Power Conservation Approaches to the Border Coverage Problem in Wireless Sensor Networks

Transcription

1 Power Conervaton Approache to the Border Coverage Problem n Wrele Senor Network Mohamed K. Watfa** and Seh Commur Abtract- Recent advance n wrele communcaton and electronc have enabled the development of low cot, lowpower, multfunctonal enor node that are mall n ze and communcate n hort dtance. A enor node are typcally battery operated, t mportant to effcently ue the lmted energy of the node to extend the lfetme of the enor network. In th paper, the border coverage problem n enor network rgorouly analyzed. Unlke prevou work n th area, we provde dtrbuted algorthm that allow the electon and actvaton of an optmal border cover for both D and D enor network. We alo provde elf healng algorthm a an optmzaton to our border coverage algorthm whch allow the border cover to adaptvely reconfgure and repar telf n order to mprove t own performance. Border coverage crucal for optmzng enor placement for ntruon detecton and a number of other ueful applcaton. Mathematcal a well a expermental proof are provded to valdate the correctne and effcency of our algorthm. Keyword: Ad-Hoc and Senor Network, Survellance, Intruon Detecton, Border Coverage, Energy Savng. I. INTRODUCTION Wrele enor network (WSN) have been under development for many year and are about to gan wdepread ue a technology mprove, prce drop, and new applcaton are developed []. Algorthm for wrele enor network mut have low communcaton overhead, rely a much a poble on local nformaton, adapt to falure and change n network condton, and produce reult n a tmely fahon. Gven the requrement to mnmze the power, t derable to elect the bare eental number of enor node dedcated for the tak whle all other enor node hould preferably be n the hbernaton or off tate. Even though target trackng ha been wdely tuded for enor network wth large node and dtrbuted trackng algorthm are avalable [], ntruon detecton n ad hoc network wth mcro enor node poe dfferent challenge due to communcaton, proceng and energy contrant. *Submtted to ICWN 06 **Correpondng author and Preenter Th work ha been upported by U.S. Department of Defene, Army Reearch Offce through grant # DAAD Mohamed K. Watfa wth the School of Electrcal and Computer Engneerng at the Unverty of Oklahoma n Norman, OK 709. He can be reached at (405) 5-40 (tel), (405) 5-44 (fax); or at mwatfa@ou.edu (by emal). Seh Commur an Aocate Profeor n the School of Electrcal and Computer Engneerng at the Unverty of Oklahoma n Norman, OK 709. H emal d commur@ou.edu Border urvellance one of the major applcaton of enor network. The border repreent the phycal extent of the regon to be montored and dependng on the applcaton, t requred to ene the ntruon nto the montored regon or ext from the montored regon of the object beng montored. In a typcal deployment of enor node, enor node are dtrbuted acro the entre regon of nteret and t neceary to determne a mnmal et of enor node that can adequately montor the border. Thu, t neceary to fnd a calable and energy effcent oluton to the border coverage problem. Such a oluton would extend the calablty of wrele enor network and enable the montorng of one of the larget nternatonal border []. The full coverage problem, whch verfe f every pont n the regon of nteret covered by at leat one actve enor, ha been tuded n a varety of context. Our prevou work [4, 5] focued on the full coverage problem n D and D regon and provded algorthm to locate redundant enor node n the regon and deactvate them ung mple geometrc technque. In [6], Gupta and Da degn and analyze algorthm for elf organzaton of a enor network to reduce energy conumpton. In partcular, they develop the noton of a connected enor cover and degn a centralzed approxmaton algorthm that contruct a topology nvolvng a near optmal connected enor cover. The work n [7] and [8] conder a large populaton of enor node, deployed randomly for area montorng. The goal to acheve an energy-effcent degn that mantan area coverage. A the number of enor node deployed greater than the optmum requred to perform the montorng tak, the oluton propoed to dvde the enor node nto djont et, uch that every et can ndvdually perform the area montorng tak. Shakkotta et al. n [9] conder an unrelable enor grd-network and derve neceary and uffcent condton for the coverage of the regon and connectvty of the network n term of the tranmon radu, enng radu, and falure rate of the enor node. In [0], Leka et al. formulate coverage problem to addre the qualty of ervce (urvellance) provded by a enor network. In partcular, they addre