Maximum Network Lifetime in Wireless Sensor Networks with Adjustable Sensing Ranges

Transcription

1 1 Maximum Nework Lifeime in Wirele Senor Nework wih Adjuable Sening Range Mihaela Cardei, Jie Wu, Mingming Lu, and Mohammad O. Pervaiz Abrac Thi paper addree he arge coverage problem in wirele enor nework wih adjuable ening range. Communicaion and ening conume energy, herefore efficien power managemen can exend nework lifeime. In hi paper we conider a large number of enor wih adjuable ening range ha are randomly deployed o monior a number of arge. Since arge are redundanly covered by more enor, in order o conerve energy reource, enor can be organized in e, acivaed ucceively. In hi paper we addre he Adjuable Range Se Cover (AR-SC) problem ha ha a i objecive finding a maximum number of e cover and he range aociaed wih each enor, uch ha each enor e cover all he arge. A enor can paricipae in muliple enor e, bu um of he energy pen in each e i conrained by he iniial energy reource. In hi paper we mahemaically model oluion o hi problem and deign heuriic ha efficienly compue he e. Simulaion reul are preened o verify our approache. Keyword: wirele enor nework, energy efficiency, enor cheduling, linear programming, opimizaion. I. INTRODUCTION Wirele enor nework (WSN) coniue he foundaion of a broad range of applicaion relaed o naional ecuriy, urveillance, miliary, healh care, and environmenal monioring. One imporan cla of WSN i wirele ad-hoc enor nework, characerized by an ad-hoc or random enor deploymen mehod [9], where he enor locaion i no known a priori. Thi feaure i required when individual enor placemen i infeaible, uch a balefield or diaer area. Generally, more enor are deployed han required (compared wih he opimal placemen) o perform he propoed ak; hi compenae for he lack of exac poiioning and improve faul olerance. The characeriic of a enor nework [1] include limied reource, large and dene nework, and a dynamic opology. An imporan iue in enor nework i power carciy, driven in par by baery ize and weigh limiaion. Mechanim ha opimize enor energy uilizaion have a grea impac on prolonging he nework lifeime. Power aving echnique can generally be claified in wo caegorie: cheduling he enor node o alernae beween acive and leep mode, and adjuing he ranmiion or ening range of he wirele All auhor are wih he Deparmen of Compuer Science and Engineering, Florida Alanic Univeriy, Boca Raon, FL {mihaela@ce., jie@ce., mlu@, mpervaiz@}fau.edu. Mihaela Cardei i he correponding auhor (phone: ; fax: ). node. In hi paper we deal wih boh mehod. We deign a cheduling mechanim in which only ome of he enor are acive, while all oher enor are in leep mode. Alo, for each enor in he e, he goal i o have a minimum ening range while meeing he applicaion requiremen. In hi paper we addre he arge coverage problem. The goal i o maximize he nework lifeime of a power conrained wirele enor nework, deployed for monioring a e of arge wih known locaion. We conider a large number of enor, deployed randomly in cloe proximiy of a e of arge, ha end he ened informaion o a cenral node for proceing. The mehod ued o exend he nework lifeime i o divide he enor ino a number of e. Uing he propery ha enor have adjuable ening range, he goal i o e up minimum ening range for he acive enor, while aifying he coverage requiremen. Beide reducing he energy conumed, hi mehod lower he deniy of acive node, hu reducing inerference a he MAC layer. The conribuion of hi paper are: (1) inroduce he Adjuable Range Se Cover (AR-SC) problem and he mahemaical model, () deign efficien heuriic (boh cenralized and diribued) o olve he AR-SC problem, uing linear programming and greedy echnique, and () analyze he performance of our approache hrough imulaion. The re of he paper i organized a follow. In ecion II we preen relaed work on enor coverage problem. Secion III define AR-SC problem and ecion IV preen our heuriic conribuion. In ecion V we preen he imulaion reul and ecion VI conclude our paper. II. RELATED WORK In hi paper we addre he enor coverage problem. A poined ou in [1], he coverage concep i a meaure of he qualiy of ervice (QoS) of he ening funcion and i ubjec o a wide range of inerpreaion due o a large variey of enor and applicaion. The goal i o have each locaion in he phyical pace of inere wihin he ening range of a lea one enor. A urvey on coverage problem in wirele enor nework i preened in [4]. The coverage problem can be claified in he following ype [4]: (1) area coverage [5], [1], [1], [14], [16], where he objecive i o cover an area, () poin coverage [], [], [6], where he objecive i o cover a e of arge, and () coverage problem ha have he objecive o deermine he maximal uppor/breach pah ha ravere a enor field [1].

