Analytical Models of Floating Content in a Vehicular Urban Environment

Contents lists available at ScienceDirect

Ad

Hoc

Networks

journal homepage: www.elsevier.com/locate/adhoc

Analytical

models

of

floating

content

in

a

vehicular

urban

environment

Gaetano

Manzo

a

,

Marco

Ajmone

Marsan

b

,

Gianluca

A.

Rizzo

c ,∗

a_{UniversityofBern&HES-SOValais,Switzerland}

b_{IMDEANetworksInstitute,Spain&PolitecnicodiTorino,Italy} c_{HES-SOValais,Switzerland}

a

r

t

i

c

l

e

i

n

f

o

Articlehistory: Received28June2018 Revised26December2018 Accepted14January2019 Availableonline15January2019

a

b

s

t

r

a

c

t

Among the many proposed opportunistic content sharing schemes, Floating Content (FC) is of special in- terest for the vehicular environment, not only for cellular traffic offloading, but also as a natural commu- nication paradigm for location-based context-aware vehicular applications. Previously published results on the performance of vehicular FC have mostly focused on the conditions under which content persists over time in a given region of space, without addressing other important aspects of vehicular FC perfor- mance, such as the effectiveness with which content is replicated and made available, and the system conditions that enable good FC performance. This work presents a first analytical model of FC perfor- mance in vehicular networks in urban settings. It is based on a new synthetic mobility model (called District Mobility Model - DMM), and it does not require a detailed knowledge of the road grid geome- try. We validate our model extensively, by comparison against numerical simulations based on real-world traces, and we prove our model accuracy under a variety of mobility patterns and traffic conditions. Our analytical and simulation results provide evidence of the effectiveness of theFC paradigm in realistic ur- ban settings over a wide range of traffic conditions.

© 2019 Published by Elsevier B.V.

1. Introduction

After manyyears of mostlyacademic interest, vehicular com-munications and networksare now one of the top priorities for several industrial segments. On the side of telecommunication equipmentmanufacturers, the5G PPP(5th GenerationPublic Pri-vatePartnership)hasidentifiedautomotiveasoneofthemost im-portant verticalsectorsfortheapplicationof5Gtechnologies [1] , and the IEEE LAN/MAN Standard Committee (IEEE 802) has de-veloped a specific standard for Wireless Access in Vehicular En-vironments(WAVE),labeled802.11p [2] .Ontheside ofcar manu-facturers, attentionis highto both connected cars providing ser-vices based on the inclusion of the vehicle in a smart environ-ment whereother vehicles, aswell asroadside units,are ableto exchange data withthe car, and to the extraction of value from thehugeamountofdatageneratedeverydaybyeachvehicle.This surgeofindustrialattentiontovehicularcommunicationsissurely duetothe fastprogressinautonomousvehicles,along the5

lev-∗ _{Correspondingauthor.}

E-mailaddresses:[email protected](G.Manzo),[email protected]

(M. Ajmone Marsan), [email protected], [email protected]

(G.A.Rizzo).

elsidentified by thetaxonomy anddefinitionintheSAE (Society ofAutomobileEngineers)document J3016 [3] ,butalsoto the in-creasingvolumeofsensordatathatcanbecollectedwithincars.

Whiletelecommunicationoperatorsaretryingtoproposetheir networksandtheirtechnologies(andinparticular5G)asthe solu-tionforallvehicularnetworkingapplications,someoftheplanned vehicular communication services impose very demanding QoS (QualityofService)targets.Accordingtotheclassificationof3GPP TS 23.203 [4] , V2X (vehicle to anything) messages fall into QCI (QualityClassIndicators)classes3,75 and79,andcan tolerate a delayof up to 50 ms with packet loss rate up to 10−3 _{or 10}−2_.

However,criticalapplications,suchasremotedriving,canimpose much stricter requirements, with latenciesas low as 10 ms (see 3GPPTS22.261 [5] ).Thisimpliesthat theuseofa telecommuni-cationsinfrastructuretocarry thehugevolume ofdatagenerated by carsforall the applicationsof interestmaynot be the wisest choice,considering thedifficultyinmeetingthespecifiedQoS tar-gets.Thisbecomesevidentifweconsiderthefactthatmostofthe dataofinterestaregeneratedatthenetworkedge(insidethe vehi-cleorclosetoit)andaredirectedtousers(othervehiclesor road-sideunits) inproximityofthevehicle.In sucha context, offload-ingvehiculardatatransfers(atleastthosewiththecharacteristics above)fromthetelecommunicationsinfrastructuretoadirectV2V

https://doi.org/10.1016/j.adhoc.2019.01.003

nificantportionoftheperformance studies onFC hasfocusedon theconditionsunderwhichcontentpersistsintheAZ(i.e.,“floats” fora “largeenough” amount oftime).Hyytiä et al. [6] introduces the criticality condition, a sufficient condition for the content to float indefinitely with very high probability, under various mo-bilitymodels. Desta et al. [8] introduces an analytical model for contentpersistenceforthecaseofoutdoorpedestrianmobilityin largeopenspaces,suchasacitysquare.However,foranypractical application, content persistence over time within the AZ is only a necessary condition for FC viability. Indeed, when the content persists,itisessential tocharacterizehowoftentheFC paradigm managestodeliverthe contentto theintended users.Despiteits importance,fewworksconsiderthisissue.Forsetupswherenodes move according to a random direction mobilitymodel, Ali et al. [9,10] characterizeanalyticallythesuccessprobability(i.e.,the prob-abilityofdeliveringcontenttousersintransitintheAZ)asa func-tionofsystemparameters.Alietal. [10] showshoweverthatthose resultsare heavily inaccurate when appliedto realistic scenarios, andinparticulartovehicularsetups.SeveralworksconsiderFCin thevehicularenvironment(seee.g., [11,12] ).However,theseworks still focuson content persistence within theAZ. Hence, howthe featuresofvehicularmobilitypatternsimpactFCperformance,and howtoengineeravehicularapplicationrelyingonFCforachieving atargetperformance,areissuesthatremaintodatestillopen.

This work ismotivatedby the needto better characterizethe FCperformance invehicularurbanenvironments.Wefocuson ur-banscenarios (whichin aEuropean contextmeansbuilt environ-mentswithapproximately50–100inhabitantsperhectare),andwe proposeananalyticalapproachtoperformanceevaluationofFCin vehicular scenarios. Our approach is based on mapping the pat-ternsofvehicularmobilityinan urbanenvironment intoa modi-fiedversion ofthe randomwaypoint (RWP)mobilitymodel. This allows the derivation of a performance model that is not based onanyspecific roadgrid geometry.Numerical assessmentsofour resultson measurement-basedmobilitytraces show that our ap-proach accurately models vehicular FC performance over a wide rangeofvaluesofsystemparameters.Throughsimulationson real-worldvehicular traces,we perform a firstcharacterization ofthe relationbetweenland uses (office/commercial, industrial,or resi-dential),their induced vehicularmobilitypatterns,andFC perfor-mance. As expected, we observethat in each scenario such fea-turesof mobilityasthelocal meannode density, themean con-tactrate,thedegreeofmixingbetweendifferenttrafficflowsplay a crucialrole indetermining the optimal AZdimensioning strat-egy,theachievablesuccessprobability,andthetypeofapplications whichFCcansupport.Moreover,andcontrarytowhatwas previ-ouslyshownin theliterature, ourresultssuggest that inrealistic urbanvehicularsettingsthe criticalityconditionisnota good in-dicatoroftheprobabilityforthecontenttofloatovertime.

Thekeycontributionsofthispaperarethefollowing:

• We propose a new synthetic mobility model (called District MobilityModel-DMM), whichcapturesthemain characteris-tics of urban mobility, withoutrequiring asinput the specific geometryoftheroadgrid;

outthecasesinwhichitcanbeinaccurate.

Therestofthispaperisorganizedasfollows.In Section 2 ,we presentthesystemmodel,andin Section 3 we presentthemain analytical results on FC success probability.In Section 4 ,we nu-mericallyassesstheaccuracyofouranalyticresults,andwe evalu-atethembysimulationonarealisticsetup.In Section 5 ,webriefly discusspreviouswork,andfinally Section 6 concludesthepaper.

