outperform mechansms that always beamform to all recevers. The paper proposes two multcast beamformng mechansms, one that splts power equally among all beams and one that allows for asymmetrc power allocaton. Both mechansms use composte beamformng, where a mult-lobe beam pattern that serves multple recevers s composed of multple sngle-lobe beam patterns. Reference [7] mproves upon ths work and provdes an optmal soluton for the equal power splt and two dfferent heurstcs for the (NP-hard) asymmetrc power allocaton mechansm. Both [4] and [7] consder the fnte horzon problem but do not take channel varatons and opportunstc schedulng nto account. The same problem s addressed n [5] usng adaptve beamformng rather than composte beamformng. Adaptve beamformng may provde better antenna gans than composte beamformng, n partcular n multpath envronments, but at the same tme determnng the optmum beamformng pattern s more complex. Opportunstc Multcast Beamformng: There s very lttle exstng work that jontly takes opportunstc multcast schedulng and multcast beamformng nto account. [8] provdes a theoretcal analyss of the optmum user selecton rato for opportunstc multcast beamformng usng extreme value theory. Once the user group s determned, the optmal beamformng pattern s the one that maxmzes the mnmum SNR among the users that are served. The algorthm s desgned for ndependent and dentcally dstrbuted users (.e., homogeneous scenaros) for the nfnte horzon multcast problem. We show that ths approach s not sutable for the fnte horzon multcast problem, especally for heterogeneous user dstrbuton.

3 3 Our paper dffers from pror work n that t addresses fnte horzon opportunstc multcast beamformng n homogeneous and heterogeneous scenaros and explctly takes nto account recever state (.e., the amount of data already receved). III. SYSTEM MODEL We consder a wreless network wth a sngle BS (or access pont) and a set T of multcast recevers, wth T = N. We assume the channels between BS and the recevers are ndependent dscrete memoryless channels. Let G denote the set of all possble vector channels from the BS to the recevers. The probablty that at a gven tme nstant the channel vector C has channel gans g G s gven by P (C = g). Let C, g, and G denote the correspondng channel nstance, gan, and set of possble channels for recever. As s common for opportunstc schedulng, we assume that the BS has perfect knowledge of the current channel nstance, but for any future channel nstances only the channel dstrbuton s known. The BS uses composte beamformng wth K antenna elements. These generate K antenna patterns that are optmzed to produce one strong sngle-lobe beam that covers a sector of approxmately 36 K and that together cover the whole azmuth of 36. A composte beam s a mult-lobe beam pattern composed of several sngle-lobe beams that are transmtted smultaneously [4]. Each sngle-lobe beam k has a certan beam weght α k. Ths weght corresponds to the fracton of the total transmt power allocated to that beam, and thus determnes the SNR at the recevers covered by the beams. To ensure that the total radated power remans unchanged, we have the constrant k α k =. Let k sngle-lobe beam that covers recever and let γg,slb be the strongest denote the SNR at that recever when usng that sngle-lobe beam when the channel gan s g. Then the SNR of that recever for a mult-lobe beam s γ g = α k γ,slb g. We consder a tme-slotted model. In each tme slot the BS transmts data to the recevers usng a certan modulaton and codng scheme (MCS) and beamformng pattern. For MCS m M, the number of bts transmtted n a slot s R m and the correspondng packet recepton probablty for an SNR of γ s p m (γ). Note that we assume that recever wll only be served when a mult-lobe beam s used wth α k. A. Problem Formulaton The BS has a block of data of sze B (n bts) to transmt to all recevers. An erasure code s appled to the data before transmsson, so that each data packet s useful for each recever that receves t, as long as that recever has obtaned less than B bts so far. The optmzaton problem s thus for the BS to select at each tme slot the mult-lobe beam pattern wth correspondng weghts as well as the MCS that mnmzes the expected completon tme. Optmal choce of beam pattern and MCS Note that our heurstc