Enabling P2P One-view Multi-party Video Conferencing

Transcription

1 Enablng P2P One-vew Mult-party Vdeo Conferencng Yongxang Zhao, Yong Lu, Changja Chen, and JanYn Zhang Abstract Mult-Party Vdeo Conferencng (MPVC) facltates realtme group nteracton between users. Whle P2P s a natural delvery soluton for MPVC, a peer often does not have enough bandwdth to delver her vdeo to all other peers n the conference. Recently, we have wtnessed the popularty of one-vew MPVC, where each user only watches full vdeo of another user. One-vew MPVC opens up the desgn space for P2P delvery. In ths paper, we explore the feasblty of a pure P2P soluton for one-vew MPVC. We characterze the vdeo source rate regon achevable through vdeo relays between peers. For both homogeneous and heterogeneous MPVC systems, we establsh tght unversal vdeo rate lower bounds that are ndependent of the number of peers, the number of vdeo sources, and the specfc vewng relatons between peers. We further propose P2P vdeo relay desgns to approach the maxmal vdeo rate regon. Through numercal smulatons, we verfed that the derved lower bounds are ndeed tght bounds, and the proposed bandwdth allocaton algorthm can acheve a close-to-optmal peer upload bandwdth utlzaton. Our results demonstrate that P2P s a promsng soluton for one-vew MPVC. Insghts obtaned from our study can be used to gude the desgn of P2P MPVC systems. Index Terms Vdeo Conference, P2P, Relay. INTRODUCTION The prolferaton of vdeo-capable consumer electronc devces and the penetraton of ncreasngly faster resdental network accesses paved the way for the wde adopton of Mult-Party Vdeo Conferencng (MPVC), whch facltates realtme group nteracton between users. Peer-to-Peer (P2P) s a natural delvery soluton for MPVC where users transmt ther voce and vdeo drectly among themselves. The major challenge for P2P MPVC s that users alone may not have enough upload bandwdth to transmt ther voce and vdeo data. Skype [9] offers MPVC servce to ts pad premum customers. Our recent measurement study [2] shows that n a Skype MPVC, whle voce s stll transmtted usng P2P, vdeo of a user s frst uploaded to a server, then relayed to all other users n the conference. Ths desgn choce s due to the fact that n an all-vew MPVC, where each user watches vdeos of all other users, the aggregate vdeo upload workload ncreases quadratcally wth the number of users, whle the aggregate upload capacty avalable on users only ncreases lnearly. Pure P2P s obvously not a self-scalable soluton for all-vew MPVC. Hybrd peer-asssted solutons have been studed recently [3], [4]. Another concern for all-vew MPVC s that, even though servers can provde abundant upload bandwdth, the downlnk of a user mght not be able Yongxang Zhao s wth the school of Electrcal and Infomaton, Bejng Jaotong unversty, CHINA. E-mal:yxzhao@bjtu.edu.cn. Yong Lu s wth ECE Department of Polytechnc Insttute of New York Unversty, USA. Emal: yonglu@poly.edu. Changja Chen s wth Bejng Jaotong unversty, CHINA. E- mal:changjachen@sna.com JanYn Zhang s wth Research Insttute of Chna Moble Bejng, CHINA. Emal:zhangjanyn@chnamoble.com to sustan hgh-qualty vdeo streams from all other users. More recently, Google+ [8] offers a free one-vew MPVC servce: each user can only choose one user to watch at hgh vdeo qualty, and receves all other users vdeos at the mnmum vdeo qualty. Our measurement study shows that Google+ s one-vew MPVC s stll mplemented as a pure server-based soluton: a user chooses a dedcated server as her MPVC proxy, uploads her voce and vdeo data to the proxy, and downloads voce and vdeo data of other users from the proxy. Such a server-centrc backhaul desgn not only ncurs hgh server cost, but also totally gnores the network and geographc localty of users n a conference. Users located far away from servers are forced to traverse long network paths wth large delay and low throughput, leadng to poor user conferencng experence. In onevew MPVC, the aggregate vdeo download workload s reduced to be proportonal to the number of users. The aggregate peer upload bandwdth can now keep up wth the aggregate vdeo upload workload. It s therefore temptng to develop a pure P2P soluton for one-vew MPVC. Such a soluton not only elmnates the server cost, but also can explore user localty better to acheve shorter delay and hgher throughput, whch s crtcal to facltate realtme user nteractons. In addton, P2P MPVC s an attractve soluton to set up ad-hoc MPVC not subject to centralzed management and montorng. P2P relay desgn n MPVC s more complcated than n vdeo streamng. In P2P vdeo streamng, a set of peers watchng the same vdeo source form a swarm and relay vdeo to each other. Due to the common vdeo nterest, vdeo relay between peers are mostly drven by ther. A user can dynamcally choose her full-qualty vdeo source based on her current nterest. By default, the system wll send full vdeo of the user currently speakng to users not specfyng ther nterests.

2 2 bandwdth avalablty. In P2P MPVC, peers have dverse vewng nterests. Each peer s a potental vdeo source watched by other peers, and at the same tme s watchng another source. The vewng relatons between peers are ntrnscally entangled. More challengngly, peers vewng nterests are drven by varous conference dynamcs, such as user voce and gesture actvtes, appearance of new objects, and topc swtchng, etc. Vdeo relays between peers have to be adaptve to the entangled and dynamc vewng relatons. In ths paper, we explore the feasblty of P2P one-vew MPVC by characterzng ts capacty regon through analyss and numercal smulatons.. We assume that voces and small vdeos of peers are delvered usng some tradtonal P2P technque and only focus on the P2P delvery of full vdeos between peers. 2 We further assume that peers n the same conference are cooperatve and relay vdeos for each other. To mantan good delay performance, P2P vdeo relay s lmted to two-hops. The contrbutons of our study s four-fold: ) We propose a P2P relay framework for one-vew MPVC. We characterze the vdeo rate capacty regon for homogeneous and heterogeneous onevew MPVC. We study the optmal P2P relay desgn to maxmze the aggregate vdeo qualty. We also propose rate allocaton schemes to acheve the max-mn farness between vdeo sources. 2) We establsh several unversal vdeo rate lower bounds for P2P one-vew MPVC that are ndependent of the vewng relatons between peers. For homogeneous one-vew MPVC wth normalzed peer upload bandwdth of, we show that each source s guaranteed to acheve the vdeo rate of 5/6, that s also ndependent of the sze of MPVC and the number of sources. 3) For heterogeneous MPVC wth the normalzed average peer upload bandwdth of, we show that the guaranteed vdeo rate for source wth upload bandwdth u s mn(u, γ), where γ = + S max( 2 3, ) wth beng the number of peers and S beng the number of sources. The lower bound can be mproved to 3/4 f all sources upload bandwdth s above the average. We further show that the derved lower bounds are tght for homogeneous and heterogeneous systems. 4) We develop peer bandwdth allocaton algorthms that effcently utlze peers upload bandwdth to approach the maxmal vdeo rate regon. Through smulatons, we verfed that the derved lower bounds are ndeed tght bounds, and our bandwdth allocaton algorthm can acheve a close-tooptmal peer upload bandwdth utlzaton. We brefly descrbe the related work n Secton 2. The P2P relay framework for one-vew MPVC s presented n 2. The bandwdth avalable for full vdeo dstrbuton s the total upload capacty mnus the upload bandwdth utlzed for transmttng voce and small vdeos. Secton 3. In Secton 4, we establsh the unversal vdeo rate lower bound for homogeneous one-vew MPVC. We study the vdeo rate capacty regon for heterogeneous MPVC n Secton 5. Two optmal P2P MPVC desgns are studed to maxmze the aggregate vdeo qualty and acheve the max-mn farness respectvely. We also derve the guaranteed max-mn capacty for heterogeneous one-vew MPVC. In Secton 6, we present a P2P relay bandwdth allocaton algorthm to approach the maxmal vdeo rate regon. In Secton 7, we demonstrate the tghtness of the derved lower bounds and the effcency of the proposed bandwdth allocaton algorthm through numercal smulatons of randomly generated one-vew MPVC scenaros. The paper s concluded wth future work n Secton 8. 2 RELATED WORK Whle P2P has been wdely adopted for fle sharng [3], [8] and vdeo streamng [7], [], only very lmted efforts have been attempted for P2P MPVC n the research communty. Chu et al. [5] proposed an End-System- Multcast archtecture to support vdeo conferencng applcatons, where multcast functonalty s pushed to the edge. Lennox and Schulzrnne [2] proposed a full-mesh conferencng protocol wthout a central pont of control. Luo et al. [5] proposed to ntegrate applcaton layer multcast wth natve IP multcast n P2P conferencng systems. In [6], all users watchng the same source form a chan and relay vdeo to each other. Akkus et al. [] extends ths dea to relay vdeo encoded n multple layers. Recently, Chen et al. [4] proposed hybrd solutons to employ helpers to maxmze the utlty n P2P conferencng swarms, where helpers assst sources n relayng vdeo streams to recevers. Ponec et al. [6] then extended ths soluton to support mult-rate conferencng applcatons wth scalable codng technques. Lang et al. [4] studed optmal bandwdth sharng n mult-swarm conferencng systems. Both ntra-swarm and nter-swarm peer bandwdth allocaton algorthms are proposed to maxmze the system-wde utlty. None of the prevous study nvestgate the mpact of vewng relatons on the achevable vdeo capacty regon. Accordng to a recent study [9], a user on average watches one or two vdeos of other users snce t s dffcult for a user to smultaneously keep track of three or more vdeo sources. It s often more preferable for a user to watch hgh qualty vdeos of a couple of users of nterests, rather than watch lousy vdeos of all users. Our work s motvated by the recent trend of one-vew MPVC. We establsh unversal vdeo rate lower bounds and propose P2P relay algorthm to acheve the maxmal capacty regon. Our study demonstrate that t s promsng to develop a pure P2P soluton for one-vew MPVC.