2 An imporan mehod for exending he nework lifeime for he area coverage problem i o deign a diribued and localized proocol ha organize he enor node in e. The nework aciviy i organized in round, wih enor in he acive e performing he area coverage, while all oher enor are in he leep mode. Se formaion i done baed on he problem requiremen, uch a energy-efficiency, area monioring, conneciviy, ec. Differen echnique have been propoed in lieraure [5], [1], [1], [14], [16] for deermining he eligibiliy rule, ha i, o elec which enor will be acive in he nex round. In [14], he auhor addreed area coverage when enor can adju heir ening range. For applicaion ha require more ringen faul-olerance or for poiioning applicaion, k-coverage migh be a requiremen. In [8], he goal i o deermine wheher a given area aifie he k-coverage requiremen, when each poin in he area of inere i covered by a lea k enor. Boh uniform and non-uniform ening range are conidered, and he k- coverage propery i reduced o he k perimeer coverage of each enor in he nework. A differen coverage formulaion i given in [1]. A pah ha he wor (be) coverage if i ha he propery ha for any poin on he pah, he diance o he cloe enor i maximized (minimized). Given he iniial and final locaion of an agen, and a field inrumened wih enor, auhor [1] propoed cenralized oluion o he wor (be) coverage baed on he obervaion ha wor coverage pah lie on he Voronoi diagram line and be coverage pah lie on Delaunay riangulaion line. The work mo relevan o our approache are [] and []. Paper [] inroduce he arge coverage problem, where dijoin enor e are modeled a dijoin e cover, uch ha every cover compleely monior all he arge poin. The dijoin e coverage problem i proved o be NP-complee, and a lower bound of for any polynomial-ime approximaion algorihm i indicaed. The dijoin e cover problem [] i reduced o a maximum flow problem, which i hen modeled a mixed ineger programming. Thi problem i furher exended in [], where enor are no rericed o paricipaion in only dijoin e, ha i, a enor can be acive in more han one e. The coverage breach problem i inroduced in [6], addreing he cae when enor nework have limied bandwidh. The objecive of he problem i o organize he enor in dijoin e, uch ha each e ha a given bounded number of enor and he overall breach i minimized. The overall breach i meaured a he number of arge uncovered by he enor e. Our paper i an exenion of he maximum e cover problem addreed in [], for he cae when enor node can adju heir ening range. Our goal i o reduce he ening range of he acive enor, while mainaining he coverage requiremen. Thi mehod ha a double impac: fir i reduce energy conumpion, and econd i reduce inerference a he MAC layer. Senor wih adjuable ening range are available commercially [11], [14]. Compared wih [], in hi paper we are alo concerned wih deigning a diribued and localized algorihm (ee ecion IV- B.) for he AR-SC problem. Diribuion and localizaion are imporan properie of a node cheduling mechanim, a i adap beer o a calable and dynamic opology. III. PROBLEM DEFINITION Le u aume ha N enor 1,,..., N are randomly deployed o cover M arge,,..., M. Each enor ha an iniial energy E and ha he capabiliy o adju i ening range. Sening range opion are r 1, r,..., r P, correponding o energy conumpion of e 1, e,..., e P. We aume a bae aion (BS) locaed wihin he communicaion range of each enor. One mehod o compue he enor - arge coverage relaionhip i o conider ha a enor cover a arge if he Euclidean diance beween he enor and arge i no greaer han a predefined ening range. The formal problem definiion i given below: Definiion 1: Targe Coverage Problem [] Given M arge wih known locaion and an energy conrained WSN wih N enor randomly deployed in he arge viciniy, chedule he enor node aciviy uch ha all arge are coninuouly oberved and nework lifeime i maximized. The approach we ued in hi paper i o organize he enor in e, uch ha only one e i reponible for monioring he arge, and all oher