the problem of fndng maxmal path of lowet and hghet obervablte n a enor network. The coverage concept wth regard to the robot ytem wa ntroduced by Gage []. He defned three type of coverage: blanket coverage, barrer coverage, and weep coverage. In th paper, the problem of determnng the mnmum number of enor node for coverng the boundare of a target regon addreed. Unlke the full coverage problem, here the prmary nteret n the detecton of the unauthorzed movement of an object acro the boundary. A recent work [] conder the determnaton of hole n the coverage area of a enor. In th work, the author ntroduced a new technque for detectng hole n coverage by mean of homology, an algebrac topologcal nvarant. In

2 [] Carbunar, et. al. tudy the problem of detectng and elmnatng redundancy n a enor network wth a vew to mprovng energy effcency, whle preervng the network coverage. They alo examne the coverage boundary detecton by reducng t to the computaton of Vorono dagram. Unlke any other related work n th feld, we frt provde optmal two dmenonal and three dmenonal technque for the deployment of a enor network for border coverage of a gven regon. The auxlary problem of electng the mnmal ubet of prevouly deployed actve enor node for border coverage then addreed. The energy effcency of a WSN tuded n the context of the deployment of enor node and the border coverage obtaned. The mnmum number of enor node requred for border coverage ued to pecfy a meaure of optmalty that can erve a a metrc for the energy effcency of a WSN. The border coverage algorthm developed ued to determne the avng that can be obtaned by deactvatng a ubet of node whle tll mantanng the coverage of the regon. Fault-tolerance crucal for many ytem and becomng vtally mportant for computng- and communcaton- baed ytem a they become ntmately connected to the world around them, ung enor node and actuator to montor and hape ther phycal urroundng. We alo extend our algorthm to allow the border cover to elf heal telf n order to cover ome border hole reultng from node falure or death and therefore, ncreae t own performance. The propoed technque are analyzed mathematcally and the algorthm are demontrated through numercal example. The ret of the paper organzed a follow. The border coverage problem formulated n ecton. A determntc enor deployment to guarantee border coverage of a regon provded n ecton. In ecton 4, dtrbuted algorthm for electng a optmal ubet of node that le on the border of the regon of nteret are tuded. Numercal mulaton reult that valdate the propoed algorthm are preented n ecton 5. Poble extenon to our developed algorthm and concluon are ummarzed n ecton 6. II. PROBLEM FORMULATION In th ecton, the noton of enng regon and border coverage are frt defned. The border coverage and optmzaton problem are then formulated. Let O be the output of a enor S that capable of enng a phenomenon P. Let the enng radu of the enor node S be R. It aumed that each enor node aware of t own locaton, the locaton of the boundare of the regon to be montored and the locaton of t neghbor. Th aumpton not too trngent and t can be atfed by communcaton between adjacent node n the network on tartup. Defnton.: The phenomenon P located at y R ad to be detected by enor S located at X R f and only f there ext a contant threhold δ uch that δ f the phenomenon P preent, O ( y) = ε otherwe. The quantty δ n the defnton above the gnal threhold and pecfc to the type of enor ued. The enng regon of enor S located at X( x,y,z ) the collecton of all pont where the phenomenon P can be detected by the enor S,.e. A = { y R P detectable by S }. In th paper, wthout lo of generalty, we wll retrct the enng regon of S to be a cloed ball centered at R. In the cae of D, the enng regon aumed to be a dk of radu R. The enng boundary (crcle) of enor S, n th cae, denoted by Cr. Mot of the reearch work thu far aume mplfed boolean enng model (Crcular dc) for coverage for protocol degn and evaluaton. In th model all event wthn the crcular dc are aumed to be detected wth probablty. Th mplfed model clearly not applcable to all type of enng meaurement however nce the enor node wll be deployed n large number and there a need for mulaton and theoretcal analy, lead reearcher n the area of wrele enor network ue th model n ther reearch [-9]. We alo aume that any two node S and S j X can drectly communcate wth each other f ther Eucldean dtance le that the communcaton range R c. Although a network can be rendered uele f t looe t connectvty, we characterze the ytem lfetme by jut obervng the reultng border coverage. Zhang and Hou [4] howed that f the communcaton