3 3 TABLE Notatons Notaton Defnton N set of peers n the conferencng system S N set of vdeo-actve peers (sources) I N S set of vdeo-dle peers (pure vewers) G s set of vewers of source s S G (S) s G s S vewers of source s who are also sources G (I) s G s I vewers of source s who are pure vewers r s rate of vdeo generated by source s u total upload bandwdth of peer u (s), S upload bandwdth of a source peer S allocated to the sub-conference t s hostng u (w) u (h) B s (W ) B s (H) 3 P2P ONE-VIEW MPVC 3. One-vew MPVC upload bandwdth of peer allocated to the sub-conference t s watchng upload bandwdth of peer allocated to the common helper bandwdth pool b.w. contrbuted to swarm s by ts vewers b.w. contrbuted to swarm s by helpers We consder an one-vew mult-party vdeo conference, where, at any gven tme, each peer only watches one full vdeo generated by another peer. We further assume a peer can swtch among vdeos of other peers, the vewng relatons between peers are tme-varyng. A snapshot of vewng relatons among all peers n the conference s defned as an one-vew MPVC scenaro. As enumerated n Table, the whole set of peers s denoted by N, wth n = N be the total number of peers. In a specfc scenaro, peers can be classfed nto two classes: the vdeo-actve peers, denoted by S, whch are the peers beng watched by some other peers, and the vdeo-dle peers, denoted by I, whch are the peers not watched by any other peer. We call each vdeoactve peer s S a vdeo source, and use G s to denote the subset of peers watchng the vdeo of s. We say peers n G s partcpate n a sub-conference hosted by s. Snce each peer watches exactly one vdeo, {G s, s S} forms a partton of N, and we have n = s S G s, where G s s the number of peers n sub-conference s. Snce a peer watchng s can also host her own subconference,we further partton the vewers of s nto two subsets:g (S) s G s S, the subset of vewers who are hostng ther own sub-conferences; and G (I) s G s I, the subset of vewers who are pure vewers. Apparently, peer upload bandwdth allocaton and peer perceved vdeo qualty depend on the vewng relatons between peers. Fg. (a) shows a watchng scenaro for the same MPVC: peer 2 and 3 watchng peer ; peer watchng peer 2; peer 4 watchng peer 3. Fg. (b) plots one feasble bandwdth allocaton scheme: for G, peer transfers one vdeo sub-stream to peer 4 at rate.5, and peer 4 relays the receved sub-stream to peer 2 and 3, peer transfers another sub-stream at rate.25 drectly to peer 2 and peer 3 wth ts remanng.5 bandwdth; for G 2 and G 3, peer 2 and peer 3 upload ther (a) vew relatons (b) P2P relay Fg.. One-vew MPVC Example /8 /8 /8 7/8 /8 /2 /2 /2 4 /8 2 3 /8 (c) P2P relay 2 streams drectly to ther vewers at rate respectvely. Under ths bandwdth allocaton scheme, the receved vdeo rates on peer,2,3,4 are (, 3/4, 3/4, ). In ths bandwdth allocaton scheme, even though peer 4 s not nterested n vdeo of peer, t helps peer upload vdeo wth all ts upload bandwdth. Fg. (c) shows another bandwdth allocaton scheme whch enables all peers watchng rate reach 7/8. In ths scheme, the vdeo sources 2 and 3 reserve /8 of bandwdth to relay the vdeo t s watchng. For general one-vew MPVC, the frst natural queston to ask s: gven peer upload bandwdth profle, what are the maxmal vdeo source rates that can be supported under a specfc vewng scenaro? It s expected that dfferent vewng relatons between peers wll lead to dfferent supportable vdeo source rates. It s also temptng to ask the second queston: what are the maxmal vdeo source rates that can be supported under all possble vewng scenaros? We wll provde answers to these questons usng analyss and smulatons n the followng sectons. 3.2 P2P Vdeo Relay In ths secton, we formally ntroduce the P2P bandwdth sharng model n one-vew MPVC. In P2P overlay networks, where each peer can reach all other peers, t s commonly assumed that peer upload lnks are the only bandwdth bottleneck [7], [4], []. In the rest of the paper, we adopt the assumpton that the core network s congeston-free and vdeo rates n MPVC are lmted by peer s upload bandwdth. To maxmally utlze peer upload bandwdth, we assume all peers are fully cooperatve. A peer not only can relay vdeo that she s watchng to other peers n the same sub-conference, she can also help peers watchng a dfferent source by downloadng and relayng vdeo of the source to whch she has no nterest to watch. Snce vdeo conferencng s hghly delay-senstve, to lmt the delay ncurred by relay, we also lmt P2P vdeo relay to two overlay hops,.e., vdeo can be relayed by at most one ntermedate peer from a source to all ts recevers. 3 Fg. 2 llustrates the concept of P2P vdeo relay among peers n dfferent sub-conferences. Let s frst focus on a source peer S. Wthout loss of generalty, peer s 3. It has also be shown that two-hop relay s bandwdth optmal n uplnk throttled P2P systems [4], [] 7/8

4 4 G u (s) (a) as source H u (w) j G j (b) as vewer Fg. 2. Dfferent Roles of a Peer n MPVC. H k G k (c) as helper hostng a sub-conference G, whle watchng the vdeo of another source, say peer j. In Fg. 2(a), peer dvdes ts vdeo nto multple sub-streams, wth possbly unequal rates. Then t sends these sub-streams to peers n ts own sub-conference G. Each peer s responsble for duplcatng and relayng the receved sub-stream to all other peers n G. Peers outsde of G can also help redstrbute s vdeo. We call those peers the helpers of G. Peer sends a sub-stream to a helper, who then relays the sub-stream back to peers n G. In Fg. 2(b), other than dstrbutng ts own vdeo, peer, as a vewer n sub-conference G j, s also responsble for redstrbutng the vdeo of peer j. Addtonally, peer may also act as a helper to help the sub-conferences that she s not hostng, nor watchng. In Fg. 2(c), peer helps relay the vdeo of peer k even though peer does not watch vdeo of k. Let s now examne how a source peer S allocates ts upload bandwdth of u among ts three roles n MPVC. ) As a source, peer allocates u (s) bandwdth to upload her own vdeo. u (s) conssts of the bandwdth used to upload vdeo drectly to her vewers and the bandwdth used to upload vdeo to her helpers; 2) As a vewer, peer allocates u (w) bandwdth to relay the vdeo of the source she s watchng; 3) As a helper, peer allocates u (h) bandwdth to relay the vdeo of other sources that she s not watchng. H u (h) For a peer not hostng a sub-conference,.e., I, she has dual roles: vewer and helper. She only needs to splt her upload bandwdth between u (w) and u (h). Gven the bandwdth allocaton on all peers, we can calculate the bandwdth resource avalable to each subconference. For the sub-conference hosted by peer, there are three portons of bandwdth avalable. The frst porton s the bandwdth contrbuted by peer tself and t s u (s). The second porton s the bandwdth contrbuted by ts vewers B (W ) j G u (w) j. The thrd porton of the avalable bandwdth s contrbuted by all helpers of sub-conference. In prncple, any peer not n G can be a helper of G. Instead of trackng bandwdth allocaton of each helper to each sub-conference, we buld a common helper pool H to manage bandwdth contrbuted by all helpers. More specfcally, each peer N contrbutes u (h) amount of bandwdth to the helper pool H. The total bandwdth avalable n H s therefore B (H) N u(h). The manager of the helper pool s n charge of dstrbutng B (H) to dfference subconferences. Let B s (H) be the helper bandwdth allocated to sub-conference s, then we have s S B(H) N u(h) In the followng, we treat the helper bandwdth allocated to a sub-conference s as f t s from a sngle vrtual helper wth total upload bandwdth of B s (H). As wll be shown shortly, such a centralzed bandwdth management and helper vrtualzaton can acheve the maxmal vdeo rates n MPVC. Peer upload bandwdth allocaton U {u (s), u (w), u (h), N} determnes the upload bandwdth avalable for each source to dstrbute her vdeo to her vewers. In the sub-conference hosted by source S, as shown n [3], [4], the maxmal achevable vdeo rate s: r = mn { u (s), u(s) + B (W ) + B (H) G } B(H) G 2, () where B (W ) s bandwdth contrbuted by the vewers, and B (H) s bandwdth borrowed from the helper pool. 4 CAPACITY OF HOMOGENEOUS MPVC Equaton () states that the achevable vdeo rate n each sub-conference s determned by peer upload bandwdth allocaton. In the followng two sectons, we wll study the optmal peer bandwdth allocaton to acheve hgh vdeo rates cross all sub-conferences. In partcular, we establsh several non-trval vdeo rate lower bounds ndependent of the vewng relatons between peers. In ths secton, we assume peers are homogeneous and have normalzed upload bandwdth of. The smplest bandwdth assgnment s to assgn peer bandwdth to each sub-conference at the gratuty of. Ths means that a vdeo source wll use all ts bandwdth to transfer ts own vdeo,.e., u (s) s =, s S. An dle peer I utlzes ts bandwdth ether to transfer the =, u (w) s = u (h) s stream t s watchng or to help other sub-conferences,.e., u (w) + u (h) =, and u (w) u (h) =, S. The bandwdth allocaton becomes an dle peer assgnment problem: how to assgn dle peers to sub-conferences to maxmze ther vdeo rates? Theorem : In homogeneous one-vew MPVC wth two sources, both sources can acheve the maxmum rate of. Proof: Wthout loss of generalty, suppose source has G vewers, and source 2 has G 2 vewers. Snce each source always watches another source, we must have the two sources watch each other,.e., G 2 and 2 G. Then there are G and G 2 dle peers n sub-conference and 2 respectvely. If we let each dle peer only relay vdeo she s watchng, then we have u (s) = u (s) 2 =, B (W ) = G, B (W ) 2 = G 2, and B (H) = B (H) 2 =. Accordng to Equaton, we have r = r2 =. From the proof of Theorem, we know that, to acheve hgh vdeo rates, t s mportant to have enough dle s

