Online Power-aware Routing in Wireless Ad-hoc Networks

Transcription

1 Olie Power-aware Routig i Wirele Ad-hoc Network Qu Li, aved Alam, Daiela Ru Departmet of Computer Sciece Dartmouth College Haover, NH 3755 {liqu, jaa, ru}@cdartmouthedu ABSRAC hi paper dicue olie power-aware routig i large wirele ad-hoc etwork for applicatio where the meage equece i ot kow We eek to optimize the lifetime of the etwork We how that olie power-aware routig doe ot have a cotat competitive ratio to the off-lie optimal algorithm We develop a approximatio algorithm called max-mi zp mi that ha a good empirical competitive ratio o eure calability, we itroduce a ecod olie algorithm for power-aware routig hi hierarchical algorithm i called zoe-baed routig Our experimet how that it performace i quite good INRODUCION he proliferatio of low-power aalog ad digital electroic ha created huge opportuitie for the field of wirele computig It i ow poible to deploy hudred of device of low computatio, commuicatio ad battery power hey ca create ad-hoc etwork ad be ued a ditributed eor to moitor large geographical area, a commuicatio eabler for field operatio, or a grid of computatio hee applicatio require great care i the utilizatio of power he power level i provided by batterie ad thu it i fiite Every meage et ad every computatio performed drai the battery I thi paper we examie a cla of algorithm for routig meage i wirele etwork ubject to power cotrait ad optimizatio We eviio a large ad-hoc etwork coitig of thouad of computer uch a a eor etwork ditributed over a large geographical area Clearly thi type of etwork ha a high degree of redudacy We would like to develop a power-aware approach to routig meage i uch a ytem that i fat, calable, ad i olie i that it doe ot kow ahead of time the equece of meage that ha to be routed over the etwork he power coumptio of each ode i a ad-hoc wirele ytem ca be divided accordig to fuctioality ito: () Permiio to make digital or hard copie of part or all of thi work or peroal or claroom ue i grated without fee provided that copie are ot made or ditributed for profit or commercial advatage ad that copie bear thi otice ad the full citatio o the firt page o copy otherwie, to republih, to pot o erver, or to reditribute to lit, require prior pecific permiio ador a fee ACM SIGMOBILE 7 Rome, Italy 2 ACM ISBN $5OO Card r Rv Idle Slp Power ma ma ma ma Sup V RageLAN a 2 5 WaveLAN(Mbp) Smart Spread 5 8 a 5 5 able : Power Coumptio Compario amog Differet Wirele LAN Card ([2,, 32]) For RageLAN2, the power coumptio for doze mode (which i claimed to be etwork aware) i 5mA he lat oe i Smart Spread Spectrum of Adco elemetry the power utilized for the tramiio of a meage; (2) the power utilized for the receptio of a meage; ad (3) the power utilized while the ytem i idle able lit power coumptio umber for everal wirele card hi ugget two complemetary level at which power coumptio ca be optimized: () miimizig power coumptio durig the idle time ad (2) miimizig power coumptio durig commuicatio I thi paper we focu oly o iue related to miimizig power coumptio durig commuicatio - that i, while the ytem i tramittig ad receivig meage We believe that efficiet meage routig algorithm, coupled with good olutio for optimizig power coumptio durig the idle time uch a thoe propoed by [33, 4] will lead to effective power maagemet i wirele ad-hoc etwork, epecially for a parely deployed etwork Several metric ca be ued to optimize power-routig for a equece of meage Miimizig the eergy coumed for each meage i a obviou olutio that optimize locally the power coumptio Other ueful metric iclude miimizig the variace i each computer power level, miimizig the ratio of cotpacket, ad miimizig the maximum ode cot A drawback of thee metric i that they focu o idividual ode i the ytem itead of the ytem a a whole herefore, routig meage accordig to them might quickly lead to a ytem i which ode have high reidual power but the ytem i ot coected becaue ome critical ode have bee depleted of power We chooe to focu o a global metric by maximizig the lifetime of the etwork We model thi a the time to the earliet time a meage caot be et hi metric i very ueful for ad-hoc etwork where each meage i importat ad the etwork are parely deployed 97

2 I thi paper we how that the olie power-aware meage routig problem i very hard (Sectio 3) hi problem doe ot have a cotat competitive ratio to the off-lie optimal algorithm that kow the meage equece Guided by thi theoretical reult, we propoe a olie approximatio algorithm for power-aware meage routig that optimize the lifetime of the etwork ad examie it boud (Sectio 4) Our algorithm, called the max-mi zp mi algorithm, combie the beefit of electig the path with the miimum power coumptio ad the path that maximize the miimal reidual power i the ode of the etwork Depite the dicouragig theoretical reult cocerig the competitive ratio for olie routig, we how that the max-mi zp mi algorithm ha a good competitive ratio i practice, approachig the performace of the optimal off-lie routig algorithm uder realitic coditio Our propoed max-mi zp mi algorithm