Multi-sensor activation for temporally correlated event monitoring with renewable energy sources. Neeraj Jaggi* Koushik Kar

Transcription

1 I J Sesor Neworks, Vol X, No Y, XXXX Mul-sesor ava for emorally orrelae eve morg wh reewable eergy soures Neeraj Jagg* Dearme of Eleral Egeerg a Comuer See, Wha Sae Uversy, Wha, KS 6760, USA Emal: eerajjagg@whaeu *Corresg auhor Koushk Kar Dearme of Eleral Comuer a Sysems Egeerg, Resselaer Polyeh Isue, roy, NY 80, USA Emal: koushk@esereu Absra: Fuure sesor eworks woul omrse sesg eves wh eergy-harvesg aables from reewable eergy soures, suh as solar ower hs aer fouses esg of effe algorhms for mul-sesor ava o omse overall eve ee robably resee of ueraes eve a reharge roesses We formulae he yam mulsesor ava ques a sohas omsa framework, a show ha a me-vara hreshol oly, whh maas a aroraely hose umber of sesors ave a all mes, s omal absee of emoral orrelas Moreover, he same eergy-balag mevara hreshol oly aroahes omaly resee of emoral orrelas as well, albe uer era lmg assums We also aalyse he lass of orrela-eee hreshol oles a erve he rage for eergy-balag hreshols hrough smulas, we omare he roose me-vara oly wh eergy-balag orrela-eee oles, a observe ha alhough he laer may erform beer, he erformae fferee s raher small he ases sue Keywors: mul-sesor ava; emoral orrelas; eergy sras; eergy-harvesg sesor sysems Referee o hs aer shoul be mae as follows: Jagg, N a Kar, K (XXXX Mul-sesor ava for emorally orrelae eve morg wh reewable eergy soures, I J Sesor Neworks, Vol X, No Y, xxx xxx Bograhal oes: Neeraj Jagg reeve hs BE egree Comuer See a Egeerg from Naal Isue of ehology, Rourkela, Ia, ME egree Sysems See a Auoma from Ia Isue of See, Bagalore, Ia, a PhD egree Comuer a Sysems Egeerg from Resselaer Polyeh Isue, roy, NY He s urrely wh Dearme of Eleral Egeerg a Comuer See, Wha Sae Uversy, Wha, KS Hs researh fouses eergy-effey, aa reserva, a sheulg sesor eworks, a seury a rvay wreless eworks He has serve as NSF ael revewer, rogramme ommee member a revewer for varous eraal ferees a jourals Koushk Kar has bee wh he Eleral, Comuer a Sysems Egeerg Dearme a Resselaer Polyeh Isue, roy, NY, se 00 He reeve hs Beh egree Eleral Egeerg 997 from he Ia Isue of ehology, Kaur, a hs PhD egree Eleral a Comuer Egeerg from he Uversy of Maryla, College Park, MD, 00 Hs ree work has lue he suy of sheulg, aess rol, a ower maageme wreless a sesor eworks He reeve he Career Awar from he Naal See Foua 005 a s urrely a Assoae Eor of he IEEE/ACM rasas Neworkg Coyrgh 00X Iersee Eerrses L

2 N Jagg a K Kar Irou Wreless sesor eworks are eloye for eeg eresg heomea a we rage of evrmes, lug oeas, foress, amoshere a mlary-survelle regs yally, vual sesors are heavly srae erms of resoures suh as omuaal ower a eergy he y, low-os aure of he sesg eves alg wh her mmal roessg aables reaes he ee o evelo smle bu effe algorhms for her oeras I a, he eergy usage a he sesors mus be omse o mrove erformae of he sesor ework For lg-erm morg of he argee evrmes, sesors are evse o be eloye wh rehargeable baeres, whh are aable of haressg eergy from reewable soures he evrme For sae, Helomoe (Hsu e al, 005; Raghuaha e al, 005, a solar eergy-harvesg laform, emsraes he self-susag aably of a sesg eve Moreover, he rolfera of -rehargeablebaery-base sesors auses severe evrmeal hazars a avoaes he ee for gree-ehology-base solus Sesg eves aable of harvesg solar ower (Hsu e al, 005; Jag e al, 005; Raghuaha e al, 005; Norma, 007 a oher eergy soures, lug w a vbra eergy (MroSra, 003, ulse he hgh avalably of a reewable eergy soure o eable ear-ereual oera of he sesor ework Oe of he mos mora ssues he effe oera of suh sesor eworks les he esg of ellge sore-a-use eergy-harvesg frameworks for eergy maageme (Kasal a Srvasava, 003; Kasal e al, 004 Desg of eergy-effe algorhms for sesor oeras s val owars realsa of suh frameworks Sesors are ofe, by esg, urelable a eergy effe hs s beause s que osly o bul hghly relable a eergy-effe sesors For sae, a sesor may ee o be eque wh a huge solar ael for o have hgh eergy avalably a all mes I a, omal eloyme of sesors s o feasble may ala searos suh as evrmeal morg a balefel survellae yally, sesors are small, eergy-srae eves wh low reharge raes (eee u eergy harvesg a may have o se a sgfa fra of her lfeme ave or slee sae Moreover, he vual sesors are re o falures herefore, rae, ue o os effey a feasbly of eloyme, sesors ee o eloye raomly a reualy (a a hgh esy he reg of eres o guaraee relably he sesg a ommua roesses For beer erformae, hese sesors woul ee o work ollaboravely o aheve a global ework objeve, suh as relable eve ee a reorg I he geeralse eve ee ala ha we ser hs aer, ala-sef eves, whh he sysem s requre o ee, our raomly he reg of eres a a oeally exhb emoral orrelas aross her ourrees he overall sysem objeve s o maxmse he me-average eve ee robably he sysem he sharge of a ave sesor ees he ava algorhm as well as he eve ourree roess, whle he reharge s base u haressg reewable eergy We aress he followg mul-sesor ava ques a reewable eergy base sesor sysem How shoul he sesor oes be avae so as o omse he overall eve ee robably aheve by he sysem? Sesor oes woul yally oerae uer uera oeraal s, lug ukow eergy releshme sheules, aral sysem sae forma a varyg egrees of sao-emoral orrelas he sese heomea hese faors a a ew mes o he esg of effe algorhms for sesor slee sheulg (Kar e al, 006; Jagg e al, 008, 009, rasmss (Borkar e al, 005; Zhag a Chas 005, roug (L e al, 005, ave eergy maageme (Nyao, Hossa, Rash e al, 007; Vgoro e al, 007 a rae alloa (Fa e al, 008 quess I hs aer, we use he sesor eergy moel frs roose Jagg e al (009 (for sgle-sesor ava ques o formulae a solve he mul-sesor ava ques uer emorally orrelae eve heomea he overall objeve of he esge algorhms s o guaraee hgh avalably of he ework he resee of ueraes sese heomea a eergy releshme sheules he ma rbus of hs aer lue: Formula of he mul-sesor ava ques, whle aroraely moellg he ueraes volve he eve ourrees a he reewable eergy soures Desgg ava sheules for mul-sesor sysems o maxmse he overall eve ee robably he resee of emoral orrelas Aalysg wo ffere lasses of hreshol ava oles o evaluae her erformae uer varous sysem arameers Proosg a me-vara hreshol oly a emsrag s ear-omal erformae a s robusess o he resee of ueraes he sysem he aer s orgase as follows Nex we suss he moellg of ueraes a formulae he roblem as a yam omsa ques Se Se 3 susses varous ava algorhms sere a rmarly fouses hreshol-base oles Se 4 aalyses he erformae of roose algorhms a reses smula resuls Se 5 susses relae researh reewable-eergy-base sesor sysems amg ohers Fally, we summarse he luss a fuure res Se 6 Problem formula I hs se, we elaborae u he sesor eergy moel use o haraerse he oeras of a reewable-eergybase sesg eve We he suss he eve ourree roess use o moel he eve heomea We also rese he ala-sef erformae mer, whh s laer use he erformae evalua of he esge algorhms