2. Systemmodel

Weconsiderasetofwirelessnodesmovingonaregionofthe plane,withtransmissionranger(Table 1 ).Weassume twonodes come in contact when they are in the range of each other, i.e. whentheirdistanceisnotlargerthanr(Gilbert’smodel [13] ).With C(x)wedenotethethroughputbetweentwonodeswhentheyare atadistancex,x≤rfromeachother.C(x)can bemodeledusing Shannon’scapacitylaw,oranyothermorerealisticmodel.

Inwhatfollows,wefocusonanurbanenvironment.Coherently withtypical features ofvehicles mobilityinan urban setting,we assumethatmobilitypatternsofeachnodealternatebetweentime intervalsspentmoving,andtimeintervalsspentstillatawaypoint, whichcouldmodelapauseatacrossroad,orinaparkinglot.

Morespecifically,weassumenodesmoveaccordingtothe Dis-trictMobilityModel(DMM),whichabstractsfromthedetailsofthe geometryofthestreetgrid.Itsformaldefinitionisasfollows.

Definition 1(DistrictMobilityModel). We assumethat nodes ar-rivein aregion oftheplane accordingto a Poissonprocesswith intensity

λ

.Nodes,insidethe region,moveaccordingtothe RWP mobility model with pause, with velocity v constant and equal

Table1

Tableofnotation.

Parameter Description Units

r Nodetransmissionrange m

C(x) Throughputbetweenthetwonodesatdistancex Mb/s

λ Nodesarrivalrate s−1

v Nodespeed m/s

p ProbabilityofthenextwaypointwiththeAZ

Tm Movingtimewithinanepoch s

Ts Stoppingtimewithinanepoch s

Tt Movingtime(whencrossingtheAZwithoutpause) s

R AZradius m

ν Frequencyofcontactbetweentwonodes s−1 Q Probabilityofsuccessfulcontentexchange

N MeannumberofnodeswithintheAZ

n MeannumberofnodeswithintheAZwithcontent m MeannumberofnodeswithintheAZwithoutcontent Psucc Probabilitytoobtainthecontent

Pe Probabilitytoobtainthecontentinanepoch

Pm Probabilitytoobtainthecontentduringmovingtime

Ps Probabilitytoobtainthecontentduringstoppingtime

L Contentsize MB

B Bandwidth MHz

α Coefficientofattenuation cm−1

N0 Powerofspectraldensity W/Hz

for all nodes (though our approach can be easily generalized to scenarios wherevelocity variesbetweenwaypointsaccordingtoa givendistribution).

When a node arrives, its initial waypointis chosen uniformly atrandomontheborderoftheregion.Uponreachingawaypoint, eachnodepausesforarandomduration.Then,withprobabilityp, thelocationofthenextwaypointisselecteduniformlyatrandom withintheregion.Otherwise,withprobability1−pthelocationof thenext waypointisselecteduniformlyatrandomontheborder oftheregion.Oncereachedtheborder,anodedisappearsfromthe region.

AsinRWP,intheDMMthesojourntimeofeverynodewithin theregionconsistsinasuccessionofepochs,whereeachepochis thesumofthetimespent movingbetweentwoconsecutive way-points, (themovingtime),andofthetime spentstill atthe desti-nationwaypoint(thepausetime).Thedurationofmovingtime,Tm

andpause time,Ts,areassumedto be independentrandom

vari-ables. We denotetheir pdf with fTm and fTs, respectively.We as-sume suchpdfsto bethe sameforall nodes.Whena node does not stop into the region, its sojourn time is spent only moving fromonepointtotheotheroftheborder.Weindicatesuchepoch as a traversal epoch. With Tt and fTt we denote, respectively, its durationandtheassociatedpdf.Finally,giventwonodes,ofwhich atleastone moving,thetrajectory distanceisthe minimumvalue that thedistancebetweenthe twonodes takeswhilethey arein therangeofeachother.

One of the distinctive features of vehicular mobility is being stronglyinfluenced bythegeometryoftheroadgrid,andthe re-latedconstraints onnode speed anddirection.As a consequence, existing approaches to vehicular FC modeling focus on specific geometries, such as highways, highway junctions, or Manhattan grids,derivingresultswhicharehardlygeneralizable [11] .

Different fromthe typical vehicle mobility over a street grid, theDMMisisotropic(i.e.,atanypointinspace,alldirectionscan bechosenwithequalprobability)andunconstrained(i.e.,avehicle can occupyanylocation within thearea).This isthe reasonwhy wesaythattheDMMabstractsfromthedetailsofthegeometryof theroad grid.However,when a“sufficientlylarge” portion ofthe roadgridis includedinsidean AZ(say,a fewblocks),even ifthe possible instantaneous directionsof movement are finite,for any twopointschosenatrandomonthemap(ontheroadgrid),often thereexistsatleastonepathbetweenthem(albeitusuallynoton a straight line). Hence,both withDMM andinreality the move-mentbetweentwopointsispossible.Onamacroscopicscale,and forthepurposeofmodelingcontentreplication anddiffusion dy-namicswithintheAZ,theapproximationsintroducedbytheDMM onlyimpactthemeanmovingtimebetweentwowaypoints.

2.1. Floatingcontentbasicoperation

We assumethat atatime t0,a node intheplane (theseeder)

definesa circularareaofradius R,theAnchor Zone (AZ), contain-ing the node itself. At time t0 the seeder (blue node in Fig. 1 a)

generates a piece ofcontent (e.g.,a text message, a picture). For t≥t0,every time anode withthecontent comes incontactwith

a node without it within the AZ, the content is exchanged suc-cessfully with probability Q (Fig. 1 b). We assume thetime taken toreplicatethecontentisnegligiblewithrespecttocontacttime. Nodes enteringthe AZdonot possessa copyofthe content,and thoseexiting theAZdiscardtheircopy. Hence,thetypical behav-iorofanodeconsistsinenteringtheAZwithnocontent,receiving the contentfromanothernode, distributingthecontent tonodes withno contentmetwithin theAZ, leavingtheAZanddropping thecontent.Ifthenodethenre-enterstheAZ,itwillagainreceive the same content when a node withcontent is met.As a result

ofsuch opportunistic exchange, thecontent floats(i.e., itpersists probabilisticallyintheAZevenaftertheseederhaslefttheAZ).In thisway,thecontentismadeavailabletonodestraversingtheAZ (Fig. 1 c)forthewholedurationofitsfloatinglifetime.

Afirstperformanceparameter ofFC iscontentavailabilityata giventime,i.e.theratiobetweenthe numberofnodeswith con-tentoverthetotalamountofnodesinsidetheAZatthattime. In-deed,a highvalue ofavailability iscorrelatedwithlow likelihood of content disappearance from the AZ, but also with the proba-bilityofgettingthe contentfora node entering theAZ. Thislast feature is captured by the success probability, the probability for a node entering the AZ ofgetting out ofthe AZwith a copy of thecontent. Thisisthe main performance indicator,andtheone whichismostdirectly relatedtothe performanceof applications andservicesrelyingonFC.Indeed,itisadirectindicatorofthe ef-fectivenesswithwhichthefloatingcontentisdeliveredandmade availabletonodestraversingtheAZ.

Anotherrelevantperformance parameteristhemeantime the contentfromenteringtheAZ,whichisespeciallycrucialforsafety applicationsrequiringthat themessage reachesnodesassoonas possible.

3. Ananalyticalmodelforsuccessprobability

Inthissection,weintroducetheanalyticalresultswhichrelate thesuccessprobabilitytothemainsystemparameters.

For our derivation, we consider a system in equilibrium, in whichthemeannumberof nodesinsidetheAZdoesnot change overtime.FornodesmobilitywithintheAZ,weconsideradistrict mobilitymodel,withareacoincidingwiththefloatingcontentAZ. Thesojourntime fora nodein theAZisgivenby thesumof theduration oftheepochs spent within theAZ, plus thetime to enterandexittheAZ,weightedwiththeirprobabilities:

Tsoj=

(

1−p

)

Tt+

∞

k=1

pk_[k

(

_E[T

m]+E[Ts]

)

+E[Tm]]

(1)

andN=

λ

Tsoj isthe meannumberofnodesinthe AZ, byLittle’s

law.

