Distributed Storage Management of Evolving Files in Delay Tolerant Ad Hoc Networks

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Delay Tolerant Ad Hoc Networks

Eitan Altman, Philippe Nain, Jean-Claude Bermond

To cite this version:

Eitan Altman, Philippe Nain, Jean-Claude Bermond. Distributed Storage Management of

Evolv-ing Files in Delay Tolerant Ad Hoc Networks. [Research Report] RR-6645, INRIA. 2008.

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Thème COM

Distributed Storage Management of Evolving Files

in Delay Tolerant Ad Hoc Networks

Eitan Altman — Philippe Nain — Jean-Claude Bermond

N° 6645

Eitan Altman

∗

, PhilippeNain

†

, Jean-ClaudeBermond

‡

ThèmeCOMSystèmes ommuni ants

ProjetsMaestro,Mas otte

Rapport dere her he n°6645January200935pages

Abstra t: This workfo usesona lassofdistributed storagesystemswhose ontentmay evolveovertime. Ea h omponentornodeof thestorage systemis mobile andthe set of allnodesformsadelaytolerant(adho ) network(DTN).Thegoalofthepaperistostudy e ient ways for distributing evolving les within DTNs and for managing dynami ally their ontent. Wespe ify to dynami les where not only thelatest versionis useful but alsopreviousones;werestri thowevertoleswherealehasnouseifanothermorere ent versionisavailable. There are

N + 1

mobilenodesin ludingasinglesour ewhi hat some pointsin timemakesavailableanewversionofasinglele

F

. We onsiderboththe ases when(a)nodesdonot ooperateand(b)nodes ooperate. In ase(a)onlythesour emay transmit a opy of the latest versionof

F

to anode that it meets, while in ase (b) any node may transmit a opy of

F

to anode that it meets. A le management poli y is a set of rules spe ifying when anode may send a opy of

F

to a node that it meets. The obje tiveistond lemanagementpoli ies whi h maximizesomesystemutility fun tions undera onstraintontheresour e onsumption. Bothmyopi (stati )andstate-dependent (dynami )poli ies are onsidered,where thestateof anodeis theageof the opyof

F

it arries. S enario(a)isstudiedundertheassumptionthatthesour eupdates

F

atdis rete times

t = 0, 1, . . .

. Duringaslot

[t, t + 1)

thesour emeetsanynodewithaxedprobability TheworkofE.AltmanandP.Nainwaspartiallysupportedbyeuropeanproje tIPBIONETSandby NetworkofEx ellen e EuroNF.TheworkofJ-C.Bermondwas partiallysupportedbyeuropeean proje t FETAEOLUSandbyCRCCORSOwithFran eTele om.

∗

INRIA, 2004 Route des Lu ioles, BP 93 06902 Sophia-Antipolis, Fran e. Email: Ei-tan.Altmansophia.inria.fr

†

INRIA, 2004 Route des Lu ioles, BP 93 06902 Sophia-Antipolis, Fran e. Email: Philippe.Nainsophia.inria.fr

‡

I3S/INRIA, 2004 Route des Lu ioles, BP 93, 06902 Sophia-Antipolis, Fran e. Email: Jean-Claude.Bermondsophia.inria.fr

q

. We nd the optimal stati (resp. dynami ) poli y whi h maximizes a general utility fun tionunder a onstraintonthenumberoftransmissionswithin aslot. Inparti ular,we showtheexisten eof athresholddynami poli y. Ins enario(b)

F

is updatedatrandom pointsintime. Wewill onsiderboththe asewhereatanytimenodesknowtheageofthe lethey arryandthe asewheretheyonlyknowthetimeatwhi hthelethey arrywas reatedbythesour e. UnderMarkovianassumptionsregardingnodesmobilityandupdate frequen y of

F

, we study the stability of the system (aging of the nodes) and derive an (approximate)optimalstati poli y. Wethenrevisits enario(a)whenthesour edoesnot knowthenumberofnodesand theprobabilitythat thesour emeets anodeinaslot,and wederiveasto hasti approximationalgorithm whi h weshowto onvergeto theoptimal stati poli y found in the omplete information setting. Numeri al results illustrate the respe tiveperforman eofoptimalstati anddynami poli iesaswellasthebenetofnode ooperation.

Key-words: Evolvingles; Storage systems; Delay-tolerant (adho ) networks; Perfor-man eevaluation;Optimization.

Résumé : Nous nous intéressons à une lasse de systèmes de sto kage distribué dont le ontenu évolue au ours du temps. On parle de  hiers (ou do uments) évolutifs (ou dynamiques). Chaque omposanteoun÷uddusystèmedesto kageestmobileetl'ensemble desn÷uds forme unréseau adho tolérant lesdélais. L'obje tif de et arti le est l'étude depolitiquese a esdedistributionetdegestionde hiersdynamiquesdans lesréseaux adho . Nous onsidérons des hiersévolutifs dont nonseulement ladernièreversionest utilemais aussilesversions antérieures;toutefois, nousnous restreignonsàdesdo uments pourlesquels il est susant de onserverla version laplus ré ente. Il y a

N + 1

noeuds mobilesy omprisunn÷udsour equià ertainsinstantsgénèreunenouvelleversiond'un  hierunique, appelé

F

. Deux assontanalysésselonque(a) lesn÷udsne oopérentpas ou(b) oopérentàladistributiondu hier

F

. Dansle as(a)seulelasour eestautoriséeà transmettreune opiedelaversion ourantede

F

àunn÷udqu'elleren ontrealorsquedans le as(b) haquen÷udqui possèdeune versionde

F

àlapossibilitéd'en transmettreune opieàunautren÷udqu'ilren ontre. Unepolitiquedegestiondes hiersestunensemble derèglesspé iantsilorsd'uneren ontreentredeuxn÷udsl'undesdeuxpeuttransmettre une opiede

F

àl'autre. Lebutestd'identierdespolitiquesdegestionquimaximisentune fon tiond'utilitédusystèmesousune ontrainteportantsurla onsommationdesressour es. Alafoisdespolitiquesstatiquesetdynamiquessont onsidérées,oùl'étatd'unn÷udestl'âge delaversionde

F

qu'ilpossède. Les énario(a) i-dessusestanalysésousl'hypothèseoùles misesàjourde

F

surviennentàdesinstantsdis rets

t = 0, 1, . . .

. Durant haqueintervalle

[t, t + 1)

lasour eren ontre ha undesn÷udsave uneprobabilitéxée

q

. Nous al ulons

lapolitiquequimaximiseunefon tiond'utilitédeportéegénérale,la ontrainteportantsur lenombremoyendetransmissionsdans

[t, t + 1)

. En parti ulier,nousmontronsl'existen e d'unepolitiquedynamique optimaledetypeseuil. Dansles énario(b)le hier

F

est mis àjourparlasour eàdesinstantsaléatoires. Nous onsidéronsàlafoisle asoù,à haque instant,lesn÷uds onnaissentl'âgedu hierqu'ilspossèdentetle asoùilsne onnaissent queladatede réationde e hier. Sousdeshypothèsesmarkoviennesquantàlamobilité des n÷uds et la fréquen ede mise àjour de

F

, nous établissons la ondition de stabilité dusystème (pro essusdevieillissementdesversionsde

F

à haquen÷ud) etproposons un algorithmeappro hépourle al uld'unepolitiquestatiqueoptimale. Nousrevenonsensuite aus énario(a)ensupposantquelasour enepossèdequ'uneinformationpartielledusystème (elle ne onnait ni

N

ni

q

) et développons un algorithme d'approximation sto hastique onvergeant vers la politique statique optimale obtenue dans le as d'information totale. Desrésultats numériques illustrentlesperforman esrespe tivesdes politiquesstatiques et dynamiquesainsiquelebéné equ'apportela oopérationinter-n÷uds.

Mots- lés : Fi hiersévolutifs;Systèmesdesto kage;Réseauxadho tolérantslesdélais; Evaluationdesperforman es;Optimisation.

