Heterogeneity in Distributed Live Streaming: Blessing or Curse?

HAL Id: inria-00414767

https://hal.inria.fr/inria-00414767

Submitted on 9 Sep 2009

HAL is a multi-disciplinary open access

archive for the deposit and dissemination of

sci-entific research documents, whether they are

pub-lished or not. The documents may come from

L’archive ouverte pluridisciplinaire HAL, est

destinée au dépôt et à la diffusion de documents

scientifiques de niveau recherche, publiés ou non,

émanant des établissements d’enseignement et de

or Curse?

Fabien Mathieu

To cite this version:

Fabien Mathieu. Heterogeneity in Distributed Live Streaming: Blessing or Curse?. [Research Report]

RR-OL-2009-09-001, 2009, pp.21. �inria-00414767�

Distributed Live Streaming:

Blessing or Curse?

Author(s): Fabien Mathieu (Orange Labs, Fran e) Resear h Report RR-OL-2009-09-001

Version: 1

Date: September 2009 Total number of pages: 21

Keywords: Distributed Live Streaming, Heterogeneous bandwidths, Theoreti al Transmission Delay

Abstra t Distributedlivestreaminghasbroughtalotof interestin thepastfew years. Inthehomogeneous ase(allnodeshaving thesame apa ity), manyalgorithms havebeen proposed,whi h havebeenproven almost optimal oroptimal. On theother hand, theperforman eof heteroge-neoussystemsisnot ompletelyunderstoodyet.

Inthispaper,weinvestigatetheimpa tofheterogeneityonthe a hiev-able delay of hunk-based live streaming systems. We propose sev-eral models for taking the atomi ity of a hunk into a ount. Forall thesemodels,when onsideringthetransmissionofasingle hunk, het-erogeneity is indeed ablessing, in the sense that the a hievable de-lay is alwaysfaster than an equivalent homogeneoussystem. But for a stream of hunks, we show that it an be a  urse: there is sys-temswhere thea hievabledelay anbearbitrarygreater omparedto equivalent homogeneous systems. However, if the system is slightly bandwidth-overprovisionned, optimal single hunk diusion s hemes anbeadapted to astream of hunks,leadingto near-optimal, faster thanhomogeneoussystems,heterogeneouslivestreamingsystems.

Contents 1 Introdu tion 4 1.1 Contribution . . . 4 2 Model 5 2.1 Chunk-baseddiusion . . . 5 2.2 Capa ity onstraints . . . 5 2.3 Collaborations . . . 5

2.4 Single hunk/streamof hunksdiusiondelays . . . 6

3 Related work 7 4 Single hunk diusion 8 4.1

(∞/1)

diusion . . . 8 4.1.1 Homogeneous ase . . . 8 4.1.2 Gainofheterogeneity . . . 9 4.1.3 Homogeneous lasses . . . 9 4.2

(1/1)

,

(1/c)

diusion . . . 10 4.2.1 Propertiesof

D

1

. . . 12 4.2.2 Extensionto

(1/c)

systems . . . 12 4.2.3 Example. . . 13

5 Stream of hunks diusion 15 5.1 Feasibilityofa hunk-basedstream . . . 15

5.2 Whenheterogeneityisa urse . . . 16

5.3 Whenheterogeneity an beablessing . . . 17

5.3.1 Extensiontothe

(1/c)

model . . . 18

5.3.2 Example. . . 19

1 Introdu tion

Re entyearshaveseentheproliferationoflivestreaming ontentdiusionover the Internet. In order to manage large audien e, many distributed s alable proto ols have been proposed and deployed on peer-to-peer or peer-assisted platforms[15,9,2,1,3℄. Mostofthesesystemsrelyona hunk-based ar hite -ture: thestream isdivided into smallparts, so- alled hunks,that haveto be distributed independentlyin thesystem.

ThemeasurementsperformedondistributedP2Pplatformshaveshownthat these platformsarehighly heterogeneouswithrespe t tothe sharedresour es, espe iallytheuploadbandwidth[14,6℄. However,ex eptforafewstudies(see forinstan e [13,12℄), mostofthetheoreti alresear h hasbeendevotedto the analysisofhomogeneoussystems,whereallthepeershavesimilarresour es.

At rst sight, it is not lear whether heterogeneity should be positive or negativefor a live streaming system. On the one hand, some studies on live streaming algorithms have reported a degradation of the performan e when onsideringheterogeneouss enarios[5℄. On theotherhand, onsiderthesetwo toys enarios:

Homogeneous asour einje tsalivestreamatarateofone hunkperse ond into asystemof

n

peers,ea h peer having anuploadbandwidth ofone hunk perse ond. Thenthebesta hievabledelaytodistributethestreamis

⌈log

2

(n)⌉

se onds[5℄.

Centralized sameasabove,ex eptthatonepeerhasanuploadbandwidthof

n

hunksperse ond,andtheothershavenoupload apa ity. Thenthestream anobviouslybedistributedwithin onese ond.

Thetotalavailablebandwidthisthesameinboths enarios,andthe entral-ized one an be seenasan extremely heterogeneousdistributed s enario,so thissimpleexamplesuggeststhatheterogeneityshouldimprovetheperforman e ofalivestreamingsystem.

In this paper, we propose to givea theoreti alba kground forthe feasible performan eofdistributed, hunk-based,heterogeneous,livestreamingsystems. Theresultsproposedherearenotmeanttobedire tlyusedinrealsystems,but theyaretightexpli itbounds,thatshouldserveaslandmarksforevaluatingthe performan eofsu h systems,and that anhelptounderstand ifheterogeneity isindeedablessing ora urse, omparedtohomogeneity.

1.1 Contribution

Weproposeasimpleframeworkforevaluatingtheperforman eof hunk-based live streaming systems. Several variant are proposed, depending on whether multi-sour este hniquesareallowedornot,andonthepossibleuseofparallel transmissions. Fortheproblem of theoptimal transmissionofa single hunk, we give the exa t lower bounds for all the onsidered variants of the model. Theseboundsareobtainedeitherwithanexpli it losedformulaorbymeansof simplealgorithms. Moreover,theboundsare omparedbetweenthemselvesand tothehomogeneous ase,showingthatheterogeneityisanimprovementforthe single hunkproblem. Forthetransmissionofastreamof hunks,webeginbya feasibilityresultthatstatesthatifthereisenoughavailablebandwidth,asystem

ana hievelosslesstransmissionwithinanitedelay. However,weprovidevery ontrastedresultsforthepre isedelayperforman eofsu hsystems: ontheone hand,weshowthat there arebandwidth-over-provisionedsystemsthat needa

Ω(N )

transmission delay, whereas equivalent homogeneoussystems only need

O(log(N ))

;ontheotherhand,wegivesimple,su ient onditionsthat allows

torelatethefeasiblestreamdelayto theoptimalsingle- hunkdelay.

