Maximization of Data Gathering in Clustered Wireless Sensor Networks

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1 Maxmzato of Data Gatherg Clustere Wreless Sesor Networks Taq Wag Stuet Member I We Hezelma Seor Member I a Alreza Seye Member I Abstract I ths paper we vestgate the maxmzato of the amout of gathere ata a clustere wreless sesor etwork (WSN The amout of gathere ata s maxmze by ( choosg the optmal trasmt power a ( selectg the optmal cluster hea For problem ( we f close-form solutos for the optmal or ear optmal trasmt power of cluster members (CM For problem ( we propose a ear optmal cluster hea selecto (CHS algorthm The commucato bure a computatoal complexty of CHS oly grow learly wth the sze of the cluster I the propose algorthms teratos have bee avoe orer to sgfcatly lower the complexty of the algorthms compare wth tratoal terato-base umercal optmzato algorthms makg these approaches sutable for use eergy-costrae wreless sesor etworks The optmzato ga s show to be sgfcat I INTRODUCTION I the oma of wreless sesor etworks there has bee much research effort ame at effcetly utlzg the lmte eergy at the sesor oes ]] Numerous strateges have bee vestgate to mprove the eergy effcecy of WSNs clug power cotrol moble ata sk eploymet multple ata sk eploymet ouform tal eergy assgmet a tellget sesor a relay eploymet ] As aother opto to mprove the eergy effcecy of WSNs clusterg protocols have bee broaly aopte ue to ther effectveess a smplcty I clustere sesor etworks eghborg oes are groupe as clusters Oe of the oes a cluster s selecte as the cluster hea a the remag oes are the cluster members The cluster hea s usually charge of certa local cooratos such as collectg ata from the cluster members a commucatg wth other clusters a the ata sk whle cluster members smply trasmt ata to the cluster hea The cluster hea may be selecte a raomze maer such as HD 4] or LACH 5] Such a raomze selecto of the cluster hea combe wth rotatg the cluster hea posto ca effectvely avo the early ra of the eergy of a partcular oe However t caot guaratee the optmalty of the selecto Trasmt power s also a very mportat factor that flueces the eergy effcecy of WSNs from the physcal layer (PHY 6] Power cotrol techques ca be easly aopte clusterg topologes 7] Ths work was supporte part by the Natoal Scece Fouato uer grat # CS-4857 a part by a Youg Ivestgator grat from the Offce of Naval Research # N Fg A typcal cluster topology (N =5 The eergy effcecy of WSNs s usually evaluate by etwork lfetme Network lfetme ca be efe as the tme elapse utl the frst oe the etwork epletes ts eergy Ths efto of lfetme ca avo the stuato that certa oes have very hgh power cosumpto such as the commo relay oes of may routes urg etwork optmzato However a clusterg-base etwork ue to the ueve mportace of cluster members a cluster heas the efto of etwork lfetme ees to be mofe I ths paper we efe the lfetme of a cluster as the tme urato for a cluster to fucto properly e the tme elapse utl the cluster hea es or all cluster members e whchever comes frst I ata-cetrc applcatos however lfetme tself s ot as meagful a crtero as the amout of ata gathere urg the lfetme to evaluate the performace of a cluster Thus the goal of our propose optmzato s to maxmze the amout of ata gathere urg the lfetme of a cluster uer eergy a faress costrats The optmzato parameters are the trasmt power of the cluster members a the selecto of the cluster hea II TRANSMIT POWR OPTIMIZATION A Optmzato Moel Fg shows a typcal cluster topology wth 5 cluster members (CMs a cluster hea (CH I ths paper cluster members operate uer a faress costrat The faress costrat of a CM s efe as the costrat that the cluster hea shoul collect a equal umber of formato bts from each CM That s f every CM works for tme urato T the urg T each CM trasmts D bts of ata to the cluster hea Ths faress costrat s ue to the fact that the sesor oes a cluster are usually geographcally close a thus they observe the same pheomeo a geerate the same amout of samples Cosequetly they geerate the same amout of ata for trasmsso gve every oe 4 4

