The Performance of Broadcasting with Network Coding in Dense Wireless Networks

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The Performance of Broadcasting with Network Coding in Dense Wireless Networks

Cédric Adjih, Song Yean Cho, Philippe Jacquet

To cite this version:

Cédric Adjih, Song Yean Cho, Philippe Jacquet. The Performance of Broadcasting with Network Coding in Dense Wireless Networks. [Research Report] RR-6120, INRIA. 2007. �inria-00129782v3�

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inria-00129782, version 3 - 13 Feb 2007ISSN 0249-6399

a p p o r t

d e r e c h e r c h e

THÈME 1

INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

The Performance of Broadcasting with Network Coding in Dense Wireless Networks

Cédric Adjih, Song Yean Cho, Philippe Jacquet

N° 6120

Février 2007

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Unité de recherche INRIA Rocquencourt

Domaine de Voluceau, Rocquencourt, BP 105, 78153 Le Chesnay Cedex (France)

Téléphone : +33 1 39 63 55 11 — Télécopie : +33 1 39 63 53 30

in Dense Wireless Networks

Cédri Adjih, Song Yean Cho, Philippe Jaquet

Thème1Réseauxetsystèmes

ProjetHIPERCOM

Rapportdereherhe n° 6120 Février200715pages

Abstrat: We present aprotool for wireless broadasttransmission basedon network

oding. Theprotooldoesnotneedneighborsensingandnetworktopologymonitoring. We

giveananalysisoftheperformaneoftheprotoolinanunit graphwirelessnetworkmodel

withuniformdensity,andintheaseofasinglesoure. Weshowthatevenwithonesoure,

duetodensity,network odingoerssomegains.

Inpartiularweshowthatin1dimensional(1D)networktheperformaneofthesimple

protoolbasedonnetworkodingislosetotheoptimalooding(withoutnetworkoding)

byafatorarbitraryloseto 1whenthenetworkdensityinreases. In2dimensional (2D)

networkssimulationsshowthattheratiotooptimalissimilartooutperformingMultiPoint

Relay(MPR) ooding.

Key-words: wirelessnetworks,networkoding, broadasting,multi-hop

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les réseaux sans l denses

Résumé: Nousprésentonsunprotoolepourladiusiondanslesréseauxsanslbasésur

leodagederéseau. Leprotoolen'apasbesoindedéouvertedevoisinsetni dedétetion

delatopologieduréseau. Nousdonnonsune analysedelaperformaneduprotooledans

le as d'une soure simple et dans un modèle de réseausans l qui est un graphe disque

unité ave unedensité uniforme. Nousmontrons quemême ave uneseule soure,grâe à

ladensitédesnoeuds, leodagederéseaupermet desgains. Enpartiulier,nousprouvons

quedans un réseauunidimensionnel (1D), laperformane duprotoole simplebasé sur le

odage de réseau est prohe de la performane de l'inondation optimale (sans odage de

réseau)parunfateurarbitrairementdeprohede1quandladensitéduréseauaugmente.

Dans réseaux en 2 dimensions, des simulations montrent que le rapport à l'optimal est

similairevoiremeilleur queelle del'inondationparrelaismultipoints(MPR).

Mots-lés : réseauxsansl,odagederéseau,diusion,multi-sauts

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1 Introdution

Intraditionalommuniationsystems,nodesexhangedatathroughrelayingbyintermedi-

atenodes: thepakets areforwardedby theintermediateroutersunmodiedforbothtra-

ditionaluniastand multiast ommuniation. Reently, thepioneering work of Ahlswede

et al. [1℄ hasshown that network oding, where intermediatenodesmix information from

dierent ows, an ahieve higherthroughput for multiasting than lassial routingover

networks represented as direted graphs. The works of [1,2,3,4,5,6℄ set theoretial and

pratialfoundationsfor theframework ofmanyreentworksonerning wirelessnetwork

oding (seesetion2). Thewirelessmediaisinherentlyabroadastmediawhihhasruial

advantagesineitallowsoverhearing. Theinherentadvantageouldbeexploitedby wire-

lessnetwork oding, permittinggainsbyreduingthenumberoftransmissionsrequiredto

transmitthesameamountinformation(energy-eieny). Dueto goodmathingbetween

theinheritadvantageandgaining,wirelessnetworkodinghasbeensuessfullyappliedto

uniasttraof802.11networksforinstanein[8℄,andithasalsobeenappliedtomultiast

tra,suhasin [7℄,[5℄and[10℄.