works for contnuous channels and we provde smulaton results for Raylegh fadng channels n Secton V. depend on the current nstantaneous channel, the probablty dstrbutons of the channels, and the amount of data receved by the recevers so far. When beamformng to a subset of recevers T T, the hghest expected rate to those recevers s obtaned by selectng beam weghts αk that maxmze the mnmum SNR at the recevers. Let T k = { T : k = k} T be the subset of recevers served by beam k. The mnmum SNR of recevers n T k for a sngle-lobe beam pattern and a gven channel g s γ SLB g (T k) = mn T k γ,slb g () and, as shown n [7], the optmum weghts for the mult-lobe beam pattern are thus gven by ( αk γg SLB (T k = ) ) K j=,t j γ, f T g SLB(T j ) k (), otherwse Ths results n the same mnmum SNRs for all lobes of the mult-lobe beam. Hence, all recevers n T are served wth the MCS that provdes the hghest expected rate m = arg max R m p m (αkγ g SLB (T k)). (3) m Thus, rather than optmzng over all possble beam weghts, t s suffcent to optmze over all possble subsets of recevers. (Note that the algorthm n [7] always serves all recevers assocated wth a gven beam, whle ths s no longer optmal for opportunstc multcast. Consder a scenaro where all recevers are located n the same beam. Ths s the conventonal OMS scenaro for whch t s well known that broadcastng to all users s not always optmal [].) B. Dynamc Programmng Soluton Wth ths we can formulate the problem as a stochastc shortest path problem and solve t through dynamc programmng [5]. The state s gven by the amount of data receved by the recevers so far s = [s... s N ], s B and we denote the state space by S. As all tme slots have the same duraton, the cost per slot s. When multcastng to a subset T of recevers wth an nstantaneous channel of g, the transton probablty from state s to state s s ρ T g (s, s ) = mn(s+r m e,b)=s e E ( N ) p m (γg ) e ( p m (γg )) e = where the vector mnmzaton above s element-wse. E = { e {, } N } s the set of bnary vectors of sze N and e s the th element of e, ndcatng whether recever receved the packet or not. The MCS m s calculated accordng to Equaton (3). Equaton (4) takes nto account all combnatons Gven that there s a dscrete set of rates R m, many states cannot be reached and we remove these states from the state space to speed up the computaton. (4)

4 4 of whch recevers wll receve the packet and ensures that the state of recevers wth s = B does not change. A polcy µ s : G T T T specfes the best subset of recevers to transmt to for any nstantaneous channel g when n state s. Let M bet the set of all possble mappngs. Snce the probablty of termnatng after a fnte number of steps s postve, we can use Bellman s equaton to fnd the optmal polcy µ s = arg mn P (C = g) ρ µs(g) g (s, s )D (s ). µ s M g G s S (5) The correspondng optmal expected completon tme s D (s) = mn P (C = g) ρ µs(g) g (s, s )D (s ). µ s M g G s S (6) Gven that the state space s fnte we can solve the dynamc program through value teraton, startng from the fnal state s B. Ths optmzaton problem s hard and even a much smpler verson of t wth fxed channels (.e., no opportunstc schedulng), as well as guaranteed packet delvery wthout errors s NP-hard [7]. The dynamc program has double exponental complexty. The state space has sze B N and for each state there are N G polces that map each of the channel states n G to one of the N possble mult-lobe patterns. Also G tself s exponental n N. Clearly, the dynamc program can only be solved for very small problem nstances. For ths reason, n the next secton we desgn a lower complexty heurstc. IV. HEURISTIC ALGORITHM Our Fnte-Horzon Opportunstc Multcast Beamformng (FH-OMB) heurstc has two man parts: ) gven the current nstantaneous channel, computng the next states the system could move to usng the dfferent mult-beam lobes that correspond to multcastng to the dfferent subsets of recevers, and ) estmatng the expected completon tmes from those new states. The decson taken by the heurstc s then to beamform to the subset of recevers that results n movng to the state wth the lowest expected completon tme. A. Instantaneous Beamformng Decson Let the current state be s and the current nstantaneous channel be g. Assume the estmated completon tmes D(s ) for all future states are