5 5 peers to upload n each sub-conference. For general cases wth more sources, we have the followng result. Lemma : For any one-vew MPVC scenaro, we have G (I) = ( G ) (2) S S Proof: Snce a peer s ether an dle peer or a busy peer, thus = S + I. In addton, every peer watches exactly one vdeo. Hence = S G. So we have I = S G S = S ( G ) In addton, snce {G, S} s a partton of N, and G (I) = G I, then {G (I), S} s a partton of I N. Then we have S G(I) = I = S ( G ) Based on Lemma, we present an dle peer assgnment procedure that can guarantee each sub-conference wth G users can be assgned wth G dle peers n followng procedure. ) For a sub-conference where all vewers are dle,.e. G (I) = G, t wll only use G of ts own vewers to relay vdeo and B (H) =, t also contrbutes one dle peer to the common helper pool H; 2) For a sub-conference where exactly one vewer s a source,.e. G (I) = G, t wll only use all of ts own vewers to relay vdeo and B (H) =, t does not contrbute any peer to the helper pool H; 3) For a sub-conference wth G I < G, t wll use t own G I dle peers and G G I dle peers from the helper pool H to relay ts vdeo,.e., B (H) = G G I. If we use S, S 2 and S 3 to represent the set of subconferences n case ), 2), and 3) respectvely, then the number of helpers contrbuted to H by sub-conferences n S s: H = S G (I) ( G ) = k=2,3 S k G G I = S 3 G G I = S 3 B (H), where the second equalty s due to Lemma and the thrd equalty s due to G (I) = G for any sub-conference n S 2. Ths guarantees that the prevous helper allocaton scheme s feasble. Idle peers assgned to sub-conference of source nclude ts own dle vewers n G (I), and dle peers from other sub-conferences. Theorem 2: If all peers bandwdth s one, for any gven scenaro {G, S}, the achevable vdeo rate r for any sub-conference G satsfy: r = B(H) G 2 G + G Proof: In the prevous dle peer assgnment, (3) u (s) =, B (W ) + B (H) = G, S. (4) Accordng to Equaton, the achevable rate s Snce B (H) r = + G G G 2 = B(H) G 2. B(H) G, we have r G + G 2 Let f(x) = x + x 2, f(x) s an ncreasng functon when x 2, and f(2) = 3 4 ; Thus, when G 2, r 3 4, when G =, the source send ts stream drectly to the only vewer, and r =. Theorem 2 apples to any one-vew MPVC. The lower bound of 3/4 s ndependent of the vewng relatons between peers. Ths non-trval lower bound has mportant mplcatons on the practcal mplementaton of MPVC, wthn whch a peer may jon or leave a sub-conference at her wll. It s undesrable to change the vdeo rates of sub-conferences frequently whenever the vewng relatons change. Our results suggest that t s possble to fnd a constant rate for all vdeo sources that s achevable n any possble one-vew MPVC scenaro, ndependent of the vewng relatons among peers, sub-conference szes, and even the total number of peers n the system. We name the maxmum value of such a constant source rate as the guaranteed capacty of one-vew MPVC and denote ths value as C. Theorem 3: If all peers have homogeneous upload bandwdth of, the guaranteed capacty C for any homogeneous one-vew MPVC s 5/6. Proof: In the confguraton of Theorem 2, the vdeo source uses up ts upload bandwdth to dstrbute the vdeo stream to other peers n ts sub-conference. Here we wll use a slghtly dfferent vdeo dstrbuton confguraton to acheve a hgher bound of the capacty C. In ths confguraton, all source peer wll use rate w to upload the ts own vdeo whle the remanng upload bandwdth of w s used to dstrbuton the vdeo t s watchng. On the other hand, dle peers are stll assgned to dfferent sub-conferences n the same way as n Theorem 2. An dle peer wll contrbute ts full upload bandwdth to help transmttng the vdeo assgned to t. Under ths confguraton, besdes the helper bandwdth, the busy peers n G also contrbute upload bandwdth to sub-conference. Accordng to Equaton, v (w) = w + ( w) G(S) + G G B(H) G 2 (5) Case : If all vewers of source are dle peers, and G (S) =. Accordng to the dle peer assgnment rule, =, S. Equaton (5) becomes B (H) v (w) = w + G G To have v (w) w, we need v (w) = w + G G w G ( G )w w

6 6 Snce w <, we always have v (w) w. Thus r = w. Case 2: If exactly one vewer of source s a source, and G (S) =. In ths case, B (H) =. Equaton (5) becomes v (w) = w + ( w) + G G = Thus, Thus r = w n ths case. Case 3: If more than one vewer of source are sources, and G (S) 2. Substtute G (S) = G G I and B(H) = G G (I) nto Equaton (5), we have v (w) = (2 w)( G ) + G 2 + ( w) G + G 2 G (I) Snce G G (S) 2, f we set w = 5 6, when G 4, we have v (5/6) > (2 5/6)( ) (2 5/6)( /4) = 7/8. G when G = 3, we have v (5/6) > (2 5/6)( /3) + /9 8/9 > 5 6 when G = 2, we have v (5/6) > (2 5/6)( /2) + /4 5/6 In all cases, we wll have r 5/6, therefore we conclude that C 5/6. Fnally, to show ths 5/6 s a tght bound of the guaranteed capacty, we only need to come up wth a homogeneous MPVC scenaro such that the maxmal achevable rate on all vdeo sources s only 5/6. Snce we wll use an optmzaton formulaton for the more general heterogeneous MPVC scenaro, we present t as a constructve proof n next paragraph. We construct the followng homogeneous one-vew MPVC: there are sx peers wth unt bandwdth, four of them are sources, S =, 2, 3, 4, the vewng relaton s: G = {3, 4}, G 2 = {6}, G 3 = {, 2},G 4 = {5}. Pluggng n ths scenaro to OPT II, we obtan the max-mn capacty γ = 5/6. Snce all sources have bandwdth, 5/6 s the maxmal achevable rate on all sources. Ths proves that the guaranteed capacty C for any homogeneous onevew MPVC can not be hgher than 5/6. 5 CAPACITY OF HETEROGENEOUS MPVC In the prevous secton, we assume peer upload bandwdth s homogeneous and only assgn dle peers to dfferent sub-conferences. In practce, peer upload bandwdth s heterogeneous. Peer upload bandwdth should be allocated to sub-conferences at fner granularty than. In ths secton, we study optmal peer bandwdth allocaton schemes to acheve dfferent desgn objectves n heterogeneous MPVC systems. 5. Maxmzng Aggregate Vdeo Qualty The frst desgn objectve s to maxmze the total vdeo qualty receved by all peers. We adopt a PSNR-type of vdeo qualty model [4], whch quantfes the qualty of a vdeo stream at rate r as log(r ). The optmal peer bandwdth allocaton s to maxmze the total vdeo qualty of the conference: subject to : OPT I: r u(s) max U,R,B G log(r ), (6) S + j G u (w) j G + B (H) B(H) G 2, (7) r u (s), S (8) u u (s) + u (w) + u (h), S (9) u u (w) + u (h), I () s S B (H) s N u (h), () where (7) and (8) are source vdeo rate constrants accordng to Equaton (), (9) and () are upload bandwdth constrants on sources and dle peers respectvely, and () enforces the bandwdth supply and demand balance n the common helper pool. The objectve functon s a concave functon of {r } and the constrants are all lnear. It s a convex optmzaton problem, for whch effcent centralzed and dstrbuted algorthms can be developed to solve for the optmal vdeo source rates R = {r, S} and the assocated optmal P2P relay scheme characterzed by the peer upload bandwdth allocaton U = {u (s), u (w), u (h), N} and helper bandwdth allocaton B = {B s (H), s S}. Due to the log vdeo utlty functon, the optmal soluton of OPT I acheves the weghted proportonal farness among all vdeo sources, wth the weght for a sub-conference be the number of vewers. 5.2 Achevng Max-Mn Farness Another wdely used farness metrc s the max-mn farness. Intutvely, we prefer all sources to acheve the same rate as long as t s allowed by the ndvdual source s upload capacty and the avalable bandwdth resource n the whole MPVC system. To acheve ths, we want to fnd a vdeo rate γ such that f a vdeo source s upload capacty u s less than γ, t should be able to stream ts vdeo at rate r = u, for any other source wth u γ, t should stream ts vdeo at the common rate r = γ. Under ths settng, the capacty of the system s defned as the maxmal supportable γ, whch can be solved by the followng optmzaton problem. OPT II: max γ (2) U,R,B subject to (7), (8), (9), (), () and a new set of constrants r = mn(γ, u ), S (3)