3 Mul-sesor ava for emorally orrelae eve morg Sesor eergy moel he eergy buke of a rehargeable sesor sores eergy us of a quaum he sze of he sesor eergy buke s eoe by K he resee of ueraes reewableeergy-base reharge roess s moelle usg a sohas framework We assume a sree me moel where eah me slo, a reharge eve ours wh a robably q a harges he sesor wh a sa harge amou of quaa he reharge roesses a he ffere sesor oes are assume o be eee of eah oher; however, he arameers a q are he same aross all sesors he sharge roess a he sesor ees u s ava (ava sae, as well as he sae of he ala-sef eve heomea A sesor havg zero eergy level s sa o be ave (ave sae f has bee avae (eavae he urre me slo A sesor havg zero eergy level s sa o be he ea sae he sesor exes a harge amou of δ quaa (oeraal os urg eah me slo s ave I a, f a ala-sef eve s eee by he ave sesor, he sesor exes a aal harge amou of δ quaa (ee a rasmss os We assume ha δ a δ δ We also assume ha a sesor a be avae ly f has suffe eergy o oerae suessfully for a leas e me slo, e, s eergy level s a leas δ + δ Noe ha we assume ha a sesor sharges eergy ly whe ave sae; however, he aalyss a be exee o ser -zero eergy sharge ave sae he reharge rae of a sesor (er me slo equals q Smlarly, he sharge rae of a ave sesor s gve by δ + δπ, where π eoes he seay-sae robably of ourree of a ala-sef eve urg he mes he sesor s ave, a eoes he eve ee robably of he sesor yally, he reharge rae of a sesor woul be sgfaly less ha s sharge rae he ave sae, whh eessaes he esg of a effe ava algorhm for sesor oeras Ala-sef eve heomea We moel he ala-sef eve heomea whh he sesor sysem s requre o ee a reor as a orrelae sohas roess orer o haraerse he here raomess a emorally orrelae eve ourree For examle, ser a sesor ework eloye o ee a war agas fores fres If he emeraure ay reg he fores rses above 00 F, mgh rerese he ossbly of a fores fre Now, f he emeraure a some o of me s hgher ha hs hreshol, he wh hgh robably, woul rema above hs hreshol ear fuure as well Smlarly, f he emeraure s muh below alarmg levels (he above hreshol, he s lkely o rema so he mmeae fuure hus, smar ava algorhms shoul ake o sera he sae (a orrela forma of he ala-sef eve heomea whle eg u ava sheules he exe of emoral orrela alasef eve heomea s sefe usg orrela robables a suh ha, If a eve ours urg me slo, he he ex me slo ( +, a smlar eve ours wh robably, whle o suh eve ours wh robably Smlarly, f o eve ourre urg he urre me slo, o suh eve ours he ex me slo wh robably he eve ourree roess use o moel he alasef eve heomea omrses a alerag sequee of eros where eves our (O ero a o o our (Off ero I rae, ala-sef eves woul our rarely; herefore, he Off eros are exee o be sgfaly larger ha he O eros, whh mles Neverheless, our aalyss ales o searos where < as well Noe ha se s a measure of relably of he sesor (a hee s a roery of he sesor oe, whereas he orrela robables a are measures of emoral orrelas he eve heomea (a hee are a roery of he ala-sef eve heomea, we assume ha s eee of a Fgure es he sesor sharge/reharge moel a behavour of a vual sesor urg ffere saes of he eve roess Cser a me slo suh ha a eve ourre urg me slo bu o eve ourre urg me slo Le X eoe he raom varable rereseg he umber of me slos (lug afer whh he eve ours aga he, ( Pr[ X = ] =, herefore: ( ( ( ( EX [ ]= ( = ( = = hus, he exee legh of a Off ero he eve ourree roess s gve by Smlarly, he exee legh of a O ero equals Usg Markov ha aalyss, he seay-sae robably of eve ourree equals π = ( π = π (

4 N Jagg a K Kar Fgure Eergy sharge/reharge moel of he sesor (see le vers for olours Noe: he sysem erformae ees he ava oly, reharge roess a he eve heomea 3 Performae mer Cser a sysem of N eal reewable-eergy-base sesg eves eloye a reg of eres o mor a ala-sef eve heomea 3 If a alasef eve ours urg a me slo, eah sesor eeely ees he eve wh a robably (eve ee robably Noe ha he ee robably of a sesor ees u s sae from he arge of eve ourree (Ch a Hu, 008 yal values of vual ee robables are exee o le he rage [0 05] (Ch a Hu, 008 Se we ser sesor eloymes a eve ourrees o be raom, s reasable o assume ha he ee robables of ffere sesors are eee of eah oher Also, se eah me, he eve s exee o our a a ffere loa he reg of eres, he average eve ee Fgure Sesors morg a ala-sef eve heomea (see le vers for olours robably of a vual sesor s moelle as Alhough he sesors may be loae a ffere os sae, se he eve ourrees are raom, we assume ha he eves are equally lkely o our aywhere ( a uformly srbue fash he reg of eres Smlar argumes oul be use for he eergy-harvesg (reharge roess as well whh jusfy he assum ha he sesors are eal her sesg aables a her reharge/sharge yams, over a large ero of me Now, f ou of he N sesors were ave sae urg he above me slo a a eve ourre urg he me slo, le he overall eve ee robably aheve be eoe U ( I geeral, U ( = 0 whe =0, a reases wh Fgure es he erformae of he sysem urg a arbrary me slo Noe: Se here are hree ave sesors, he eve ee robably he sysem urg me slo equals U (3 Se here are four sesors wh osve eergy, he maxmum ahevable ee robably urg me slo equals U (4 Maxmum ee robably urg ay me slo equals U (5