Therest ofthepaperis organizedasfollows. Se tion2presentsourmodel andnotation,andSe tion3presentstherelatedwork. ThenSe tion4presents thebounds forthediusion ofonesingle hunk,whileSe tion 5 onsiders the aseof astreamof hunks. Se tion 6 on ludes.

2 Model

We onsideradistributedlivestreamingsystem onsistingof

N

entities, alled peers. A sour e inje tssomelive ontentinto the system,and goalis that all peers re eivethat ontentwith aminimal delay. We assumeno limitation on the overlay graph,so any peer anpotentiallytransmit a hunk to anyother peer (fullmesh onne tivity).

2.1 Chunk-based diusion

The ontent is split into atomi units of data alled hunks. Chunk de om-position is often used in distributed livestreaming systems, be ause it allows moreexible diusion s hemes: peers an ex hange maps of the hunks they have/need, and de ide on-the-y of the best way to a hieve the distribution. The drawba k is the indu ed data quanti ation. Following a standard ap-proa h[5℄,weproposetomodelthisquanti ationbyassumingthatapeer an onlytransmita hunkifithasre eiveda omplete opyofit.

Forsimpli ity, weassumethatall hunkshavethesamesize,whi hweuse asdataunit.

2.2 Capa ity onstraints

Weassumeanupload- onstrained ontext,wherethetransmissiontimedepends only upon the upload bandwidth of the sending peer: if a peer

i

has upload bandwidth

u

i

(expressedin hunks perse ond), thetransmissiontimefor

i

to delivera hunk to anyother peer is

1

u

i

. Without lossofgenerality,weassume that thepeers aresortedde reasinglybytheirupload bandwidths,so wehave

u

1

≥ u

2

≥ ... ≥ u

N

≥ 0

. Wealsoassumethatthesystemhasanon-nullupload

apa ity(

u

1

> 0

).

Forsimpli ity, we assumethat there is no onstraint on the download a-pa ityofapeer,but wewill dis ussthevalidityofthatassumption lateron.

2.3 Collaborations

Wealso needto dene the degreeof ollaborationenabled forthediusion of one hunk, i.e. how many peers an ollaborate to transmit a hunk to how manypeerssimultaneously. Themainmodels onsideredinthispaperare:

Many-to-one (short notation:

(∞/1)

) The

(∞/1)

model allowsan arbi-trarynumberofpeersto ollaboratewhentransmittinga hunktoagivenpeer message. three peers

i

,

j

,

k

an ollaborate to transmit a hunk theyhaveto afourth peer in a time

1

u

i

+u

j

+u

k

. The

(∞/1)

model maynot be very

pra ti- al, be auseitallows

N − 1

peerstosimultaneously ollaborate forone hunk, whi h an generate syn hronization issues and hallenge the assumption that downloadis not a onstraint (the re eiving peer must handle the umulative bandwidthsoftheemitters). However,ithasastrongtheoreti alinterest,asit en ompassesmore realisti models. Thereforethe

(∞/1)

bounds anserveas landmarkfortheother models.

One-to-one(short notation:

(1/1)

) Inthe

(1/1)

model,a hunk transmis-sionisalwaysperformedbyasinglepeer: ifatsometime,threepeers

i

,

j

and

k

havea hunkandwanttotransmitit,theymustsele tthreedistin tre eivers, whi h will re eivethe messageafter

1

u

i

,

1

u

j

and

1

u

k

se onds respe tively. The onne tivityanddownloadbandwidthburdensare onsiderablyredu edinthat model. Note that

(1/1)

isin ludedin

(∞/1)

(anyalgorithm thatworksunder

(1/1)

isvalidin

(∞/1)

).

One-to-some(short notation:

(1/c)

) Themodelsaboveimpli itlyassume thatagivenpeertransmits hunkssequentially,butforte hni alreasons, pra ti- alsystemsoftentrytointrodu esomeparallelisminthetransmissionpro ess: pure serialization an lead to a non-optimal use of the sender's transmission buer, for instan e in ase of onne tivity or node failures. We propose the

(1/c)

modeltotakeparallelismintoa ount: atransmittingpeer

i

alwayssplits

itsuploadbandwidthinto

c

distin t onne tionsofequal apa ity. Wemodela pri efor theuseof parallelism,byassuming that these onne tions annotbe aggregated. Thatmeansthat apeer

i

antransmitto upto

c

re eivers simul-taneously,butitalwaysneeds

c

u

i

se ondstotransmitthemessagetoanygiven peer. Notethat anyalgorithm thatworksin the

(1/c)

model anbeemulated

in the

(1/1)

model.

2.4 Single hunk / stream of hunks diusion delays In order to study the a hievable diusion delay of the system, we propose a twostepapproa h: werst onsider the feasibledelayfor thetransmission of asingle hunk,thenweinvestigatehowthis anberelatedtothetransmission delayofastream of hunks.

Inthesingle hunk transmissionproblem,weassumethatattime

t = 0

,

n

0

opies of anewly reated hunk are delivered to

n

0

arefully sele ted distin t peers(

1 ≤ n

0

≤ N

),andwewanttoknowtheminimaldelay

D(n)

neededfor

n

opiesofthe hunktobeavailableinthesystem. Notethatasthesystemhasa non-nullupload apa ity,

n

opies analwaysbemadeinanitetime,so

D

is welldened. Themainvalueofinterestis

D(N )

(timeneededforallpeerstoget a opyofthe hunk),but

n > N

analsobe onsideredfortheoreti alpurposes (we assumethen that theextra opies are transmitted to dummy nodes with null upload apa ity). We use the notation

D

m

,

D

1

or

D

c

depending on the model used(many-to-one,one-to-oneorone-to-

c

respe tively).

In thestream of hunk problem, new hunks are reated at agiven rate

s

(expressed in hunks per se ond) and inje ted with redundan y

n

0

. In other words,every

1

s

se onds,

n

0

opies ofanewly reated hunkaredeliveredto

n

0

arefully sele tedpeers. The (possibly innite) stream is feasible if there is a diusion s heme that insures a lossless transmission within a bounded delay. It means that there is a delay su h that any hunk, after being inje ted in thesystem, isavailable to the

N

peers within that delay. Foragiven feasible stream, we all

D

˜

(or

D

˜

m

,

D

˜

1

,

D

˜

c

ifthe underlying model mustbespe ied) the orrespondingminimal a hievabledelay. Obviously,

D

isalowerboundfor

˜

D

.