known. When beamformng to T T we can calculate γg SLB (T k ), α k, and the resultng optmum MCS m usng Equatons () (3). The expected future state s (T ) s gven by s (T ) = mn ( s + R m p m (γ g ), B ) and the optmum subset of recevers T to beamform to s thus T = arg mn D(s (T )). (7) T T In contrast to the dynamc programmng formulaton we compute expected average future state rather than lookng at all combnatons of possble future states based on packet loss events. Note that ths stll requres mnmzaton over a number of completon tmes that s exponental n the number of recevers, whch can be done exhaustvely for small recevers sets. For larger recever sets, we cluster recevers accordng to ther state s and relatve qualty of the nstantaneous channel. The rate recever would obtan wth the current channel g for a sngle-lobe pattern s R() = max m R m p m (γg,slb ), and the average rate that s obtaned under all possble channels s R() = ( P (C = g ) max Rm p m (γg,slb m ) ). (8) g G The relatve channel qualty s R()/ R(). Let = ξ < ξ <... < ξ U = B be a set of state thresholds and = θ < θ <... < θ V = be a set of relatve channel qualty thresholds. We then group all recevers wth T uv = { T : ξ u s < ξ u+, θ v R()/ R() < θ v+ } where the total number of groups s UV. In Equaton (7) only optmze over subsets T T that nclude whole recever groups (.e., f one of the recevers n a group s ncluded, the whole group must be ncluded). We set the thresholds so that the recevers are dstrbuted relatvely evenly among the groups. In order to further reduce the number of schedules, a group can only be scheduled f all groups that have hgher channel state (.e., better relatve channel qualty) and at the same tme have lower recever state as well are scheduled. Wth ths, the maxmum number of combnatons of schedules whch ndcates the complexty of the FH-OMB s ( ) (V + U )! O. (9) U!(V )! The number of beamformng patterns s fxed for fxed value of U and V. 3 B. Estmatng the Expected Completon Tme The man complexty of the dynamc programmng soluton les n the calculaton of the expected completon tme. Hence, ths s what the heurstc prmarly addresses. As only the nstantaneous channel s known at the BS, we base the expected completon tme of a future state on the average channel of the recevers. Due to the shape of the rate functon, smply averagng the channel would overestmate the receve rate. Hence we frst calculate the average sngle-lobe rate of recever, R(), as gven by Equaton (8) and then set the recever s average SNR γ,slb such that max m g ( Rm p m ( γ,slb g )) ) = R(). For fxed SNRs and a contnuous rate functon, accordng to [7] the maxmum rate when multcastng to a recever set 3 From the smulatons we fnd that a reasonably low value for U and V (.e., U = V = 4) suffces n practce, leadng to a fxed number of subsets to consder for the optmzaton.

5 5 s obtaned for a mult-lobe beam pattern that encompasses the whole recever set. Analogous to Equatons () and (), for a recever subset T we can derve γ g SLB (T k ) as well as ᾱk based on the average SNRs γ,slb g calculated above. The correspondng hypothetcal average rate s gven by R(T ) = R(T k) = max R mp m (ᾱk γ g SLB (T k)). () m We have R(T ) = R(T k ) for any non-empty lobe k, snce all lobes have the same mnmum rate. Wth ths, we can now approxmate the expected completon tme as follows. For a gven state s, let T = { T : s < B} be the set of recevers that stll requre further packets and let s () max = max T s be the state of the recever(s) closest to completng. When multcastng to ths recever set at rate R(T ) gven by Equaton (), one or more of the recevers would complete after a tme τ = (B s () max)/ R(T ). Determne the set of remanng recevers T = { T : s < s () max} and set s () max = max T s to calculate τ, etc. In general, τ j = (B s (j) max)/ R(T j). () In other words, the estmaton algorthm proceeds dagonally through the state space untl httng a boundary wth s = B for one of the dmensons, then proceeds dagonally along that boundary untl httng the next one, and so on, untl reachng the fnal state. The algorthm termnates after at most N steps. The expected completon tme s gven by D(s ) = j τ j. () Accountng for opportunstc gan: When determnng τ j above, we assume that recevers n T are served frst, then recevers n