Bottlenecks and stability in networks with contending nodes

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International

Journal

of

Electronics

and

Communications

(AEÜ)

j o u r n al hom ep ag e :w w w . e l s e v i e r . c o m / l o c a t e / a e u e

Bottlenecks

and

stability

in

networks

with

contending

nodes

Tom

Coenen

a,∗

_{, Hans}

_van

_den

_Berg

b,c

_{, Richard}

_J.

_Boucherie

a

_,

_Maurits

_de

_Graaf

a,d

_,

_Ahmad

_Al

_Hanbali

e a_Department_of_Applied_Mathematics,_University_of_Twente,_Enschede,_Netherlands

b_TNO_Information_and_{Communication}_Technology,_Delft,_Netherlands

c_Department_of_Design_and_Analysis_of_{Communication}_Systems,_University_of_Twente,_Enschede,_Netherlands d_Thales_Nederland_B.V.,_Huizen,_Netherlands

e_Department_of_Management_and_Governance,_University_of_Twente,_Enschede,_Netherlands

a

r

t

i

c

l

e

i

n

f

o

Articlehistory: Received24August2010 Accepted14June2012 Keywords: Bottleneck Stability Contention

a

b

s

t

r

a

c

t

Thispaperconsidersaclassofqueueingnetworkmodelswherenodeshavetocontendwitheachother

toservetheircustomers.Ineachtimeslot,anodewithanon-emptyqueueeitherservesacustomeror

isblockedbyanodeinitsvicinity.Thefocusofourstudyisonanalyzingthethroughputandidentifying

bottlenecknodesinsuchnetworks.Ourmodelingandanalysisapproachconsistsoftwosteps.First,

consideringtheslottedmodelonalongertimescale,thebehaviorisdescribedbyacontinuoustime

Markovchainwithstatedependentservicerates.Inthesecondstep,thestatedependentservicerates

arereplacedbytheirlongrunaveragesresultinginanapproximateproductformnetwork.Thisenablesus

todeterminethebottlenecknodesandthestabilityconditionofthesystem.Numericalresultsshowthat

ourapproximationapproachprovidesveryaccurateresultswithrespecttothemaximumthroughput

anetworkcansupport.Italsorevealsasurprisingeffectregardingthelocationofbottlenecksinthe

networkwhentheofferedloadisfurtherincreased.

1. Introduction

Inspiredbywirelessadhocnetworkswhereinterference pro-hibitsneighboringnodestosimultaneouslytransmitpackets,this paperconsidersaclassofopenqueueingnetworkmodelsinwhich serverscontendforserviceslots.Ineachtimeslotnodesthathave packets available for transmission tryto obtainthechannel to transmittheirpackets.Asnodeswithineachothersinterference rangecannottransmitatthesametime,anallocationmechanism, i.e. a medium access controlprotocol, is used todecide which nodesgettheopportunitytotransmit,i.e.toserveapacket.Once aserverin anodeisallowedtotransmitapacket,itblocksthe serversinaspecifiedsetofothernodescorrespondingtoan inter-ferenceneighborhood.Uponservicecompletion,apacketeither movestoanextnodeforfurtherservice,orleavesthenetwork. The network is called stable when for each node the average service rate exceeds the averagearrival rate of packets. When multipleorlargeflowspassthrougha node,theservicerateof thenode may not suffice, making this node a bottleneck.This paperinvestigates thestability range, thearrival rates offlows atwhich nodesbecomebottlenecks, andthethroughputofthe network.

∗ Correspondingauthor.

E-mailaddress:t.j.m.coenen@utwente.nl(T.Coenen).

Thebehaviorofthesystemunderconsiderationcanbedescribed byastatedependentdiscretetime Markovchainasweassume thatineachslotthecontentionbetweennodestakesplace inde-pendentof previousoutcomes.Inspired byresultsobtainedfor loss-networks,wemakeatwostepapproximationtoanalyzethis network, see Fig.1. As a ﬁrst step, we considerthe long term averagebehavior, which neglects theeffect of theslottedtime andleadstoacontinuoustimeMarkovchain.However,withthe transitionrates ofthis chain stillbeingstate dependent, analy-sisremainscumbersomeandafurtherapproximationisneeded. Using alongtermaverageservicerate,weintroduce aproduct formnetworkapproximationwhichenablesustoﬁndthe bottle-necksinnetworksofarbitrarysizeandtopologyanddetermine themaximalthroughput.Interestingly,itturnsoutthatwhenthe loadofthenetworkisincreased,abottlenecknodecanbecome stable again as a different node becomes the bottleneck. This surprising behavior is predictedcorrectly by our product form model.

Theremainderofthepaperisorganizedasfollows.First,Section 2givesaliteratureoverview,afterwhichSection3introducesthe discretemodelandcontentionprocess.Section4describestheﬁrst approximationstepresultinginthecontinuoustimemodelwith statedependentservicerates,followedbythesecond approxima-tionstepinSection5.Section6givestheresultsforthestability analysisandSection7presentsresultsfromsimulationtoillustrate theaccuracyofthemodelpresentedinthispaper.Finally,Section 8concludesthepaper.

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Fig.1. Approximationsteps.

2. Literatureandcontribution

The stability of networks, as considered in this paper, has receivedconsiderableinterestintheliterature.Alsoinspiredby wirelessnetworks,[1]analyzesadiscretetimeslottedALOHA sys-tem.Boundsonthestabilityregionarefoundusingtheconcept ofdominance.Adifferentapproachispresentedin[2]wherethe ratestabilityandoutputratesarecalculatedforsharedresource networks.Stabilityconditionsforseparatenodesarederivedfor generalallocationfunctionsundermildassumptions.Themodel discussedin this paper however doesnot fallunder theset of allocation functions, as the overall capacity of the network is notconstant.Foranetworkofparallelserverswithcoupled ser-vice rates, necessary and sufﬁcient conditions for stability are derivedin[3].Stabilityandperformanceofnetworkswherethe serviceratedependsonthenetworkstateisalsoanalyzedin[4], where transmissions over links with a ﬁxedcapacity are con-sidered. Opposed to the work presented in these papers, the rate allocated to a server does not depend on the number of packetspresentinthequeue,butonthenumberofnodes com-peting.

Similarassumptionsregardingthecontentionbetweennodes, wherealivenodesblockothernodesasdiscussedinthispaperare madein[5].Thethroughputinamultihoptandemnetworkis con-sideredbothundersaturation,whereeachnodegeneratesitsown trafﬁcandunderasingleﬂowoverallnodes.Theauthorsconjecture thatarandomaccessschemeseverelydegradesthethroughputof thenetwork.