require iformatio about the power level of each computer i the etwork owig thi iformatio accurately i ot a problem i mall etwork However, for large etwork it i difficult to aggregate ad maitai thi iformatio hi make it hard to implemet the max-mi zp mi algorithm for large etwork Itead, we propoe aother olie algorithm called zoe-baed routig that relie o max-mi zp mi ad i calable (Sectio 5) Our experimet how that the performace of zoe-bae routig i very cloe to the performace of max-mi zp mi with repect to optimizig the lifetime of the etwork Zoe-bae routig i a hierarchical approach where the area covered by the (eor) etwork i divided ito a mall umber of zoe Each zoe ha may ode ad thu a lot of redudacy i routig a meage through it o ed a meage acro the etire area we fid a global path from zoe to zoe ad give each zoe cotrol over how to route the meage withi itelf hu, zoe-baed power-aware routig coit of () a algorithm for etimatig the power level of each zoe; (2) a algorithm computig a path for each meage acro zoe; ad (3) a algorithm for computig the bet path for the meage withi each zoe (with repect to the power lifetime of the zoe) 2 RELAED WOR We are ipired by excitig recet reult i ad-hoc etwork ad i eor etwork Mot previou reearch o ad-hoc etwork routig [7, 3, 23, 24, 26, 3, 9, 8] focued o the protocol deig ad performace evaluatio i term of the meage overhead ad lo rate o improve the calability of routig algorithm for large etwork, may hierarchical routig method have bee propoed i [2, 8, 22, 2,, 28] I [25, 6], zoe, which are the route maiteace uit, are ued to fid the route hi previou work focued o how to fid the correct route efficietly, but did ot coider optimizig power while edig meage Sigh et al [3] propoed power-aware routig ad dicued differet metric i power-aware routig Some of the idea i thi paper are exteio of what that paper propoed Miimal eergy coumptio wa ued i [29] Chag ad aiula [3] alo propoed maximizig the lifetime of a etwork whe the meage rate i kow heir mai idea, amely to avoid uig low power ode ad chooe the hort path at the begiig, ha ipired the approach decribed i thi paper We alo ue the ame formula to decribe the reidual power fractio I [2], Gupta ad umar dicued the critical power at which a ode eed to tramit i order to eure the etwork i coected Eergy efficiet MAC layer protocol ca be foud i [7, 6] he work preeted i thi paper i differet from thee previou reult i that we develop olie, hierarchical, ad calable algorithm that do ot rely o kowig the meage rate ad optimize the lifetime of the etwork A recet ad very importat body of work cocer optimizig power coumptio durig idle time rather tha durig the time of commuicatig meage [33, 4] hi work i complemetary to the reult preeted i thi paper Combied, efficiet way for dealig with idle time ad with commuicatio ca lead to powerful power maagemet olutio Related reult i eor etwork iclude [27,, 5, 9, 4, 5] he high-level viio of wirele eor etwork wa itroduced i [27, ] Achievig eergy-efficiet commuicatio i a importat iue i eor etwork deig Uig directed diffuio for eor coordiatio i decribed i [5, 9] I [4] a low-eergy adaptive protocol that ue data fuio i propoed for eor etwork Our approach i differet tha thi previou work i that we coider meage routig i eor etwork ad our olutio doe ot require to kow or aggregate the data tramitted 3 FORMULAION OF POWER-AWARE ROUING 3 he Model Power coumptio i ad-hoc etwork ca be divided ito two part: () the idle mode ad (2) the tramitreceive mode he ode i the etwork are either i idle mode or i tramitreceive mode at all time he idle mode correpod to a baelie power coumptio Optimizig thi mode i the focu of [33, 4] We itead focu o tudyig ad optimizig the tramitreceive mode Whe a meage i routed through the ytem, all the ode with the exceptio of the ource ad detiatio receive a meage ad the immediately relay it Becaue of thi, we ca view the power coumptio at each ode a a aggregate betwee trait ad receive power which we will model a oe parameter a decribed below More pecifically, we aume a ad-hoc etwork that ca be repreeted by a weighted graph G(V, E) he vertice of the graph correpod to computer i the etwork hey have weight that correpod to the computer power level he edge i the graph correpod to pair of computer that are i commuicatio rage Each weight betwee ode i the power cot of edig a uit meage betwee the two ode Suppoe a hot eed power e to tramit a meage to aother hot who i d ditace away We ue the model of [3, 4] to compute the power coumptio for edig thi Without lo of geerality, we aume that all the meage are uit meage Loger meage ca be expreed a equece of uit meage 98