3 Related work

The problem of transmitted a message to all the parti ipants (broad ast) or asubsetofit (multi ast)in apossibly heterogeneous apa ity- onstraint envi-ronmentisnotnew. Afew years ago,so- allednetworks of workstations have been the subje t of many studies [8, 4, 11, 7℄. However, most of the results presented in those studies were too generi for presenting adire t interestfor the hunk-basedlivestreamingproblem.

As far asweknow, theworkthat isprobably the losestto ourshasbeen madebyYongLiu[13℄. Forthesingle hunkproblem,Liuhas omputed

D

1

in spe i s enarios,andhegavesome(nontight)boundsforthegeneral ase. For the stream problem, he gave someinsight onthe delay distribution when the apa itiesarerandom,independentvariables. Liu'sstudyismore ompletethan oursfor spe i s enarios and implementation, but we provide tighter results forthegeneral ase,wherenoassumptionismadebut the

(∗/∗)

model.

Thereisalsotwo loselyrelatedproblemsforwhi htheoreti alanalysisand fundamentallimitationshavebeen onsidered: the hunk-based,homogeneous, livestreamingproblemandthestripe-based,possiblyheterogeneouslive stream-ingproblem.

For hunk-basedhomogeneoussystems,themain resultis thatif thepeers have enough bandwidth to handle the streamrate (

u ≥ s

), then the stream problem is feasible for the

(1/1)

model and we have

D

1

= ˜

D

1

=

1

u

log

2

(

n

N

0

)

(see for instan e [13℄). The intuitive idea is that as all peers have the same bandwidth values, they an ex hange their pla e in a diusion tree without hangingtheperforman eofthattree. Thisallowstousetheoptimaldiusion treeforea hnew hunk introdu ed inthesystem: whena hunkisaninternal node of the tree of agiven hunk

i

, he just have to be a leaf in the trees of thenextnodesuntilthediusion of

i

is omplete. Of ourse, thispermutation te hnique annotbeused inaheterogeneous ase.

The stripe-based model onsists in assuming that the stream of data an be dividedinto arbitrary smallsub-streams, alled stripes. There isno hunk limitation in that model, therefore the transmissionof data betweennodes is onlydelayedbylaten ies. Theupload apa ityisstill a onstraint,butit only impa tstheamountof streamthat apeer anrelay. Apretty ompletestudy fortheperforman eboundsofstripe-basedsystemsisavailablein[12℄. Itshows thataslongasthereisenoughbandwidthtosustainthestream(meaning,with ournotation,

n

0

+

1

s

P

N

i=1

u

i

≥ N

),thestream anbediusedwithinaminimal delay. InSe tion 5, we willshowthat thisfeasibilityresult anbeadapted to

the hunk-basedmodel, althoughthedelaytendsto explodein thepro ess.

4 Single hunk diusion

AsexpressedinŸ2.4,

D

isalowerboundfor

D

˜

,soitisinterestingtounderstand thesingle hunkproblem. Moreover,aswewillseeinthenextse tion,anupper boundfor

D

˜

analsobederivedfrom

D

on ertain onditions.

4.1

(∞/1)

diusion

We rst onsider the many-to-one assumption, where ollaboration between uploadersisallowed. Underthisassumption,we angiveanexa tvalueforthe minimaltransmissiondelay.

Theorem 1. Let

U

k

be the umulative bandwidth of the

k

best peers (

U

k

=

P

k

i=1

u

i

). Thenthe minimal transmissiondelay

D

m

isgiven by

D

m

(n) =

n−1

X

k=n

0

1

U

k

. (1)

Proof. Wesaythatagivenpeeris apable whenitownsa omplete opyofthe hunk(itis apabletotranmistthat hunk). Ifatagiventimethesumofupload bandwidthsofthe apable peers(i.e. witha omplete opyofthemessage)is

U

, thenthe minimaltime forthosepeer tosend a omplete opy ofthe hunkto anotherpeeris

1

U

. Fromthatobservation,wededu ethatmaximizing

U

during thewhole diusionisthewaytoobtainminimaltransmission. Thisisa hieved by inje ting the

n

0

primary opies of the messageto the

n

0

best peers, then propagatingthemessagepeerbypeer,alwaysusingalltheavailablebandwidth of apable peers and sele ting the target peers in de reasingorder of upload. Thisgivesthebound.

Remark in[13℄, Liuproposed

D

m

asa(loose)lowerbound for

D

1

. Indeed,

D

m

isanabsolutelowerboundforany hunk-basedsystem,be ausethediusion used makesthebestpossibleuseof theavailable bandwidthatanytime. The onlywaytogobelow

D

m

wouldbetoallowpeerstotransmitpartiallyre eived hunks,whi his ontrarytothe hunk-based mainassumption. Thus

D

m

an serveas areferen elandmarkforall thedelays onsidered here. Moreover,an appealing property of

D

m

is that it is a dire t expression of the bandwidths of the system, so it is straightforward to ompute as long as the bandwidth distributionis known.

4.1.1 Homogeneous ase

Ifallpeershavethesameuploadbandwidth

u

i

= u

,wehave

U

k

= ku

for

k ≤ N

, sothebound

D

m

be omessimplertoexpressfor

n ≤ N

:

D

m

(n) =

1

u

n−1

X

k=n

0

1

k

. (2)

D

m

(n) ≈

ln(

n

n

0

)

u

. (3)

Sointhehomogeneous ase,the

(∞/1)

transmissiondelayisinverse propor-tional to the ommon upload bandwidth, and growslogarithmi ally with the numberofpeers.

4.1.2 Gain of heterogeneity

We an omparetheperforman eofagivenheterogeneoussystemtothe homo-geneous ase: letus onsider aheterogeneoussystemwithaverage peer band-width

u

¯

, and maximum bandwidth

u

max

. As peers are sorted by de reasing bandwidth,wehave

k¯

u ≤ U

k

≤ ku

max

. From(1) ,itfollowsthat

D

u

max

m

≤ D

m

≤ D

u

m

¯

, (4) where

D

u

m

is

D

m

in a homogeneous system with ommon bandwidth

u

. In parti ular,by ombiningthepreviousequations,onegets

D

m

(n) <

1

¯

u

(ln(

n − 1

n

0

) +

1

n

0

)

. (5)

Inotherwords,theoptimaltransmissiondelayissmallerforaheterogeneous sys-temthanforahomogeneoussystemwithsameaveragepeeruploadbandwidth. Inthat sense,heterogeneity an be seenasablessing forthetransmissionof onesingle hunk.

4.1.3 Homogeneous lasses

Equation(3) anbeextendedto the asewhere there is lassesofpeers, ea h lassbeing hara terizedbythe ommonvalueof theuploadbandwidthsofits peers.