However,thefoushasoftenbeenonmultiplesoureswherethegainoriginatespreisely

from theexisteneof dierentdiretional ows, forinstane in [8℄foruniast, in [9℄(with

pakets andanti-pakets pairs),andin [5℄.

In ontrast, in this artile, we fous on broadastingwith a single soure. Our goalis

notsomuhtoahievemaximumperformanewithwirelessnetworkoding,but ratherto

illustratethatwirelessnetworkodinganahievegainsevenwithasinglesoure(asinthe

general,non-wireless,aseof[1℄),thankto density,andtogiveaninsightonthedynamis

ofnetworkodingin suhanenvironment. Thereforeourapproahfollowsthespiritofthe

algebrai gossip [11℄,but fousingontheidiosynrasiesofthewirelessmedium.

Todoso,wefousonasimpleprotoolfornetworkoding,whihdoesnotneedneighbor

sensing or network topology monitoring. We initially onsider its behavior in the one-

dimensional(1D)ase,i.ethe1Dunitdiskgraphmodel. Wedesribehowthepaketows

fromasoureevolveinwaves,andbystudyingthepropagationofthewaves,weareableto

quantifythe performaneof the simpleprotool in an unit graphwireless network model,

with an asymptoti model. Our analysis shows that, in 1D, as the density inrease the

performaneoftheprotoolisloseto theoptimal broadastbyafatorarbitraryloseto

1. Simulationsonrmtheresultsoftheasymptotimodel.

Thenweonjeturethebehaviorofthenetworkwithonesourein1Dtothatin2Dunit

diskgraphmodel. Thebehaviorin2Dshouldbeessentiallythesameasin1D(withthesame

oneptofwaves,onlywithmoreomplexwaves),withsimilarperformaneaspreditedby

themodelin1D.Simulationsresultsonrmthisinsight. Theartileisorganizedasfollows:

insetion2,wedesribethesimplenetworkodingalgorithmforwirelessbroadasting. In

setion3wedesribethewavepropagationanalysisthatwillbethebasisofourperformane

analysis. Insetion3wedeveloptheanalytialmodelsandourresults. Insetion5weshow

thesimulationresultswhih onrmandextendouranalytialresults.

RR n°6120

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2 Related Work, Framework and Algorithms

2.1 General Framework of Wireless Network Coding

Thestartingpointofnetwork oding is theelebratedwork from [1℄,showingthatoding

innetworksouldahievemaximumbroadastapaity(givenbythemin-ut),whileinthe

general ase, it is out of reah of traditional transmission methods (i.e. without network

oding).

Subsequently, ithas been shownthat a simpleformof oding, linear oding [2℄,(using

linearombinationsofdatasymbolsbelongingtoGaloiseldsF_p^),^is^suient^to^ahieve^the

boundsof[1℄. Furthermore,[3℄presentedonemethod whih doesnotrequireoordination

of (the oding at) the nodes, by introduing random linear oding and by showing that

suient eld size results in high probability of suess. With randomlinear oding, the

oding inside thenetwork isno longerpredetermined, sineit usesrandomoeientsfor

thelinearombinations. Thisapproahwasrenedby[4℄,intheontextofpaketnetworks:

theinformationisontainedinindividualpaketsandtherandomizedoeientsanthen

beinluded in a speial additionalpaket header. Relying on thefundamental broadast

nature of the wireless media, [6℄ introdued the rst pratialimplementation of network

odingin awirelessenvironment,anddemonstratedgainsforrossinguniastdatatras.

Theseworkssetthepathtopratialfoundations,whiharedesribedforinstanein[6℄,

andthatweareusingin thisartile.

Generation: rst,thepaketsaresupposedtobedividedintogenerations. Ageneration

isasetofpakets(ingeneral,fromoneorseveralsoures),thatouldbemixedtogetherby

linearombination. Intherestof thepaper,wewill assumethatallpaketsbelongto one

generationfromasoureand denoteagenerationsize thedimension, D^.

Vetors: seond, the pakets are equally sized and are divided into bloks of sym-

bols over a eld F_p ^: ôntent = (s1, s2, ..., sh)^. Âs ⁱⁿ ^[4℄, ^the ^pakets înlude â ^header

whih isthelistofoeients. Hene thepaket formatis atuallyavetoroftheformat:

(g1, g2, . . . , gD;s1, s2, . . . , sh)^.