3 meage: e = kd c, where k ad c are cotat for the pecific wirele ytem (uually 2 c 4) We focu o etwork where power i a fiite reource Oly a fiite umber of meage ca be tramitted betwee ay two hot We wih to olve the problem of routig meage o a to maximize the battery live of the hot i the ytem he lifetime of a etwork with repect to a equece of meage i the earliet time whe a meage caot be et due to aturated ode We elected thi metric uder the aumptio that all meage are importat Our reult, however, ca be relaxed to accommodate up to m meage delivery failure, with m a cotat parameter 32 Relatiohip to Claical Network Flow Power-aware routig i differet from the maximal etwork flow problem although there are imilaritie he claical etwork flow problem cotrai the capacity of the edge itead of limitig the capacity of the ode If the capacity of a ode doe ot deped o the ditace to eighborig ode, our problem ca alo be reduced to maximal etwork flow We ue the followig pecial cae of our problem i which there i oly oe ource ode ad oe ik ode to how the problem i NP-hard he maximal umber of meage utaied by a etwork from the ource ode to the ik ode ca be formulated a liear programmig Let ij be the total umber of meage from ode v i to ode v j, e ij deote the power cot to ed a meage betwee ode v i to ode v j, ad ad t deote the ource ad ik i the etwork Let P i deote the power of ode i We wih to maximize the umber of meage i the ytem ubject to the followig cotrait: () the total power ued to ed all meage from ode v i doe ot exceed P i; ad (2) the umber of meage from v i to all other ode i the ame a the umber of meage from all other ode to v i, which are give below: maximize j j j ubject to ij e ij P i () j ij = j ji (for i, t) (2) hi liear programmig formulatio ca be ca be olved i polyomial time However, we eed the iteger olutio, but computig the iteger olutio i NP-hard Figure how the reductio to et partitio for provig the NP-harde of the iteger olutio 33 Competitive Ratio for Olie Power-aware Routig I a ytem where the meage rate are ukow, we wih to compute the bet path to route a meage Sice the meage equece i ukow, there i o guaratee that we ca fid the optimal path For example, the path with the leat power coumptio ca quickly aturate ome of x x2 x x S Figure : he iteger olutio problem ca be reduced to et partitio a follow Cotruct a etwork baed o the give et he power of x i i a i for all i, ad the power of y i a i A ai2 he weight of each edge i marked o the etwork For ay et of iteger S = a, a 2,, a, we are aked to fid the ubet of S, A uch that a i A ai = a i S A ai We ca cotruct a etwork a depicted here he maximal flow of the etwork i a i A ai2, ad it ca oly be gotte whe the flow of x iy i a i for all a i A, ad for all other x iy, the flow i the ode he difficulty of olvig thi problem without kowledge of the meage equece i ummarized by the theoretical propertie of it competitive ratio he competitive ratio of a olie algorithm i the ratio betwee the performace of that algorithm ad the optimal off-lie algorithm that ha acce to the etire executio equece prior to makig ay deciio heorem No olie algorithm for meage routig ha a cotat competitive ratio i term of the lifetime of the etwork or the umber of meage et heorem, whoe proof i how i Figure 2, how that it i ot poible to compute olie a optimal olutio for power-aware routig 4 AN ONLINE MAX-MIN ALGORIHM POWER-AWARE ROUING I thi ectio we develop a approximatio algorithm for olie power-aware routig ad how experimetally that our algorithm ha a good empirical competitive ratio ad come cloe to the optimal We believe that it i importat to develop algorithm for meage routig that do ot aume prior kowledge of the meage equece becaue for ad-hoc etwork applicatio thi equece i dyamic ad deped o eed value ad goal commuicated to the ytem a eeded Our goal i to icreae the lifetime of the etwork whe the meage equece i ot kow We model lifetime a the earliet time that a meage caot be et Our aumptio i that each meage i importat ad thu the failure of deliverig a meage i a critical evet Our reult ca be exteded to tolerate up to m meage delivery failure, where m i a parameter We focu the remaiig of thi dicuio o the failure of the firt meage delivery Ituitively, meage route hould avoid ode whoe power i low becaue overue of thoe ode will deplete their bat- y 99

4 #"! )(?> =< ;: ba dc ^] fe S 65 S X Y X X2 Y2 X2 X Y X * * + +,,- $ $ % % & &' X Y X PQS Y X 7879 Y2 X2 F8F Y X G8G A8A B8BC Y D8DEN8N8N N8N8N N8N8N N8N8N N8N8N N8N8N X Y Y2 Y Y S YUY ZUZ [U[\ _U_ Ù` RSU VUV WUWX Figure 2: I thi etwork, the power of each ode i + ɛ ad the weight o each edge i he left figure give the etwork; the ceter oe i the route for the olie algorithm; ad the right oe i the route for the optimal algorithm Coider the meage equece that begi with a meage from S to, ay, S Without lo of geerality (ice there are oly two poible path from S to ), the olie algorithm route the meage via the route SX X 2X 3 X X he meage equece i X X 2, X 2X 3, X 3X 4,, X X It i eay to ee that the optimal algorithm (ee right figure) route the firt meage through SY Y 2Y 3 Y Y, the route the remaiig meage through X X 2, X 2X 3, X 3X 4,, ad X X hu the optimal algorithm ca tramit meage he olie algorithm (ceter) ca tramit at mot meage for thi meage equece becaue the ode X, X 2,, X are all aturated after routig the firt meage he competitive ratio i mall whe i large tery power hu, we would like to route meage alog the path with the maximal miimal fractio of remaiig power after the meage i tramitted We call thi path the maxmi