Impac of Human Mobiliy on Opporunisic Forwarding Algorihms Augusin Chainreau, Pan Hui, Jon Crowcrof, Chrisophe Dio, Richard Gass, and James Sco *, Universiy of Cambridge Microsof Research Thomson Research 5 J J Thomson Av. 7 J J Thomson Av. 46 quai A. Le Gallo Cambridge, CB3 FD, UK 92648 Boulogne FRANCE pan.hui@cl.cam.ac.u, augusin.chainreau@homson.ne jon.crowcrof@cl.cam.ac.u, chrisophe.dio@homson.ne james.sco@microsof.com, Inel Research 472 Forbes Avenue Pisburgh, PA 523, USA richard.gass@inel.com, Absrac We sudy daa ransfer opporuniies beween wireless devices carried by humans. We observe ha he disribuion of he iner-conac ime (he ime gap separaing wo conacs beween he same pair of devices) may be well approximaed by a power law over he range minues; day. This observaion is confirmed using eigh disinc experimenal daa ses. I is a odds wih he exponenial decay implied by he mos commonly used mobiliy models. In his paper, we sudy how his newly uncovered characerisic of human mobiliy impacs one class of forwarding algorihms previously proposed. We use a simplified model based on he renewal heory o sudy how he parameers of he disribuion impac he performance in erms of he delivery delay of hese algorihms. We mae recommendaions for he design of well founded opporunisic forwarding algorihms, in he conex of human carried devices. I. INTRODUCTION The increasing populariy of devices equipped wih wireless newor inerfaces (such as cell phones or PDAs) offers new communicaion services opporuniies. Such mobile devices can ransfer daa in wo ways - by ransmiing over a wireless (or wired) newor inerface, and by aing advanage of user s mobiliy. They form a Poce Swiched Newor. Communicaion services ha rely on his ype of daa ransfer will srongly depend on human mobiliy characerisics and on how ofen such ransfer opporuniies arise. Therefore, hey will require neworing proocols ha are differen from hose used in he Inerne. Since wo (or more) ends of he communicaion migh no be conneced simulaneously, i is impossible o mainain roues or o access cenralized services such as he DNS. In order o beer undersand he consrains of opporunisic daa ransfer, we analyze eigh disinc daa ses colleced in newors wih mobile devices. Three daa ses come from experimens we conduced ourselves. We define he inerconac ime as he ime beween wo ransfer opporuniies, for he same devices. We observe in he eigh races ha he iner-conac ime ail disribuion is slowly varying over a large range. Inside his range he iner-conac ime disribuion can be compared o a power law. We sudy he impac of hose large iner-conac imes on he acual performance and heoreical limis of a general class of opporunisic forwarding algorihms ha we call oblivious forwarding algorihms. Algorihms in his class do no use he ideniies of he devices ha are me, nor he recen hisory of conacs, or he ime of he day, in order o mae forwarding decisions. Insead forwarding decisions are based on saically defined forwarding rules ha bound he number of daa replicas or he number of hops. Based on our experimenal observaions, we develop a simplified model of opporunisic conac beween human-carried wireless devices. Our model maes several independence assumpions which are common in he lieraure of mobile ad-hoc rouing. We do no claim ha his model capures he performance of differen forwarding algorihms accuraely. Raher, i serves our purpose which is o demonsrae how he ail of iner-conac imes influences he performance of oblivious forwarding algorihms, and how hese should be configured o offer reasonable guaranees. Experimenal resuls are presened in Secion II. In Secion III, we model conac opporuniies based on our observaions and we analyze he delay ha wireless devices would experience using a class of forwarding algorihm previously sudied in he lieraure. Secion IV is dedicaed o relaed wor. The paper concludes wih a brief summary of conribuions and presenaion of fuure wor, including a discussion of he implicaions of our assumpions. A. Daa ses II. EXPERIMENTAL ANALYSIS In order o conduc informed design of opporunisic forwarding algorihms, i is imporan o analyze he frequency and duraion of conacs beween human carrying communicaing devices. Ideally, an experimen would cover a large user base over a large ime period, as well as include daa on connecion opporuniies encounered weny-four hours a day. We examined wo ypes of daa ses. Firs, we use publicly available races measuring conneciviy beween cliens and access poins (APs) in several wireless newors (using WiFi or GSM echnology); conacs beween he cliens were deduced from he races following an assumpion ha we discuss below. Second, we colleced our own races of direc conacs recorded using small porable wireless radio devices (imoes)
ha were disribued o differen groups of people. We found a few oher races of direc conacs and we have included hem for comparison wih ours. In oal, here are eigh daa ses, he characerisics of each of hem are summarized in Table I. ) AP-based daa ses: UCSD 2 and Darmouh 3 races mae use of WiFi neworing, wih he former including clien-based logs of he visibiliy of access poins (APs), while he laer includes SNMP logs from he access poins. The duraions of he differen logs races are hree and four monhs respecively. Since we required daa abou device-o-device ransmission opporuniies, he raw daa ses were unsuiable for our experimen and required pre-processing. For boh daa ses, we made he assumpion ha mobile devices seeing he same AP would also be able o communicae direcly (in adhoc mode). Consequenly a lis of ransmission opporuniies was deduced for each pair of devices, which corresponds o he ime inervals for which hey share a leas one AP. The races from he Realiy Mining projec 4 a he MIT Media Lab include records of visible GSM cell owers, colleced by cellphones disribued o sudens and faculy on he campus during 9 monhs. We have assumed, as above, ha wo devices are in conac whenever hey are conneced wih he same cell ower. Unforunaely, he assumpions we have made for all hese daa ses inroduces inaccuracies. On he one hand, i is overly opimisic since wo devices aached o he same (WiFi or GSM) base saion may sill be ou of range of each oher. On he oher hand, he daa migh omi connecion opporuniies, such as when wo devices pass each oher a a place where here is no insrumened access poin. Anoher poenial issue wih hese daa ses is ha he devices are no necessarily co-locaed wih heir owners a all imes. Despie hese inaccuracies, hese races are a valuable source of daa spanning many monhs and including housands of devices. In addiion, considering wo devices conneced o he same base saion as being poenially in conac is no alogeher unreasonable. These devices may indeed be able o communicae locally hrough he base saion. 2) Direc conac daa ses: In order o complemen he previous races, we did our own experimen using Inel imoes, which are embedded devices similar o Crossbow moes, excep ha hey communicae via Blueooh. We programmed he imoes o log conac daa every 2s for all visible Blueooh devices (including imoes as well as oher Blueooh devices such as cell phones). Each conac is represened by a uple (MAC address, sar ime, end ime). The experimenal seings are described in deail in ; an anonymized version of our daa is now available o oher research groups in he CRAWDAD 2 archive. We include in his paper he resuls from hree imoe-based experimens. The firs obained daa from welve docoral sudens and faculy comprising a research group a he Universiy of Cambridge. The second experimen included a group of hiry seven paricipans in Hong Kong seleced in such a way ha hey do no belong o he same wor or social group, See www.xbow.com 2 See crawdad.cs.darmouh.edu and in paricular ha none of hem nows each oher. The hird experimen was conduced during he IEEE INFOCOM 25 conference in Miami where imoes were carried by 4 aendees for 4 days. The conacs colleced by imoes are classified ino wo groups: he sighing of anoher imoe is classified as an inernal conac, while he sighing of oher ypes of Blueooh devices is called an exernal conac. The exernal conacs are numerous and hey provide a measure of he wireless neworing opporuniies presen a ha ime. Inernal conacs, on he oher hand, represen he daa ransfer opporuniies among paricipans, if hey were all equipped wih devices which are always-on and always-carried. In addiion o our own experimen, we found wo daa ses wih direc conacs, and included hem for comparison: A research group from he Universiy of Torono has colleced direc conac races using 23 Blueooh-enabled PDAs disribued o a group of sudens. These devices performed a Blueooh inquiry every 2 seconds and his daa was logged. This mehodology does no require devices o be in range of any AP in order o collec conacs, bu i does require ha he PDAs are carried by he paricipans, and ha he paricipans eep hem charged. The daa se we use comes from an experimen ha lased 6 days. The races from he Realiy Mining projec 4 include direc Blueooh sighings, recorded every 3 seconds by each paricipan s cell phone. B. Definiions We are ineresed in how he characerisics of ransfer opporuniies impac daa forwarding decisions. In his paper, we focus on how ofen such opporuniies occur, bu no in heir duraion. We decided no o analyze how much daa can be ranspored during a ransfer opporuniy, because his srongly depends on he wireless echnology used. Laer in our analysis (see Secion III), we will assume ha all conacs las a single ime slo and we will address wo exreme cases corresponding o a lower and upper bounds of he amoun of daa ha could be ransferred in each connecion opporuniy. We define he iner-conac ime as he ime elapsed beween wo successive conac periods for a given pair of devices. Iner-conac ime characerizes he frequency wih which daa can be ransferred beween newored devices; i has rarely been sudied in he lieraure. Two remars mus be made wih regard o his definiion: Firs, he iner-conac ime is compued once a he end of each conac period, as he ime inerval beween he end of his conac and he beginning of he nex conac wih he same device 3. An alernaive opion would be o compue he remaining iner-conac ime seen a any ime : for each pair of devices, i is he ime i aes afer, before hese devices mee again (a formal definiion is given in Secion III). Inerconac ime and remaining iner-conac ime have differen disribuions, which are relaed, for a renewal process, via a classical resul nown as he waiing ime paradox (see p.47 in 5). A similar relaion holds for saionary processes, in 3 Noe ha we did no include he ime beween he beginning of he experimen and he firs conac for a pair, nor he ime beween he las conac of a pair and he end of he experimen.