2 has the same source cog rate We assume that a suffcet amout of ata has bee geerate a store the source oes buffers before the commucato proceure begs Moreover ths paper a perfect tme vso multple access (TDMA scheulg s assume Therefore o multple access terferece (MAI s cosere ths paper Gve the faress costrat a Shao s chael capacty theorem we have P BT log( + P D { N } ( where B s the bawth P s the trasmt power of cluster member s the trasmsso stace from cluster member to the cluster hea s the path loss expoet P s the atve whte gaussa ose (AWGN power o the lk from oe to the cluster hea a N eotes the umber of cluster members Rearragg q ( the trasmt power of CM s P ( D BT P ( That s the trasmt power of CM s eterme by ts operato tme urato T a the total trasmtte ata D Moreover the eergy costrats at the CMs are T (P + P CT { N } ( where P CT eotes the crcut power cosumpto at the CMs a the trasmt powers are costrae by q ( At the cluster hea the eergy cosumpto comes from recevg the ata from cluster members Therefore the eergy costrat at the cluster hea s P CR T (4 where P CR s the crcut power cosumpto to receve ata B Problem Formulato The resultg maxmzato of the total gathere ata gve the eergy costrats a the faress costrat ca be formulate as m D st C : T > C : P CR T (5 C : T P + P CT ] C 4 : P ( D BT P where { N } I ths moel costrats C a C are the eergy costrats at the cluster hea a cluster members respectvely C 4 results from the faress costrat Our goal s for gve resual eerges a commucato evromets to f the optmal cluster member operato tme uratos T T T N ] a trasmt powers P that maxmze the total ata gathere at the cluster hea Problem (5 s a mmum feasble set test problem wth varables T P a D 8] Assume that the optmal soluto s D T a P where P =( D /BT P +ɛ ɛ > It s obvous that D T a ˆP where ˆP =( D /BT P + ɛ /ɛ > s also a optmal soluto as D T a ˆP satsfy the costrats Therefore we ca always have the optmal trasmt power P ( D /BT P from the rght ha se Thus we ca replace the equalty costrat C 4 wth equalty The resultg problem moel s m D st C : T > C : P CR T (6 C : T P + P CT ] C 4 : P =( D BT P The optmal solutos of the above problem are arbtrarly close to the optmal solutos of problem (5 We eterme a terato-free soluto to ths problem the ext subsecto C Trasmt Power Optmzato Algorthm Problem (6 s a typcal mmum feasble set test of a quascovex optmzato problem whch ca be realy solve through teratve umercal methos 8] However performg hures of teratos s prohbtvely complex eergylmte WSNs a a smple soluto wth few or o teratos s esrable I ths secto we propose a smple terato free soluto to f the ear optmal trasmsso tme uratos a trasmt powers of each CM a cluster Whe we fx the eergy of the cluster members ( a crease the eergy of the fuso ceter ( from zero to fty we observe that as creases the maxmum ata gathere at the CH creases utl t reaches a saturato lmt That s there are two regos of maxmum ata gathere D as creases amely the CH-costrae rego a the CM-costrae rego I the CH-costrae rego the cluster hea eergy s the lmtg factor the total amout of ata collecte whle the CM-costrae rego the resual eergy at the CMs becomes the lmtg factor Our goal s to f a terato-free soluto for T that maxmzes D both the CH-costrae a the CM-costrae regos After obtag the optmal T the correspog optmal trasmt power P of the CMs ca be easly eterme CH-costrae rego: I the CH-costrae rego the lmtg factor o the amout of ata collecte s the cluster hea eergy I ths case the cluster hea a all cluster members shoul use up ther eergy Otherwse ay cluster member havg eergy left cates that at least oe of the cluster members lmts the crease of the collecte ata a the cluster s operatg the CM-costrae rego Therefore both the cluster hea a the cluster member eergy costrats are actve That s T = P CR (7 T ( D BT P + P CT ]= (8

3 From (8 we have ( T P CT log + P a = D BT = T log ( T D P B = T log ( T P log (T D B (9 where W ( s Lambert W fucto ] Although the Lambert W fucto ca be calculate effcetly usg umercal methos t s stll prohbtvely complcate to calculate sesor oes However t s possble to further smplfy (6 whe P CT P I ths case T P CT l ( +P CT l l P P CT (7 b = T log ( P + 6 l where step a follows by assumg that the sgal-to-ose rato s much greater tha a /T P CT Stepb follows from Borchart s algorthm 9] whch proves the followg approxmato that s the learzato of ths fucto arou zero: T log (T 6T / l Therefore we have T N = T log ( P + l 6 N log ( P + l 6 = /(log (/ P+6/ l N /(log (/ P+6/ l Thus from (7 a ( we have T /(log (/ P+6/ l P CR N /(log (/ P+6/ l ( ( The above approxmato s accurate arou zero The optmal trasmt power for oe the CH-costrae rego ca be easly calculate by P = T P CT ( The maxmum collecte ata from each oe the CHcostrae rego s the D BT log( + P ( CM-costrae rego: I the CM-costrae rego the maxmum possble total ata gathere from each cluster member as s eterme by the mmum value of the followg sequece D = m{d D D N } (4 where the operator m{ } returs the mmum elemet of a sequece a D are the values of the followg ucostrae maxmzato