path he performace of max-mi path ca be very bad, a how by the example i Figure 3 Aother cocer with the max-mi path i that goig through the ode with high reidual power may be expeive a compared to the path with the miimal power coumptio oo much power coumptio decreae the overall power level of the ytem ad thu decreae the life time of the etwork here i a tradeoff betwee miimizig the total power coumptio ad maximizig the miimal reidual power of the etwork We propoe to ehace a max-mi path by limitig it total power coumptio he two extreme olutio to power-aware routig for oe meage are: () compute a path with miimal power coumptio P mi; ad (2) compute a path that maximize the miimal reidual power i the etwork We look for a algorithm that optimize both criteria We relax the miimal power coumptio for the meage to be zp mi with parameter z to retrict the power coumptio for edig oe meage to zp mi We propoe a algorithm we call max-mi zp mi that coume at mot zp mi while maximizig the miimal reidual power fractio he ret of the ectio decribe the max-mi zp mi algorithm, preet empirical jutificatio for it, a method for adaptively chooig the parameter z ad decribe ome of it theoretical propertie he followig otatio i ued i the decriptio of the maxmi zp mi algorithm Give a etwork graph (V, E), let Figure 3: he performace of max-mi path ca be very bad I thi example, each ode except for the ource S ha the power 2 + ɛ, ad the weight of each edge o the arc i he weight of each traight edge i 2 Let the power of the ource be he etwork ca ed 2 meage from S to accordig to max-mi trategy by takig the edge o the arc (ee the arc o the top) But the optimal umber of meage follow the traight edge with black arrow i ( 4) where i the umber of ode Fid the path with the leat power coumptio, P mi by uig the Dijktra algorithm Fid the path with the leat power coumptio i the graph If the power coumptio > z P mi or o path i foud, the the previou hortet path i the olutio, top 2 Fid the miimal u tij o that path, let it be u mi 3 Fid all the edge whoe reidual power fractio u tij u mi, remove them from the graph 4 Goto Figure 4: max-mi zp mi-path algorithm P (v i) be the iitial power level of ode v i, e ij the weight of the edge v iv j, ad P t(v i) i the power of the ode v i at time t Let u tij = Pt(v i) e ij P (v i be the reidual power fractio after ) edig a meage from i to j Figure 4 decribe the algorithm I each roud we remove at leat oe edge from the graph he algorithm ru the Dijktra algorithm to fid the hortet path for at mot E time where E i the umber of edge he ruig time of the Dijktra algorithm i O( E + V log V ) where V i the umber of ode he the ruig time of the algorithm i at mot O( E ( E + V log V )) By uig biary earch, the ruig time ca be reduced to O(log E ( E + V log V )) o fid the pure max-mi path, we ca modify the Bellma-ford algorithm by chagig the relaxatio procedure he ruig time i O( V E ) 4 Adaptive Computatio for z A importat factor i the max-mi zp mi algorithm i the parameter z which meaure the tradeoff betwee the maxmi path ad the miimal power path Whe z = the algorithm compute the miimal power coumptio path Whe z = it compute the max-mi path We would like to ivetigate a adaptive way of computig z > uch that max-mi zp mi that will lead to a loger lifetime for

5 Chooe iitial value z, the tep δ Ru the max-mi zp mi algorithm for ome iterval P 2 Compute P t for every hot, let the miimal oe be t 3 Icreae z by δ, ad ru the algorithm agai for time P 4 Compute the miimal P t amog all hot, let it be t 2 5 If ome hot i aturated, exit 6 If t < t 2, the t = t 2, goto 3 7 If t > t 2, the δ = δ2, t = t 2, goto 3 Figure 5: Adaptive max-mi zp mi algorithm the etwork tha each of the max-mi ad miimal power algorithm Figure 5 decribe the algorithm for adaptively computig z P i the iitial power of a hot P t i the reidual power decreae at time t compared to time t P P t Baically, give a etimatio for the lifetime of that ode if the meage equece i regular with ome cyclicity he adaptive algorithm work well whe the meage ditributio are imilar a the time elape We coducted everal imulatio experimet to evaluate the adaptive computatio of z I a firt experimet we geerated the poitio of hot i a quare field radomly uig the followig parameter he cope of the etwork i, the umber of hot i the etwork i 2, the power coumptio weight for tramittig a meage are e ij = d 3 ij, ad the iitial power of each hot i 3 Meage are geerated betwee all poible pair of hot ad are ditributed evely Figure 6 (top) how the umber of meage tramitted util the firt meage delivery failure for differet value of z Uig the adaptive method for electig z with z iit =, the total umber of meage et icreae to 2, 27, which i almot the bet performace by max-mi zp mi algorithm I the ecod experimet we geerated the poitio of hot evely ditributed o the perimeter of a circle he radiu of the circle i 2, umber of hot 2; the weight formula: e ij = d 3 ij; ad the iitial power of each hot i Meage are geerated betwee all poible pair of the hot ad are ditributed evely he performace accordig to variou z ca be foud