User Populaion Cambridge Hong Kong Infocom Torono UCSD Darmouh MIT BT MIT GSM Device imoe imoe imoe PDA PDA Lapop/PDA Cell Phone Cell Phone Newor ype Blueooh Blueooh Blueooh Blueooh WiFi WiFi Blueooh GSM Conac ype direc direc direc direc AP-based AP-based direc AP-based Duraion (days) 3 5 4 6 77 4 246 246 Granulariy (seconds) 2 2 2 2 2 3 3 Devices paricipaing 2 37 4 23 273 6,648 # of inernal conacs 4,229 28 22,459 2,82 95,364 4,58,284 54,667 572,9 # of inernal conacs/pair/day.42 3.4.35.34.8.22.23 Recorded exernal devices 48,84 97 N/A N/A N/A N/A N/A # of exernal conacs 2,44 6,88 5,723 N/A N/A N/A N/A N/A TABLE I COMPARISON OF DATA COLLECTED IN THE EIGHT EXPERIMENTS. he heory of Palm Calculus (see p.5 in 6). We choose o sudy he firs definiion of iner-conac ime seen a he end of a conac period, as he second gives oo much weigh o large iner-conac imes. In oher words he definiion we have chosen is he mos conservaive one in he presence of large values. Second, he iner-conac ime disribuion is influenced by he experimen s duraion and is granulariy (i.e. he ime elapsed beween wo successive scanning for he same device). Iner-conac imes ha las more han he duraion of he experimen canno be observed, and iner-conac imes close o he duraion are less liely o be observed. In a similar way, we canno observe he iner-conac imes ha las less han he granulariy of measuremen (which ranges from wo o five minues for differen experimens). Anoher measure of he frequency of ransfer opporuniies ha could be considered is he iner-any-conac ime, i.e. for a given device, he ime elapsed beween wo successive conacs wih any oher device. This measure is very much dependen on he densiy of wireless devices during he experimen, as i characerizes ime ha devices spend wihou meeing any oher device. This measure was sudied for mos of hese daa ses in. We do no presen furher resuls here. C. Iner-conac ime characerizaion We sudy he empirical disribuion of he iner-conac imes obained for all experimens shown in Figures and 2. For all plos, an empirical disribuion of he iner-conac imes was firs compued separaely for each pair of devices ha me a leas wice. I is hard o sudy he characerisics of he disribuions for all pairs individually, because here are many such disribuions, and ha some of hem may only include a few observed values. This is why we follow a wo-sep approach: Firs, we presen he disribuion obained when all pairs disribuions are combined, each wih an equal weigh, in a disribuion ha we call he aggregaed disribuion. Second, we use a parameric model moivaed by his firs par and esimae he parameer of he individual disribuion for each pair. ) Aggregaed disribuion: Figure presens he aggregaed disribuion for differen daa ses. All plos show he complemenary cumulaive disribuion funcion, using a loglog scale. For imoe experimens, (i) indicaes ha he daa se shown is obained using inernal conacs only, while (e) indicaes ha he daa se shown includes only exernal conacs. For he firs wo imoe experimens (labeled Cambridge and Hong Kong) we presen only one case here (corresponding respecively o inernal and exernal conacs). They are shown in Figure (lef), which also includes he disribuion obained among pairs of experimenal devices in he race from he Universiy of Torono. Disribuions belonging o he imoe based experimen a Infocom are shown in Figure (middle), where disribuions associaed wih inernal and exernal conacs have been ploed separaely for comparison. Figure (righ) presens he disribuion of iner-conac ime compued using races from oher experimens han ours. Le us firs noe ha, alhough iner-conac imes are shor in mos cases, he occurrence of large iner-conac imes is far from negligible: in he hree imoe based experimens, 7 o 3% of iner-conac imes are greaer han one hour, and 3 o 7% of all iner-conac imes are greaer han one day. In he Torono daa ses, 4% of iner-conac imes las more han a day, and 8% more han a wee. These large inerconac imes are even more frequen in he races colleced a UCSD, Darmouh and MIT, he mos exreme case being he MIT race using Blueooh sighings, where up o 6% of he iner-conac ime observed are above one day. The variaion beween daa ses is significan. I can be expeced given he diversiy of communicaion echnologies and populaion sudied, as well as he impac of experimenal condiions (granulariy, duraion). Bu here are also common properies ha we now discuss in more deail. We now concenrae on he region beween minues and one day. In his region, all daa ses exhibi he same characerisics: he CCDF is slowly varying, i is lower bounded by he CCDF of a power law disribuion, ha may in some cases be a close approximaion. This conradics he exponenial decay of he ail which characerizes he mos common mobiliy models found in he lieraure (see Secion IV), and we prove in he nex secion ha his can have a significan impac on he performance of opporunisic neworing algorihms. To jusify he above claim, we sudied he quanile-quanile plo comparison beween he empirical disribuion found and hree parameric models (exponenial, log-normal, and power law). An example is shown in Figure 2 (lef) for he disribuion based on inernal conacs during he Infocom
... PIner-conac > PIner-conac > PIner-conac >... Torono Cambridge (i) Hong Kong (e) PL (slope.6, granulariy 2s). 2min min hour 3h 2h day wee monh Time Infocom (i) Infocom (e) PL (slope.35, granulariy 2s). 2min min hour 3h 2h day wee monh Time MIT BT Darmouh UCSD MIT GSM PL (slope.2, Granulariy s). 2min min hour 3h 2h day wee monh Time Fig.. Aggregaed disribuion of he iner-conac ime in eigh daa ses experimens: imoe-based experimens a Cambridge and Hong Kong, and Torono experimen (lef), imoe-based experimen a INFOCOM (middle), daa colleced a UCSD, Darmouh and MIT (righ). monh wee exponenial model (median=979) lognormal (median=6.9, var=4.23) power law (slope =.33) using he median using order saisics 438 pairs in oal.2 Quanile (empirical disribuion) day 2h 3h hour min heavy ail index.8.6.4.2 heavy ail index.8.6.4.2 2min 2min min hour 3h 2h day wee monh Quanile (parameric disribuion)..2.3.4.5.6.7.8.9 pair s ran, divided by he oal number Infocom (i) HKG (e) Darmouh MIT GSM Infocom (e) UCSD MIT BT Fig. 2. imoe-based experimen a Infocom: Quanile-quanile plo of comparison beween he aggregaed disribuion of he iner-conac ime and hree parameric models (lef), esimaion of he heavy ail index of he power law applied separaely for each pair (middle), summary of resuls obained in all daa ses (righ). experimen. All parameric models have been se o ae he same median value as he empirical disribuion. We also normalize he power law o fi he granulariy =2 seconds, and he log-normal disribuion such ha he logarihm of boh he empirical variable and he model have he same variance. No surprisingly, we observe ha he hree models deviae significanly from he empirical findings for values above one day. As expeced he exponenial disribuion is very far from he empirical one, he quanile for he log-normal disribuion deviaes from he empirical case by a non negligible facor. The power law disribuion, in conras, remains close o he empirical one for values up o 8 hours, and i seems o be he mos appropriae model o apply. In oher daa ses, he power law may someimes no mach he empirical findings as well as in his example, bu among hese hree models i is always he closes o he empirical disribuion. For values above one day, we expec models wih addiional parameers (e.g. following a Weibull disribuion) o improve he mach wih he empirical disribuion, bu ha is beyond he scope of his paper. The mos noable difference we observe beween daa ses is ha he fi wih a power law is beer for he daa ses ha conain he larges number of poins, such as in Figure (middle) and (righ). We also observe ha he slope of he power law ha is a lower bound on he range minues; day is differen beween daa ses: his is.6 for he imoe experimens a Cambridge and Hong Kong, as well as for Torono daa ses,.35 for he imoe based experimen a Infocom, and.2 for races colleced in UCSD, Darmouh and MIT. In all cases, i is below. The value of his slope, which is also called he heavy ail index, is criical for he performance of opporunisic forwarding algorihms (see he analysis in Secion III), and we discuss i furher below. Figure (middle) shows ha he disribuion is almos unchanged if one consider inernal or exernal conacs. The same observaion was made for oher imoe experimens, excep for he experimen conduced in Hong Kong where, as expeced, very few inernal conacs were logged. Some variaions of he heavy ail index have been observed depending on he ime of he day. 2) Individual disribuions for each pair: So far we have sudied he aggregaed disribuion where all pairs have been combined ogeher, and we found ha i can be approximaed by a power law for values up o day. In his secion, we assume ha his claim can be made individually for all pairs, alhough he parameer of his power law, also called he heavy ail index, may be differen among hem. This approach allows us o sudy he heerogeneiy beween pairs via a single
