Impact of Human Mobility on the Design of Opportunistic Forwarding Algorithms

Transcription

1 Impac of Human Mobiliy on he Design of Opporunisic Forwarding Algorihms Augusin Chainreau, Pan Hui *, Jon Crowcrof *, Chrisophe Dio, Richard Gass, and James Sco, Thomson Research 46 quai A Le Gallo Boulogne FRANCE augusinchainreau@homsonne chrisophedio@homsonne *Universiy of Cambridge Inel Research 5 JJ Thomson Avenue Cambridge, CB3 0FD, UK panhui@clcamacu, joncrowcrof@clcamacu richardgass@inelcom, jameswsco@inelcom Absrac Sudying ransfer opporuniies beween wireless devices carried by humans, we observe ha he disribuion of he iner-conac ime, ha is he ime gap separaing wo conacs of he same pair of devices, exhibis a heavy ail such as one of a power law, over a large range of value This observaion is confirmed on six disinc experimenal daa ses I is a odds wih he exponenial decay implied by mos mobiliy models In his paper, we sudy how his new characerisic of human mobiliy impacs a class of previously proposed forwarding algorihms We use a simplified model based on he renewal heory o sudy how he parameers of he disribuion impac he delay performance of hese algorihms We mae recommendaion 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 - ransmiing over a wireless (or wired) newor inerface, and carrying from locaion o locaion by heir user (while sored in he device) 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 fundamenally differen neworing proocols han 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 ae an experimenal approach We analyze six disinc daa ses, hree of which we have colleced ourselves We define he iner-conac ime as he ime beween wo ransfer opporuniies, for he same devices We observe in he six races ha he iner-conac ime disribuion follows a heavy ailed disribuion on a large range of values Inside his range he iner-conac ime disribuion can be compared o he one of a power-law We sudy he impac of hose heavy ailed iner-conac imes on he acual performance and heoreical limis of a general class of opporunisic forwarding algorihms ha we call naive forwarding algorihms Algorihms in his class do no use he ideniy of he devices ha are me, nor he recen hisory of he conacs, or he ime of he day, in order o mae forwarding decision Insead forwarding decision are based on forwarding rules saically defined ha bound he number of daa replicaes, or he number of hops Based on our experimenal observaions, we develop a simplified model of opporunisic conac beween human-carried wireless devices I is based on several independence assumpions which are usually me, a leas implicily, in he lieraure of mobile ad-hoc rouing We do no claim ha his model is saisfacory o have accurae performance of differen forwarding algorihms I raher serves our purpose which is o demonsrae how heavy ail iner-conac imes influence he performance of naive forwarding algorihms in opporunisic ransmission condiions and how hese forwarding algorihms should be configured o offer reasonable performance guaranee Our 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 wors The paper concludes wih a brief summary of conribuions and presenaion of fuure

2 wor, including a discussion of our assumpions II EXPERIMENTAL ANALYSIS A Daa ses In order o conduc informed design of forwarding algorihms beween devices carried by humans, i is imporan o sudy daa on he frequency and duraion of conacs beween hem 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, wih a granulariy measured in seconds We examined wo ypes of daa ses Firs, we used races made available o us by people who have performed previous measuremen exercises Three daa ses emerged, namely from UCSD, Darmouh Universiy 2 and he Universiy of Torono 3 We complemened hese races wih hree of our own experimens These six experimens use differen user populaions as well as differen wireless echnologies The characerisics of he six daa ses, explained below, are shown in Table I ) Exernal daa ses: UCSD and Darmouh mae use of WiFi neworing, wih he former including clienbased 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 ad-hoc mode), and creaed a lis of ransmission opporuniies by deermining, for each pair of devices, he se of ime regions for which hey shared a leas one AP Unforunaely, his assumpion inroduces inaccuracies On one hand, i is overly opimisic, since wo devices aached o he same access poin may sill be ou of range of each oher On he oher hand, he daa migh omi connecion opporuniies, since wo devices may pass each oher a a place where here is no insrumened access poin, and his conac would no be logged In addiion, he UCSD daa se is more exhausive han he Darmouh one, since i logs all reachable APs for each clien a each ime slo, while he Darmouh daa only logs he associaed AP Anoher issue wih hese daa ses is ha he devices are no necessarily co-locaed wih heir owner a all imes (ie hey do no always characerize human mobiliy) Despie hese inaccuracies, he WiFi races are a valuable source of daa, since hey span many monhs and include housands of