problems: { } T P CT D = max BT log( + (5 P The above ucostrae maxmzato problems are coucte over T a ther aalytcal solutos ca be fou as T = W ( l (P CT P l P (P CT P P +P CT P (6 The D = m{bt log( + T P CT } (8 P where exact (6 or approxmate (7 values of T ca be use Wthout loss of geeralty assume D D { N } The we have D = D T = T (9 Moreover T { N } ca be ay values that satsfy the followg costrats: T ( D T P CR BT P + P CT ] = ( Aother mportat observato s that the CM-costrae rego a crease the cluster hea eergy caot crease the total umber of bts collecte from the CMs Thus oce the cluster eters the CM-costrae rego that s whe the eergy at the CH s greater tha a crtcal value Ê a crease the recever eergy becomes reuat The value of Ê ca be eterme by the followg equato Ê = P CR T ( where T = T as efe by (9 a { ]} T T = m arg BT log( + P CT P =D ( { N } where D s from (8 Clearly the resultg T { N } also satsfy the costrat set ( Base o the prevous aalyss the trasmt power optmzato algorthm that fs the ear optmal soluto to moel (6 s summarze Fg As show Fg the frst step s to eterme the coto of the cluster of terest (CH-costrae rego or CM-costrae rego; oce the operatg coto s eterme the results from sectos II-C a II-C ca be use rectly III CLUSTR HAD SLCTION I ths secto we propose a smplfe metho to etfy the optmal cluster hea whch oly volves the calculato of (4 a ( each terato The smplfe algorthm s base o a lear approxmato arou zero of the followg fuctos: D = BT log( + ( /T P CT /( P

4 No (CH-costrae Calculate T usg ( Calculate P usg ( Gve P P CT CR Calculate ˆ usg ( ˆ? Yes (CM-costrae Calculate T D usg ( (8 Calculate P usg ( The CHS algorthm ca be strbute as follows (assumg each oe kows the staces to ts eghbors: frst each oe uses oe broacast to form the other oes of ts resual eergy a each oe ca the f ts ow D through ( The oe wth the most resual eergy (assume t s oe broacasts to the other oes to eclare ts D ( The rest of the oes wll compare the receve D ( wth ther ow D (jj IfD (j >D ( the oe j wll otfy oe of ts D (j Otherwse oe j oes ot take ay acto At last oe wll compare the receve formato a use oe broacast to form the rest of the oes about the selecte cluster hea that proves the largest D Note that the CHS algorthm oly ees to be execute oce at the begg of the cluster formato Fg The trasmt power optmzato algorthm flowchart Frst we have the followg observato: D has a earlear relatoshp wth the cluster hea eergy the CHcostrae rego The learty s stregthee whe P CT P a T Proof: From ( the optmal T s a the eergy of the cluster hea has a approxmately lear relatoshp the CH-costrae rego Therefore to show that D has a ear-lear relatoshp wth the CH-costrae rego we oly ee to show the learty of D a T for { N }: D = BT log( + T P CT P = D = BT log( PCT P P ] + log( + ( P+PCT T a = D BT log( PCT P + log( P ] log(t + log( b = D BT PCT T log( PCT P P P CT ] + log( P CT + 6 l The approxmato a becomes accurate whe P CT P a the approxmato b becomes accurate whe T As show by the above ervato D a T have a ear-lear relatoshp whe P CT P a T Let D ( eote the lear approxmato of the maxmum ata collecte for the cluster wth oe as the cluster hea The proceure for the propose cluster hea selecto algorthm s straghtforwar: for each oe calculate D (; the choose the oe wth the largest D ( as the cluster hea I the propose cluster hea selecto (CHS algorthm D ( ca be expresse as { D Ê D ( = D D Ê (Ê < Ê ( where D ca be obtae from (4 a Ê ca be calculate from ( IV RSULTS I the trasmt power optmzato algorthm the ma result s the approxmato mae o the optmal operato tme assgmet T through ( the CH-costrae rego Compare wth the solutos of T the CM-costrae rego (whch are exact optmal solutos ( proves a ear optmal approxmato Therefore the effectveess of the approxmato ees to be evaluate We assume that 5 sesor oes are uformly place wth a sk wth a raus of m cetere at the cluster hea The path loss expoet s =4 The crcut powers are P CR =mw a P CT = mw The sgal bawth s B = KHz The AWGN power s 65 BmW a s equal o all lks Fg shows the maxmum ata gathere whe all CMs have raom resual eergy chose from a uform strbuto betwee J a 5 J The cluster hea has eergy varyg from to 5 J Ths setup guaratees that the cluster works the CH-costrae rego We compare four scearos: ( the optmal soluto to problem (6 through umercal methos; ( the propose aalytcal approxmato (terato free soluto; ( each oe has a equal trasmsso tme urato costrat wth umercally optmze trasmt power a tme urato; (4 each oe has a equal trasmt power costrat wth umercally optmze trasmt power a tme urato The