parameer; some of hese resuls also measure he accuracy of he above assumpion for each pair. Esimaor for he heavy ail index Le us consider a pair of nodes, he sample of he iner-conac imes observed for his pair will be denoed by X,...,X n, is order saisics by X ()... X (n), and is median value by m. All imes will be given in seconds. If we assume ha his sample follows a power law wih granulariy 2s and heavy ail index α, we have: P X x = (x/2) α, such ha an esimaor of α based on he sample s median m is given by: ˇα = ln(2) ln(m) ln(2) More generally one can consider all order saisics X (i) ha fi in he range minues; day and esimae α based on each of hem. This creaes a collecion of esimaors for he value of α, as follows: { } ln(n) ln(n i) ln(x (i) ) ln(2) 6 X (i) 864, i < n. We denoe by α and α respecively he minimum and maximum value in his se above. This is equivalen o ploing he empirical CCDF for his sample in a log-log scale, and bounding his CCDF from above and below by wo sraigh lines ha go hrough probabiliy a ime value 2s. The slopes of hese lines would be equal respecively o α and α. In conras o ˇα, hese wo esimaors are no cenered around he value of α, and hey do no converge o his value when he sample becomes large. They raher serve he purpose of a heurisic analysis; hey characerize some bounds ha are verified by each pair. Noe also ha, inuiively, he difference α α indicaes how he condiional disribuion of he sample in his range differs from a pure power law. In Figure 2 (middle), we plo he values of ˇα and he inerval α, α for all pairs of imoes during he experimen conduced a Infocom. One can expec ha he heavy ail index aes differen values among pairs, as some paricipans are more liely o mee ofen han ohers. We iniially raned all pairs according o heir value for ˇα, in he decreasing order. Alhough we have compued hese values for all pairs we only draw he inerval α; α for pairs chosen arbirarily according o heir ran (one every 4), in order o eep he figure readable. As shown in Figure 2 (middle) esimaions of α for differen pairs may indeed vary beween.5 and. Beween hese wo exreme values, which are very rarely observed, esimaes for almos all pairs lies beween. and.7 depending on he esimaor. Noe ha all esimaes of α are smaller han ; he only excepions are he upper esimae α for hree pairs (i.e. less han.2% of pairs in his case). The median based esimae lies in.2 ;.4 for half of he pairs, he lower esimaes (resp. he upper esimae) lies in.4 ;.32 (resp..32 ;.5) again for half of he pairs. These resuls have hree major implicaions: Firs, he heerogeneiy among pairs implies differen possible values for α, which are cenered around he value already observed when sudying he aggregae disribuion (i.e..33). Second, he difference beween he median esimaor and he heurisic bounds we defined above remains wihin.25 excep in a few correlaion of order.5 -.5-2 3 4 5.5 -.5-2 3 4 5 Fig. 3. Correlaion coefficiens for he sequence of iner-conac imes: for all pairs of imoes in he Infocom daa se (lef), for all pairs of devices in MIT GSM daa se (righ). cases. Las, he upper esimae α almos never goes above, which esablishes ha he iner-conac ime disribuion for each pair is lower bounded in his range by a power law wih a heavy ail index smaller han. The same resuls have been obained for oher daa ses, and hey are summarized in Figure 2 (righ). For each daa se indicaed, we show he disribuion of values obained among pairs for he hree esimaors defined above. Each esimaor sands for one box-plo: from lef o righ, α, ˇα, α; he hic par indicaes he values found in 5% of he pairs, he hin par conains he region where 9% of he pairs are found. In he Hong Kong and Darmouh daa ses, where conacs are sparser, iner-conac ime samples for each pair conains fewer values. As a consequence, he difference beween esimaors can grow significanly. We even observe ha α goes slighly beyond for % of he pairs in Hong Kong daa se, alhough his could be an arefac of our conservaive esimae. Correlaion: We sudy he auo-correlaion coefficiens o see how he value of he iner-conac ime may depend on he previous values for he same pair. The resuls are shown in Figure 3 for all order up o 5. Since he iner-conac ime disribuion usually has no finie variance, we compued he correlaion coefficien on he values of he logarihm of he iner-conac imes. Noe ha a correlaion coefficien was compued for each pair, we presen for all order he average value we observed among all pairs, as well as he inerval conaining 5% and 9% of he cenered values (respecively, in he hic box and he hin bar). In he Infocom daa se, he variaion of he coefficien among pairs is quie imporan, alhough mos pairs remain reasonably non-correlaed (he hic box remains always less han.3 away from zero). Overall we observe a slighly negaive correlaion on average over all pairs, which reduces as grows. Correlaion coefficiens are smaller when he daa se is large (as seen for example in he MIT GSM race shown here, as well as for all oher long races). This ends o indicae ha hese coefficiens pairs would be closer o zero if he imoe experimen could be done wih a longer duraion, and ha he sample of iner-conac imes colleced for each pair was bigger. Based on he above resuls, we assume in he nex secion ha he iner-conac ime disribuion follows a power law for each pair. To simplify he analysis we assume in addiion ha
he coefficien is he same for all pairs, ha he sequence of iner-conac imes is i.i.d. (i.e. correlaion coefficien are null) and ha hey are independen beween pairs. This simplificaion allows us o characerize he performance of forwarding algorihms quie generally. Some of he resuls we presen can be exended o saionary ergodic sequences, or correlaion beween pairs bu ha is lef for fuure wor. III. FORWARDING WITH POWER LAW BASED OPPORTUNITIES We now analyze he impac of our findings on he performance of a class of forwarding algorihms. We define firs our absrac model of he opporunisic behavior of mobile users based on our experimenal observaions. A. Assumpions and Forwarding Algorihms ) Conac process model: We consider a sloed ime =,,.... For a given pair of devices (d, d ), le us inroduce is conac process (U (d,d ) ) defined by: { U (d,d ) if d and d = are in conac during slo, oherwise. For he pair (d, d ) we consider he sequence of he ime slos T (d,d ) < T (d,d ) <... < T (d,d ) <... ha describes all he values of N such ha U (d,d ) =. We do no include in his model he conac ime represening he duraion of each conac, assuming ha each conac sars and ends during he same ime slo. This is jusified here by he fac ha we are ineresed in a model accouning for consequences of large values of he iner-conac ime. I was observed (see ) ha he conac ime disribuion may also be approximaed by a power law bu over a range ha is much smaller han he range for he iner-conac ime. Under his condiion, he ime τ (d,d ) = T (d,d ) + T (d,d ) for any d, d and is he iner-conac ime afer he h conac of his pair. We suppose in our model ha i has he same law as X, which follows a power law wih heavy ail index α > : PX = α for all =, 2,.... () Noe ha X is no bounded bu is finie almos surely. I may easily be seen ha X has a finie mean if and only if α >. In addiion we assume ha he conac process (U (d,d ) ) of each node pair (d, d ) is a renewal process, and ha conac processes associaed wih differen pairs are independen. In oher words, he iner-conac imes in he sequence ) are i.i.d., for all (d, d ), and sequences belonging o differen pairs are independen. We come bac o hese assumpions laer in Secion V. Noe ha hese assumpions are shared explicily or implicily by mos of he analyses