devices In addiion, considering ha wo devices, conneced o he same AP, are poenially in conac is no alogeher unreasonable, as hese devices could indeed communicae hrough he AP, wihou using endo-end conneciviy The Universiy of Torono have colleced races from 20 Blueooh-enabled PDAs ha were disribued o a group of sudens These devices performed a Blueooh inquiry each 00s 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 requires ha he PDAs are carried by subjecs and ha hey have sufficien baery life for hem o paricipae in he daa collecion Daa may be colleced over a long period if devices are recharged The daa se we use comes from an experimen ha lased 6 days 2) imoe-based experimens: In order o complemen he previous races, we buil our own experimen using Inel imoes, which are embedded devices similar o Crossbow moes, bu wih he ey feaure (for our experimens) ha hey communicae via Blueooh We programmed he imoes o log conac daa 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 4; an anonymized version of our daa will be made available o oher research groups on demand Three imoe-based experimens were conduced The firs included eigh researchers and inerns woring a Inel Research in Cambridge The second obained daa from welve docoral sudens and faculy comprising a research group a he Universiy of Cambridge Compuer Lab The hird experimen was conduced during he IEEE INFOCOM 2005 conference in Miami where 4 imoes where carried by aendees for 3 o 4 days imoes conacs were classified ino wo groups: imoes recording he sighings of anoher imoes are classified as inernal conacs, while sighings of oher ypes of Blueooh devices are called exernal conacs The exernal conacs are numerous and include anyone who has an acive Blueooh device in he viciniy of he imoe carriers, hereby providing a measure of acual wireless neworing opporuniies presen a ha ime The 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 wwwxbowcom

3 User Populaion Inel Cambridge Infocom Torono UCSD Darmouh Device imoe imoe imoe PDA PDA Lapop/PDA Newor ype Blueooh Blueooh Blueooh Blueooh WiFi WiFi Duraion (days) Granulariy (seconds) Devices paricipaing Number of inernal conacs,09 4,229 22,459 2,802 95,364 4,058,284 Average # Conacs/pair/day Recorded exernal devices N/A N/A N/A Number of exernal conacs,73 2,507 5,79 N/A N/A N/A TABLE I COMPARISON OF DATA COLLECTED IN THE SIX EXPERIMENTS 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 We decided no o aemp o analyze how much daa can be ranspored for each of hem, as his srongly depends on facors such as he ransmission proocol, he anennas used, and oher facors ha could be modified o provide improved ransmission performance In our analysis in Secion III, we address wo exreme cases corresponding o a lower and upper bounds of he amoun of daa ha may be ransferred in each connecion opporuniy We define he iner-conac ime as he ime elapsed beween wo successive conacs of he same devices Iner-conac ime characerizes he frequency wih which paces can be ransferred beween newored devices; i has rarely been sudied in he lieraure Two remars mus be made a his poin: 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 nex conac wih he same devices 2 Anoher opion would be o compue he remaining iner-conac ime seen a any ime, ie a ime, for each pair of devices: he remaining iner-conac ime is he ime i aes afer, before a given pair of devices me again (a formal definiion is given in Secion III) Iner-conac 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, or inspecion paradox (see p47 in 5) A similar relaion holds for saionary process, in he heory of Palm Calculus (see p5 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 values of iner-conac imes In oher words he definiion ha was chosen is he mos 2 Iner-conac saring afer he las conacs recorded for his pair of devices were no included conservaive one in he presence of large values Second, he iner-conac ime disribuion is influenced by he duraion and he granulariy of he experimen 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, iner-conac imes ha las less han he granulariy of he measuremen (which ranges from wo o five minues among differen experimens) canno be observed Anoher measure of he frequency of ransfer opporuniies, ha could be considered, is he iner-any-conac, ie for a given device, he ime elapsed beween wo successive conacs wih any oher device This measure is very much dependen on he deploymen of wireless devices and heir densiy during he experimen, as i characerizes ime ha devices spend wihou meeing any oher device This measure was sudied for mos of hese daases in 4, we do no presen furher resuls here, due o a lac of space C Iner-conac ime characerizaion We plo he iner-conac ime disribuion for all six experimens in Figure -2 For he wo firs imoe experimens (labeled Inel and Cambridge) he disribuion of iner-conac were compued using all pairs of wo imoes Therefore i conains only values associaed wih inernal conacs (However, we observe he exac same properies for exernal conacs) Daa from Torono experimen were also colleced beween pairs of experimenal Blueooh devices Disribuions for hese hree daases are ploed in Figure (lef) Disribuions belonging o he imoe based experimen a Infocom is shown in Figure (righ), where inerconacs belonging o boh inernal and exernal conacs have been ploed separaely for comparison Figure 2 presens he disribuion of iner-conac compued using races from WiFi experimens All plos describe he ail disribuion funcion, in log-log scale

