Maximizing the Probability of Delivery of Multipoint Relay Broadcast Protocol in Wireless Ad Hoc Networks with a Realistic Physical Layer

0.1 MB·English

by  François Ingelrest

Maximizing the Probability of Delivery of Multipoint
Relay Broadcast Protocol in Wireless
Ad Hoc Networks with a Realistic Physical Layer
Franc¸oisIngelrestandDavidSimplot-Ryl
7
IRCICA/LIFL,UniversityofLille1.
0
0 CNRSUMR8022,INRIAFuturs,France.⋆
2 {Francois.Ingelrest, David.Simplot}@lifl.fr

n
a
J

6 Abstract. Itisnowcommonlyacceptedthattheunitdiskgraphusedtomodel
1 thephysicallayerinwirelessnetworksdoesnotreflectrealradiotransmissions,
and that the lognormal shadowing model better suits to experimental simula-

] tions.Previousworkonrealisticscenariosfocusedonunicast,whilebroadcastre-
I
N quirementsarefundamentallydifferentandcannotbederivedfromunicastcase.
Therefore,broadcast protocolsmustbeadaptedinorder tostillbeefficientun-
.
s derrealisticassumptions.Inthispaper,westudythewell-knownmultipointrelay
c
protocol(MPR).Inthelatter,eachnodehastochooseasetofneighborstoactas
[
relaysinordertocoverthewhole2-hopneighborhood.Wegiveexperimentalre-

1 sultsshowingthattheoriginalmethodprovidedtoselectthesetofrelaysdoesnot
v givegoodresultswiththerealisticmodel.Wealsoprovidethreenewheuristics
4
inreplacementandtheirperformanceswhichdemonstratethattheybettersuitto
9
theconsideredmodel.Thefirstonemaximizestheprobabilityofcorrectrecep-
0
tionbetweenthenodeandtheconsideredrelaysmultipliedbytheircoveragein
1
the2-hopneighborhood.Thesecondonereplacesthecoveragebytheaverageof
0
7 theprobabilitiesofcorrectreceptionbetweentheconsideredneighborandthe2-
0 hopneighborsitcovers.Finally,thethirdheuristickeepsthesameconceptasthe
/ secondone,buttriestomaximizethecoveragelevelofthe2-hopneighborhood:
s
2-hopneighborsarestillbeingconsideredasuncoveredwhiletheircoveragelevel
c
: isnot higher thanagivencoverage threshold, manyneighborsmaythusbese-
v
lectedtocoverthesame2-hopneighbors.
i
X
r
a 1 Introduction
Nowadays,wirelessnetworkinghasbecomeanindispensabletechnology.However,the
mostdeployedtechnology,knownasWiFi,istoorestrictive,asusersmuststaynearto
fixed access points. Therefore, the latter must be sufficiently deployed and correctly
configuredtoofferagoodqualityofservice.Moreover,thereexistssomemoreunusual
situations where an infrastructure may be unavailable (e.g., rescue areas). The future
ofthistechnologyprobablyliesinwirelessadhocnetworks,whicharedesignedtobe
⋆Thiswork waspartiallysupported by agrant fromCPERNord-Pas-de-Calais/FEDERTAC
COM’DOM,INRIAresearchactionIRAMUSandCNRSnationalplatformRECAP.
functionalwithoutanyinfrastructure.Theyaredefinedtobecomposedofasetofmo-
bileorstatichostsoperatinginaself-organizedanddecentralizedmanner,whichcom-
municatetogetherthankstoradiointerfaces.Hostsmaybeeitherterminalsorrouters,
dependingontheneedsofthesystem,leadingtoacooperativemulti-hoprouting.
