Dependable k-coverage algorithms for sensor networks

Inst rument ati on and Measur em ent Technol ogy C onferen ce – I MT C 2007 War saw, Pol and, May 1- 3, 2007 D e p e nda b l e k- c ov e ra ge a l g or i t hm s f or s e ns or ne t w or ks Gy ul a Si m o n 1,3 , Mi kl ó s M ol ná r 2 , Lá s zl ó Gön czy 3 , Ber nard Co us in 2 1 De pa rtme nt o f Co mpute r S cie n ce , U nive rsity of Pan non ia, H -8200, Veszp rém, Egy ete m u. 10, H un gar y 2 IRIS A, Cam pus de B eaulieu, 35 042 R ennes Cedex, F rance 3 Dept . of Me asureme nt and I n fo r m ation Sy stems, B udapest U ni ve rsit y of Technolo gy an d Ec onomics H -1117 B udapest, XI. Magy a r tudóso k krt. 2., H ungary Emai l: {simon ,gonczy }@m it.bme.hu , {molnar , bcousin} @irisa.fr Abst rac t – Re dunda nt sensing capa bilit ies are often re qu ired in sensor netw ork a pplic ation s due to var io us re asons, e.g . robu stnes s, fault tolera nce , or increa sed ac cura cy. At the same time high se nsor redundan cy offe rs the p ossib ility of i ncreas ing netw ork lifet ime by schedul ing s lee p interva ls fo r som e se nsors and stil l prov idin g contin uous se rvice w ith help of the re main ing ac tive sensor s. In th is paper ce ntral ized an d d istribu ted alg orithm s are p ropo sed to solv e the k- covera ge sensi ng p roble m and max imi ze ne tw ork life time . W hen phys ically po s sib le, the propo se d robus t Con trol led G ree dy Slee p Algori thm prov ides gu ara ntee d service indepe nde ntly of n od e and commu nication e rrors in the ne tw ork. The perf orman ce of the algorithm i s illustr ated and compa re d to resul ts of a rando m solution by simu latio n examp l es. Keywor ds – senso r netw or k, k-cove rage, de pendable , s lee p- schedul ing. I. INTRODU CTI ON Wir eless sen s or n et work s ar e co n str uct ed fr o m s m al l, au t on om ous s en s or s an d ar e u t il iz ed f or va r i ou s m ea sur em ent purpos es . Th e individual pow er capacity of th e senso r s is v e r y limite d (us i ng ba tte ries it m ay be only some day s) b ut the e xpe c t ed us ef ul lif e time of the netw ork is requi r ed to be i n t he r a nge o f w ee ks o r mo nt hs , d e pe ndi ng o n t h e appl ic at io n. To ac hie v e netw o rk longe v ity l ow dut y cyc le o pe ratio n is util ize d . I n co nti nuou sly operat ing s e nso r netwo rks r edu ndant se nsors are deploy ed from w hich only a little sub se t is activ e at a time ; t he maj o r par t o f senso r s is turned o ff an d t h us sav es energy . In redu n da nt de nse se nsor netwo rks vario us sc h edul ing algo r ithms a r e use d to c o ntr ol en er gy con ser va t i on. In sen sor n etwor ks u sed t o a ust er ely mon it or an area in spac e it may also be a requ i reme n t t hat m ul tip le se nsor s be ab le to prov ide me as ureme n ts f rom e ac h po int i n spac e . T h is pr opert y m a y ei t h e r be n ecessa r y be ca use o f t h e a p pl i ed mea s ur em en t tech n o l og y, sa fet y or p er f or man ce r eason s or to satis fy accu racy requ ire me n ts w ith re lativ ely low -quality sen s or s. Hi gh red un dan cy p r es en t in th e n et wor k i s n ecess a r y to ac hiev e this go al. I n gene r al t his c lass o f p rob lems can b e tr ea t ed as th e k-cover ag e pr oblem , wh er e cover ag e mean s the ab ility of a se nso r to pe rfo r m me as u reme nts o v e r a ce r tai n ar ea . Wh il e in a d en se