4 0 0 PX>x PX>x 00 Torono imoe Inel imoe Cambridge PL wih slope for an exernal addresses for an imoe PL wih slope min 0 min h 3 h 8 h Time day wee monh min 0 min h 3 h 8 h Time day wee monh Fig Tail disribuion funcion of he iner-conac ime in six experimens: imoe-based experimen a Inel and Cambridge, and Torono experimen (lef), imoe-based experimen a INFOCOM (righ) The mos ineresing region is he middle of he graphs, as he lefmos and righmos pars show arifacs due o he granulariy and duraion of he experimens as explained above In his region, all six disribuions show he same characerisics: hey exhibi an heavy ail, ha can be approximaed or lower bounded by he ail of a power law, over a large range of value This common propery is raher surprising given he diversiy of he six daa ses The mos noable difference is ha he mach wih a power law, as evidenced by he sraighness of hecurve,isbeerforhedaaseshaareshownin Figures (righ) and 2, which conain he larges number of conacs Figure (righ) proves ha he disribuion is almos unchanged if one consider inernal or exernal conacs The same resuls was shown for he wo oher imoe experimens and are presened in 7 A power law is characerized by is coefficien reflecing he slope of he line on log-log graphs; we show laer ha his coefficien is criical for he performance of he forwarding algorihms presened in Secion III For he imoe-based experimens a Inel and Cambridge, and he daa colleced in Torono, he ail is lower bounded by a power law wih coefficien 09 for he range 2 min; day The disribuion for he imoebased experimen a Infocom is remarably close o a power law wih coefficien 04 on he range 2 min, 6h The ail from Darmouh daa can be approximaed by a power law wih a coefficien of 03 on he range 0 min; wee The ail from UCSD daa can be also compared wih a power law wih coefficien 03, bu over a more limied range 0 min; day PX>x min 0 min h Darmouh PL wih slope 03 UCSD PL wih slope 03 3 h 8 h day Time wee monh Fig 2 Tail disribuion funcion of he iner-conac ime: daa colleced in Darmouh and UCSD A ail ha can be compared or lower bounded by a power law means he ail disribuion funcion decreases slowly over his range This conradics he exponenial decay ha is implied by many mobiliy models in he lieraure (see Secion IV) As a resul, opporunisic neworing algorihms which have been designed around exponenial models mus be re-evaluaed in he ligh of our observaions (see nex secion) In he imoe-based races, 8 o 25% of iner-conac imes are greaer han one hour, and 2 o 3% are greaer han one day In

5 he Torono race, 3% las more han a day, and 7% las more han one wee Similarly in he Darmouh race, we find ha large iner-conac imes are far from negligible: 20% las more han a day, 0% las more han a wee In he UCSD race, 5% las more han a day, and 4% las more han one wee While he WiFi experimens have longer duraions, longer iner-conac imes may be affeced by he more limied mobiliy of lapops or PDAs as heir users may no carry hem all he ime However, his is a characerisic of how users of wireless devices behave and his should be aen ino consideraion in he design of forwarding mechanisms Wha remains is ha he same paern a heavy ail ha can be compared o he one of a power law seems o apply o all experimens despie he fundamenal differences in mehodology and in experimenal environmens III FORWARDING WITH POWER LAW BASED OPPORTUNITIES We now sudy analyically he impac of our findings on he performance of a class of forwarding algorihms We define firs our model of he opporunisic behavior of mobile users ha is based on our experimenal observaions A Assumpions and Forwarding Algorihms ) Conac process model: We consider a sloed ime =0,, For a given pair of devices (d, d ),leus inroduce is conac process (U (d,d ) ) 0 defined by: { U (d,d ) if d and d = are in conac during slo, 0 oherwise For he pair (d, d ) we consider he sequence of he ime slos T (d,d ) 0 <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 7) ha he conac ime disribuion is also heavy ailed, bu i aes smaller values, by several orders of magniude, han he ones of he iner-conac ime Under his condiion, he ime τ (d,d ) = T (d,d ) + for any d, d and 0 is he iner-conac ime afer h conac of his pair We suppose in our model ha i is disribued according o he law X, haisa power law disribuion wih coefficien α: T (d,d ) P X = α for all =, 2, () In paricular, variable X is no bounded bu i is almos surely finie In addiion we assume ha he conac process (U (d,d ) ) 0 of each pair is a renewal process, and ha conac processes associaed wih differen pairs are independen In oher words, he