Theorem2. Weassumehere thatwehave

l

lasses withrespe tivepopulation sizeand upload bandwidth

(n

1

, u

1

)

,...,

(n

k

, u

l

)

,with

u

1

> . . . > u

l

and

n

i

≫ 1

(largepopulation sizes). If

n

0

≤ n

1

,thenwehave

D

m

(N ) ≈

1

u

1

ln(

n

1

n

0

) +

l

X

i=2

ln(1 +

n

i

u

i

P

i−1

j=1

n

j

u

j

)

u

i

. (6)

Proof. be ause the minimal delay is obtained by transmitting the messageto thebest peersrst,in the lass s enario,theoptimaltransmissionmustfollow the lassorder,beginningbythe

(n

1

, u

1

)

lassandendingbythe

(n

l

, u

l

)

lass. Sointheminimaldelaytransmission,the

n

0

initialmessagesareinsertedinthe rst lass and in arst phase, it will only be disseminated within that lass. A ordingtoEquation(3), after about

1

u

1

ln(

n

1

n

0

)

se onds, allpeersofthe rst lasshavea opyofthemessage.

Then,forthegeneri termofEquation(6),wejustneedto onsiderthatthe time

D

i−1→i

neededto llupa lass

i

,

2 ≤ i ≤ l

,after allprevious lassesare already apable.

D

i−1→i

isgivenbyEquation(1),with

n

0

=

P

i−1

j=1

n

j

(previous lassestotalsize)and

n =

P

i

D

i−1→i

=

P

(P

i

j=1

n

j

)−1

k=

P

i−1

j=1

n

j

1

U

k

=

P

n

i

−1

k=0

U

P

i−1

1

j=1

nj +k

=

P

n

i

−1

k=0

₍

P

i−1

1

j=1

n

j

u

j

)+ku

i

≈

_u

1

i

ln(1 +

n

i

u

i

P

i−1

j=1

n

j

u

j

)

.

Bysumming

D(i)

for

2 ≤ i ≤ l

,oneobtainstheEquation(6).

Remark ifwehave

n

1

u

1

≫ n

i

u

i

forall

2 ≤ i ≤ l

( asewherethetotalupload apa ityoftherst lassisfargreaterthanthe apa itiesoftheother lasses), wehaveasimplerapproximationfor

D

m

:

D

m

(N ) ≈

1

u

1

ln(

n

1

n

0

) +

l

X

i=2

n

i

P

i−1

j=1

n

j

u

j

. (7)

Inparti ular, ifwe onsider,following[13℄, atwo- lasss enario,these ond lassbeingmadeoffree-riders (

u

2

= 0

),Equation(3)simpliesinto:

D

m

(n) ≈

1

u

ln(

min(n, n

1

)

n

0

) +

max(n − n

1

, 0)

N u

. (8)

The rst lass gets themessage after alogarithmi time, while itis linear for thefree-rider lass.

4.2

(1/1)

,

(1/c)

diusion

In the diusion s heme used for Theorem 1, all apable peers ollaborate to-gether to transmit the hunk to one single peer. Obviously, this approa h is not sustainablebe ause of the underlying ost for syn hronizing an arbitrary greatnumberof apablepeersmaybeimportantanyhowandofthedownload bandwidththatthere eiverpeer musthandle.

Inpra ti e,manysystemsdonotrelyonmulti-sour es apabilities anduse one-to-onetransmissionsinstead.Weproposenowto onsidertheminimaldelay

D

1

forthe

(1/1)

,and ompareitwiththebound

D

m

.

Contrary to

D

m

, for whi h a simple losed formula exists,

D

1

is hard to express dire tly. However,itis stillfeasible to omputeitsexa tvalue, whi h isgivenbyAlgorithm1.

TheideaofAlgorithm1isthatifone omputesthetimeswhenanew opy of the hunk an be made available, greedy dissemination is always optimal for a single hunk transmission: at any time when a hunk opy ends, if the re eiver of that opy is not the best peer missing the hunk, it redu es the usablebandwidthandthereforein reasesthedelay. Sothealgorithmmaintains a time- ompletion list that indi ates when opies of the hunk an be made underabandwidth-greedyallo ation. Indetails:

•

at line1,the ompletion timelistis initiatedwith

n

0

valuesof

0

(the

n

0

primary opies);

•

line 1 hooses thelowest ompletion available ompletion timeand allo- atesthe orresponding hunk opytothebestnon- apablepeer

i

;

•

at line1, the orrespondingvalue

D

1

(i)

isremoved,withoutmultipli ity;

Algorithm1Algorithmto ompute

D

1

Input: Asetof

N

uploadbandwidths

u

1

≥ ... ≥ u

N

An integer

n

0

(numberofinitial opies)

Amaximumvalue

n

max

Output:

D

1

(n)

for

n ← 1

to

n

max

1:

L ←−

zeros

(n

0

× 1)

2: for

i ← 1

to

n

max

do 3:

D

1

(i) ←− min(L)

4:

L = L \ {D

1

(i)}

5: if (

i ≤ N & u

i

> 0

)then 6:

L

i

= D

1

(i) + {

1

u

i

, . . . ,

n

max

−i

u

i

}

7:

L = L ∪ L

i

8: endif 9: endfor 10: return

D

1

Remark in [13℄,Liu proposed asnowball approa h for omputing afeasible delay. Thedieren ebetweenLiu'salgorithmandoursisthatLiuusedagreedy s heduling based on the time when a peer is to start a hunk transmission, whileweusethetimewhenitisabletonish atransmission. Asaresult,our algorithm givestheexa tvalueof

D

1

, but thepri eis that the orresponding s hedulingis notpra ti al: it needsall peer to syn hronizea ordingto their respe tivenishdeadlines,whileLiu'salgorithmonlyrequiresthatready peers greedilysele tadestinationpeer. AlsonotethatalthoughAlgorithm1provides theexa tvaluefor

D

1

, thea tual behaviorof thedelayis di ultto analyze. Inthefollowing,weproposetogiveexpli itboundsfor

D

1

.

Conje ture 1. Thefollowing boundshold for

D

1

:

D

m

≤ D

1

<

n

0

U

n

0

+

D

m

ln(2)

(9)

This onje ture expresses the fa t that the pri e for forfeiting the multi-sour es apa ities(leavingthemany-to-onemodelfortheone-to-onemodel)is adelayin reasethatisuptoafa tor

1

ln(2)

andsome onstant.