5 Mul-sesor ava for emorally orrelae eve morg wo examles of feasble uly fus are rove below: Examle : Le sesors be ave a a eve ours urg me slo he, he robably ha he eve ges eee by a leas e ave sesor s gve by U ˆ ( = ( Noe ha he overall eve ee robably U ˆ ( s zero whe o sesor s ave urg he me slo, a reases as he umber of ave sesors reases from 0 o N However, hs rease he eve ee robably exhbs mshg reurs wh rese o he umber of ave sesors I oher wors, he uly fu U ˆ ( s a -ereasg a srly ave fu, wh U ˆ (0 = 0 Fgure 3 es he shae of hs uly fu for varous values of sesor eve ee robably Noe ha lm U( = Fgure 4 los hs uly fu U ( as a fu of umber of ave sesors We observe ha hs uly fu ( U ( s also a -ereasg a ave fu Noe ha we o o exlly assume he ef of he uly fu U ( our aalyss, a our resuls aly o all alas where he erformae a be exresse usg a -ereasg a ave uly fu U, ( lug he above examles Fgure 4 Geeralse uly fu rereseg he majoryes rule (see le vers for olours Fgure 3 Overall eve ee robably exhbs mshg reurs wh rese o he umber of ave sesors uer uly fu U ˆ ( (see le vers for olours Examle : Le ou of he N sesors be ave urg me slo If a eve ours urg me slo, eah ave sesor eeely ees he eve wh robably Le eoe he umber of sesors whh (orrely ee he eve urg me slo he sesor sysem elares eve eee f >, e, more ha half of he ave sesors ee he eve hus, eve ee robably urg me slo equals Pr[ > ] We have: Pr[ = ] = ( ( [0 ] ( Pr > = ( ( (3 = + Usg Cher s bou (Mowa a Raghava, 995, whe >05, hs majory-es rule fu s lower boue by U ( as: Pr > U ( = e (4 he goal of he sysem s o maxmse s eve ee aably over me Le Π eoe he ava algorhm (or oly emloye by he sesor sysem Le Π eoe he umber of ave sesors he reg urg me slo whe he sesor sysem oeraes uer oly Π Le Π Π Π eoe he veor [,, ] Le x be he aor varable eog he ourree of a eve urg me slo, e, x = f a eve ourre urg me slo ; 0 oherwse he, he erformae of oly Π, eoe U ( Π, s gve by: U Π ( xu = ( Π= lm x = he es roblem s ha of fg he ava oly ˆΠ suh ha Π ˆ = arg max U ( Π 3 Ava algorhms/oles A hreshol oly wh arameer m s haraerse as follows A avalable sesor (e a sesor wh eergy level δ + δ s sheule for ava a me slo f he umber of sesors sheule for ava urg hs me slo s less ha he hreshol m; oherwse, he sesor s move o ave sae ul he ex es sa (5

6 N Jagg a K Kar A ew es s ake f he hreshol arameer hages, f ay ave sesor rus ou of eergy a moves o he ea sae or f a sesor ha was revously he ea sae beomes avalable hrough baery reharge hus, a hreshol oly wh a hreshol of m res o maa he umber of ave sesors he sysem as lose o m as ossble (however ever exeeg m I vew of he emorally orrelae aure of he ala-sef eve heomea, smar hreshol oles mgh emloy wo ffere hreshol arameers: ossbly a larger hreshol urg he O eros, a a smaller hreshol urg he Off eros We ser wo ffere hreshol oles, amely me-vara hreshol oly ( a orrela-eee hreshol oly (CP he algorhm s oblvous o he emoral orrela forma a emloys a sa hreshol arameer a all mes O he oher ha, he CP algorhm emloys ffere hreshol arameers urg he O a Off eros, resevely: me-vara hreshol oly (: Durg eah me slo, a hreshol of m ave sesors s argee from a se of avalable sesors A algorhm s smler o use rae Se he hreshol arameer oes o vary over me, requres mmal sae maeae overhea a he sesg eves Correla-eee hreshol oly (CP: A hreshol of m s emloye me slo f he eve ourree roess s kow o be he O ero, e, f a ala-sef eve was eee he revous me slo by ay of he ave sesors Oherwse, a hreshol of ( m s argee me slo hus, CP algorhm ales a me-varyg hreshol as oose o a sa hreshol emloye by he algorhm Iuvely, he CP algorhm res o serve he eergy a he sesors urg he Off eros orer o be able o use more juously urg he O eros Noe ha he algorhm s a seal ase of he CP algorhm wh = m Noe ha all he above ava algorhms are smler o eloy a sesor ework se hey requre mmal sae forma a a be realse base ly u loal forma Of arular eres are he ava algorhms whh aheve eergy balae he reewableeergy-base mul-sesor sysem seay sae For sae, a algorhm wh arameer m aheves eergy balae f he average reharge rae he sysem equals he average sharge rae he sesor sysem whe he hreshol arameer m s ale Smlarly, mulle CP hreshol ars ( m, oul aheve eergy balae he sesor sysem he algorhms whh aheve eergy balae he sesor sysem aheve beer erformae, as we show he ex se Iuvely, hs s smlar o maag he serve rae a queueg sysem o be equal o he arrval rae, hus ahevg he maxmum ossble ulsa 4 Performae aalyss he erformae aheve by a ava oly Π s measure usg equa (5 I hs se, we erve a uer bou ahevable erformae Se 4 We he aalyse he erformae of varous hreshol oles Se 4 a rese smula resuls Se 43 4 Uer bou omal erformae Se he omal erformae s fful o haraerse, we oba a uer bou We laer omare he erformae of our roose ava algorhm wh rese o hs bou Le, ψ be he aor varable eog wheher he sesor was ave urg me slo, e, ψ, = f he sesor was ave urg me slo ; 0 oherwse Lemma : For all sesors N : lm xψ =, x = q π δ+ δ Proof: As, he oal umber of ala-sef eves ha our urg me [ ] sasfes lm P, x = = π Le,, = [ Pr[ Pr[ ψ, = ] P =Pr[ x = ψ =] he: = [ Pr[ ψ, = ] Pr[ x =, ψ, = x = ]Pr[ x = ] + Pr[ x =, ψ, = x = 0]Pr[ x = 0]] = [ Pr[ ψ, = ] Pr[ x = ψ, =, x = ]Pr[ ψ, =, x = ] + Pr[ x = ψ, =, x = 0]Pr[ ψ, =, x = 0]] ψ =, x =] + ( (,, = Pr[ x = ψ, =] Pr[ ψ =, x = 0]] Pr[ x = 0 ψ =], ψ, ψ, [Pr[ x = =] + Pr[ x = 0 =]] = he equaly above follows se + (6 +, a hee Also, se he eve ourree roess s eee of sesor ava saes, we have use he followg equaly above: Pr[ x = ψ, =, x = ] =Pr[ x = x =] = Le eoe he umber of me slos whh he sesor was ave, oerag uer some saary oly Π urg me [0 ] Le L, eoe he eergy level of sesor a me he exee eergy level of sesor a