Analyticresultsforamultihopnetworkwithtwocontending queuesarepresentedin[6].UsingthetheoryofRiemann–Hilbert boundaryvalueproblems,thegeneratingfunctionofthe station-arydistributionisobtained.In[7]someperformancemeasuresof thissystemareanalyzed,focussingonthecomputationalissues thatoccur.Evenforsuchasmallnetworkasconsideredinthese papers,acomplexanalysisisneededtoobtainanalyticalresults. Theapproachwepresentisapplicableforgeneralsizenetworks, howeverwedonotobtainresultsonthestationarydistribution, butonstabilityandthroughput.

Theoptimalthroughputanetworkcansupport,oftenreferredto asthecapacityofthenetwork,isdiscussedin[8],whichhowever doesnotfocusonmulti-hopnetworks.Thisaspectisaddressed in[9],whereforasinglemulti-hopflowanewcapacitylimitis derived.Theseresultsarelimitingresultsforlargenetworks.More detailedmodelsarediscussedin[10]foratandemandlattice net-workwithsaturatednodes.Theycalculatetheoptimalofferedload, preventingpacketlossinanetworkwithhiddennodes.Thisworkis extendedformultiplecrossingflowsin[11].Insteadoffocussingon thespecificparametersoftheMACprotocol,aspresentedinthese papers,wetakeahigherlevelview,providingvaluableinsightsfor generalnetworks.

Nexttolimitingthecapacityofanetwork,contentionbetween nodeshasanimpactonthefairnessofprotocols,asintheequality inrateallocatedtonodesorthethroughputofﬂows.In[12]the authorsdescribethebordereffectsinaCSMA/CAnetworkandits impactonfairness.Thestabilityandthroughputforaweightedfair queueingmodelwithsaturatednodesisdiscussedin[13]showing thatthethroughput,whiletakingintoaccountthetopology,routing andrandomaccessintheMAClayer,inthissettingdoesnotdepend

ontheloadintheintermediatenodesaslongasthenetworkis stable.

Differentaspectsofimportanceforthestabilityandthroughput ofnetworkshavealsoreceivedmuchattention.Focussingonthe impactofrouting,[14]investigatesthestability andthroughput ofstaticwirelessnetworkswithslottedtime.Theauthorsshow that routing hasa large impactonthe stability properties and thataslongastheintermediatequeuesinanetworkarestable, thethroughputdoesnotdependonthetrafﬁcgeneratedatthese intermediatenodes.In[15]thefocusisonthecalculationofthe interferencetonoiseratioandshowtheinﬂuenceofthenetwork sizeandthedatarateonthisratioandlinkthistothethroughput ofthenetwork.

Thecontributionofthis paperis thatwe providea compre-hensiblemodelthatveryaccuratelypredictsthebottlenecksand maximal throughputof a network, which alsois applicablefor networkswithunstablenodes.Theresultsprovideinsightinthe impactthatcontentionbetweennodeshasontheperformanceof thenetwork,withouttheneedofacomplexanalysis.

3. Discretetimemodel 3.1. Generalmodel

Consideranetworkconsistingofnqueueswithinfinitebuffers. Duetocontentionbetweennodesnotallnodescantransmittheir packetsatthesametime.WedefinethecontentionsetI(i),i=1,..., n,ofanodeiasthesetofnodesblockedfromtransmissionwhen nodeiistransmitting.Atypicalexampleisthesetofnodeswithina certaininterferencerange.However,forthemodelthereneednot bearelationbetweenthenetworkstructureandthecontention set.Thewaycontentionbetweennodestakesplacewillbe elabo-rateduponbelow.AsetofJtrafficflowsf(tj),j=1,...,J,travelover

multihoppaths,denotedbytheorderedsetstj,fromnodetj(1)to

tj(mj),whereweassumethatnoloopsaremadewithinapath,i.e.

pathsaresimpleandpacketsautomaticallyfollowtheirpath. Traf-ficconsistsofequallysizedpacketsthataretransmittedonepacket pertimeslot.AnexampleofsuchanetworkisdepictedinFig.2.In thisexampleanetworkof8nodesisdepicted.Inthefigurethere arethreeflows:f(t1)fromnode1throughnode2tonode3(i.e.we

havethatt1={1,2,3}),f(t2)fromnode1throughnodes4and6to

node8andf(t3)fromnode7throughnodes6and5tonode3.

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firstcomefirstserved.Anodeiscalledstablewhenitsaverage ser-vicerateexceedstheaveragearrivalrateofpacketsatthenode, andanetworkiscalledstablewhenallitsnodesare.Anodethatis unstableiscalledabottlenecknode.Theaveragenumberof pack-etsofaflowthatreachthedestinationnodepertimeunitisthe throughputofthisflow,whichislimitedbytheservicerateofthe bottlenecknodesofthenetwork.Themaininterestinthispaper isthethroughputoftheflowsandtheidentificationofbottleneck nodes.

Anodeiscalledalivewhenithaspacketstotransmitandthus participatesinthecontention.Ineachtimeslot,allalivenodes con-tendtobeallowedtotransmitapacket.Theprobabilityofanode beingallowedtotransmitdependsonthesetofnodes contend-ing,whichwefocusoninthefollowingsection.Ineachtimeslot anodeiseithernotcontending,blockedorallowedtotransmit.In eachtimeslotthisprocessisrepeated,whereweassumethe selec-tionofnodesbeingallowedtotransmittobeindependentbetween timeslots.

Letpidenotetheprobabilitythatnodeiisaliveandletbea

livelinessvector,suchthati=1ifnodeiisaliveandi=0

other-wise.Thesetofall2n_possible_liveliness_vectors_is_denoted_by_˘.

Theprobabilitythatalivelinessvectoroccursisdenotedbyq.

Theprobabilitythatnodeitransmitsunderlivelinessvectoris denotedbyri,.Thenetworkcanberepresentedbyadiscretetime

Markovchainwiththequeuelengthsateachnodeasthestateofthe system.Actually,aspacketsareforwardedtoanextnode depend-ingontheﬂowtheybelongto,alsothetypeofthepacketsinthe queueneedstobeincludedinthestatedescription.However,in oursteadystatedescriptionthesetypeswillnotplayaroleandare thereforeomittedfromthestatedescription.Thetransition prob-abilitiesdependonthestateofthesystemviathelivelinessvector only,i.e.thenumberortypeofpacketsinaqueuedoesnotaffect theprobabilityofanodetransmittinginaslot,unlessitisempty.