1 Introdu tion

Mu h work hasbeendevoted forthe study of Delay Tolerant Networks (DTNs). Most of theworkonproto oldesignhasfo usedontheuseofmobilityinordertorea honeormore dis onne teddestinations. Theproto olsarebasedondistributionoftheletorelaynodes soastoin reasethesu essfuldeliveryprobability[2,3,4,10,12℄.

Insu h appli ations, theDTN be omes adistributed storage systemthat ontains opies ofale that isbeingtransmitted. Inthis paperwefo us onaspe ial typeof lethat we all"dynami le"or"evolvingle". Bythatwemeanalewhose ontentmayevolveand hangefrom timeto time. One(orvarious)sour eswishto makealeavailable tomobile nodes,andtosendupdatesfrom timetotime. Someexamplesare:

ˆ a sour e has ale ontaining update information su h asweather fore ast ornews headlines. The le hanges in rementally from time to time with new information updates;

ˆ a sour e wishes to make ba kups of some dire tories and to store them at another nodesin ordertoin reasethereliability;

ˆ somesoftwareupdatesorpat hesmaybedistributedregularly.

Severalformatsofdynami leshavebeenstandardized:

ˆ the RSS(Real SimpleSyndi ation [5℄)familyof Web feed formatsused to publish frequently updated ontent su h as blog entries, news headlines, and pod asts in a standardizedformat. Updates anoriginatefrom varioussour es;

ˆ another format alled the Atom Syndi ation Format has been adopted as IETF Proposed StandardRFC4287.

We spe ify to dynami les where not only the latest versionis useful but also previous ones; werestri t howeverto les where ale has nouse ifanother morere entversionis available. For example, onsider anevolvingle ontainingthe weather fore astforseven onse utivedays. Ifauserneedstheweatherfore astforthenextdaythen anyversionof thele fromthe sixlast daysis useful. Themorere entthe leis, themorea uratethe requestedinformationis. Furthermore,havinga essto agivenlemakesallpreviousles irrelevantto theuser.

Thegoalof ourpaperistostudy e ientwaysfordistributingevolvingleswithin DTNs andfor managingdynami ally their ontent. Theobviouswayto providethemost up-to-dateinformation isto useepidemi routing(e.g. see [12℄)for ea h newversionof

F

. This however onsumesalot ofnetworkresour es.

Westartwithageneraldes riptionofthemodel. Moredetailswillbegiveninthesubsequent se tions. There are

N + 1

mobile nodes in ludingone sour enode. From nowon anode designatesanymobilenodeotherthan thesour e. Atsometimeepo hsthesour e reates anupdated versionof a le

F

. Whenthe sour emeets a node it may de ideto transmit tothis nodea opyof

F

. Similarly,whentwonodesmeetthenodewhi h arriesthemore re entversionof

F

maytransmit a opyof this versionto the other node. When anode re eivesamorere entversionof

F

thantheoneitwas arrying(if any)itdeletesat on e theoldestversionofF.

Thesettinginwhi honlythesour emaytransmit(a opyof)

F

toanothernodeis alled thenon- ooperative setting,while inthe ooperativesetting anymobilenodemaytransmit toanyothernode. Weassumethattransmissionsarealwayssu essful.

Thereisautility

U (k)

asso iatedwithanodeinstate

k

,wherethestateofanodeisdened astheageofthe opyof

F

, ifany, this node arries. A le management poli y, orsimply apoli y, is a set of rules spe ifying whether the sour e and anode, ortwonodes, should ommuni ate whenever they meet. A poli y is stati (resp. dynami ) if the de ision to transmitdoesnot(resp. does)dependonthestateofthemobilenodes. Theobje tiveisto ndalemanagementpoli ythatmaximizestheexpe tedsystemutilitygivena onstraint ontheexpe tednumberof ommuni ationstakingpla ein aslot.

Se tion 2addresses thenon- ooperativesetting. Timeis slottedand thereis axed prob-ability

q

that anypair of mobile nodes meets in aslot. Atthe beginning of ea h slot the sour e reates anew version of

F

, sothat ea h node arryinga opy of

F

knowsthat its statehasin reasedbyoneunit. A opyof

F

rea hingage

K + 1

(

K < ∞

)isimmediately deleted. Wendtheoptimalstati poli y(Proposition1)andshowthatthereisanoptimal dynami poli yofathresholdtype(Proposition2)whi hwefully hara terize(Proposition 3). Theperforman eoftheoptimalstati anddynami poli iesare ompared(Figures1-4) fortwodierentutilityfun tions(

U (k) = 1

and

U (k) = 1/k

).

Se tion3investigatesthe ooperativesetting. Wedevelopa ontinuous-timemodelinwhi h mobile nodes meet at random times and le

F

is updated by the sour e also at random times. We onsider both thesituation when(i) nodes know theage ofthe le they arry (Se tion3.1)andthe ase(ii)wheretheyonlyknowthedate atwhi hthelewas reated bythe sour e(Se tion 3.2). In thesituation where anode onlydeletes ale ifitrepla es it by a more re ent version (i.e.

K = ∞

), we study the stability of the network, in a Markovianframework,forbothsettings(i)(Proposition 4) and(ii)(Proposition 5) above. Here,stabilityrefersto thestateorageofea h nodebeingalmost surelynite. Underthe morerestri tiveassumptions where node meeting timesand update times aremodeledby independentPoissonpro esses,wederiveamean-eldlike approximationfortheexpe ted numberofnodes in state

k ≥ 1

inthe asewhereastati poli y is enfor ed. Wethenuse thisresulttoquantify inFigures8-9thebenetofhavingnodesto ooperate.

The deployment of optimal poli ies derived in Se tions 2-3 requires that the sour e has a omplete information on the network (node mobility, number of nodes). In Se tion 4

we releasethis assumption. We fo us on thenon ooperativesetting and restri tto stati poli ies, and weassume that the sour e does notknowthe numberof nodes

N

and does notknowthemeeting probability

q

. Byusing thetheoryof sto hasti approximations,we onstru t an algorithm whi h onverges to the optimal stati poli y found in Se tion 2. Se tion5 on ludes thepaper.

Remarkonthenotation: by onvention

P

j

k=i

· = 0

and

Q

j

k=i

· = 1

if

i > j

.

IR

+

denotesthe setofallnonnegativerealnumbers.

2 Non- ooperative nodes

Inthisse tionwe onsiderthes enariowherenodesdonot ooperateandmayonlyre eive le

F

fromthesour e. Nodesarelabeled

1, 2, . . . , N

. Attimes

t = 0, 1, . . .

thesour e reates anew versionof le

F

. Inthe following, aslot denotes any time-period

[t, t + 1)

,

t ≥ 0

, and slot

t

standsfor the time-period

[t, t + 1)

. There is aprobability

q(i) > 0

that node

i = 1, . . . , N

meetsthesour einaslot. Wedenethemeetingtimesbetweenthesour eand

anode asthe su essiveslotsat whi h theymeet. Themeeting times ofea h nodewhi h thesour eformasequen eofindependentandidenti allydistributed(iid)randomvariables (rvs)and allmeeting timepro essesareassumed tobemutually independent. Forsakeof simpli ity,weassumethatalltransmissionsbetweenthesour eandthenodesinitializedin aslotare ompletedbytheendofthisslot. Thisimpliesthatthetransmissiontimeof

F

is smallw.r.t. thedurationofaslot.

When anode re eives an updated version of

F

it deletes at on e the previous versionof

F

it was arrying, ifany. Wedenethe ageof aversionof

F

asthenumberof slots that haveelapsedsin ethis versionwasgenerated bythe sour e. Weassumethat a versionof

age

K + 1

or more is uselessand that a node deletes at on e ale that has rea hed age

K + 1

. Therefore,theageofaversionof

F

variesbetween

1

(the versionwasgenerated in

the urrentslot) and

K

(the versionwasgenerated

K − 1

slots ago). Wefurther assume

that

K < ∞

(seeRemark2.1).

Thestateofanodeisdened astheageoftheversionof

F

it arries,ifany. Anodeis in state

0

ifitdoesnot arryanyversionof

F

. Anodeinstate

K

attheendofaslotswit hes tostate

0

atthebeginningofthenextslot.