7 Mul-sesor ava for emorally orrelae eve morg me (assumg ha he sesor o lose ay harge ue o s eergy buke beg full whe a harge quaum arrve s gve by EL [, ]= L,0 + q [ δ + δp ], where P s he seay-sae robably P,, e: P, ψ =, P = lm (7, Noe ha, P P Se he sesor eergy level s always -egave, we have EL [, ] 0 Smlfyg, vg by, a akg he lm as, we have: q lm (8 δ + δp Noe ha f ay harge was los ue o he sesor eergy buke beg full, he fra he RHS equa (8 woul erease From equa (7, we have: lm xψ = lm Pr[ x = ψ =] ψ,,, = = = lm P, ψ, = = P herefore, usg equa (8, we have: Le lm xψ =, = lm x π = (9 P P = lm π Pq π δ + δp U B eoe he RHS equa (0 he: U P qδ = > 0 B π [ δ+ δp ] hus, U s a -ereasg fu of U B B q π δ + δ (0 P From equa (6, Now, he lemma follows: Lemma : he erformae aheve by ay saary ava oly Π s uer boue as: N q U( Π U ( π δ+ δ Proof: Le f a be measurable fus fe ae a se R Suose f a are egrable R, 0 a >0 If φ s vex a erval ag he rage of f, R he Jese s equaly (Wheee a Zygmu, 977 saes: φ R f φ( f R R R Reall ha eoes he umber of sesors he ave sae urg me slo Se U ( s ave, subsug φ = U (, f = a = x he above equa, Jese s equaly he sree-sae sae mles: U x ( U x = = x x = = Se U ( s uous, we have: U ( Π= lm U ( x = lmu x = x U = x = x U = = x = N x ψ, U = = x = N x ψ =, U = x = = lm = lm = lm = x = x, = lm x ψ, = U = Nlm x = N q U π δ + δ he las equaly follows from he fa ha all sesors are eal he las equaly follows from Lemma a se U ( s -ereasg 4 Performae of hreshol oles We frs aalyse he erformae of he algorhm Se 4 a show ha aheves ear-omal erformae for a aroraely hose hreshol he eergy-balag hreshol arameer m s suh ha whe hs hreshol s emloye, he average reharge rae equals he average sharge rae he sesor sysem For smly of aalyss, we assume fe sesor eergy buke sze (e K We also assume ha whe K, a a eergy-balag hreshol arameer m s emloye he sysem, he hreshol of m a be always me (e wh robably Laer, we jusfy hs assum by showg ha for fe values of K, he robably ha a eergy-balag hreshol s o me s of he orer o K ( Aex A hus, for suffely large sesor eergy buke sze K, he erformae aheve ψ

8 N Jagg a K Kar by he eergy-balag algorhm s auraely haraerse by he resuls Noe ha se he graulary of sharge, geeral, s of he orer of mll joules (or less (Jurak e al, 00, a he sesor s baery aay s of he orer of klo joules (or more, s reasable o assume ha K s suffely large We aalyse he erformae of he CP algorhm Se 4 4 me-vara hreshol oly m Cser a emloyg a hreshol of m, eoe Π Noe ha he algorhm wh arameer m s he same as a CP algorhm wh hreshol ar (m, m Cser a O ero of legh followe by a Off ero of legh he eve ourree roess From equa (, we have: E [ ] =, E [ ] = he exee amou of eergy gae by he sesors he sysem hrough reharge urg hs O/Off yle, eoe E, s gve by: Nq E = NqE[ + ] = π ( ( Assumg ha he hreshol of m s always me, he exee amou of eergy se by he sesors he sysem urg hs O/Off yle, eoe E, s gve by: [ + ] + δ [ ] E = mδ E m E mδ m δ = + ( π ( (3 he eergy-balag algorhm, eoe Π, emloys a hreshol suh ha he average reharge rae equals he average sharge rae he sesor sysem Se a O/Off yle forms a reewal erval for he eve ourree roess, he eergy-balag hreshol, eoe m, emloye by Π, a be erve usg he equa E = E hus, we have: δ δ m + δ + δ π = Nq m = δ + δ π ( π ( ( m Nq = Nq π (4 Assumg ha he ale hreshol s always me, he erformae aheve by he ava algorhm Π s gve by: U ( Π xu m ( ( = = lm = U m x = (5 * Le U eoe he uer bou o maxmum ahevable erformae for ay ava algorhm, gve by Lemma Also, le β = δ he, he eergy-balag algorhm δ Π aheves he followg erformae bou: Lemma 3: ( β + Nq U Π U U * = δ + δπ β + π Proof: Se U ( s a ave -ereasg fu, U( X U( Y we have for X Y Subsug, X Y = Nq N q X a Y = o δ, we ge: + δπ π δ+ δ X U( Π = U( X U( Y Y = U ( + ( δ + δπ π δ δ * β + β + π = * U (6 Noe ha X Y se π (beause + he values of X a Y above oul ffer subsaally whe he orrela robables are large Noe ha he absee of emoral orrelas, whe = = = π, from equa (6, Π aheves omal erformae * (equal o U Corollary : Π aheves omal erformae he absee of emoral orrelas From Lemma 3, Π aheves omal erformae as β I rae, se rasmss os ( δ s yally muh larger ha he sesg os ( δ, β = δ δ s exee o be suffely large hus, Π aheves lose o omal erformae I fa, from Lemma 3, * aheves erformae o U We beleve βπ ha he uer bou gve by Lemma s gh whe = = 05, a s loose oherwse Neverheless, he bou s suffely lose as see from resuls Se 43

9 Mul-sesor ava for emorally orrelae eve morg 4 Correla-eee hreshol oly Cser a CP emloyg a hreshol ar of (m,, m, eoe Π CP Noe ha he CP algorhm wh arameer m= s he same as he algorhm wh a hreshol of m We ly ser he eergy-balag hreshol ar ( m, suh ha whe hey are emloye, he emloye hreshol s always me (e wh robably Frs, we erve bous he erformae of eergy-balag CP oles a he omme he rage of eergy-balag hreshol ars ( m, Cser a O ero of legh followe by a Off ero of legh he eve ourree roess Reall ha he CP algorhm emloys a hreshol of m me slo f he eve ourree roess s kow o be he O ero, e, f a ala-sef eve was eee he revous me slo by ay of he ave sesors Oherwse, a hreshol of s argee me slo herefore, a hreshol of s emloye urg he frs me slo of he O ero Smlarly, a hreshol of s emloye urg he las - me slo of he Off ero, as ee Fgure 5 Durg he oher me slos, a hreshol of m or s emloye eeg u wheher a eve was eee or o urg he revous me slo Fgure 5 Noe: hreshols emloye by CP algorhm urg a reewal erval Se o eve was eee urg he Off ero, a hreshol of s emloye urg he frs me slo of he O ero hereafer, urg he O ero, a hreshol of m s emloye f he eve was eee urg he revous me slo Oherwse, a hreshol of s emloye Smlarly, urg he frs me slo of he Off ero, a hreshol of m or s emloye eeg u wheher (or o a eve was eee urg he revous me slo (he las me slo of he O ero hereafer, a hreshol of ges emloye urg he Off ero, se o eve ours (or s eee urg he Off ero Le s = Um, ( y = U( a z = s y Le u eoe he overall eve ee robably aheve urg me slo of he O ero he, u = y, a: u = yu( m + ( y U( = ys+ ( y y = y+ yz = y( + z Smlarly: u3 = uu( m + ( u U( = ys( + z + [ y( + z] y = y( + z+ z a so hus: ( u = y + z+ z + + z, [ ] (7 he me average eve ee robably urg he O/Off yle s gve by: u = y = [ ( z ( z z ( z z z ] 3 y z z z z = z z z z y z( + z+ z + + z ( z = y z = z ( z z y yz ( z = z z ( ( z ( ( z (8 From equa (, E [ ]= Se a O/Off yle forms a reewal erval for he eve ourree roess, he me average eve ee robably aheve he sysem s gve by: yz ( z m, y U ( ΠCP = z (9 y yz z Noe ha for he eergy-balag ar ( m,, m=, z = U( m U( = 0 m, a U( Π CP = U( I oher wors, m, Π CP reues o Π whe m= Nex, we ser he hoes a rages of he eergybalag hreshol ar ( m, he exee amou of eergy gae by he sesors he sysem hrough reharge urg he O/Off yle, eoe E, s gve by equa ( Le v eoe he exee amou of eergy se by he sesors urg me slo of he O ero he, v δ + δ, a: ( = ( ( δ δ v = um+ u + Smlarly: ( ( δ δ v = u m+ u +, [ ] (0 Le w eoe he exee amou of eergy se by he sesors urg me slo of he Off ero he, w = u m+ ( u δ, a w = w3 = = w = δ Assumg ha he emloye hreshol s always me, he oal eergy se by he sesors he sysem urg he O/Off yle, eoe E, s gve by: s