range at leat twce the enng range, then complete coverage of a convex area mple connectvty among the node. Aumng the communcaton range at leat twce the enng range ( Rc R ), the theorem n [9] could be ealy extended to handle border coverage of the regon a well. Defnton.: Let R be a ubet of the (D or D) pace. The pont p ad to be near R f every neghborhood of p contan a pont from R.e. ε > 0, x Ball( p, ε ) and x R. In the defnton above, Ball ( p, ε ) mean the et of all pont whoe Eucldan dtance from p le than ε. Defnton.: The et of all pont n R and near R called the cloure of R and denoted by cl( R ).e. cl( R ) = ( R ) { All pont near R}. Defnton.4: The border of a regon R denoted by B( R ) defned a the et of all pont that are common to R and t complement.e. B( R) = cl( R) cl( R) where R the complement of the regon R.e. all the pont that don t belong to R. An ntruder any object that ubject to detecton by the enor network a t croe the border. A reaonable aumpton made that no ntruder aware of the locaton of the deployed enor node. A regon ad to be border covered f and only f an ntruder alway detected a t croe the border of the regon. Defnton.5a: A et of enor node C Border ad to be a border cover of a regon R f every pont on the border of R

3 belong to the enng regon of at leat one enor n C ;.e. p B( R ), p S for ome S Border C. Border Accordng to defnton.5a, a regon border covered by a et of enor node but that doen t necearly mean that the number of enor node border coverng R mnmum nce there mght ext ome redundant enor node. Defnton.5b: A et of enor node CBorder,Reduced ad to be a reduced border cover of a regon R f p B( R ), p S for ome S C and no Border,Reduced proper ubet of C Border,Reduced a border cover of R..e. CBorder,Reduced S l, for any Sl CBorder,Reduced, not a border cover of R. Defnton.6: A enor node called a redundant enor node f t enng regon completely covered by t neghborng enor node. Deactvatng a redundant enor won t affect the full coverage of the regon of nteret. Defnton.7a: A enor node called a border redundant enor node f t nterectng enng regon wth the border completely covered by t neghborng enor node. Defnton.7b: A enor node called a non-border enor node f t enng regon doe not nterect the boundary of the regon of nteret. From defnton.7a and.7b, t can be ealy een that the deactvaton of a border redundant enor or a non-border enor wll not affect the overall border coverage of the regon of nteret.(fgure (b)). (a) (b) Fgure : (a) Example of a redundant enor (dahed crcle). (b)example of a border redundant enor (dahed crcle) and a non-border enor (black haded crcle). Ung defnton..7, the border coverage problem analyzed n th paper by dvdng t nto the followng two ub problem: I. Optmal Deployment for Border Coverage: Fnd the mnmum number of enor node and ther placement for border coverage of a gven regon R. II. Optmal Selecton for Border Coverage: Gven a dene deployment of enor node n a regon R, fnd a mnmum ubet of node that guarantee border coverage of R. III. OPTIMAL DEPLOYMENT A crtcal ue n WSN the deployment and organzaton of the enor network. Although many cenaro aume random deployment, uch a deployment not optmum and therefore a lot of energy wated due to multple actve node n a gven regon. When flexblty n deployment ext, t advantageou to fnd an optmum border deployment of the enor node o that border coverage can be acheved ung a mnmum number of node. In th ecton, theorem for optmal deployment of the enor node are developed. Thee theorem provde lower bound on the number of node needed to border cover both -dmenonal and -dmenonal regon of nteret.. Optmal D Deployment for Border Coverage In the D deployment problem, the mnmum number of enor node modeled a dk and ther locaton for border coverage of a gven rectangular regon R are to be determned. Whle the regon to be border covered aumed to be a rectangular regon, the algorthm could be ealy extended to border cover any arbtrary hape of a regon wth mnor modfcaton. Lemma..: Conder a rectangular regon R of length L and wdth W. The lower bound on the number of enor node needed to acheve border coverage of R L W ( + ) where repreent the operaton R R of fndng the leat upper bound nteger. Proof: The optmal way to deploy the enor node to acheve border coverage of the regon to deploy the enor node acro the permeter of the entre regon uch that any adjacent enor node that are on the ame row or column L are tangent to each other. the leat number of R enor node to cover a lne of length L. For a rectangular regon of length L and wdth W, the permeter can be L W optmally covered by ( + ) enor node. R R However, uch a cover wll have overlappng enng coverage at the vertce of the rectangle a hown n Fgure. A better way would be to elect the next poton of the center uch that t crcle nterect the lat crcle n t boundary nterecton. Snce the enor node have equal enng rad and ther center le on the boundary of the R rectangle, then n the bet cae cenaro, of the boundary lne wll be covered. The bet enhancement on the frt placement of enor node would be mnmzng the number of enor by one on three boundary edge (Fgure ). So, the lower bound on the number of enor node to cover a D rectangular regon each wth enng radu R L W ( + ). R R (a) (b) Fgure : (a) The optmal deployment of 0 enor node modeled a crcle n D to border cover a rectangular regon. (b) Illutraton of the bet poble way to mnmze the number of enor node coverng the border. If we aumed that the regon to be montored large n comparon to the enng regon of an ndvdual enor node, then the neceary and uffcent number of node to

4 cover a rectangular regon would mply L W be ( + ). R R. Optmal D Deployment for Border Coverage The three dmenonal optmal enor deployment for border coverage far more complex than that of the two dmenonal cae. It addreed by determnng the mnmum number of enor node requred to cover the boundary of a cubcal regon of nteret. Snce the coverage regon of a enor node modeled a a cloed ball, the border coverage problem requre the determnaton of all the pont on the urface of the cube that are covered by the enor node. To addre th ue, the nterecton of the enng regon and a boundary plane frt defned. Th defnton wll then be ued to determne the leat number of enor node requred for border coverage. Defnton..: A great crcle on a phere the nterecton of that phere wth a plane pang through the center of the phere. Lemma..: The center of all the optmal deployed enor node mut le on one of the face of the cube. Proof: It clear that each enor cover a maxmum area when the coverage regon le n the plane pang through the center of the phere repreentng the enng regon. Thu, mnmzng the number of enor requred to cover the urface of the regon to be montored equvalent to maxmzng the coverage area of each enor. Th poble only when the center of all the enor node le on the urface of the regon to be montored. Theorem..: The optmal deployment locaton of enor node modeled a phere to border cover a D cubcal regon the locaton of the phere whoe center form a lattce of pacng Λ=.7R on each face of the cube. Proof: In a two-dmenonal pace, the optmal coverng of a regon wth crcle obtaned when the center of the crcle le at the vertce of a hexagonal lattce. If the dtance between adjacent vertce n th cae one unt, then the entre regon can be covered by cope of a dc whoe coverng radu R cover = = (mnmum enng radu that wll cover the regon R). Such a lattce alo perodc and completely reduced. Moreover, the thckne of π the cover θ = =.09 (average number of enor node that cover a pont n the pace). Thu, the deployment wll be optmal f the pacng between the center of adjacent R dc equal Λ=. So, the optmal deployment to cover the border of a three dmenonal cubcal regon by placng the phere at center of the lattce on each face of the cube wth pacng Λ=.7R. Theorem..: Conder a cubcal regon R of de a ( a uffcently large n comparon to R ). An approxmaton on the lower bound on the number of enor node of enng radu R to acheve border coverage of R.09a.845a mn = R N Proof: Our approach baed on the problem of coverng a quare by crcle whch ha been tuded by Kerhner [5] and Verblunky [6] where N c, the leat number of crcle of unt radu whch can cover a quare wa determned. They proved that there a contant c uch that for all uffcently large a, a + ca < Nc < a + 8a+ 6. Snce there are 6 face of a cube to be covered and we are ntereted n fndng the lower bound of the number of enor node needed, then c =. Let R be the enng radu of each enor therefore a bac lower bound a + a Nmn = ( ). But nce phere can cover or R even face of the cube at the ame tme, the optmal way would be f the face nterect the phere n em great crcle reultng n mnmzng number of node by approxmately 5 a. Snce we are concerned wth a lower R.09a.845a mn = R bound N a vald lower bound on the three dmenonal border cover. IV. OPTIMAL SELECTION In practce, gven an extng dtrbuton of enor node, t often neceary to mnmze the number of node that reman actve whle tll achevng border coverage of the entre regon. In th ecton, an algorthm developed where the node make local decon on whether to leep or jon the et of actve node. The two-dmenonal and threedmenonal cae for electng an optmum border cover of a gven regon are tuded. A meaure of optmalty alo propoed to compare the performance of the border coverage of a gven enor network wth the optmum coverage obtaned n ecton. The border coverage algorthm preented n th ecton ha the followng key feature:. It a decentralzed algorthm that depend only on the local tate of the enng neghbor.. It provde guaranteed degree of border coverage.. It could ealy be extended to handle dfferent degree of coverage. 