3.2. Contention

Multiplenodescanonlybetransmittingsimultaneouslyinthe sametimeslotwhentheyareoutsideofeachotherscontention set.Ifmultiplenodeswithineachotherscontentionsetarealive, thecontentionprotocoldecideswhichnodesmaytransmit.The probabilitythatanodeisallowedtotransmitapacketinthe follow-ingslotcanbedeterminedwhenthecontentionsets,theprotocol inuseandthecompetingnodesareknown.Weassumeanideal contentionprotocol,wherenocollisionswilloccurandhenceno packetswillbelost.

Aswearenotinterestedinthedetailsofthecontentionprotocol butonlythecorrespondingprobabilitiesfornodestotransmit,we willuseasimpleprotocolgivingeachnodeaninitialequal prob-abilityofwinningacontention.Forotherprotocols,transmission probabilitiescanalsobecalculated.Using

||=

n

i=1

i, (1)

i.e.||equalsthenumberofalivenodesand(takingemastheunit

vectoroflengthn,withallzerosexcepta1onlocationm), ˜

(k)=−

m∈I(k):m=1

em (2)

asthelivelinessvectorremainingafteranodekblocksallnodes initscontentionset(asitwonthecontention),theprobabilityri,

ri,=

⎧

⎪

⎨

⎪

⎩

0 for i=0 (1+

k/=i:k=1 ri, ˜(k))/|| for i=1 (3)

withri,0=0,where0denotesalivelinessvectorwithnoalivenodes.

Thiscanbeseenasfollows:withequalprobabilityof1/||any non-emptynode(sonodeiitselforanyotheralivenodek)winsthe directcontention.Assumingnodekwinsthecontention,itblocks allnodesinitscontentionregion,reducingthelivelinessstateto

˜

(k),afterwhichallremainingnodescontendagain.Anynodethat didnotwinthecontention,butwasnotblockedhencecancompete againandmightwinthenewcontention,withprobability1/| ˜(k)|. Thisprocesscontinuesuntilallnon-emptynodeseitherareallowed totransmitorareblocked.

Asanexample,considerthenetworkasdepictedinFig.2with contentionsetschosensuchthatnearbynodescontend:I(1)={2,

4},I(2)={1},I(4)={1,5,6},I(5)={4,6},I(6)={4,5,7},I(7)={6}.

Assumethatall6nodeshavepacketstotransmit(asnodes3and 8donottransmitpacketstheyareneveralive),sothat=(1,1, 0,1,1,1,1,0).Theprobabilityr4, thatnode4 willbeallowed

totransmitbydirectlywinningthecontentionis1/||=1/6.Iffor examplenode1winsthecontention,node4isblocked,asitisin itscontentionset.As ˜(1)=(0,0,0,0,1,1,1,0),wegetr4, ˜(1)=0

as ˜4(1)=0.Thesameholdsifnode5or6winsthecontention,

asr4, ˜(5)=0and r4, ˜(6)=0.Ifnode2or7wins thecontention,

node4stillcouldbeallowedtotransmit.Theprobabilitythatnode 4winscontentionafternode2haswonthecontentionisgiven byr4, ˜(2),where ˜(2)=(0,0,0,1,1,1,1,0).Thisprobabilitycan

becalculatedbycalculatingr4,,butwith=(0,0,0,1,1,1,1,0),

showingtherecursion.Asaftereachstep,butwiththenewvalue of,thenumberofzeroesinthelivelinessvectorincreases,the recursionwillstopwhen=(0,0,0,0,0,0,0,0).Forthisexample, theprobabilitythenodesmaytransmitaregivenby[19/48,29/48, 0,14/48,20/48,19/72,53/72,0].Wefurtheranalyzethisnetwork inSection7.2.

Notethattheoverallprobabilityofbeingallowedtotransmit is not equal for all nodes. A similar analysisto obtainri, can

bedoneforanynetwork,withanycontentionsetsandprotocol. More extensivecalculationswillbe neededforlargernetworks withdifferenttopologies,buttheprinciplewillnotchange.Inthe remainderofthispaper,wewillassumethatthecontentionregions andprotocolareknown,suchthatallconditionalratesri, ofthe

nodescanbecalculated.

4. Approximationstep1:continuoustime

Weareinterestedinthelongtermaveragebehaviorofthe net-work,especiallythethroughputandstabilityissues.Considering thesystemonahigherlevelandalargertimescale,thediscrete characterduetothetimeslotsfadesandthemodelcanbeseen asacontinuoustimeMarkovprocess.Thestateofthesystem con-sistsofthenumberandtypeofpacketsateachqueue,butasthe stateofthesystemonlyinﬂuencesthetransitionratesthroughthe livelinessofthenetwork,wedonotfocusonthequeuelengths. Theﬂowapacketbeingservedbelongstodeterminesthedirection inwhichitwillbeforwarded.Weincorporatethisintothemodel asdescribedbelow.Inthefollowingwewilldenoteparameters usedforthecontinuoustimeapproximationbyaddingahattothe equivalentparameterintheoriginaldiscretetimemodel.

Whenaqueuehaspacketsavailableandthelivelinessisgivenby ,theprobabilityofapacketbeingsentisgivenbyri,.Onaverage,

thenumberofpacketssentperslotunderstatehenceisri,.For

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understateofthenodeusingtheexponentialdistributionwith rate ˆri,=ri,.Theprobabilityofanodebeingaliveornotdepends

onthearrivalrateofpacketsandtheservicerateatthenode.We ﬁrstfocusonthearrivalrateofpackets.