enor are in leep mode. Beide deermining he e cover, we are alo concerned wih eing he ening range of each acive enor. The goal i o ue a minimum ening range in order o minimize he energy conumpion, while meeing he arge coverage requiremen. Nex we formally define he Adjuable Range Se Cover (AR-SC) problem, ued o olve he arge coverage problem. Definiion : AR-SC Problem Given a e of arge and a e of enor wih adjuable ening range, find a family of e cover c 1,c,..., c K and deermine he ening range of each enor in each e, uch ha (1) K i maximized, () each enor e monior all arge, and () each enor appearing in he e c 1,c,..., c K conume a mo E energy. In AR-SC definiion, he requiremen o maximize K i equivalen wih maximizing he nework lifeime. The ening range of a enor deermine he energy conumed by he enor when ha e i acivaed. If a enor paricipae in more han one e, hen he um of energy pen ha o be a mo E. AR-SC problem i NP-complee, by rericion mehod [7]. Maximum Se Cover [] i a pecial cae of AR-SC problem when he number of ening range P =1and when he ime a enor i acive i conidered o be he energy conumed. Figure 1 (a) how an example wih four enor 1,,, 4 and hree arge,,. Each enor ha wo ening range r 1,r wih r 1 <r. In hi example we aume a node ening area i he dik cenered a he enor, wih

3 r 1 r (a) Fig. 1. Example wih hree arge T = {,, } and four enor S = { 1,,, 4 } (b) 4 1 (a) (b) 4 a radiu equal o he ening range. We ue a olid line o denoe range r 1 and a doed line for range r. The coverage relaionhip beween enor and arge are alo illuraed in Figure 1 (b): ( 1,r 1 )={ }, ( 1,r )={, }, (,r 1 )= { }, (,r )={, }, (,r 1 )={ }, (,r )={, }, ( 4,r 1 ) = {, } and ( 4,r ) = {,, }. The doed line in Figure 1 (b) how he addiional arge covered by increaing he ening range from r 1 o r. Noe ha a circular ening area i no a requiremen for our oluion; we are ju concerned wih idenifying which enor cover each arge. In hi paper, a enor can be par of more han one cover e. Le u conider for hi example E =, e 1 =.5, and e =1. Each e cover i acive for a uni ime of 1. One oluion for he AR-SC problem ue he e cover illuraed in he Figure. Thi oluion ha five differen e cover, and maximum lifeime 6, obained for example wih he following equence of e cover: C 1, C, C, C 4, C 5,andC 4. Afer hi equence, he reidual energy of each enor become zero. If enor node do no have adjuable ening range, hen we obain a lifeime 4 for a ening range equal o r. Senor can be organized in wo diinc e cover, uch a {( 1,r ), (,r )} and {( 4,r )}, and each can be acive wice. The number of ime a e cover i acive depend on he reidual energy value. Therefore, hi example how a 5% lifeime increae when uing adjuable ening range. IV. SOLUTIONS FOR THE AR-SC PROBLEM In hi ecion we preen hree heuriic for olving he AR-SC problem. In ecion IV-A we formulae he problem uing ineger programming and hen olve i uing relaxaion and rounding echnique. In ecion IV-B we propoe a greedy heuriic, where boh cenralized and diribued (localized) oluion are given for compuing he e cover. The cenralized heuriic are execued a he BS. Once he enor are deployed, hey end heir coordinaion o he BS. The BS compue and broadca back he enor chedule. In he diribued and localized algorihm, each enor node deermine i chedule baed on communicaion wih one-hop neighbor. 1 4 (c) (e) Fig.. Five e cover: C 1 = {( 1,r 1 ), (,r )}, C = {( 1,r ), (,r 1 )}, C = {(,r 1 ), (,r )}, C 4 = {( 4,r ), and C 5 = {( 1,r 1 ), (,r 1 ), (,r 1 )} A. Ineger Programming baed Heuriic In hi ubecion we fir formulae he AR-SC problem uing ineger programming in ecion IV-A.1 and hen preen he LP-baed heuriic in ecion IV-A.. 1. Ineger Programming Formulaion of he AR-SC Problem Given: N enor node 1,..., N Marge,,..., M P ening range r 1, r,..., r P and he correponding energy conumpion e 1, e,..., e P iniial enor energy E he coefficien howing he relaionhip beween enor, radiu and arge: a ipj =1if enor i wih radiu r p cover he arge j. For impliciy, we ue he following noaion: i: i h enor, when ued a index j: j h arge, when ued a index p: p h ening range, when ued a index k: k h cover, when ued a index Variable: (d)

