SENSOR networks are wireless ad hoc networks used for. Minimum Energy Fault Tolerant Sensor Networks

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1 Mnmum Enrgy Fault Tolrant Snsor Ntworks Ptar Djukc and Shahrokh Vala Th Edward S Rogrs Sr Dpartmnt of Elctrcal and Computr Engnrng Unvrsty of Toronto, 0 Kng s Collg Road, Toronto, ON, MS 3G4, Canada -mal:{djukc,vala}@commutorontoca Abstract W dvs a schm whch can provd rlabl transport srvcs n snsor ntworks and gv an algorthm whch mnmzs th nrgy us of our schm W us a dstrbutd snk whr nformaton arrvs at th snk va multpl proxy nods, calld prongs n ths papr Th sndr nod uss Forward Error Corrcton (FEC) rasur codng to ncod ach packt nto multpl fragmnts and transmts th fragmnts to ach of th prongs ovr a path whch s dsjont from th paths to th othr prongs Th rasur codng allows th snk to rconstruct th orgnal packt vn f som of th fragmnts ar lost W us a cross-layr dsgn whr hghr ntwork layrs us nformaton about packt loss and nrgy consumpton to dstrbut th load n th ntwork W show that th sourc can dstrbut th load so that nrgy consumpton s mnmzd Th optmzaton taks fault tolranc nto account wth a bound on th probablty of packt loss Th xtra fragmnts ncras both th rlablty and th nrgy n th ntwork Howvr, w show wth smulatons that t s possbl for th snsor to dcras nrgy us n th ntwork by usng th dvrsty avalabl wth multpl ntwork paths Indx Trms Snsor ntworks, nrgy-awar communcatons, wrlss communcaton, fault tolranc, ntwork montorng I INTRODUCTION SENSOR ntworks ar wrlss ad hoc ntworks usd for montorng and nformaton gathrng Thy consst of many small, slf-organzd nods that form an ad hoc ntwork that rports to a common snk at th dg of th ntwork Snsor ntworks ar usd to obsrv natural phnomna such as ssmologcal and wathr data, collct data n battlflds, and montor traffc n urban aras W show a typcal snsor ntwork n Fg Th snk snds a qury whch s dssmnatd throughout th snsor ntwork wth floodng Th qury rqusts a subst of nods to snd thr collctd nformaton Th nods, whch hav th rqustd nformaton, snd t to th snk by formng a routng tr wth th root at th snk rcvr Howvr, collcton of nformaton wth a sngl snk may not b approprat for snsor ntworks In a snsor ntwork wth a sngl snk, th top lvl of th tr contans rlatvly fw nods, compard to th total numbr of nods So, most of th nrgy s consumd by th fw nods clos to th snk Thrfor, th nods closr to th snk wll run out of battry powr arlr than othr nods A nod closr to th snk may also b a bottlnck For xampl, a routng protocol whch taks fault tolranc (rlablty) nto account [] would not hav too many optons n choosng th last hop Qurs Rcvr Qury rsponss Fg Th convntonal snsor ntwork dsgn Th snk has only on rcvr whch conncts to th ntwork W propos a snsor ntwork archtctur whch assums that th snk s dstrbutd throughout th snsor ntwork Th snk uss a numbr of rcvrs, calld prongs, whch connct to t wth rlabl and hgh bandwdth lnks W assum that f a packt arrvs at a prong, t wll b dlvrd ntact to th snk Ths crats a hrarchcal archtctur as shown n Fg Th fgur shows a numbr of snsor nods and a snk wth four rcvrs As llustratd n th fgur, th nods can b connctd drctly to a prong or wth multpl hops through othr snsor nods Th advantag of ths archtctur s that t dstrbuts th load on th last hop among a largr st of nods than th archtctur wth a sngl rcvr dsgn Th prsnc of multpl prongs also allow us to ncras th numbr of dstnct paths from vry nod n th ntwork to th snk Ths allows th snsor nod to us path dvrsfcaton to ncras fault tolranc (rlablty) n th ntwork In path dvrsfcaton, ach nod n th ntwork (a sourc) snds packts ovr multpl dsjont paths On way to ncras rlablty would b to snd cops of th sam packt ovr th multpl paths Howvr, ths would b vry nffcnt Instad, path dvrsfcaton ncrass rlablty ffcntly wth Forward Error Corrcton (FEC) Th snsor ncods ach packt of M fragmnts nto M + K fragmnts wth an rasur cod [] Th fragmnts ar thn dstrbutd ovr th paths and smultanously snt to th snk Th snk can rconstruct th packt f t rcvs mor than M fragmnts W us multpaths n th ntwork layr, as opposd to a schm that may us multpaths n th physcal layr