i Figure 7 (top) By uig the adaptive method, the total umber of meage et util reachig a etwork partitio i, 588, which i much better tha the mot cae whe we chooe a fixed z 42 Empirical Evaluatio of Max-mi zp mi Algorithm We coducted everal experimet for evaluatig the performace of the max-mi zp mi algorithm I the firt et of experimet (Figure 6), we compare how z affect the performace of the lifetime of the etwork I the firt experimet, a et of hot are radomly geerated o a quare For each pair of ode, oe meage i et i both directio for a uit of time hu there i a total of ( ) meage et i each uit time, where i the umber of the hot i the etwork We experimeted with other etwork topologie Figure 7 (top) how the reult he maximal meage tramitted he maximal meage tramitted he parameter z x he parameter z Figure 6: he effect of z o the maximal umber of meage i a quare etwork pace he poitio of hot are geerated radomly I the top graph the etwork cope i, the umber of hot i 2, the weight are geerated by e ij = d 3 ij, the iitial power of each hot i 3, ad meage are geerated betwee all poible pair of the hot ad are ditributed evely I the bottom graph the umber of hot i 4, the iitial power of each ode i, ad all other parameter are the ame a the top graph obtaied i a rig etwork Figure 7 (bottom) how the reult obtaied whe the etwork coit of four colum where ode are approximately aliged i each colum he ame method ued i experimet varie the value of z hee experimet how that adaptively electig z lead to uperior performace over the miimal power algorithm (z = ) ad the max-mi algorithm (z = ) Furthermore, whe compared to a optimal routig algorithm, max-mi zp mi ha a cotat empirical competitive ratio (ee Figure 8 (top)) Figure 8 (bottom) how more data that compare the maxmi zp mi algorithm to the optimal routig trategy We computed the optimal trategy by uig a liear programmig package 2 We ra 5 experimet I each experimet a etwork with 2 ode wa geerated radomly i a etwork pace he meage were et to oe gateway ode repeatedly We computed the ratio of the lifetime of the max-mi zp mi algorithm to the optimal lifetime Figure 8 how that max mi zp mi perform better tha 8% of optimal for 92% of the experimet ad perform withi more tha 9% of the optimal for 53% of the experimet Sice the optimal algorithm ha the ad- 2 o compute the optimal lifetime, the meage rate are kow he max-mi algorithm doe ot have thi iformatio

6 he maximal meage tramitted he maximal meage tramitted 25 x he parameter z 24 x he ratio betwee the max mi ad the optimal olutio he umber of ode i the etwork 9 8 umber of experiemet he parameter z the ratio of the lifetime i max mi ad the optimal lifetime (%) Figure 7: he top figure how the effect of z o the maximal umber of meage i a rig etwork he radiu of the circle i 2, the umber of hot i 2, the weight are geerated by e ij = d 3 ij, the iitial power of each hot i ad meage are geerated betwee all poible pair of the hot ad are ditributed evely he bottom figure how a etwork with four colum of the ize Each area ha te hot which are radomly ditributed he ditace betwee two adjacet colum i he right figure give the performace whe z chage he vertical axi i the maximal meage et before the firt hot i aturated he umber of hot i 4; the weight formula i e ij = d 3 ij; the iitial power of each hot i ; meage are geerated betwee all poible pair of the hot ad are ditributed evely Figure 8: he top graph compare the performace of max-mi zp mi to the optimal olutio he poitio of hot i the etwork are geerated radomly he etwork cope i, the weight formula i e ij = d 3 ij, the iitial power of each hot i, meage are geerated from each hot to a pecific gateway hot, the ratio z i he bottom figure how the hitogram that compare max-mi zp mi to optimal for 5 experimet I each experimet the etwork coit of 2 ode radomly placed i a * etwork pace he cot of meage i give by e ij = d 3 ij he hot have the ame iitial power ad meage are geerated for hot to oe gateway hot he horizotal axi i the ratio betwee the lifetime of the max-mi zp mimax-mi algorithm ad the optimal lifetime, which i computed off-lie vatage of kowig the meage equece, we believe that max-mi zp mi i practical for applicatio where there i o kowledge of the meage equece 43 Aalyi of the Max-mi zp mi Algorithm I thi ectio we quatify the experimetal reult from the previou ectio i a attempt to formulate more preciely our origial ituitio about the tradeoff betwee the miimal power routig ad max-mi power routig We provide a lower boud for the lifetime of the max-mi zp mi algorithm a compared to the optimal olutio We dicu thi boud for a geeral cae where there i ome cyclicity to the meage that flow i the ytem ad the how the pecializatio to the o cyclicity cae Suppoe the meage ditributio i regular, that i, i ay period of time [t, t + δ), the meage ditributio o the ode i the etwork are the ame Sice i eor etwork we expect ome ort of cyclicity for meage tramiio, we aume that we ca chedule the meage tramiio with the ame policy i each time lice we call δ I other word, we partitio the time lie ito may time lot [, δ), [δ, 