sen sor n et wor k th er e may b e s ev er al eq ually goo d so lutio n s to the g enera l k-co verage prob l em, th e en erg y con ser vat ion cr it er i on narr ow s t h e r ange o f th e accep t a bl e solu t i on s. In ma ny appli catio ns a r equi red co verage is s atisfacto ry as wel l, so lar ger c over ag e d oes n ot pr ovid e h i gh e r p er for m an ce. Thus f indi ng a n ‘ec o no mical ’ s o lutio n to t h e k-c ove rag e pro b lem w ith s m all n umb er of part icip at ing nod es at t he s am e time re su lts ene rgy conse rv atio n and thus lo nge r netw o r k lif e time as w ell. Ef f ectiv e sc heduli n g al go r it hm is requ ired t o organize t he alte rnation of active (awake ) an d sleepi ng sensor sets to p rov ide continuo us se rvice of t h e n etw o rk. In this paper new r ob ust algori thms a re pr opo se d to provide depe nda ble k-cove r age and pr olo nged netw o rk lif e time. I n Se c tio n II t h e m ai n p rev ious res ults a re sum mar ized . Sect i on I I I intr od uces a cen tr a li z ed al g ori thm and its f ully distrib ute d v ariant (C o n tro lle d G r ee dy S l eep Algorit hm) to prov ide rob ust k-cove ra ge w hile minim izing th e n um ber of a wak en sen sor s at the s a m e tim e. In Sect i on IV n ew qu a li t y of s er v i ce m et ri cs ar e intr od uced an d sim ula ti on resu lts a re p r es ente d to illus t rate the c apab ili ties of the pr opos ed a lg or i t h m s . II. PREVIO US R ES ULTS Because of th eir fragi lity and power de priv atio n th e dependabil ity of sensor networks is an import ant and ho t resea rch topi c. Th ere are se ver al pro pos it ions to ens ur e fau lt tol eran ce a t diff erent le vels [ 1]. S inc e th e sen so rs ar e performing bot h sensi ng and co mmuni cation tasks the m ain problems of s ensor n etworks are assoc iated to these tw o activities [14] . The sensor n etwork s hould be capable o f taking me asurem ent in the obs erved ar ea and t rans mitting t he measured va lues to sink nod es. The k-cover age prob lem is ass oci at ed to th e measu r emen t fu nc tion ality [1 0 ]: ev ery po in t of the target are a mus t be co v ered b y at le ast k se nsors ( k is determined by t he applicat ion). I t also has se vera l imp li catio ns to co nn e ctiv ity iss ues [ 8]. The life-tim e of the network is ge neral ly prolonged by scheduling sleep int ervals for s ome sensors, meanwh ile th e continuous se rvice is provided by the active sensors (s ee examples in [5] , [6], [7]). The lifeti me long evity and th e netwo rk o perab i lity req uir e effic ient tra de- o ffs , re al ized b y different sch eduling algori thms, wh ich can m ainly b e divid ed into two main groups : random a nd coordin ated schedul ing algorithms [2] . A distributed, r andom sle eping algorit hm was proposed in [3] wh e re nodes m ake local dec isions on wh ether to sleep or to jo in a forwar ding ba ckbone, to ensur e communic ations. Ea ch node bas es its decision on an estimat e of how many of its neighbours w ill benef it from its bein g awake and the amount of e nergy available to it. In [4] the authors propos e a rando mized, si mple scheduling for dense an d mostly sle eping sensor netwo rks . They suppose that th ere are many re dund ant sensors in t h e targ et ar ea and on e c an compu te the requir ed ( ide nti ca l) d ut y cycle for ind ividual s ensors. In th e propos ed Rando mized Independent Sl