4 4 c k, boolean variable, for k =1..K; c k =1if hi ube i a e cover, oherwie c k =. x ikp, boolean variable, for i =1..N, k =1..K, p =1..P ; x ikp =1if enor i wih range r p i in cover k, oherwie x ikp =. Maximize c 1... c K ubjec o K k=1 ( P p=1 x ikpe p ) E for all i =1..N P p=1 x ikp c k for all i =1..N, k =1..K N i=1 ( P p=1 x ikp a ipj ) c k for all k =1..K, j =1..M x ikp {, 1} and c k {, 1} Remark: 1) K repreen an upper bound for he number of cover ) The fir conrain, K j=1 ( P p=1 x ikpe p ) E for any i =1..N, guaranee ha he energy conumed by each enor i i le han or equal o E, which i he aring energy of each enor. ) The econd conrain, P p=1 x ikp c k for any i = 1..N and k =1..K, aure ha, if enor i i par of he cover k hen exacly one of i P ening range are e. 4) The hird conrain, N i=1 ( P p=1 x ikp a ipj ) c k for any k =1..K and j =1..M, guaranee ha each arge j i covered by each e c k.. LP-baed Heuriic In hi ubecion we propoe a heuriic o olve he AR-SC problem. In ecion IV-A.1 we preened he Ineger Programming (IP) baed formulaion. Since IP i NP-hard, we propoe o ue a relaxaion and rounding mechanim. We fir relax he IP o Linear Programming (LP), olve he LP in polynomial ime, and hen round he oluion in order o ge a feaible oluion for he IP. Relaxed Linear Programming: Maximize c 1... c K ubjec o K k=1 ( P p=1 x ikpe p ) E for all i =1..N P p=1 x ikp c k for all i =1..N, k =1..K N i=1 ( P p=1 x ikp a ipj ) c k for all k =1..K, j =1..M x ikp 1 for all i =1..N, k =1..K, and p =1..P c k 1 for all k =1..K LP-baed Heuriic 1: olve he LP and ge he opimal oluion x ikp and c k : e x ikp =and c k =for all i =1..N, k =1..K, p = 1..P : or c k in nonincreaing order c 1, c,..., c K 4: for all variable c k aken from he li in nonincreaing order do 5: if c k > hen 6: / ry o build a e cover if c k > / 7: or x ikp, i =1..N, p =1..P in nonincreaing order 8: for all x ikp do 9: if x ikp cover new arge and enor i ha a lea e p energy a he beginning of eing up he cover c k hen 1: e up he range of enor i o r p, x ikp =1 11: ele 1: x ikp = 1: end if 14: end for 15: if all arge are covered by x ikp having value 1 hen 16: / we formed a valid e cover / 17: e c k =1 18: updae reidual energy of any enor i wih range r p in c k : E i = E i e p 19: ele : e c k =and ree x ikp =for any i =1..N and p =1..P 1: end if : end if : end for 4: reurn he oal number of e cover K k=1 c k The heuriic ar in line 1 by olving he relaxed LP ha oupu he opimal oluion x ikp and c k. We round hi oluion in order o ge a feaible oluion x ikp and c k for he IP. We ue a greedy approach, by giving prioriy o he e cover wih a larger c k. When adding enor o a cover c k, prioriy i given o he enor wih larger x ikp. We or value c k in he nonincreaing order. In line 8..14, we add enor o he curren e cover k, by adding fir he enor wih higher x ikp value. If, laer, he ame enor wih a larger range i encounered, he new range eing i ued if new arge are covered and if he enor ha ufficien energy reource for hi eing. If all he arge are covered by he eleced enor in hi e cover, hen we e c k =1. Oherwie, forming he curren e cover wa unucceful, c k =, and all of e k member are removed ( x ikp =for any i =1..N and p =1..P ). The complexiy of hi algorihm i dominaed by he linear programming olver. The be performance i O(n ) uing Ye algorihm [15], where n i he number of variable. In our cae n = K(1 NP), wherep uually a mall number. B. Greedy baed Heuriic In hi ubecion we propoe wo greedy oluion for he AR-SC problem. The cenralized oluion i given in ubecion IV-B.1 followed by a diribued and localized oluion in ubecion IV-B.. 1. Cenralized Greedy Heuriic In hi ubecion we preen a cenralized greedy heuriic. We ue he following noaion: T ip : he e of uncovered arge wihin he ening range r p of enor i.