iner-conac imes in he sequence (τ (d,d ) ) 0 are iid, 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 verified, or implicily assumed, in mos of he analysis 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 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 a single daa uni of a given size can be sen beween devices in a single ime slo where hey are in conac he long conac case : where wo devices in conac can exchange an arbirary amoun of daa during a single ime slo These wo cases represen a lower and an upper bounds for he evaluaion of bandwidh The number of daa uni ransmied in a conac (wheher shor or long) is defined as a daa bundle he long and he shor case differ from a queuing sandpoin In he long conac case, he queue is empied any ime a desinaion is me In he shor conac case, only one daa uni is sen and herefore, daa can accumulae in he memory of he relay device Noe ha our model is no aing ino accoun explici geographical locaions and movemen of devices, as i assumes direcly ha he processes of conacs beween hem are given 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 Before describing forwarding algorihms ha we consider, le us inroduce for any pair of device (d, d ), he remaining iner-conac ime, observed a ime slo : I is denoed by R (d,d ) and defined as { =min R (d,d ) and U (d,d ) = } 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

6 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 of a bundle We call such algorihms naive, alhough 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 copies for iself The firs algorihm uses minimal resources bu can incur very long delays and does no ae full advanage of he ad-hoc newor capaciy The second algorihm, ha was iniially proposed in 8, 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 class of algorihms ha play on he number of relay devices o maximize he chance of reaching he desinaion in a bounded delay 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 sen message ha are sill waiing o be delivered, and could be oudaed Some sraegies, based on ime-ous, buffer managemen, limi on he number of hops and/or duplicae copies have been proposed (see 8, 9, 0) o minimize replicaion and baclog B Analysis of he wo-hop relaying algorihm Having described he class of naive algorihms we are considering in his wor, we now inroduce he wohop relaying algorihm, and evaluaes is performance for he model of power law iner-conacs ha we have described Resuls are generalized o he class of naive 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 iid, disribued uniformly in a bounded region This resul depends srongly on he mobiliy process of devices Auhors of assumed an exponenial decay of he iner-conac ime The same resul has been proven for devices following for insance he random way-poin mobiliy model 2 and 2 have shown ha 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 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, as 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 We have for D = (d) (s) : (i) If α<2, lim E D =+ (ii) If α>2, and we assume ha all conacs are long, lim E D = D <+ and we have R D 2 R where R = 2 + EX2 2EX (iii) If α > 3, and we assume ha all conacs are shor, for a saionary poin process ( (s) ) 0 wih

7 inensiy λ<(n )/ R, here exiss a saionary regime where he delay of a bundle has a finie expeced( value D verifying ) for a consan σ R 2 + EX2 EX D σr N λ R 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 wo-hop relaying sraegy uses muliple roues o ranspor bundles belonging o his pair; ha is because any oher conaced device may ac as a relay Le us denoe by (s) he ime when he h bundle is creaed in he source for his desinaion This bundle is ransmied o he firs relay ha is me by s afer ime (s) The relay chosen is r ) = argmin r s ; and his R(s,r (s) ransmission occurs a ime (r) = (s) +min r s R (s,r ) (s) The bundle is hen delivered o desinaion d a ime = (r) + R (r,d) Noe ha in his case: (r) (d) D =min r s R(s,r) (s) + 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 T (d,d ) E =T (d,d ) R (d,d ) = E X(X +)/2 <, forany 0 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) ) 0 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 + E X 2E X Lemma 2 can hen be applied o his process, wih ( (s) ) 0 which is independen from i; his proves lim E min r s R (s,r) EX2 +EX (s) 2EX If we consider he collecion of random variables ((R (r,d) ) 0 ) r s, he condiion (i) of Lemma 2 is me As ( (r) ) 0 and (r ) 0 depend only on ( (s) ) 0 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) = +EX2 (r) 2EX Using (2), we have R (r,d) (r) 2 ( + EX2 EX ) D =min r s R(s,r) + R (r,d), and (s) (r) lim E D ( + E X 2 E X ) 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) 0,wehavefor any A, ha may be chosen arbirary large: A(A +) I 2 {X A} T (r,d) =T (r,d) 0 min(r (r,d),a) AX 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 2EX As his inequaliy holds for A arbirary large, and α< 2, we have: lim E R (r,d) = + The collecion of processes ((R (r,d) ) 0 ) r s verifies condiion (b) of Lemma 2 As ( (r) ) 0 and (r ) 0 are independen of his collecion, we can herefore deduce ha lim E R (r,d) =+ hence lim E D =+ (r) Proving (i), forα : In his case, for any device r, he Marov chain defining (R (r,d) ) is recurren null, so ha Orey s heorem (see 5 p3) ells us : lim P R (r,d) = i =0 for all i In paricular, for any A arbirary large, lim P R (r,d) <A =0 and lim P R (r,d) A = We have, E R (r,d) AP R (r,d) A As a consequence, 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 resul (i) for shor conacs can hen be deduced as he delay in his case, ha may include some queuing, is always greaer Proving (iii) : This resul is in fac an exension of a mehod proved in 3, where he wo-hop relaying sraegy was analyzed for a mobiliy model assuming brownian moion The argumen is he following: Le us focus on he queue belonging o a sourcedesinaion pair, in a given relay We denoe arrival of daa bundles in his queue by We inroduce W ha is he remaining load in he queue when he bundle

8 arrives W is equal o he ime needed o ransmi o d all he daa presen in he queue when arrives The sequence (W ) follows he following recurrence equaion: W + =(W + s ( + )) +, where s denoes he addiional ime, ha is added o W,o deduce he ime a which pace leaves his queue This equaion is exacly one of a single server queue of cusomer arriving a ime ( ) 0, requesing service (s ) 0 The difficuly comes from he fac ha (s ) 0 is no an iid sequence as i depends on he value of (W ) 0 : In fac if W = 0, s is equal o R r,d Oherwise, he ime when all bundles unil are delivered is a ime of conac T r,d j for a cerain j, and s corresponds o an addiionnal iner-conac ime T r,d j+ T r,d j, ha is independen from he res, and follows law X The ey argumen ha we use here was firs proposed in 2: he law of X is sochasically smaller han he one of R r,d, hence we can show ha he sequence (s ) is sochasically smaller han he sequence R,where R 0 is an iid sequence wih he disribuion R, ha is he one of R r,d in seady sae W is a monoone funcion of he values of he sequence s, hence we can show ha he following sequence ( ( W ) 0, defined by recurrence as W = W + R + ( )), is sochasically greaer han (W ) 0 This allows us o prove he sabiliy of he queue if he arrivals in he queue follow a process wih inensiy λ ER As we have assumed α>3, he second momen of he law of R is finie (as can be seen from he expression of π) Once can hen use Kingman s bound (see (9) in 4) o show ha under his sabiliy condiion, he 2σ expeced value of W verifies: E W R 2(/λ ER) We can apply he same proof o all he N nodes in he newor, so ha he inensiy of he creaion of daa bundle in he source can be λ(n ) 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: For a value of α ha is greaer han 2 in he long conac case and 3 in he shor conac case, he algorihm converges o a finie expeced delay, as in he case of an exponenial decay By opposiion For α smaller han 2, he wo-hop forwarding algorihm will no converge o a finie expeced delay, as he delay ha can be expeced grows wihou bound wih ime This remains rue even for long conac case, where daa exchange are unlimied during conacs, and queuing in relay devices have herefore no impac on he delay experienced In oher words, he region α>2 (α >3 in he shor conac case) may be hough as he sabiliy region of he wohop relaying algorihm C Generalizaion In his secion we characerize he region of sabiliy (defined as he value of α for which an algorihm achieves a bounded delay) for he general class of naive algorihms We conduc he following proof in he long conac case only To do so, 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 conacs processes belonging o hese relays are independen, he source may 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 following lemma: Lemma Le (R (d,d ) ) 0,,(R (dm,d m)) ) 0 be remaining iner-conac imes for m differen pairs of devices (d i,d i ) i m We suppose m> and ha + m <α<2, hen E and E = = E min(r (d,d ),,R (dm,d m) ) R (d,d ) R (dm,d m) =+ < Proof: To illusrae he proof wih simple argumen, we rea he case where m =2 We suppose 3 2 <α<2 and we prove for any wo pairs of devices (d, d ) and (e, e ) ha E min(r (d,d ),R (e,e ) ) < The general case is reaed in Appendix B We decided no o include i direcly in he ex, as i involves many addiionnal noaion, wih almos he same echnique As α >, Lemma 3 (ii) can be applied The produc chain (R (d,d ),R (e,e ) ) 0 hen admis he following saionary disribuion: π(i, j) = (i+) α (j+) α (c ) where c 2 = i 0 (i +) α E min(r (d,d ),R (e,e ) ) We have, by symmery min(i,j) i,j i α j = 2 α i,j = 2 j = min(i, j) (c ) 2 (i +) α (j +) α i,j min(i,j) i α j I α {i j} j α ( j i= i α ) The funcion x x α is non-increasing on 0; +, as α>, hence we have for any i : i α i i x α dx < + as α<2, hence