eeping algor ithm, time is div ided into peri ods . At t he be g i nni n g o f ea c h pe r i od , ea c h se n s o r de ci d e s whe t h er to go to sleep (wit h prob ability p co mpu ted fr om the d ut y cyc le) or no t, th us the l ife ti me of th e netw ork is in cre as ed b y a factor clos e to 1 /p. This solut ion is v ery simpl e and d oes not require communi cation betwe en sensors. The draw back of th e proposition is tha t ther e is no gu arant ee for coverag e nor fo r network conn ectivity . Furthermor e, since the s leeping fac tor is the s ame fo r al l sens ors , this so lu tio n c anno t adapt to inhomogeneous or mobil e sensor set ups. To handle the basic covera ge problem th e authors in [2] propose a Role-A lternating, Cov erage- preservin g, Coordinated Sle ep Algorit hm (RACP) . Each sensor sends a message p eriodically t o its ne ighbourhood con taining it s location, res idual energy and other control in form ation. A n explicit acknow ledgm ent-base d election a lgorith m permits t o decide the sleep /awake status . The coordin ated sleep is m ore robust and red uces th e duty cycle o f se nsors c ompare d to t h e random sleep algorit hm, and i t guaran tees 1-cover age in th e network. In this solut ion the topolo gy can a ffect th e behaviour; thus the sensors can adapt their s leeping t o t he needs . The p rice of the pe rf or man ce is th e sig nif ic an t communic ation overhe ad increasing power consu mption. In [9] t he asy mptotic be ha vio r of cove ra ge in la r ge-sc ale senso r netwo rks is studied. For the k - cov er a g e pr oble m , for m ul at ed a s a d e ci s i on pr obl em , pol yn om i al -t ime al g ori th m s (in ter m s of th e n um ber of sen sor s) ar e pr esen ted in [10]. A c ompre h e nsive s tud y on bo th cove ra ge and con n ect i vi t y i ssu es ca n b e foun d in [1 1]. In this pape r a new coo r dinated a lgorithm is p ropo se d th at is a ble t o g uar ant ee k - cover age in th e n et wor k wh er e it is phy sically pos sib le and a t t he s ame ti me c a n pro v ide pro lo nged netw o r k lif etime . The pro po s e d algori t hm take s into acc ount bo th th e po we r status a nd the se nsing ass ig nme nt o f the s e nso rs. Fi rst , a c e ntrali ze d s o lutio n w ill b e pro po se d, w hic h will b e app roxima te d b y a dist rib ute d algo rit hm . T he la tte r so lutio n is mo r e f easib le in pract ic e due to its lo w commu n ica tio n o ve rh ead . III. SCHEDULING ALGORIT HM A. Background The se nsing assig nment in a se n sor netw ork can b e r epr es en t ed by a bi par tit e gr aph () E S R G , ∪ , wh er e th e two di sjoi n ts set s of ver ti ces r ep r esen t th e nodes S and ge og r aph ic al r eg i on s R (see Fig. 1). T h e regions are def ined by th e s ubs et o f sen s or s th at can mon itor th em. Gen e ra l ly , th e y cover th e whol e m ea sur ed ar ea an d ar e di sjoin t. In G th er e is an edg e e bet ween reg i on R r ∈ and sensor S s ∈ if and o nly if s (co mpletely ) cov ers regio n r . The simp le k-c overage probl em is to find a su b -grap h () E S R G ′ ′ ′ , ∪ wher e S S ⊆ ′ so th at for al l ver t i ces R in G ′ th e d egr ee i s at l ea st k . Th e min imal k- coverag e prob lem is to f ind a non- redundant sub- gr a ph G ′ that so lves the k -cover a g e pr oblem . A gra ph G ′ is no n-red u nda nt if there e xis ts no () E S R G ′ ′ ′ ′ ′ ′ , ∪ an d S S ′ ⊂ ′ ′ tha t