Broadcastingisoneofthemostimportantcommunicationtask inthose networks,
as it is used for many purposessuch as route discovery (e.g., OLSR [1]) or synchro-
nization.Inastraightforwardsolutiontobroadcasting,hostsblindlyrelaypacketsupon
firstreceptiontotheirneighborhoodinordertofullycoverthenetwork.However,due
to known physical phenomena, this leads to the broadcast storm problem [2]. More-
over,thisisatotallyinefficientalgorithm,becausemostoftheretransmissionsarenot
neededtoensuretheglobaldeliveryofthepacket,andahugeamountofenergyisthus
unnecessarilywasted.Manyotheralgorithmshavebeenproposedinreplacement.Some
ofthemarecentralized(aglobalknowledgeofthenetworkis needed),whiletheoth-
ersarelocalized(hostsonlyneedtoknowtheirlocalneighborhoodtotakedecisions).
Obviously,thelatterbetterfittoadhocnetworksandtheirdecentralizedarchitecture.
Alltheproposedbroadcastschemeshavealwaysbeenstudiedunderidealscenario,
where the unit disk graph is used to model communications between hosts. In this
model, two hosts can communicatetogether if the distance between them is no more
thanagivencommunicationradius,andpacketsarealwaysreceivedwithoutanyerror.
Recently, this modelhas been highly criticized as it does notcorrectly reflect the be-
havior of transmissions in a real environment[3]. Indeed,signal strengthfluctuations
haveasignificantimpactonperformance,andthuscannotbeignoredwhendesigning
communicationprotocolsforadhocandsensornetworks.Unfortunately,thishasbeen
thecaseuntilnowforbroadcastprotocols.
Inthispaper,weconsiderthewell-knownmultipointrelayprotocol(MPR)[4],used
forbroadcastinginadhocnetworks,underamorerealisticscenariowheretheproba-
bilityofcorrectreceptionofapacketsmoothlydecreaseswiththedistancebetweenthe
emitterandthereceiver(s).Wethusreplacetheunitdiskgraphmodelbythelognormal
shadowing model [5] to simulate a more realistic physical layer, and provide experi-
mental results. As they demonstrate the need for a more suitable algorithm, we also
propose several modifications to MPR in order to maximize the delivery ratio of the
broadcastpacket,whileminimizingthenumberofneededretransmissions.Byexperi-
mentation,weshowthatthesenewversionsaremuchmoreefficientthanthe original
oneundertheconsideredrealisticscenario.
Theremainderofthispaperisorganizedasfollows:wefirstprovidethedefinitions
needed by our models, while in Sec. 3 a detailed description of MPR is proposed.In
Sec. 4, we provideananalysisof the behaviorofthe originalalgorithmused in MPR
withtherealisticphysicallayer.WethendescribeinSec.5newalgorithmsthatbetter
fitthelatter.WefinallyconcludeinSec.6andgivesomedirectionsforfuturework.
2 Preliminaries
ThecommonrepresentationofawirelessnetworkisagraphG=(V,E),whereV isthe
set of vertices(the hosts, or nodes)andE ⊆V2 the set of edgeswhichrepresentsthe
availablecommunications:thereexistsan orderedpair(u,v)∈E if the nodev is able
tophysicallyreceivepacketssentbyu(inasingle-hopfashion).Theneighborhoodset
N(u)ofthenodeuisdefinedas{v:(u,v)∈E∨(v,u)∈E}.Thedensityofthenetwork
isequaltotheaveragenumberofnodesinagivencommunicationarea.Eachnodeuis
assignedauniqueidentifier(thiscansimplybe,forinstance,anIPoraMACaddress).
Weassumethatnodesareawareoftheexistenceofeachneighboringnodewithina
distanceof2hops(wecallthisa2-hopknowledge).Inadhocnetworks,theneighbor-
hooddiscoveryisgenerallydonethankstosmallcontrol(HELLO)messageswhichare
regularlysent by each host. A 2-hopknowledgecan easily be acquiredthanksto two
roundsofHELLOexchanges:nodescanindeedinserttheidentifiersoftheirneighbors
intheirownbeaconmessages.
Inourmathematicalmodel,theexistenceofapair(u,v)∈E isdeterminedbythe
consideredphysicallayermodelanddependsonseveralconditions,the mostobvious
onebeingthedistancebetweenuandv.Inthemostcommonlyusedmodel,knownas
theunitdiskgraphmodel,abidirectionaledgeexistsbetweentwonodesifthedistance
between them is not greater than a given communicationradius R (it is assumed that
allnodeshavethesamecommunicationradius).Inthismodel,thesetE isthensimply
definedby:
E ={(u,v)∈V2|u6=v ∧ dist(u,v)≤R}, (1)
dist(u,v)beingtheEuclideandistancebetweennodesuandv.