9 j i α i= min(i,j) i,j j 0 x α dx = j2 α 2 α i α j 2 α j j ( α 2 α j2 α ) ( ) j j2 2α 2 2 α This proves ha he expecaion is finie if α> 3 2 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 lim E D = 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: lim E D = D <+ (iii) if α, for a newor conaining a finie number of devices, and any forwarding algorihm, including flooding, we have lim E D =+ 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, wherem 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 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 (R (s,r) sored sequence Corollary, which is a simple variaion of Lemma shown in Appendix B, 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 his relay 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 is 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 p3), we hen have ha lim P R (s,r) = i =0for all i when ends o infiniy In paricular for any A arbirary large, we have lim P R (d,d ) A =0,soha 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 conac beween devices Noe: By comparison, he resul (iii) applies o any case ha includes shor conacs as well as long conacs A newor conaining N devices admis forwarding algorihms ha achieve a bounded expeced delay for any α>+ N/2 : flooding (ha may use up o N 2 relays) is one of hese, bu i no he only one, as a forwarding algorihm using only N/2 relays is sufficien D Summary, Discussion A his sage, we have esablished he following resuls for he class of so-called naive forwarding algorihm defined in III-A, in he long conac case : For α > 2 any algorihm from he class we considered achieve a delay wih finie mean

10 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 build a naive forwarding algorihm ha achieves a delay wih finie mean This requires ha a number of m duplicae copies of he daa are produced and forwarded, where m mus be greaer han α, and he newor mus 2 conain a leas α devices If α <, none of hese algorihms, including flooding, can achieve a ransmission delay wih a finie expecaion IV RELATED WORK Our opporunisic communicaion model is relaed o boh Delay-Toleran Neworing and Mobile Ad-Hoc Neworing 3 Research wors on MANET and DTN confirm he imporance of he problem we address, as several proposiion were made o use mobile devices as relays for daa ranspor Such an approach was considered o enable communicaion where no conemporaneous pah may be found 8, o gaher efficienly informaion in a newor of low power sensors 5, 6, 7, or o improve he spaial reuse of dense MANET, 2, 8, 3 All hese wors prove ha he mobiliy model ha is assumed has a srong impac on he performance of he algorihms proposed 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 many mobiliy models found in he lieraure is ha he ail of he iner-conac disribuion decays exponenially In oher words, for hese models, he iner-conac ime is ligh ailed This is he case for iid locaion of devices in a bounded region (as assumed in ), or in he case of he popular random way-poin model as demonsraed in 8 I was shown in a recen aricle 3 ha, by opposiion, 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) The mos relevan wor is he algorihm proposed by Grossglauser and Tse in, furher analyzed in 283 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, as bandwidh 3 wwwdnrgorg and wwwieforg/hmlcharers/mane-charerhml migh be unlimied a each conac, and he focus of he analysis is on he delay incurred by he daa ranspored V SUMMARY, CONCLUSION AND FUTURE WORK We sudy a scenario where mobiliy of newored devices and he opporunisic connecion wih oher devices are used o ransfer daa We observe from six experimenal races ha he disribuion of he inerconac ime seen beween wo devices in an opporunisic neworing environmen exhibis a heavy ail over a large range of value, ha can be compared o a power law wih a coefficien less han one This observaion is in conras wih he exponenial decay assumpion made implicily by mobiliy models used o dae in ad-hoc neworing We prove he following major resul: Naive forwarding algorihm may deliver daa wih a bounded expeced delay in he case of ligh ailed iner-conac imes, as well as when mobiliy of devices implies power law iner-conac wih coefficien greaer han Bu all of hese algorihms have indeed an infinie expeced delay when mobiliy implies power law wih coefficien smaller han Some of he implicaions of our findings are: ) Curren mobiliy models (eg random way-poin, uniformly disribued locaions) do no exhibi characerisics found in our six daa ses New mobiliy models are herefore required 2) Lile wor has been done in he area of informed design of opporunisic forwarding algorihms his remains an area for sudy Suiable direcions for wor migh involve he sharing of recen conac informaion beween devices, leading o a more careful selecion of poenial relay devices which are liely o have a shor pah o he desinaion, while also being independenly moving as compared o oher chosen relay devices We will invesigae hese various aspecs of forwarding in opporunisic newors We also inend o perform more human mobiliy experimens The power law naure of he iner-conac ime disribuion seems well esablished for muual sighings of devices carried by humans during he woring day, or in a conference Bu parameers in he environmen or in he naure of ransfer opporuniies (including special devices, or infrasrucure) cerainly affec he shape of he disribuion These new experimenal daa will allow us o revisi hree assumpions made in his paper: Firs, approximaing he iner-conac ime by a power law does no seem o be valid over an unlimied range Experimenal resuls indicae ha he iner-conac ime