10 N Jagg a K Kar s = + = = = δ+ ( m δu + δ+ δ = + ( m δ u = E v w ( = δ + δ + δ + δ δ δ ( = + ( + ( m + u ( m u = δ δ δ y yz + ( m ( δ + δ z z ( m δ y z ( z ( z ( Usg equa (, he exee amou of eergy se by he sesors he sysem urg he O/Off yle uer, oly Π m, eoe E, s gve by: CP δ E = E[ E ] = + ( δ + δ + s ( yz z ( m y ( δ+ δ z z ( z ( m δ y z δ = + ( δ + δ ( ( m y δ + δ z + ( δz+ δ z z δ + ( δ + δ ( m y δ + δ δz+ δ + z z ( he las equaly follows se 0 z < he bes eergybalag CP algorhm maxmses he erformae objeve: m, m, ( CP maxu Π (3 subje o sras: E = E a m he above s a -vex omsa roblem whh s o smle o solve Hee, we fous fg a rage for he hreshol arameers m a, usg eergy balae Usg equa ( a E =E, we ge: Nq δ + ( δ + δ π ( ( m y δ+ δ δz+ δ + z z o smlfy he exress, le us roue he Nq followg sas Le =, ( π = δ a δ + δ 3 = he, we have: ( m y δz+ ( 3 δ z z Smlfyg, we ge: ( m y z ( δ + 3 z z Le (4 y z f( m, = δ+ 3 z z Se, hus: y z z f( m, y δ + 3 z + z = y δ+ 3 z = y δ+ 3 3 z δ+ δ = y δ+ 3 z δ + δ y δ + 3 y = U( ( δ + ( δ + δ 3 Usg equas (4 a (5, we have: ( ( m f( m, mf( m, ( f( m, ( + 3 m + f( m, ( + 3 m + U ( ( δ + ( δ + δ 3 (5 (6 Se he eergy gae he sysem s he same for boh he CP a he algorhms, a se he CP algorhm emloys a larger hreshol urg he O eros a a relavely smaller hreshol urg he Off eros, he

11 Mul-sesor ava for emorally orrelae eve morg hreshol mus be lower ha he eergy-balag hreshol m gve by equa (4 o sasfy eergy balae urg a reewal erval Base u he above aalyss, we have he followg resul: Lemma 4: Ay eergy-balag CP ar ( m, mus sasfy he followg equales: Nq = a m δ + δπ ( + 3 ( δ + ( δ + δ m + U ( 3, * erformae o U ereases a all values of hreshol he erease erformae s more vsble a hgher hreshol values hus, a rease he egree of emoral orrelas worses sysem erformae uer he algorhm Fgure 6 erformae he absee of emoral orrelas (see le vers for olours Nq where = π ( δ, = δ + δ a = 3 Nex, we rese smula resuls for a CP algorhms 43 Smula resuls We evaluae he erformae of varous ava algorhms usg sree-eve smula of he sesor sysem he smula oe s wre C rogrammg laguage he sysem arameers use are N =6, q =05, =, δ =, δ =4, =05 a K =000 Noe ha he arameers are hose suh ha he average reharge rae of a sesor s lower ha s sharge rae he ave sae Also, he eve ee a rasmss os ( δ s moelle o be hgher ha he ava os ( δ Exermes are erforme for a rage of ee robably ( 0 06, orrela robables ( 05, 099 a he rao of eve ee versus ava os ( β 8 Alhough resuls are resee for N =6sesors, we observe smlar erformae res for oher values of N as well he uly fu use s U ( = ( Fgure 6 es he erformae of he algorhm wh =05 a =05 (o emoral orrelas * U eoes he uer bou o maxmum ahevable erformae gve by Lemma Π orress o he erformae of he algorhm a varous values of he hreshol arameer From equa (4, he eergybalag hreshol m =4 Se = = 05, Π aheves omal erformae hs ase, as suggese by Corollary Nex, we roue emoral orrelas he alasef eve heomea wh = = x, x {06,07,08,09,099} Fgure 7 es he erformae of algorhm a varous values of hreshol I all hese ases, m =4 We observe ha he bes erformae s obae a m= m a sasfes he erformae bou gve by Lemma 3 I a, we observe ha as he egree of emoral orrelas reases, he rao of algorhm Fgure 7 erformae wh varyg egree of emoral orrelas (see le vers for olours Fgure 8 es he erformae of Π algorhm for varous values of ee robably, wh = = 09 Noe ha he uer bou U *, as well as he eergy-balag hreshol m, vares wh he eergy-balag hreshol m s o a eger hese searos, a he erformae for he eares-roue eger value s ee he fgure As reases, he eergy-balag hreshol m ereases, whle he erformae of Π aroahes he uer bou, as suggese by Lemma 3 Fgure 9 shows he erformae of Π algorhm for varous values of β wh = = 06 a =05 he hreshol m ereases wh a rease β However, for all values of β, Π aheves erformae >96% of * maxmum ahevable erformae (gve by U