Wheneverthenodesonamultihoppathtj precedinganode

tj(i)arestable,thearrivalratefromthisﬂowwillbej,theexternal

arrivalrateoftheﬂow.Thetotalarrivalrateoftrafﬁcaiatnodeiis

givenby ai=

j:i∈tj

j(i) (4)

wherej(i)is thearrivalrateatnodeiforﬂow f(tj).Whenthe

networkisstable,thissimpliﬁestoai=

j:i∈tjj.Whenthereare

unstablenodesinthenetwork,thearrivalrateofpacketsateach queuecanbedeterminedasfollows.Duetothemultihopfeed for-wardstructureofthenetworkwehavethatthearrivalratej(i)is

determinedbyitsprecedingnodes.Ifoneormoreofthe preced-ingnodesareunstable,theaveragearrivalrateforthenodesafter thebottleneckonthispathwilldependontheservicerateofthe unstablenodes.Theprobabilityp_t_j_(i−1)t_j_(i)thataservedpacketat nodetj(i−1)continuestonodetj(i),thepacketisofﬂowf(tj),is

givenby ptj(i−1)tj(i)=

j(tj(i−1))

a_t_j_(i−1) . (5)

Thearrivalratej(tj(i))fromﬂowf(tj)atnodetj(i)isgivenby

j(tj(i))=min(j(tj(i−1)),ptj(i−1)tj(i)ˆrtj(i−1)), (6)

wherej(tj(1))=j,theexternalarrivalrateofpacketsattheﬁrst

nodeinpathtj.Thiscanbeseenasfollows:eitherthepreceding

nodecanserveallitsincomingtrafﬁc,oritsservicerateistoolow. Inthelattercase,thefractionoftheservicerateofnodetj(i−1)that

isusedforﬂowf(tj),equaltoptj(i−1)tj(i),determinesthearrivalrate

atthenextnodeforthisﬂow.Here ˆrtj(i−1)denotestheaveragestate

independentservicerateofnodetj(i−1),whichwillbedetermined

inthenextsection.Assumingthisrateisknown,Eqs.(5)and(6)give asystemofequationsthatcaneasilybesolved,givingthearrival rateperﬂowateachnode.Weusethesearrivalratesintheanalysis ofthelivelinessofthesystem,whichinﬂuencestheservicerateof thenodes.

5. Approximationstep2:productformnetwork

TheMarkov chainwith statedependentservice rates is not amenibleforanalysis.Foranetworkwithonlytwoqueuesin tan-dem,thisequalsthemodelpresentedin[6]underdeterministic servicetimes.Evenforsuchasmallnetwork,acomplexanalysis isneededtoobtainanalyticalresults.Therefore,foranarbitrary network,weapproximatethecontinuoustimeapproximationby obtainingan appropriatestate independentservicerateforeach nodetoanalyzethebehaviorofthenetwork.

Thestateindependentservicerate ˆriisobtainedbyconsidering

thelongtermaveragepercentageoftimethesystemisinastate withlivelinessvector.Theprobabilityofnodeibeingaliveisgiven by

ˆpi=min(

ai

ˆri

,1). (7)

Fortheﬁnalapproximationstep,let ˆq denotethesteadystate

probabilitythatthelivelinessvectoris(tobecalculatedlater)

andassumethestateindependentaverageservicerateofanodei inthenetworktobegivenby,

ˆri=

∈˘

ˆri,ˆq

ˆp_i (8)

WeobtainEq.(8)byconsideringalargetimescaleandweighingthe servicerateoverthepossiblelivelinessofthesystem,i.e.by uncon-ditioningontheliveliness,butconditioningonthenodebeingalive. Thestateindependentservicerate ˆricanbeseenastheaveragerate

atwhichanodeservicespackets,giventhatitisalive.

Theorem1. Thesteadystateprobability ˆqthatthesystemisina

statewithlivelinessvectorisgivenby ˆq= n

i=1 (1− ˆpi)(1−i)ˆpii. (9)

Proof. Summarizingtheabove,wehavethefollowing assump-tionsforthestateindependentcontinuoustimeapproximation: 1.TheexternalarrivalprocessoftrafﬁcatqueuesisaPoisson

pro-cess.

2.Thereisinﬁnitewaitingspaceatallthequeues.

3.Theservicetimeatthequeueshasanexponentialdistribution andisindependentofthestateofthesystemandarrivalprocess. 4.Aftercompletionofserviceatqueueiapacketinstantaneously movestothenextqueuekwithprobabilitypik,k=1,...,n,for

additionalserviceorwithprobabilitypi0thepacketcompletes

serviceandleavesthesystem,wherewehavethat

n_k=0p_ik₌1. Theroutingprobabilitiesareindependentofthehistoryofthe system.

Anetworkforwhichtheassumptions(1)–(4)holdisaproduct formnetwork(c.f.[16]).Hence,theprobabilityofacertainstateof thesystemoccurringistheproductoftheprobabilitiesofnodes containingacertainnumberofpackets.Asthestateofthesystem directlyimpliesacertainliveliness,alsothelivelinessvectorcanbe foundastheproductofthelivelinessofseparatenodes,showing (9)holds.

WewillnowuseEqs.(4)–(9)asanapproximationforthe dis-cretetimemodel.Thisrathercoarseapproximationwillprovide quiteaccurateresults,asweareinterestedintheinﬂuenceofthe loadonaveragebehaviorofthenetwork.

6. Stability

Theaverageservicerate ˆriatwhicheachnodeoperates

deter-minestheloadunderwhichthenetworkisstable.Aspresented earlier,theaverageservicerateofanodeiisgivenby(8)andthe probabilitythatanodeisaliveby(7).Writingouttheexpression for ˆq andinserting(7)into(8),weobtainnequations,with2n

unknowns,whicharetheaiand the ˆri.Assumingthatallnodes

arestable,thatiswhenallai< ˆri,thearrivalrateateachnodeis

known.Thevaluesfor ˆricanhencebecalculatedforastablesystem.

However,itisstilltobedeterminedforwhichvaluesofj(andthus

ai)thenetworkisstable.

AspresentedinSection4,thearrivalrateforacertainﬂowjat nodetj(i)isgivenby(6)andthetotalarrivalrateby(4).Usingthen

Eq.(4)forai,itispossibletosolvethesystemof2nunknown

vari-ables,whichentailssolvingpolynomialsofdegreesthatincrease exponentiallywiththenetworksize. Solutionscan beobtained numerically,however,usingforinstancethefollowingalgorithm toobtainthevaluesof ˆri.

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1.Setallvalues ˆrito1,i=1,...,n 2.Calculatej(k),j=1,...,Jandk=tj(1),...,tj(mk) 3.Calculateai=

j:i∈tjj(i),i=1,...,n 4.Calculate ˆpi=min(ai/ˆri,1),i=1,...,n 5.Calculate ˆq=

n_i₌₁(1− ˆpi)(1−i)ˆp_ii,∈˘ 6.Calculatenew ˆri=

∈˘(ˆri,ˆq/ˆpi),i=1,...,n

7. Calculatethedifference

i= ˆri(new)− ˆri(old),i=1,...,n

8. Repeatstep2till7untilconvergenceoccurs,thatis|

|≤ıforan appropriatevalueofı.

WehavenumericallyestablishedthatAlgorithm2convergestoa uniquesolution ˆriforanyvaluesofj,j=1,...,J.Forananalysisof

thealgorithmwereferAppendixA.