2 Rcvr, q, b,m Path, q, b,m, q, b,m 8, q 8, 8 b,m 8 4, q 4, 4 b,m 8 3 8, q 8, 8 b,m 8 4 9, q 9, 9 b,m 9 7, q 7, 7 b,m 7 6, q 6, 7 b,m 7 6 Path 8 7 9, q 9, 9 b,m 9 7, q 7, 7 b,m 7 Path 3 9 9, q 9, 9 b,m 9 Fg Our snsor ntwork dsgn Th snk has many rcvrs scattrd throughout th ntwork Th sourc uss multpl dsjont paths to snd nformaton to th snk Ths mprovs load-balancng and ncrass rlablty Th approach of usng rasur cods wth multpl paths to ncras rlablty was shown to b ffctv prvously n [3], [4], [] W show n [3] that rasur cods ar not suffcnt to ffcntly ncras th rlablty Ths s bcaus a sngl path may b mor lkly to fal on ts own than to fal togthr wth othr dsjont paths Ths mans that th sourc nds to us mor party fragmnts wth a sngl path than wth multpl dsjont paths n ordr to achv th sam lvl of rlablty In ffct, th smultanous us of multpl paths taks advantag of th dvrsty avalabl n dns snsor ntworks In ths papr, w show that t s possbl to dstrbut th fragmnts on th paths so that th nrgy consumpton s mnmzd W propos an algorthm that allows a snsor to mnmz th nrgy us whl mantanng th rlablty n th ntwork Our algorthm uss standard lnar programmng tchnqus Our smulatons show that path dvrsfcaton uss two typs of dvrsty, whch dcras nrgy us n th snsor ntwork Th frst typ of dvrsty coms from th us of multpl paths, whch ncrass rlablty and dcrass nrgy rqurd to achv rlabl transmssons In ths cas, th nrgy s dcrasd wth mor ffcnt FEC Th scond typ of dvrsty coms from th varaton n nrgy rqurd to transmt nformaton on ach path As th varaton ncrass, th snsor can ncras ntwork rlablty wth lss ffcnt FEC and stll dcras th nrgy us by usng paths wth lowr nrgy consumpton II SENSOR NETWORK MODEL W poston our dsgn wthn th contxt of xstng snsor ntworks W assum that our ntwork oprats wth a systm smlar to TnyDB [6], whch s a qury basd systm whr th qurs ar floodd through th ntwork Th qury opraton s also a mchansm usd to fnd routs to th snk Th snk snds a qury to th ntwork whch asks a subst of nods to rspond Th qurs ar usually of th typ All nods wth tmpratur hghr than C rspond Each qury s floodd through th ntwork wth som optmzatons Fg 3 Our snsor ntwork dsgn Th snk snds qurs by floodng th ntwork Th qury carrs nformaton about th condtons on vry hop n th ntwork Ths mprovs load-balancng and ncrass rlablty For xampl, f th snk knows that a subst of nods cannot answr a qury t dos not snd th qury to that subst of nods As th qury propagats through th ntwork, th nods also buld a routng tabl of routs to th snk A qury may also spcfy f th rsponss should b aggrgatd at on of th snsor nods Ths allows for rmoval of rdundant nformaton Normally, th qury s snt from th sngl snk, but n our cas w chang th systm to snd th qury from th dffrnt prongs of th snk at th sam tm Fg 3 llustrats an xampl for th qury/rout dscovry opraton Th snk floods th ntwork by sndng th qury through ach of th prongs Th qury rcords th path t took on th way to th sourc nod as wll as th condtons on ach hop of th path Th condtons ar gvn as rlablty (q ) and nrgy paramtrs (M (), (b) ) on ach hop Th nformaton about th rlablty and nrgy on ach hop s usd n th algorthm to mnmz th total nrgy Each snsor nod rcords ts nformaton n th qury and forwards th qury to th nxt st of nods Th snsor nods also kp