7 7 OPT II s no longer a smple convex programmng problem due to the non-lnear constrants n (3). We developed the followng algorthm to obtan the soluton of OPT II.We dvde the soluton space of γ accordng to the bandwdth dstrbuton of all vdeo sources. Specfcally, we frst sort the upload bandwdth of vdeo sources n a non-decreasng order and denote the sequence as {b, b 2,, b S }, wth b b j, < j. We then condense the lst nto a strctly ncreasng lst {c, c 2, c m } by removng redundant values. Let c =, and c m+ =, the soluton space for γ can be dvded nto m + ntervals: [c k, c k+ ), k m. When casted nto nterval k, OPT II becomes a lnear programmng problem beng replaced by a set of lnear constrants: { u, S such that u r = c k (4) γ, S such that u > c k We can solve OPT II teratvely, startng from nterval untl the frst nterval k where the optmal soluton of OPT II satsfes γk < c k+. Then γk s the fnal soluton of OPT II: γ = γk, and the optmal source rates are r = u f u γ and r = γ f u > γ. Theorem 4: The optmal source rates obtaned n solvng OPT II s max-mn far. Proof: Accordng to the defnton of max-mn farness, an allocaton vector X s max-mn far f and only f X, when sorted n non-decreasng order, s lexcographcally maxmal among all feasble allocaton vectors sorted n non-decreasng order. We prove the theorem usng contradcton argument. Let s assume the optmal source rates R of OPT II s not max-mn far, then there must exst another source rate vector R whch s lexcographcally larger than R. In other words, f we sort both R and R nto non-decreasng order, there exsts an ndex k such that r = r for =,, k, and rk+ < r k+. Wthout loss of generalty, we sort peer d n non-decreasng order of ther upload capacty, let w be the peer d such that u w < γ and u w+ γ, then we know that R = {u,, u w, γ,, γ }. For any peer, w, ts vdeo rate s constraned by ts own upload capacty. In any other feasble soluton, ncludng R, the hghest possble vdeo rate for peer s stll u. In other words, the frst w components of R (when sorted n non-decreasng order) s upper bounded by {u,, u w }, wth component-wse vector comparson. So we must have k w. If k == w, then rw+ > γ, and a new vector R {u,, u w, rw+,, rw+} s a feasble source rate vector, and rw+ > γ s a better soluton for OPT II. Ths contradcts wth the fact that γ s the optmal soluton of OPT II. If k > w, then R = {u,, u w, γ,, γ, rk+, }. Then for source peer k +, we can reduce ts vdeo rate by an amount of = r k+ γ 2, and contrbute the saved upload bandwdth on peer k + to the helper pool to ncrease the rate of source w + through k. Specfcally, let ɛ = mn { u w+ γ, k =w+ G }. By allocatng ɛ G helper bandwdth to sub-conference, wth w + k, we ncrease the vdeo rates of subconferences from w+ to k by ɛ. Then the newly acheved vdeo rates vector s R 2 = {u,, u w, γ + ɛ,, γ + ɛ, r k+, }. Consequently, γ + ɛ s a better soluton of OPT II than γ. Ths agan contradcts wth the fact that γ s the optmal soluton of OPT II. In concluson, there s no feasble vdeo source rate vector whch s lexcographcally larger than R. The optmal source rates obtaned n solvng OPT II s max-mn far. Defnton Max-mn Capacty: we defne the optmal soluton γ of OPT II as the max-mn capacty of a heterogeneous one-vew MPVC scenaro. 5.3 Lower Bound of Max-mn Capacty Whle the max-mn capacty γ for each one-vew MPVC scenaro can be teratvely solved for the correspondng optmzaton problem OPT II, smlar to the homogeneous case, t s mportant to obtan lower bounds of γ for heterogeneous systems that s ndependent of specfc watchng relatons, and even better, ndependent of conference szes. We normalze peers upload bandwdth such that the average peer bandwdth s unt one. Then we have N u =. We frst establsh a lower bound for γ as a functon of the number of sources and the number of peers n MPVC, but ndependent of the vewng relatons among peers. Theorem 5: For any one-vew MPVC wth N peers and S sources, for any vewng scenaro, we have γ + S Proof: We prove t by constructng a peer and helper bandwdth allocaton scheme that leads to γ + S. Specfcally, for each source peer S, ts vdeo rate s r = mn(u, γ ), and ts bandwdth allocaton scheme s u (s) = r, u (w) =, u (h) = u r,.e., each source peer only reserves upload bandwdth of ts own source rate r, and contrbutes the remanng bandwdth to the helper pool. For each dle peer I, the bandwdth allocaton scheme s u (w) =, u (h) = u,.e., each dle peer contrbutes all ts upload bandwdth to the common helper pool. Under such a bandwdth allocaton, source frst uploads ts vdeo to the helper pool usng ts reserved upload bandwdth u (s) = r, then the helpers wll duplcate and relay a copy to each peer n G. The total helper bandwdth needed by sub-conference s B (H) = G r. To make the bandwdth allocaton scheme feasble, the demand of helper bandwdth should be less than the supply of helper bandwdth, that s r G u r j. S N j S