so lves the sim ple k - cove ra ge p rob lem. Fi g. 1. An exampl e of the targ et are a c overe d by fo ur sens ors (s 1 , …, s 4 ). Th e sen s in g di sk s and t he se n si n g r e gio n s (R 1 , … , R 11 ) are al so show , alon g with th e c orr e sp on di ng bi pa rt it e gr a ph B. Assump ti ons Fo r the s ake of simplicity we as s ume that t he c o ve rage o f each sen sor can be m odel ed by a sen sin g di sk an d a co rr espo nding sensing rad ius (w e use a constan t se nsing radiu s bu t it’s no t es se ntia l) . Wit hi n its se n si ng radi us a se nso r is ab le to pe rfo r m meas u re me nts , w hile o uts ide t he sensing ci rcle th e se nsing perfo rman ce may degrade (but not n ecess a r il y). It ’ s a l so a ssu m ed th at th e comm un i ca ti on r a diu s i s a t lea st tw ic e of the se nsi ng r adius . In mo s t prac tic a l cas e s it’ s a se nsib le assump t io n and it au to mat ically pr o v ide s netw o rk- w ide communication if 1-c ove ra ge in sensing is prov ided [7]. Ge neral ly , f rom t he as su mptio n it a lso follo w s that ne tw o rk co n nectiv ity is highe r th an k w hen se nsing k-co ve rage is provide d [8]. s 2 s 3 s 4 R 1 R 2 R 3 R 4 R 5 R 6 R 7 R 8 R 9 R 10 R 11 s 2 s 1 s 1 s 3 R 1 R 2 R 3 R 4 R 5 R 6 R 7 R 8 R 9 R 10 R 11 s 4 C. Th e cen tral ized k- cov era g e a lgo ri thm In t his v e rsio n a c e nt ral sc hedule r un it is c h arge d w ith t he co n trol o f awake /doze s tates o f se nso rs. The sc hed u ler coll ect s t h e sen sor s t a te in forma t i on p eri od i ca l ly, ca l cu l at es a specia l dr ow sin ess fact or f or ea c h sen so r and send s to sleep a su bs et o f s en sor s, d ep e n din g on th e ir dr owsi n es s fa ct or . The algo rit hm is the f o llow in g: 1. Run th e n et work for a p er i od of T 2. Wak e u p all s ensor s an d coll ect st a t e in for ma t i o n 3. Com pu t e dr owsin ess fact or for ea ch n ode. 4. Se le c t the nod e w it h t he la rge st po sitiv e d ro w s iness f ac to r. Se nd t his nod e to sl ee p. 5. Rep ea t St eps 3-4 whil e p ossi bl e (i . e. th er e is at l ea st o ne nod e w ith p o sitiv e d row s iness f ac to r). Th e dr owsi n es s fa ct or of a n ode s wi t h c ur r en t en er g y E s is def ined as ()      − ∀ > Φ Φ = ∑ ∈ ∀ otherwise 1 0, if , 1 r s r E D r R r r s s δ α , (1) wher e ()    = otherwise 0 and between in edge an is there if 1 , r s G s r δ (2) An d α is a p o sitiv e co nsta nt (e .g. 2 = α ), and r Φ is the cove ra ge ratio of region r de f ined as f o llo ws :      − > − = Φ otherwise 1 if 1 K c K c r r r (3) wher e c r is the d egr ee of n ode r in G. T he co verage ratio r Φ is po sitiv e if the regio n is o v e r-co v ere d, i.e . mo r e t han k sen sor s co ve r r eg i on r . r Φ is ne ga tiv e if regio n r is not o ve r- cov er ed : in thi s ca s e th e op er a ti on of a l l s ens or s co ver in g r is es s entia l. The d r ow sin ess fact or D s takes into acco unt the e nergy of sen sor s : th e sma ll er th e e n erg y of a sen sor th e lar g er i ts dr owsiness. Nega ti ve dr owsin ess in d icat es th at th e sen sor is not al lo w ed to sle ep. A sen sor p art i ci pa tin g in man y r eg i on s th at ha ve l ow ov er - co ve rage is lik ely to par ticip ate i n mo re po ssib le so lutions th an s en s or s co ver in g r e g i on s al s o c o ver ed b y m an y ot h er sen sor s. Th us a h eur is ti c pr op er t y is in cl u de d