Whenthesour emeets node

i

, withprobability

a

k

(i)

it transmitsto itthenewestversion of

F

if that node is in state

k

(

k = 0, 1, . . . , K

). We assume that the transmission is always su essful. The de ision by the sour e to transmit to a node is independent of allpastde isionsmadebythesour eandisalsoindependentofallmeeting timepro esses. Introdu e

p

k

(i) := q(i)a

k

(i)

theprobabilitythatnode

i

instate

k

re eivesthenewestversion of

F

in aslot. Dene

p

c

k

(i) := 1 − p

k

(i)

. Atequilibrium, let

π

k

(i)

bethe probability that

k

attheendofaslot. Wehave

X

k

=

N

X

i=1

π

k

(i),

k = 0, 1, . . . , K,

(1) with

P

K

k=0

X

k

= N

.

Forea h

i = 1, . . . , N

,theprobabilities

{π

k

(i)}

K

k=0

satisfytheChapman-Kolmogorov equa-tions

π

0

(i)

= π

0

(i)p

c

0

(i) + π

K

(i)p

c

K

(i)

(2)

π

k

(i)

= π

k−1

(i)p

c

k−1

(i),

k = 2, . . . , K,

(3)

1

=

K

X

k=0

π

k

(i).

(4)

There is oneadditional equilibrium equation given by

π

1

(i) =

P

K

k=0

π

k

(i)p

k

(i)

whi h we willnot onsidersin eit anbederivedbysummingupequations(2)-(3). Equations(2)-(4) denealinearsystemof

K + 1

equationsand

K + 1

unknowns.

From now on wewill assume that

p

0

(i) > 0

(i.e.

a

0

(i) > 0

sin e we haveassumed that

q(i) > 0

)forall

i

asotherwisethesolutionto(2)-(4)maynotbeunique. Thenon-uniqueness

ofthesolution orrespondstosituationswherethesteady-stateofnode

i

will dependupon itsinitial state(e.g. take

p

1

(i) = 1

and

p

k

(i) = 0

for

k 6= 1

),adegenerated situation that aneasilybehandledandthat wewillnot onsider fromnowon. Solvingfor(2)-(4)gives

π

0

(i) =

Q

K

k=1

p

c

k

(i)

D

i

,

π

k

(i) =

p

0

(i)

Q

k−1

l=1

p

c

l

(i)

D

i

(5) for

k = 1, . . . , K

,

i = 1, . . . , N

,with

D

i

:= p

0

(i)

K

X

k=1

k−1

Y

l=1

p

c

l

(i) +

K

Y

k=1

p

c

k

(i).

(6) Hen e,by(1),

X

0

=

N

X

i=1

Q

K

k=1

p

c

k

(i)

D

i

,

X

k

=

N

X

i=1

p

0

(i)

Q

k−1

_l=1

p

c

_l

(i)

D

i

(7) for

k = 1, . . . , K

.

Intheparti ular asewhere

p

k

(i) = p

k

forall

i

,

k

then

X

0

=

N

K

Y

k=1

(1 − p

k

)

D

,

X

k

=

N p

0

k−1

Y

l=1

(1 − p

l

)

D

(8)

for

k = 1, . . . , K

, where

D := p

0

K

X

k=1

k−1

Y

l=1

(1 − p

l

) +

K

Y

k=1

(1 − p

k

).

(9)

Ifwefurtherassumethat

p

k

= p

for

k = 0, 1, . . . , K

then

X

0

= N (1 − p)

K

,

X

k

= N p(1 − p)

k−1

, k = 1, . . . , K.

(10)

Remark2.1 (

K = ∞

) Formulas(7) hold if

K = ∞

(i.e. nodesnever delete the le they arryunless theyre eive anew versionfromthe sour e)providedthat

D

i

in(6)isnite for every

i

as

K ↑ ∞

. Thisissoif

lim

k↑∞

p

k

(i) > 0

for

i = 1, . . . , N

(Hint: applyd'Alembert's riterion totheseries

P

k≥1

Q

k−1

l=1

p

c

l

(i))

). Notefrom(7 )that

X

0

= 0

if

K = ∞

.

Remark2.2 (Intermittentlyavailablenodes) Thesituationwherenodesare intermit-tently available an be handled by repla ing

p

k

(i)

by

r(i)p

k

(i)

with

r(i)

the probability that node

i

isavailable inaslot.

2.1 Performan e metri s

There are several performan e metri s of interest whi h an bederived from (8). One of theseistheexpe tednumber opiesof

F

givenby

X =

K

X

k=1

X

k

= N − X

0

.

(11)

Another one is the expe ted age of the opies given by

(1/N )

P

K

k=1

kX

k

. Of parti ular interestistoevaluatethepower onsumption. Sin ethepower onsumption,denotedas

Q

, isproportionaltotheexpe tednumberof transmissionsduringaslot,wewilldene itas

Q = γX

1

.

(12)

Withoutlossofgeneralityweassumefromnowonthat

γ = 1

.

2.2 Energy e ient le management poli ies

Until the end of Se tion 2 we will assume that nodes are homogeneousin the sense that

q(i) := q

and

a

k

(i) := a

k

forall

i, k

with

q > 0

. Weassumethat

a

0

> 0

. Wedene

p

k

:= qa

k

sothat

p

= q a

with

a

= (a

0

, . . . , a

K

)

. Inthissetting

{X

k

}

K

Inthis frameworktheexpe tedageofthe opiesandthepower onsumptionintrodu edin Se tion2.1aredenotedby

Age(p)

and

Q(P)

,respe tively,so astostresstheirdependen y ontheve tor

p

. Morepre isely,

Age(p) =

1

N

K

X

k=1

kX

k

,

(13)

Q(p) =

X

1

.

(14)

A le managementpoli y is anyde isionve tor

a

= (a

0

, . . . , a

K

) ∈ (0, 1] × [0, 1]

K

, where were all that

a

k

is the( onditional) probabilitythat the sour etransmits

F

to anode in state

k

whenitmeets su hanode. An equivalentdenition ofalemanagementpoli y is anyve tor

p

= (p

0

, . . . , p

K

) ∈ (0, q] × [0, q]

K

sin e

p

= qa

. Unless otherwisementionedwe willworkwiththelatterdenition.

Ourobje tiveisto nd anoptimallemanagementpoli y

p

whi h maximizesthesystem utility given a power onsumption onstraint. More pre isely, let

U (k)

be the utility for havingaleofage

k

inthesystem. Weassumethatthemapping

U : {0, 1, . . . , K} → IR

+

is non-in reasing. Without lossof generality we assume

U (0) = 0

. The system utility is denedas

C(p) =

K

X

k=1

X

k

U (k).

(15)

If

U (k) = 1

for all

k > 0

then

C(p) = X

, given in (11). We will assume that

U

is not

identi allyzeroasotherwisethesystemutilityisalwayszero.

Theoptimization problemisthefollowing:

P:Maximize

C(p)

overtheset

(0, q]×[0, q]

K

given

Q(p) ≤ V

,where

V

isapositive onstant. WewillsolvePin twodierentsettings: thestati setting wheremanagementpoli iesare restri ted to poli ies of the form

p

= (p, . . . , p)

with

p ∈ (0, q]

, and the dynami setting wheretheoptimizationismadeoverallve tors

p

= (p

0

, . . . , p

K

) ∈ (0, q] × [0, q]

K

.

2.2.1 Stati optimal poli y

Inthestati setting,problemPbe omes(see(10)):

P:Maximize

C(p) := N p

P

K

k=1

(1 − p)

k−1

U (k)

over

p ∈ (0, q]

giventhat

N p ≤ V

.