Transmission: atanypointoftime,anodeofthenetworkhasalistof vetors,linear

ombinations of initial soure pakets. When the node transmits, it generates a random

linearombinationofthevetorsv0, v1, ..., vk ^it^urrently^has: P

iαivi ^(where^the(αi)^are

randomoeientsofF_p^),^and^transmit^it^by^wirelessbroadasting.

Deoding: oneanodehasreeivedD^linearlyindependentvetors,itisabletodeode theD^pakets^of^thegeneration.

Withinthislassialframework,thetwonalquestionsare: whendoesthesouresenda

paket? and: whendotheothernodes(non-soure)maketransmissions? Thistransmission

sheduling istheheartofthealgorithm.

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Algorithm1: NCAL1

Soure sheduling: poisson soure;thesouretransmitssequentiallytheD ^vetors

1.1

(pakets)ofagenerationwithanexponentialinterarrival.

Nodes'start and stoponditions: thenodesstarttransmittingwhentheyreeive 1.2

therstvetor. Theyontinuetransmittinguntiltheyhaveenoughvetorstoreover

theD ^soure^pakets. ^Then^they^still^transmitôneâdditional^paket^(after ûsual

exponentialdelay).

Nodes'sheduling: poissonretransmission;thenodesretransmitlinear 1.3

ombinationsofthevetorsthattheyhave,withanexponentialinterarrival

2.2 Algorithms for Transmission Sheduling

2.2.1 Single soure

Thetransmission sheduling used in ourartile is the algorithm NCAL1. Essentially, the

nodestransmitrandomombinationsofthevetors,withexponentialinterarrival,untilthey

havesuientvetors. This algorithmanbeomparedto thealgorithmsused in [5℄,[10℄

and[11℄forinstane. Thealgorithm1isaversionofRandomLinearCoding(RLC)withpush

forthewireless ontext(hene, withfousonspatial gossip). It is alsoavariantof Blind-

Forwarding with NetworkCoding (BF-NC)from[10℄. Finally,thealgorithm1omparesto

the algorithm NC3 of [5℄, exept that we use asimple stop ondition. By hoie, in this

artile,wedidnotattempttouseopportunisti([6℄)mehanismsalsopresentedintheited

works.

Inalgorithm1,theintentoftheadditionalpaket transmissionat1.2istogiveneigh-

bors onemorehanetogetthelasttransmissionompletingtheD ^dimension^spae.

However, thestopondition in NCAL1 still doesnotguarantee that allinformation is

disseminated to all nodes, hene for referene we used another algorithm, NCAL2 whih

assumesreeption reports andstoptransmittingonlywhenallneighborhaveallpaketsto

restorealloriginalpakets.

Algorithm2: NCAL2

Soure sheduling: asin 1.1.

2.1

Nodes'start and stoponditions: asin 1.2,thenodesstarttransmittingwhen 2.2

theyreeivetherstvetorbut theyontinuetransmittinguntilthemselvesand

their neighborshaveenoughvetorstoreovertheD ^soure^pakets.

Nodes'sheduling: asin 1.3 2.3

RR n°6120

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2.2.2 Extension to multiple soures

Thetwo algorithm(NCAL1 and NCAL2) anbeextended formultiple soures. Considering

paketsfromdierentsouresarein thesamegeneration,synhronizationisneessaryand

isnotaddressedhere.

3 Casading Wave Model

Figure1: Wavemodel

The entral fous of this artile is the study the behavior of the above algorithms in

wireless networks with are modeled with disk-unit graphs. The dynami behavior of the

networkisrepresentedingure1: thenodes(whitedisks)arerandomlydistributedinaline

topology,thesoureisoriginatingnewvetors,(additional dimensionsof thevetorspae)

and these new dimensionspropagate in the network as waves. We represented the eet

ofsome suessivetransmissions t0, t1, t2, t3 ândt4^. ^The êet ôfônetransmission anbe summarized by howit inreases (or does not inrease)the dimension spae of the nodes.