Proof inthe homogeneous ase. The left part of the inequality only expresses that

D

m

is anabsolute lowerbound for hunk-based diusion. Forthe right part,asstatedbyEquation(2) ,wehave

D

m

(n) =

P

n−1

n

0

1

ku

≥

1

u

ln(

n

n

0

)

. Onthe otherhand,asstatedforinstan e in[13℄,

D

1

isgivenby

D

1

(n) =

1

u

⌈log

2

(

n

n

0

)⌉

. Wededu e

D

1

(n)

<

1

_u

(log

2

(

n

n

0

) + 1) =

n

0

n

0

u

+

1

u

ln(

n

n0

)

ln(2)

≤

n

0

U

_n0

+

D

m

ln(2)

To ompletetheproof,weshouldshowthatifwestartfromahomogeneous system and add some heterogeneity into it, the bounds of Equation (9) still holds. Thisis onrmedbyourexperiments,whi hshowthatthehomogeneous s enarioistheonewherethe

n

0

U

_n0

+

D

m

that themoreheterogeneousasystemis, themorethebehaviorof

D

1

is lose to

D

m

. We aim at providing a omplete, rigorous proof of Conje ture 9 in a nearfuture work.

Remark alesstight,yet easierto prove,relationshipbetween

D

1

and

D

m

is

D

1

<

n

0

U

n

0

+ 2D

m

. (10)

Thisinequality omesfromthefa tthat atanygivenmoment,thequantityof raw data presentin thesystem(the sumofthe omplete hunks opiesand of thepartiallytransferred hunks)isnomorethantwi etheamountof omplete opies: this is straightforwardby noti ing that for ea h partially downloaded opy,one anasso iatea omplete, distin t,one(ownedby thesenderofthat opy). Theadditive onstant

n

0

U

_n0

insuresthataquantity

2n

0

ofdataispresent in the system. The

2

fa tor omes from the fa t that after atime

2

U

n

, araw quantityofatleast

2n

(morethan

n

omplete opies)be omesatleast

2(n + 1)

(morethan

n + 1

omplete opies).

Inrestofthepaper,however,weprefertousethe onje turedEquation(9) insteadofEquation(10)be auseofitstightness.

4.2.1 Properties of

D

1

Mostofthepropertiesobservedforthe

(∞/1)

modelhaveanequivalentin the

(1/1)

model. Thisequivalent anbeobtainedusingConje ture1. Forinstan e,

thegainofheterogeneityisgivenby ombiningEquations(5) and(9):

D

1

(n) <

1

¯

u

+ D

¯

u

1

(n)

. (11) In other words, up to some onstant, an heterogeneous system is faster than an equivalent homogeneous system. However, this onstant means the delay an a tually be higher. For instan e, onsider the four peer system

with

(u

1

, u

2

, u

3

, u

4

) = (1.6, 0.8, 0.8, 0.8)

and

n

0

= 2

. It is easy to verify that

D

1

(4) = 1.25

for that parti ular system, whereas for the equivalent

homoge-neoussystem(allpeers'bandwidthsequaltoone)wehave

D

1

(4) = 1

. Thisisa goodillustrationofthefa tthatbe auseofquanti ationissues,heterogeneity isnotalwaysablessinginthe

(1/1)

model.

4.2.2 Extensionto

(1/c)

systems

All theresultsofthe

(1/1)

systems anbestraightforwardlyextendedto

(1/c)

ones. Rememberthat the onlydieren e is that insteadof beingableto sent one opyto one hunkevery

1

u

i

se onds,apeer

i

anfet hupto

k

peers with the hunk every

c

u

i

se onds. In fa t the only reason we have studied

(1/1)

separately wasthat

(1/1)

is a ful rum model, more ommonly used than the generi

(1/c)

one,sowewantedto highlightitin order to learly separatethe impa t of disabling multi-sour e apabilities and from thepossibility of using parallelism.

As the reasonings are mostly the same than for the

(1/1)

, we propose to dire tly statethe results. First, the exa tvalueof

D

c

anbe omputedby a slightmodi ationofAlgorithm1: allthatisneededistorename

D

1

to

D

c

and repla etheline 1by

L

i

= D

c

(i) + {

c

u

i

, . . . ,

c

u

i

|

{z

}

c

times

,

2c

u

i

, . . . ,

2c

u

i

|

{z

}

c

times

, . . .

. . . ,

⌈

n

max

−i

c

⌉c

u

i

, . . . ,

⌈

n

max

−i

c

⌉c

u

i

|

{z

}

c

times

}

.

Then,therelationshipbetween

D

c

and

D

m

isgivenbythefollowing onje -ture:

Conje ture 2. Thefollowing boundshold for

D

c

:

D

m

≤ D

1

< c

n

0

U

n

0

+

c

ln(1 + c)

D

m

. (12)

This onje tureexpressesthefa tthat thepri eforusingmono-sour eand

c

-parallelism, ompared totheoptimalmulti-sour es-enabledmodel, isadelay in reasethat is up to afa tor

c

ln(1+c)

(andsome onstant). It isvalidated by experien e,andprovedinthehomogeneous ase,whereasaboundfullyproved forthegeneral aseis

D

m

≤ D

1

< c

n

0

U

n

0

+ (c + 1)D

m

. (13)

Lastly,theso- alledgainofheterogeneityisstillonlyguaranteeduptosome onstant:

D

c

(n) < c

n

0

U

n

0

+ log

c

(

n

n

0

) < c

1

¯

u

+ D

¯

u

c

(n)

. (14) 4.2.3 Example

Inorder toillustratetheresultsgiveninthat se tion,weproposeto onsidera systemof

N = 10

4

peers that are fet h with

n

0

= 5

initial opies ofa hunk. Weproposethethreefollowingdistribution:

•

ahomogeneousdistribution

H

0

;

•

a heterogeneous distribution

H

1

with

3

bandwidth lasses, and a range fa torof

10

betweenthehighestandthelowest lass;

•

a heterogeneous distribution

H

2

with

3

bandwidth lasses, and a range fa torof

100

.

The details of the size and upload apa ity of ea h lass are expressed in Table 1. The numberswere hosensothat the average bandwidthis

1

in the three distributions,sowe an say theyareequivalentdistributions,ex eptfor theheterogeneity.