track of ach path that arrvs at a rcvr prong If a nod nds to snd back a rply to th qury, t slcts a st of arc-dsjont paths to th dstrbutd rcvrs on whch t can snd th rply For xampl, n Fg 3 nod 3 can us path or 3 to th rcvr on th rght and path to th nod on th lft In ths cas nod 3 should slct paths and 3 Snc th slcton schm s suboptmal, t may happn that th sourc may not communcat wth all of th prongs of th snk Each snsor nod may slct th paths wth th shortst hop count, or wth th hghst rlablty Th shortst path mtrc s not th bst mtrc for snsor ntworks du to th poor qualty of lnks [] shows that th shortst path mtrc rsults n th nods usng a fw long hops, whch dcrass rlablty Th slcton mthods, whch tak rlablty nto account, prform much bttr Nvrthlss, th contrbuton of ths papr s to analyz th ntwork onc th paths hav Two arc-dsjont paths hav no vrtcs or dgs n common

3 x = M + K M Data K Party M K x S Snsor x x n P P P n Qury: q, (b),m () Z M M + K 3 4 Fg 4 Transmsson to a dstrbutd snk P,,P n ar th prongs of th snk Th snsor transmts x,,x n fragmnts of nformaton to ach of th n prongs of th dstrbutd snk Th snk can rconstruct th orgnal nformaton f t rcvs mor than M fragmnts n total bn slctd W not that th sz of th packts snt by th nods may b qut small, compard to packt szs usd n rgular ntworks Howvr, our schm can stll work on th nods whch aggrgat data from othr snsor nods Ths srvs a dual purpos Frst, th aggrgaton nods can lmnat th rdundant data rportd by th snsors wth codng tchnqus [7] Thrfor, th total amount of data, whch s transmttd to th snk, s rducd Scond, th aggrgaton nods hav mor data to snd than th snsors, whch maks subdvson of packts on th aggrgaton nods mor practcal III ANALYTICAL MODEL AND ASSUMPTIONS In ths scton, w gv a mathmatcal modl for path dvrsfcaton W assum that a snsor nod gnrats packts of sz bm bts Th packt s splt nto M fragmnts, ach wth sz b, and an addtonal K party fragmnts wth sz b ar gnratd usng a lnar rasur cod [] Th sourc nod thn transmts th fragmnts ovr multpl paralll paths as shown n Fg 4 Th sourc transmts x fragmnts on path whr =,,n and n s th total numbr of avalabl paths, and: n x = x T = M + K, () = whr x =[x,x,, x n ] T s th allocaton vctor of fragmnts, and s a vctor of all s A Fault tolranc n th ntwork Th dstnaton nod nds to rcv a total of M fragmnts n ordr to rconstruct th packt W us random varabls Z to ndcat th numbr of fragmnts rcvd on path So, th probablty that th packt can b rconstructd s gvn as: [ n ] P succ =Pr Z >M () = P succ s a functon of x th allocaton of fragmnts on ach path, q =[q,,q n ] T th vctor ndcatng th probablty that a fragmnt wll b succssfully transmttd on ach path, Z Z Z n and K th numbr of party fragmnts W wll us P succ and P succ (x, q,k) ntrchangably n th rst of th papr W masur th ffctvnss of th schm, n trms of th ovrhad ntroducd by th rasur cod, as: η(x,k)= xt K x T = M M + K whr η(x,k) s th ffcncy w can achv for a gvn rlablty lvl Th schm s mor ffctv as η(x,k) approachs W us a lowr bound on η(x,k) n th mnmzaton of nrgy B Enrgy Consumpton n th Ntwork W assum that ach snsor has th ablty to masur th avrag amount of nrgy t uss to transmt a bt of nformaton (k) and th amount of avalabl nrgy on th snsor E (k), whr w hav ndxd th snsor as snsor k on path In ordr to smplfy th optmzaton, w assum that (k) and E (k) do not chang durng th transmsson of a sngl packt Th nformaton about th nrgy consumpton s dssmnatd durng th qury procss Th qury contans th nformaton about th pr-bt nrgy rqurd to transfr a packt btwn th sourc and th dstnaton on a path, (b) Th pr-bt nrgy consumpton on a path can b dtrmnd by addng up th nrgy rqurd to transfr a bt at vry nod on th path (b) = n k=0 (k), whr n