4. It handle dfferent enng range. 4. A D Dtrbuted Border Cover Selecton Algorthm In order to olve the border coverage problem for a twodmenonal regon of nteret, t aumed that the regon to be montored a rectangle pecfed by t vertce V, V, V, andv 4. It alo aumed that all the enor node are aware of the locaton of the vertce,.e. the enor

5 node are aware of the extent of coverage that requred. The border coverage algorthm can be appled to any hape of boundary but the regon of nteret aumed to be a rectangular regon for the ake of eae of preentaton. We wll tart by gvng ome defnton and aumpton that wll ad u n developng an algorthm to elect a border cover. Let B repreent the boundary of the regon to be covered. B 4 = B, j where, j, j=, j B the egment connectng vertce V and V j. Wthout lo of generalty, uppoe the boundary edge are ordered a B, B, B, and B a hown n fgure. 4 4 Fgure 5: The orderng baed on the mappng functon. Defnton 4..: We call Seg j = [ x j, y j] the ucceor of Seg = [ x, y] denoted by Seg Seg j f the followng condton are atfed: Seg j Seg = Seg j x > x y > y j & j there no other tartng pont n Seg..e. j p Seg j, p x k for omek, j Fgure 6: Example of a egment and t ucceor,[ A, B] [ C, D]. Fgure : The border cover of a rectangular boundary. Defnton 4..: An nterecton egment the porton of the boundary B covered by the enng regon of a enor node and repreented by the cloed nterval [x, y] uch that: z x, y, [ x, y] an nterecton egment [ ],.., n uch that z A and x, y Cr. A egment Seg [ x, y ] = repreented by t tart pont x and end pont y. Example of nterecton egment are hown n Fgure 4. Fgure 4: Interecton egment [A, B] and [C,D]. Snce our algorthm depend on the concept of orderng, we defne a mappng: ϕ : B [0,] baed on the dtance metrc from the nearet orgn.e. d( x,v ) + d(v,v ) x B j, ϕ(x) = where P the P total length of the permeter of the rectangular boundary and d(v,v j ) the dtance along the boundary of the regon.e. for example d(v,v ) = B + B. A pecal cae hould be taken for the enng regon coverng the orgn vertex v, where the reultng nterecton egment dvded nto ub-egment each of whch mapped eparately. Theorem 4..: Conder the et of egment S = Seg, Seg,..., Seg where Seg = x, y [0,]. { } [ ] m Aume that no two egment are contaned n each other.e. Seg, Seg S, Seg Seg, j. A egment Seg j j ( [ x, y] ) = covered by m Seg f and only f the = followng hold: a. nteger ab,,..., k m x Segaand y Segb b. Seg Seg... Seg. a b k. Proof: Sega Segb Sega Segb = [ xa, yb] k Therefore, Seg = [ x, y ]. = a and. Further from (a), x < x, a b y k > y [ x, y] [ xa, yk] = Seg Seg. On the = a = other hand, uppoe that the egment [x, y] covered by the egment Seg,..., Seg m. Snce the egment covered, there ext ome egment Seg a uch that x Seg a. Smlarly, there ext at leat one egment k uch that y x, y. Thu, condton (a) ealy atfed. Now, f [ k k] > y, then [ x y] ya, Sega and condton (b) trvally atfed. Otherwe, there ext a egment Seg b uch that Sega Segb. If th wa fale, then t mean that ya yk for k m. Th would then mply that there ext pont n the nterval ( ya, y ] that are not covered, thereby contradctng the aumpton that the egment [x, y] covered. If yb > y, then condton (b) proved. Otherwe, repeatng the proce, we obtan nteger a k uch that Sega Segb... Segk and y k > y. Therefore, condton (a) and (b) together mply that the egment [x, y] covered by the collecton of egment Seg, Seg,..., Seg. m k m a

6 Theorem 4.. ndcate that a enor node completely border redundant f each egment n the parttonng of t nterecton egment by t neghbor nterecton egment ha a ucceor and the end pont are alo covered. Therefore, to check f a enor S 0 a border redundant enor and therefore could be deactvated wthout affectng the overall border coverage, one ha to frt fnd all the neghborng enor node that le on the border of the regon of nteret. For each neghborng enor, fnd the reultng nterecton egment (or egment) wth the boundary lne and check f S 0 porton of border coverage completely covered by t neghborng enor node. We can do that by ung theorem 4... An algorthm preented that llutrate the tep n th proce. D Dtrbuted Border Coverage Algorthm