8 8 That s (r G + r ) u = (5) N S Snce r γ = r ( G + ) S + S, the left-hand sde of (5) = + S ( G + ) S ( + S ) = + S Thus, γ s a feasble soluton of OPT II and γ + S Whle the prevous lower bound depends on the number of sources and vewers, t s stll desrable to establsh lower bounds of γ whch apples to any one-vew MPVC. We call the maxmum of such lower bounds the guaranteed max-mn capacty of one-vew MPVC. Theorem 6: The guaranteed max-mn capacty of onevew MPVC s 2/3. Proof: We frst prove 2/3 s a guaranteed lower bound by constructng a specfc bandwdth allocaton scheme to acheve γ = 2/3 n any one-vew MPVC. Frstly, we allocate peer bandwdth as follows: u (s) = mn(u, ), u (w) =, u (h) u (w) = u u (s) ; S (6) =, u (h) = u, I (7) In (6), a vdeo source wth bandwdth larger than one reserves bandwdth of one to transfer ts own vdeo, and contrbutes the remanng bandwdth to the helper pool; a vdeo source wth bandwdth less than one uses up all ts bandwdth to transfer ts own vdeo. In (7), dle peers contrbute all ther bandwdth to the common helper pool. Secondly, we assgn the helper bandwdth to each subconference as B s (H) bandwdth needed s s S = G s u (s), s S. The total helper B s (H) = G s u (s) = u (s) s S s S s S = N = N u s S u (h), u (s) = S (u u (s) ) + u I where the second equalty s due to the total number of revewers s, the thrd equalty s due to the total user upload bandwdth (after normalzaton) s N, the last equalty s due to the prevous upload bandwdth allocaton on sources and dle peers. The sequence of equaltes show that the total helper bandwdth needed equals to the total helper bandwdth contrbuted. Ths bandwdth allocaton scheme s feasble. Now we calculate the acheved vdeo rates under ths peer and helper bandwdth allocaton scheme. For a source wth u >, accordng to Equaton, we have r = + G G G G 2 = G G 2 As dscussed n secton 4, r 3/4. For a source wth u, accordng to Equaton, we have v = u + G u G ( G u ) G 2 = G u G 2 (8) If G =, v = u. If G 2, let f(x) = x + u x 2, then df(x) dx = x 2 2u x 3 = x 3 (x 2u ). f(x) s an ncreasng functon when x 2u. For any gven u, v s an ncreasng functon of G when G 2, and the mnmal value s /2 + u /4 when G = 2. For a source wth upload capacty 2/3 u, v / /4 = 2/3, so the acheved vdeo rate r = mn(u, v ) 2/3. Fnally, for a source wth u < 2/3, we automatcally have u < /2 + u /4 v, the acheved vdeo rate r = mn(u, v ) = u,.e, the vdeo rate s constraned by the source upload capacty. In concluson, wth the proposed bandwdth allocaton scheme, the acheved vdeo source rates are: r = u, f u < 2 3 ; r 2 3, f 2 3 u < ; r 3 4, f u. Thus, γ = 2 3 s a lower bound of γ for any one-vew MPVC. The guaranteed max-mn capacty s at least 2/3. Now we prove the guaranteed max-mn capacty cannot be hgher than 2/3 by constructng one-vew MPVC wth γ 2/3. We construct the followng one-vew MPVC: there are 2m + peers (m s a postve nteger). Among these peers, there are m peers actng as vdeo sources, and one source peer has bandwdth of + ɛ, wth ɛ be a small postve value, each other vdeo source s bandwdth s one. In addton, there s a super peer whose bandwdth s 2m( ɛ). The remanng m peers upload bandwdth s zero. Each source watches vdeo of another source, and no two sources watch the same source. Each dle peer watches one source, and no two dle peers watch the same source. Fnally, the super peer watches the source wth upload bandwdth of + ɛ/2. All sources wth bandwdth one has two vewers, the source wth bandwdth of + ɛ/2 has three vewers. The bandwdth allocaton scheme to maxmze γ s: each vdeo source uploads ts vdeo to the super peer at rate ɛ and the super peer relays t to each of the two vewers n the sub-conference. Each vdeo source also uploads drectly to each of ts vewer at rate ɛ/2. The acheved vdeo rate of each sub-conference s ɛ/2, whch s less than each source s upload capacty. The average upload bandwdth of the conference s 2m( ɛ)+m+ɛ/2 2m+. Then the max-mn

9 9 capacty γ s the acheved vdeo rate normalzed aganst the average upload bandwdth γ (m, ɛ) = (2m + )( ɛ/2) 2m( ɛ) + m + ɛ/2 Snce lm m,ɛ γ (m, ɛ) = 2/3, we can construct a sequence of one-vew MPVCs wth max-mn farness capacty approachng 2/3 from the above. So the guaranteed max-mn farness capacty for arbtrary one-vew MPVC s 2/3. Wth Theorem 5, and Theorem 6, we mmedately have Corollary : For any one-vew MPVC wth peers and S sources, under any vewng scenaro, we have { γ max + S, 2 } 3 When provng theorem 6, we notced that for a source wth bandwdth greater than, ts vdeo rate can be larger than 3/4. Ths suggests that f all sources have upload bandwdth above the average upload bandwdth, the max-mn capacty can be made large. Corollary 2: If the bandwdth of all vdeo sources s larger than one, we have γ 3/4. Proof: Smlar to the proof n Theorem 6, we assume each vdeo source allocates bandwdth of one n ts own sub-conference, and contrbutes the remanng bandwdth to the helper pool. The bandwdth of each dle peer s contrbuted to the helper pool. The total bandwdth n the helper pool s S. Snce S ( G ) = S G S = S. Thus each sub-conference can be assgned wth G helper bandwdth,.e, B (H) = G. From Equaton, we have r = + G G G G 2 = G G 2 As dscussed n secton 4, r 3/4, S. So we have γ 3/4. 6 P2P MPVC RELAY DESIGN In the prevous two sectons, we characterzed the vdeo rate capacty regon for one-vew MPVC. Now we propose peer bandwdth allocaton algorthms to acheve a feasble vdeo source rate vector wthn the capacty regon. Instead of squeezng all peers upload bandwdth to acheve the maxmal vdeo rates, we focus on supportng a gven vdeo rate vector wth the mnmum peer upload bandwdth through effcent bandwdth allocaton. The saved peer bandwdth provdes a cushon to absorb the mpacts of peer churn and network bandwdth varatons ncurred n practcal MPVC systems. 6. Desgn Gudelnes As dscussed n Secton 3.2, a peer allocates ts upload bandwdth among ts three dfferent roles n a subconference: source, vewer, and helper. To develop effcent bandwdth allocaton algorthm, let s frst examne how dfferent roles contrbute to the acheved vdeo rate. From (), f source s not constraned by ts own upload capacty, the acheved vdeo rate can be rewrtten as r = u(s) G + B(W ) G + B(H) G ( G ), (9) where a unt bandwdth from ether the source or a vewer ncreases the vdeo rate by / G, but the contrbuton of a unt helper bandwdth s dscounted by a factor of ( / G ). The dscount reflects the overhead of employng a helper. Specfcally, whenever swarm employs a helper, source has to frst stream some vdeo to the helper so that t can relay vdeo back to the vewers of swarm. Snce the helper tself s not a vewer n swarm, the bandwdth used to stream vdeo to t does not drectly contrbute to the acheved vdeo rate n swarm. The overhead s nversely proportonal to G and decreases wth the sze of the sub-conference beng helped. An effcent bandwdth allocaton should maxmally avod helper bandwdth overhead. Ths leads to the frst gudelne: G: A sub-conference should maxmally utlze bandwdth avalable on ts source and vewers before usng helpers. A peer should always allocate ts bandwdth to the sub-conference she s hostng or vewng before contrbutng bandwdth to the helper pool. To avod helper bandwdth overhead, an dle vewer s bandwdth can only be used by the sub-conference she s vewng, but the bandwdth of a vdeo source can be utlzed by two sub-conferences: the sub-conference that she s hostng and the sub-conference that she s vewng. To preserve bandwdth allocaton flexblty, we propose the second gudelne: G2: A sub-conference wth target vdeo rate r frst draws bandwdth r from ts source, then t should maxmally utlze bandwdth avalable on ts dle vewers before drawng addtonal bandwdth from ts source and busy vewers. From (9), the helper bandwdth overhead s a decreasng functon of the sub-conference sze. Between the two sub-conferences that a vdeo source can upload to wthout overhead, the one wth the smaller number of vewers would ncur hgher helper bandwdth overhead f t uses bandwdth from the helper pool. To reduce the system-wde helper bandwdth overhead, we have the thrd gudelne: G3: If a source has surplus bandwdth over ts target vdeo rate, between the two sub-conferences that she s hostng and vewng, she should allocate the surplus bandwdth frst to the sub-conference wth the smaller number of vewers. 6.2 Bandwdth Allocaton Algorthm Now we present our bandwdth allocaton algorthm based on the three gudelnes. We adopt two-level h-