T, etc. Ths gnores that recever sets wll be selected based (also) on ther nstantaneous channels. As a consequence, R(T ) s a conservatve estmate of the actual rate at whch ths recever group s served, snce they are more lkely to be served when ther channel s good. We refne Equaton () to take nto account opportunstc gan as follows. We assume that recevers n groups T and T are served durng τ + τ. If the channels of the recevers n T \ T are good, group T wll be served, otherwse group T wll be served. Hence, recevers n T see better average channels (snce some of the beam weght α that was requred for recevers n T \ T can now be used for other beams) whereas there s no change for recevers n T. We remove the worst fracton τ /(τ + τ ) of channel combnatons of the recevers n T \ T and update ther average channels accordngly. We then recompute Equatons () and () and obtan a new τ. Smlarly, the calculaton of τ s based on recever groups T and T 3, and so on. The completon tme s then calculated as D(s ) = j τ j. Note that ths s stll a conservatve estmate of the opportunstc gan. Example and dscusson: To provde an ntuton for the completon tme estmaton, we dscuss an example for a two-recever case n Fg.. In a $ " ( B!"#$%&'( + - (!"#$)*%&'( Boundary!"#$)&'( +, (!"#$%&'( ()! " #*( ( Actual path! % #&.(/-,(! " #&.(/-( B [B,B] Fg.. Completon tme estmaton wth the FH-OMB heurstc. two-user scenaro, there are only three possble beamformng patterns servng recever sets {}, {}, or both {, }. For {} and {}, sngle-lobe beamformng patterns wth maxmum array gan to the respectve recever are used, whereas for {, } the mult-lobe beam that equalzes the SNRs of the recevers s chosen. In the latter case, both recevers are served at the same rate and have the same packet loss probablty. For each of the average future states s ({}), s ({}), and s ({, }) we compute the expected completon tme. Consder, for example, s ({}). Snce both users have not yet fnshed, we calculate the number of tme slots τ requred for the frst recever to have B bts. In the example ths s recever. We then compute τ requred for the second user to complete, based on the sngle-lobe pattern to that user only. Usng τ, we can recompute the frst segment to obtan τ that partally accounts for opportunstc gan. τ = τ snce there s no opportunstc gan for a sngle recever. The actual path that s taken (shown wth dotted lnes) depends on the actual nstantaneous channel condtons at future states and s generally shorter than the sum of the estmated path segments. An mportant observaton s that determnng the exact completon tme s not mportant. What s mportant s to have approxmately the rght relatve dfferences among completon tmes of nearby states (n ths case s ({}), s ({}), and s ({, })), such that the rght nstantaneous beamformng decsons are taken. As a consequence, t s possble to use average channels nstead of all possble channel combnatons, wthout ncurrng a substantal drop n performance. V. SIMULATION RESULTS In ths secton, we present smulaton results to analyze the performance of the algorthms. We frst nvestgate a smple scenaro wth two recevers and a two-state channel to compare the optmal dynamc programmng soluton (Dyn-Prog) and the fnte horzon opportunstc multcast beamformng heurstc (FH-OMB) and gan nsghts nto the optmum strategy and fundamental tradeoffs. We then nvestgate more realstc scenaros wth mult-path Raylegh fadng channels, larger number of recevers, and larger block szes. For these, we do not provde dynamc programmng results as the run tme s prohbtve due to the algorthm s complexty. The mult-path Raylegh fadng channel corresponds to the ITU Pedestran B path loss model n [6]. For all the scenaros, we use a subset of 3 MCSs gven n the LTE specfcaton for the MHz LTE downlnk model (wth modulaton schemes QPSK, 6-QAM, and 64-QAM, and code rates from.885 to.958). The $ % (