Thefollowingresultholds:

Proposition 1(Optimal stati poli y)

If

N q ≤ V

then

p

⋆

_{= q}

isthe optimal solution; otherwise

p

⋆

_{= V /N}

isthe optimal solution or, equivalently,

p

⋆

_{= min(q, V /N )}

It is enough to show that the mapping

p → C(p)

is stri tly in reasing in

(0, q)

. Dene

U (K + 1) = 0

. Wehavefrom(15)and(10)

C(p)

=

K

X

k=1

(U (k) − U (k + 1))

k

X

l=1

X

l

= N

K

X

k=1

(U (k) − U (k + 1))(1 − (1 − p)

k

),

Hen e,

dC(p)/dp = N

P

K

k=1

(U (k) − U (k + 1))k(1 − p)

k−1

> 0

for

p ∈ (0, q)

, sin e

U

is

non-in reasingandnotidenti allyzero(whi h ne essarilyimpliesthat

U (K) > 0

).

2.2.2 Dynami optimal poli y

Let us introdu e the new de ision variables

x

k

= 1 − p

k

for

k = 1, . . . , K

and

x

K

=

(1 − p

K

)/p

0

. Notethat

1 − q ≤ x

k

≤ 1

for

k = 1, . . . , K

and

x

K

≥ (1 − q)/q

withequality

ifandonlyif

p

0

= p

K

= q

. Let

x

= (x

1

, . . . , x

K

)

. Introdu etheset

E

=

x : x ∈ [1 − q, 1]

K−1

_{× [(1 − q)/q, ∞) .}

Anyve tor

x

∈ E

is alled apoli y. Denethemappings

F (x) =

K

X

k=1

U (k)

k−1

Y

l=1

x

l

,

G(x) =

K+1

X

k=1

k−1

Y

l=1

x

l

andlet

H(x) := F (x)/G(x)

. Notethat

F (x)

doesnotdependonthevariable

x

K

. From(9)

D = p

0

G(x)

,andsoby(8)

C(p) = N H(x)

and

Q(p) =

N

G(x)

.

InthisnewnotationproblemPbe omes

max

x∈E

H(x)

subje ttothe onstraint

G(x) ≥ C

,

with

C := N/V

.

Anadmissiblepoli y isanypoli ysu h that

G(x) ≥ C

.

Denition2.1 (Threshold poli y)

A poli y

x

= (x

1

, . . . , x

K

) ∈ E

is a threshold poli y if either

x

k

= 1

or

x

k+1

= 1 − q

for

k = 1, . . . , K − 2

and ifeither

x

K−1

= 1

or

x

K

= (1 − q)/q

.

Anythreshold poli y

x

= (x

1

, . . . , x

K

)

is su h that

x

1

≥ . . . ≥ x

K−1

. Morepre isely, itis easilyseenthat athresholdpoli yifeitherofType I orofTypeII with

Type I:for

k = 1, . . . , K

x

k

(α) := (1, . . . , 1, α, 1 − q, . . . , 1 − q, (1 − q)/q)

(16)

where

1 − q ≤ α < 1

isthe

k

-thentry; Type II:

x

K

(β) := (1, . . . , 1, β)

with

β ≥ (1 − q)/q.

(17)

Intermsofthelemanagementpoli y

p

= (p

0

, . . . , p

K

) ∈ (0, q] × [0, q]

K

,TypeIthreshold poli y

x

k

(α)

, uniquelytranslatesinto

p

_k

(α) := (q, 0, . . . , 0, 1 − α, q . . . , q, q)

(18)

where

1−α ∈ (0, q]

isthe

(k +1)

-stentry(

k = 1, . . . , K

)(asalreadyobserved

p

0

= p

K

= q

in (18)sin ethisistheonlysolutionoftheequation

(1−p

K

)/p

0

= (1−q)/q

when

0 ≤ p

0

, p

K

≤ q

with

p

0

6= 0

). Inparti ular

p

1

(1 − q) = (q, . . . , q)

.

Anylemanagementpoli y

p

K

(β) = (p

0

, 0, . . . , 0, p

K

)

(19)

with

(1 − p

K

)/p

0

:= β

orrespondstotheuniqueType IIthresholdpoli y

x

K

(β)

.

Proposition 2(Optimality ofthreshold dynami poli y)

Undertheassumption thattheutilityfun tion

U : {1, . . . , K} → IR

+

isnon-in reasingthere existsanoptimal threshold poli y.

Proof. Assumethattheoptimalpoli y

x

isnotathresholdpoli y. Hen e,thereexistsa

k

,

1 ≤ k ≤ K − 1

,su h thateither

x

k

< 1

and

x

k+1

> 1 − q

if

k 6= K − 1

or

x

K−1

< 1

and

x

K

> (1 − q)/q

if

k = K − 1

.

Assume rst that

x

1

· · · x

k−1

6= 0

. Let us show that one an always nd

ǫ

k

> 0

and

ǫ

k+1

> 0

su h that

x

′

k

:= x

k

+ ǫ

k

< 1

,

x

′

k+1

= x

k+1

− ǫ

k+1

> 1 − q

if

k 6= K − 1

(resp.

x

′

k+1

= x

k+1

− ǫ

k+1

> (1 − q)/q

if

k = K − 1

) and

G(x) = G(x

′

₎

, where

x

′

₌

(x

1

, . . . , x

k−1

, x

′

k

, x

′

k+1

, x

k+2

, . . . , x

K

)

. Set

δ

k

:= x

′

k

x

′

k+1

− x

k

x

k+1

= ǫ

k

x

k+1

− ǫ

k+1

x

k

− ǫ

k

ǫ

k+1

. The identity

G(x

′

_{) = G(x)}

is equivalentto

x

1

· · · x

k−1

(ǫ

k

+ δ

k

A

k

) = 0

that is

ǫ

k

+ δ

k

A

k

= 0

,with

A

k

:= 1 + x

k+2

+ x

k+2

x

k+3

+ · · · + x

k+2

· · · x

K

. Theequation

ǫ

k

+ δ

k

A

k

= 0

rewrites

ǫ

k+1

= ǫ

k

1 + A

k

x

k+1

A

k

(x

k

+ ǫ

k

)

.

So,we annd

ǫ

k

and

ǫ

k+1

smallenoughsothattheysatisfythe onditions. Observethat

ǫ

k

+ δ

k

A

k

= 0

with

ǫ

k

> 0

yields

δ

k

< 0

sin e

A

k

> 0

.

Letusnallyshowthat

F (x

′

_{) > F (x)}

whi hwill ontradi ttheoptimalityof

x

. Wehave

F (x

′

_{) − F (x)}

x

1

· · · x

k−1

= ǫ

k

U (k + 1) + δ

k

[U (k + 2)

+x

k+2

U (k + 3) + · · · + x

k+2

· · · x

K−1

U (K)]

=

(ǫ

k

+ δ

k

A

k

− δ

k

x

k+2

· · · x

K

)U (k + 1)

+δ

k

[U (k + 2) − U (k + 1)

+x

k+2

(U (k + 3) − U (k + 1)) + · · ·

+x

k+2

· · · x

K−1

(U (K) − U (k + 1))]

=

−δ

k

x

k+2

· · · x

K

U (k + 1) + δ

k

[U (k + 2)

−U (k + 1) + x

k+2

(U (k + 3) − U (k + 1)) + · · ·

+x

k+2

· · · x

K−1

(U (K) − U (k + 1))]

(20)

wherewehaveusedtheidentity

ǫ

k

+ δ

k

A

k

= 0

toderive(20). Sin e

U

isnon-in reasingand

δ

k

< 0

asnoti edearlier,wededu ethattheright-handsideof(20)isstri tlypositive,and

therefore

F (x

′

_{) > F (x)}

.

Assumenowthat

x

1

· · · x

k−1

= 0

. Thismayonlyhappen when

q = 1

sin e

1 − q ≤ x

k

≤ 1

for

k = 1, . . . , K

. Let

l ∈ {1, . . . , k − 1}

bethesmallestintegersu hthat

x

l

= 0

.

Iftheoptimalpoli yissu hthat

x

l

= 0

thenthevalueof

x

l+1

, . . . , x

K

areirrelevantsin e

x

l

= 0

impliesthat

X

l+1

= · · · = X

K

= 0

sothatboththe ostandthe onstraintwill not dependonthevaluesof

x

l+1

, . . . , x

K

. Assumeforinstan ethat

x

l+1

= · · · = x

K

= 0

sothat poli y

x

isoftheform

x

= (x

1

, . . . , x

l−1

, 0, . . . , 0)

. Itthisisnotathresholdpoli ythenone annd

k

′

_{∈ {1, . . . , l − 2}}

su hthat

x

k

′

< 1

and

x

k

′

+1

> 1 − q = 0

. We anthendupli ate

thesameargumentusedtoestablish(20)with

k

repla edby

k

′

. Sin e

x

1

· · · x

k

′

−1

6= 0

from thedenitionof

l

we on ludethat

F (x

′

_{) > F (x)}

. This ompletestheproof.