5 5 B ip : he conribuion of enor i wih range r p. B ip = T ip /e p. B ip : he incremenal conribuion of he enor i when i ening range i increaed o r p. B ip = T ip / e p, where T ip = T ip T iq and e p = e p e q.the range r q i he curren ening range of he enor i, hu r p >r q. Iniially, all he enor have aigned a ening range r =and he correponding energy i e =. C k :heeofenorinhekh cover. T Ck : he e of arge uncovered by he e C k. The algorihm elec enor in a greedy fahion, baed on heir conribuion value. A conribuion parameer B ip i aociaed wih each (enor, range) pair. For breviy, in cae of no ambiguiy, we wrie (i, p) inead of ( i,r p ). Inuiively, a enor ha cover more arge per uni of energy hould have higher prioriy in being eleced in a enor cover. We are uing he incremenal conribuion parameer B ip,defined a he beginning of hi ubecion, a he elecion deciion parameer. In our algorihm, we are concerned no only wih elecing he enor of each e cover, bu alo wih deermining heir ening range. Inuiively, a maller ening range i preferable a long a he arge coverage objecive i me, ince energy reource are conerved, allowing he enor o be operaional longer. Our algorihm repeaedly conruc e cover, a long a each arge i covered by a lea one enor wih enough energy reource. In forming a e cover, enor are eleced repeaedly, giving prioriy o he enor wih highe conribuion. We aume ha iniially all he enor have been aigned he range r =. If a enor i i eleced baed on i conribuion B ip, i ening range i increaed o r p. Once he e cover i formed (e.g. all arge are covered by he eleced e of enor), he enor wih a ening range greaer han zero form he e of acive enor, while all oher enor wih ening range r will be in leep mode. Aume ha a enor (i, p) wih he highe conribuion B il i eleced o be added o he curren e cover. Then he enor i updae i ening range from r p o r l. For each enor node x ha cover a lea one arge in T il, we updae T xu = T xu T il and B xu for any range r u greaer han he curren ening range of x. Noe ha alhough here are P ening range for each enor, we mainain conribuion value only for hoe ening range for which ufficien reidual energy i available. For example, if he reidual energy E x of he enor x aifie he relaion e q E x <e q1, hen we conider only he conribuion B xu for u q. We preen nex he Cenralized Greedy Algorihm ha repeaedly conruc e cover a long a each arge i covered by a lea one enor node wih ufficien reidual energy. Cenralized Greedy Algorihm 1: e he reidual energy of each enor i o E, E i = E : aign o each enor i a range r = having he correponding energy e = : k = 4: while each arge i covered by a lea on enor (i, p) and E i >e p do 5: /* a new e cover will be formed */ 6: k = k 1 7: T Ck = { j j =1..m} 8: for each enor i compue B ip and T ip, for all ening range ha can be e up wih he curren reidual energy 9: while T Ck do 1: /* more arge have o be covered */ 11: elec he enor (i, p) wih he highe conribuion value B il 1: increae enor i ening range from r p o r l 1: T Ck = T Ck T il 14: for all (x, u) uch ha T xu T il do 15: /* updae he uncovered arge e and he incremenal conribuion */ 16: updae T xu = T xu T il 17: updae B xu = T xu / e u 18: end for 19: end while : for all (i, p) C k do 1: updae he reidual energy of enor i, E i = E i e p : end for : end while 4: oupu he number of e cover k The complexiy of Cenralized Greedy Algorihm i O(MN P E e 1 ). The number of ieraion of he while loop (line 4..) i upper-bounded by N E e 1, correponding o he cae when all he arge are covered by all enor wih range r 1. The complexiy of he inner while loop (line 9..19) i upperbounded by MNP.. Diribued and Localized Heuriic In hi ubecion, we exend he algorihm inroduced in ubecion IV-B.1 o a diribued and localized verion. We ue he noaion inroduced in he previou ubecion. By diribued and localized we refer o a deciion proce a each node ha make ue of only informaion for a neighborhood wihin a conan number of hop. A diribued and localized algorihm i deirable in wirele enor nework ince i adap beer o dynamic and large opologie. The diribued greedy algorihm run in round. Each round begin wih an iniializaion phae, where enor decide wheher hey will be in an acive or leep mode during he curren round. The iniializaion phae ake W ime, where W i far le han he duraion of a round. Each enor mainain a waiing ime, afer which i decide i au (leep or acive) and i ening range, and hen i broadca he li of arge i cover o i one-hop neighbor. The waiing ime of each enor i depend on i conribuion, and i e up iniially o W i =(1 BiP B max ) W where B max i he large poible conribuion, defined a B max = M/e 1,whereM i he number of arge.