6 Dyn Prog 5 5 Channel varaton (δ) Average rate 5 5 Dyn Prog Tme slot (slots) Fg.. Completon tme n a homogeneous scenaro wth ncreasng channel varablty. Fg. 3. Average throughput over tme for a homogeneous scenaro wth channel varablty σ =.5dB. Fg. 4. Expected completon tme for Dyn-Prog (left) and FH- OMB (rght) for σ =.5dB. correspondng transmt rates range from 5Mbps to 95Mbps. A tme slot has a duraton of ms. The man performance metrc s completon tme,.e., the number of tme slots needed for all recevers to receve B kbts. We compare the performance of Dyn-Prog and the FH-OMB heurstc wth two alternatve mechansms: ) Algorthm: uses a mult-lobe beam pattern that covers all recevers wth s < B, maxmzes the mnmum SNR across all lobes, and serves the recevers wth the optmum MCS m for that SNR as gven n Equatons () (3). Ths scheme s presented n [7] and t s shown to be optmal for constant channels wth fxed SNR. ) Algorthm: For, we sort the recevers wth s < B accordng to ther nstantaneous channel qualty, gven by the sngle-lobe SNR γg,slb. Let T be the recever set that ncludes the recever wth the best channel (that hasn t fnshed yet), T be the set of the two recevers wth the two best channels, etc. The algorthm then determnes the recever set to beamform to as T = arg max T j R m p m (γg,slb ). T j The optmum recever set s the one wth the hghest overall sum rate for all recevers that have not yet fnshed. Ths algorthm corresponds to the one proposed n [8] and works well for homogeneous recever sets. A. Smple Scenaro In ths secton, we present the results for a smple scenaro wth N = recevers and block sze B = kbts. Each recever has two possble nstantaneous channels (g = {H, L }), such that G = {H H, H L, L H, L L } wth P (C = H ) = P (C = L ) =.5. We analyze a homogeneous scenaro and a heterogeneous scenaro. ) Homogeneous Scenaro: In ths scenaro recevers have the same set of channels (H = H = H, L = L = L ). We nvestgate the mpact of channel varablty, σ = γh SLB γslb L,.e., the dfference between the hgh gan channel and the low gan one. (For example, the left most pont of Fg. has γh SLB = db, γ SLB = 9dB, σ = db and the rght most pont has γh SLB L = 8dB, γslb L = 4.7dB, σ =.7dB). γslb H and γslb L values are chosen such that wth sngle-lobe beamformng the recevers would acheve the same average rate and hence we can compare relatve rate changes as the varablty ncreases. As shown n Fg., both and the FH-OMB heurstc perform almost as good as the optmal Dyn-Prog. As both recevers have the same channel dstrbuton, dfferences n recever state are lkely to cancel out over tme and maxmzng the nstantaneous sum rate as does s a good strategy. Only when one recever s close to fnshng and the other recever s laggng further behnd may t be benefcal to favor the laggng recever nstead. Note that the graph also shows 95% confdence ntervals but due to the large number of smulaton runs they are very small. For small channel varablty (σ < 3dB), the maxmum sum rate s acheved by servng both recevers for any of the channel combnatons, hence and have the same performance. Once the channel varablty s ncreased beyond ths pont, beamformng only to the recever wth a good channel when the other recever has a bad channel (H L, L H ) provdes hgher throughput than beamformng to both recevers. Hence, s unnecessarly conservatve by always servng both recevers and ts completon tme ncreases substantally as the channels become more varable. Snce n such a homogeneous scenaro maxmzng sum throughput s almost always the rght strategy, even slghtly outperforms FH-OMB for hgher channel varablty. Due to ths varablty, recever states may dffer enough so that FH-OMB s conservatve completon tme estmate prevents t from opportunstcally explotng good channels as aggressvely as. Ths can be seen n more detal n Fg. 3, whch shows average system throughput per tme slot (averaged over all smulaton runs and over both recevers, where recevers that fnshed have throughput) for the scenaro wth channel varablty σ =.5dB. Throughput of FH-OMB starts out the same as that of Dyn-Prog and, but drops off slghtly once recever states becomes more heterogeneous and one recever s close to fnshng. Fg. 4 shows the completon tme estmates for the dynamc programmng algorthm (left) and the FH-OMB heurstc (rght) for the same scenaro (.e., σ =.5dB). FH-OMB s completon tme estmate based on average channels underestmates completon tme when the channel s more varable, but the relatve dfferences n estmated completon tme for the dfferent states for the two algorthms are very smlar. FH-OMB s completon tme estmaton algorthm thus leads to the rght beam-formng decsons n most cases. The performance gap s due to