Withqweindicatethemeanfractionofsojourntimeinwhich anodemoves:

q=

(

1−p

)

Tt+∞k=1pk

(

kE[Tm]+E[Tm]

)

Tsoj

Notethatwhenptendsto1,qiswellapproximatedby E[Tm]

E[Tm]+E[Ts]. In orderto derive an analytical expression for success proba-bility, we start with the following two results, which determine thefrequency withwhichany twonodes comewithin the range of each other in the AZ, andthe content availability (and hence themeannumberofnodeswithcontentintheAZ).

Lemma 1 (Frequency of contacts). In the district mobility model, consideringaregionwithareaA,themean frequencyofcontacts be-tweenanytwonodes,withmeanspeedvandtransmissionrange ra-diusr,is

ν

= 2rq

v

[2

(

1−q

)

+1.27q]

A (2)

Fig.1. BasicoperationofFloatingContent.1a)Seeder(blue)definestheAZ.1b)Opportunisticmessageexchangebetweennodes.1c)NodesgoingoutoftheAZ(red)discard thecontent.

Lemma 2 (Content availability). For R_>>r, when the criticality condition

NTsoj

ν

>1 (3)

issatisfied,thestationarymeannumberofnodesintheAZwith con-tent,n¯_,isgivenby

n=N¯− 1 Tsoj

ν

Q

Themeannumberofnodeswithoutcontent,denotedwithm¯,is

m= 1 Tsoj

ν

Q

(4)

Fortheproofof Lemma 2 wereferto Appendix B .

We now present our main result, which relates the success probabilitytotheprobabilityforagenericnodetogetthecontent duringtheepochsofitssojourntimeinsidetheAZ.

Theorem 1 (Success Probability). In the district mobility model, whenR>>r andthe criticality condition(3) is satisfied,the prob-ability fora node to get thecontent duringitssojourn timewithin theAZ,instationaryregimeis

Psucc=

(

1−p

)

Pt+

p

1−p

(

1−Pe

)

[pPe+Pm

(

1−p

)

] (5)

wherePeistheprobabilitythatanodegetsthecontentduringan

epoch(otherthanthefinalone),givenby

Pe=Pm+

(

1−Pm

)

Ps (6)

Ps is the probability of getting the content during the pause time,

givenby

Ps=

_+∞

0

1−e−ντn¯Q

_f

Ts

(

τ

)

d

τ

(7)

Pm is theprobabilityofgettingthecontentduring themovingtime,

givenby

Pm=

2R

v

0

1−e−ντn¯Q

_f

Tm

(

τ

)

d

τ

. (8)

WherethepdfofTmisgivenby

fTm

(

τ

)

=

4

τ

v

2

π

R2

arccos

τ

v

2R−

τ

v

2R

1−

τ

v

2R

2

(9)

Ptistheprobabilityofgettingthecontentduringthetransitionepoch.

Itsexpressionisthesameas(8),with fTt insteadof fTm:

fTt

(

τ

)

=

τ

v

2

2R√4R2₋

τ

2

v

2 (10)

Fortheproofof Theorem 1 wereferto Appendix C .

Note thatthe epochin whichthenode exits theAZcoincides withthetime spentmoving towardstheborderoftheAZ, asthe

nodeisassumedtodisappearoncereachedtheborder.Hencefor thefinalepochPe=Pm.

In realistic settings, every time two nodes come in range of eachother,asubstantial amountoftimeisrequiredbyeach node todetectthepresenceoftheothernode,andtoestablishthe con-nection.Tosuchtimeoverhead(whichisintherangeofafew sec-ondsinWi-FiDirect,upto10−12sinBluetooth [14] ,and sensi-blyinferiorinDSRC [15] ),itaddsupthetimeforcontenttransfer, whichisafunctionofcontentsizeLandofthecapacitybetween thetwonodesduringthecontacttime.

Thefollowingresultprovidesawaytoincludesucheffectsinto theperformance modelofvehicularFC. Let

τ

0 bethe meantime

takenbytwonodesincontacttoestablishaconnectionand coor-dinateforcontenttransfer.

Theorem2. When r<<R,and Ts is exponentially distributed with

mean 1/

μ

,theprobabilityof successfulcontentexchangeQ is given by

Q=2

(

1−q

)

qh

s

0

r +q

2hr0

r +

(

1−q

)

2_exp

₂

μ

τ

0+ L C0

(11) where

• hs

0 (respectively, hr0)is themaximum trajectory distance beyond

whichnosuccessfulcontenttransfercantakeplace,whenonlyone nodeis moving(respectively,whenbothnodesaremoving).hs

0 is

thesolutionofthefollowingequation:

L=

2√r2₋₍hs₀)2 v

τ0

C

(

hs

0

)

2+

r2₋

(

_hs

0

)

2−

v

t

2

dt (12)

• hr

0 is theexpectation of the maximum trajectorydistance hrwith

respect to the distributionof therelative speed vr between two

movingnodes:

hr

0=

2v

0

hr

(

v

r

)

fr

(

v

r

)

d

v

r

wherefr(vr) is thepdf oftherelativespeedbetween twomoving

nodes:

fr

(

v

r

)

=

v

r

v

2

π

₁₋

(

v

_r/

v

₋₁

)

2

andwherehr_(v

r)isthesolutionofthefollowingequation:

L=

2√r2₋₍hr_(vr))2 vr

τ0

C

(

hr

(

v

r

))

2+

r2₋

(

_hr

(

v

r

))

2−

v

rt

2

dt

(13)

• C0isthemeanthroughputbetweentwostaticnodes:

C0=

2

r2

r

0

xC

(

x

)

dx

Fortheproofof Theorem 2 ,pleasereferto Appendix D . Note that Theorem 2 assumesthat on a contactbetweentwo nodes, the time for the content to be successfully transferred is not affected by other alreadyongoing content transfers involving the two nodes. This is a reasonable assumption for setups with low node densities, i.e., setups where it is unlikely that a node is simultaneously in contact with several other nodes, and with r<<R,whilebringingtooptimisticperformanceestimationsinall othercases.Finally,notethatresultsinthissectionhavebeen de-rived underthe non-spatialapproximation,by which,forthe sake ofanalyticaltractability,nodeshavebeenassumedtobeuniformly distributedwithintheAZ. In Section 4 ,wenumericallyassessthe impactofsuchassumptionsonthemodelaccuracy.

4. Simulationandvalidation

In this section, we illustrate the numerical validation of our model,andthenumericalassessmentoftheperformance ofFCin a vehicular environment, based on two scenarios. In a first sce-nario,we considernodes moving accordingtothe DMM,andwe compare simulationresults withtheanalytical predictions ofour model.OurgoalistocharacterizeFCperformanceasafunctionof themainsystemparameters,andtovalidate Theorem 1 by assess-ing the impact ofthose effects, such asclustering ofnodes with contentandbordereffects,whicharenotincludedinourmodel.

In a second scenario, inorder toperform a more realistic as-sessment,weconsiderasetupwheremobileusersmodelvehicles onthestreetsoftwoEuropeancities,movingaccordingtoasetof measurement-basedmobilitytraces.

In both scenarios, simulations are performedusing the VEINS framework [16] ,basedonOMNET₊₊ [17] fornetworksimulation, andonSUMO [18] forroadtrafficsimulation.

Unlessotherwisestated,we assumethatcontentistransferred (withprobability Q₌1) everytimetwo nodescomeinthe range of each other, regardless of the amount of time spent in range. This corresponds to the case in which the content size is small enough, for content transfer time, to be much smaller than the typicalmeancontacttime inurbanscenarios.Thisisthecase, for instance, of such applications as traffic warning, accident warn-ing, ortextual advertisements.We assume Ts tobe exponentially

distributed, with mean 1/

μ

. In what follows, we model C(x) us-ingShannon’scapacitylaw(thoughanyother,morerealisticmodel canbeused):

C

(

x

)

=Blog2

1+Px−α N0

where

α

istheattenuationcoefficient,N0thepowerspectral

den-sity of the additive white Gaussian noise, andB is the available bandwidth. We donot includethe effects offadingand interfer-ence.Notehoweverthatourapproachcaneasilybeadaptedto in-corporatesucheffects.