UsingAlgorithm2,theservicerateofallnodescanbecalculated foranysetofﬂowsthroughthenetwork.Thecorrespondingarrival ratesatthedestinationnodesoftheﬂowsgivethethroughputof thenetwork.Wheneverthenetworkisstable,thetotal through-putwillequal

jj.Forageneralnetwork,thecalculationofthe

throughput,independentofthetopologyofthenetwork,involves solvingnequationsinnunknowns.UsingAlgorithm2,thearrival rate(s)canbechosenarbitrarily.Todeterminethestabilityrangeof thenetwork,weseparatelyconsidereachflowinthenetwork. Fix-ingthearrivalratesofallbutoneflow(suchthatthesystemwith theseflowsisstable),thereexistsavalueopt fortheremaining

ﬂowsuchthatak= ˆrkforatleastonek∈1,...,n,whichprovidesthe

maximalthroughputoptofthisﬂow.Nodekisthenthebottleneck

ofthenetwork.Inthismannerthestabilityrangeofthenetwork canbecalculated(examplesareshowninthefollowingsection). 7. Examplesandvalidation

7.1. Multihoptandemnetwork

Inthefollowingweanalyzeamultihoptandemnetwork.When consideringageneralnetwork,theanalysisofthestabilityregion involvesconsideringﬂowsseparately.First,weshowhowfora spe-ciﬁccontentionprotocolthetransmissionprobabilitiesri,canbe

calculatedinthisnetwork,whichcorrespondstoasingle multi-hoptransmissioninanetwork.Next,weusesimulationtovalidate resultsobtainedbyouralgorithmfordifferentsizesofthenetwork. Somesurprisingresultsareobtained,whicharecorrectlypredicted byourmodel.

work.Asindirectlyallnodesinthenetworkinﬂuenceeachother, thetotallengthofthenetworkhasanimpact.Thisimpactwhen allnodesarealiveisshown,usingacontentionprotocolselecting anodetotransmitwithequalprobabilityamongallalivenodes.

Considerthetandemnetworksuchthatnodescannottransmit andreceiveatthesametime.Anodethatisallowedtotransmit henceblocksitsdirectneighbor(s).Whenallnnodesarealive,each nodehasaprobability1/nofobtainingthechanneldirectlyand blockingitsneighbor(s).Theremainingnodescontinuecontending forthechanneluntiltheyareeitherblockedorallowedtotransmit. Therateri,1(n)foranodeatpositioniinafullyalivetandemnetwork

oflengthncanbecalculatedusing

ri,1(n)= 1 n

_i₋₂

k=1 r_i−k−1,1(n−k−1)+1+ n

k=i+2 ri,1(k−2)

. (10)

Therighthandsideof(10)followsfromthenodewinningthe con-tention:iftheﬁrstnodein thenetwork winsthecontention,it blocksthesecondnodeandtheremainingn−2nodescompete, withnodeinowatpositioni− 2.Otherwise,inasimilarmanner, anodebefore(butnotaneighboring)nodeiwinsthecontention, nodeiwinsthecontentionitself,eitherofnodei’sneighborswins thecontentionoranodekbehindnodeiwinsthecontention.Each oftheseeventsoccurswithaprobabilityof1/n,togethergivingthe recursiveformula.

Note that amultihop tandemnetwork (in thissetting) with nodesthat arenotalive canbedecomposedintomany smaller multihopnetworks.Forafullyalivetandemnetworkwherenodes cannottransmitandreceiveatthesametime,Table1showsthe ratesfordifferentlengthsofthenetwork.

Theorem3. Forthemultihoptandemnetworkwithallalivenodes, the rate allocatedto the nodes converges when the networksize increases,whereinparticular

lim n→∞r1,1(n)=1− 1 e and nlim→∞r2,1(n)= 1 e. (11)

Proof. FortheproofwerefertoAppendixA.2.

OtherlimitsareobservedinTable1,showingthattheborder effects fadefor themiddle nodesas thelengthof thenetwork increases,inaccordancewith[12].Thisbordereffectalreadystarts tofadefornetworksofsize12.

Wenotethatthecalculationoftheratesri,forthelinear

set-tinghasthepleasantpropertythattherateofacertainnodeiunder livelinessisonlydependentonthenumberofnodesthatarealive anddirectlyconnectedtoeachother.Whenconsideringdifferent Table1

Transmissionprobabilityforafullyalivetandemnetwork.

Size/node 1 2 3 4 5 6 7 8 9 10 11 12 1 1 – – – – – – – – – – – 2 0.5 0.5 – – – – – – – – – – 3 0.6666 0.3333 0.666 – – – – – – – – – 4 0.625 0.375 0.375 0.625 – – – – – – – – 5 0.6333 0.3667 0.4667 0.3667 0.6333 – – – – – – – 6 0.6319 0.3681 0.4444 0.4444 0.3681 0.6319 – – – – – – 7 0.6321 0.3679 0.4488 0.4262 0.4488 0.3679 0.6321 – – – – – 8 0.6321 0.3679 0.4481 0.4297 0.4297 0.4481 0.3679 0.6321 - – – – 9 0.6321 0.3679 0.4482 0.4291 0.4334 0.4291 0.4482 0.3679 0.6321 – – – 10 0.6321 0.3679 0.4482 0.4292 0.4328 0.4328 0.4292 0.4482 0.3679 0.6321 – – 11 0.6321 0.3679 0.4482 0.4292 0.4329 0.4322 0.4329 0.4292 0.4482 0.3679 0.6321 – 12 0.6321 0.3679 0.4482 0.4292 0.4329 0.4323 0.4323 0.4392 0.4292 0.4482 0.3679 0.6321

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0 2 4 6 8 10 12 14 16 18 20 0.3 0.35 0.4 0.45 0.5 0.55 0.6 Arrival rate

Average queue length

Node 1 Node 2 Node 3

Fig.3.Simulatedaveragequeuelengthina3hopnetworkwiththeinstabilityratescalculatedbythemodel.

contentionsetsandprotocolornetworklayout,thisproperty how-evermaynolongerbepresent.

Tovalidatetheresultspresented in thispaper, a simulation modelhasbeenconstructedthatmimicsthebehaviorofthe dis-cretetimenetworkunderconsideration.Thearrivalandprocessing ofthepacketsismodeled,withasimulationforeachparameter settinglastingonemillionsimulatedtimeslotsafterawarmup periodof100.000slots.Theresultsarecomparedwiththe stabil-ityrangesandthethroughputofthenetworkcalculatedwiththe stateindependentcontinuoustimeapproximation,usingthe pro-videdalgorithm.Forsomesettings,weprovidetheexactderivation oftheresults.