This model, while being well spread, cannot be considered as realistic. Indeed, it
isassumedthatpacketsarealwaysreceivedwithoutanyerror,aslongasthe distance
between the emitter and the receiver is smaller than the communication radius. This
totally ignoresrandomvariationsin thereceivedsignalstrength,whileit wasdemon-
stratedthattheirimpactisreallysignificant.
These fluctuations generate erroneous bits in the transmitted packets. If the error
rateissufficientlylow,thesebitscanberepairedthankstocorrectioncodes.However,
ifitistoohigh,thenthepacketmustbedroppedandanewemissionmustbedone.This
supposestheexistenceofanacknowledgementmechanism(ACKpackets)thatcannot
beusedinbroadcastingtasksduetothereallyhighnumberofemitters.Ourworkthus
only relies upon the probability of correct reception, which is influenced by a lot of
factors (e.g.,power of emission, distance with the receiver(s),presence of obstacles).
Wesupposethatallnodeshavethesametransmittingradius,sothepowerofemission
doesnothavetobetakenintoaccounthere.
Toconsiderthesignalfluctuations,wechangeGintoaweightedgraphwhereeach
edge(u,v)∈E holdstheprobabilityp(u,v)ofcorrectreceptionbetweenthetwonodes
uandv.Todeterminetheseprobabilities,wechosetoconsiderthelognormalshadowing
model[3]inoursimulations.WeusedanapproximatedfunctionP(x)describedin[6]:
(x)2a
1− R if0<x≤R,
 2
P(x)=(2RR−x)2a ifR<x≤2R, (2)
2
0 otherwise,
Unit disc graph model
1 Lognormal shadowing model
0.8
Probability of correct reception 00..46
0.2
0
0 50 100 150 200
Distance
Fig.1.Unitdiskgraphandlognormalshadowingmodels(R=100,a =4).
a being the power attenuation factor, and x the considered distance. Fig. 1 illustrates
thismodelwithR=100anda =4.
We assume thateachnodeuis ableto determinethe probabilityp(u,v)ofcorrect
reception of a packetthat would be sent to a neighborv. The gain of this knowledge
maybesimplyachievedthankstobeaconmessages:basedonthequantityofcorrectly
receivedHELLOpackets,visabletodetermineanapproximatedvalueofp(u,v).Node
vmaythenincludethisvalueinitsownbeaconmessages.
Oneof themajorcriticisms ofthe unitdisk graphmodelis thatitdoesnotmodel
thepresenceofobstaclesbetweennodes.Thelognormalshadowingmodelneithercon-
siders them, but we argue that it is sufficient enough for simulations. The most im-
portantfactoristheweightingofedgesbyreceptionprobabilities,themethodusedto
distributethe latteris notimportantto compareprotocolsin generalcases. A realistic
modelwouldbemandatorytosimulateexistingsituationsandtoextractexactvalues.
Butinrealcases,anobstaclewoulddecreasetheprobabilityheldbythecorresponding
edge and would thus be detected by nodeswhen countingHELLO messages (if such
a methodis used).Thismeansthatin those cases, the broadcastalgorithmwoulduse
‘real’probabilitiesanditsbehaviorwouldbeadaptedtothesituation.
Thetwopreviousphysicalmodelsintroducetwodifferentbehaviors:
– In the unitdisk graphmodel,one hasto maximizethe lengthof each hopso that
a single emission is able to reach as many mobiles as possible. The quantity of
neededemittersisthusgreatlyreduced.
– In the lognormal shadowing model, maximizing the length of each hop leads to
smaller probabilities of correct reception, but minimizing them leads to a lot of
spentenergy.
Somepapershavealreadybeenpublishedaboutroutinginarealisticenvironment.