of currenly proposed mobiliy models. This is because i is ypically very difficul o analyze models where dependence may arise beween differen devices or beween successive evens occurring wih one or more devices. Even if we do no explicily model he conac ime (each conac lass one ime slo), we need o ae ino consideraion (τ (d,d ) he fac ha a conac may las long enough o ransmi a significan amoun of daa. We hen inroduce wo siuaions: he shor conac case : where only one daa uni can be sen beween he wo devices during each conac. he long conac case : where we assume ha all queues in he wo devices can be compleely empied during each conac. These wo cases represen a lower and an upper bound for he evaluaion of bandwidh. The number of daa unis ransmied in a conac (wheher shor or long) is defined as a daa bundle 4. The long and he shor case differ from a queuing sandpoin. In he long conac case, as soon as a daa uni has arrived in a node, i can be sen o all oher nodes ha are me. In he shor conac case, only one daa uni is sen a once and, herefore, daa can accumulae in he memory of a relaying device. Noe ha our model does no ae ino accoun explici geographical locaion or movemen of devices; raher, i direcly describes he processes of conacs beween devices. The resuls of his secion exend o any mobiliy model which creaes independen conac processes for all pairs of devices, ha follow his same law. For any pair of devices (d, d ), le us inroduce he remaining iner-conac ime observed a ime slo : I is an ineger denoed by R (d,d ) and defined as { } R (d,d ) = min and U (d,d ) =. As he conacs for each pair are supposed o follow a renewal process, R (d,d ) is a homogeneous Marov Chain. As shown in Appendix A, i is recurren, and ergodic if and only if α >. 2) Forwarding algorihms: We are ineresed in a general class of forwarding algorihms, which all rely on oher devices o ac as relays, carrying daa beween a source device and a desinaion device ha migh no be conemporaneously conneced. These relay devices are chosen purely based on conac opporunism and no using any sored informaion ha describes he curren sae of he newor. The only informaion used in forwarding is he ideniy of he desinaion so ha a device nows when i mees he desinaion for a bundle. We call such algorihms oblivious, hey could be in realiy quie complex and, as we will see, very efficien in some cases. The following wo algorihms provide bounds for he class of algorihm described above: wai-and-forward: The source wais unil is nex direc conac wih he desinaion o communicae. flooding: a device forwards all is received daa o any device which i encouners, eeping a copy for iself. The firs algorihm uses minimal resources bu can incur very long delays and does no ae full advanage of he adhoc newor capaciy. The second algorihm, iniially proposed in 7, delivers daa wih he minimum possible laency, bu does no scale well in erms of bandwidh, sorage, and baery usage. In beween hese wo exreme algorihms, here is a whole range of algorihms ha differ in he number of relays 4 In DTN sandards, a bundle usually denoes a large objec wih a collecion of daa unis.
used o maximize he chance of reaching he desinaion wih a delay as small as possible, while avoiding flooding. The mos imporan reason no o flood is o minimize memory requiremens and relaed power consumpion in relay devices, and o delee he baclog of previously relayed message ha are sill waiing o be delivered, and could be oudaed. A number of sraegies, based on ime-ous, buffer managemen, limi on he number of hops and/or duplicae copies have been proposed (7, 8, 9) o minimize replicaion and baclog. B. Analysis of he wo-hop relaying algorihm Having described he class of oblivious algorihms we are considering in his wor, we now inroduce he wo-hop relaying algorihm, and evaluae is performance for he model of power law iner-conac imes ha we have described. Resuls are generalized o all oblivious algorihms in he following secion. ) Descripion: The wo-hop relaying algorihm was inroduced by Grossglauser and Tse in. This forwarding algorihm operaes as follow: when a source has a bundle o send o a desinaion, i forwards i once o he firs devices ha i mees. This firs device is eiher a relay device or he desinaion iself. If i is he desinaion, he bundle is delivered in one hop; oherwise he device acs as a relay and sores he bundle in a queue corresponding o his desinaion. Bundles from his queue will be delivered when he relay device mees he desinaion. Bundles for he same desinaion are delivered by a relay device in a firs-come-firs-served order. As queuing may occur in he devices ha ac as relays, in he shor conac case, he forwarding process of bundles sen by he source o a relay needs o be of lower inensiy han he bundles sen by his relay o he desinaion. This is he case in he implemenaion proposed in and we mae he same assumpion below. We choose his algorihm o sar our sudy of he impac of power law iner-conac imes on opporunisic forwarding for he following hree reasons: In he shor conac case, his algorihm was shown o maximize he capaciy of dense mobile ad-hoc newors, under he condiion ha devices locaions are i.i.d., disribued uniformly in a bounded region. The mobiliy process of he devices is an imporan parameer. The auhors of assumed ha each node moves a each ime slo o an i.i.d. posiion, which implies ha he iner-conac ime is geomerically disribued. This resul holds more generally for any Marovian evoluion (such as a random wal) defined on a finie domain. I has also been shown when devices move according o he random way-poin mobiliy model (see he analysis of Secion 3 in ). and have shown ha daa experiences a finie expeced delay under hese condiions. 2) Analysis: We consider N mobile devices which ransmi daa according o he wo-hop relaying algorihm described above. Insead of he mobiliy model used in we assume ha conacs beween devices follow he model ha we have inroduced in he beginning of his secion. To ensure sabiliy in he relay s queuing mechanism, we assume ha he source s is no sauraed: bundles are creaed a s during a sequence of ime slos, denoed by ( (s) ) Z. The same assumpion is made for he long conac case alhough sabiliy of he queue occupancy is no an issue in his conex as he queue is empied afer each conac wih he desinaion. We have he following resul, ha is a consequence from he regeneraive heorem (or Smih s formula). Theorem For a pair of source-desinaion devices (s, d), le (s) be he ime when he h bundle is creaed a s o be sen o d, and le (d) be he ime when i is delivered o d. Le D = (d) (s), we have, saring from any iniial condiion: (i) If α < 2, lim ED = +. (ii) If α > 2 and we assume ha all conacs are long, lim ED = D < + and we have R D 2 R where R = 2 + EX2 2 EX. (iii) If α > 2 and we assume ha all conacs are shor, when each source sends daa o a unique and disinc desinaion, wih rae λ < N 2EX, hen he delay of a bundle has finie expecaion. Proof: We sudy firs he case of long conacs, where any amoun of informaion may be exchanged when a conac occurs beween wo devices. We analyzed here a single source-desinaion pair. The wohop relaying sraegy uses muliple roues o ranspor bundles belonging o his pair; ha is because any oher conaced device may ac as a relay. This bundle is ransmied o he firs relay ha is me by s afer ime (s). Le r be his relay, we ) have r = argmin r s R(s,r ; and his ransmission occurs a ime (r) (s) = (s) +min r s R (s,r ). The bundle is hen delivered o desinaion d a ime (d) D = (d) (s) (s) = (r) = min + R(r,d) (r) r s R(s,r) (s). We can rewrie: + R (r,d) (r). (2) Le us firs esablish he posiive resul (ii) ha he wo-hop relaying sraegy achieves a delay wih finie mean if α > 2. Proving (ii) : In his case, E X 2 is finie, and E T (d,d ) =T (d,d ) R (d,d ) = EX(X + )/2 <, for any pair (d, d ) of devices. By Smih s formula (see (5) in he appendix), we have lim E R (d,d ) = EX2 +EX 2EX. The process (min r s R (s,r) ) is aen as a minimum of a finie number of independen processes, corresponding o pairs {(s, r) r s}, which all have he same law. Hence, lim E min r s R(s,r) E X 2 + EX 2EX Lemma 2 can hen be applied o his process, wih ( (s) ) which is independen from i; his proves lim E min r s R(s,r) E X 2 + EX. (s) 2EX If we consider he collecion of random variables ) ) r s, he condiion (i) of Lemma 2 is me. As ((R (r,d).