7 7 Completon Tme (slots) Dyn Prog 5 5 SLB Avg SNR of the recever (γ db) Average rate 5 5 Dyn Prog Tme slot (slots) Fg. 5. Completon tme n a heterogeneous scenaro wth ncreasng average SNR of the better recever γ SLB (.e., recever ). Fg. 6. Average throughput over tme for a heterogeneous scenaro γ SLB = 4.dB. Fg. 7. Expected completon tme for Dyn-Prog (left) and FH- OMB (rght) for γ SLB = 4.dB s s s s s Fg. 8. State space vsts for at γ SLB = 4.dB s Fg. 9. State space vsts for at γ SLB = 4.dB. the fact that FH-OMB s completon tme estmate s slghtly less round than the true estmate, makng t appear more benefcal to stay close to the dagonal where both recevers have the same state. It s nterestng to note that the completon tme ncreases for db σ.5db and then decreases agan. When channel varaton s low, both recevers are lkely to fnsh at approxmately the same tme. The hgher σ, the more lkely t becomes that one recever fnshes earler than the other, whch ncreases completon tme gven by the maxmum of the ndvdual completon tmes. When ncreasng σ even further, completon tmes reduce snce wth a good channel, only very few tme slots are needed to complete. There s a sgnfcant probablty that one of the recevers wll fnsh very early, and the system can then serve the remanng recever at a hgher rate wth the correspondng sngle-lobe beam. ) Heterogeneous Scenaro: For the heterogeneous scenaro, we fx the γh SLB = db and γl SLB =.4dB of the frst recever. For the second recever, we vary γh SLB between db and 3dB and γl SLB between.4db and 8.6dB, so that the two recevers become more and more heterogeneous as the channel values for the second recever ncrease. As the γh SLB and γl SLB ncrease, completon tme decreases for all algorthms. performs close to optmal for the frst three data ponts where recevers are suffcently homogeneous and the optmum strategy s to beamform to the recever wth hgh channel gan when one recever has hgh channel gan and the other recever has low channel gan. Here, s agan too conservatve. The jump n s completon for the next data pont s due to the fact that from ths pont on the good channel of the better recever s so good that favors the recever exclusvely n that case and only serves both recevers when the good recever has a low channel gan. In contrast, s strategy to balance 5 75 s Fg.. State space vsts for Dyn- Prog at γ SLB = 4.dB s Fg.. State space vsts for FH- OMB at γ SLB = 4.dB. the rates and forego opportunstc gan becomes closer and closer to optmal as the scenaro becomes more heterogeneous and from an average SNR of γ SLB 7dB on s the optmal strategy. The weak performance of can be explaned from Fg. 6, where acheves hgh throughput untl the frst recever fnshes at less than approxmately 8 tme slots. The second recever s stll far from fnshng as evdenced by the throughput curve flattenng out around 3 tme slots. FH-OMB performs close but s sub-optmal compared to Dyn- Prog, snce the expected completon tme s slghtly naccurate. The comparson n Fg. 7 shows that the expected completon tme of FH-OMB algorthm (rght) s less round than that of Dyn-Prog (left). Thus FH-OMB s more conservatve and t sacrfces hgher nstantaneous rates to ensure that the relatve dfference n recever state does not dverge too much. To provde further nsghts nto the behavor of the algorthms we show the state space vsts n Fg. 8. As expected keeps the two recevers very close to the dagonal where both recevers have the same amount of data, and slght devatons from the dagonal are only due to packet loss. n contrast makes quck progress untl the second recever fnshes and for the remanng tme only has the frst recever to serve. In fact, the steps wth whch the good recever makes progress wth can clearly be seen n Fg. 9. FH-OMB serves recevers smlar to Dyn-Prog early on but then becomes too conservatve as the good recever progresses and beamforms more to the laggng recever to balance recever states. B. Multpath Raylegh Fadng In ths secton, we show smulaton results for a flat multpath Raylegh fadng channel, where the channel does not change wthn a tme slot. The Doppler shft for the Raylegh channel s set to Hz, correspondng to a slow fadng channel