maxmum amout of ata collecte s show Fg From Fg we ca see that the propose trasmt power optmzato algorthm proves a close approxmato to the umercal optmzato maxmzg the amout of ata collecte a CH-costrae cluster The propose trasmt power optmzato algorthm acheves a sgfcat ga compare to the cluster wth equal power a equal trasmsso tme urato costrats For stace whe = J the cluster usg the propose trasmt power optmzato algorthm gathers a average of tmes the maxmum gathere ata bts the equal tme urato case a 89 tmes the umber of the maxmum gathere ata bts the equal trasmt power case Also show the fgure s that the aalytcal approxmato becomes more accurate as the cluster hea eergy ecreases whch agrees wth the aalyss secto II

5 The performace of the propose CHS algorthm s evaluate a the results are show Fg 4 The resual eergy of the oes s geerate usg a raom varable s whch s uformly strbute betwee a through =s { N } (4 where N { } a the oes are place wth a sk wth a raus m followg a uform strbuto I Fg 4 the term lear approxmato meas that the propose CHS algorthm by ( s use to select the cluster hea; whle the term aalytcal approxmato refers to a brute force cluster hea selecto approach whch uses the propose trasmt power optmzato algorthm each terato to select the optmal cluster hea The term Optmal refers to a brute force cluster hea selecto approach whch uses a umercal soluto of moel (6 through a teror-pot metho each terato Moreover the performace of a raom selecto a a worst case selecto of the cluster hea s prove After cluster hea selecto the propose trasmt power optmzato algorthm s use to cofgure the trasmt powers of the CMs all cases Fg 4 shows that the propose CHS algorthm (lear approxmato proves a sou approxmato to the optmal cluster hea selecto terms of performace That s the maxmum amout of ata collecte by the cluster usg the CHS algorthm s almost as much as that of a cluster usg the umercally optmze cluster selecto whle the CHS algorthm avos the teratos the umercal optmzato Therefore the propose CHS algorthm has great potetal WSN applcatos Also show Fg 4 the aalytcal approxmato metho also proves a sou performace although compare wth the propose CHS algorthm t has slghtly worse performace a hgher complexty The propose CHS algorthm has a sgfcat performace ga over the raom selecto a worst case selecto For example whe there are te oes the cluster (N =9 the cluster usg the propose CHS algorthm wth lear approxmato ca collect 9 tmes the average umber of maxmum ata bts collecte by the clusters wth the raom selecto a 54 tmes the average umber of maxmum ata bts collecte by the clusters wth the worst case selecto V CONCLUSIONS I ths paper for trasmt power optmzato we propose a terato-free algorthm for maxmzg the amout of ata gathere by a cluster throughout ts lfetme coserg eergy costrats a a strct ata faress costrat Moreover we evelope a cluster hea selecto (CHS algorthm to eterme the optmal cluster hea that proves the largest amout of collecte ata for a cluster The performace ga by usg the optmal trasmt power a the optmal cluster hea selecto has bee show to be sgfcat RFRNCS ] I F Akylz W Su Y Sakarasubramaam a Cayrc A Survey o Sesor Networks I Commucatos Magaze pp 4 Aug D (bts 45 x Aalytcal approxmato Optmal qual tme urato qual power 4 5 (J Fg The maxmum ata gathere by clusters operatg the CHcostrae rego D (bts 6 x Optmal Lear approxmato Aalytcal Approxmato Raom selecto Worst case 5 5 Number of oes Fg 4 The maxmum ata gathere by clusters wth fferet cluster hea selecto strateges ] A J Golsmth a S B Wcker Desg Challeges for ergy- Costrae A Hoc Wreless Networks Wreless Commucatos I Aug ] C Zhao M Perllo a W Hezelma Geeral Network Lfetme a Cost Moels for valuatg Sesor Network Deploymet Strateges I Tras o Moble Computg pp ] O Yous a S Fahmy Dstrbute clusterg a-hoc sesor etworks: A hybr eergy-effcet approach Proceegs of the r Aual Jot Cof of the I Computer a Commucatos Socetes (INFOCOM 4 5] W Hezelma A Charakasa a H Balakrsha A Applcato-Specfc Protocol Archtecture for Wreless Mcrosesor Networks I Tras o Wreless Commucatos pp Oct 6] C S Taek a A Golsmth Degrees of Freeom Aaptve Moulato : A Ufe Vew Proceegs of Vehcular Techology Coferece Sprg I (VTS 7] S Cu A Golsmth a A Baha Jot moulato a multple access optmzato uer eergy costrats Proceegs of Global Telecommucatos Coferece I (GLOBCOM 4 4 8] S P Boy a L Vaeberghe Covex Optmzato st e Cambrge UK: Cambrge Uversty Press 4 9] RWDoerflerDea Recog: Calculatg wthout strumets Housto: Gulf Publshg Compay 99 ] M Wrght Soluto of the quato ze z = a Bull Amer Math Soc vol 65 pp