Considerthemultihoptandemnetworkforn=3.Theaverage serviceratesatwhichthenodesoperatearegivenby(using(8) and(9)) ˆr1=(1− ˆp2)+ 1 2ˆp2(1− ˆp3)+ 2 3ˆp2ˆp3 (12) ˆr2=(1− ˆp1)(1− ˆp3)+ 1 2ˆp1(1− ˆp3)+ 1 2(1− ˆp1)ˆp3+ 1 3ˆp1ˆp3 ˆr3=(1− ˆp2)+ 1 2(1− ˆp1)ˆp2+ 2 3ˆp1ˆp2.

Obviously,thesecondnodewillbethebottleneckofthenetwork as ˆr2 issmallerthan ˆr1 and ˆr3,asitistheonlynodecontending

withtwoneighbors.Whennode2isunstable,wehavethat ˆp2=1.

Todetermineatwhatarrivalratethiswilloccur,weusethat =ai= ˆr2,sothat ˆp1= ˆr2 ˆr1 and ˆp3= ˆr2 ˆr3 . (13)

CombiningEqs.(12)and(13)with ˆp2=1weﬁndthat ˆp1= ˆp3=

(9−√57)/2, resultingin thecritical arrivalrate of= ˆr2=8−

√

57.Fromthis valueofonthesecondnodewillbeunstable. Ifweincreasethearrivalrateevenmore,thefirstnodewillalso becomeunstable.Thethirdnodehoweverwillalwaysremain sta-ble,asitsserviceratewillalwaysbehigherthantheservicerate atthesecondnode,whichdeterminesthearrivalrateatthethird node.Tofindfromwhichvalueofonthefirstnodewillalsobe unstable,wesubstitute ˆp1= ˆp2=1in(12)whichleadsto ˆp3=0.6,

andtherateatwhichnode1becomesunstableequals= ˆr1=0.6.

Alsonotethattherateofthesecondnodehasnowfallentoavalue of ˆr2=0.4,sothatthethroughputofthenetworkhasdecreased.

Forthethreenodetandemnetwork,Fig.3showstheaverage queuelengthatthethreenodesforincreasingloadofthesystem andFig.4showsthethroughputofthesystem.Thecalculated val-uesofarrivalratesforwhichqueuesbecomeunstablearedepicted asdottedverticallinesintheﬁgures.

AscanbeseeninFigs.3and4,thearrivalratesatwhichthe firstandsecondnodebecomeunstablecoincidewiththe calcu-latedvalues.Additionalsimulationsforthearrivalratesnearthe onescausinginstabilityofnodeswereperformedtoconfirmthe results,butarenotshowninthefigurestomaintainreadability.The throughput,whichreachesamaximumof8−√57≈0.4501when thesecondnodebecomesunstable,decreasesafterthisvalue.This decreaseinthroughputiscausedbythedecreaseinservicerate atthesecondnode,asthefirstnodebecomesmorehighlyloaded. Thiscausesthefirstqueuetobealivealargerfractionofthetime, blockingthesecondnode.Thethroughputsettlesat0.4afterthe firstnodehasbecomeunstableatanarrivalrateof0.6,whichisin agreementwiththevaluescalculated.

0.3 0.32 0.34 0.36 0.38 0.4 0.42 0.44 0.46 0.48 0.5 0.3 0.35 0.4 0.45 0.5 0.55 0.6 Arrival rate Throughput

Simulated throughput Calculated throughput

(7)

0 2 4 6 8 10 12 14 16 18 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 Arrival rate

Average queue length

Fig.5. Simulatedaveragequeuelengthina5hopnetworkwiththeinstabilityratescalculatedbythemodel.

Next, considering a larger network with 5 hops, one might expectthatitisthesecondnodethatbecomesthebottleneck.Using thepresented modelandsetting ˆp2=1howevershowsthat no

realvalued solutionexists,meaning thatnode2 cannotbethe nodetobecomeunstableﬁrst.It actuallyisthethirdnodethat becomesthebottleneckﬁrstatanarrivalrateof0.4323,whichis themaximumthroughputofthenetwork.Increasingthearrival rateto0.4448causesthesecondnodetobecomeunstableaswell. Increasingthearrival ratefurther, thethird nodebecomes sta-bleagain.Thepresentedmodelalsodeterminesthearrivalrateat whichthisoccursbymakingasmalladjustmenttotheequations. Asthethirdqueuewillbecomestableassoonasitsaverageservice rateislowerthanthesecondqueue’srate,wenowset ˆr2= ˆr3.As

bothqueuesarestillunstablewehavethat ˆp2= ˆp3=1andthat

ˆp4= ˆr2/ˆr4and ˆp5= ˆr2/ˆr5.Usingthestandardequationsforthe ˆri’s

andsetting ˆp1=/ˆr1,wesolvethesystemtoobtain=0.4803

and ˆr2= ˆr3=0.4306.Finallyincreasingthearrivalrateto0.6108

causestheﬁrstnodetobecomeunstable,resultinginathroughput of0.3892.Simulationofthenetworkunderconsiderationprovided theresultsaspresentedinFigs.5and6wheretheverticallines showthecalculatedvaluesforwhichnodesbecome(un)stable.

Thatitisqueue3thatistheﬁrstnodetobecomeunstablecan becalledsurprising.Whenallqueuesarealive,theaverageservice rateofqueue2islowerthanthatofqueue3.However,whenqueue 1and/orqueue5areempty,thethirdqueuehasthelowestrate (seeTable1fora3–5nodenetwork).AscanbeseeninFig.5,the averagequeuelengthatnodes1and5arelowfortheloadwhen queue2and3arealreadyreachinginstability.Thisindicatesthat theyfrequentlywillnotbealive,whichisinthedisadvantageofthe thirdnode,makingitthebottlenecknode.However,asthearrival rateincreases,nodes1and5willbealivemoreoften,whichis

beneﬁcialfornode3,resultinginthequeuebecomingstableagain. Surprisingasthisbehaviormaybe,itispredictedcorrectlybythe model.