Itis a tuallypossibletond thebestdynami lemanagementpoli y in expli itform,as nowshown.

Proposition 3(Best dynami le managementpoli y)

Assume that the utility fun tion

U : {1, . . . , K} → IR

+

is non-in reasing. The following results hold:

(a) if

N q < V

the optimal lemanagementpoli y is

p

1

(1 − q) = (q, . . . , q)

;

(b) if

N q

qk+1

< V ≤

N q

q(k−1)+1

for some

k = 1, . . . , K

,the optimal le managementpoli yis

p

k

(q(C − k)) = (q, 0 . . . , 0, 1 − q(C − k), q, . . . , q)

(see(18));

( ) if

V ≤

N q

q(K−1)+1

any lemanagementpoli y

p

K

(C − K) = (p

0

, 0, . . . , 0, p

K

)

su hthat

Proof. Sin ewehaveshownin Proposition2thatthereexistsanoptimalthresholdpoli y, weonlyneedtofo usonthresholdpoli iesasdenedin(16)-(17). Easyalgebrashowthat

G(x

k

(α))

=

k +

α

q

,

k = 1, . . . , K

(21)

G(x

K

(β))

=

K + β,

(22)

sothat

G(x

1

(α

1

)) ≤ · · · ≤ G(x

K−1

(α

K−1

)) ≤ G(x

K

(β))

for all

α

1

, . . . , α

K−1

∈ [1 − q, 1)

,

β ≥ (1 − q)/q

. Fromthis wededu ethat there arethree dierent asesto onsider(re all

that

C = N/V

): (a)

C < G(x

1

(1 − q)) = 1/q

orequivalently

V > N q

; (b)

G(x

k

(1 − q)) ≤ C < G(x

k+1

(1 − q))

orequivalently

N q

qk+1

< V ≤

N q

q(k−1)+1

; ( )

C ≥ G(x

K

((1 − q)/q))

orequivalently

V ≤

N q

q(K−1)+1

.

Case(a): Inthis aseanythresholdpoli ysatisesthe onstraint,sothattheoptimalpoli y isthepoli ywhi hmaximizesthe ost

H(x)

. ItisshowninLemma3intheappendixthatfor

ea h

k = 1, . . . , K

,themapping

x

k

→ H(x)

isnon-in reasingforany

x

= (x

1

, . . . , x

K

) ∈ E

.

Therefore, poli y

x

1

(1 − q) = (1 − q, . . . , 1 − q, (1 − q)/q)

is optimal, orequivalently (see (18))thelemanagementpoli y

p

1

(1 − q) = (q, . . . , q)

is optimal.

Case (b): Assume that

G(x

k

(1 − q)) ≤ C < G(x

k+1

(1 − q))

for some

1 ≤ k ≤ K − 1

. By Lemma 3in theappendixweseethat thebest thresholdpoli y istheonewhi h saturates the onstraint,namelypoli y

x

k

(α)

su hthat

G(x

k

(α)) = C

,thatis

α = q(C − k)

. By(21) thispoli yisuniqueandisgivenby

x

k

(q(C − k))

. Equivalently(see(18)),theoptimalle managementpoli yis

p

k

(q(C − k))

.

Case ( ): Inthis asethere is noType I poli y whi h satisesthe onstraint

G(x) ≥ C

. AmongallTypeIIpoli iessatisfyingthis onstrainttheonewiththesmallest

K

-thentryis thepoli ysu hthat

G(x

K

(β)) = C

,thatis(see(19))poli y

x

K

((C−K)) = (1, . . . , 1, C−K)

. We on ludeagainfrom Lemma 3that this isthe optimalpoli y. Equivalently(see (19)), anylemanagementpoli y

p

K

(C − K) = (p

0

, 0, . . . , 0, p

K

)

su hthat

(1 − p

K

)/p

0

= C − K

isoptimal. This on ludes theproof.

Remark2.3 (Non-uniquenessofbest poli yin ase ( )) If the onstraint is strong in the sense that no Type I poli y an meet it ( ase ( ) of Proposition 3) then any le managementpoli y

p

= (p

0

, 0, . . . , 0, p

K

)

su hthat

(1 − p

K

)/p

0

= C − K

isoptimal,thereby showingthat

p

0

and

p

K

arenotuniqueand, inparti ular, neednotbeequallikein ases(a) and(b)of Proposition3.

2.3 Numeri al results

Inallguresthenode populationisset to 100(

N = 100

)andthe maximumageof opies

ofFissetto5(

K = 5

).

Let

p

⋆

s

(resp.

p

⋆

d

) be the stati (resp. dynami ) le management poli y whi h solvesthe optimizationproblemPasfoundinProposition1(resp. Proposition3). Figures1-4 dis-playthemappings

q →

P

K

k=1

U (k)X

k

under bothpoli ies

p

⋆

s

and

p

⋆

d

( orresponding urves are referred to as stati  and dynami , respe tively), for two dierent utility fun tions

(

U (k) = 1

,

U (k) = 1/k

)andfortwodierentvaluesofthe onstraint

V

(

V = 10, 20

). These

resultsshowthat the useof the optimal dynami poli y mayyield substantial gains (e.g.

for

U (k) = 1

gainof

≈ 22%

forall

q ≥ 0.2

see Fig. 2;gainof

≈ 45%

for

q

loseto1see

Fig. 1. Gainishalvedfor

U (k) = 1/k

.). Thegainisan in reasingfun tionof themeeting probability

q

.

Figures5-7 displaythemappings

q → Age(p

⋆

s

)

and

q → Age(p

⋆

d

)

for

V = 20

and

V = 50

,

respe tively, where we re all that

Age(p)

is the expe ted age of opies of

F

under poli y

p

(see(13)). These resultsshowthatthebest dynami poli y

p

⋆

d

forproblem Pmayyield a higher expe ted age than the orresponding optimal stati poli y

p

⋆

s

(see Figure 6 for

q ≥ 0.2

).

0

20

40

60

80

100

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

static

dynamic

Figure1:

q →

P

5

0

10

20

30

40

50

60

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

static

dynamic

0

10

20

30

40

50

60

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

static

dynamic

0

10

20

30

40

50

60

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

static

dynamic

Figure2:

q →

P

5

k=1

X

k

underoptimalstati /dynami poli y: V=10(N=100,K=5)

0

10

20

30

40

50

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

static

dynamic

Figure 3:

q →

P

5

0

5

10

15

20

25

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

static

dynamic

Figure 4:

q →

P

5

k=1

X

k

/k

underoptimalstati /dynami poli y: V=10(N=100,K=5)

2.75

2.8

2.85

2.9

2.95

3

3.05

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

static

dynamic

Figure5:

q → {Age(p

⋆

s

), Age(p

⋆

d

)}

: V=10(N=100,K=5)

2.5

2.6

2.7

2.8

2.9

3

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

static

dynamic

Figure6:

q → {Age(p

⋆

s

), Age(p

⋆

d

)}

: V=20(N=100,K=5)

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

3

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

static

dynamic

Figure7:

q → {Age(p

⋆

s

), Age(p

⋆

d

)}

: V=50(N=100,K=5)

3 Cooperative nodes

Inthisse tionweassumethat nodes ooperatein thesensethatwhentwonodesmeetthe node(in ludingthesour e)withthemostre entversionof

F

maysenda opytotheother node. Whenanodere eivesanewversionof

F

itdeletesaton etheversionitwas arrying, ifany. Whenthesour e reatesanewversionof

F

itdeletesaton ethepreviousversion. Theidentityofthesour eis

0

and nodesarelabeled

1, 2, . . . , N

. Weobservethesystemat dis retetimes

{t

n

}

n≥0

,where

t

n

isthetimeofthe

n

thevent. Aneventiseitherthemeeting ofthe sour ewithanode,the meetingof twonodesorthe reationof anew versionof

F

bythe sour e. Let

{ξ

i,j

n

}

n

,

i 6= j

, and

{ζ

n

}

n

be

{0, 1}

-valuedrvswhere

ξ

i,j

n

= 1

,

j 6= 0

,if

node

i

meetsnode

j

attime

t

n

,

ξ

i,0

n

= 1

ifnode

i

meetsthesour eattime

t

n

,and

ζ

n

= 1

if thesour e reatesanewversionof

F

attime

t

n

. Weassumethat

ζ

n

+

P

i,j

ξ

i,j

n

= 1

forall

n

(onlyoneeventattime

t

n

). Dene

κ

n

:= ({ξ

i,j

n

}), ζ

n

)

for

n ≥ 1

.