Finally,withrespecttotheexpectedvaliditytimeofthefloated content,wegroupvehicularservicesrelyingonFCintotwobroad categories.The“nearreal-time” categorytypically floatsmessages with a very short validity in time (a few minutes). Examples aresituatedintroductions,orinfrastructure-lessridesharing,where passing cars inform neighboring pedestrians of their availability and their planned trip. Services in the “delay tolerant” category, instead,areassociatedtoeventswithslowerdynamics,andhence they requirelongerfloating lifetimesof contentinside theAZ(of theorderofone-twohours).

4.1. Baselinescenario

In order to assess the accuracy of our model, we performed asteady-stateanalysisofthesystem. Eachsimulationrun started withan empty AZ. Then, afterwaitinglong enoughfor the tran-sient on node population to be exhausted, we assumedthat the closest node to thecenter ofthe AZgeneratesa message, andit startsreplicating it opportunistically. Finally, once exhausted the transienton content population inthe AZ, we measured content availability within the AZ, averaged over two hours. Being this a steady-state analysis, we only retained data forthose contents whichfloat.Moreover,forthewelfareofhomogeneityinmeasured data, we retained only simulation results for which the content floatsforthewholesimulationtime.Notethat,sincethecriticality conditionwassatisfiedinall simulationsetups, thevast majority ofthecontentfluctuatedforthewholesimulationtime.

Inaddition,foreachsimulation,we measuredtheaverage suc-cessrateoverthesimulationtime.Theaveragesuccessrateis de-finedasthe fractionofnodeswhichleave theAZwitha copyof thecontent duringthesimulation time, anditis an estimatorof thesuccessprobabilityforthegivencontent.

Unlessotherwisestated,weconsideredanAZradiusR=50m, arrivalrate

λ

₌0_.1s−1_{, anda meanmoving time9s. The}

proba-bilitypforanode toremainintheAZafterapausehasbeenset to0.9.

In Figs. 2 and 3 , respectively, we plot the values of success probabilityandavailability derived fromsimulations,asfunctions oftheratiobetweenthetransmissionrangeandtheAZradius,for differentvaluesofmeanpausetime1/

μ

,andhenceofthefraction ofmovingnodesq.

In both plots, we compare simulation results with analytical predictions.

As the plots show, both performance parameters increase monotonicallyasafunctionofr/R.Indeed,botharedirectlyrelated tocontactrate, andfrom Lemma 1 we knowthat contactrateis directlyproportionaltor/R.

Therangeofvaluesofr/Rwaschosenasfollows.Valuesofr/R below0.1do not allow thecontent to floatformore than a few minutes(thecriticalityconditionisnotsatisfied).Whenr/R>1 in-stead,evenatverylownode densities,nodeswithintheAZform astronglyconnectedcomponent,bringingsuccessprobabilityvery closetoone.

Fig.2. Successprobabilityovertransmissionradius,with95%confidenceintervals.

Fig.3. Availabilityovertransmissionradius,with95%confidenceinterval.

movingwithin theAZ, thusdecreasingthe overallcontactratein theAZ.Aswecanseefromtheplots,thenetresultisanincrease ofsuccessprobabilityandsuccessrate.

Wealsoseethatthemarginalbenefitofincreasingr/Rishigher forhighvaluesofq,indicatingthat themoremobile nodesthere are,the higher are theadvantages ofa large transmission range. Fig. 3 showsthatavailabilityhasasimilardependencywithrespect tor/Randqassuccessprobability.Thefigurealsoshowsthatthere isnodirectproportionalitybetweenavailabilityandsuccess proba-bility,andthattypicallyavailabilityislowerthansuccess probabil-ity(indeed,successprobability istheavailabilityofnodesleaving theAZ, whichtypicallyhavemorechancesofpossessingthe con-tentthantheaveragenodeintheAZ).Hence,astheplotssuggest, anddespite beingoften chosen as the main FC performance pa-rameter,availabilityis,ingeneral,apoorindicatorofsuccess prob-ability.Wecanalsoseethat,increasingthemeanamountoftime spentmovingbyanodeallowsdecreasingtheavailabilityrequired forachievingagivenvalueofsuccessprobability.

Finally, we analyzed the impactof finite content size and ca-pacity between two nodes on content diffusion and on FC per-formance. We assumedcontent ofsize L=5 MB, toemulate the exchange ofmultimedia content(e.g.,a video), attenuation

α

₌3 (typicalofurban scenarios),B=1 MHzanda transmitterSNR of 5 dB (which models setups with high interference, common in crowded city centers, e.g. at the 2.4GHz frequency). As for mo-bility,we haveassumeda speed of10m/s,andmeanpause time of10s, inorderto emulatethemobilityinurban scenarioswith intense traffic. The initial delaybefore content exchange

τ

0 was

Fig.4. ProbabilityofsuccessfulcontentexchangeQvstransmissionrange,forthreevaluesofinitialdelayτ0.Simulationsarewitha95%confidenceintervalof4%.

a change inmobility(from staticto moving, or viceversa) takes place, nor itincludes border effects.Indeed, the loss ofaccuracy increaseswithincreasing

τ

0becausethoseeffectsaremorelikely

toimpactthosecontactswhichlastlonger.

The effects offinite contentsize andlimited channel capacity onFCperformancearealsovisiblein Figs. 2 and 3 ,whereweplot success probability andavailability,fora content ofsize 4MB.As expected, theimpact of thesefactors on performance is relevant for values ofmean contact time less than orequal to the mean time takenbyacontacttransfer,whileslowlydecreasingwith in-creasingtransmissionrange.

4.2. Assessmentinurbanscenarios

In orderto test our model ina realistic context, andto char-acterize those conditions in which FC is feasible in a vehicu-lar environment, we considered two European city scenarios, in which nodes move on an urban road grid, and according to measurement-basedvehicularmobilitytraces.

Specifically, we consider a section of the area of Luxembourg City,andofCologne,Germany.TheLuxembourgscenario(Fig. 5 a) consistsofansurfaceof155.95km2_{,whichincludes thecity}

cen-ter as well as part of its surroundings. The street grid and the measurement-basedmobilitytracesforthisscenariofora24h pe-riod have been derived from [19] . The Cologne scenario extends overanareaof400km2_{,(Fig. 5 b)inwhichtheroadgridwas}

de-rivedfrom [21] ,whilemobilitytracesarederivedfromthe TAPAS-Cologne project [20] .Given the heterogeneity of the urban envi-ronment, inorder to capturethe effects on FC performance of a specific road structure and of the resulting mobilitypatterns, in bothsettingswechosethreedifferentlocationsasAZcenters.One is set inthe city center(downtown, area “C”),another in a resi-dential area (area“R”),anda third intheindustrial district(area “I”).

Inordertotake intoaccounttemporal fluctuationsinthe mo-bilitypatternsofthesecities,weconsideredtwodifferenttime in-tervalsoverthecourseofthe24h.Afirstone,from7AMto9AM, corresponds toaperiodofpeak traffic,duetopeople commuting towork.Asecond timeintervalwaschosenearlyinthemorning, from2.30AMto 5.30AM.Beingthistypicallyaperiodofverylow cardensity,itusually representsaworst-casescenarioforFC per-formance. Forbothtime intervals, we chose aduration that isof thesameorderofmagnitudeofthetypicalvalidityofmessagesin suchservicesastrafficcongestionwarnings,oraccidentwarnings.