6 6 The waiing ime can change during he iniializaion phae, when broadca meage are received from neighbor. If a enor i receive a broadca meage from one of i neighbor, hen i updae he e of uncovered arge T ip and e up i ening range o he malle value r u needed o cover hi e of arge. The enor conribuion value i alo updaed o B iu.ifall i arge are already covered by i neighbor, hen i e up i ening range o r =. The waiing ime W i of he enor i i alo updaed o (1 Biu B max ) W. A he end of i waiing ime, a enor broadca i au (acive or leep) a well a he li of arge i cover. If i ening range i r hen hi enor node will be in he leep mode, oherwie i will be acive during hi round. A differen enor have differen waiing ime, hi erialize he enor broadca in heir local neighborhood and give prioriy o he enor wih higher conribuion. Thee enor decide heir au and broadca heir arge coverage informaion fir. In hi algorihm we ue a dicree ime window, where d i he lengh of he ime lo. Thu, he ime window W ha W d ime uni. If he waiing ime of wo enor i and j are oo cloe, i.e. W i W j <d, hen he enor ha are neighbor o boh i and j canno ell from whom he meage wa received, hu hey will no updae heir uncovered arge e. We aume enor node are ynchronized and he proocol ar by having he bae aion (BS) broadca a ar meage. If, afer he iniializaion phae, a enor i canno cover one of he arge in he e T ip and i waiing ime reached he value zero, hen i end hi failure informaion o BS. In our algorihm, we meaure he nework lifeime a he ime unil BS deec he fir failure. Nex we preen he Diribued Greedy Iniializaion,ha i run by each enor i, i =1..N during he iniializaion phae: Diribued Greedy Iniializaion ( i ) 1: compue he waiing ime W i and ar imer : while W i and T ip do : if meage from neighbor enor i received hen 4: updae T ip and e-up he ening range o he malle value r u needed o cover T ip 5: if T ip == hen 6: e i ening range o r 7: break 8: end if 9: updae i conribuion o B iu 1: updae he waiing ime W i o (1 Biu B max ) W 11: end if 1: end while 1: /* aume i ening range wa e up o r u */ 14: if r u == r hen 15: i broadca i leep ae deciion 16: reurn 17: end if 18: if E i <e u hen 19: i repor failure o BS, indicaing he arge i canno cover due o he energy conrain : end if 1: i broadca informaion abou he e of arge T iu i will monior during hi round : reurn The complexiy of he Diribued Greedy Iniializaion procedure i O( W d NMP). Thi correpond o he cae when i receive meage from N neighbor, each d ime. The updae for each meage ake O(MP). V. SIMULATION RESULTS In hi ecion, we evaluae he performance of LP-baed and greedy-baed heuriic. We imulae a aionary nework wih enor node and arge randomly locaed in a 1m 1m area. We aume enor are homogeneou and iniially have he ame energy. In he imulaion, we conider he following unable parameer: N he number of enor node. In our experimen we vary N beween 5 and 5. M he number of arge o be covered. I varie beween 5 o 5. P ening range r 1,r,...,r P.WevaryP beween 1 and 6, and he ening range value beween 1m and 6m. Energy conumpion model e p (r p ). We evaluae nework lifeime under linear (e p =Θ(r p )) and quadraic (e p = Θ(rp )) energy conumpion model. Time lo d in he diribued greedy heuriic. d how he impac of he ranfer delay on he performance of he diribued greedy heuriic. We vary d beween. and.75. In he fir experimen in Figure, we compare he nework lifeime compued by LP-baed, cenralized greedy and diribued greedy heuriic when we vary he number of enor. We conider 1 arge randomly deployed, and we vary he number of enor beween 5 and 1 wih an incremen of 5. Each enor ha wo adjuable ening range, m and 6m. The energy conumpion model i linear. Nework lifeime reul reurned by he heuriic are cloe and hey increae wih enor deniy. When more enor are deployed, each arge i covered by more enor, hu more e cover can be formed. In he econd experimen in Figure 4, we udy he impac of he number of adjuable ening range on nework lifeime. We conider 4 arge randomly diribued and we vary he number of enor beween 1 and 5 wih an incremen of 1. We le he large ening range equal o 6m for all cae. We compare he nework lifeime when enor uppor up o 6 ening range adjumen: r 1 =6m, r =5m, r =4m, r 4 =m, r 5 =m, andr 6 =1m. A cae wih P ening range, where P =1..6, allow each enor node o adju P ening range r 1, r,..., r P. Noe ha P =1i he cae when all enor node have a fixed ening range wih value 6m.