Twodierents enariiwillbe onsiderdependingontheamountofinformationavailableto thenodes. Ins enarioI we assumethat the nodes knowthe age of the le they arry (if any). Morepre isely,theageof theversionof

F

arriedbynode

i

is equalto

k ≥ 1

ifthe sour ehasupdatedle

F k − 1

timessin ethisversionwas reated.

Ins enarioIIweassumethatthenodesonlyknowthedateat whi htheversionof

F

they arry(ifany)hasbeen reatedbythesour e.

3.1 S enario I: Nodes know the age of the le they arry

With a slight abuseof terminologywewill say that node

i

is of age

k

ifthe versionof

F

thatnode

i

it arriesisofage

k

.

Let

Y

i

n

∈ {0, 1, 2 . . .}

betheageof node

i

justbefore the

n

theventtakespla eat time

t

n

.

By onvention

Y

i

n

= 0

ifnode

i

doesnot arryanyversionof

F

. Weintrodu etheadditional

{0, 1}

-valuedrvs

{a

i,j

n

(k, l)}

and

{a

i

n

(k)}

,where

a

i,j

n

(k, l) = 1

if

nodeiinstate

k

re eivesa opyof

F

fromnode

j

instate

l

iftheymeetat

t

n

and

a

i

n

(k) = 1

ifthe sour etransmitsthelatest versionof

F

to node

i

in state

k

iftheymeetat

t

n

. We assumethat

a

i,j

n

(k, l) = 0

if

l ≥ k

sin e learlyanodehasnointeresttore eiveaversionof

F

whi hiseither identi alorolderthantheoneit arries.

Let

θ

i,j

(k, l) := P (a

i,j

n

(k, l) = 1)

and

θ

i

(k) := P (a

i

n

(k) = 1)

.

Thefollowingre ursionshold(

i = 1, . . . , N

):

Y

n+1

i

= Y

n

i

+ (1 − Y

n

i

)ξ

i,0

n

a

i

n

(Y

n

i

) +

N

X

j=1

j6=i

(Y

n

j

− Y

n

i

) ξ

i,j

n

a

n

i,j

(Y

n

i

, Y

n

j

) + ζ

n

.

(23)

Toemphasizethefa t that

θ

i,j

(k, l) = 0

if

l ≥ k

wewillrewrite(23)as

Y

i

n+1

= Y

n

i

+ (1 − Y

n

i

)ξ

n

i,0

a

i

n

(Y

n

i

) +

N

X

j=1

j6=i

(Y

i

n

− Y

n

j

) 1

Y

j

n

−Y

n

i

<0

ξ

i,j

n

a

i,j

n

(Y

n

i

, Y

n

j

) + ζ

n

.

(24)

Denetheve tors

Y

n

= (Y

1

n

, . . . , Y

n

N

) ∈ E := {1, 2, . . .}

N

, AssumptionsA1: (1)

{ζ

n

}

n

,

{ξ

i,j

n

}

and

{ξ

i,0

n

}

n

aremutually independentrenewalsequen eswith ommon probability distribution

r := P (ζ

n

= 1)

,

q

i,j

:= P (ξ

i,j

n

= 1)

and

q

i

:= P (ξ

i,0

n

= 1)

, respe tively,for

i, j = 1, . . . , N

,

i 6= j

; (2)

r > 0

and

q

i

> 0

for

i = 1 . . . , N

;

(3) the probability that two nodes ommuni ate when they meetonly depends ontheir identity andstate,namely

P (a

i,j

n

(Y

n

i

, Y

n

j

) = 1|{Y

m

, κ

m

}

m≤n

) = θ

i,j

(Y

n

i

, Y

n

j

)

for

i, j =

1, . . . , N

,

i 6= j

;

(4) giventhatthesour emeetsanode,theprobabilitythat it ommuni ateswithitonly dependsonthenodeidentityandstate,thatis,

P (a

i

n

(Y

n

i

) = 1 | {Y

m

, κ

m

}

m≤n

) = θ

i

(Y

n

i

)

for

i = 1, . . . , N

;

(5) If

K = ∞

thenfor ea h

i = 1, . . . , N

there exist anite integer

M

i

and

θ

i

> 0

su h

that

θ

i

(k) ≥ θ

i

for

k ≥ M

i

.

Similartothenon- ooperativesettingwedene

K

asthemaximumageaversionof

F

an rea h. This means that aversionof

F

whi h rea hes age

K + 1

is deleted at on e bythe nodethat arriesit. Werststudythe asewhere

K = ∞

.

3.1.1 Stability analysiswhen

K

is innite

In this setting the age of a node is unbounded. The result below establishes onditions underwhi htheMarkov hain

{Y

n

}

n

isstable,thatis onditionsunderwhi h

Y

n

onverges indistributiontowardan(a.s.) niterandomvariableas

n

goesto innity.

Proposition 4(Stability of

{Y

n

}

n

) AssumethatA1holds.

{Y

n

}

n

isatime-homogeneous,aperiodi , irredu ibleandpositivere urrentMarkov hainon

Proof. Onlythepositivere urren epropertydoesnottriviallyfollowfromA1(1)-(4). We willshowit byapplyingFoster's riterion(seee.g. [7℄)to

{Y

n

}

n

. ConsidertheLyapounov fun tion

f : E → IR

+

denedby

f (y) =

P

N

i=1

y

i

with

y

= (y

1

, . . . , y

N

)

. Weneedtoshowthat

(i)thereexistaniteset

F ⊂ E

anda onstant

ǫ > 0

su hthat

E[f (Y

1

)−f (y) | Y

0

= y] ≤ −ǫ

for

y

∈ E − F

and(ii)

E[f (Y

1

) − f (y) | Y

0

= y] < ∞

for

y

∈ F

.

Under A1(5)dene

M

0

:= max{M

1

, . . . , M

n

} < ∞

and

θ := min(θ

1

, . . . , θ

N

) > 0

so that

θ

i

(k) ≥ θ

for

k ≥ M

0

and

i = 1, . . . , N

.

DenetheLyapounovfun tion

f (y) :=

P

N

i=1

y

i

. Wehavefrom(24)

E[f (Y

1

) − f (y) | Y

0

= y]

=

N

X

i=1

(1 − y

i

)q

i

θ

i

(y

i

) +

N

X

i=1

N

X

j=1

j6=i

(y

j

− y

i

) 1

y

j

−y

i

<0

q

i,j

θ

i,j

(y

i

, y

j

)

+N r

≤

N

X

i=1

(1 − y

i

)q

i

θ

i

(y

i

) + N r

(25) for

y

∈ E

.

Fix

ǫ > 0

. Let

M

beanyniteintegersu hthat

M ≥ max{M

0

, (ǫ+N (r+1))/(θ min{q

1

, . . . , q

N

})}

.

Let

G

bea subsetof

E

dened by

G := {y = (y

1

, . . . , y

N

) ∈ E : max{y

1

, . . . , y

N

} > M }

. Fix

y

∈ G

. Let

i

⋆

besu hthat

y

i

⋆

= max{y

1

, . . . , y

N

}

,sothat

y

i

⋆

> M

.