In Table 2 welistsomeofthemainparametersofmobilityfor thethreeAZs,foran AZradiusof260m.Thesevaluesshow that, withineachcity,thethreedistrictsdiffersubstantiallyintermsof meannode density, meansojourn time, and mean contactrates. Morespecifically,inbothcities,theindustrialdistrictandthecity centerhavea sensiblyhighernode densityandcontactratethan the residential district.Indeed, asthe maps in Fig. 5 show, typi-callytheresidentialzoneismainlyatransitzoneforvehicles,with littlelocaltraffic, andwithfew main roadscarryingthe majority ofthetraffic ofthearea.Thisimplies,onaverage,shortersojourn timesandhencefewer opportunities forcontentreplication with respecttootherareasofacity.Despitethesedifferences,however, weobservedthatmeanpausetime,meanmovingtime,andmean speed are essentially the same for the three districts in Luxem-bourgCity(andequaltoabout15s,25s,and

v

=14m/s, respec-tively),andtheydonotvarysignificantlyinthecourseofthe24h. TheColognescenarioischaracterized,ineach AZ,byasimilar ar-rivalrateasthe correspondingAZs inLuxembourgcity.However, duetodifferencesin roadgrid geometry(andinparticular toan averageblock sizelargerthaninLuxembourgCity),itisalso char-acterizedbyaloweraveragestoppingtime,ahigheraveragespeed (16m/s),longermovingtime(30s),andalowermeancontactrate. The latterfeature is probably relatedto the fact that the blocks, beinglargeronaverage,makeitmoredifficultfornodeson oppo-sitesidesofablocktocomeintherangeofeachother.Notethat, forthecomputationofthemeanpausetime,weassumedthatcars whichareparkeddonotparticipateincontentexchange.

Ineachsimulation run,theadoptedseeding strategy isas fol-lows.Among allnodespassingintheAZinthefirst5minofthe giventime interval,wechooseatrandomanode asseeder.Then, fromthemomentinwhichtheseederenterstheAZ,wesimulate theprocessofcontentreplicationandfloatingforthewhole dura-tionofthetimeinterval.SuchstrategymodelsthoseFCapplication contextsinwhich asinglevehicleisthe producerofthecontent. Thisisthecase, forinstance,ofinfrastructure-less carsharing,in whicheachcarofferingaliftadvertisesitsavailability(plus, possi-bly,itsapproximatelocationandtripplan),orofparking manage-mentapplications,inwhichacarleavingaparkinglocation adver-tisesthelocationoftheavailableparkingspot.

Fig.5. TheareasofLuxembourgCityandCologneconsideredinourevaluations.Ineacharea,theyellowcirclescorrespondtothreeAZwithR=260m,locatedinthecity center(C),intheindustrialdistrict(I),andinaresidentialdistrict(R).

Table2

MainmobilityparametersintheLuxembourgandColognescenarios,forthethreeAZs inFig.5forlow-density(2.30AM−5.30AM)andhigh-density(7AM−9AM)traffic pe-riods.AZradiusR=260m,transmissionranger=20m.

CityScenario Sojourn Contact N.ofnodes Arrival Time Rate intheAZ rate Tsoj ν(N−1) N λ

Luxembourg C lowtraffic 335.7 0.04 29.5 0.087 peaktraffic 368.4 1.16 489.2 1.327

I lowtraffic 412.2 0.05 38.7 0.0938

peaktraffic 440.8 0.72 512.1 1.161

R lowtraffic 189.6 0.0063 1.9 0.010

peaktraffic 192.4 0.043 12.9 0.067

Cologne C lowtraffic 192.2 0.037 20.7 0.107

peaktraffic 245.8 0.75 294.9 1.257

I lowtraffic 258.3 0.029 32.9 0.128

peaktraffic 311.8 0.81 326.5 1.051

R lowtraffic 124.3 0.0011 2.6 0.021

peaktraffic 136.1 0.018 10.7 0.079

themodelin Section 3 .The factthat ourresults maintaina high accuracyon very different roadgrid scenarios, andmobility pat-ternssuggeststhat thedistrictmobilitymodeliseffectivein cap-turingthoseaspectsofvehicularmobility,whichhaveasignificant impactonFCperformanceinurbansettings.

In theindustrial area, thenode densityandmobilityare such thateven withasmalltransmission range(26 m)success proba-bilityisveryhigh(above0.8inLuxembourg,andslightlylowerin Cologne).AsforthecitycenterAZ,despitehavingonlymarginally lowernode density,itexhibitsamarkedlysmaller success proba-bilityatlow valuesofr/R.Similarly towhat seen inthebaseline scenario, in scenarios with smaller sojourn times, increasing the ratiobetween transmission range and AZradius brings to larger marginalincreasesinsuccessprobability.Finally,inbothcitiesthe plotsindicatethat,evenatpeaktraffic,the residentialarea isthe most critical for FC performance, requiring a large ratio r/R to achieve highsuccess probabilities.Note that inpractical settings, valuesofr/Rcloseto onedefeat thepurposeofhaving anAZ, as thevastmajorityofusers whoshould getthecontent arewithin transmissionrange.

Wealsonote that thehighertraffic “fluidity” inCologne,with lessfrequentstops,andwithaslightlylessdenseroadgrid,brings ineach districttogenerallyinferiorperformance intermsof suc-cessprobabilityandavailability.Thehighertrafficregularitybrings however also to a higher level of accuracy between model and simulations,forasamenumberofsimulations,asaveragesof sys-temparameters andofmobilityparametersare more representa-tive ofthe actual mobilityfeatures intheconsidered observation window.

Fig.6. Successprobabilityovertransmissionradius,inthethreeAZs,fortheLuxembourgandColognescenarios.95%confidenceinterval.

Fig.7. SuccessProbabilityoversimulationtime.SeveralseedingstrategiesinthecitycenterofLuxembourgCityathigh-densitytraffic(7AM−9AM).AnchorZoneradius 260m,transmissionradius160m.Att=0thereare94nodeswithintheAZ.Meanoversamplepopulationof20(eachpoint)every300s.

the fractionsofnodesthat receive thecontentare 75%,50%, 25% orjustonenode.Weclearlyseethattheseedingstrategyonly af-fects the duration of the transientbefore the success probability stabilizes toslightlymorethan0.9. Asimilar,butmorenoisy be-haviorisobservedin Fig. 8 ,wherewereport theevolutionofthe availabilityversustimeinthesameconditions.

Fig.8. Availabilityoversimulationtime.SeveralseedingstrategiesinthecitycenterofLuxembourgCityathigh-densitytraffic(7AM−9AM).AnchorZoneradius260m, transmissionradius160m.Att=0thereare94nodeswithintheAZ.Meanoversamplepopulationof20(eachpoint)every300m.

Fig.9. Empiricaldistributionfunctionofclustersizeversusr/RfortheindustrialAZintheLuxembourgscenario,forthelow-densitytrafficperiod.

gettingoutoftheAZwiththecontentisaffectedbythecompound effectofmanysuchfluctuationsalong thewhole sojourntime of a userin the AZ. As it can be seen from Fig. 7 ,such compound effectbrings to smoothingout fluctuationsinsuccess probability overtime.

One ofthemainfeatures ofmobilitywhicharenot accounted for by our model is node clustering, which typically brings to a higher rate of contacts, and hence to a larger amount of op-portunitiesforcontent exchange withrespect toscenarios which arehomogeneous inspace.In particular,giventhat ourapproach is based on mapping mobility features to a synthetic mobility model,such astheDMM,which tendstodistributenodesevenly in the AZ, we expect our approach to yield substantially pes-simisticestimations ofFCperformance inscenarioswitha signif-icantamountofclustering. Toclarifythisaspect,wehave consid-ered,for aspecific AZ (the industrialAZ inthe Luxembourg sce-nario), the empirical distribution of cluster size. In order to be conservative,we haveconsidered the low-density traffic time in-terval(2.30AM−5.30AM), whenclustering phenomenashouldbe attheirdaily minimumforthe consideredarea. Fig. 9 showsthe empirical distribution of cluster size for three values of the ra-tior/R. We seethat the amount of clusteringis significant even at small values of transmission range, and that the cluster size (i.e., the number of vehicles in the cluster), as well as the

rel-ative number of clusters, increase with transmission range, as expected. This observation, together with the high model accu-racyatevery value oftransmissionrangeobservedin Fig. 6 , sug-geststhatourapproachisrobustagainst veryhighlevels ofnode clustering.

InordertogetinsightsontheseperformancefeaturesofFC,we analyzedtheevolution overtimeofthemeancontent availability foreachofthethreeAZsintheLuxembourgscenario.In Fig. 10 we plot(inred)themeancontentavailabilityovertimefor30 simu-lationruns.