2δ), [2δ, 3δ), Note that δ i the lifetime of the etwork if there i o cyclical behavior i meage tramiio We aume the ame meage are geerated i each δ lot but their equece may be differet Let the optimal algorithm be deoted by O, ad the maxmi zp mi algorithm be deoted by M I M, each meage i tramitted alog a path whoe overall power coumptio i le tha z time the miimal power coumptio for that meage he iitial time i he lifetime of the etwork by algorithm O i O, ad the lifetime by algorithm M i M he iitial power of each ode i: P, P 2, P 3,, P ( ), P he remaiig power of each ode at O by ruig algorithm O i: P O, P 2O, P 3O,, P O, P O he remaiig power of each ode at M by ruig algorithm M i: P M, P 2M, P 3M,, P M, P M Let the meage equece i ay lot be m, m 2,, m, ad the miimal power coumptio to tramit thoe meage be P m, P m2, P m3,, P m 2

7 heorem 2 he lifetime of algorithm M atifie M O z + δ ( PkO z Proof We have P k = P km + M M Pm k PkM) P Mmk = P M where M M i the umber of meage tramitted from time poit to M P Mmk i the power coumptio of the k-th meage by ruig algorithm M We alo have: P k = P ko + M O P Omk = P O where M O i the umber of meage tramitted from time poit to O P Omk i the power coumptio of of the k-th meage by ruig algorithm O Sice the meage are the ame for ay two lot without coiderig their equece, we ca chedule the meage uch that the meage rate alog the ame route are the ame i the two lot (thik about divide every meage ito may tiy packet, ad average the meage rate alog a route i algorithm O ito the two coecutive lot evely) We have: ad M O So we have:, ad P Omk = M O M M P O = P M = M δ P Mmk = P Omk = O δ j= P ko + O δ M δ P km + j= P O = P M P Mmkj P Omk P Mmkj P Omk P Mmkj i the power coumptio of the k-th meage i lot j by ruig algorithm M We alo have the followig aumptio ad the miimal power of P mk For ay j M δ ad k, we have oly oe correpodig l, he, P Mmkj z P ml ad P Omk P mk P O P M P ko + O δ P km + z M δ P mk P mk (3) hu, We have: P km + z M δ P mk P ko + O δ M O z + δ ( PkO z Pm k PkM) P mk heorem 2 give u iight ito how well the meage routig algorithm doe with repect to optimizig the lifetime of the etwork Give a etwork topology ad a meage PkO, Pm k ditributio, O, δ, are all fixed i Equatio 3 he variable that determie the actual lifetime are PkM ad z he maller PkM 3 i, the better the performace lower boud i Ad the maller z i, the better the performace lower boud i However, a mall z will lead to a large PkM hi explai the tradeoff betwee miimal power path ad max-mi path heorem 2 ca be ued i applicatio that have a regular meage ditributio without the retrictio that all the meage are the ame i two differet lot For thee applicatio, the ratio betwee δ ad Pm k mut be chaged r to Pm, where k Pm k i the miimal power coumptio for the meage geerated i a uit of time heorem 3 he optimal lifetime of the etwork i at t mot SP P h where t P h P h SP SP ad Ph SP are the life time of the etwork ad remaiig power of hot h by uig the leat power coumptio routig trategy P h i the iitial power of hot h Proof t OP = t SP P h P h P h SP P h P SP m = P h( P h Ph SP t SP ) 5 ZONE-BASED ROUING Although it ha very ice theoretical ad empirical propertie, max-mi zp mi algorithm i hard to implemet o large cale etwork he mai obtacle i that maxmi zp mi require accurate power level iformatio for all the ode i the etwork It i difficult to collect thi iformatio from all the ode i the etwork Oe way to do it i by broadcat, but thi would geerate a huge power coumptio which defeat our origial goal Furthermore, it i ot clear how ofte uch a broadcat would be eceary to keep the etwork data curret I thi ectio we propoe a hierarchical approach to power-aware routig that doe ot ue a much iformatio, doe ot kow the meage equece, ad relie i a feaible way o max-mi zp mi We propoe to orgaize the etwork tructurally i geographical zoe, ad hierarchically to cotrol routig acro the zoe he idea i to group together all the ode that are i geographic proximity a a zoe, treat the zoe a a 3 hi i the remaiig power of the etwork at the limit of the etwork 3

8 etity i the etwork, ad allow each zoe to decide how to route a meage acro 4 he hot i a zoe autoomouly direct local routig ad participate i etimatig the zoe power level Each meage i routed acro the zoe uig iformatio about the zoe power etimate I our viio, a global cotroller for meage routig maage the zoe hi may be the ode with the highet power, although other cheme uch a roud robi may alo be employed If the etwork ca be divided ito a relatively mall umber of zoe, the cale for the global routig algorithm i reduced he global iformatio required to ed each meage acro i ummarized by the power level etimate of each zoe We believe that i eor etwork thi value will ot eed frequet update becaue obervable chage will occur oly after log period of time he ret of thi ectio dicue () how the hot i a zoe collaborate to etimate the power of the zoe; (2) how a meage i routed withi a zoe; ad (3) how a meage i routed acro zoe () ad (3) will ue our max-mi zp mi algorithm, which ca be implemeted i a ditributed way by lightly modifyig