8 8 for recevers movng at walkng speed. Recevers are randomly dstrbuted wthn the coverage area. The BS transmt power s set to 43dBm. Wth ths, a cell edge recever that s 5m from the BS s able to receve a packet wth the lowest MCS wth an average probablty of 3%. The block sze B s set to 64kbts. We study the mpact of ncreasng the number of recevers N from to 64 wth dfferent number of beamformng lobes (.e., K = {, 4, 8, 6}), agan for a heterogeneous and a homogeneous scenaro. Note that due to the hgh complexty of Dyn-Prog, we only compare the performance of the FH- OMB heurstc wth that of and Number of recevers (N) Fg.. Random recever dstrbuton, K = 8, B = 64kbts Number of beamformng lobes (K) Fg. 3. Random recever dstrbuton, N = 3, B = 64kbts. ) Random Recever Dstrbuton: We frst dscuss a heterogeneous scenaro, where N = {, 4, 8, 6, 3, 64} recevers are randomly dstrbuted wthn the cell area of radus 5m and for K = 8 beamformng lobes. The performance depends sgnfcantly on the specfc recever dstrbuton, n partcular for smaller numbers of recevers. For up to 6 recevers, performs almost as good as FH-OMB snce there s a hgh probablty that there s one recever wth a sgnfcantly worse channel than the others (see Fg. ). As the number of recevers ncreases, a hgher number of recevers see smlar channel condtons and as n the prevous two-channel scenaro, the performance of degrades snce t does not explot opportunstc gan. However, n ths heterogeneous scenaro ths effect occurs manly for N > 3 recevers, where s performance s sgnfcantly worse than that of and FH-OMB. performs worse than for small N for the same reason as above. The scenaro s so small that the recevers are all very heterogeneous. As homogenety ncreases for hgher network denstes, explotng opportunstc gan becomes more mportant and outperforms. FH-OMB performs well for all szes of the recever set. Its state-based completon tme estmaton results n the rght tradeoff between opportunstc gan and multcastng gan and provdes the lowest completon tmes of all approaches. It consstently outperforms by 9% to 9%. The performance gan over ranges from % to 76%. Next, we look at the mpact of varyng K for a fxed N = 3. When ncreasng the number of beamformng lobes K, the array gan of the sngle lobe beam ncreases as well. In the specfc antenna confguraton that s chosen for our smulaton, the array gans for K = {, 4, 8, 6} are.9, 3.4, 6.6 and.4, respectvely. (Note that the array gan s not lnear n K.) Therefore, as observed from Fg. 3, completon tme decreases wth ncreasng K wth respect to the achevable gan. FH-OMB outperforms both and for all K. However, ncreasng K has a more sgnfcant mpact on the completon tme of than on and FH-OMB. For low K and a wder beamwdth, s lmted by the recever wth the lowest SNR n each beam. (Also, a sgnfcant amount of the radated energy does not cover any recever.) As K ncreases, fewer and fewer recevers are covered by a beam and n the extreme case of a sngle beam per recever, manages to perfectly balance the SNRs at the recevers (.e, no energy s wasted by havng a hgher than necessary SNR at any recever). Hence, s performance becomes closer and closer to FH-OMB. In contrast, may stll beamform to a few recevers wth hgh SNRs so that those fnsh frst, before servng recevers wth lower SNRs. In short, n heterogeneous scenaros wth suffcent K, that favors the weaker recevers by multcastng to all the recevers performs better than that captalzes n maxmzng opportunstc gan. CDF Fg. 4. CDF of completon tme for random recever dstrbuton. N = 6, K = 8, B = 64kbts. CDF.5 Fg. 5. CDF of completon tme for random recever dstrbuton. N = 64, K = 8, B = 64kbts. To shed more lght on the behavor of the algorthm, we show the CDF of completon tme for the smulaton runs for K = 8 and wth N = 6 and N = 3 recevers n Fg. 4 and Fg. 5, respectvely. In Fg. 4, and have relatvely smlar completon tme as FH-OMB n % of the smulaton runs. Ths happens n scenaros where all recevers are dstrbuted qute close to the BS and thus all recevers have a relatvely homogeneous good average channel qualty. When recevers are dstrbuted sparsely wthn the cell radus, wth hgh probablty they have dfferent average channel qualtes. Under ths scenaro, performs badly snce t opportunstcally serves the better recevers frst and therefore results n hgher completon tmes than both FH- OMB and. Here, FH-OMB recevers fnsh at 465 slots for the worst scenaros, whereas and both requre 57 and 84 slots, respectvely. When the number of recevers ncreases, Fg. 5 shows that no longer has most of ts completon tme close to FH-OMB n most of the smulaton runs. In fact, around % of s completon tme s smlar to due to the lmted number of beamformng lobes (K = 8), whch leads to low mult-lobe beam s SNR. s partcularly bad n scenaros where many of the recevers are relatvely far from the BS (and thus more homogeneous). The worst case completon tme of FH-