in D s to inc rease the l if e time o f the netw ork: s e nso rs p art icip a ting i n regio ns o nly slig htly ove r-c ov e red hav e large r dro w siness . Th e d r owsi n ess fact or for ea ch sen sor in cl u d es th e s um of t he co ve rage ratio s o f the regio n s the se nso r is ab le to ob s erve . Th i s pr o per t y en f or ces t h e sen sor s in cri t i cal p ositi on s to go to sl eep wh en ever it i s p ossi ble, t o con ser ve th eir e n er g y for times w h e n t hei r pa rtic ipa tion w ill b e co me inev itable . Al th ou gh th e cen tr al iz ed a l g ori th m sol ves t h e k -co v er a g e pr ob lem an d con ser ves t h e en e r gy o f th e n et work a t th e same time , it ass ume s n etw ork-w ide inf o rmatio n dis t ributio n. In a large netw ork it w o uld mean ex ce ssiv e amo unt o f me ss ages transfe r and thus t he so luti on w ould impose an impo r tant com m u n i cat i on over h ea d. In st e a d, an appr oxi mat ion of t h e algo rit hm is p ro po se d that use s inf o rmatio n loc ally av ailab l e in th e n ei gh borh ood o f a n ode . D. Th e d istr ibu te d k-co vera ge alg or ithm The f o llow ing r o b us t, f ault to le ra nt, dis trib ute d a lgor ith m so lves the k-c o verage p r ob lem us i ng lo cally availab le in for m at i on onl y an d th u s it s com m un i ca ti on over h ead is lo w . The algo rit hm is b ase d on t he f o llow in g o bs erv atio ns : To pe rfo r m app r ox im ately the s ame sche dul i ng as i t w as sh own in the cen tra li z ed a l g o r ith m, a sen sor s can go t o sl eep if its ne ig hbo rs w ith la rge r dro ws ines s f acto r dec ide d the ir st a t e for th e n ext per i od an d s has no cr it i cal (n o t ove r co ve red) regio n to mo n ito r. Fo r this , e ac h se nso r s hou ld kn ow th e d r owsi n ess fact or o f t h e n eigh bor s an d th e d eci si on of n ei gh b or s with lar ger fa ctor . To min im iz e th e loca l comm u ni ca ti on , a comm un ica ti on d ela y (ST D) ca n b e ass o c iate d w ith eac h se nso r. T his de lay is inve rse ly pr opor ti on a l with th e dr ow si n es s fa ctor . So t h e s en sor s wit h lar ge fact or b r oa dca s t th eir deci si on ear li er. O nl y th e a wa ke state de c is io n s h ou ld b e b road cas te d e xplic itly , in t his w a y th e com m un i cat i o n over h ead can be m ini ma l. Controlled G re edy S leep (CG S) A lgorithm 1. Run th e n et work for a p er i od of T 2. Wake up a ll se n sors 3. No des w ith en e rgy enough f o r at least o ne mo re pe ri od b road cas t lo c al H e llo me ss age s c ontain ing nod e loca t i on s. Ba sed on recei ved H el lo m essa g es n o de s b uilds up its lo cal se t o f alive nei g hb o r no de s ( S s ) wit h t hei r loc atio n s. 4. Each n ode s calcul ates its dr ow sin ess factor D s (see E q . 1 ). In st ea d of t h e gl oba l gr aph G us e t he lo c ally kn own s u bgr aph () s s s s E S R G , ∪ . R s and E s are def ined in Not e 1. 5. Base d o n D s each n ode sel ect s a Sh out Ti m e D ela y (ST D ). S ma ll dr o wsin ess mean s lar g e ST D, lar ge dr owsin ess mean s small STD. 6. Each n ode s b roadcasts i ts S TD s an d star ts coll ecti n g oth er n odes ’ A wak e Messa g es ( AMs ). Fr om the r ecei ved AMs each node buil ds a List of Awake Nodes (LAN). 