10 erarchy for bandwdth management. At the top level, a centralzed tracker manages the helper pool shared by all sub-conferences. It keeps track of the bandwdth contrbuted by peers n sub-conferences wth surplus bandwdth, and allocates helper bandwdth to subconferences wth bandwdth defct. At the bottom level, the bandwdth allocaton among peers n each subconference s coordnated by the vdeo source. Source of sub-conference mantans the followng states: ) r : the target vdeo rate for sub-conference ; 2) B (H) : the helper bandwdth borrowed from the helper pool, ntalzed to. 3) A : acheved vdeo rate under current allocaton, ntalzed to. 4) L j : bandwdth on peer j G that has not been allocated, ntalzed to u j. Bandwdth allocaton s carred out n four stages: vdeo source bandwdth allocaton at target rate r ; dle peer bandwdth allocaton; busy peer bandwdth allocaton; bandwdth allocaton to/from helper pool. Bandwdth allocaton n all sub-conferences are coordnated such that bandwdth allocaton n any sub-conference advances to stage k only after all sub-conferences fnsh the allocaton n stage k. Stage : Vdeo source allocates r bandwdth to send out the vdeo stream t produces. The remanng bandwdth of vdeo source s updated as L = u r. Accordng to Equaton (), the acheved vdeo rate s A = r / G. Stage 2: Vdeo source utlze dle vewers bandwdth to grow the achevable vdeo rate from A. The detaled algorthm s shown n Algorthm. Lne pcks up an Algorthm Idle Vewer Bandwdth Allocaton : for each dle peer p G (I) do 2: x = mn ((r A ) G, L p ) 3: A = A + x / G 4: L p = L p x 5: f A r then 6: break 7: end f 8: end for 9: B (H) = B (H) + p G (I) L p dle vewer p n local sub-conference G. Lne 2 uses ths peer s bandwdth to ncrease the vdeo rate. (r A ) G s the amount of bandwdth needed to mprove vdeo rate from A to r. Lne 3 and lne 4 update the acheved vdeo rate A and the unallocated bandwdth L p. Lne 5, 6 and 7 break loop f the target rate r s acheved. Lne 9 allocates the unallocated dle vewer s bandwdth to the helper pool. Stage 3: In ths stage, we allocate the bandwdth on busy peers to sub-conferences n whch the target vdeo rate has not been acheved. Accordng to gudelne G3, a busy peer should frst upload to the smaller subconference between the one she s vewng and the one she s hostng. To acheve ths, we conduct bandwdth allocaton for sub-conferences n the non-decreasng order of ther szes. The bandwdth allocaton wthn each subconference follows Algorthm 2. Ths process allocates Algorthm 2 Busy Vewer Bandwdth Allocaton : for each peer p G (S) {} do 2: x = mn ((r A ) G, L p ) 3: A = A + x / G 4: L p = L p x 5: f A r then 6: break 7: end f 8: end for 9: B (H) = B (H) + p G (S) {} L p bandwdth on the vdeo source of sub-conference and all other vewers who act as vdeo source for other subconferences. The allocaton s smlar to Algorthm and s self-explanatory. Stage 4: In ths stage, a sub-conference that has not acheved ts target rate usng bandwdth on ts source and vewers borrows bandwdth from the helper pool. Accordng to (9) to mprove vdeo rate of G from A to r takng nto account the helper bandwdth overhead, the needed helper bandwdth s B (H) = (r A ) G 2. G Each sub-conference wth bandwdth defct wll request bandwdth B (H) from the common helper pool. In the helper pool, f sum of the requested helper bandwdth s not bgger than the aggregate helper bandwdth B (H) contrbuted by bandwdth surplus sub-conferences, the centralzed tracker wll allocate to each sub-conference the requested helper bandwdth. Otherwse, the targeted vdeo rate vector s not supportable, and the tracker can proportonally reduce the helper bandwdth allocaton to sub-conferences. Through teratve bnary search, the bandwdth allocaton algorthm can also be used to dynamcally approach the max-mn capacty γ defned n OPT II. We frst set the search nterval to be [γ l, γ h ], wth γ h = max s S u s, and γ l beng the lower bounds obtaned n Secton 4 and 5. Specfcally, for a homogeneous MPVC wth normalzed upload bandwdth of, we set γ l = 5 6 ; for a heterogeneous MPVC wth normalzed average upload bandwdth of, we set γ l = max( 2 3, + S ). From the analyss n Secton 4 and 5, the vdeo rate vector determned by γ l : {r s = mn(u s, γ l ), s S}, s always achevable. Usng γ l as the startng pont, we teratvely fnd the maxmal γ that can be acheved by our bandwdth allocaton algorthm. At each teraton, we check whether the vdeo rate vector determned by γ = (γ l + γ h )/2 s achevable. If yes, the search range shrnks to [γ, γ h ]; otherwse,the search range shrnks to

11 [γ l, γ]. Ths process fnshes untl the range s smaller than a pre-defned threshold ɛ. The bnary search pseudocode s presented n Algorthm 3. Algorthm 3 Approachng Capacty through Bnarysearch : procedure MAX-γ(S, N, {u, N}, {G s, s S}) 2: normalze u such that ū = ; 3: f u homogeneous then 4: γ l = 5 6 5: else 6: γ l = max ( ) 2 3, + S 7: end f 8: whle (γ h γ l ) > ɛ do 9: γ (γ h γ l )/2, r s = mn(u, γ), s S : ok=bandwdth-allocaton ({r s, s S}) : f ok== then 2: γ l γ 3: else 4: γ h γ 5: end f 6: end whle 7: return γ 8: end procedure 7 NUMERICAL EVALUATION In ths secton, we present numercal results to demonstrate the tghtness of the derved lower bounds and the effcency of the proposed bandwdth allocaton algorthm. We adopt three types of performance measures. The frst one s the dfference between the acheved vdeo rates and the optmal vdeo rates. The second one s the average vdeo qualty perceved by all users. Usng PSNR vdeo qualty model, the average vdeo qualty s: V = N log(w ) = s S G s log(r s ), (2) where w s the vdeo rate receved by vewer, r s s the vdeo rate of source s, and w = r s, G s. The thrd measure s the bandwdth utlzaton n the conference. Frst of all, the aggregate receved vdeo rate cross all sub-conferences should be less than the sum of upload bandwdth on all peers. Secondly, the vdeo rate of a sub-conference s lmted by the bandwdth of ts vdeo source. Even f there s abundant bandwdth avalable, the aggregate receved vdeo rate n sub-conference hosted by s s lmted by G s u s. We defne the upload bandwdth utlzaton as N B = w mn( N u, s S G (2) s u s ) 7. Homogeneous One-vew MPVC We frst study the tghtness of the derved unversal lower bounds at dfferent system szes by varyng from 6 to 4 wth step-sze 4. For each, we generate γ * Number of vdeo source (a) varyng # of peers Fg. 3. Capacty of Homogeneous MPVC CDF of γ * =6 = = γ * (b) varyng # of sources, random vewng scenaros: we frst select a random number of peers as vdeo sources, then each peer randomly selects a source to watch. For each scenaro, we frst calculate ts max-mn capacty γ usng the optmal algorthm OPT II. The CDF dstrbuton of γ s plotted n Fg. 3(a). The mnmum of γ s 333 5/6. At all system szes, more than 9% scenaros have maxmn capacty greater than.9. Note that the maxmum achevable vdeo rate s at most. Ths ndcates that whle 5/6 s a unversal lower bound ndependent of vewng relatons between peers, for most vewng scenaros, the achevable vdeo rate s pretty close to the upper bound of. As the systems sze grows, less scenaros can acheve the maxmum rate of. For each scenaro, we also use the bnary search algorthm presented Secton. 6.2, denoted as the BA algorthm, to teratvely approach the capacty. We also calculate the dfference between the acheved rate γ by the BA algorthm wth the optmal value γ and fnd the maxmum error s smaller than 3. To nvestgate the mpact of the number of sources, we fx at and vary the number of vdeo sources S from 2 to. For each S, we generate, random vewng scenaros and calculate the max-mn capacty for each scenaro. The results are presented as boxplot n Fgure 3(b). For each S, the central mark n the box s the medan, the edges of the box are the 25th and 75th percentles, the whskers extend to the most extreme data ponts not consdered outlers, and outlers are plotted ndvdually. When S = 2, as proved n Theorem, the maxmal rate of s acheved. As S ncreases, the medan value decreases and the varance ncreases. The lowest medan value and the hghest varance appear at S = 6, where the number of possble vewng scenaros s the largest. As S ncreases further, the medan ncreases and the varance decreases. When S =, each peer s a source and only has one vewer. Each source sends vdeo drectly to her vewer to acheve the maxmum rate of. 7.2 Heterogeneous One-vew MPVC To smulate heterogeneous system, we randomly set peer upload capacty accordng to the dstrbuton lsted n the Table 2, whch s obtaned from a measurement study n [2]. The average peer upload bandwdth s