For each node S, form the et of neghbor, N ().Do the followng: Step : Fnd the nterecton egment Seg Fnd Seg the nterecton egment of S wth the boundary of the regon of nteret and map t to [0, ]. If Seg = or a pont then S a non border node. Ele go to tep. Step : Non contanment property Let Seg be the et of egment coverng Seg and ntally et to. For every par of node S j, S k n N () Fnd the common nterecton egment Seg j = [ x j, y j] and Segk = [ xk, yk] rep. If the end pont appear n ncreang order a x j, xk, yk, y j.e. Segk Seg j and can be gnored. Update Seg to nclude Seg j.e. Seg = Seg { Seg j} Step : Check for endpont coverage Check that, Seg = ( x, y ) and Seg = ( x, y ) n l f f f l l l Seg x x y and x y y. f f l l If true go to tep 4. Step 4: Check for ucceor Check that, for each element Seg = ( x, y ) n m m m Seg Seg, Seg = ( x, y ) Seg Seg and m n. m n n n n If th condton atfed, the boundary nterectng egment Seg of the gven enor completely covered and S declared a a border redundant enor node and can be deactvated wthout affectng the overall border coverage. The algorthm guarantee that every pont on the boundary of the target regon covered by at leat one enor. The optmal et of enor node alo elected. The computatonal complexty of the redundancy electon algorthm developed n th ecton depend on n = = N max N( ),the maxmum number of node n the neghbor et of any enor n the network and n, the total number of enor node n the network. The complexty of the border coverage algorthm therefore Ο ( N ). Snce we have n enor node to be checked, then the complexty Ο (. nn ). For large network, the number of neghbor of any enor mall compared to the ze the network ( N n) o the computatonal complexty of the algorthm for uch large network of order n ( Ο ( n) ) where n the total number of enor node n the network. The key to our algorthm that t dtrbuted and low n computatonal complexty. 4. A D Dtrbuted Border Cover Selecton Algorthm The three dmenonal optmal enor border coverage problem dfferent than the two dmenonal cae. We wll approach t from a dfferent angle and try to tranform t to optmal complete coverage of the enor node n a D plane. The followng lemma would be the key to addreng the three dmenonal border coverage problem. Lemma 4..: The problem of D border coverage of a cube by enor node modeled a D ball equvalent to the problem of complete coverage of a D plane by enor node modeled a crcle. Proof: Accordng to the defnton of border coverage, each pont on the border hould be covered by at leat one enor. The border B(R) of the cubcal regon R repreented by 6 face (D plane). Frt, f each face of the cube Ba B(R),a=,,,4,5 and 6 completely covered by a et of crcle Cra = { Cra,.. Cran} and f D the dc bounded by the crcle Cr, then p Ba, p D a for ome Cra Cra. The - dmenonal border coverage tranformed to fndng the phere whoe nterecton wth the boundary are thee crcle. p B( R), p A for ome Cr = A B( R). Now, f we have a et of enor node that border cover a D cubcal regon, takng the nterecton of the phercal enng regon of the enor node wth each face of the cube wll reult n the formaton of crcle whch completely cover the D plane. So, the -dmenonal border coverage problem tranformed to the -dmenonal full coverage problem. Lemma 4.. ndcate that a enor node S 0 border covered f t crcle of nterecton wth the boundary plane completely covered by t neghbor. To check f a crcle completely covered by the neghborng crcle, we could ue our prevou reult developed n [4, 5]. Therefore, to check f S 0 border redundant; one ha to frt fnd all the crcle obtaned by the nterecton of S0 Bm, m =..6. For each crcle Cr k, fnd all the nterecton pont that le wthn D k. If all thee nterecton pont are covered, then the crcle Cr k are covered. Then, by lemma 4.., S 0 border redundant and can be deactvated wthout affectng the overall cubcal border coverage. Gven the regon to be montored, one could ealy fnd the number of enor node requred and ther locaton for