7. After STD s eac h node s m ake s a de cisio n based upo n the r ecei ved A Ms: − if the k-co ver age p r oble m can be so lved us ing o nly n odes pr esen t in th e LAN an d n odes wi t h S TD lar g er th an ST D s th en go t o sl eep − oth er w i s e stay aw ake an d br oad c a st an A M t o in for m o the r no d e s. Note 1: For each n ode t h e co ver ed r eg i on s ar e r epr esent ed by a set of squares, as illust rated in Fig. 2. (I n the e xamp le th er e ar e 24 r egi on s, whi ch can be incr ea sed t o pr ovid e bett e r app ro xim at io n). L e t’s de note t he no de ’s lo catio n by ( x s , y s ), an d th e cen ter o f r egi on i by () i i y x ∆ ∆ , in t he no de ’s lo cal co o rdina te sy s te m. As a pes simis tic ap p rox i matio n, in G s th er e is an edg e bet ween n ode w p l a ced a t l oca t i on ( x w , y w ) an d r egi on i if () ( ) 2 2 2 w i s w i s y y y x x x R − ∆ + + − ∆ + > ∆ − (4) The sq ua re mo de l may s eem a rude appro xim at io n of th e se nsi ng d isk , but s i nce t he s e n si ng dis k mo de l is i nhe r en tly a rat he r impe rf ec t estimat io n o f the rea l s ensi n g a rea o f a se nso r, the r e is no p o int to p ut grea t effo r t to ac curately app ro xim ate i t. T he p ro po se d solutio n is pe ss imist ic, i. e . cer t ain area s ma y be un n eces sa r ily ov er - co ver ed. Fi g. 2. Possib le app roxim atio n of the se ns ing dis k of a senso r s by a s et o f re gi ons R s No te 2: Eac h n od e go es t o slee p in a g ree dy ma nn e r: if the cove ra ge p rob lem can be solved with th e alre ady known a wa k e n ei gh bor s ( wi t h hi gh e r dr owsi n ess fa ct or ) in th e LA N (these nodes already ha ve dec ided o n their sle eping st atus) and s o me of the neig h bo r s w ith low er drow sines s fac tor (t hes e no de s w ill de c ide their sle e p/aw ake s tatu s late r) the n th e n ode gr eed i l y el e cts t o sl eep. No te 3: The node s make t heir dec isio n base d upo n rece iv ed H e llo, STD , and Aw ake me ssages . T h us th e algo rit hm is i nsensi tiv e to lo st me s sage s in the se n se th at th e cove ra ge is alw a y s prov ided (if and where it is po ssib le). Na t ura ll y, c omm u n ica ti on p r oblem s m a y ca u se n odes t o st a y aw ake un nec es s arily and thus t hey sho rte n the lif etime of the netw ork b ut do not af fec t t he q uality of s erv ic e . No te 4: T he e le ctio n algo rit hm to lerate s no de f ailures as we ll. T he o nly situ atio n a f aili ng nod e ca n ca us e p rob lem is w hen t he nod e die s rig ht af te r tra n s mitti ng its S TD me ss ag e . In t h i s ca s e n od es wi t h h i gh e r dr owsi n ess fa ct or ma y in corr ectl y rel y on th e pr esen ce of t h e fa il ed n ode. No te 5: The c o mmu nicat io n ov erhead o f the algo rithm is low. In each cycle e v er y n ode br oa dca s t s on l y at most thr e e mes s age s ( tw o if it t he no de w ill go to s le ep , th ree o the rw is e). In ad ditio n to t his , nod es mus t st ay aw ake in o rde r t o co mplete the e lec tio n p ro ce ss. Du ring t his e xtra T e time n od es con su m e en er gy . Th e com m un i cat i on an d a wake- t ime over h ead can be n eglect ed i f T is si g nif ica n tly lo nger t h an T e , w hich is t rue i n mo st p rac tic a l c ase s . IV. RESULTS A. Quality of Ser vice M e trics: The f irst (f un ct io nal) requi re ment is co v e rage: t he netw o rk must maint ai n k-co ve rage at the la rge st p o ss ib le part o f the ar ea . Th e sec on d (n on - fun ct i ona l) req u ir em en t is