( (r) ) and (r ) depend only on ( (s) ) and conacs processes belonging o oher pairs han {(r, d) r s}, hey are independen from ( he collecion ) above, and we have lim E R (r,d) +EX = 2 (r) 2EX. Using (2), we have R (r,d) (r) 2 ( + EX2 EX ) D = min r s R(s,r) + R (r,d), hence (s) (r) lim ED ( + E X 2 EX ). Noe ha his resul holds if he law of X is replaced by any law ha admis a finie second momen. Proving (i), for < α < 2 : As α >, Smih s Formula (5) holds in his case for any funcion f verifying he inegrabiliy condiion. Le r denoe any device differen from s. For convenience, le us denoe X = T (r,d) T (r,d), we have for any A, ha may be chosen arbirarily large: A(A + ) I {X A} 2 T (r,d) =T (r,d) min(r (r,d), A) A X. These variables are posiive; hey all have a finie expecaion by comparison wih he righ erm. This proves he inegrabiliy condiion required in (5) for he funcion f(x) = min(x, A), hence we obain lim E min(r (r,d), A) A(A+) 2 PX A EX A2 A α 2.EX. E R (r,d) As his inequaliy holds for arbirary large A, and α < 2, we have: lim = +. The collecion of processes ((R (r,d) ) ) r s verifies condiion (b) of Lemma 2. As ( (r) ) and (r ) are independen of his collecion, we can herefore deduce ha lim E R (r,d) = + hence lim ED = +. (r) Proving (i), for α : In his case, for any device r, he Marov chain defining (R (r,d) ) is recurren null, so ha Orey s heorem (see 5 p.3) implies : lim P R (r,d) = i = for all i In paricular, for any A, lim P R (r,d) < A = and lim P R (r,d) A =.. As a consequence, We have, E R (r,d) A P R (r,d) A and because he resul holds for any arbirary A, we have lim E R (r,d) = +. This holds for any device r. Anoher applicaion of Lemma 2 wih condiion (b) allows us o prove lim E = +. R (r,d) (r) The shor-conac case: The resul (i) follows from he long-conac case, as he delay in he shor conac case is always larger. The proof of (iii) is a lile more complex bu follows from classical resuls on Palm Calculus in discree ime and random wals, i may be found in Appendix B. To summarize, we have idenified wo regions where he behavior of he wo-hop relaying algorihm would differ, under he power law iner-conac ime assumpion: When α is greaer han 2, he algorihm converges o a finie expeced delay, as in he case of an exponenial decay. When α is smaller han 2, he wo-hop forwarding algorihm does no converge o a finie expeced delay, as he delay ha can be expeced, saring from any iniial condiion, grows wihou bound wih ime. This remains rue even for he long conac case, where daa exchange is unlimied during conacs, and queuing in relay devices has herefore no impac on he delay experienced. In oher words, he region α > 2 may be hough of as he sabiliy region of he wo-hop relaying algorihm. C. Generalizaion In his secion we characerize he sabiliy region (defined as he values of α for which an algorihm achieves a bounded delay) for he general class of oblivious algorihms. We conduc he following proofs in he long conac case only. We furher assume, when α > and herefore ha a seady sae exiss, ha he sysem has reached is saionary behavior; oherwise, when α, we sar from any iniial condiion. We generalize he wo-hop relaying algorihm as follows. Insead of sending a single copy of a given daa uni o a unique relay, he source will send m copies of each daa uni: one o each of he firs m relays ha i mees. As we have assumed ha he conac processes belonging o hese relays are independen, he source may hereby reduce he oal ransmission delay by increasing is probabiliy o pic a relay wih a small delay o he desinaion among he m relays o which i has forwarded he message. This observaion is made rigorous in he following lemma: Lemma Le (R (d,d ) ),..., (R (dm,d m )) ) be remaining iner-conac imes for m differen pairs of devices (d i, d i ) i m. We suppose ha hey have reached heir seady sae. We suppose m > and ha + m < α < 2, hen E R (d,d ) =... = E R (dm,d m ) = + and E min(r (d,d ),..., R (dm,d m ) ) <. Proof: As α >, Lemma 3 (ii) holds: A unique saionary disribuion exiss for he produc chain R (d,d ),..., R (dm,d m ), given as he produc of he saionary disribuion for each componen. Hence, ( ) m P min(r (d,d ),..., R (dm,d m ) ) > i = P R (d,d ) > i ( m c (α )) (i + ) m (α ). The expecaion of he minimum is herefore finie as soon as m (α ) < or, equivalenly, α > + m. This resul shows ha for α smaller han 2, he expeced ime o mee he desinaion is infinie. However, he expeced ime for he desinaion o mee a group of m devices may have a finie expeced value, provided ha α > and ha m is large enough. This observaion is he ey componen in he
nex resul, which proves ha using a wo-hop relaying sraegy wih m relays is sufficien o exend he sabiliy region o any value of α >. This heorem also proves ha he case α <, which is observed in mos daa ses, is of a quie differen naure, as even unlimied flooding does no achieve a bounded delay. We commen on his difference furher in Secion V. Theorem 2 Le us consider a source desinaion pair (s, d) and (s), (d), D defined as in Theorem. We assume ha all conacs are long. (i) if α > 2, here exiss a forwarding algorihm using only one copy of he daa, wih a finie expeced delay, such ha, saring from any iniial condiion. lim ED = D < +. (ii) if < α < 2, m N is chosen such ha α > + m, and he newor conains a leas N 2m devices, here exiss an algorihm using m relay devices such ha, in seady sae: ED = D < +. (iii) if α, for a newor conaining a finie number of devices, and any forwarding algorihm, including flooding, we have saring from any iniial condiion lim ED = +. Proof: Proving (i) is jus a reminder of he resul of Theorem. The wo hop relaying algorihm may be chosen and i achieves a finie expeced delay. Proving (ii): Le us assume ha α > +/m and N 2m, where m N. The forwarding algorihm ha we consider in his case is a wo-hop relaying algorihm using m differen relays. STEP : A bundle is creaed a ime in he source (denoed as device s). I is firs ransmied o he m firs devices ha are me. We esimae firs he ime when each of hese m relays are all conaced and have received he bundle. Le us consider he collecion of remaining iner-conac ime wih all he oher devices (R (s,r) ) r s. This collecion conains N variables. If we consider a version of his collecion, sored for each ime, in he increasing order, he ime o conac m differen devices a ime is he m h value of his sored sequence. Corollary 2, which is a simple variaion of Lemma shown in Appendix C, ells ha his variable is of finie expeced value if α > + /(N m + ). This las assumpion is auomaically verified as N m + = N m m by assumpion. STEP 2 : A ime, a copy of he bundle is presen in each of he m relays, ha we denoe r,..., r m. We now consider he vecor (R (r,d),..., R(rm,d) ) which describes he imes needed for each of hese relays o ge in conac wih he desinaion. The ime lengh elapsed unil he pace is delivered o he desinaion is aen as he minimum of his values. An applicaion of Lemma ells us ha his ime has a finie expeced value. As a consequence he overall delay, from he ime of creaion of he bundle in he source, o he delivery a he desinaion, is he sum of wo variables wih finie expecaions. I is hence of finie expeced value. Proving (iii): Le us consider in his case, for a source s and any oher device r in he newor, he remaining ime R (s,r) a ime unil he nex conac. As α <, all of his sequences of random variables are irreducible null recurren Marov chains. By Orey s heorem (5 p.3), we hen have ha lim P R (s,r) = i = for all i when ends o infiniy. In paricular for any arbirary large A, we have lim P A =, so ha R (d,d ) P min r s R(s,r) A = P r s{r (s,r) A}. Consequenly, E min r s R (s,r) diverges for large. As a consequence, saring from any iniial condiion, he ime for a source o reach any oher device is of infinie expecaion as imes increases. No forwarding algorihm, no maer how redundan, can hen ranspor a pace wihin a finie expeced delay, using only opporunisic conacs beween devices. Noe: By comparison, he resul (iii) applies o any case ha includes shor conacs as well as long conacs. Generally, a newor conaining N devices admis forwarding algorihms ha achieve a bounded expeced delay for any α > + N/2. One example of hose is flooding (ha may use up o N 2 