Fig.10. Percentageofnodeswithcontent,meanfractionofnodeswithcontent,andmeanavailabilityoverfloatingtimeforthethreeAZsareasintheLuxembourgscenario. Ineachrunadifferentseedernodehasbeenchosen.Thelowtraffictimeintervalis2.30AM−5.30AM,whilethepeaktraffictimeintervalis7AM−9AM.Transmission radiusis100m.

of the effective likelihood for a node inside the AZto possess a copyofthecontentatagiventimefromcontentgeneration.

In all configurations, the criticality condition (3) is satisfied. However,inthelowtrafficperiodwithR₌260m,onlyinthe in-dustrial district the content floats (in mostof the cases) forthe wholedurationoftheinterval(Fig. 10 a).InthecitycenterAZ, in-stead, (Fig. 10 f), despite achievingrapidly a high availability (0.7 within the first 10 min), on average content floats for 30 min, while after an hour content has almost invariably disappeared. Similarbehaviorisobservedintheresidentialarea,wherethe con-tentfloatsforatmost10min(notshown).

Summarizing, in low traffic periods,even withrelatively large ratios r/R, content floats for a few minutes at most, making FC suitable onlyfor“nearrealtime” applications,withshortlifetime. In theresidential zone, atall times ofthe day,a large AZradius is required to float the content with high probability for more than 30 min.In low traffic, extendingthe AZ radiuswhile keep-ing constant the transmission radius has an overall positive im-pactonavailability,andonfloatinglifetime,inallpartsofthecity (Fig. 10 b,hande).

Whencontentfloatsforthewholetwohours,wenoteastrong correlation betweenthe evolution ofavailability ofdifferent con-tentovertime(Fig. 10 a,b,candh).Inaddition,weseeavailability taking onlya finite set ofvalues overtime. This is an indication oftheexistenceoffewclustersofvehicles,withlittleinter-cluster exchanges.Onlytheincreaseofthenumberofvehiclesarrivingin

theAZ(at around4000s fromthebeginning offloatingtime, in thelow traffic time interval,see Fig. 11 ) increases theexchanges betweentheseclusters(sharpincreasein Fig. 10 a,bandh).

Finally,fromtheseresults,itclearlyemerges how the critical-ityconditionis a poorindicator of the actual feasibilityof FC in realistic scenarios. However, they also show that, at least in the considered urban settings, by appropriately settingthe AZradius (withinreasonablelimits,i.e.notextendingittothewholecity)it isalways possibleto setcontent tofloatfor aconsistent amount oftime.

4.3.Limitationsofourapproach

Fig.11. MeannumberofnodesinthethreelocationsoftheLuxembourgScenariooversimulationtime.

Fig.12. ResidentialscenarioinLuxembourg,includingaportionofahighway.

considerthesetup in Fig. 12 . Itconsistsof anarea of thecity of Luxembourg,in thetime interval (7 AM,9 AM). As one can see, itislocated ina residentialarea,butitalsoincludesasmall por-tionofahighway.Theconsideredtransmissionradiusisr₌26m, andhence it is such that the highway vehicles have little or no chancesofexchangingcontentwiththeresidentialnodes.In addi-tion,highwayvehicles spend avery smallamountoftime in the AZ.Asaresult,theyhardlygetthecontent,andwhentheygetit, they oftengo out before beingable toexchange it. Despitetheir modestcontributiontocontent diffusion,highwayvehicles, being numerous,giveanimportantcontributiontothemobility parame-tersonwhichourmodelisbased.Theresultisahighdiscrepancy betweenthesuccessratiopredictedbyourmodel(0.741)andthe valuearisingfromsimulations(0.13).

5. Relatedwork

A schemerelatedtoFC inVANETS isGeocasting (e.g [22] and referencestherein).SimilarlytoFC,itaimsatdeliveringamessage toatargetregionthroughopportunisticreplications.However, typ-icallythe messagesource residesout of thetarget region. Hence Geocastingfocusesmoreongettingcontent tothatregion, rather than(asdoneinFC)proposingstrategiestomake surethat,once intheregion,itpersistsintheregionovertime,anditisdelivered totheintendedrecipientswithagiventargetprobability.

The work in [12] considers a setup in which AZ size and lo-cation is based exclusively on application requirements (such as the region within which thecontent is of interest tousers), and it proposes an algorithm which dynamically tunes content repli-cationwithintheAZ,inordertoensureatargetavailabilitywhile tryingtominimizethenumberofreplications.Beingtiedtoafixed sizeAZ,suchanapproachcaneasilyleadtoinfeasibilityorto con-tentdisappearance fromtheAZ, asitisonlyabletodecreasethe numberofcontentreplicationswhentheyaremorethantheones requiredtoachievetheperformancetarget.

In [9] , authors consider settings where users move according toRandomDirectionmobilitymodel(RDMM)withoutpauses.The analytical model they propose allows deriving accurate estimates ofavailabilityandofsuccessprobabilityforvariousconfigurations oftheAZ,aswellasoftheprobabilityforanodetogetthecontent asa function of the time spent in the AZ. Such approach for FC performance modelingis tightlydependent onthe assumptionof uniformity in space of userdistribution, andon the geometrical propertiesoftheAZandofthemobilitypatterns.Assuch,itdoes notlenditselftogeneralizationstoheterogeneoussettings.

Ali etal. [23] , Rizzo etal. [24] consider a campus setup, and proposean analyticalmodelofFCbasedon aPoissonJumps mo-bility model,which captures exchangeswithin user clusters, and betweenclusters.Theproposedmodelisbasedontheassumption that on-the-flyexchangesbetween nodeson themove are negli-gible, in a way that results are not easily generalizable to other contexts,suchasvehicularones, wheresuchassumptiondoesnot hold.ForFC performance in vehicularenvironments, Nakanoand Miyakita [11] developsan analytical modelfor themean floating lifetime on a two-lanes highway. Di Maio etal. [25] considers a setupwherea SoftwareDefinedNetworking(SDN)controller col-lects mobilityinformation, such asvehicle speed andposition. It proposes a general strategy for optimizing the AZradius for ve-hicular applications inurban environments. However, so far such strategy could not rely on accurate modelsof FC performance in vehicularsettings.

con-Table3

Accuracyoftheapproachpresentedinthispaperandofthetwomainexistinganalytical approaches,intermsof%differenceinsuccessprobabilitywithrespecttosimulation values,forthreedifferentscenariosinLuxembourgCityinahigh-densitytrafficperiod (7AM−9AM).Contentsize:4MB,connectiontimeτ=5s,bandwidth4Mhz,coefficient attenuation3,SNR5dB.Anchorzoneradius260m,transmissionradius160m.

Approach Mobility Successprobability,%differencewrtsimulation

CityCenter Industrial Residential

Thispaper DMM −2.18 −1.45 −2.34

[26] DMM 18.63 18.04 18.53

[9] RandomDirection 29.69 20.22 28.10

tent transfer time, of contact time and of finite channel capac-ityandofthe connectionsetup time. Indeed,thelatterhasbeen shown experimentally to be key forthe practical feasibility of a given wireless technology for FC. These new results allow more realistic andaccurateestimations ofFCperformance inurban ve-hicular settings.Allthesenewresultshavebeenassessed numer-ically by simulation.Inorder toappreciatethe impactofthe de-velopments presented in this work with respect to the state of the art, we have considered the three areas of the Luxembourg scenario duringthe peak traffic period (7AM-9AM). In these se-tups, we have compared the success rate arising from simula-tions withthesuccessprobabilityderived fromtheapproach pre-sentedin thispaper.Inaddition,we haveconsidered thesuccess probability obtainedfromtheonlytwo otherresults whichcould be applied toa realistic, non-homogeneoussetup,that isthe ap-proach in [26] and in [9] . As the latter has been derived under theassumption ofnodesmoving accordingtotheRandom Direc-tion (RD)mobilitymodel, weparametrized it bysubstituting the PDF of the number of contacts during the sojourn time with a Poisson Distribution withmean equal to the experimental mean value.

Asitcanbeseenfrom Table 3 ,inallofthethreescenariosthe accuracy of theapproach presented inthis paperis substantially betterthananyotherexistingresult.