Amongstthem,DeCoutoetal.[7]andDravesetal.[8]investigatethequestionofrout-
ingmetricsforunicastprotocolsinwirelessnetworkswitharealisticphysicallayer:the
keyinsightinmostofthisworkisthathop-countbasedshortest-pathroutingprotocols
result in transmissions over long links. While this reducesthe hop-countof routes, it
alsodecreasesthereceivedsignalstrengthatthereceiveroftheselinks,leadingtovery
highlossratesandlowend-to-endthroughput.Thesepapersalsoproposeotherrouting
metricswhichincorporatelink-quality(e.g.,intermsoferror,congestion).
To the best of our knowledge, this paper is the first one to consider broadcasting
overarealisticphysicallayer.Broadcastfundamentallydiffersfromunicast,andleads
toadifferenttradeoffbetweenthelengthofeachhopandthenumberofrelays.Indeed,
inabroadcastprocess,anodecanrelyontheredundancyintroducedbyotheremitters.
Furtherrelaysmaythusbeselectedwithoutdecreasingthefinaldeliveryratio.Thisis
not possible in routing, as a given emitter is the only one able to transmit the packet
to the nexthop.The redundancyof broadcastingmustbe fully consideredin orderto
improvetheperformanceoftheunderlyingprotocol.
3 Related Work
As stated in Sec. 1, the easiest method for broadcasting a packet is to have all nodes
forwarditatleastoncetotheirneighborhood:thismethodisknownasblindflooding.
However,suchasimplebehaviorhashugedrawbacks:toomanypacketsarelostdueto
collisionsbetweenneighboringnodes(thiscanleadtoapartialcoverageofthenetwork)
and far too much energy is consumed. Many other solutions have been proposed to
replaceit,andanextensivereviewofthemcanbefoundin[9].
Amongallthesesolutions,wehavechosentofocusonthemultipointrelayprotocol
(MPR)describedin[4]forseveralreasons:
– Itisefficientusingtheunitdiskgraphmodel.
– Itisusedinthewell-knownstandardizedroutingprotocolOLSR[1].
– Itcanbeusedforothermiscellaneouspurposes(e.g.,computingconnecteddomi-
natingsets[10]).
Inthisalgorithm,itisassumedthatnodeshavea2-hopknowledge:theyareawareof
theirneighbors(1-hopdistance),andtheneighborsoftheseneighbors(2-hopdistance).
Its principle is as follows. Each node u that has to relay the message must first elect
some of its 1-hop neighbors to act themselves as relays in order to reach the 2-hop
neighborsofu.Theselectionisthenforwardedwithinthepacketandreceiverscanthus
determineiftheyhavebeenselectedornot:eachnodethatreceivesthemessageforthe
firsttimechecksifitisdesignatedasarelaynodebythesender,andifitisthecase,the
messageisforwardedaftertheselectionofanewrelayingsetofneighbors.Avariant
existswherenodesproactivelyselecttheirrelaysbeforehavingtobroadcastapacket,
andselectionissentwithinHELLOmessages.
Obviously,the tricky part of this protocollies in the selection of the set of relays
MPR(u) within the 1-hop neighbors of a node u: the smaller this set is, the smaller
thenumberofretransmissionsisandthemoreefficientthebroadcastis.Unfortunately,
findingsuchasetsothatitisthesmallestpossibleoneisaNP-completeproblem,soa
greedyheuristicisproposedbyQayyumetal.,whichcanbefoundin[11].Considering
anodeu,itcanbedescribedasfollows:
1. Place all 2-hop neighbors (considering only outgoing links) in a set MPR′(u) of
uncovered2-hopneighbors.
v4
u
v3 w4
v1 v2
w3
w1 w2
Fig.2.ApplyingMPRatnodeu:MPR(u)={v1,v3}.
2. Whilethereexistsa1-hopneighborvwhichistheonlycommonneighborofuand
somenodesinMPR′(u):addvtoMPR(u),removeitsneighborsfromMPR′(u).