The diusion delays aredisplayedin Figure 1. Forea h bandwidth distri-bution,wedisplayed:

•

theoptimaldelay

D

m

, givenbyEquation(1);

•

thedelays

D

1

and

D

4

ofthe

(1/1)

and

(1/4)

models, givenbythe Algo-rithm1anditsmodiedversion;

H

0

(Homogeneous)

H

1

(Lightly-skewed)

H

2

(Skewed)

C

1

(100%, 1)

(33%, 2.22)

(30%, 2.92)

C

2

(33%, 0.56)

(40%, 0.292)

C

3

(33%, 0.222)

(30%, 0.0292)

Table1: Relativesize andupload apa ityofthe lassesof

3

bandwidth distri-butions

10

0

10

1

10

2

10

3

10

4

0

5

10

15

20

n

t

4n

0

U

n0

+

4

ln(5)

D

m

D

4

n

0

U

n0

+

1

ln(2)

D

m

D

1

D

m

(a)

H

0

(homogeneous distribu-tion)

10

0

10

1

10

2

10

3

10

4

0

5

10

15

20

n

t

4n

0

U

n0

+

4

ln(5)

D

m

D

4

n

0

U

n0

+

1

ln(2)

D

m

D

1

D

m

(b)

H

1

(Lightly-skewed distri-bution)

10

0

10

1

10

2

10

3

10

4

0

5

10

15

20

n

t

4n

0

U

n0

+

4

ln(5)

D

m

D

4

n

0

U

n0

+

1

ln(2)

D

m

D

1

D

m

( )

H

2

(Skeweddistribution)

•

theupperbounds for

D

1

and

D

4

givenbyConje tures1and2.

Fromtheobservedresults,we ansaythefollowing:

•

the delays in rease logarithmi ally for the onsidered distributions (or equivalently,the hunkdiusiongrowthsexponentiallywithtime),as pre-di tedbyEquation(6). Notethat thislogarithmi behaviorisonlyvalid for no tooskewed distribution: theexisten eof a highly dominant lass mayindu eanasymptoti allinearbehavior( dEquation(7)and(8));

•

Conje tures 1and 2 (pri e of mono-sour e diusion and pri e of paral-lelism) are veried. Of ourse, we also onfrontedthese onje turesto a lotofdistributionsnotdis ussedin thispaper(powerlaws,exponentially distributed, uniformly distributed, with free-riders,...) and they were veriedinall ases);

• D

1

and

D

4

looks like simple fun tions. This omes from the fa t that weusedbandwidth lasses,sosimultaneous arrivalsof new opies is fre-quent. Nevertheless,

D

1

and

D

c

alwayslook lesssmooththan

D

m

even for ontinuousdistributions, be ause thearrival ofnew hunks,whi his veryregularinthe

(∞/1)

model,ismoreerrati inthe

(1/∗)

models;

•

Delaysare fasterin

H

2

thanin

H

1

,and fasterin

H

2

thanin

H

0

. Thisis thegainofheterogeneity.

5 Stream of hunks diusion

The issue brought by the stream of hunks problem, ompared to the single hunk problem, is that ea h hunk is in ompetition with theothers forusing thebandwidthof thepeers: when apeeris devoted to transmittingonegiven hunkit annotbeusedforanotherone

1

. Therefore

D

isalowerbound for

D

˜

, but itis notne essarytight. In thisse tion, we proposeto see how

D

˜

anbe estimated.

5.1 Feasibility of a hunk-based stream

A rst natural question, before studying

D

˜

, is to know whether the stream problemis feasibleornot. Byadapting aresultfrom[12℄,we ananswerthat question.

Theorem3. Ane essary,foranydiusionmodel,andsu ient,forthe

(∞/1)

and

(1/1)

models, onditionfor the streamproblem tobefeasible is

n

0

+

1

s

N

X

i=1

u

i

≥ N

(15)

Proof. Theproofis dire tlyderivedfrom Theorem

1

in[12℄ anditsproof 2

. 1

Anex eptionisthe

(1/c)

model,howeverwebelievethattransmittingdierent hunkin parallelisnotveryee tive,atleastw.r.t.delay.

2

Equation(15)isne essarybe auseitexpressesthebandwidth onversation laws: the total bandwidth of the whole system (sour e and peers) must be greaterthanthe

N s

bandwidthneededforthe

N

peersto getthestream.

We then haveto showthat the onditionis su ient for the

(1/1)

model.

As

(∞/1)

an a t like

(1/1)

(the multi-sour e apa ity is not an obligation),

this will provethe result for

(∞/1)

aswell. Inthe proof in [12℄, theauthors onstru tsasolutionwhereea hpeerre eivesfromthesour eastripewhoserate is proportional to that peer's bandwidth. It is then in harge of distributing that stripe to all other peers. To adapt this to a hunk-based s enario, we follow the same idea: ea h peer will be responsible for a partof the hunks. Wejusthavetodistributethe hunksfromthesour etothepeersa ordingto as heduler that ensuresthat theproportion of hunks sentto agivenpeer is asproportionalaspossibleto itsuploadbandwidth(forinstan e,forea hnew hunk,sendittothepeersu hthatthedieren ebetweenthebandwidthand the hunk responsibility repartitionis minimal). Note that there issituations ( ase

2

in the proofin [12℄) where thesour e must distribute some hunks to allthepeers. Inthosesituations,a apa ity

1

ofthesour eisdevotedtoinitial allo ation,whiletheremaining

n

0

− 1

apa ityisusedlikeavirtual

(N + 1)

th

peer(sointhose ases,thesour emayhavetohandleold hunksinadditionof inje tingnewones).

Theorem 3basi ally states that if thebandwidth onservation is satised, any hunk-basedsystemisfeasible. Butwhiletheproposedalgorithmis delay-optimalinastripe-basedsystem,theresultingdelayisterribleina hunk-based system: if

n

isthelabelofthelastpeerwithanon-nulluploadbandwidth,the hunks for whi h

n

is responsible (they represent a ratio

u

n

U

n

of the emitted hunks) needsatleastadelay

N−1

u

n

tobetransmitted. Infa t,itmayneedup to

2

N−1

u

n

: be auseof quanti ation ee ts, it may re eive anew hunk before it hasnishedthedistribution of thepreviousone. This transmissiondelay is lowerfor allother hunks,so the (loose) bound that anbederived from the feasibilitytheoremis

˜

D ≤ 2

N − 1

u

n

,for

u

n

= min

u

i

>0

(u

i

)

. (16)

5.2 When heterogeneity is a urse

One may think that the bound of Equation (16) is just a side-ee t of the onstru tion proposed in [12℄, whi h is not adapted to hunk-based systems. Maybeinpra ti e,aslongasthefeasibility onditionisveried,

D

˜

is omparable to

D

? Thisideaiswrong,asshownbythefollowingsimpleexample: foragiven

0 < ǫ <

1

2

onsidera hunk-basedsystemoftwopeerswithuploadbandwidths

u

1

= 1 − ǫ

and

u

2

= ǫ

respe tively,

n

0

= 1

,

s = 1

. Wehave

D

m

= D

1

=

1

1−ǫ

(

u

1

re eivesthepeer and transmitsit to

u

2

). Equation(15) is veried so the system is feasible. However, when onsidering the stream problem,

u

1

alone has notthe ne essarybandwidth to support the diusion. Therefore at some point, thesour e is for ed to give a hunk to

u

2

, whi h need

1

ǫ

for sending a hunk. Thereforewene essarilyhave

D ≥

˜

1

ǫ

, so theminimal a hievabledelay an bearbitrary great. As a omparison, in theequivalent homogeneous ase (

u

1

= u

2

=

1

Then again, one ould argue that this ounter-example of heterogeneity's e ien yissomehowarti ial,asonly

2

peersare onsidered andtheavailable bandwidthis riti al. Thefollowingtheoremprovesthe ontrary.