7 LP-baed Cenralized Greedy Diribued Greedy Cenralized Diribued d=. Diribued d=.5 Diribued d=.75 Lifeime 5 5 Lifeime Number of Senor Number of Senor Fig.. Nework lifeime wih number of enor Fig. 5. Nework lifeime for differen value of he ime lo d Lifeime ening range 5 ening range 4 ening range ening range ening range 1 ening range Lifeime Linear 5 arge Quadraic 5 arge Linear 5 arge Quadraic 5 arge Number of Senor Number of Senor Fig. 4. Nework lifeime for differen ening range value Fig. 6. Linear and quadraic energy model Simulaion reul indicae ha adjuable ening range have grea impac on nework lifeime, epecially when increaing P from 1 o, or 4. When increaing P from 4 ening range o 5 or 6 ening range, he nework lifeime increae a a lower rae. From P = 1 o P =, he nework lifeime increae wih more han e cover on average. Thi imulaion reul alo juify he conribuion of hi paper, howing ha adjuable ening range can grealy conribue o increaing he nework lifeime. In Figure 5 we compare he nework lifeime produced by cenralized and diribued greedy algorihm. We meaure he nework lifeime when he number of enor varie beween 1 and 5 wih an incremen of 1 and he number of arge i 5. Each enor ha 6 ening range wih value 1m, m, m, 4m, 5m, and6m. The energy conumpion model i linear. We changehe lenghof he ime lod in he diribued greedy algorihm o d =.,.5, and.75. Nework lifeime produced by he cenralized algorihm i longer han ha produced by he diribued algorihm. Thi happen becaue he cenralized greedy heuriic ha global informaion and can alway elec he enor wih he greae conribuion. Alo, if here i a ie beween he conribuion of differen enor, he cenralized greedy heuriic can break he ie arbirarily, wihou any addiional co. In he diribued heuriic, breaking a ie i a he expene of backoff ime, and here i alo no guaranee of no conflic. A conflic occur when enor broadca a he ame ime baed on heir conribuion. Then, here migh be enor ha work inead of going o he leep ae, even if he arge wihin heir ening range are already covered. A illuraed in Figure 5, he ranfer delay alo affec he nework lifeime. The longer he ranfer delay i, he maller he lifeime. In Figure 6 we udy he impac of wo energy model on he nework lifeime compued by he diribued greedy heuriic when we vary he number of enor beween 4 and, and he number of arge i 5 or 5. Each enor ha P =ening range wih value 1m, m, and m. The wo energy model are he linear model e p = c 1 r p,and quadraic model e p = c rp. In hi experimen we defined conan c 1 = E/( P r=1 r p) and c = E/( P r=1 r p ), where E =1i he enor aring energy. For boh energy model, he imulaion reul are conien and indicae ha nework lifeime increae wih he number of enor and decreae a more arge have o be moniored. In Figure 7, we give an example of coverage produced by cenralized and diribued heuriic. We aume a 1m 1m area, wih 4 enor and arge. Each enor ha P = ening range wih value 1m, m, and m. We ue olid line o repreen r 1 = 1m, dahed line for r =mand doed line for r =m. Weueda linear energy model. The fir graph repreen he enor and arge random deploymen. Figure 7 (b) and (c) how e cover produced by he cenralized and diribued greedy heuriic. The acive enor are blackened and he line ype indicae he ening range value.