7 border coverage. However, f the enor node are already deployed and a ubet of thee enor node elected to keep actve, then the meaure of optmalty a meaure of exce energy pent n montorng the regon a compared to an optmum deployment of the enor node. A network wth a lower meaure of optmalty would reult n leer expendture of energy n montorng the regon. V. SIMULATION RESULTS The theoretcal development n ecton -4 are valdated through numercal example n th ecton. The cae of random deployment of enor node tuded and compared to the optmum deployment for border coverage. Both D and D cae are condered and the number of enor node requred for border coverage tuded. The number of enor node requred to cover a D regon of ze 0 unt by 0 unt (or a D regon of ze 0x0x0) condered. Random deployment, optmal deployment and optmal electon of the node for border coverage are tuded for dfferent value of the enng radu. The optmzaton to our border algorthm alo teted and reultng border coverage lfetme of the network analyzed. In the t experment, the optmum D coverage algorthm ued to fnd the optmum border cover of regon 0x0 unt when enor node are randomly deployed. The node have a enng radu of unt and ntally dfferent number of node are randomly deployed n th regon ung a unform dtrbuton. It can be een that the average optmalty meaure of our border electon algorthm.8 and the node that were actve n the optmum border cover reulted n average avng of 98.4% (Fgure 8). In Fgure 9, the requred number of enor node wth dfferent rad ung random deployment, optmal D Border deployment and D Border electon algorthm are compared. In the nd experment, we evaluate the border coverage percentage of the regon when the enor node are randomly deployed and the border coverage electon algorthm appled. A we vary the number of deployed node, we evaluate the border coverage of the regon ung the border cover obtaned (Fgure 0). It notced that after a pecfc threhold value, the border coverage percentage alway one. The reaon that random deployment of the enor node doe not guarantee border coverage of the regon below that threhold. In the rd experment, we evaluate the ytem lfe tme. The metrc ued n evaluatng ytem lfetme the border coverage lfetme. The overall border coverage lfetme the contnuou operatonal tme of the ytem before the border coverage drop below t pecfed threhold (for example 0.9). In Fgure, the ytem lfetme evaluated aumng the each enor node ha a lmted energy upply (00 Joule) and when t run out of energy t deactvated. The node deployment dente are 00 and 600 repectvely. The power conumpton of Tx (tranmt), Rx (receve), Idle and Sleepng mode are 400mW, 000mW, 80mW, 0mW repectvely. A tme pae, enor node node wll be deactvated due to lack of energy and wll leave ome coverage hole n the border of the regon. If 00 enor node were deployed, after approxmately 600 econd, the border coverage percentage ung the orgnal network wll drop below 0.9. However, ung the border electon algorthm t need about 00 econd to drop below the threhold. If we ncreae the number of deployed node to 600, the cot for calculatng the border cover wll ncreae and thu after approxmately 690 econd the border coverage percentage wll go below 0.9. In both experment, the border coverage lfe tme of the network ung our border electon algorthm much better that that ung the orgnal network. In Fgure, the degree of border coverage analyzed before and after runnng the algorthm. We dvde the border n 000 unformly dtrbuted grd pont and check how many enor node cover each center of each grd cell. It can be een that the degree of border coverage decreaed and a mnmum ubet of node only actvated to cover the border of the regon. In the 4 th experment, we do the ame comparon that wa done n the rd experment however wth elf healng optmzaton added (Fgure ). The dea behnd our elf healng enhancement to our algorthm that a border enor node wll end a HELP meage to t neghborng border node (.e. neghbor node that alo nterect the border of the regon) when t energy level goe below a pecfc value (t about to de). The neghborng node, that wll receve the HELP meage, wll actvate them elve and wll cover a much border coverage a poble. We notce that the ytem lfe tme (border coverage lfe tme) much better than the cae f we had tarted wth the orgnal et of deployed node. The trength of our developed algorthm evdent through th mulaton reult where the algorthm allow the enor network to adaptvely reconfgure and repar telf n order to mprove t own performance. A we ncreae the number of deployed node, the elf healng border coverage algorthm perform better nce actvatng a border ubttute et wll reult n better percentage of border coverage and therefore the border coverage lfetme of the network ncreaed. In Fgure, 000 node were deployed, and after runnng the border electon algorthm, 974 node were deactvated reultng n avng of 98.7%. Mnmzng the number of enor node actve to border cover a regon of nteret wll reult n mnmzng the energy conumed by the whole enor network and thu ncreang the lfe tme of the network a demontrated n the mulaton reult. Fgure 8: The optmalty meaure of the border electon algorthm. Fgure 9: Comparon between Random Deployment, Optmal Deployment, and BSCA (Border Selecton Cover Algorthm).