12 Fgure 8 N Jagg a K Kar Π erformae wh varyg ee robably (see le vers for olours Fgure 0 CP erformae wh small ee robably (see le vers for olours Fgure 9 able Π erformae wh varyg β (see le vers for olours Performae for CP hreshol ars (m, m = 5 m = 6 m = 7 = = = Noe: he omal erformae s bol Π erformae s resee als Fgure 0 shows he erformae of varous CP algorhms wh = = 09 a =03 Noe ha whe m =, he CP algorhm orress o a algorhm, wh m= =5 orresg o Π As s rease from o 3, he erformae of CP algorhm reases a all values of m However, as s rease furher, he erformae of CP algorhm ereases, arularly a larger values of m Noe ha CP algorhm wh =0 s o feasble, as he sysem woul ever be able o ee a eve e he hreshol of s ale, a hee woul always rema he sae wh o sesors ave hereafer I s observe ha he erformae of CP algorhm wh ( m, ars (6, 4 a (7,3 s slghly beer ha he erformae of Π ( m =5 However, he erformae ga s <%, as able shows he erformae of CP algorhm wh 6 s observe o be serably lower ha ha of Π From Lemma 4, whe =4, eergy-balag ar ( m, shoul sasfy: 4 m 6, a whe =3, eergybalag ar ( m, shoul sasfy: 3 m 796 Boh he ( m, hreshol ars (6, 4 a (7,3 sasfy he s oule Lemma 4 Hee, whle evaluag CP algorhms, we shoul fous he rage of hreshols ( m, as gve Lemma 4 hs fa also suggess ha he eergy-balag CP hreshol ars erform beer ha oher CP hreshol ars Fgure es he erformae of varous CP algorhms wh = = 09 a greaer ee robably =05 I hs ase, Π orress o m= = 4 We observe ha he bes CP erformae s aheve for m =4 a =4, a hee Π aheves he bes erformae hs ase he CP erformae res for hs searo are ee more eal Fgure I Fgure a, he hreshol he ar ( m, s ke fxe, a m s vare from o 6 he x-axs Smlarly, Fgure b, he hreshol m s ke fxe a s vare he wo fgures orres o wo ffere D rojes (ross-ses of he 3D erformae lo ee Fgure For he fxe hreshol of, he value of m orresg o eak erformae ereases as s rease Smlar behavour s observe for he fxe hreshol of m as well We observe ha for fxe hreshol =4, he CP erformae reases wh a rease m from o 4, a laer ereases as m s rease furher Smlar erformae re s observe for fxe hreshol m =4, a for oher values of m a he erformae of CP algorhm wh ( m, ars (5, a (4,3 s also que lose o he bes ahevable erformae (however he erformae s slghly lower ha ha of Π From Lemma 4, he rage for eergy-balag hreshol ar ( m, s gve by: m 595 whe =, 3 m 466 whe =3 a 4 m 4 whe =4

13 Mul-sesor ava for emorally orrelae eve morg Noe ha here s o eergy-balag hreshol ar ( m, suh ha >4 We observe ha all he hreshol ars whh aheve erformae lose o omal belg he rage for eergy-balag hreshol ar erve Lemma 4 Noe ha all he above searos, he eak erformae of algorhm s aheve a he hreshol of m gve by equa (4 Also, he erformae mroveme, f ay, usg a CP algorhm over ha of Π s observe o be <% yal searos I a, woul be easer o emloy a me-vara hreshol of m whe omare wh a me-varyg hreshol ar ( m, real sesor ework eloymes hus, he algorhm wh a eergy-balag hreshol s suable mos raal searos Noe ha somemes, he hreshol m gve by equa (4 mgh o be a eger value I suh ases, a robabls ava sheulg oul be use For sae, le I be a eger suh ha I < m < I + he, eah me slo, a hreshol of I oul be hose wh robably a a hreshol of I + oul be hose wh robably ( suh ha he exee hreshol (a hee he seay-sae me average hreshol equals m Ru-me sass: I s worh og ha s o ossble o kee all he sesors ave, arularly whe a large hreshol s ale, ue o he eergy sras I he searo orresg o Fgure 0, we observe ha whe he eergy-balag hreshol of 5 s ale, fve sesors are ave for 9999% of he me However, whe a hreshol of 0 s ale, he fra of me sesors ha are ave for 0 0 s ee Fgure 3(a hus, emloyg large hreshol ly aems o avae a large umber of sesors, f ossble Iee hs exlas why he erformae urves are o ave for hreshol values Fgures 6, 7 a We also observe he eergy levels of a arbrarly hose sesor over a ero of me For Π, he sesor s eergy level s 0% of buke sze K for more ha 9% of he me, as show Fgure 3(b O he oher ha, wh a hreshol of 0, he sesor s eergy level s observe o be <0% of buke sze K for 9997% of he me, resulg he behavour show Fgure 3(a Smlarly, Fgures 3( a 3( e he ru-me sass erms of umber of ave sesors for CP hreshols of (6, 4 a (8, 6, resevely I boh hese ases, he sesor s eergy level s <0% of buke sze K for more ha 98% of he me However, he argee hreshol ar of (6, 4 s aheve more ofe ha he ar (8, 6 For he eergy-balag CP ar (6, 4, usg Fgure 5 a equa (8, he hreshol of 6 ges ale 486% of he me, whle a hreshol of 4 ges ale 574% of he me From Fgure 3(, hese argee hreshols are aheve almos always For he CP ar (8, 6, he argee hreshols are o aheve for a large fra of he me, as show Fgure 3( Noe ha eve hough he ar (6, 4 les he rage of eergy-balag hreshol ar gve by Lemma 4, he sesors se mos of her me wh suffe eergy hs suggess ha Π maas a more sable eergy level a he sesors ha he CP algorhm Fgure CP erformae wh large ee robably (see le vers for olours Fgure CP erformae res eergy-balag CP hreshol ars ( m, aheve ear-omal erformae: (a fxe hreshol a (b fxe hreshol m (see le vers for olours (a (b

14 N Jagg a K Kar Fgure 3 Ru-me sass erms of umber of ave sesors a sesors eergy levels a seay-sae: (a uer a argee hreshol of 0, (b uer eergybalag hreshol, ( uer eergy-balag CP hreshol ar (6, 4 a ( uer CP hreshol ar (8,6 (see le vers for olours (a Imlemea mehasm: We brefly esrbe a smle mehasm whh oul be use o mleme he hreshol oles ( or CP a srbue maer A he begg of every me slo, eah sesor exhages hello message wh all oher oes Eah hello message as he seer oe s l ( l [ N] a s urre eergy level E l Sesor wh l oul rasm m-slo l o avo hael e Afer all he messages have bee exhage, eah sesor sors he eergy levels of all oes eseg orer a heks wheher s eergy level s amg he o m eergy levels he sysem If yes (o, he sesor avaes (eavaes self urg hs me slo Here, m s he argee hreshol a orress o for algorhm I he ase of CP algorhm, he hello message also as a b o ae wheher he seer eee a eve urg he revous me slo f was ave Usg he bs reeve from all sesors, a base u he es rule for eve ee (Examles a Se 3, he sesors ee wheher a eve was eee he sysem If a eve was eee, he sesors arge a hreshol of m urg hs me slo, oherwse a hreshol of s argee m 5 Relae work (b ( ( Zhao a Ye (008, 009 ser he roblem of qukes ee mulle O/Off roesses, a usg a esheore framework, hey show ha he jo swhg a ee rule havg a hreshol sruure aheves omal erformae uer a ml Here, he swhg hreshol a he ee hreshol are hose aroraely base u he hael s a he esre ee relably Nyao, Hossa a Fallah (007 roose robabls slee a wake-u sraeges for a sgle solarowere sesor oe, where he objeve s o reue akerog a ake-blokg robables a suy her erformae as well as he hoe of omal arameers usg a game heore aroah L e al (005 rese a le roug algorhm mul-ho wreless eworks wh reewable eergy soures Here, eah oe he ework s assume o have he kowlege of s shor-erm eergy releshme sheule, a he roose algorhm s show o aheve a omeve rao whh s asymoally omal wh rese o he umber of ework oes Baerjee e al (007, Baerjee a Khera (007, Gazaas e al (008 a Nyao, Hossa a Fallah (007 have sere he yam ava ques he ex of eergy-harvesg sesor sysems Baerjee e al (007 ser he sgle sesor, a by moellg as a lose hree-queue sysem, hey oba Nor s equvale of he sysem o evaluae he sruure of he omal-rae rol oly Baerjee a Khera (007 ser he sesor ava roblem uer a sesor eergy moel smlar o Kar e al (006, a emsrae he omaly of hresholbase oles for a broa lass of uly fus a sae yams Gazaas e al (008 ser he resoure alloa roblem he resee of rehargeable oes,