6. Conclusions

In thiswork,we presented andevaluated an analyticalmodel forthe prediction ofFC performance invehicular urban environ-ments. The modeldoes not require a detaileddescription of the road grid geometry, but only a few key parameters of mobility, stillprovingtobequiteaccurate.Indeed,thevalidationof analyti-calpredictionsagainstsimulations,basedbothonsynthetic mobil-itypatternsandonreal-worldvehiculartraces,showedverygood matches.Hence,ourworkprovidesthefirstaccuratetool to engi-neeravehicularapplicationbasedonFCinarealisticsetting.

Inaddition,throughsimulationandanalysis,wehaveevaluated thefeasibilityoftheFCparadigminrealisticurbansetupsundera variety ofsystemparameter values,traffic conditions,and mobil-itypatterns.Ourresultssuggestthat,inEuropeancitycontexts,in anycitydistrictandatanytimeofthedayitisalwayspossibleto findareasonablesizeoftheAZsuchthatthecontentfloatsforthe wholetargetdurationwithveryhighprobability.Asforapplication performance,ourresultsindicatethatinindustrialandcitycenter districts the FC paradigmis able to satisfy tight application-level performance requirementsat anytime ofthe day. Conversely, in residentialdistrictsandinareawithlowdensityofinhabitants,FC isbestsuitedforthediffusionofinformationwithrelativelyshort timeofvalidity,orforthoseapplications(suchasproactive mecha-nismsfortrafficcongestionavoidance)forwhichreachinglessthan 50%ofthetotalamountofnodeswithin theareaofinterestdoes notcompromiseapplication-levelperformance.

Acknowledgments

This research was supported by the CONTACT project CORE/SWISS/15/IS10487418, funded by the National Research Fund Luxembourg (FNR) 164205 and the Swiss National Science Foundation (SNSF)projectno. 164205 .

AppendixA. Frequencyofcontacts

Proof. Letusconsidertwo nodesa andb withinthe region,and thecaseinwhichp=1,i.e.nodesneverexittheregion.Letus as-sumethatnodea movesaccordingtothedistrictmobilitymodel, whilenodebisstatic.Weconsiderthelocationofbaswellasthe initiallocationofatoberandomwithintheregion.

Letusdivideatimeofanepochofnodeaintointervalsofsame duration

τ

, with

τ

much smaller than the expected value of Tm

andofTs,andsmallenough toassumethat, ineachinterval,a is

eithermovingorstatic.Themeanfractionofintervalsinwhicha movesishenceq.qisalsotheprobabilitythataismovingduring anintervalofduration

τ

.Letuscomputenowthemeannumberof contactswhichnodeaexperimentsinatimeinterval[0,

τ

]during whicha ismoving. Notethat since a movesalong astraight line duringa moving time, itcannot meet b more than once.Hence, theexpectednumberofcontactsbetweenthetwonodesisequal to 1 which multiplies the probability of finding b in the surface sweptby aduringthegiventime interval.Asthe locationofb is uniformlydistributedwithintheregion,suchprobabilityisthe ra-tiobetweentheareaofthesurfacesweptbya,andthetotalarea A.

Notethatweassumeaneventofcontactoccursatthefirsttime instantinwhichtwonodesareinrangeofeachother.Henceinthe computationofthecontactsoccurredin[0,

τ

],ifnodebisalready inrangeofaatt=0wedonotconsideritasacontactoccurred inthegiventime interval.Asa consequence,forthecomputation ofthe above probability we haveto subtract from thetotal area sweptbyainthetimeinterval,theareacoveredbyaatt=0.The resultingexpressionoftheprobabilitypcthatthetwonodescome

incontactduring[0,

τ

],whenamoves,is

pc= 2r

v

τ

A (A.1)

In every interval, a moves with probability q. Hence, the mean number of contacts during[0,

τ

] is pcq, and the contact rate is

pcq/

τ

. Finally, the meantime betweencontacts is the inverse of

themeancontactrate.

Let’snowconsiderthecaseinwhicha andbbothmove with-out everexiting the considered area. In each interval

τ

, thetwo nodes both move with probability q2_{. For computing the mean}

timebetweencontacts,weconsidertheequivalentsetupinwhich a moves at a relative speed vr while b is still. Due to the

2rq

v

(

1.27q+2

(

1−q

))

Theinverseofthisquantity isthe frequencyofcontactsbetween anypairofnodesinoursystem,andwedenoteitwith

ν

.Ifthere areNnodesinthe area,theoverall contactrateishence

ν

N

(

N− 1

)

_/2.

Whenp<1,inequilibriumthemeannumberofnodesdoesnot vary.Thatmeans,whenoursystemisinequilibriumstate,

ν

isalso themeancontactratepercoupleofnodes.

AppendixB. Nodedensity

Proof.We write the balance equations for the mean number of nodes with (and without) content. The mean number of nodes withcontentistheresultofnodeswithcontentgoingoutfromAZ, andofnodeswithoutcontentcomingincontactwithnodeswith content,andsuccessfullyexchangingit.IfT_soj isthemeansojourn timeintheAZ,andifweassumethatnodeswithcontentare uni-formlydistributedwithinthe AZ, n(t)/Tsoj is therateatwhich,at

timet,nodeswithcontentdecidetogetoutoftheAZ.Therateat whichnodesacquirecontentisdeterminedbythefrequencywith whichanodewithcontentcomesincontactwithanode without it,withintheAZ.ThemeanrateofcontactswithintheAZisgiven by

ν

(themeanrateatwhichagivenpairofnodescomesin con-tact)multiplied bythe numberof node pairswithin the AZat t,

N(t)(N(t) −1)

2 .Ofthesecontacts,those whichincreasen(t)arethose

inwhichonlyonenodeofthetwohasthecontent,andforwhich thecontentistransferred successfully.As pn

(

t

)

=_Nn(₍t_t) ₎ thefraction

ofnodeswithcontentintheAZ,andasQistheprobabilitythata contenttransferissuccessful,themeanrateatwhichn(t)increases attisgivenby

2pn

(

t

)(

1−pn

(

t

))

Q

N

(

t

)(

N

(

t

)

−1

)

2

ν

(B.1)

Since N(t)1, we can approximate (B.1) with

ν

n(t)m(t)Q. Hence wehave

ν

n¯m¯Q− n¯ Tsoj =

0 (B.2)

Overtime, mincreasesdueto arrivalsofnewnodesinto theAZ, anditdecreasesbecauseofnodeswithoutcontentleavingtheAZ, andasnodeswithoutcontent comein rangeofnodes with con-tent.Hence

λ

−

ν

n¯m¯Q− m¯ Tsoj

=0 (B.3)

ByLittle’slawandbalanceequationsweget (4) .

AppendixC. Successprobability

Proof.Westartby analyzingtheprobabilityforoneormore suc-cessful content exchanges during a single epoch in the sojourn timeof anode in theAZ. In thestationaryregime, forthe given mobilitymodel,we assume that contentis uniformly distributed amongthe populationofnodes.Thisimpliesthat atagiventime instanttheprobabilityforanode tohavethecontentisthesame

queue is Poisson [28] . Hence, in stationary state, N(t) is Poisson withintensityN.Thereforethe expectedcontactrateforasingle nodeatt(wheretheexpectationiswithrespecttothecontactrate betweenapairofnodes)isalsoPoissondistributed,withintensity

ν

N. Notethat this holdsbecause thetwo processes (contactrate forapairofnodes,andnumberofnodesintheAZ)are indepen-dent.

Thankstotheuniformcontentdistribution,theprobabilityfora nodetohavethecontentis n

N.Theprobabilityforanodeofhaving

anunsuccessfulcontact(i.e.,togetincontactwithanodewithout content, orto fail in transferring a piece ofcontent) istherefore

m N +

n

N

(

1−Q

)

=1− n

NQ.Theprobability foramoving nodetoget

thecontentduringTm,conditionedtohavingjcontactsisthe

com-plement ofthe probability that j contactsbring to no transferof content.Thatis,

P[succ,m

|

Tm,j]=1−

1− n NQ

j

(C.1)

Bythelawoftotalprobability,

P[succ,m

|

Tm]=

∞

j=1

P[succ

|

Tm,j]Poiss_NνTm

(

j

)