Here,initialtransmissiont0^and^the^suessivetransmissiont1^from^the^soure,^inrease^the

dimensionoftheneighborsofthesoure(whihbeame1^). Transmissiont2^from^a^neighbor

ofthesoure,propagatesfurther alinearombinationoft0 ^andt1^. ^This^proess^ontinues,

and d transmission from the soure in generalpropagates as seen in gure 1. Hene we

have a asade of waves. Note that this ours in our ase, beause we haveone soure

andaline topology, whereanyvetorthat anodereeivemustbealinearombinationof

thevetorsalreadyreeivedatanyleftneighbor. Preiseanalysisofthepropagationofthe

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waveisparamounttotheharaterizationoftheperformaneofalgorithm1. Suhanalysis

ispresentedinsetion4.

4 Analysis of performane

4.1 Wave propagation model

In this setion, we provide an analysis of the propagation waves in the network oding

shemedesribedin setion2.2.1. Eahwaveisdenedbytheborderbetweenareaswhere

nodes have reeived samevetor spae ombination of pakets, asshown on gure 1. In

partiular thek^th ^waveîndiates ^the ^limit^between^the ârea^where ^nodes ^have^reeived â

vetorspaeof k^dimension^with ^theâreaôf^the^nodes ^whih ^have^reeivedâ^vetor^spae

ofk−1 ^dimension.

In1D, thewavepositionsare determinedby theposition of theborderon thepositive

realaxis: xk(t)îs ^theâbsissaôf^thek^th^waveât ^timet^, ^withôriginx= 0ât ^the^loation

ofthesoure. Wealsoassumethattheorderingxk(t)> xk+1(t)âtâll^times. ^The^modelîs

thefollowing:

Soure transmission: thesouretransmitspakets atrateλν ^where ν ^is ^the^node

densityandλ <1 ^(average^neighbor^size^isM = 2ν^).

Neighborsofthesoure: notethatthenewreatedwavesatuallydepartfrompo-

sitionx= 1^,^sine^theâll^the^diret^neighborsôf^the^soure^reeiveêverytransmission ofthesoure.

Node retransmission: eah node retransmitsat rate1^. ^Hene ^note ^that ⁱⁿ ^a^area

oflengthunitandtimeunit,therelaynodestransmitat aPoissonrateofdensityν^.

Wave propagation: whenarelay nodeat position y ^transmits ^a^vetor, ^two^ases

an our. If there atually exists a losest wave border xi ^suh ^that xi > y ^and xi−y <1^,^then^thetransmissionmovesthewaveborderatpositiony+ 1^. ^Otherwise

transmissionhasnoeet onexisting waves.

Notie that wavepropagation may break the order of the xk^, ⁱⁿ ^whih ^ase ^the ^indexing

wouldneedto beadjusted.

4.2 Atual performane

From now we restrit our analysis to 1D unit disk graph model. Let onsider anode at

distane x ^from ^the ^soure ^node. ^Let∆(x, Q) ^be^the ^average ^time ^between ^the ^rst ^wave

and thelast wave. Forexampleifx≤1 ^we ^have∆(x) = ^2Q_λ ^, ^sine ^whenneighboringthe soure,thenodeexperieneanaveragetimeof

2

λM ^between^two^wavegenerations(i.e. two datapakettransmissionsbythesoure)wherethegenerationsize DisMQ.

Theorem1 The average number of paket transmission by a node at distane x ^to ^the

soureis∆(x, Q) + 1^.

RR n°6120

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Proof: The node starts to shedule enoded paket when it reeives the rst wave and

ontinueaslongasit hasnotreeivedthelastwavewhih arrivesin average∆(x, Q)^time

unit later. Sine the enoded paket generation is Poisson of rate 1, the average number

ofenoded paket transmittedduring∆(x, Q)îsêxatly∆(x, Q)^. ^Therefore^the^numberôf

paketsheduledis∆(x, Q) + 1^. ^The+1^denotes^the^last^paket^whih^is^sheduled^during

the∆(x, Q)^period^but ^is^atuallytransmitted aftertheperiodaordingtoalgorithm.

Corollary1 Thetotalaveragenumberofpakets(dataandenoded)transmittedbyallthe

nodesbetween distane0anddistaney ^is^equal ^toQM+^M₂ Ry

0(T(x, Q) + 1)dx^.

Proof: The term QM ^denotes ^the ^data ^pakets transmitted by the soure. The other paketsaretheenodedpaketgeneratedbytherelaynodeswherethedensityν= ^M₂^.