15 Mul-sesor ava for emorally orrelae eve morg roose a oly whh eoules amss rol a ower alloa ess a show ha aheves asymoally omal erformae for suffely large baery aay o maxmum ower rao Ava sheulg a sysem of rehargeable sesors has also bee sere revously Kar e al (006 a Jagg e al (008 Kar e al (006 ser a sesor eergy moel where eah sesor oul be avae ly u omlee reharge of s baery, a a be eavae ly u omlee sharge Saal orrela s roue he reharge a sharge ervals of he olloae sesors a a eergy-balag hreshol-base ava oly s esge whh s robus o he resee of orrelas a aheves erformae greaer ha 75% of he omal erformae Jagg e al (008 ser sesor sysems where vual sesors oul be avae or eavae a all eergy levels Saal orrela s roue he reharge a sharge roesses of olloae sesors a a eergy-balag hreshol-base ava oly s esge whh s robus o he resee of orrelas a aheves asymoally omal erformae wh rese o he sesor eergy buke sze hs aer ffers from he above ha fouses he ueray moellg of he ala-sef eve heomea self sea of roug orrelas he sharge roesses (or ervals of olloae sesor oes I a, hs aer fouses he emorally orrelae aure of eve heomea, omare wh saal orrela sef reame Jagg e al (008 Reely, Jagg e al (009 have sere suh moellg of eve heomea for a sgle-sesor sysem a have esge algorhms whh erform lose o omal I hs aer, however, we ser sesor sysems wh mulle sesor oes a fous esgg algorhms o solve he mul-sesor ava ques uer emorally orrelae eve heomea I he sgle-sesor searo, he es of he sesor ly lues whe o avae or eavae a for how lg, whereas he ase of mul-sesor sysems, he es ques s a jo es amg he sesor oes he sesors make a olleve es o eerme how may sesors o avae a ay me, a hee he resuls Jagg e al (009 o o smly exe o hs searo I a, hs aer exlly sers a robably of ee for eah sesor oe, ras wh Jagg e al (009, where he sesor s ee robably s mlly assume o be 6 Summary a luss We have sere he mul-sesor ava ques for a reewable-eergy-base sesor sysem he resee of emorally orrelae eve heomea Parularly, we fouse hreshol-base ava oles a erve erformae bous for, whh emloys a sa, eergy-balag hreshol he sysem Assumg suffely large eergy buke sze K, he roose algorhm aheves omal erformae wh o emoral orrelas a ear-omal erformae he resee of emoral orrelas he eve heomea We also aalyse he erformae of CP algorhms, whh emloy a me-varyg hreshol he sysem, a erve feasble rages for eergy-balag CP hreshol ars he algorhm s smler, volves mmum sae maeae overhea a woul be suable o eloy mos raal searos Alhough, geeral, algorhm a lea o subomal erformae, he erformae mroveme gae by emloyg CP algorhm s observe o be sgfa he searos sere We have sere a frs-orer orrela moel o moel he emoral orrelas he eve heomea Oe of he fuure res woul be o exlore he exes of hese resuls o hgher-orer orrela moels, ossbly wh mulle eve roesses, a o show he omaly (or ear-omaly of hreshol oles suh geeral searos Aoher re of fuure researh lues he sera of false alarms geerae by he sesors urg he Off eros a fg a ava oly whh maxmses he ee of geue eves, whle keeg he false alarm rae low or wh a sefe bou Akowlegemes hs work was suore ar by Naal See Foua uer Awar , a by he US Army Researh Offe uer DEPSCoR ARO Gra W9NF Referees Baerjee, a Khera, AA (007 Sesor oe ava oles usg aral or o forma, Proeegs of he 5h Ieraal Symosum Moelg a Omza Moble, A Ho, a Wreless Neworks (WO, Arl, Lmassol, Cyrus, 7 Baerjee,, Pahy, S a Khera, AA (007 Omal yam ava oles sesor eworks, Proeegs of he Ieraal Cferee Commua Sysems Sofware a Mleware (COMSWARE, Jauary, Bagalore, Ia, 8 Borkar, VS, Khera, AA a Prabhu, BJ (005 Close a oe loo omal rol of buffer a eergy of a wreless eve, Proeegs of he 3r Ieraal Symosum Moelg a Omza Moble, A Ho, a Wreless Neworks (WO, Arl, reo, Ialy, 3 9 Ch, -L a Hu, YH (008 Omal arge ee wh loalze fus wreless sesor eworks, Proeegs of he IEEE Global eleommuas Cferee (GLOBECOM, November, New Orleas, LA, USA, 5 Fa, K-W, Zheg, Z a Sha, P (008 Seay a far rae alloa for rehargeable sesors ereual sesor eworks, Proeegs of he 6h ACM Cferee Embee Nework Sesor Sysems (SeSys, Ralegh, NC, USA, 39 5