relays); ha is no he only one, as a forwarding algorihm using only N/2 relays is sufficien. D. Summary A his sage, we have esablished he following resuls for he class of oblivious forwarding algorihms defined in III-A, in he long conac case : For α > 2 any algorihm from he class we considered achieves a delay wih finie mean. If < α < 2, he wo-hop relaying algorihm, inroduced by, is no sable in he sense ha he delay incurred has an infinie expecaion. I is however sill possible o design an oblivious forwarding algorihm ha achieves a delay wih finie mean. This requires ha m duplicae copies of he daa are produced and forwarded, where m mus be greaer han α, and he newor mus conain 2 a leas α devices. If α <, none of hese algorihms, including flooding, can achieve a ransmission delay wih a finie expecaion. In oher words, we have characerized he performance of all hese algorihms in he face of exreme condiions (i.e. heavy ailed iner-conac imes). The las case where α < corresponds o he mos exreme siuaion, and he resul we provide in his case seems a firs unsaisfacory: none of he algorihms we have inroduced can guaranee a finie expeced delay. To mae he maer worse, his case where α < seems o be ypical of he iner-conac ime disribuion in he minues; day range for all he scenarios we have previously sudied empirically. This overall implies ha he expeced delay for all he scenarios we have discussed before should be a leas of he order of one day. Noe ha his was shown for any forwarding algorihms used, and even when queuing delay in relay devices are negleced. In fac his is a
negaive resul, and we come bac o inerpre i and discuss is implicaions in Secion V. IV. RELATED WORK Our opporunisic communicaion model is relaed o boh Delay-Toleran Neworing and Mobile Ad-Hoc Neworing 5. Research wors on MANET, DTN, and more recenly Poce Swiched Newors 2 confirm he imporance of he problem we address, as several proposiions were made o use mobile devices as relays for daa ranspor. Such an approach has been used o enable communicaion where no conemporaneous pah may be found 7, o efficienly gaher informaion in a newor of low power sensors 3, 4, 5, and o improve he spaial reuse of dense MANET,. All hose wors have proved ha he mobiliy model used has a srong impac on he performance of he algorihms hey use. We did no find any previous wor sudying he characerisics of iner-conac ime for users of porable wireless devices. However, we have idenified relaed wor in he area of modeling and forwarding algorihms. A common propery of he mos common mobiliy models is ha he ail of he iner-conac ime disribuion decays exponenially. In oher words, for hese models, he inerconac ime is ligh ailed: This is he case for i.i.d. locaion of devices in a bounded region (as assumed in ), or more generally for any random wal defined on a finie region, using a comparison argumen. I is also he case for he popular random way-poin model as demonsraed in he Secion 3 of. This fac is commonly used in he analysis of mobile newors o esimae he delay of a scheme. I was shown recenly ha devices moving according o a Brownian moion in a bounded region, exhibi heavy ailed iner-conac ime, wih a finie variance (corresponding in our analysis o he case α > 2) (see 6 and he associaed echnical repor). The mos relevan wor is he algorihm proposed by Grossglauser and Tse in, furher analyzed in. The wo-hop relay forwarding algorihm was iniially inroduced o sudy how he mobiliy of devices impacs he capaciy of he newor. Our wor sars from very differen assumpions. Mos noably, we do no model he bandwidh limiaion due o inerference, as we focus only on he delay induced by mobiliy. However, some of he resuls ha we show could be used o characerize he delay obained in such conexs. V. CONCLUSION We have analyzed several newor scenarios for opporunisic daa ransfer among mobile devices carried by humans, using eigh experimenal daa ses. For all daa ses, we observe ha he iner-conac ime beween wo devices can be approximaed by a power law in he minues; day range. We prove in a simple model he following major resuls: power law condiions may be addressed wih finie expeced delay by oblivious forwarding algorihms as long as he heavy ail index of he power law is greaer han. When he heavy ail index is smaller han, he expeced delay canno 5 www.dnrg.org and www.ief.org/hml.charers/mane-charer.hml be bounded for any forwarding algorihm of ha ype, even when one ignores he queuing occurring in each relay device. We have measured a heavy ail index smaller han in all daa ses. As a consequence, he expeced delay is a leas of he order of one day. These observaions bring new pracical recommendaions o evaluae he performance of forwarding algorihms. Mos of he mobiliy models commonly used oday are characerized by a ligh ailed iner-conac ime disribuion for any pair of nodes. This propery has been used in he pas o esimae he delay in hese newors. Our empirical findings of inerconac ime disribuions, in conras, are well approximaed by a power law for values up o one day. Some mobiliy models can in heory be modified o accoun for his las propery, his may be a fuure research direcion. Anoher complemenary direcion, which is chosen in his paper, is o direcly model opporuniies beween devices insead of heir geographical locaions. This approach has he advanage ha i can be direcly compared wih a growing ses of real-life conneciviy races, now publicly available. We believe ha his is a pracical soluion, a leas for some of he issues o be addressed in opporunisic neworing. More generally our resuls are dealing wih he feasibiliy of forwarding in opporunisic newors and heir consequence requires furher aenion. A leas hree differen direcions may be followed. Firs, i migh be ha reasoning wih expeced value of delay is no suiable, since he possible occurrence of a long delay is unavoidable, whaever forwarding algorihm is used. Applicaions for such newors should herefore be designed o cope wih his aspec of opporunisic communicaion. Second, noe ha we did no model he general case where conac processes for a pair of nodes are heerogeneous or conains significan correlaion. I is sill possible ha a finie expeced delay exiss in a more complex model ha reproduces accuraely he saisical properies of our daa ses. This direcion is appealing bu i requires he removal of one of he modeling assumpions ha we have made and which are common for mos of he resuls currenly nown in his area. I also necessiaes o design a forwarding algorihm ha differeniae beween nodes; some schemes of ha ype have been only recenly proposed 7, 8, 9. Third, one can invesigae how o add connecion opporuniies in a mobile newor, using special devices or parial infrasrucure, ha could in some cases be already available. We are currenly woring on performing more human mobiliy experimens, using differen ype of devices, and diverse sociological groups, in order o follow he direcions we menion above. One of our long erm goal is o sudy he properies of he acual raffic creaed by users in an opporunisic daa newor. VI. ACKNOWLEDGMENTS We would lie o graefully acnowledge Dave Koz and Trisan Henderson a Darmouh College, Geoff Voeler a