our defiitio of the max-mi zp mi path he max mi algorithm ued i (2) i baically the Bellma-Ford algorithm, which ca alo be implemeted a a ditributed algorithm 5 Zoe Power Etimatio he power etimate for each zoe i cotrolled by a ode i the zoe hi etimatio meaure the umber of meage that ca flow through the zoe Sice the meage come from oe eighborig zoe ad get directed to a differet eighborig zoe, we propoe a method i which the power etimatio i doe relative to the directio of meage tramiio he protocol employed by the cotroller ode coit of pollig each ode for it power level followed by ruig the max-mi zp mi algorithm he retured value i the broadcated to all the zoe i the ytem he frequecy of thi procedure i iverely proportioal to the etimated power level Whe the power level i high, the power etimatio update ca be doe ifrequetly becaue meage routed through the zoe i thi period will ot chage the overall power ditributio i the etire etwork much Whe the power level i low, meage tramiio through the zoe i likely to chage the power ditributio igificatly Without lo of geerality, we aume that zoe are quare o that they have four eighbor poited to the North, South, Eat, ad Wet 5 We aume further that it i poible to commuicate betwee the ode that are cloe to the border betwee two zoe, o that i effect the border ode are part of both zoe I other word, eighborig zoe that ca commuicate with each other have a area of overlap (ee Figure 9 (top)) he power etimate of a zoe ca be approximated a follow We ca ue the max-mi zp mi algorithm to evalu- 4 hi geographical partitioig ca be implemeted eaily uig GPS iformatio from each hot 5 thi method ca eaily be geeralized to zoe with fiite umber of eighborig zoe S &&' A B SB A SC B AB A B C BC 9 6 "" ## 4 $$ %% 3 $$ %% 9 4!! 6 5 Figure 9: hree zoe, A, B, ad C SB, SC are the ource area of B ad C, ad A, B are the ik area of A ad B AB ad BC are overlap border area he right figure how how to coect the local path i zoe B with the local path i zoe C he umber ext to each ode i the umber of path paig through that ode i the power evaluatio procedure he vertical tripe are the ource ad ik area of the zoe chooe for the meage graularity P = ; repeat{ Fid the max-mi zp mi path for meage ed the meage through the zoe P = P + } util (ome ode are aturated) retur P Figure : A approximatio algorithm for zoe power evaluatio ate the power level, fid the max-mi zp mi path, imulate edig meage through the path, ad repeat util the etwork i aturated i choe to be proportioate to the power level of the zoe More preciely, coider Figure 9 (top) o etimate the power of zoe B with repect to edig meage i the directio from A to C, let the left part of the overlap betwee A ad B be the ource area ad the right part of the overlap betwee B ad C the ik area he power of zoe B i the directio from A to C i the maximal umber of meage that ca flow from the ource ode to the ik ode before a ode i B get aturated hi ca be computed with the max-mi zp mi algorithm (ee Figure ) We tart with the power graph of zoe B ad augmet it We create a imagiary ource ode S ad coect it to all the ource ode We create a imagiary ik ode ad coect all the ik ode to it Let the weight of the ewly added edge be he max-mi zp mi algorithm ru o thi graph determie the power etimate for zoe B i the directio of A to C 52 Global Path Selectio C D 4

9 Give power-level for each poible directio of meage tramiio, it i poible to cotruct a mall zoe-graph that model the global meage routig problem Figure 2 how a example of a zoe graph A zoe with k eighbor i repreeted by k + vertice i thi graph 6 Oe vertex label the zoe; k vertice correpod to each meage directio through the zoe he zoe label vertex i coected to all the meage directio vertice by edge i both directio I additio, the meage directio vertice are coected to the eighborig zoe vertice if the curret zoe ca go to the ext eighborig zoe i that directio Each zoe vertex ha a power level of Each zoe directio vertex i labeled by it etimated power level computed with the procedure i Sectio 5 Ulike i the model we propoed i Sectio 33, the edge i thi zoe graph do ot have weight hu, the global route for edig a meage ca be foud a the max-mi path i the zoe graph that tart i the origiator zoe vertex ad ed i the detiatio zoe vertex for the meage We would like to bia toward path electio that ue the zoe with higher power level We ca modify the Bellma-Ford algorithm (Figure ) to accomplih thi Give graph G(V, E), aotated with power level p(v) for each v V Fid the path from to t, = v, v,, v k, v k = t uch that mi k i= p(vi) i maximal for each vertex v V [G] do If edge (, v) E[G] the d[v], π[v] ele d[v], π[v] NIL d[] for i to V [G] do for each edge (u, v) E[G] ad u do if d[v] < mi(d[u], p[u]) the d[v] mi(d[u], p[u]) π[v] u retur π[t] Figure : Maximal miimum power level path A C Figure 2: Four zoe are i a quare etwork field he power of a zoe i evaluated i four directio, left, right, up, ad dow A zoe i repreeted a a zoe vertex with four directio vertice he power label are omitted from thi figure 53 Local Path Selectio 6 For quare zoe k = 4 + a how