11 disribuion may be differen for very large ime scale However, given ha he ail is impaced by he experimen duraion, i is difficul o say if he ail of he disribuion is a characerisic of he user mobiliy or a side-effec of he experimenal mehodology If his differen behavior occurring a very large ime scale is confirmed, i may avoid he unbounded expecaion of he delay observed in our model One should neverheless eep in mind ha he range of values where he naure of he disribuion may change may be far above he delays ha mos newor applicaions can olerae Our model also assumes ha he conac process of a pair of devices follows a renewal process wih a given iner-conac ime disribuion I may be possible o ge rid of his assumpion, assuming ha he iner conac imes follow locally in ime a saionary process ha may exhibi some memory in he sequence disribuion Anoher exciing direcion is o approximae he iner-conac imes wih phase-ype disribuion We have indeed already observed ha he parameer of he disribuion of he iner-conac may change wih he ime of he day 7 Similarly, we assumed ha conac processes for differen pairs of devices are independen, and all described by he same disribuion This is cerainly a major simplificaion These assumpions are usual, eiher explicily or implicily, in he modeling lieraure of mobile ad-hoc newors However, hey migh no be very realisic and his remains a large area of invesigaion for opporunisic communicaion The conac process of wo people is cerainly differen depending on he communiies ha hey have in common explicily (wor group, insiuion, friends), or implicily (nearby neighbors) VI ACKNOWLEDGEMENTS We would lie o graefully acnowledge Dave Koz and Trisan Henderson a Darmouh College, Geoff Voeler a Universiy of California San Diego, and Eyal de Lara and Jing Su a Universiy of Torono 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, and he paricipans of he hree experimens we conduced REFERENCES M McNe and G M Voeler, Access and mobiliy of wireless PDA users, Compuer Science and Engineering, UC San Diego, Tech Rep, T Henderson, D Koz, and I Abyzov, The changing usage of a maure campus-wide wireless newor, in MobiCom 04: Proceedings of he 0h annual inernaional conference on Mobile compuing and neworing, 2004, pp J Su, A Chin, A Popivanova, A Goel, and E de Lara, User mobiliy for opporunisic ad-hoc neworing, in Proceedings of he 6h IEEE Worshop on Mobile Compuing Sysems and Applicaions (WMCSA 04), 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 (WDTN-05), Aug P Bremaud, Marov Chains, Gibbs Field, Mone Carlo Simulaion and Queues Springer-Verlag, F Baccelli and P Bremaud, Elemens of Queuing Theory, 2nd ed Springer-Verlag, A Chainreau, P Hui, J Crowcrof, C Dio, R Gass, and J Sco, Poce swiched newors: Realworld mobiliy and is consequences for opporunisic forwarding, Universiy of Cambridge, Compuer Lab, Tech Rep UCAM-CL-TR-67, Feb 2005 Online Available: hp://wwwclcamacu/techrepors/ucam-cl-tr-67hml 8 A Vahda and D Becer, Epidemic rouing for pariallyconneced ad hoc newors, Universiy of California San Diego, Tech Rep CS , Jul X Chen and A Murphy, Enabling disconneced ransiive communicaion in mobile ad hoc newors, in Proceedings of he Worshop on Principles of Mobile Compuing, Aug 200, pp J A Davis, A H Fagg, and B N Levine, Wearable compuers as pace ranspor mechanisms in highly-pariioned ad-hoc newors, in ISWC 0: Proceedings of he 5h IEEE Inernaional Symposium on Wearable Compuers, 200, p 4 M Grossglauser and D Tse, Mobiliy increases he capaciy of ad hoc wireless newors, IEEE/ACM Transacions on Neworing, vol 0, no 4, pp , G Sharma and R Mazumdar, Scaling laws for capaciy and delay in wireless ad hoc newors wih random mobiliy, in IEEE Inernaional Conference on Communicaions, vol7,jun 2004, pp X Lin, G Sharma, R R Mazumdar, and N B Shroff, Degenerae delay-capaciy rade-offs in ad hoc newors wih brownian mobiliy, mar 2005, preprin, School of ECE, Purdue Univerisy Online Available: hp://minecnpurdueedu/ linx/paper/i05pdf 4 J F C Kingman, Inequaliies in he heory of queues, J Roy Saisical Sociey, Series B, vol 32, pp 02 0, S Jain, R C Shah, G Borriello, W Brunee, and S Roy, Exploiing mobiliy for energy efficien daa collecion in sensor newors, in WiOp, March 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 0h inernaional conference on Archiecural suppor for programming languages and operaing sysems, 2002, pp 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 03: Proceedings of he 4h ACM inernaional symposium on Mobile ad hoc neworing & compuing, 2003, pp G Sharma and R R Mazumdar, Delay and capaciy rade-off in wireless ad hoc newors wih random way-poin mobiliy, mar 2005, preprin, School of ECE, Purdue Univerisy Online Available: hp://minecnpurdueedu/ linx/paper/i05pdf