Whenthenodeisnotneighborofthesoure,i.e. x >1^,^we^have^the^expression∆(x, Q) =

2Q

λ + (x−1) (sl−sf)^. ^where sf îs^theâverage^slownessôf^rst^wavepropagationandsl îs

theaverageslownessofthelast wave.

Theorem2 Wehavesf ≥ _M² ^andsl≤ ₍₁^16λ

−λ²)M

Beauseλ <1 ^the^time^gap^between^the^rst^wave^and^the^last ^wave^isproportionalto

β

M^. ^WhenM → ∞^the^time^gap^beomes^zero,^thus^the^wavepropagationproessdoesnot bringmuheetontheperformane.

Corollary2 The total average number of paket transmittedby nodes between distane 0

anddistaney isequal toQM+ (^2Q_λ + 1)^{M y}₂ +O(^y₁⁻¹

−λ)^whenM → ∞^.

Corollary3 Theaveragenumberofretransmissionsperdatapaketbetweendistane0and

distaney ^is^equal ^to1 + (²_λ+_Q¹)^y₂+O(₍₁^y⁻¹

−λ)QM)^.

If thenetworkis limited todistane y ^to ^the^soure: N = ^yM₂ ^assuming ^the^soure ^is^the

leftmost node on the network, we have a retransmission ratio equal to 1 + (_λ² + _Q¹)_M^N + O(₍₁ ^N

−λ)QM²)^whih^isarbitrarilyloseto2_M^N ^whenN, M→ ∞^andλ→1^.

4.3 Wave propagation slowness

Inthis setion we useν = ^M₂ ^theûniform ^densityôf ^node ^perûnit ^lengthⁱⁿ ^the ^1Dûnit

graphmodel.

4.3.1 Propagation ofthe rst wave

Theorem3 Therst wavemovesataverage speedvf = ^ν₂ ^length^unit ^per^time^unit.

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Proof: the relay nodes that move the rst wave are those whih are betweenx1(t)−1

and x1(t)^. ^This ^nodes ^reate ^a transmission rateof ν transmissions per time unit. Eah transmissionmovesthewavebyanaveragetranslationof

1

2 ^unit^length.

Corollary4 The average slowness of the rst wave satises sf ≥ ²_ν ^time^unit ^per ^length

unit.

Proof: wehavetheinequalitysf ≥_v¹

f

sinetheinversefuntionisonvex.

4.3.2 Propagation ofthe last wave

Theorem4 Thelastwave propagatesatan average slownesssmaller than

8λν

1−λ² ^time^unit

perlengthunit.

The proof of this theorem is based on lemmas that study an upper bound of the atual

system.

4.3.3 The upper-bound system

Byupperboundsystemweonsiderasystemwherewavespositionsyk(t)^are^always^smaller

thanatualwaveposition: yk(t)< xk(t)^for^allt^andk^. ^We^desribe^suh^system.

Letℓ^beâ^number^suh^that ¹_ℓ <1−λ^. ^Weût^the^realâxisⁱⁿ êqualîntervalôf^length

1

ℓ^. ^We^will^fore^the^wave^positionsyk ^of^theupper-bound systemto beintegermultipleof

1 ℓ^.

(unhanged)Thesouretransmitspaketsatrateλν^where ^M₂ ^is^the^node^density^and λ <1

Thenewreatedwavesareat absiss1−¹_ℓ^.

(unhanged) Relay nodes transmit at a Poisson rate of ν ^per ^length ^unit ^and ^time

unit.

When a relay node transmits at absiss y ∈]^k⁻_ℓ¹,^k_ℓ] ^for k < ℓ^, ît ^moves â ^wave ât

position

ℓ−1

ℓ ^to^position1 +^k⁻_ℓ¹^. ^If^there^was^no^wave^at^position ^ℓ⁻_ℓ¹^,^then^the^relay

nodetransmissionhasnoeet onwavepositions.

When a relay node transmits at absiss y ∈]^k⁻_ℓ¹,^k_ℓ] ^for k ≥ ℓ^, ît ^moves â ^wave ât

position

k

ℓ ^to^position1 +^k⁻_ℓ¹^. ^If^there^was^no^wave^at^position ^k_ℓ^,^then^the^relay^node

transmissionhasnoeet onwavepositions.

Notie that thewavesin theupper bound systemmoveless frequently than in theatual

system,andwhentheymove,itisonashorter distane. Thereforethisisanupper-bound

system.

RR n°6120