th e long lif e -time o f the ne tw o rk. T he pe rfo r ma nce o f the netw ork c a n be c har acte rized w ith the size of th e full y covered regions. A po s sib le me t ric k Θ can be t h e k-cov erage-rati o def i ned as A A k k = Θ , (5) wher e k A is t he are a of t h e k-c ove red regio ns a nd A is the tot a l ar ea o f th e tar get s pace. I f th e r egion s ha ve app ro xim ately t he s ame size t he n a no the r si mila r me tr ic k Θ ′ can b e de f in ed to a pp roximate k Θ : N N k k = Θ ′ , (6) wher e N k is t h e n um ber of t h e k - cove r e d r egi on s wh il e N is th e t ot al num ber of r eg i on s in th e tar get sp a ce. In a c ritic a l app lica tio n t he k -c ov e rage mu st b e mai ntai ned as lo n g as po ssib le and w it h k Θ a s h i g h a s p o s s i b l e . I f i n a degrading netw o rk k Θ is no t sat is f ac to ry any more it m ay still b e impo rtant to ma i ntai n a hig h v alue f or 1 − Θ k , 2 − Θ k , etc , e.g. i n o r de r to p r ov ide fu ll co n nectiv ity in the re mai ni ng netwo rk. Th e k- life tim e ) ( λ k L of a n etwor k c an be d efi n ed a s th e max imu m o pe rat io nal ti me o f the netw ork w ith λ > Θ k , wher e 1 0 ≤ < λ (c lo s e to 1 in p ract ic e ). B. Sim ula tion resu lts The propo sed CGS algorithm and the random k-co ver age algo rit hm [ 4] w e re simu late d i n P r ow ler, a p r ob abilis ti c se nso r ne tw o rk simu l ato r [12] . The simu lato r pa r ame te rs we re s e t to mo de l t he B e rkeley MI CA mo tes’ MA C lay er [1 3] . Th e r a d i o pr op a gati on m od el in clu des r eal i sti c ef fect s , e.g . f ad i ng, c o llisio n s a nd lo st mes s age s. The tes ts w e re pe rfo rmed w ith a w e ll c o ntr olle d setu p co nta ini n g 100 node s place d uni fo rml y on a gri d, as can be see n in F i g. 3, sh owi n g Pr owl e r ’ s ma in dis p la y . T h e di s ta n c e of adj acent node s o n th e g rid w as 10 m , t he sensing ra dius w as 15 m, a nd t he c ommu nica tio n r ad ius w as app rox im ately 40 m ( se e the par amete r setti ngs i n Fig. 4) . In the simu lat io n the i niti al e n e rgy of all se n so rs w as se t to 20 u nits a n d i n eac h per iod a wak e sen sors con sum ed 1 uni t of en er gy . Th e peri od T w as se t to 1 hou r and t h e required cov erage w as 3. In Fig. 5 t he perfo rmance of the random k-co v erage and th e C GS a l g or i th m s can be c om pa r ed. Th e p l ot s sh ow k- cove ra ge ratio s k Θ for k = 1 , 2 ,3, as a f unc ti o n o f time. ∆ x i ( x s , y s ) ∆ y i Fi g.3. Prowle r sim ul ating the 10x 10-g rid ne two rk. Node 5 and 87 (big dots ) are transm itting AM mess a ges. Sm a ll LEDs i n bo xes show no des th at h ave alre ady transm i tted A M and w ill stay awake d uring the ne xt pe riod. The nu mb er s i n di ca t e th e a ctu a l en er g y r es er v e o f t h e n o de s . Fi g.4. Simul ation p ara mete rs in P rowle r. The pl ot ill ustr ates a poss ible (ran dom ) s ign al pow er vs. d ist ance f unc tion In th e exp er iment s th e r an d om al gori th m ’ s sl eep in g prob ability was set to p sleep = 0.4 and p sleep = 0.25 . In the first cas e the lif e time o f the rando m and CG S n e tw o rks w ere ap pr ox i ma t ely th e same, bu t whil e th e CGS al g ori thm managed to provide th e required co ve r age, t he rando m algorithm’s 3-c o verage ratio w as only around 90%. During the se rv ice d eg radatio n phase (af ter time i nsta nt 26) the CG S still p r ov ided muc h be tte r cov erage. W h e n p slee p was set t o 0.25 the random algo rithm imp rov ed its perfo r mance (app r ox . 