12 APPENDIX A Preliminary Resuls ) Independen composiion and limi expecaion: The following lemma will be useful o prove limi resul on compound process: Lemma 2 Le ((F (i) ) N ) i I be a finie collecion of sequences of real valued random variables verifying, lim E F (i) = l, wherel R {+ }, and (a) i,, E F (i) R, and l R, or (b) i,, E R {+ } and l =+ F (i) Le ( ) N and (i ) N be wo N valued processes, independen from F, such ha lim =+ as We hen have lim E F (i) = l Proof: Le us firs develop he following expecaion E F (i) = jp i = i, =, F (i) = j i I 0 j 0 = jp i = i P = P F (i) = j i I 0 j 0 = P i = i P = E F (i) i I 0 (3) If we suppose (a), wehavel<+ and ε >0, 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 2M E F (i) l can be bounded from above by P i = i P = E l i I 0 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), wehavel =+ and A >0, T,( >T = E F (i) 2(A +)) ( ) Le M = sup max E F (i), 0 K s: i I, T >K = P T max ( /2, /M ), and E F (i) = i I P i = i i I i I 0 P i = i P = E ( M T P = + P = E >T F (i) P i = i ( + 2 ) (2(A +) A ) F (i) 2) Remaining iner-conac: Because he conac process (U (d,d ) ) 0 is a renewal process, he sequence (R (d,d ) ) 0 of inegers is an Homogeneous Marov Chain in N such ha: { R (d,d ) + = ) R(d,d if R (d,d ) > 0, R (d,d ) + = i wih prob P X = i if ) R(d,d =0 (4) This Marov Chain is clearly irreducible and aperiodic as P X => 0, 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 we have (i) If α>, (R (d,d ) ) 0 is recurren posiive (ii) If α>, he chain (R (d,d ),R (e,e ) ) 0 is ergodic and admis he following saionary disribuion: π(i, j) = (i+) α (j+) α (c ) where c 2 = i 0(i +) α (iii) If α, (R (d,d ) ) 0 is recurren null Proof: Le us inroduce re 0 he ime for R (d,d ) o reurn in he sae 0 From he srucure of he Marov chain (4), saring from sae 0, we can easily deduce ha E 0 re 0 =E X Ifα>, wehavee X < +, proving (i),andifα,wehavee X =+, 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 0 (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 Smih s formula for α>: For any devices d and d, he process (R (d,d ) ) 0 is regeneraive wih respec o he delayed renewal sequence (T (d,d ) ) 0 If we assume

13 α>, wehavee X < +, hence he iner-even of he sequence (T (d,d ) ) 0 admis a finie mean We now in his case (see 5 p48) ha T (d,d ) E lim E f(r (d,d ) =T (d,d ) f(r (d,d ) ) 0 ) = E T (d,d ) T (d,d ) 0 T (d,d ) for any f verifying E f(r (d,d ) ) < =T (d,d ) 0 (5) B Proof of Lemma Lemma is a generalizaion of he mehod presened in III-C Le us sar by he following remar: j for β<α, i β α (β α)+jβ α+ (6) β α + i= Indeed, he funcion x x β α is non-increasing on 0; +, hence we have for any i 2, i β α i i j i β α + i= x β α dx < + hence (6) follows from j x α dx =+ jβ α+ β α + As all processes of conacs beween devices are independen, he saionary disribuion of he produc of m Marov Chains is given ( by he he produc measure ) Hence we have ha E min R (d,d ),,R (dm,d m) is min(i equal o,,i m) (c ) m i,,i m (i +) α (i m+) In paricular α his value is finie if we can prove g(m, α, ) <, where g is defined as g(m, α, β) = (min(i,,i m ))β (i i ) α (i m ) α,,i m We will prove more generally ha if α>+/m, and β, heng(m, α, β) < For m =, his is rue as for α>2 and β, g(,α,β)= i iβ α < + More generally, g(m, α, β) is bounded by min(i 2,,i m) m (i ) β α (i 2 ) α (i m ) α i 2,,i m i = a m + b m (min(i 2,,i m m ))β+ α (i i 2 ) α (i m ) α 2,,i m a m + b mg(m,α,β+ α) a + b g(,α,β+(m )( α)) } {{ } = i iβ+(m )( α) α a m,b m,a m,b m are finie consan real number, ha depends on α, β, andm They can be compued using (6); a and b can also be compued, by developing he recurrence equaion, bu he exac values of hese consans have no imporance for he resul of he heorem The resul follows as β +(m )( α) α β 2 + } {{ } } m + {{ mα } < <0 For any real numbers (x,,x m ),andi 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 (i, (x,,x m )) = min(x,,x m ) We have Corollary Le (R (d,d ) ) 0,,(R (dm,d )) m ) 0 be he remaining iner-conac imes for m differen pairs of devices (d i,d i ) i m We suppose ha α> (m j+)+ m j+, ( ) hen E ord j, (R (d,d ),,R (dm,d m) ) < Proof: Following he same sep han for he previous proof we can easily show ha his expecaion is given by : ord (j, (i,,i m )) (c ) m (i i +) α (i m +) α,,i m By symmery we can show ha his sum is upper bounded by (c ) m m i,,i m i j I {ord(j,(i,,im))=ij} (i +) α (i m +) α Considering he number of choices of j elemens in m, we can show by anoher symmery argumen, c 2 i,,i m ord(j,(i,,i m)) (i +) α (i m+) α I {ord(j,(i,,i m))=i j,i j i j+,,i j i m} wih c 2 = m (j )!(m )! In he erm of his sum, i j is always he maximum of i,,i j,i j and he minimum of i j,i j+,,i m The produc of erms corresponding o i,,i j is aen only for hem wih values in {, 2,,i j } ; i could be overall upper bounded by c j, by compleing each sum We can hen show ha c 2 c j min(i j,,i m ) (i i j ) α (i m ) α j,,i m } {{ } g(m j+,α,) (c ) m (m )! The resul is hen proved, once we remember, from he proof of Lemma, ha g(m j +,α,) < + if α + m j+