8 (b) (a) Fig. 7. Se cover example, where and are inacive (leeping) and acive enor, repecively and are arge. (a) Senor and arge deploymen. (b) Se cover produced by he cenralized greedy heuriic. (c) Se cover produced by he diribued greedy heuriic. The imulaion reul can be ummarized a follow: Given he number of arge and he ening range value, he nework lifeime oupu by our heuriic increae wih he number of enor deployed. Nework lifeime increae wih he number of adjuable ening range. Greaer impac i oberved when increaing P from 1 o mall value (P 5). Afer ha he increae in he nework lifeime converge a a lower rae. Even if he wo cenralized oluion perform beer han he diribued oluion (longer nework lifeime), uing a diribued and localized heuriic i an imporan characeriic for a oluion in wirele enor nework environmen. Tranfer delay ued for inernode communicaion in he diribued greedy heuriic affec he nework lifeime. Smaller ranfer delay reul in longer nework lifeime. For boh linear and quadraic energy model, nework lifeime increae wih he number of enor and decreae a more arge have o be covered. (c) VI. CONCLUSIONS In hi paper we propoed cheduling model for he arge coverage problem for wirele enor nework wih adjuable ening range. The problem addreed in hi paper i o deermine maximum nework lifeime when all arge are covered and enor energy reource are conrained. In hi paper we inroduced he mahemaical model, propoed efficien heuriic (boh cenralized and diribued and localized) uing ineger programming formulaion and greedy approache, and verified our approache hrough imulaion. In our fuure work we will inegrae he enor nework conneciviy requiremen. Mainaining conneciviy among he eleced enor ha an advanage in faciliaing he exchange of informaion beween enor and he bae aion. ACKNOWLEDGMENT Thi work i uppored in par by NSF gran CCR 9741, CNS 445, CNS 476, and EIA 186. REFERENCES [1] I. F. Akyildiz, W. Su, Y. Sankaraubramaniam and E. Cayirci, A Survey on Senor Nework, IEEE Communicaion Magazine, pp 1-114, Aug.. [] M. Cardei, D.-Z. Du, Improving Wirele Senor Nework Lifeime hrough Power Aware Organizaion, ACM Wirele Nework, Vol 11, No, May 5. [] M. Cardei, M. Thai, Y. Li, and W. Wu, Energy-Efficien Targe Coverage in Wirele Senor Nework, IEEE INFOCOM 5, Mar. 5. [4] M. Cardei, J. Wu, Energy-Efficien Coverage Problem in Wirele Ad Hoc Senor Nework, acceped o appear in Compuer Communicaion, pecial iue on Senor Nework. [5] J. Carle and D. Simplo, Energy Efficien Area Monioring by Senor Nework, IEEE Compuer, Vol 7, No (4) [6] M. Cheng, L. Ruan, and W. Wu, Achieving Minimum Coverage Breach under Bandwidh Conrain in Wirele Senor Nework, IEEE INFO- COM 5, Mar. 5. [7] M. R. Garey and D. S. Johnon, Compuer and Inracabiliy: A guide o he heory of NP-compleene, W. H. Freeman, [8] C.-F. Huang and Y.-C. Teng, The Coverage Problem in a Wirele Senor Nework, ACM MobiCom, pp , Sep.. [9] S. Megerian, M. Pokonjak, Wirele Senor Nework, Wiley Encyclopedia of Telecommunicaion, Edior: John G. Proaki, Dec.. [1] S. Meguerdichian, F. Kouhanfar, M. Pokonjak, and M. Srivaava, Coverage Problem in Wirele Ad-Hoc Senor Nework, IEEE Infocom, pp , 1. [11] Phooelecric Senor, hp:// [1] D. Tian and N. D. Georgana, A Coverage-Preerving Node Scheduling Scheme for Large Wirele Senor Nework, Proc. of he 1 ACM Workhop on Wirele Senor Nework and Applicaion,. [1] X. Wang, G. Xing, Y. Zhang, C. Lu, R. Ple, and C. D. Gill, Inegraed Coverage and Conneciviy Configuraion in Wirele Senor Nework, Fir ACM Conference on Embedded Neworked Senor Syem,. [14] J. Wu and S. Yang, Coverage and Conneciviy in Senor Nework wih Adjuable Range, Inernaional Workhop on Mobile and Wirele Neworking (MWN), Aug. 4. [15] Y. Ye, An o(n l) Poenial Reducion Algorihm for Linear Programming, Mahemaical Programming, Vol 5, pp 9-58, [16] H. Zhang and J. C. Hou, Mainaining Sening Coverage and Conneciviy in Large Senor Nework, NSF Inernaional Workhop on Theoreical and Algorihmic Apec of Senor, Ad Hoc Wirele and Peer-o-Peer Nework, Feb. 4.