s th numbr of nods on path Th vctor of pr bt nrgy consumpton n th ntwork s gvn by b =[ (b),(b),, (b) n ] T So, th total amount of nrgy spnt to transmt th M +K fragmnts s gvn by: E Total (x, b )=bx T b (4) Th snsor nod obtans th vctor b prodcally whn t rcvs th qury from th snk So, t should also prform th mnmzaton prodcally, whn t rcvs th updats Th qury also contans nformaton about th maxmum numbr of fragmnts, M (), that can b transmttd on ach path bfor th nrgy on th path runs out: M () = mn k n { E (k) b (k) (3) } () W dnot wth M th vctor of maxmum numbr of fragmnts that w can transmt on ach path, M = [M () (),, M n ] T,M() IV MINIMUM ENERGY FOR FAULT TOLERANT SENSOR NETWORKS W mnmz th total consumd nrgy ( th sum of nrgy us across ach path n th ntwork) wth a gvn bound on rlablty and ffcncy n th ntwork Th mnmzaton fnds th optmum numbr of party fragmnts k ndd to satsfy th rlablty bounds, as wll as th allocaton of

4 fragmnts on ach path x: Mnmz: E Total (x, b )=bx T b x,k (6a) Subjct to: P succ ɛ (6b) η(x,k) δ (6c) x T k = M (6d) 0 x M (6) whr s pontws comparson and k s n lowr cas snc t s now a varabl Th rlablty constrant (6b) s th guarant that th packt can b rconstructd at th rcvr nod wth som mnmum probablty ɛ Th ffcncy constrant (6c) puts a bound on th maxmum numbr of fragmnts K that can b usd to achv th rlablty, k K max = M( δ)/δ Th last two constrants tak nto account that th total numbr of fragmnts s M + k and that th maxmum numbr of fragmnts, whch can b transmttd on ach path (du to nrgy constrants), s gvn by M % Enrgy Incras TABLE I VALUES OF s(ɛ) AND c(ɛ) ɛ log 0 ( ɛ ɛ ) s(ɛ) c(ɛ) Enrgy Ovrhad vs Mnmum Rlablty q =07 q =08 q =09 W transform th optmzaton (6) nto a lnar program n two stps Frst, w us th Posson cumulatv dstrbuton functon (cdf) to calculat th lowr bound for th ntwork rlablty P succ n (6b) W approxmat th loss of conscutv fragmnts on ach path to b ndpndnt and dntcal to ach othr and approxmat P succ (x, q,k) wth: Log-odd Mnmum Rlablty ɛ Fg Enrgy Ovrhad vs Mnmum Rlablty ɛ Q(x, q,k) P succ, (7) whr Q(x, q,k)= k λ(x) [λ(x)] j, (8) j! j=0 ln(q) = [ln(q ),,ln(q n )] T and λ(x) = x T ln(q) Th approxmaton of P succ wth th Posson cdf follows from th rsults of [8] Th approxmaton (7) allows us to rplac P succ n (6b) wth Q(x, q,k) Snc, n practc w would only b ntrstd n farly hgh valus of Q(x, q,k), w can conclud that th bound n (7) s tght For xampl, f Q(x, q,k)=0999, th rror can b at most 0 3 Scond, w lnarz th Posson approxmaton and convrt th mnmzaton nto a lnar program: Mnmz: E Total (x, b )=bx T b x,k (9a) Subjct to: x T ln(q) s(ɛ)k c(ɛ) (9b) 0 k K max (9c) x T k = M (9d) 0 x M, (9) whr s(ɛ) and c(ɛ) ar constants rlatng th Posson cdf wth ɛ W show som valus for s(ɛ) and c(ɛ) n Tabl I Th constrant (9b) s drvd by lnarzng (7) W gv mathmatcal dtals of th lnarzaton n Appndx I V RESULTS W smulatd a ntwork n whch thr s a larg numbr of snsors and fw snks In ach smulaton, th sourc prforms 000 packt transmssons to th prongs of th snks Th ) chang bfor vry transmsson and th snsor has prfct knowldg of ths changs In our frst smulaton, w modl th channl as a bnary symmtrc channl (BSC) wth th avrag probablty of fragmnt succss q Fg shows th ffct that th ncras n rlablty has on nrgy consumpton W plot th prcnt ncras n nrgy consumpton from th mnmum nrgy whn th mnmum rlablty ɛ changs Th mnmum nrgy s achvd whn th snsor dos not us any party fragmnts to snd th nformaton to th snk, th optmzaton s constrand wth (6) only In ths cas, P succ = q W show ɛ n th log-odd format whr w plot ɛ as log(ɛ/( ɛ)) Th log-odd scal allows us to map th st [0, ] unformly to th st [, ], so that w can obsrv th asymptotc ffct whn ɛ 0 or ɛ W can s from Fg that wth lss than thr tms th ncras n total nrgy consumpton, w can mak th mnmum rlablty ɛ>0999 vn for channls whr th probablty of fragmnt loss s 0% on avrag In our scond smulaton, w modl th rlablty on th path wth a Markov-Chan modl, whr th channl s fragmnt succss rat s changng through th followng stats condtons n th ntwork (q and (b)