9 9 OMB s at 675 slots, whle and requre 8 and 4 slots, respectvely Number of recevers (N) Fg. 6. Cell edge recever dstrbuton, K = 8, B = 64kbts Number of beamformng lobes (K) Fg. 7. Cell edge recever dstrbuton, N = 3, B = 64kbts. ) Cell Edge Recever Dstrbuton: In ths scenaro, recevers are all dstrbuted close to the cell edge n the range of 9m to m and thus form a relatvely homogeneous group. Whle such a scenaro s less realstc than the one presented n Secton V-B, t s ncluded to llustrate the performance degradaton of the algorthm n more homogeneous scenaros. Note that ths performance s also ndcatve of the performance n heterogeneous scenaros wth very hgh user denstes, where many recevers are at the cell edge (see Fg. ). Here, the performance dfferences are much more drastc and performs worse than the other schemes already for N > 8 (see Fg. 6). For 3 recevers, FH-OMB outperforms by 59%. Although maxmzng nstantaneous throughput s the rght strategy for homogeneous scenaro, FH-OMB stll manages to slghtly outperform the algorthm by about %. Despte the homogenety of the scenaro, the slght dfferences among the recevers requre a more sophstcated mechansm that does take states nto account. Smlar to the scenaro wth heterogeneous recever dstrbuton n Secton V-B, completon tme mproves wth ncreasng K (see Fg. 7) due to the hgher effectve SNR for each beam. 3) Remarks: Summarzng, we can observe that n the more realstc Raylegh fadng scenaro, the performance gans of our FH-OMB heurstc are much more pronounced than n the smple scenaros wth two recevers and two channel states. Compared to, these gans ncrease as the number of recevers ncreases snce does not explot opportunstc gan. On the other hand, FH-OMB s partcularly benefcal over n heterogenous users scenaros because ts balanced tradeoff between mult-user dversty and multcast gan results n lower completon tmes. VI. CONCLUSION In ths paper we studed opportunstc multcast beamformng for the fnte horzon problem, where a base staton has a fxed amount of erasure-coded data to transmt to multple recevers. We model the problem as a dynamc programmng problem to obtan the optmal soluton. Due the hgh complexty of ths approach, we desgn a heurstc algorthm, FH- OMB, that captures the characterstcs of the optmal soluton and provdes a performance that s very close to t. We evaluate FH-OMB s performance both for a dscrete channel model as well as multpath Raylegh fadng. It outperforms other schemes based on maxmzng the mnmum SNR and broadcastng to all recevers (), as well as greedly maxmzng sum rate (). It mproves performance by up to 76% over the former and up to 9% over the latter for heterogeneous scenaros wth Raylegh fadng. For homogeneous scenaros, these gans are up to % and %, respectvely. Smlar (though slghtly lower) gans are obtaned for the smpler scenaro wth a dscrete channel model. For future work, we plan to address the mpact of delayed and mperfect channel knowledge as well as optmze the feedback overhead. ACKNOWLEDGEMENT Ths paper was supported n part by the Madrd Communty through the MEDIANET project (S9-TIC468) and Spansh MINECO/MICINN grant TEC-9688-C-. REFERENCES [] U. Kozat, On the throughput capacty of opportunstc multcastng wth erasure codes, n IEEE INFOCOM, Apr. 8. [] T.-P. Low, M.-O. Pun, Y.-W. P. Hong, and C.-C. J. Kuo, Optmzed opportunstc multcast schedulng (OMS) over wreless cellular networks, IEEE Transactons on Wreless Communcatons, vol. 9, no., pp. 79 8, Feb.. [3] G. H. Sm, B. Rengarajan, and J. Wdmer, Adaptve modulaton for fnte horzon multcastng of erasure-coded data, n COMSNETS, 3. [4] K. Sundaresan, K. Ramachandran, and S. Rangarajan, Optmal beam schedulng for multcastng n wreless networks, n ACM MobCom, 9. [5] E. Aryafar, M. Khojastepour, K. Sundaresan, S. Rangarajan, and E. 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