95 %) b ut t he de gr ad atio n o f the netw ork w as muc h mo re ab ru pt: at ti me ins ta nt 3 4 al l se nso rs w ere c o mple te ly drained ( 0 1 2 3 = Θ = Θ = Θ ), w hile at t his i nst ant f o r the propos ed algo rithm % 81 3 = Θ , % 86 2 = Θ an d % 95 1 = Θ . (a) (b) ( c) Fi g. 5. Deg radatio n of QoS charac te ri st ics of the CGS (a) and the random al gori thm w ith p sleep = 0.4 (b ) and p sleep = 0.25 (c) , fo r the senso r netw or k shown in Fig. 3. Th e n um ber of a wa k e se n s or s i s sh own , a s a fu n cti on of time , in F ig . 6 . W ith p sleep = 0. 2 5 m uc h m or e sen s or s wer e awa ke i n th e ran d om n etwor k th an in th e CG S n etwor k . To prov ide s imi lar ene r gy sav ings s imi lar to t he CG S algo rit hm, th e p sleep = 0. 4 s et t in g wa s a p pr opr ia te for th e ra n dom al g ori th m. Wi th thi s s et t ing , h owever , th e co ver age pr oper ti es were muc h wo rse, acc o rdin g to Fig. 5 . CGS , how e ver, provide s co n stan t go o d quality se rvice an d lo ng netw o rk lif e time at t he s ame ti me. Fi g. 6. The numbe r of a w ake sens o rs as a f unc tion of time for the ra ndom an d CGS al gorit hms V. SUMMARY Algorit hm s w ere propo se d th at solve the k-coverage pro b lem a n d ca n p rov ide pro lo nge d netw ork lif e time . W e sh owed t h at th e pr opose d p er iod i c r e sc h e d u l in g of t h e sl eepi n g and a wak e nod es sa ves en er gy in th e n et work and ex te nds o v e rall netw ork lif e time . The c o nt ro lled sc hed u l e al g ori th m guar antees k- c o ver a ge i n th e wh ole n etwor k when ever th e topol og y of t h e net wor k per m it s i t. Th e cen tral iz ed ver s ion of th e al g ori th m req uir es n et wor k- w ide co mmun icat ion an d t hus it is not f easib le in larg e n e t wor ks. A d i st r ibu t e d a p pr oxim ati on wa s p r opos ed t h at use s o nly loc ally av ailab le info r mat io n. T he Co n tro lle d Greedy Sleep Algo rithm requires o nly a few messages to be broadc asted f rom ev er y node in eac h perio d, thus t he e nergy wast ed on th e com m u ni ca ti on over h ead i s sma ll , com par ed to the gain in the total ene rgy savin g in the netw o r k, suppos ing th e p eri od of t h e s ched ulin g is su ffici en tl y lar ge. The CG S algo r it hm is rob ust and f ault to le r an t: the algorithm p rov ides t he r equ ired co ver age n etw o rk-wide (if pos sib l e) in depe n dently of node failures o r eve n hi gh amo unt of lo s t mes s age s. T he al go r it hm w as c o mpared to t he rando m k-cov erage algorithm a nd w as prov ed to be superior in two se nse s: w hile it is po ssib le , the CG S algo rith m gu a rante e s the required co v erage all ove r the netw o r k. Also, the de gradatio n cu rve is muc h ge ntle r, t hus t he netw ork s erv ic e is pro v ided fo r a lo nge r time . ACKNOWLEDGMENT This rese arc h was p artially supported by “ Dependable, inte ll ige n t netw orks ” (F -30 /05 ) p ro je c t of th e H un ga ria n- Fre n ch I nte rgo vernme n tal S &T C o o pe ratio n P rogr am a nd by th e Hun gari an Gov er n m ent und er contr a ct NKFP 2- 00018/2005 . RE FE REN C E S [1] F. Ko us ha n far, M. Po tk on ja k a nd A . 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