5 % Enrgy Incras Enrgy Ovrhad vs Mnmum Rlablty paths 3 paths 4 paths paths Log-Odd Effcncy ɛ Total Enrgy Log-Odd Effcncy vs Enrgy Varanc Enrgy Varanc Enrgy Consumpton vs Enrgy Varanc Log-Odd Mnmum Rlablty ɛ Enrgy Varanc Fg 6 Enrgy Ovrhad vs Mnmum Rlablty ɛ Fg 7 Effcncy vs Enrgy Varanc q =[07, 08, 09], to gv an avrag fragmnt succss rat of q =07 Fg 6 shows th ffct of th ncrasd numbr of prongs on nrgy consumpton W obsrv from Fg 6 that as th numbr of prongs ncrass th nrgy us dcrass Indd, th dstrbutd snk allows th snsor to tak advantag of th dvrsty avalabl n th ntwork, whch dcrass th nrgy consumpton Th dvrsty coms from th varaton n th rlablty of th paths and so th snsor can ncras th rlablty by usng bttr paths wthout ncrasng th numbr of party fragmnts K Fg 7 llustrats th ffct of varanc of nrgy consumpton on th ffcncy η Effcncy η and total nrgy E Total ar plottd for th fragmnt allocaton that has th mnmum nrgy consumpton W can s that th ffcncy dos not dpnd on th mnmum rlablty alon Th snsor can dcras th nrgy consumpton wth transmssons whch us mor party fragmnts; ths transmssons dcras ffcncy Ths s bcaus th snsor can snd most of th fragmnts on th path wth th lowst nrgy and stll achv th dsrd rlablty by ncrasng th numbr of party fragmnts usd n th transmsson Th ncrasd varanc of nrgy consumpton mans that thr s a hghr lklhood that a path wth low nrgy consumpton also has a fragmnt succss rat that allows th snsor to achv th rlablty bound APPENDIX I LINEARIZATION OF NETWORK RELIABILITY W transform Q(x, q,k) nto a lnar functon by notng that Q(x, q, k) s a monotoncally dcrasng functon of λ(x) It can b asly shown that: Q(x, q,k) x = λ λ(x) [λ(x)]k x k! < 0 (0) whch mans that Q(x, q,k) s a dcrasng functon of λ(x) So, for a gvn k thr xsts α ɛ (k) such that: λ(x) α ɛ (k) Q(x, q,k) ɛ P succ ɛ () Whn w plot α ɛ (k) for a rang of ɛ, w notc that for k>, α ɛ (k) s almost a lnar functon So, w approxmat α ɛ (k) as: α ɛ (k) s(ɛ)k + c(ɛ) () In our smulatons, w hav usd th last-squar mthod to obtan th slop s(ɛ) and constant c(ɛ) W show som of th valus, for s(ɛ) and c(ɛ), n Tabl I W hav xamnd ths approxmaton mor closly n [3] whr w hav shown that th ncras n mnmum nrgy n (9) ovr th non-lnar optmzaton (6) s almost nglgbl At th sam tm, th lnar constrant dcrass th complxty of th optmzaton n th ordr of Θ(M + K max ) REFERENCES [] A Woo, T Tong, and D Cullr, Tamng th undrlyng challngs for rlabl multhop routng n snsor ntworks, n SnSys 03 ACM Prss, 003 [] M O Rabn, Effcnt dsprsal of nformaton for scurty, load balancng, and fault tolranc, Journal of th Assocaton for Computng Machnry, vol 36, no, pp , Aprl 989 [3] P Djukc, Optmum rsourc allocaton n multpath ad hoc ntworks, MASc Thss, Unvrsty of Toronto, 003 [4] A Tsrgos and Z J Haas, Analyss of multpath routng-part I: Th ffct on th packt dlvry rato, IEEE Trans Wrlss Commun,vol3, no, pp 38 46, January 004 [], Analyss of multpath routng, part : Mtgaton of th ffcts of frquntly changng ntwork topologs, IEEE Trans Wrlss Commun, vol 3, no, pp 00, March 004 [6] J Ghrk and S Maddn, Qury procssng n snsor ntworks, IEEE Prvasv Computng, vol 3, no, pp 46, January-March 004 [7] R Crstscu, B Bfrull-Lozano, and M Vttrl, On ntwork corrlatd data gathrng, n INFOCOM 004 [8] R J Srflng, Som lmntary rsults on Posson approxmaton n a squnc of Brnoull trals, SIAM Rvw, vol 0, no 3, pp 67 79, July 978