16 N Jagg a K Kar Gazaas, M, Georgas, L a assulas, L (008 Asymoally omal oles for wreless eworks wh rehargeable baeres, Proeegs of he Wreless Commuas a Moble Comug Cferee (IWCMC, Augus, Cree, Greee, Hsu, J, Kasal, A, Frema, J, Raghuaha, V a Srvasava, M (005 Eergy harvesg suor for sesor ework, Proeegs of he5h IEEE IPSN Demsra, Los Ageles, CA, USA Jagg, N a Kar, K (009 Mul-sesor eve ee uer emoral orrelas wh reewable eergy soures, Proeegs of he 7h Ieraal Symosum Moelg a Omza Moble, A Ho, a Wreless Neworks (WO, Jue, Seoul, Korea, 9 Jagg, N, Kar K a Krshamurhy, A (008 Near-omal ava oles rehargeable sesor eworks uer saal orrelas, ACM rasas Sesor Neworks, Vol 4, No 3 Jagg, N, Kar, K a Krshamurhy, A (009 Rehargeable sesor ava uer emorally orrelae eves, Srger Wreless Neworks, Vol 5, No 5, Jag, X, Polasre, J a Culler, D (005 Pereual evrmeally owere sesor eworks, Proeegs of he 4h Ieraal Symosum Iforma Proessg Sesor Neworks (IPSN, 5 7 Arl, Berkeley, CA, USA, Jurak, R, Ruzzell, AG a O'Hare, GMP (00 Rao slee moe omza wreless sesor eworks, IEEE rasas Moble Comug, Vol 9, No 7, Kasal, A, Poer, D a Srvasava, M (004 Performae aware askg for evrmeally owere sesor eworks, ACM SIGMERICS Performae Evalua Revew, Vol 3, No, 3 34 Kasal, A a Srvasava, M (003 A evrmeal eergy harvesg framework for sesor eworks, Proeegs of he Ieraal Symosum Low Power Elers a Desg, Seoul, Korea, Kar, K, Krshamurhy, A a Jagg, N (006 Dyam oe ava eworks of rehargeable sesors, IEEE/ACM rasas Neworkg, Vol 4, No, 5 6 L, L, Shr, N a Srka, R (005 Asymoally omal ower-aware roug for mulho wreless eworks wh reewable eergy soures, Proeegs of he IEEE INFOCOM, Marh, Mam, FL, USA, 6 7 MroSra (003 Mrosra ws avy ra for self owere wreless sesor eworks (ress release Avalable le a: h://wwwmrosraom/ews/arle-9asx Mowa, R a Raghava, P (995 Raomze Algorhms, Cambrge Uversy Press, Cambrge Nyao, D, Hossa, E a Fallah, A (007 Slee a wakeu sraeges solar-owere wreless sesor/mesh eworks: erformae aalyss a omza, IEEE rasas Moble Comug, Vol 6, No, 36 Nyao, D, Hossa, E, Rash, MM a Bhargava, VK (007 Wreless sesor eworks wh eergy harvesg ehologes: A game-heore aroah o omal eergy maageme, IEEE Wreless Commuas Magaze, Vol 4, No 4, Norma, BC (007 Power os for wreless sesor eworks, IEEE Aerosae a Eler Sysems Magaze, Vol, No 4, 4 7 Paouls, A a Plla, SU (00 Probably, Raom Varables a Sohas Proesses, MGraw Hll Raghuaha, V, Kasal, A, Hsu, J, Frema, J a Srvasava, M (005 Desg seras for solar eergy harvesg wreless embee sysems, Proeegs of he 4h Ieraal Symosum Iforma Proessg Sesor Neworks (IPSN - Seal rak Plaform ools a Desg Mehos for Nework Embee Sesors (SPOS, 5 7 Arl, Berkeley, CA, USA, Vgoro, CM, Gaesa, D a Baro, AG (007 Aave rol of uy ylg eergy-harvesg wreless sesor eworks, Proeegs of he IEEE SECON, Jue, Sa Dego, CA, USA, 30 Wheee, RL a Zygmu, A (977 Measure a Iegral: A Irou o Real Aalyss, Marel Dekker, New York, NY Zhag, F a Chas, S (005 Imrovg ommua eergy effey wreless eworks owere by reewable eergy soures, IEEE rasas Vehular ehology, Vol 54, No 6, 5 36 Zhao, Q a Ye, J (008 Whe o qu for a ew job: Qukes ee of serum oorues mulle haels, Proeegs of he IEEE Mlary Commua Cferee (MILCOM, November, Sa Dego, CA, USA, 6 Zhao, Q a Ye, J (009 Qukes hage ee mulle - roesses, Proeegs of he IEEE Ieraal Cferee Aouss, Seeh, a Sgal Proessg (ICASSP (ve, ae, awa, Arl, Noes hs aer exes he resuls ha aeare WO 009, Seoul (Jagg a Kar, 009 Eve hough we have sere a Beroull reharge roess, he resuls hs aer oul easly be exee o he ase of orrelae reharge roess, se we ly ser he average reharge rae he aalyss (a mos ases, over a fe horz 3 Noe ha we ser a sgle-eve roess wh emoral orrelas hs aer he sera of mulle-eve roesses, wh ossble orrelas amgs hem, s bey he soe of hs work a oul be sere for fuure exess

17 Mul-sesor ava for emorally orrelae eve morg Aex A: Effe of bouary s erformae wh eergy-balag hreshol We show ha for a eergy-balag hreshol oly, he robably ha he emloye hreshol s o me s of he orer o K, where K s he sesor eergy buke sze For smly of exos, we erform hs aalyss for he ava algorhm Π Smlar resuls are exee o hol for a eergybalag CP algorhm as well Cser a O/Off yle of legh for he sesor sysem oerag uer he eergy-balag algorhm Π, wh hreshol arameer m= m Le R eoe he umber of eergy quaa reeve by he sesor sysem hrough reharge, a D eoe he umber of eergy quaa se by he sesor sysem urg hs yle From equas ( a (3, we have: Nq ER [ ]= π ED ( mδ [ ]= + m δ ( π ( M Le,,, eoe he leghs of M seuve O/Off yles Le ( R, D,,( RM, DM eoe he umber of eergy quaa reeve a se, resevely, urg hese M yles Noe ha R : M are (eee a eally srbue raom varables Smlarly, D : M are raom varables Also, R a D are eee for all : M Le us efe M raom varables, Z = D R, M E[ Z ]= E[ D] E[ R]=0, M Var[ Z] = E Z = E D + E R E[ D] E[ R] Se ED [ ] = ER [ ], we have: Var[ Z ] =Var[ D] + Var[ R], M M Z = Le Z = M Noe ha: ( Var[ D] + Var[ R] EZ [ ]=0; Var Z = M From Chebyshev s equaly (Paouls a Plla, 00, we have: Var Z Pr Z E Z ε hs mles: ε Var[ D] + Var[ R] Pr Mε Z ε (A Le us assume ha eah of he sesors ha a eergy level of K (or of he orer OK ( a he begg of he frs yle A he e of Mh yle, he hreshol of m wll o be me f a ly f a leas N m+ sesors have ru ou of her eergy a hs M M me, e, ly f D R >( N m+ O( K he above = = M eve s he same as he eve Z >( ( = N m+ O K or ( ( > N m + Z O K Usg equa (A, we oba: M ( N m+ O( K M(Var[ D] + Var[ R] Pr Z M ( N m+ O K ( (A Le us assume ha for some M, a he e of Mh yle, he hreshol of m s o me Le eoe he umber of me slos afer he Mh yle for whh he esre hreshol s o me ( δ + δ We have: E [] he oal umber of eves exee q o our M yles s gve by ME[ ] π herefore, he fraal loss erformae for he ava oly Π s ( δ+ δ Um ( δ+ δ he robably ha he hreshol qme[ ] U ( m qme [ ] M ( Var[ D] + Var[ R] of m s o me a he e of Mh yle s ( N m+ O( K from equa (A Assumg K N, he exee fraal loss uly of Π, eoe τ, sasfes: ( δ δ( D R ( [ ] ( + Var[ ] + Var[ ] τ = o N m+ qe O K K (A3 Se equa (A3 hols for all M, follows ha f K s suffely large, he eergy-balag hreshol of m = m wll almos always be me he erformae aheve by Π s U( m o K, from equa (5