12 2 TABLE 2 Bandwdth dstrbuton Uplnk(kbps) Probablty class 28.2 class class 3.25 class Mbps. We vary the number of peers from 6 to 2 wth step-sze of 2. For each, we randomly generate 4, vewng scenaros by lettng each peer randomly choose another peer to watch. Totally 6, random vewng scenaros are generated. Fgure 4(a) plots the CDF dstrbuton of max-mn capacty obtaned by OPT II and our bandwdth allocaton algorthm (labeled wth + S )ū BA). We also plot the lower bound of max ( 2 3, for each scenaro. We can see that the BA curve s very close to the OPT II curve. Ths suggests that the BA algorthm s very effcent n approachng the maxmn capacty bound n heterogeneous systems. In the fgure, there s a large gap between the max-mn capacty and lower bound. Ths s because the lower bound s ndependent of vewng scenaros and s always below peer s average upload bandwdth. But OPT II and BA algorthms work on specfc vewng scenaro, and the obtaned γ reflects the obtaned maxmal vdeo source rate, whch can go well beyond the average upload rate f a vdeo source wth hgh upload bandwdth has just one or few vewers. In Fgure 4(a), we also plot the average vewng rate among all peers. In addton to OPT II, BA and the lower bound, we also consder OPT I defned n (6), the bandwdth allocaton optmzed drectly for vdeo qualty. The average curves of OPT I, OPT II, and BA algorthm are clustered together, and the gap between them and the average rate curve of the lower bound s smaller than the max-mn capacty gap. Fgure 4(b) shows the relatve performance dfference of OPT I, BA algorthm and lower bound compared wth OPT II (the relatve dfference between x and y s defne as x y y ). We frst consder the max-mn capacty obtaned by BA. By the curve labeled as γ of BA, the BA algorthm can acheve 93% of optmal max-mn capacty wth 9% probablty. For the average vewng rate, the dfference between the BA algorthm and OPT II s farly small. Snce OPT I s optmzed for the vdeo qualty, the average rate obtaned by OPT I can be hgher than OPT II. The relatve performance of the lower bound s the worst. The average rate of the lower bound s wthn 75% of OPT II wth 8% probablty. Fgure 4(c) plots the average vdeo qualty V obtaned by dfferent algorthms. The curve of OPT I, OPT II and BA algorthms are almost dentcal. The performance of the lower bound s worse than the other three algorthms, wth the relatve dfference less than 8%. Fnally, Fgure 4(d) compares the peer bandwdth utlzaton B as defned n (2). The utlzaton of OPT I, OPT II, TABLE 3 Heterogeneous MPVC, Random Sources =6 =8 = =2 S =2.6838/ S =3.776/ / S =4 255/ /693.77/54 - S =5 98/ 222/ / /66 S =6./78 6/277 48/43 9/387 S = /28 483/442 79/22 S = / /432 S = /6 7/777 S = /9 S = /2 BA algorthm are all very close to one. Ths suggests that those algorthms have effcently utlzed upload bandwdth avalable on sources and vewers to acheve hgh vdeo rates, and there s not much space for further qualty mprovement. But for the lower bound curve, snce t s not optmzed for specfc vewng scenaro, the bandwdth utlzaton s stll far from the perfect case. Ths suggests that the space for bandwdth allocaton optmzaton for ndvdual vewng scenaros s often necessary and rewardng. To nvestgate the mpact of and S, we cluster 6, random vewng scenaros based on the, S tuple. For each scenaro, we normalze the average vdeo vewng rate wth the average upload bandwdth. For each N, S cluster, we calculate the mean of the normalzed average vewng rate for all scenaros n that cluster. Table 3 presents results for N, S clusters wth at least 2 random scenaros. For each tem of the table, left number represent the mean of the normalzed average vewng rate and rght number represent the number of samples. the Each column corresponds to one system sze. Dfferent from the homogeneous case, at all smulated system szes, the average vdeo rate ncreases as the number of sources ncreases. Ths s because the acheved vdeo rate n each sub-conference s lmted by both the source upload bandwdth and the bandwdth avalable to ths sub-conference. When the number of sources s smaller, each source wll have more vewers. If a weak peer s chosen as a source, t wll degrade the vdeo qualty on more peers. Consequently, the acheved average vdeo rate wll be lower. To elmnate the mpact of weak sources, we repeat the prevous experments wth an addtonal requrement that each source must have upload bandwdth larger than the average bandwdth. Specfcally, we frst generate the peer bandwdth accordng to Table 2, choose only peers wth bandwdth larger than the average bandwdth as sources, then let each peer randomly choose a source to watch. Accordng to Corollary 2, we now use max ( 3 4, + S )ū as the lower bound. The results are plotted n Fg. 5. When we requre all sources have

13 3 CDF of average vdeo rate.6 γ * of BA lower bound OPT II.4 Ave BA.2 Ave lower bound Ave OPT I Ave OPT II Average vdeo rate(kbt/s) CDF of relatve dfference γ * of BA lower bound Ave BA Ave OPT I Ave lower bound.5.5 Relatve dfference CDF of vdeo qualty.6.4 OPT II lower bound.2 BA OPT I Vdeo qualty CDF of bandwdth utlzaton optmal II lower bound BA opt I Bandwdth utlzaton (a) acheved vdeo rates (b) relatve performance (c) average vdeo qualty (d) bandwdth utlzaton Fg. 4. Performance of Heterogeneous MPVC wth 6, random vewng scenaros. CDF of average vdeo rate.6.4 Ave BA Ave lower bound.2 Ave OPT I Ave OPT II Average vdeo rate(kbt/s) (a) acheved vdeo rates CDF of relatve dfference Ave BA Ave OPT I Ave lower bound.5.5 Relatve dfference (b) relatve performance CDF of average vdeo qualty OPT II lower bound BA OPT I Average vdeo qualty (c) average vdeo qualty CDF of bandwdth utlzaton optmal II lower bound BA opt I.7.9 Bandwdth utlzaton (d) bandwdth utlzaton Fg. 5. Performance of Heterogeneous MPVC when each vdeo source s bandwdth s larger than the average bandwdth. capacty hgher than the average upload bandwdth, the source uplnk wll no longer be the bottleneck. To acheve the max-mn farness, all sub-conferences wll acheve the same rate. So the max-mn capacty acheved by OPT II s exactly the same as the average vewng rate of all peers. In Fg. 5(a), we only plot the average rates acheved by dfferent algorthms. If we compare Fg. 4(a) and 5(a), we do acheve hgher average vewng rates when all sources are bandwdth-rch. But the correspondng max-mn capacty γ s lower than those acheved n Fgure 4(a). Ths s because when there s no requrement on source bandwdth, sub-conferences hosted by weak sources are lmted by source upload bandwdth, strong sources can potentally acheve hgher rates and push up the max-mn capacty γ. Fg. 5(b) plots the relatve performance on the average rate of BA, OPT I and lower bound compared wth OPT II. Fg. 5(c) compares the average vdeo qualty acheved by dfferent algorthms. In Fg. 5(a), 5(b) and 5(c), the new lower bound curves are closer to OPT and BA curves than n Fg. 4(a), 4(b) and 5(c). Comparng Fg. 4(d) and 5(d), bandwdth utlzaton mproves when sources are no longer bottleneck. The lower bound curve n Fg. 5(d) s pece-wse constant. Ths s because the bandwdth + S ). utlzaton defned n (2) s now exactly max ( 3 4, For the smulated scenaros, there are only lmted number of, S tuples satsfyng + S > 3 4, e.g, 8, 2,, 3, etc., leadng to fve dscrete values of B. Fnally, we revst the mpact of the number of sources when sources are bandwdth-rch. As presented n Table 4, opposte to Table 3, at all smulated system szes, when the number of sources ncreases, the vdeo rate decreases. TABLE 4 Heterogeneous MPVC, Strong Sources =6 =8 = =2 S =2.9992/ / / /326 S =3.988/34.979/ /37.992/223 S =4.9546/25.972/ /75.977/826 S =5.955/29.95/84.956/ /48 S = /4.9393/7 S = /33 Ths s because when the sources are no longer the bottleneck, the acheved vdeo rate n a sub-conference s only determned by the bandwdth avalable to ths sub-conference. When the number of sources s larger, the number of peers n each sub-conference s smaller. Wth heterogeneous peer upload bandwdth, the average bandwdth wthn each sub-conference has larger varance. Sub-conferences wth less bandwdth have to borrow bandwdth from the helper pool and ncur helper bandwdth overhead. Consequently the acheved vdeo rate decreases. 7.3 Helper Overhead of BA Algorthm The desgn objectve of the BA algorthm s to acheve target vdeo rates wth mnmum peer upload bandwdth. The major consderaton of the BA desgn gudelnes n Secton 6 s to maxmally avod helper bandwdth overhead. In ths secton, we study the helper bandwdth overhead ncurred by our BA algorthm. We defne the aggregate helper bandwdth overhead rato