3. While the set MPR′(u) is not empty, repeatedlychoose the 1-hopneighborv not
presentinMPR(u)thatcoversthegreatestnumberofnodesinMPR′(u).Eachtime
a new node is added to MPR(u), removeits neighborsfrom MPR′(u). In case of
tie,choosethenodewiththehighestdegree.
AnexampleofthisheuristicisgiveninFig.2,startingwithMPR(u)=0/.Thenode
v istheonlyoneabletoreachw ,soitisaddedtoMPR(u)andnodesw andw are
1 1 1 2
removedfromMPR′(u).Noothermandatory1-hopneighborofuexists,sootherrelays
areselectedaccordingtothenumberofnodesinMPR′(u)theycover.Nodesv andv
2 4
coveronlyonenodeinMPR′(u),whilenodev coversatthesametimew andw ,so
3 3 4
v ischosenandaddedtoMPR(u).ThesetMPR′(u)beingempty,noothernodesare
3
selected.WefinallyhaveMPR(u)={v ,v }.
1 3
Beingthebroadcastprotocolusedin OLSR,MPRhasbeenthesubjectofmiscel-
laneousstudiessince itspublication.Forexamplein [12],authorsanalyzehowrelays
areselectedandconcludethatalmost75%ofthemareselectedinthefirststepofthe
greedyheuristic,sothatimprovingthesecondstepisnotreallyuseful.Thisconclusion
seemscorrect,aslongastheunitdiskgraphmodelisused.
4 OriginalGreedy Heuristic
4.1 Graphsgeneration
Inthissection,weprovideresultsabouttheperformanceofMPRoverourconsidered
realisticphysicallayer,thelognormalshadowingmodel.Wechosenottouseageneral
purpose simulator in order to focus on the area of our study: we thus implemented
algorithmsandmodelsinourownsimulator,sothatwehadtodecidehowtogenerate
‘realistic’graphsconsideringtherealisticmodel.
We chosetoconsiderthemethodcitedinSec. 2,whichisbasedonHELLOmes-
sages.Neighborhoodinformationisstoredinatablewhichisregularlycleanedinorder
toremovetoooldentries.Anentryistoooldwhenthecorrespondinghosthasnotsig-
naled itself since a given amount of time, that we denote by x. Beacon messages are
regularly sent by each host to signal itself. Let us denote by y the time between two
50
100 Unit disc graph model
Lognormal shadowing model
40
80
Percentage of receiving nodes 4600 Percentage of transmitting nodes 2300
20 10
Unit disc graph model
Lognormal shadowing model
0 0
15 20 25 30 35 40 45 50 15 20 25 30 35 40 45 50
Average density Average density
(a)Receivingnodes. (b)Transmittingnodes.
Fig.3.PerformanceofMPRwiththetwoconsideredphysicalmodels.
HELLOmessages(wehavex>y).Anodeuseesaneighborvifithasreceivedatleast
oneHELLOmessageduringthelastyseconds.Theprobabilityp (u,v)forthisevent
n
tooccurisequalto:
x
p (u,v)=1−p(u,v)y. (3)
n
Foreachdirectionaledge,arandomnumberisthusdrawntodetermineifitexists.
Thisway,whenanodeuisawareoftheexistenceofaneighborv,itcandecidetosend
messagestothelatter.Ofcourse,ucannotbeensuredthatitsmessageswillreachv.We
caneasilyconcludethatlongedgeshavea highprobabilityto beunidirectionalwhile
shortedgeshaveahighprobabilitytobebidirectional.
Alltheresultswereobtainedwith thefollowingparameters.Thenetworkis static
and always composedof 500 nodesrandomlydistributedin a uniformmannerovera
squarearea whose size is computedin orderto obtaina givenaveragedensity.Edges
arecreatedusingthemethodpreviouslydescribed,andforeachmeasure,wetookthe
average value obtained after 500 iterations. We fixed the communicationradiusto be
equalto 75 in both physicalmodels. An idealMAC layer is consideredto isolate the
intrinsicpropertiesoftheselectedrelays:collisionsofpacketscouldskewbothresults
andanalyses.