Universiy of California San Diego, Eyal de Lara and Jing Su a Universiy of Torono, and he Realiy Mining projec a MIT, for providing heir daa. We would lie o han also Vincen Hummel and Ralph Kling a Inel for heir suppor, Mar Crovella for his insighful commens on our findings, Lauren Massoulié for his advice on some of he proofs, and he paricipans of he hree experimens we conduced. REFERENCES P. Hui, A. Chainreau, J. Sco, R. Gass, J. Crowcrof, and C. Dio, Poce swiched newors and he consequences of human mobiliy in conference environmens, in Proceedings of ACM SIGCOMM firs worshop on delay oleran neworing and relaed opics, 25. 2 M. McNe and G. M. Voeler, Access and mobiliy of wireless PDA users, Compuer Science and Engineering, UCSD, Tech. Rep., 24. 3 T. Henderson, D. Koz, and I. Abyzov, The changing usage of a maure campus-wide wireless newor, in MobiCom 4: Proceedings of he h annual inernaional conference on Mobile compuing and neworing, 24, pp. 87 2. 4 N. Eagle and A. Penland, Realiy mining: Sensing complex social sysems, Journal of Personal and Ubiquious Compuing, 25. 5 P. Bremaud, Marov Chains, Gibbs Field, Mone Carlo Simulaion and Queues, s ed. Springer-Verlag, 999. 6 F. Baccelli and P. Bremaud, Elemens of Queuing Theory, 2nd ed. Springer-Verlag, 23. 7 A. Vahda and D. Becer, Epidemic rouing for parially-conneced ad hoc newors, UCSD, Tech. Rep., 2. 8 X. Chen and A. Murphy, Enabling disconneced ransiive communicaion in mobile ad hoc newors, in Proceedings of he Worshop on Principles of Mobile Compuing, 2, pp. 2 27. 9 J. A. Davis, A. H. Fagg, and B. N. Levine, Wearable compuers as pace ranspor mechanisms in highly-pariioned ad-hoc newors, in ISWC : Proceedings of he 5h IEEE Inernaional Symposium on Wearable Compuers, 2, p. 4. M. Grossglauser and D. Tse, Mobiliy increases he capaciy of ad hoc wireless newors, IEEE/ACM Transacions on Neworing, vol., no. 4, pp. 477 486, 22. G. Sharma and R. R. Mazumdar, Delay and capaciy rade-off in wireless ad hoc newors wih random way-poin mobiliy, 25, preprin, School of ECE, Purdue Universiy. Online. Available: hp://min.ecn.purdue.edu/ linx/paper/i5.pdf 2 J. Sco, P. Hui, J. Crowcrof, and C. Dio, Haggle: a neworing archiecure designed around mobile users, in Proceedings of The Third IFIP WONS Conference, 26. 3 S. Jain, R. C. Shah, G. Borriello, W. Brunee, and S. Roy, Exploiing mobiliy for energy efficien daa collecion in sensor newors, in Proceedings of IEEE WiOp, 24. 4 P. Juang, H. Oi, Y. Wang, M. Maronosi, L. S. Peh, and D. Rubensein, Energy-efficien compuing for wildlife racing: design radeoffs and early experiences wih zebrane, in ASPLOS-X: Proceedings of he h inernaional conference on Archiecural suppor for programming languages and operaing sysems, 22, pp. 96 7. 5 T. Small and Z. J. Haas, The shared wireless infosaion model: a new ad hoc neworing paradigm (or where here is a whale, here is a way), in MobiHoc 3: Proceedings of he 4h ACM inernaional symposium on Mobile ad hoc neworing & compuing, 23, pp. 233 244. 6 X. Lin, G. Sharma, R. R. Mazumdar, and N. B. Shroff, Degenerae delay-capaciy radeoffs in ad-hoc newors wih brownian mobiliy, IEEE/ACM Trans. New., vol. 4, no. SI, pp. 2777 2784, 26. 7 H. Dubois-Ferriere, M. Grossglauser, and M. Veerli, Age maers: efficien roue discovery in mobile ad hoc newors using encouner ages, in MobiHoc 3: Proceedings of he 4h ACM inernaional symposium on Mobile ad hoc neworing & compuing, 23, pp. 257 266. 8 A. Lindgren, A. Doria, and O. Schelén, Probabilisic rouing in inermienly conneced newors, SIGMOBILE Mob. Compu. Commun. Rev., vol. 7, no. 3, pp. 9 2, 23. 9 J. Leguay, T. Friedman, and V. Conan, Evaluaing mobiliy paern space rouing for DTNs, in Proceedings of IEEE Infocom, 26. 2 S. Asmussen, Applied Probabiliy and Queues, 2nd ed. Springer-Verlag, 23. A. Preliminary Resuls APPENDIX ) Independen composiion and limi expecaion: Lemma 2 Le ) N ) i I be a finie collecion of sequences of real valued random variables verifying, lim E F (i) = l, where l R {+ }, and (a) i,, E F (i) R, and l R, or(b) i,, E F (i) R {+ } and l = +. Le ( ) N and (i ) N be wo N valued processes, independen from F, such ha lim = + a.s.. We hen have lim E F (i ) = l ((F (i) Proof: Le us firs develop he following expecaion E F (i ) = jp i = i, =, F (i) = j i I j = jp i = i P = P F (i) = j i I j = P i = i P = E F (i) i I (3) If we suppose (a), we have l < + and ε >, T s.. ( > T = E F (i) l < ε 2 ). E Le M = sup i I, T F (i) l, here exiss K s.. > K = P T ε and hence 2 M E F (i ) l can be bounded from above by P i = i P = E l i I i I i I P i = i F (i) (M T P = + P = E >T P i = i (ε/2 + ε/2) ε. F (i) ) l Le us now suppose (b), we have l = + and A >, T, ( > T = E F (i) 2 (A + )). ( ) Le M = sup max E F (i),. K s..: i I, T > K = P T max(/2, /M ), and E F (i ) = i I P i = i P = E F (i) P i = i ( M T P = i I + ) P = E F (i) >T P i = i ( + 2 ) (2(A + )) A. i I
2) Remaining iner-conac: Because he conac process (U (d,d ) ) is a renewal process, he sequence (R (d,d ) ) of inegers is an Homogeneous Marov Chain in N such ha: { R (d,d ) + = R (d,d ) if R (d,d ) >, R (d,d ) + = i wih prob. P X = i if R (d,d ) =. (4) This Marov Chain is clearly irreducible and aperiodic as P X = >, i is recurren as X is almos surely finie. The following lemma characerizes is properies, which depend on he value of α, based on classical resuls from he heory of Marov chains. Lemma 3 For any devices d, d, e, e such ha (d, d ) (e, e ), we have (i) If α >, (R (d,d ) ) is ergodic. (ii) If α >, he chain (R (d,d ), R (e,e ) ) is ergodic and admis he following saionary disribuion: π(i, j) = (i+) α (j+) α (c ) where c 2 = i (i + ) α. (i + 2) (α ) c (α ) such ha, we have in seady sae P R (d,d ) > i (iii) If α, (R (d,d ) ) is recurren null. (i + ) (α ) c (α ) Proof: Le us inroduce re he ime for R (d,d ) o reurn in he sae. From he srucure of he Marov chain (4), saring from sae, we can easily deduce ha E re = EX. If α >, we have EX < +, proving (i), and if α, we have EX = +, proving (iii). By (i), we now ha he Marov chain R (d,d ) is recurren posiive, hence i admis a saionary disribuion. I is easy o chec, from is regeneraive srucure, ha i is given by: π(i) = c (i + ) α where c = / i (i + ) α. The same resul holds for R (e,e ). As hese wo Marov Chains are independen, one can hen chec easily ha he produc Marov chain (R (d,d ), R (e,e ) ), which is irreducible and aperiodic, admis a saionary disribuion given by he produc of he measure. I is hence ergodic. In seady sae we have: P R (d,d ) > i = π(i) = (j + ) α c j>i j>i As he funcion x (x + ) α is non-increasing, we have: i+(x + ) α dx j>i hus (i + 2) (α ) α (j + ) α (j + ) α j>i which complees he proof for (ii). i. (x + ) α dx (i + 2) (α ) α Smih s formula for α > : For any devices d and d, he process (R (d,d ) ) is regeneraive wih respec o he delayed renewal sequence (T (d,d ) ). If we assume α >, we have EX < +, hence he iner-even of he sequence ) admis a finie mean. We now in his case (see 5 p.48) ha T (d,d ) E lim E f(r (d,d ) =T (d,d ) f(r (d,d ) ) ) = (T (d,d ) for any f verifying E E T (d,d ) T (d,d ) T (d,d ) f(r (d,d ) =T (d,d ) B. Queuing wih a Process of Service Insan ) <. In consras wih classical queueing sysems, he nodes of a mobile newor only serve bundles from a given queue when hey are in conac wih he corresponding desinaion. In his secion, we exend some well nown resuls on queues o handle his consrain. Le us consider a queue receiving cusomers according o a poin process a = { a Z }, ha may be served only a some service insan, which follow a process s = { s m m Z }. We mae he following assumpions: A cusomers arriving a ime joins he queue and can be served saring from +. A each service insan, one cusomer from he queue is served, excep if he queue is empy. Hence, his sysem behaves as if he ime slo were divided in wo pars: in he firs half of he ime slo, a cusomer from he queues is served if he slo is a service insan; in he second half, new cusomers join he queue. Le us inroduce Q() he number of cusomers presen afer he firs half of he ime-slo is compleed. The process Q follows he recursion: (5) Q() = max(, Q( ) + N a ( ) N s ()), (6) where N a (resp. N s ) denoes he couning measure associaed wih he poin process a (resp. s). ) Saionariy, Lile s law: Resuls are shown in he θ saionary ergodic framewor (see 6). We assume here ha θ is a measurable mapping Ω Ω, which preserves he probabiliy measure (i.e. P θ = P) and is ergodic (all θ invarian evens have probabiliy or ). A poin process is called saionary wih respec o θ if is couning measure verifies: N(θ(ω), C) = N(ω, C + ) where C Z, and ω Ω. We define is inensiy as EN(). The nex resul follows closely he proof of he sabiliy regime for a single server queue (see 6 p.83-87). I shows ha, under a simple sabiliy condiion, he sysem admis a seady sae ha is saionary in a srong sense (compaible wih he shif θ). The expeced delay of a cusomer hrough his sysem is hen given by a generalized Lile Formula. Noaion: Following he usual convenion of Palm calculus (here, in discree ime), we denoe by P a he probabiliy measure P under he condiion ha poin process a has a poin in =. We number cusomer wih he convenion ha cusomer = denoes he las cusomer ha arrived sricly before. We denoe by V he sojourn ime of cusomer.