7.2. Generaleightnodenetwork

ConsiderthenetworkasdepictedinFig.2.Notethatanysetof interferingnodescanbeused,mimickingthebehaviorofanyaccess controlprotocol,i.e.tomimicanRTS/CTSprotocolallnodeswithin transmissionrangeofthesendingandreceivingnodecanbeused asthecontentionset.Toavoidtrivialresultswesettheinterference rangesforthisexampletobe(onlyshowingthenodesthatneedto transmit)I(1)={2,4},I(2)={1},I(4)={1,5,6},I(5)={4,6},I(6)={4,

5,7_},I(7)=_{6}.Firstﬂowf(t1)issetup,withrate1=0.1.Obviously

thenetworkcanhandlethisﬂow.Second,ﬂowf(t3)issetup,with

rate3=0.1aswell.Again,thenetworkremainsstable(notethat

eventhoughbothﬂowshavenode3asendpoint,thisdoesnotcause problemsasweassumeperfectreceptionofalltransmissions).Now ﬂowf(t2)isinitiatedandtheopenquestioniswhichratecanbe

achievedforthisﬂow.Thearrivalratesoftrafﬁcatthenodes,as longasthenetworkisstable,isgivenby

Node 1 2 3 4

Arrivalrate 2+0.1 0.1 0.2 2

Node 5 6 7 8

Arrivalrate 0.1 2+0.1 0.1 2

andtheprobabilitiesqofallpossiblelivelinessvectorscaneasily

becalculated.UsingthesevaluesinEqs.(7)and(8)givesasetof8 equationswith9unknowns(allthe ˆriand2),whichcanbesolved

whenitisknownwhichnodebecomesthebottleneck.Using2= ˆri

0.3 0.32 0.34 0.36 0.38 0.4 0.42 0.44 0.46 0.48 0.5 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 Arrival rate Throughput

Simulated throughput Calculated throughput

(8)

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.3 0.35 0.4 0.45 0.5 0.55 0.6

Arrival rate flow 2

Throughput

Flow 1 Flow 2 Flow 3 Flow 1 sim Flow 2 sim Flow 3 sim

Fig.7.Simulatedandcalculatedthroughputinthe8nodenetworkwiththeinstabilityratescalculatedbythemodel.

fori=1,4,6andsolvingshowsthatitisnode4thatbecomesthe bottleneckatanarrivalrateof2=0.3789.Increasingthearrival

rate2furthercausesnode1tobecomeunstableaswell,

influ-encingthethroughputofthefirstflow.Usingthemodel,this is calculatedtohappenatanarrivalrateof2=0.5092.Fig.7shows

thethroughputoftheseparateﬂowsforanincreasingarrivalrate ofthesecondﬂow.Boththevaluescalculatedbythemodelandthe simulationresultsareshown.

Numericalevaluationshowsthatthemodelgivesveryaccurate predictionsofthethroughput,wheretheerrorateachcalculated pointstaysbelow1%.Theloadatwhichnode1and4become unsta-blecanberecognizedasthepointswheretheslopeofthegraph changes,wherethesimulationagainshowsthatthisisatthepoint predictedbythemodel.

8. Conclusion

Inspiredbywirelessadhocnetworks,whereinterference lim-itsthecapacity,networkswithcontendingnodesareanalyzedin thispaper. Eachtime slot,nodes competetotransmita packet fromtheirqueue, where awinning nodeblocksothernodes in itsneighborhood.The timeslot systemisapproximatedin two steps.First,byconsideringthelongrunaveragebehaviorofthe discretetimesystem,acontinuoustimemodelisobtained.Asthe secondstep,appropriate stateindependentserviceratesforthe nodesinthenetworkaredetermined.Combiningrelationsbetween thearrivaland serviceratesof thenodes,bottlenecknodesare identiﬁedwhichdeterminethethroughputofamultihopwireless network.Usingthetworathercoarseapproximationsteps,we pro-poseaproductformnetworkapproximation.Takingadvantageof thepropertiesofproductformnetworks,equationsforthe liveli-nessvector(whethernodeshavepacketsintheirqueuesornot)and theaverageserviceratesofthenodesarederivedandsolvedusing asimplealgorithm.Surprisingly,thecontinuousapproximationfor thelongtermaveragebehaviorturnsouttogiveaccurateresults concerningthestabilityandthroughputofthenetwork.Other per-formancemeasures,asthequeuelengthandwaitingtime,havenot beenconsidered.

Our approach provides very accurate resultsfor the lowest arrivalrateofaflowatwhichoneofthenodesbecomesunstable, thusgivingthemaximalthroughputfor thisflow.Also, increas-ingthe arrival rate further, instabilityof the restof the nodes isanalyzed.Ourmodelcorrectlypredicts surprisingbehavior in amultihoptandemnetwork,whereaqueueatfirstturningout tobethebottleneck,returnedtostabilityagainafterincreasing thearrivalrate.Theapproachpresentedisapplicableforgeneral networks,withvariouscontentionsettingsandprotocols.Using simulationsofthediscretetimesystem,resultswerecompared

withthecontinuoustimemodel,showingthatthemodelprovides veryaccurateresults.

AppendixA.

A.1. AnalysisofAlgorithm2

ToanalyzetheconvergenceofAlgorithm2,weconsiderthe sep-aratestepsandtherecursion.Theinitialvalueof ˆri=1corresponds

toanetworkwithoutcontention,immediatelygivinganindication whetherthenetworkisstableornot.Tocalculateallj(k)’sinstep

2,Eqs.(6),(4)and(5)needtobecombined,givingJmequations withequallymanyunknownvariableswhichcanbesolved.From thesevalues,obviouslysteps3–6canbecalculated,leadingtothe recursion.

Letg(r)denotethefunctionthatcalculates thenewvalueof rusingthestepsdescribed.Thefunctiong(·):Rn_→_Rn_is_a

con-tinuousfunctionontheconvexcompactsubset[0,1]n_._Following

Brouwersﬁxedpointtheorem(c.f.[17]),weconsidertheequation g(r)=r,whichhasasolution,whichwe needtoshowtobethe uniqueﬁxedpoint.Toachievethis,weusetheContraction Map-pingTheorem(CMT,c.f.[17]),sayingthattheequationg(r)=rhas auniquesolutionifandonlyif

• Thefunctiong(·)maps[0,1]n_to_[0,_1]n

• ThereisaconstantG<1suchthat||g(x)−g(y)||≤G||x−y||forall x,y∈[0,1]n

First,thealgorithmneedstobeshowntomapanystartingvalue forrtoanothervalueofrthatiswithinthepossiblerangeof[0,1]n_.

Forthistobethecase,weneedthat 0≤

ˆri,ˆq≤ ˆpi.