4.2 Experimentalresults
WeprovideinFig.3(a)thedeliveryratioofMPRusingthetwoconsideredphysicallay-
ers.Whenusingtheunitdiskgraphmodel,atotalcoverageofthenetworkisachieved
as MPR is a deterministic algorithm.However,this is no more the case with the log-
normalshadowingmodelduetothemultipleerrorsoftransmission:thedeliveryratio
isunder70%foreachconsidereddensity,andisaslowas55%foradensityd=15.
Thispoorperformancecanbeexplainedbythefactthat,ashighlightedbyBusson
et al. in [12], the chosen relays are located at the limit of the communication range,
wheretheprobabilityofcorrectreceptionislow.Thisisconfirmedinourexperiments,
70
60
50
Distance 40
30
20
10
0
15 20 25 30 35 40 45 50
Average density
Fig.4.Averagedistancebetweenanodeanditsrelays.
asillustratedbyFig.4:theaveragedistancebetweenanodeanditsmultipointrelaysis
almostequalto68,whilethemaximalcommunicationrangeis75.Moreover,[12]also
states that 75%of the relaysare chosen duringthe first step: this meansthat, when a
relaydoesnotcorrectlyreceivethemessage,thereisariskof75%thatthisrelaywas
the only one able to reach an isolated node,which will thus notreceive the message,
potentiallyleadingtoapartitionofthenetwork.
WealsoprovideinFig.3(b)thepercentageofnodeswhichcorrectlyreceivedand
thenrelayedthemessage.Itisinterestingtonotethatthispercentageisdifferentwith
the two models. Indeed,as onlynodeswhich receivedthe message are taken into ac-
count,onewouldhaveexpectedtoobservethesamevaluesinbothcases.Thismeans
thattheneedednumberofrelayingnodesdoesnotlinearlyvarywiththenumberofcov-
erednodes:obviously,onlyafewrelaysareneededtocoverahighnumberofdifferent
nodes,butalargernumberisneededtocoverthelastfewremainingones.
5 New Heuristics forMPR
As illustrated in the previoussection, the originalgreedyheuristic used by Quayyum
etal.in[4]isnotsuitableforarealisticphysicallayer.Anaveragedeliveryof70%is
indeednotsufficientformostofapplications,andanalternativesolutionmustthusbe
used.
In this section, we propose miscellaneous replacement heuristics in order to im-
provetheperformanceof MPR. Theyaimatmaximizingtheaveragecoverage,while
minimizingthenumberofneededrelays(andthustheenergyconsumption).Inallour
proposals,thefirststepoftheoriginalheuristicwhichallowsisolated2-hopneighbors
tobecoverediskept(itismandatory),onlythesecondstepisreplaced.
WekeepnotationsintroducedinSec.3.Thus,consideringanodeu,thesetMPR(u)
containsthemultipointrelayschosenbyu,whilethesetMPR′(u)contains2-hopneigh-
borsofunotyetcovered.
5.1 Firstproposal:Straightforwardapproach
Aspreviouslyexplained,thelowdeliveryratioofMPRiscausedbythetoohighdis-
tancebetweenanodeanditsrelays.Thelatterhavinglittlechancetocorrectlyreceive
u v
(cid:1)
w
v (cid:2)
(cid:0)
w
(cid:1)
w
(cid:0)
Fig.5.Acasewherethenodeuhastoselectitsmultipointrelaysbetweenitsneighborsv andv
1 2
(MPR(u)=0/,MPR′(u)={w1,w2,w3}.
thebroadcastpacket,theyalsohavelittlechancetobeabletorelaythispacketandthus
tocoverthe2-hopneighborsoftheemitter.
A first and straightforward idea could be, when choosing a relay, to balance the
coverageitoffersanditsprobabilitytocorrectlyreceivethepacket.Thus,ateachstep
consideringanodeu,ascorecanbecomputedforeachpotentialrelayv.Thenodewith
thehighestscoreisselectedandplacedinMPR(u).Wedenotebyc (v)theadditional
u
coverageofferedbyvtou:
c (v)=|MPR′(u)∩N(v)|. (4)
u
Thescoreobtainedbyvatagiveniterationforanodeu,denotedbys (v),isthus
u
definedby:
s (v)=c (v)×p(u,v). (5)
u u
Insimpleterms,theadditionalcoverageofferedbyvisweightedbyitsprobability
to correctly receive the broadcast packet. In Fig. 5, the score s (v ) of v is equal to
u 1 1
3×p(u,v ).