Lemma 4 If a, s are wo saionary poin processes wih respec o θ, wih respecive inensiies λ, µ such ha λ < µ, (i) There exiss an iniial condiion, Q < a.s., such ha he queue process verifies: Q() = Q θ. (ii) In his saionary regime: E a Ṽ = + λ E Q (iii) If he queue sars empy, limsup EV + λ E Q Proof: We define he sequence of variables indexed by T Q T = max (N a (,..., ) N s ( +,..., )). T Clearly his sequence is posiive, non-decreasing, and verifies ( Q T+ θ = max, Q ) T + N a () N s (). (7) I hen admis an a.s. limi, denoed by Q, verifying: ( Q θ = max, Q ) + N a () N s (). This limi may ae infinie values. Noe ha { Q = } is θ invarian and θ is ergodic, i hen has probabiliy or. In oher words eiher his limi is a.s. infinie or i is a.s. finie. We can rewrie (7) as: Q T+ θ = Q ( ) T min QT, N s () N a (), such ha ( ) E min QT, N s () N a () = E QT Q T+ θ = E QT Q T+. Proof: We recall he classical resul on random wals (see p.27 in 2): for (Z ) i.i.d., EZ <, E (Z + )2 <, we have E max (Z +... + Z ) <. (8) Le us prove firs, for any ν > λ: E max (N a(,...,) ν) <. > Le us denoe by S, S 2, he sequence of poins of he process a ha belongs o {,, 2,... }. They may be seen as he resul of a random wal S n = X +... + X n, where variables (X ) are i.i.d. and follow he iner-even disribuion. The above expecaion may be rewrien E max n> (n ν S n) = ν E max n> (Y +... + Y n ). where Y = ν X. Noe ha E (Y + )2 ν 2 < and EY <, proving by (8) ha he above expecaion is finie. Nex, we prove for any ν < µ, E max ( ν N s(,...,)) > = ν E = E max n> (Z +... + Z n ) max n> (S n ν n) <. as Z = X ν, EZ < and E (Z + )2 E (X )2 <. To conclude, we choose ν such as λ < ν < µ, and we have: max > (N a(,...,) N s (,...,)) E By monoone convergence, we deduce ( ) = E max E min Q, Ns () N a () (N a(,...,) ν + ν N s (,..., )) > E max Assume Q (N a(,..., ) ν) + max ( ν N s(,...,)). is a.s. infinie, he minimum above is hen > > always given by he second erm, which implies ha EN s () N a () = µ λ. By he converse inducion, µ > λ = Q Corollary If a and s saisfy condiions of Lemma 5, < a.s. ; which proves (i). E Q < and E a Ṽ <. (ii) is an applicaion of he Campbell-Mece egaliy: Proof: According o he proof of Lemma 4, E Q = E I {a }I {a +V } Q = max (N Z = λ a (,..., ) N s ( +,...,)). I { } I {+v } P a V = v The resul is hen following he above lemma. v> Z = λ v>(v )P a V = v = λ(e a V 3) Proof of Theorem in he shor-conac case: Le us ) summarize resuls from he above subsecions: In a queue wih arrival a and service insan s, cusomers experienced We have a.s. V Ṽ, which proves (iii). a finie expeced delay if () a and s are renewal processes, 2) Expeced queue lengh: (2) he sabiliy condiion is verified and (3) iner-serviceinsan disribuion has a finie variance. Noe ha condiions (2) and (3) are necessary. In he following, we presen a Lemma 5 Assume a and s are wo renewal poin processes, scheme insuring ha all queues implemened in he mobile - wih inensiies λ < µ, nodes verify condiions (),(2) and (3). I may be improved a - such ha iner-even disribuion F a has a finie mean, he cos of an addiionnal effor o weaen assumpion (). - and he iner-even disribuion F s has a finie variance, Each source devices s mainain a se of N source queues hen E max (N corresponding o each oher device. We assume ha bundles a(,..., ) N s (,...,)) <. > are creaed in each of hese queues according o a renewal process wih inensiy λ < p 2 EX wih p >. When anoher
device d is me during an odd ime slo, a bundle from he queue associaed wih d is served, if his queue is no empy. The device d may be he desinaion for s, bu, oherwise, he bundle is enering a relay queue (see below). For echnical reason we also assume ha wih a small probabiliy p, aen independenly, an independen blocing occur, and no bundle a all is sen by he source during his conac. All devices (including all sources) mainain, in addiion, N relay queues, each one corresponding o a given desinaion. When a bundle is received during an odd imeslo (as described above), i is enering he relay queue corresponding o is desinaion. If anoher device d is me during an even ime slo, a bundle for desinaion d is sen, unless he corresponding queue is empy. Le us prove ha bundles experience finie expeced delay in each of hese queues: Each source queue receives and serves bundles according o saionary processes ha saisfy (), (2) and (3). A relay queue saisfy (2) and (3); unforunaely, he arrival process in his queue is no a renewal process. Neverheless, he same resul holds by a comparison. All arrival imes of a bundle in his relay queue are included in a quasi-sauraed renewal process (ha include all meeing imes wih he source corresponding o he desinaion of he queue, wihou independen blocing). Noe ha he expeced delay in a relay queue is always no larger han he expeced delay in he same queue wih a quasi-sauraed arrival process. One chec easily ha his las case verify all condiions (), (2) and (3), proving ha he expeced delay is finie in boh cases. We deduce ha all sources can ransmi o heir desinaion a a rae smaller han (N ) ( p) 2EX, such ha bundles experienced a finie expeced delay. As p may be chosen arbirarily, he same resul holds for any rae smaller han N 2 EX. probabiliy may be rewrien as: m ( ) p ( p) m p m j+ m =m j+ m =m j+ ( ) This proves ha, for c 2 = m c (α ) =m j+ m P M j > n c 2 (n + ) (m j+)(α ), ( ). m which implies ha EM j < as soon as α > + m j+. C. Proof of Corollary of Lemma For any real numbers (x,...,x m ), and i m, le us denoe by ord(i, (x,..., x m )) he i h elemen of he sequence afer i is reordered in he increasing order. In paricular ord(, (x,..., x m )) = min(x,...,x m ). We have Corollary 2 Le (R (d,d ) ),..., (R (dm,d m )) ) be he remaining iner-conac imes for m differen pairs of devices (d i, d i ) i m. We suppose ha α > + m j+, ( ) hen E ord j, (R (d,d ),...,R (dm,d m ) ) <. ( ) Proof: Le M j = ord j, R (d,d ),...,R (dm,d m ) { P M j > n = P # i } R (di,d i ) > n m j +. This is he probabiliy ha a leas m j + evens occur on a collecion of m variables. Noe ha all hese evens are independen, and each of hem occurs wih he same probabiliy p (n+) (α ) c (α ). As a consequence, he above