=1−exp

−N

ν

TmQ ¯

n

¯

N

For the derivation of the pdf of the moving time fTm, note that Tm has adifferent distributionwhen it refers tothe first moving

time of anode inan AZ, when it refers to thelast moving time inthe AZ, andwhenthe moving time isthefirst andthelast of thenodepathinsidetheAZ(i.e.,thenodedoesnotstopinsidethe AZ),andinalltheother cases.However,whenthemeannumber ofwaypointsonthepathofanodeislargeenough,wecanassume allthesedistributiontobethesameasinthosecasesinwhichthe nodeisneitherenteringnorexitingtheAZ.Thedistributionofthe transitionlengthLisgivenby [29]

fL

(

l

)

= 4l

π

R2

arccos l 2R−

l

2R

1−

l

2R

2

Then, given that Tm=L_v, fT_m,d

(

τ

)

=

v

fL

(

τ

v

)

. By the law of total

probability, the probability of success during a moving time is hence

Pm=

2R

v

0

1−e−ντn¯Q

_f Tm

(

τ

)

d

τ

TheprobabilityofsuccessduringatransittimePtisderivedinthe

samewayasPm.ThepdfofthetransittimeTt isderivedfromthe

pdfofthechordlengthcforacircleofradiusR.Asthislastpdfis givenby

c

2R√4R2_−c2

andgiventhatTt= c_v,weget

fTt

(

τ

)

=

τ

v

2

Inasimilarmanner,theprobabilityforastaticnodetogetthe contentduringTsis

Ps=

_+∞

τ=0

1−e−ντn¯Q

_f

Ts

(

τ

)

d

τ

. (C.2)

Finally,

Pe=Pm+

(

1−Pm

)

Ps (C.3)

The probability ofgettingthecontent duringthesojourn timein theAZ, conditionedtonostopsintheAZisPt.Theprobability of

no stopsis1₋p.Fork_>0 stopsintheAZ, theconditional prob-abilityis[1−

(

1−Pe

)

k

(

1−Pm

)

],andtheprobabilityofsuchevent

is pk

(

_1−p

)

_.

The probabilityofgettingthecontent duringthe sojourntime intheAZishencegivenby

Psucc=

(

1−p

)

Pt+

∞

k=1

pk_[1−

(

_1−P

m

)(

1−Pe

)

k] (C.4)

whichgives (5) .

AppendixD. Probabilityofsuccessfulcontenttransfer

Proof. Whenr<<Randwhentwo nodesareinrange,achange ofstate(i.e.fromstilltomoving,andviceversa)isrelativelyrare. Henceweconsideronlythefollowingthreecases:

• bothnodesmoveforthewholedurationofcontacttime;

• one node isstill, andtheother movesforthewhole duration ofcontacttime;

• bothnodesarestillforthewholedurationofcontacttime.This is equivalent to not counting aspart of the contact time the timeintervalswhenbothnodesareinrange,butoneismoving (beforeand/orafterstopping).

Let us consider first the case in which one node is still and theothermoves. Lett=0bethetimeatwhichthemovingnode comesinrangeofthestillnode.Themovingnodewillmovealong asecantlineofacircleofradiusr,centeredatthelocationofthe stillnode,andhisthedistanceofthechordfromthecenterofthe circle.

At anyinstant t duringwhichthe moving nodeis inrange of thestillnode,let

v

t≤2

r2_−h2_{be thelength ofthepathwithin}

thecircleuntiltimet,where2

r2_−h2_{isthechord length.Then}

thedistanceofthemoving nodeattimetfromthestill nodecan bederivedfromstandardtrigonometricresults:

d

(

t,h

)

=

h2

0+

r2_−h2

0−

v

t

2

The total amount of (useful, i.e. not control) bits which can be transferred duringthe contacttime, is the integral of the ca-pacityof thelink betweenthetwo nodes,over thecontacttime. If

τ

0 is the time taken by two nodes to sense their proximity

andto get readyforcontent exchange,such total amount ofbits whichcanbetransferredduringacontactwithtrajectorydistance his

B

(

h

)

=

2√r2₋h2 v

τ0

C

(

d

(

t,h/r

)

)

dt (D.1)

A successfulcontenttransfer impliesthatB(h)≥L.Onecan easily seethatB(h)ismonotonicdecreasingwithh.Hencethemaximum trajectory distanceh0 isthe one satisfying (12) .As forthe

statis-ticsofhintheRMM,whennodesarenottooclosetotheborder of theAZ, the assumption that all directionsare equally likelyis a goodapproximation,andh(the trajectorydistanceofa moving node withrespecttoastill node,conditionedto theeventofthe

twonodescomingincontactofeachother)can bemodeled asa uniformrandomvariableintheinterval[0,r].Hence,the probabil-ityofsuccessfulcontent transferconditioned tohavingone node moving andthe other still,is equal tothe CDFof hevaluated in h0(v).

The derivationfor thecase withtwo moving nodes issimilar tothecasewithonlyonemovingnode,consideringonenodestill andthe other moving atthe relative speedgiven by the compo-sition of the two speeds. Letfr be the pdf of the relative speed

betweenthetwo nodes.In ourcase, inwhichall nodeshavethe samespeed,frcanbeobtainedwithsimplegeometricalarguments,

anditisgivenby

fr

(

v

r

)

=

v

r

v

2

π

₁₋

(

v

_r/

v

₋₁

)

2

Now,wehave

B

(

h

)

=

2v

0

fr

(

v

r

)

B

(

h

|

v

r

)

d

v

r

WhereB(h|vr)isgivenby (D.1) with

v

=

v

r.

Finally,let usconsider thecasein whichboth nodes arestill. Giventhe propertiesof the DMM,all positions ofa node within the transmission rangeof the other are equally likely (excluding againeffectsduetotheborderoftheAZ).Inthiscase, themean throughputbetweenthetwonodesis

C0=

r

0

C

(

x

)

2

π

xdx

π

r2

If Ts is distributed as an exponential, thanks to the memoryless

property of the exponential distribution, the time spent still by the home node, counted from the arrival of the second node, is still exponential withmean 1/

μ

. Hence the duration ofthe con-tacttime istheminimumofthetimespentstill bythenodes.By the properties of the exponential distribution, such contacttime isdistributedasanexponentialrandomvariablewithmean 1

2μ.If

1−exp

(

−1

2μ

τ

)

is theCDFofsuch distribution, theprobability of

successfulcontent transfer is exp− 1

2μ

(

τ

0+CL₀

)

. Putting it all

to-gether,wehave (11) .

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icsand TelecommunicationsDepartment ofthe Politec-nicodiTorinoinItaly,andapart-timeresearch profes-soratIMDEANetworks Institute inLeganes, Spain.He obtaineddegrees in EE from the Politecnico di Torino andtheUniversityofCalifornia,LosAngeles(UCLA).Since 1974hehasbeenatPolitecnicodiTorino,inthe differ-entrolesofanacademiccareer,withaninterruptionfrom 1987to1990,whenhewasafullprofessoratthe Com-puterScienceDepartmentoftheUniversityofMilan.He hasbeendoingresearchinthefieldsofdigital transmis-sion,distributedsystemsand networking. Hehas pub-lishedover350 papers inthe leadingconferences and journalsofhisresearchareaaswellastwobooks.MarcoAjmoneMarsanhasbeen amemberoftheeditorialboardandofthesteeringcommitteeoftheACM/IEEE TransactionsonNetworking.Heisamemberoftheeditorialboardsofthejournals ComputerNetworksandPerformanceEvaluationofElsevierandofACMTOMPECS. Hewas inthe organizingcommitteeofseveral leadingnetworkingconferences, andgeneral chairofINFOCOM 2013.HeisaFellowoftheIEEE,and amember oftheAcademiaEuropaeaandoftheAcademyofSciencesofTorino.Hereceived ahonorarydegreeinTelecommunicationNetworksfromthe BudapestUniversity ofTechnologyandEconomics.HewastheVice-RectorforResearch,Innovationand TechnologyTransferatthePolitecnicodiTorino,andtheDirectoroftheIstitutodi ElettronicaeIngegneriadellInformazioneedelle TelecomunicazionioftheItalian NationalResearchCouncil.HewastheItaliandelegateintheICTandIDEAS com-mitteesofFP7.