Theorem4. Let

n

0

≥ 1

,

V ≥ 0

,and

s > 0

bexed. There exist

(1/1)

systems ofsize

N

that verify the following:

•

the sour ehas apa ity

n

0

;

• U

N

=

P

N

_i=1

u

i

≥ N s + V

(the system is feasible and the peers have an

ex essbandwidth ofatleast

V

);

• ˜

D

1

= Ω(N )

.

Rememberthatfor anhomogeneoussystem,thetworst onditionsimply

˜

D = O(log(N ))

: for thesystems onsidered by the theorem, heterogeneityis

indeeda urse,althoughthebandwidthisover-provisioned!

Proof. The idea is exa tly the same than for the two-peers example: having peers with averylow upload bandwidth and showing that the systemhas to use them from time to time. Here we assume

N > n

0

+ 1

and we onsider a systemwithsour e apa ity

n

0

andthefollowingbandwidthdistribution:

• u

1

= (N − n

0

− 1)s

,

• u

i

=

n

0

_N

+V +1

₋₁

s

for

2 ≤ i ≤ N

.

By onstru tion,thetworst onditionsareveried. However,

n

0

s + u

1

< N s

, soonlythesour e and

u

1

do notsu eto distribute thestream. This means that atsomepoint,at leastonepeer

i > 1

must sendat leastone hunk toat leastoneotherpeer,whi htakes

1

u

i

=

N

−1

s(n

0

+V +1)

= Ω(N )

.

5.3 When heterogeneity an be a blessing

Thereisatleastone asewhereweknowforsurethat

D = D

˜

evenfor heteroge-neoussystems: if

D(N ) ≤

1

s

,thenthesystem anperformtheoptimaldiusion of a hunk before the next oneis inje ted in the system. There is no ompe-tition between dierent hunks. For instan e, in the

(∞/1)

model, we have

D

m

(N ) ≤

1

_¯

_u

ln(

_n

N

0

)

(Equation(5)),soif

u ≥ ln(

¯

N

n

0

)s

,wehave

D

˜

m

= D

m

(n)

. Of ourse,thisimpliesatremendousbandwidthover-provisioningthatmakes this resultof littlepra ti alinterest. However, theidea anlead tomore rea-sonable onditions,asshownbythefollowingtheorem.

Theorem 5. For a given

(∞/1)

system, if one an nd an integer

E

that veries:

1. the

(∞/one)

single- hunk transmission delay of the sub-system made of

the peers

E, 2E, . . . , ⌊

N

E

⌋E

issmallerthan

E

s

, 2.

u ≥ s + E

¯

U

_E−1

N

, thenwehave

D

˜

m

≤ 2

E

s

.

The se ond ondition is about bandwidth provisioning, whereas the rst onditionis alledthenon-overlapping ondition( ftheproofbelow). Of ourse,

Chunk i+E

Chunk i

Chunk i-E

Peers

Intra-diffusion

Inter-diffusion

i/s

_D

(i+E)/s

E/s

Figure 2: Prin ipleoftheintra-then-inter hunk distribution

Proof. Theideaisto onstru tas hedulingalgorithmthatprote tsea h hunk, sothatit anbeoptimallydiused,atleastforafewmomentsafteritisinje ted. For that purpose, we split the peers into

E

groupsof peers

G

1

, . . . , G

E

, su h thatgroup

G

g

ontainsallpeers

i

thatverify

i ≡ g (mod E)

. Thenweusethe followingintra-then-inter diusion algorithm,whose prin iple is illustratedin Figure2. Foragiven hunk

i

,wedothefollowing

•

the sour e inje ts the hunk

i

to the

n

0

best peer of thegroup

G

g

that veries

i ≡ g (mod E)

. If

n

0

> |G

g

|

, theextra opies are givento peers from othergroups;

•

hunk

i

is diused as fast as possible inside the group

G

g

. This intra-diusion endsbefore thenext hunk

i + E

issentto

G

g

;

•

assoonastheintra-diusionisnished(we all

D

g

therequiredtime),all peers of

G

g

diuse the hunk

i

to the other groups(inter-diusion). Of ourse ea h peer of

G

g

must easeto parti ipatetothe inter-diusionof

i

atthemomentwhere itisinvolvedintheintra-diusionof

i + E

.

If thealgorithmworks,thediusion delayofea h hunkisboundedby

2

E

s

( f Figure 2), whi h proves the theorem. This requires rst that the intra-diusion of hunk

i

isnishedbefore hunk

i + E

isinje ted (non-overlapping ondition). Theslowergroupis

E

,sothe onditionisveriedisthesingle- hunk transmissiondelayof

G

E

is smallerthan

E

s

. Thenwemust guaranteethat

G

g

hasenoughavailablebandwidthfordiusingthe hunktotheothergroups. The peersof

G

g

ansendaquantity

E

s

P

|G

g

|−1

k=0

u

g+kE

of hunk

i

, ountingboththe intra and inter diusions. This leadsto thebandwidthprovisioning ondition

E

s

P

|G

g

|−1

k=0

u

g+kE

≥ N − n

0

. By noti ing that

P

|G

g

|−1

k=0

u

g+kE

≥

U

E

N

− U

E−1

, wegetthebandwidthprovisioning onditionofthetheorem.

5.3.1 Extensionto the

(1/c)

model

theequivalentofTheorem5forthe

(1/c)

model(in luding

c = 1

)isthe follow-ing:

1. the

(1/c)

single- hunk transmission delay of the sub-system made of the peers

E, 2E, . . . , ⌊

N

E

⌋E

issmaller than

E

s

, 2.

u ≥ s(1 +

¯

c

E

) + E

U

_E−1

N

, thenwehave

D

˜

c

≤ 2

E

s

.