1
5.2 Secondproposal:Cleverapproach
The previousheuristic,while beingmoresuitable fora realistic environmentthanthe
originalone,stillhasanobviousflaw:itstilltakesintoaccountadditionalcoveragein
a too simple way. One can thuseasily imaginea situationwhere a verydistant1-hop
neighbor would offer an additional coveragesuch that the latter would compensate a
low probability of correct reception. In this case, this neighbor would be selected as
relay while its probabilityto correctlyreceivethe packet,and thusto be able to relay
it,wouldbeverylow.Onecanalsoimagineasituationwherethedistancebetweenthe
relayandthe2-hopneighborsitcoverswouldbe veryhigh,suchthatthe re-emission
ofthisrelaywouldhavelittlechancetoreachthese2-hopneighbors.
Weproposetoextendtheconceptusedinthefirstproposal,bytakingintoaccount
theprobabilitiesofcorrectreceptionbetweenthepotentialrelayandthe 2-hopneigh-
borsitcovers.Wethusreplacetheadditionalcoverageofferedbyarelaybytheaverage
probabilityofcorrectreceptionby2-hopneighbors.Wethusobtain:
100 50
Original heuristics
Heuristics 1
Heuristics 2
80 40 Heuristics 3, threshold=0.5
Percentage of receiving nodes 4600 Percentage of transmitting nodes 2300
20 Original heuristics 10
Heuristics 1
Heuristics 2
Heuristics 3, threshold=0.5
0 0
15 20 25 30 35 40 45 50 15 20 25 30 35 40 45 50
Average density Average density
(a)Receivingnodes. (b)Transmittingnodes.
Fig.6.Performanceofthedifferentheuristicsusingthelognormalshadowingmodel.
i=|(cid:229)cu(v)|
s (v)=p(u,v)× (p(v,w)/|c (v)|). (6)
u i u
i=1
Thisway, multipointrelaysofferinga low coverageintermsof probabilitieshave
littlechancetobeselected.InFig.5,thescores (v )ofv isnowequaltop(u,v1)×
u 1 1
((p(v ,w )+p(v ,w )+p(v ,w ))/3).
1 1 1 2 1 3
5.3 Thirdproposal:Robustnessapproach
Inthepreviousproposals,assoonasa2-hopneighborhasanon-nullprobabilitytobe
covered,itisremovedfromMPR′(u).Thisremovalisdoneevenwithaverylowprob-
ability,whichinthiscasemaybemeaningless.Itcanbemoreinterestingtoconsidera
2-hopneighborascoveredwhenitsprobabilitytocorrectlyreceivethebroadcastpacket
isoveragiventhreshold,inordertoincreasethedeliveryratio.
Wethusproposetokeepthescorecomputationusedinthepreviousheuristic,while
modifyinghow2-hopneighborsareremovedfromMPR′(u).Forsucha2-hopneighbor
wofu,itsremovalfromMPR′(u)isdoneonlyifitscoveragelevelt (w)isoveragiven
u
threshold.Thevalueoft (w)isgivenby:
u
i=|MPR(u)|
(cid:213)
t (w)=1− p(v,w), (7)
u i
i=1
p(v,w) being equal to 1−p(v,w). In simpler terms, the coverage level of a 2-hop
i i
neighborisequaltoitsprobabilitytocorrectlyreceivethepacketfromatleastoneof
thechosenrelays.
StillconsideringFig.5,ifthenodesv andv areselectedasrelays,thenthecover-
1 2
agelevelt (w )ofw isequalto1−(p(v ,w )×p(v ,w )).Severalrelayscanthusnow
u 3 3 1 3 2 3
beselectedtocoverthesamesetof2-hopneighbors,inordertoincreasethedelivery
ratio.