From(25)andA1(5)wend

E[f (Y

1

) − f (y) | Y

0

= y]

≤

N

X

i=1

q

i

θ

i

(y

i

) −

N

X

i=1,i6=i

⋆

y

i

q

i

θ

i

(y

i

) − y

i

⋆

q

_i

⋆

θ

_i

⋆

(y

_i

⋆

) + N r

≤

−M θq

i

⋆

+ N (r + 1)

≤

−M θ min{q

1

, . . . , q

N

} + N (r + 1)

≤

−ǫ,

(26)

byusingthedenition of

M

,whi hshowspart(i)ofFoster's riterion. Dene

F := G

c

_{= {y = (y}

1

, . . . , y

N

) ∈ E : max{y

1

, . . . , y

N

} ≤ M }

. The set

F

is anite

and,morever,

E[f (Y

1

) − f (y) | Y

0

= y] < N r < ∞,

∀y ∈ F,

byusing(25),whi hshowspart(ii)ofFoster's riterionand ompletestheproof.

Wewillshowina ompanionpaperthatthestabilityof

{Y

n

}

n

anbeinvestigatedinamu h moregeneralframeworkthantheMarkovianframework.

3.1.2 Quantitative performan e when

K

is innite

Wemakeadditionalassumptionsin order to ompute

X

k

, theexpe ted numberofles of

age

k ≥ 0

insteady-state. Weassumethatthesour eandnode

i = 1, . . . , N

(resp. anypair

ofnodes

i

and

j

,

i 6= j

)meeta ordingto aPoissonpro ess withrate

λ > 0

and thatthe sour e reatesanewversionof

F

atea ho urren eofaPoissonpro esswith rate

µ > 0

. These

N (N + 1)/2 + 1

Poissonpro essesareassumedtobemutually independent.

Weassumethat

θ

i

(k) := a

k

and

θ

i,j

(k, l) := b

k,l

forany

i, j, k, l

. Inotherwords,whentwo nodes(i.e. sour e or nodes)meet theprobabilitythat atransmissiono urs onlydepends onthenodesstateandnotonthenodesidentity.

Last,weassumethat thereexistaniteinteger

M

0

anda onstant

θ > 0

su hthat

a

k

≥ θ

for all

k ≥ M

0

, sothat thesystem is stable by Proposition 4(Hint: apply Proposition 4

with

q

i

= q

i,j

= λ/ν

and

r = µ/ν

where

ν := λN (N + 1)/2 + µ

).

Let

X

k

(t)

benumberofnodesinstate

k ≥ 0

attime

t

. Set

X

k

(t) := E[X

k

(t)]

. WehavetheKolmogorovequations

dX

0

(t)

dt

= −λa

0

X

0

(t) − λ

X

l≥1

b

0,l

E[X

0

(t)X

l

(t)]

(27)

dX

1

(t)

dt

= −µX

1

(t) + λ

X

l≥0,l6=1

a

l

X

l

(t) + λ

X

l≥0,l6=1

b

l,1

E[X

l

(t)X

1

(t)]

(28)

dX

k

(t)

dt

= µX

k−1

(t) + λb

0,k

E[X

0

(t)X

k

(t)] + λ

X

l≥k+1

b

l,k

E[X

l

(t)X

k

(t)]

−λ

k−1

X

l=1

b

k,l

E[X

k

(t)X

l

(t)] − (λa

k

+ µ)X

k

(t),

k ≥ 2.

(29)

Let

X

k

:= lim

t→∞

X

k

(t)

(a.s.) and

X

k

:= lim

t→∞

E[X

k

(t)] = E[X

k

]

where the latter

equality omes from the bounded onvergen e theorem (Hint:

0 ≤ X

k

(t) ≤ N

). From (28)-(29)wend

a

0

X

0

=

−

X

l≥1

b

0,l

E[X

0

X

l

]

(30)

µX

1

=

λ

X

l≥0,l6=1

a

l

X

l

+ λ

X

l≥0,l6=1

b

l,1

E[X

l

X

1

]

(31)

µX

k−1

+ λb

0,k

E[X

0

X

k

] + λ

X

l≥k+1

b

l,k

E[X

l

X

k

] = λ

k−1

X

l=1

b

k,l

E[X

k

X

l

] + (λa

k

+ µ)X

k

,

k ≥ 2.

(32)

Wededu efrom (30)that

X

0

= b

0,l

E[X

0

X

l

] = 0

forall

l ≥ 1

,sothat (31)-(32)be ome

µX

1

=

λ

X

l≥2

a

l

X

l

+ λ

X

l≥2

b

l,1

E[X

l

X

1

]

(33)

µX

k−1

+ λ

X

l≥k+1

b

l,k

E[X

l

X

k

] = λ

k−1

X

l=1

b

k,l

E[X

k

X

l

] + (λa

k

+ µ)X

k

,

k ≥ 2.

(34)

Remark3.1 Itisnotsurprising that

X

0

= 0

sin e, when

K = ∞

,any node willre eivea versionof

F

withprobability oneatsomepointintimeandforthis timeonwardswillnever beempty. Thisimpliesthat

X

0

= 0

.

Wewill onsidertwo ases.

Case (a):

b

k,l

= 0

forall

k, l

. This ase orrespondsto thenon- ooperativesetting studied inSe tion 2when

K = ∞

. Wend(Hint: use

P

k≥1

X

k

= N

)

X

k

=

N

Q

k

j=2

µ+λa

µ

j

P

j≥1

Q

j

i=2

µ

µ+λa

i

,

k ≥ 1.

(35)

If we perform the hange of variable

µ/(µ + λa

i

) = 1 − p

i−1

in (35) we retrievethe or-responding results (8) found in the dis rete-time setting with

K = ∞

(see Remark 2.1), therebyshowingthatthismodelisthe ontinuous-timeanalogofthedis rete-timemodel.

Case (b):

a

k

= a > 0

and

b

k,l

= b > 0

forall

k

,

l

. Be ause oftheterms

E[X

k

X

l

]

equations

(31)-(32) annot be solved. To solve them we will assume that ov

(X

k

, X

l

)

is negligible

for

k 6= l

so that

E[X

k

X

l

] ≈ X

k

X

l

. We onje ture that this approximation (referred to

as the mean-eld approximation  see e.g. [1℄) is a urate for large

N

(the mean-eld approa hin [6,Theorem3.1℄ doesnotapply hereand annotthereforebeused tovalidate theseapproximations).With thisapproximationandtheuseoftheidentity

P

k≥1

X

k

= N

, equations(31)-(32)be ome

bX

2

₁

− X

1

(bN − a − µ/λ) − aN = 0

(36)

bX

2

_k

− X

k

bN − a − µ/λ − 2b

k−1

X

l=1

X

l

!

+ µX

k−1

/λ = 0

(37)

for

k ≥ 2

. Theuniquenonnegativerootof(36)is

X

1

=



D

1

+

q

D

2

1

+ 4abN



/2b,

(38)

whilefor

k ≥ 2

wegetfrom (37)

X

k

=



D

k

+

q

D

2

k

+ 4bµX

k−1

/λ



/2b

(39) with

D

k

:= bN − a − µ/λ − 2b

P

k−1

l=1

X

l

. Equations(38)-(39) dene a re ursive s heme allowingthe omputationof

X

k

forany

k

.

3.1.3 Quantitative performan e when

K

is nite Inthis asethesystemis alwaysstable sin e

|Y

i

n

| ≤ K

forall

i = 1, . . . , K

,

n ≥ 1

.

When

K < ∞

thesteady-stateequationssatisedby

X

0

, . . . , X

K

are

µX

K

= λa

0

X

0

+ λ

K

X

l=1

b

0,l

E[X

0

X

l

]

(40)

λa

0

X

0

+ λ

K

X

l=2

a

l

X

l

+ λb

0,1

E[X

0

X

1

] + λ

K

X

l=2

b

l,1

E[X

l

X

1

] = µX

1

(41)

µX

k−1

+ λb

0,k

E[X

0

X

k

] + λ

K

X

l=k+1

b

l,k

E[X

l

X

k

] = λ

k−1

X

l=1

b

k,l

E[X

k

X

l

] + (λa

k

+ µ)X

k

(42) for

k = 2, . . . , K − 1

,and

µX

K−1

= λ

K−1

X

l=1

b

K,l

E[X

K

X

l

] + (λa

K

+ µ)X

K

.