Theﬁrstinequalityisobvious,forthesecondonewenotethat ˆri,=

0forallsuchthati=0andthat ˆri,≤1.Thisgivesthat

ˆri,ˆq ≤

:i=1 ˆq =

:i=1 n

j=1 (1− ˆpj)1−jˆp j j = ˆpi

:i=1

j/=i (1− ˆpj)1−jˆp j j = ˆpi,

(9)

contractionmappingtheorem.

Thesecondpartismoreinvolved.Weprovideacompleteproof foratwonodenetworkandindicatewhythesecondconditionis conjecturedtoholdforlargernetworks.

Whenfollowingthestepsofthealgorithmforatwonode tan-demnetwork,wehavethat

a(1)=(1)= and a(2)=(2)=min(, ˆr1)

ˆp1 = min

ˆr1 ,1

and ˆp2=min

min(, ˆr1) ˆr2 ,1

ˆr1 = 1− 1 2ˆp2 and ˆr2=1− 1 2ˆp1.

Notethatas0≤ ˆpi≤1wehavethat ˆri∈[(1/2),1].Firstassuming

wearedealingwithastablenetwork,thearrivalrateatbothnodes equals.Wenow(bysubstitutingpi)havethefunctionalvector

g(ˆr)=

1− 2ˆr2 ,1− 2ˆr1

.

Thisgives,forx=(x1,...,xn),

||g(x)−g(y)||2₌

2x2y2

2 (x2−y2)2+

2x1y1

2 (x1−y1)2,

and for this to be smallerthan ||x−y||2 _we _need _to _have _that

(/2xiyi)2<1.Asweassumedastablenetwork,wehavethat<xi,

sothat 2x2y2 < 1 2yi≤ 1

sinceyi∈[(1/2),1]andsoindeedthesecondconditionholds

prov-ingthatforastablesystemthealgorithmconverges.Ifthesystem wouldbeunstable,wehavethat

g(ˆr)=

1−min((min(, ˆr1)/ˆr2),1) 2 ,1− min((/ˆr1),1) 2

,

wherethefollowingsituationscanoccur:≥ ˆr1or ˆr2≤< ˆr1.In

theﬁrstcasewehavethat

g(ˆr)=

1−1₂min

ˆr1 ˆr2 ,1

,1 2

whichwithintwostepsofthealgorithmleadstog(ˆr)=(1/2,1/2) andthusconvergestothisuniquesolution.Inthesecondcasewe havethat g(ˆr)=

1₂,1− 2ˆr1

, ||g(x)−g(y)||2₌

2x1y1

2 (x1−y1)2

and(/2x1y1)2<1asshownearliercompletingtheproofthatthe

algorithmconvergesforthistwonodenetwork.

g(r) =

1−min((min(,r1)/r2),1) 2 +min((min(,r1)/r2),1)min((min(min(,r1),r2)/r3),1) 6 , 1−min((/r1),1) 2 − min((min(min(,r1),r2)/r3),1) 2 +min((/r1),1)min((min(min(,r1),r2)/r3,1) ) 3, 1−min((min(,r1)/r2),1) 2 + min((/r1),1)min((min(,r1)/r2),1) 6

. As we have that g(p)=(1− (p2/2)+(p2p3/6), 1−(p1/2)−(p3/2)+(p1p3/3), 1−(p2/2)+(p1p2/6)), starting in

([(1/2),1],[(1/3),1],[(1/2),1]),g(·)willalsoprojectonthisrange. FortheCMTtohold,weﬁrstconsiderthestablesystemagain,so that<ri.Inthiscasewehavethat

g(r)=

1−1₂_r 2+ 1 6 2 r2r3 ,1−1₂_r 1− 1 2 r3 + 1 3 2 r1r3 , 1−1 2 r2 + 1 6 2 r1r2

.

Checkingwhether||g(x)−g(y)||<||x−y||provestobecumbersome, evenforsuchasmallnetwork.Thereforewenumericallyanalyzed thefunctionh(x,y)=||g(x)−g(y)||(||x−y||)−1 whichprovedtobe smallerthanoneforallvaluesofxandy.Asinthetwonode net-work,itiseasytoshowthatforaninstablenetwork,eitherthereis anobviousdirectconvergencetotherates(2/3,1/3,2/3)or conver-genceisprovenbyusingpartsoftheapproachforthestablecase. Wepostulatethatforanynetworkasimilaranalysiswillshowthat thealgorithmconstitutesacontraction,andthusconverges. A.2. ProofofTheorem3

Theformulafortherateri,1(n)ofanodeonpositioniinann

nodenetworkthatisfullyaliveisgivenby

nri,1(n)= i−2

k=1 ri−k−1,1(n−k−1)+1+ n−2

k=i ri,1(k) (14)

asdescribedinthepaper.Duetosymmetryofthenetworkwealso havethat

ri,1(n)=rn−i+1,1(n) i=1,...,n.

Therateofanodecannever exceedone,but willbeoneifthe nodeistheonlyalivenodewithinitsinterferenceregion,i.e.its neighborsarenotalive.Theminimalrateofanodeis1/naswith thisprobabilityitwinsthecontentionoverallothernodes.

Inthefollowingweomitthe1denotingthefullyalivenetwork. Toﬁndanexpressionforri(n),notethat

nri(n)−(n−1)ri(n−1)=ri(n+2) + i−2

k=1 [ri−k−1(n−k−1)−ri−k−1(n−k−2)]

andlettingci(n)=ri(n)−ri(n−1)thisgives

ci(n)= 1 n

_i₋₂

k=1 ck(n+k−i)−ci(n−1)

.

(10)

As−1≤ci(n)≤1foranyvalueofiandn,wehavethat ci(n)≤ 1 n[(i−2)−ci(n−1)]≤ 1 n(i−1) sothatfor each iwehave that lim

n→∞ci(n)=0, proving thatri(n)

convergesforn→∞. Fori=1thisleadsto c1(n)=− 1 nc1(n−1) whichgives c1(n)= (−1)n−1 n! , r1(n)= n

i=1 (−1)i−1 i! . Similarly,wehavethat

c2(n)= (₋₁₎n n! , r2(n)= n

i=1 (₋₁₎i i! . Takingthelimitshowsthat

lim n→∞r1(n)=1− 1 e, n→∞limr2(n)= 1 e

Unfortunately,forlargervaluesofi,noniceexpressionsarefound forci(n)orri(n),butthelimitingvaluescanbecalculatedusingthe

sameapproach.TheresultsarepresentedinTable1.

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Bottlenecks and stability in networks with contending nodes

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