i Figure 2 D B C A D B Give a global route acro zoe, our goal i to fid actual route for meage withi a zoe he max-mi zp mi algorithm i ued directly to route a meage withi a zoe If there are multiple etry poit ito the zoe, ad multiple exit poit to the ext zoe, it i poible that two path through adjacet zoe do ot hare ay ode hee path have to be coected he followig algorithm i ued to eure that the path betwee adjacet zoe are coected (ee Figure 9 (right)) For each ode i the overlap regio, we compute how may path ca be routed locally through that ode whe zoe power i evaluated I order to optimize the meage flow betwee zoe, we fid path that go through the ode that ca utai the maximal umber of meage hu, to route a meage through zoe B i the directio from A to C we elect the ode with maximum meage weight i the overlap betwee A ad B, the we elect the ode with maximum meage weight i the overlap betwee B ad C, ad compute the max-mi zp mi path betwee thee two ode 54 Performace Evaluatio for Zoe-baed Routig he zoe-baed routig algorithm doe ot require a much iformatio a would be required by max-mi zp mi algorithm over the etire etwork By givig up thi iformatio, we ca expect the zoe-baed algorithm to perform wore tha the max-mi zp mi algorithm We deiged large experimet to meaure how the zoe-baed algorithm doe relative to the max-mi zp mi algorithm (I the followig experimet, we oly coider the power coumptio ued for the applicatio meage itead of the cotrol meage hu we ca compare how much the performace of our zoe-baed algorithm i cloe to that of the maxmi zp mi algorithm without the ifluece of the cotrol meage) We dipere, ode radomly i a regular etwork pace (ee Figure 3) he zoe partitio i decribed i the figure Each zoe ha averagely 4 ode Each ode ed oe meage to a gateway ode i each roud (A roud i the time for all the ode to fiih edig meage to the gateway) he zoe power evaluatio protocol i executed after each roud By ruig the max-mi zp mi algorithm, we ra the algorithm for about 4 meage before oe of the hot got aturated By ruig the zoe-baed routig algorithm, we got about 39 meage before the firt meage caot be et through he performace ratio betwee the two algorithm i term of the lifetime of the etwork i 945% Without the zoe tructure, the umber of cotrol meage o the power of each ode i every iformatio update i, ad they eed to be broadcated to ode I zoe-baed algorithm, the umber of cotrol meage i jut the umber of the zoe, 48 here, ad they are broadcated to 24 zoe after the zoe power evaluatio Ad the zoe-baed routig dramatically reduce the ruig time to fid a route i our imulatio I aother experimet, we dipere 24 eor to a quare field with ize he eor are ditributed radomly i the field Each eor ha a iitial power of 4 he power coumptio formula i e ij = d 3 ij he etwork 5

10 field i divided by 5*5 quare each of which correpod to four zoe i four directio (left, right, up ad dow) he zoe-baed algorithm achieved 96% of the lifetime of the max-mi zp mi algorithm 6 CONCLUSION We have decribed a o-lie algorithm for power-aware routig of meage i large etwork dipered over large geographical area I mot applicatio that ivolve ad-hoc etwork made out of mall had-held computer, mobile computer, robot, or mart eor, battery level i a real iue i the duratio of the etwork Power maagemet ca be doe at two complemetary level () durig commuicatio ad (2) durig idle time We believe that optimizig the performace of commuicatio algorithm for power coumptio ad for the lifetime of the etwork i a very importat problem It i hard to aalyze the performace of olie algorithm that do ot rely o kowledge about the meage arrival ad ditributio hi aumptio i very importat a i mot real applicatio the meage patter are ot kow ahead of time I thi paper we have how that it i impoible to deig a o-lie algorithm that ha a cotat competitive ratio to the optimal off-lie algorithm, ad we computed a boud o the lifetime of a etwork whoe meage are routed accordig to thi algorithm hee reult are very ecouragig We developed a olie algorithm called the max-mi zp mi algorithm ad howed that it had a good empirical competitive ratio to the optimal off-lie algorithm that kow the meage equece We alo howed empirically that maxmi zp mi achieve over 8% of the optimal (where the optimal router kow all the meage ahead of time) for mot itace ad over 9% of the optimal for may problem itace Sice thi algorithm require accurate power value for all the ode i the ytem at all time, we propoed a ecod algorithm which i hierarchical Zoe-baed power-aware routig partitio the ad-hoc etwork ito a mall umber of zoe Each zoe ca evaluate it power level with a fat protocol hee power etimate are the ued a weight o the zoe A global path for each meage i determied acro zoe Withi each zoe, a local path for the meage i computed o a to ot decreae the power level of the zoe too much Ackowledgmet hi work ba bee upported i part by Departmet of Defee cotract MURI F ad DARPA cotract F , ONR grat N , NSF CAREER award IRI , NSF award IS-99293, Hoda corporatio, ad the Sloa foudatio; 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