(43)

Similarlyto thesettingwhere

K = ∞

letusbriey onsiderthefollowing ases.

Case (a):

b

k,l

= 0

forall

k, l

. This ase orrespondsto thenon- ooperativesetting studied inSe tion 2when

K < ∞

. Easyalgebrayields

X

0

=

N

Q

K

i=2

µ+λa

µ

i

R

,

X

k

=

N (λa

0

/µ)

Q

k

i=2

µ+λa

µ

i

R

,

k = 1, . . . , K

where

R := (λa

0

/µ)

P

K

j=1

Q

j

i=2

(µ/(µ + λa

i

)) +

Q

K

i=2

(µ/(µ + λa

i

))

.

Case (b):

a

k

= a > 0

and

b

k,l

= b > 0

for all

k, l

. Similarlyto Case (b) in Se tion 3.1.2

we assume that ov

(X

k

, X

l

) ≈ 0

, or equivalently

E[X

k

X

l

] ≈ X

k

X

l

, for all

k 6= l

, sin e withoutthis assumption itdoesnot seempossible to solve(40)-(43). We onje ture that thisassumptionisjustiedwhen

N

,thenumberofnodes,islarge.

From(41)wendthat

X

1

isgivenby(38). From(42)wendthat

X

k

isgivenby(39)for

k = 2, . . . , K − 1

. Itremainsto al ulate

X

0

and

X

K

. Equation(43)gives

X

K

=

µX

K−1

λ



a + b

P

K−1

l=1

X

l



+ µ

(44) sothat from(40)

X

0

=

µX

K

λ



a + b

P

K

l=1

X

l

 .

(45)

Remark3.2 The asewherethe rateatwhi hthesour emeetsanodeisdierentfromthe rate atwhi h twonodesmeetisastraightforwardextensionofthe analysisinSe tions3.1.2 and3.1.3.

3.1.4 Numeri al results

Wewanttoquantifythe impa tofnode ooperationonthesystemperforman ewhenthe sour ehaslimitedpowerresour es. Wewantto optimizethesystemutility

P

K

k=1

U (k)X

k

under a onstraint,denoted by

V

, on theexpe ted numberof transmissionsby the sour e between the reation of two onse utive version of

F

. To this end, we will assume that the sour etransmits to anynodethat it meets with the probability

a = a

⋆

, where

a

⋆

_:=

min(1, (1 + ρ)V /N ρ)

isthestati poli ythatsolvesproblemP(Proposition1). Let

q

ρ

bethe

probabilitythatthesour emeetsagivennodebetweentwo reationsofanewversionof

F

.

Wehave

q

ρ

= λ/(λ + µ) = ρ/(1 + ρ)

thankstothePoissonassumptions. Inallexperiments

reported below we set

N = 20

,

V = 2

and

K = 5

. Figures 8-9 display the mapping

q

ρ

→

P

K

k=1

U (k)X

k

(with

U (k) = 1

in Fig. 8 and

U (k) = 1/k

in Fig. 9) fortwo values

ofthe probability

b

. The value

b = 0

orrespondsto thenon- ooperativesetting( ase(i); urvereferredtoasnon ooperative)andthevalue

b = 0.05

orrespondstothe ooperative setting( ase(ii)). For

b = 0.05

theresultshavebeenobtainedbothbysimulationsandfrom the approximationformulas (38)-(39). Note that theapproximationdevelopedin Se tion 3.1.2works wellwhen

b = 0.05

(resultsare notasgoodas

b

in reases). Oneobservesthat the ooperativesettingoutperforms thenon ooperativesettingevenwhen theprobability of ooperation

b

issmall,andthatthegainofnode ooperationin reaseswith

q

ρ

.

3.2 S enario II: Nodes only know the date of reation of their le

Thestateofanode(in ludingthesour e)at time

t

isnowdened asthedateof reation oftheversionof

F

thisnodeis arrying,ifany,attime

t

. Morepre isely,let

Y

˜

i

n

bethedate of reation(bythesour e)oftheversionof

F

thatnode

i = 1, . . . , N

is arryingjustbefore the

n

thevent takes pla e, namely at time

t

n

−

. By onvention

Y

˜

i

0

5

10

15

20

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

b=0.05: simulation

b=0.05: approximation

noncooperative

Figure8:

q

ρ

→

P

5

k=1

X

k

(

a = a

⋆

,

b ∈ {0, 0.05}

,

N = 20

,

V = 2

,

K = 5

)

0

2

4

6

8

10

12

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

b=0.05: simulation

b=0.05: approximation

noncooperative

Figure9:

q

ρ

→

P

5

k=1

X

k

/k

(

a = a

⋆

,

b ∈ {0, 0.05}

,

N = 20

,

V = 2

,

K = 5

)

arryanyversionof

F

at time

t

n

−

. Similarly,let

Y

˜

0

n

bethedateof reationofthe urrent versionof

F

thatthesour eis arryingattime

t

n

−

. Without lossofgeneralityweassume thattherstversionof

F

is reatedbythesour eattime

t = 0

.

Dene

˜

Z

n

i

:= ˜

Y

n

0

− ˜

Y

n

i

,

i = 0, 1, . . . , N, n = 1, 2, . . .

(46) andlet

˜

Z

n

:= ( ˜

Z

n

1

, . . . , ˜

Z

n

N

) ∈ IR

N

+

,

Z

˜

:= { ˜

Z

n

}

n

.

(47) Therv

Z

˜

i

n

is theagedieren ebetweenthemostre entversionof

F

andtheversionof

F

that node

i

is arrying(if any) attime

t

n

−

. Inparti ular

Z

˜

0

n

= 0

sin e thesour ealways arriesthemostuptodate versionof

F

.

We introdu e the

{0, 1}

-valued rvs

{˜

a

i,j

n

(y, z)}

where

˜

a

i,j

n

(y, z) = 1

if node

i

in state

y

re eivesale from node

j

in state

z

given that theymeetat time

t

n

. We further assume that

˜

a

i,j

n

(y, z) = 0

if

y ≥ z

, namelyanode annot re eiveale from anothernodethat is

olderthantheoneit arries.

Thefollowingdynami shold

˜

Y

i

n+1

= ˜

Y

n

i

+

N

X

j=0

j6=i

( ˜

Y

j

n

− ˜

Y

n

i

) ξ

n

i,j

˜

a

n

i,j

( ˜

Y

n

i

, ˜

Y

n

j

),

i = 1, . . . , N,

(48) and

˜

Y

n+1

0

= ˜

Y

n

0

+ (t

n

− ˜

Y

n

0

)ζ

n

.

(49)

Subtra ting(48)from(49)andusingthedenition (46)yields

˜

Z

i

n+1

= ˜

Z

n

i

+

N

X

j=0

j6=i

( ˜

Z

j

n

− ˜

Z

n

i

) ξ

i,j

n

a

˜

i,j

n

( ˜

Y

0

n

− ˜

Z

n

i

, ˜

Y

0

n

− ˜

Z

n

j

) + (t

n

− ˜

Y

n

0

)ζ

n

.

(50) AssumptionsA2:

(1) the su essivemeeting times betweenany pairof nodes (in ludingthe sour e) form a Poisson pro ess with rate

λ

. These

N (N + 1)/2

Poisson pro esses are mutally independent;

(2) thesu essivedates of reationbythe sour eofanew versionof

F

followaPoisson pro esswithrate

µ > 0

;

(3) the

N (N + 1)/2 + 1

Poissonpro essesintrodu edin(1)-(2)abovearemutually

inde-pendent;

(4) forevery

n ≥ 1

,

i = 1, . . . , N

,

j = 0, 1, . . . , N

,

i 6= j

,thereexistsamapping

ˆ

a

i,j

n

: IR →

{0, 1}

with

ˆ

a

i,j

n

(x) = 0

for

x ≤ 0

,su hthat

a

˜

i,j