Proof. Theproofis almost thesamethanfor theprevioustheorem. The only dieren earethefollowing:

•

regardingthediusionalgorithm,ea hpeermuststarttheinter-diusion at the moment it is notinvolved in the intra-diusionany more(in the

(∞/1)

model,allpeersnishatthesametime,butnotheresobandwidth

wouldbewastedifallpeerswait fortheendoftheintra-diusion);

•

also,whenapeerhasnotthetimetotransmita hunk

i

toothergroups be-foreitshouldbeinvolvedintheintra-diusionof hunk

i + E

,itstaysidle untilthat moment,foravoidingtointerferewiththenextintra-diusion;

•

asa result, apossiblequantity of bandwidthmay be wasted during the diusion of

i

. However, the orresponding quantity of data is bounded by

c|G

g

|

, whi h leads to the supplementary

c

E

term in the bandwidth provisioning ondition. 5.3.2 Example

H

0

(Homogeneous)

D

D

˜

(

s = .9

)

D

˜

(

s = .5

)

(∞/1)

7.70

N/A

(1/1)

11

(1/4)

20

H

1

(Lightly-skewed)

D

D

˜

(

s = .9

)

D

˜

(

s = .5

)

(∞/1)

3.72

8.16

9.72

(1/1)

5.40

16.51

11.40

(1/4)

9.00

53.44

19

H

2

(Skewed)

D

D

˜

(

s = .9

)

D

˜

(

s = .5

)

(∞/1)

2.70

6.04

6.96

(1/1)

4.11

14.88

10.11

(1/4)

6.86

51.30

16.86

Table 2: Delay performan e examples for the three bandwidth distributions des ribedinTable1

in orderto illustrateprevioustheoremswith realnumbers,we onsider the three s enariosused in Se tion 4.2.3. Table 2gives thesingle hunk diusion delays, as well as the upper bounds for

D

˜

in slightly overprovisioned (

s =

0.9

) and a well overprovisioned(

s = 0.5

)s enarios. Note that we hoose the parameterssothataproperinteger

E

anbefoundin all ases.

Ourmain ndingsarethefollowing:

•

for the

(∞/1)

model, it iseasy to ndan integer

E

lose to

sD

m

. This

leadstoagooddelayperforman e,whi hforthedistribution

H

2

isbetter thanthedelayofthehomogeneous ase;

•

for

(1/1)

and

(1/4)

, the

c

E

term in the overprovisioning ondition an requiretopi kahighvalueof

E

forthat onditionto beveried,leading to large delays. This is espe ially noti eable for the

(1/4)

model and

s = 0.9

;

•

asaresult,forthesemono-sour emodels,thebounds arenotbetterthat the known streaming delays in the homogeneous. Of ourse, this is not a proofthat heterogeneity isa urse in that ase: it mayexist diusion s hemes that a hieves lower streaming delays. But su h s hemes may behard to nd(and heterogeneity maybe onsideredasa urse in that sense).

6 Con lusion

Weinvestigatedtheperforman eofheterogeneous, hunk-based,distributedlive streamingsystems. Westartedbystudyingthetransmissionofonesingle hunk and showedthat heterogeneoussystemstendsto produ e faster dissemination than equivalent homogeneous systems. We then studied the transmission of a stream of hunks, where heterogeneity an be a disadvantage be ause the oordinationbetween on urrent hunkdiusions ismore omplexthanforthe homogeneous ase. Althoughthereisexampleswhere thefeasibledelay anbe arbitrarylong,wegavesu ient onditionstolinkthefeasiblestreamdelayto thesingle- hunktransmissiondelay. Be auseofquanti ationee ts,however, the obtained bounds may require the bandwidth to be highly heterogeneous and/oroverprovisionedinordertobe ompetitivewithhomogeneouss enarios, espe iallyforthemono-sour emodels.

A knowledgment

ThisworkhasbeensupportedbytheCollaborativeResear hContra tMardiII between INRIA andOrange Labs, and by the EuropeanCommission through theNAPA-WINE Proje t,ICT Call1FP7-ICT-2007-1,GrantAgreementno.: 214412.

Referen es

[1℄ TVants,http://tvants.en.softoni . om/. [2℄ Sop ast,http://www.sop ast. om/. [3℄ UUseein ., http://www.uusee. om/.

[4℄ M. Banikazemi, V. Moorthy, and D. K. Panda. E ient olle tive om-muni ationonheterogeneousnetworksofworkstations. InICPP'98: Pro- eedingsofthe1998InternationalConferen eonParallel Pro essing,pages 460467,Washington,DC, USA,1998.IEEE ComputerSo iety.

[5℄ T. Bonald, L. Massoulié, F. Mathieu, D. Perino, and A. Twigg. Epi-demi live streaming: optimal performan e trade-os. In SIGMETRICS '08: Pro eedingsofthe 2008ACMSIGMETRICSinternational onferen e on Measurement and modeling of omputer systems, pages325336, New York,NY,USA,2008.ACM.

[6℄ D. Ciullo, M. Mellia, M.Meo, andE. Leonardi. Understanding P2P-TV systemsthrough real measurements. In GLOBECOM, pages 22972302. IEEE,2008.

[7℄ P.Fraigniaud,B.Mans,andA.L.Rosenberg.E ienttrigger-broad asting in heterogeneous lusters. J. Parallel Distrib. Comput., 65(5):628642, 2005.

[8℄ N.G.Hall,W.-P.Liu,andJ.B.Sidney.S hedulinginbroad astnetworks. Networks,32(4):233253,1998.

[9℄ X.Hei,C.Liang,J.Liang,Y.Liu,andK.W.Ross. InsightsintoPPLive: Ameasurementstudyofalarge-s aleP2PIPTVsystem.InPro .ofIPTV Workshop,International World WideWebConferen e,2006.

[10℄ R.Kumar,Y.Liu,andK.W.Ross.Sto hasti uidtheoryforP2P stream-ingsystems. InINFOCOM,pages919927,2007.

[11℄ R.Libeskind-Hadas,J.R.K.Hartline,P.Boothe,G.Rae,andJ.Swisher. On multi ast algorithms for heterogeneous networks of workstations. J. Parallel Distrib. Comput.,61(11):16651679,2001.

[12℄ S.Liu,R.Zhang-Shen,W.Jiang,J.Rexford,andM.Chiang.Performan e boundsforpeer-assistedlivestreaming. InSIGMETRICS'08: Pro eedings ofthe2008ACMSIGMETRICSinternational onferen eonMeasurement andmodeling of omputer systems, pages 313324,New York,NY, USA, 2008.ACM.

[13℄ Y. Liu. On the minimum delay peer-to-peer video streaming: how real-time an it be? In MULTIMEDIA '07: Pro eedings of the 15th inter-national onferen e on Multimedia, pages127136,New York,NY, USA, 2007.ACM.

[14℄ S. Saroiu, P. K.Gummadi, and S. D. Gribble. A measurement study of peer-to-peerlesharingsystems. InPro eedings ofMultimediaComputing andNetworking(MMCN) 2002,SanJose,CA,USA,January2002. [15℄ X.Zhang,J.Liu,B.Li,andT.Yum.Coolstreaming/donet: Adata-driven