14 4 as: O H S (u(s) + u (w) N w + u (h) ) + I (u(w) + u (h) ), where n the second term, the numerator s the total vdeo rate receved by all peers, and the denomnator s the total upload bandwdth consumed on all peers. If there s no bandwdth overhead, the total vdeo receve rate should equal to the total vdeo upload rate. We generate heterogeneous bandwdth settngs randomly accordng to the bandwdth dstrbuton n Table 2. The number of users n the conference s 2. For each bandwdth settng, we generate 2 random vewng scenaros among peers. For each vewng scenaro, we frst obtan the maxmal value of γ usng the BA algorthm. Then we set the target vdeo rate vector as {r s = mn(u s, γ), s S}, wth γ rangng from.5γ to γ. At each γ, we run our bandwdth allocaton algorthm and calculate the ncurred helper bandwdth overhead. The dstrbuton of O H s shown n Fgure 6. In the fgure, almost all ncurred overhead rato s less than 25%. Ths demonstrates that our bandwdth allocaton algorthm s robust aganst random bandwdth settngs and vewng relatons. The overhead rato ncreases as the vdeo rate vector s pushed closer to the capacty bound. Ths s because to push all sub-conferences to acheve hgher vdeo rates, sub-conferences wth weak source and vewers have to borrow bandwdth from the helper pool, thus ncur hgher helper bandwdth overhead. CDF of overhead γ *.6 γ *.7 γ * γ *.9 γ * Overhead Fg. 6. CDF of Helper Bandwdth Overhead 8 CONCLUSION AND FUTURE WORK In ths paper, we explore the desgn space of pure Peerto-Peer one-vew Mult-party Vdeo Conferencng. We proposed a P2P relay framework for one-vew MPVC. Through analyss, we characterzed the vdeo rate capacty regon of P2P one-vew MPVC. We showed capacty of MPVC for both homogeneous system and heterogeneous system. We further showed that all the derved lower bounds are tght. We developed peer bandwdth allocaton algorthms that effcently utlze peers upload bandwdth to approach the maxmal vdeo rate regon. γ * Almost all proofs n ths paper are constructve and can be appled nto real mplementaton drectly wth few modfcatons. The capacty study here can be generalzed to study k-vew MPVC where each user watches full vdeos of k, k, users. One straghtforward way s to decompose a k-vew MPVC nto k parallel one-vew MPVCs, and on each peer, equally partton ts upload bandwdth nto k shares, one for each one-vew MPVC. Then mmedately the lower bounds obtaned n ths paper can be appled to each one-vew MPVC after beng scaled down by a factor of k. It wll be nterestng to nvestgate how much gan one can obtan by consderng k-vews jontly. Another mmedate extenson s to study the capacty of server-asssted P2P MPVC, where a server can provde addtonal bandwdth to dssemnate users vdeos. To analyze ts capacty, we can treat the server as a super peer wth abundant bandwdth and randomly assgn a source for t to vew, then the derved lower bounds automatcally apply. Snce our derved lower bounds are normalzed wth the average peer upload bandwdth, the mpact of the server assstance s quantfed as the ncrease n the average peer+server upload bandwdth. The lower bounds demonstrate that t s possble to mantan stable vdeo qualty on all sources n face of dynamc peer churn and vewng relaton changes. We wll refne our algorthms to mnmze the dsruptons to P2P vdeo relays upon peer churn and vewng relaton changes. ACKNOWLEDGMENTS Ths work was supported n part by the Natonal Scence Foundaton of Chna under Grant No and the fundamental research funds for the central unverstes. It s also partally supported by US Natonal Scence Foundaton under contract CNS and CNS REFERENCES [] I. E. Akkus, O. Ozkasap, Cvanlar, and M. Reha. Mult-objectve Optmzaton For Peer-to-Peer Multpont Vdeo Conferencng Usng Layered Vdeo. In Packet Vdeo 27, 27. [2] A. R. Bharambe, C. Herley, and V. N. Padmanabhan. Analyzng and Improvng a BtTorrent Network Performance Mechansms. In INFOCOM, 26. [3] BtTorrent. Homepage. [4] M. Chen, M. Ponec, S. Sengupta, J. L, and P. Chou. Utlty Maxmzaton n Peer-to-peer Systems. In Proceedngs of ACM SIGMETRICS, 28. [5] Y. Chu, S. G. Rao, S. Seshan, and H. Zhang. Enablng Conferencng Applcatons on the Internet usng an Overlay Multcast Archtecture. In Proceedngs of ACM SIGCOMM, 2. [6] M. Cvanlar, O. Ozkasap, and T. Celeb. Peer-to-peer multpont vdeoconferencng on the Internet. Sgnal Processng: Image Communcaton, 2(8):4 27, 25. [7] M. Dschnger, A. Haeberlen, K. P. Gummad, and S. Sarou. Characterzng Resdental Broadband Networks. In Internet Measurement Conference, 27. [8] Google+. Homepage. [9] M. Hossen and N. D. Georganas. Desgn of a Mult-sender 3D Vdeoconferencng Applcaton over an End System Multcast Protocol. In ACM Multmeda, 23.

15 5 [] Y. Huang, T. Z. Fu, D.-M. Chu, J. C. Lu, and C. Huang. Challenges, desgn and analyss of a large-scale p2p-vod system. In Proceedngs of the ACM SIGCOMM, 28. [] R. Kumar and K. Ross. Optmal Peer-Asssted Fle Dstrbuton: Sngle and Mult-Class Problems. In IEEE HOTWEB, 26. [2] J. Lennox and H. Schulzrnne. A Protocol for Relable Decentralzed Conferencng. In NOSSDAV, 23. [3] J. L, P. A. Chou, and C. Zhang. Mutualcast: An Effcent Mechansm for Content Dstrbuton n a P2P Network. In Sgcomm Asa Workshop, 25. [4] C. Lang, M. Zhao, and Y. Lu. Optmal Bandwdth Sharng n Mult-Swarm Mult-Party P2P Vdeo Conferencng Systems. IEEE/ACM Transactons on Networkong, 9(6), 2. [5] C. Luo, W. Wang, J. Tang, J. Sun, and J. L. A Multparty Vdeo Conferencng System over an Applcaton-Level Multcast Protocol. In IEEE Transactons on Multmeda, volume 9, pages , 27. [6] M. Ponec, S. Sengupta, M. Chen, J. L, and P. Chou. Mult-rate Peer-to-Peer Vdeo Conferencng: A Dstrbuted Approach Usng Scalable Codng. In Proceedngs of ICME, 29. [7] PPLve. Homepage. [8] S. Sen and J. Wang. Analyzng peer-to-peer traffc across large networks. In IEEE/ACM Transactons on Networkng, volume 2, pages , 22. [9] Skype. Homepage. [2] Y. Xu, C. Yu, J. L, and Y. Lu. Vdeo Telephony for End-consumers: Measurement Study of Google+, Chat, and Skype. In Proceedngs of Internet Measurement Conference, November poly.edu/faculty/yonglu/docs/mc2tech.pdf. Changja Chen s a Full Professor wth Bejng Jaotong Unversty. He receved hs M.S. degree from the Electroncs Insttute of Chnese Academy n 982 and Ph.D degree from Unversty of Hawa n 986, respectvely. Hs general research nterests nclude modelng, desgn and analyss of communcaton networks. Hs current research nterests nclude Peer-to-Peer systems, overlay networks, and network measurement. Yongxang Zhao s an assocate professor wth the Electrcal and Informaton Engneerng School at the Bejng JaoTong Unversty (BJTU),Chna He joned BJTU as an assstant professor n March, 22. He receved hs Ph.D. degree from Electrcal and Informaton Engneerng School at BJTU, n March 22. He receved hs master and bachelor degrees n the feld of Communcaton and Electronc system from BJTU, n 992 and 998, respectvely. Hs general research nterests nclude modelng and desgn communcaton networks. Hs current research nterests nclude Peer-to-Peer systems, overlay networks, and cloud computng. Yong Lu s an assocate professor at the Electrcal and Computer Engneerng department of the Polytechnc Insttute of New York Unversty (NYU-Poly). He joned NYU-Poly as an assstant professor n March, 25. He receved hs Ph.D. degree from Electrcal and Computer Engneerng department at the Unversty of Massachusetts, Amherst, n May 22. He receved hs master and bachelor degrees n the feld of automatc control from the Unversty of Scence and Technology of Chna, n July 997 and 994 respectvely. Hs general research nterests le n modelng, desgn and analyss of communcaton networks. Hs current research drectons nclude Peer-to-Peer systems, overlay networks, network measurement, onlne socal networks, and recommender systems. He s the wnner of ACM/USENIX Internet Measurement Conference (IMC) Best Paper Award n 22, IEEE Conference on Computer and Communcatons (INFOCOM) Best Paper Award n 29, and IEEE Communcatons Socety Best Paper Award n Multmeda Communcatons n 28. He s a member of IEEE and ACM. He s currently servng as an assocate edtor for IEEE/ACM Transactons on Networkng, and Elsever Computer Networks Journal. Janyn Zhang s a network research project manager n the Department of Network Technology at Chna Moble Communcatons Corporaton Research Insttute(CMCC). He joned CMCC as a project manager n August 28. He receved hs Ph.D. degree from Computer Scence and Technology School at BUPT n July 27. He receved hs master degree n the feld of Computer Communcaton from Chna Electroncs Technology Group Corporaton 54 th Research Insttute n March 22. He receved hs bachelor degree n the feld of mcroelectroncs from Xdan Unversty n July 997. Hs general research nterests nclude feature nteracton, new generaton Internet and telecom servces, new generaton telecom network, and network vrtualzaton. Hs current research s focused on Peer-to-Peer systems, multmeda servces, IPTV and OTT vdeo, and Web real-tme communcaton. He has served as an edtor of Q9 n the ITU-T SG3 from 29 to 2.