8 REFERENCES Fgure 0: Varyng the number of deployed node and the reultng border coverage after runnng the electon algorthm. Fgure : The border coverage lfe tme of the regon a tme pae. Fgure : Degree of border coverage before and after runnng the algorthm Fgure : An example of the border electon algorthm. Actve node, before and after runnng the algorthm are hown. VI. CONCLUSIONS In th paper, the border coverage problem n wrele enor network (WSN) wa formulated and analyzed. Algorthm were propoed to compute the mnmum number of enor node requred for border coverage of a gven regon. A meaure of optmalty wa alo propoed that compare a gven border deployment of WSN wth optmum deployment. Part of our future work to ue the algorthm developed n th paper for trackng applcaton. [] K. Sohrab, J. Gao, V. Alawadh and G. J. Potte, Protocol for elf-organzaton of a wrele enor network, IEEE Peronal Communcaton Magazne, pp. 6-7, October 000. [] M. Chu, H. Hauecker, and F. Zhao, Scalable nformaton-drven enor queryng and routng for ad hoc heterogeneou enor network, Internatonal Journal of Hgh-Performance Computng Applcaton, vol. 6, no., 00. [] Extreme cale Wrele Senor Networkng. Techncal report, [4] M.Watfa, S.Commur, Optmal -Dmenonal Senor Deployment Strategy, Conumer Communcaton & Networkng Conference, La Vega, 006. [5] M.Watfa, S.Commur, A Coverage Algorthm n D Wrele Senor Network, Internatonal Sympoum on Wrele Pervave Computng, Thaland, 006. [6] H. Gupta, S.R. Da, and Q. Gu, Connected enor cover: Self-Organzaton of enor network for effcent query executon, Proceedng of MobHoc 0, Annapol, Maryland, USA, pp , June 00. [7] M. Carde, D. MacCallum, X. Cheng, M. Mn, X. Ja, D. L, and D.-Z. Du, Wrele enor network wth energy effcent organzaton, Journal of Interconnecton Network, vol., pp. -9, 000. [8] S. Sljepcevc and M. Potkonjak, Power effcent organzaton of wrele enor network, Proceedng of IEEE Internatonal Conference on Communcaton, vol., pp , 00. [9] S. Meguerdchan, F. Kouhanfar, M. Potkonjak and M. B. Srvatava, Coverage problem n wrele ad-hoc enor network, Proceedng of the 0 th IEEE INFOCOM, pp , 00. [0] K. Leka, E. Latnen, and J. Lahteenmak, Rado coverage optmzaton wth genetc algorthm, Proceedng of IEEE Internatonal Sympoum on Peronal, Indoor, and Moble Rado Communcaton (PIMRC), vol., pp.8-, 998. [] D. W. Gage, Command control for many-robot ytem, Proceedng of the 9 th Annual AUVS Techncal Sympoum, pp. -4, 00. [] Robert Ghrt and Abubakr Muhammad, Coverage and hole-detecton n enor network va homology, 4 th Internatonal Conference on Informaton Proceng n Senor Network (IPSN'05), Lo Angele, CA, Aprl 5-7, 005. [] B. Carbunar, A. Grama and J. Vtek. Dtrbuted and dynamc vorono overlay for coverage detecton and dtrbuted hah table n ad-hoc network, 0 th Internatonal Conference on Parallel and Dtrbuted Sytem, pp , Newport Beach, 004. [4] H. Zhang and J. C. Hou, Mantanng enng coverage and connectvty n large enor network, Ad Hoc & Senor Wrele Network, an Internatonal Journal, vol., UIUCDCS-R-00-5, June 00. [5] R. Kerhner, The number of crcle coverng a et, Amer. J. Math., vol.6 pp , 99. [6] S. Verblunky, On the leat number of unt crcle whch can cover a quare, J. London Math. Socety vol.4, pp.64-70, 949.