Distributed Topology Control for Power Efficient Operation in Multihop Wireless Ad Hoc Networks

Transcription

1 Dstrbuted Topology Control for Power Effcent Operaton n Multhop Wreless Ad Hoc Networks Roger Wattenhofer L L Paramvr Bahl Y-Mn Wang Mcrosoft Research CS Dept. Cornell Unversty Mcrosoft Research Mcrosoft Research Redmond, WA 9 Ithaca NY Redmond, WA 9 Redmond, WA 9 rogerwa@mcrosoft.com ll@cs.cornell.edu bahl@mcrosoft.com ymwang@mcrosoft.com Abstract The topology of wreless multhop ad hoc networks can be controlled by varyng the transmsson power of each node. We propose a smple dstrbuted algorthm where each node makes local decsons about ts transmsson power and these local decsons collectvely guarantee global connectvty. Specfcally, based on the drectonal nformaton, a node grows t transmsson power untl t fnds a neghbor node n every drecton. The resultng network topology ncreases network lfetme by reducng transmsson power and reduces traffc nterference by havng low node degrees. Moreover, we show that the routes n the multhop network are effcent n power consumpton. We gve an approxmaton scheme n whch the power consumpton of each route can be made arbtrarly close to the optmal by carefully choosng the parameters. Smulaton results demonstrate sgnfcant performance mprovements. I. INTRODUCTION The lfetme of a wreless network that s operatng on battery power s lmted by the capacty of ts energy source. For ncreasng longevty of such networks and thus ncreasng ther usefulness, t s mperatve that we fnd ways of ether ncreasng battery power or alternatve tether-less sources of energy that nodes n a wreless network can use. A complementary approach to tacklng the network longevty problem s to develop energy-effcent algorthms and mechansms that optmze the use of the battery power whle mantanng network connectvty. Generally speakng, a node n a wreless network ndependently explores ts surroundng regon and establshes connectons wth other neghborng nodes that are wthn ts transmsson and recepton range. In establshng these local connectons, t s desrable to choose only those local connectons that wll guarantee overall global network connectvty whle satsfyng dfferent and often contradctory performance metrcs such as overall throughput, network utlzaton, and power dsspaton. Unlke wred networks, each node n a multhop wreless network can potentally change ts set of one-hop neghbors and consequently the overall network topology by smply changng ts transmsson and receve power. Wthout proper topology control algorthms n place a randomly connected multhop wreless ad hoc network may suffer from poor network utlzaton, hgh end-to-end delays, and short network lfetme. Although the problem doman s farly clear, there has been only a lmted amount of work n the general area of topology control and network desgn for ncreasng network longevty. Hu [] descrbes a dstrbuted, Delaunay trangulaton-based algorthm for choosng logcal lnks and as a consequence carryng out topology control. In choosng these lnks he follows a few heurstc gudelnes such as not exceedng an upper bound on the degree of each node and choosng lnks that create a regular and unform graph structure. He does not take advantage of adaptve transmsson power control. Ramanathan and Rosales- Han [] descrbe a centralzed spannng tree algorthm for creatng connected and b-connected statc networks wth the objectve of mnmzng the maxmum transmsson power for each node. Addtonally, they descrbe two dstrbuted algorthms, that adjust the node transmt power to mantan network connectvty. Ther reasonng and algorthms are based on smple heurstcs and consequently do not guarantee network connectvty n all cases. Rodoplu and Meng [] propose an ngenous dstrbuted topology control algorthm that guarantees connectvty of the entre network. Ther algorthm reles on a smple rado propagaton model for transmt power roll-off as =d n ;n. Usng ths they acheve the mnmum power topology, whch contans the mnmum-power paths from each node to a desgnated master-ste node. Other researchers workng n the feld of packet rado networks, wreless ad hoc networks, and sensor networks have also consdered the ssue of power effcency and network lfetme but have taken dfferent approaches. For example, Hou and L [] analyze the effect of adjustng transmsson power to reduce nterference and hence acheve hgher throughput as compared to schemes that use fxed transmsson power []. Henzelman et al. [] descrbe an adaptve clusterng-based routng protocol that max-

2 mzes network lfetme by randomly rotatng the role of per-cluster local base statons (cluster-head) among nodes wth hgher energy reserves. In ths paper we descrbe our approach to tacklng the network longevty problem. Specfcally, we descrbe a novel dstrbuted cone-based topology control algorthm that ncreases network lfetme whle mantanng global connectvty wth reasonable throughput n a multhop wreless ad hoc network. Network lfetme s ncreased by determnng the mnmal operatonal power requrement for each node n the network whle guarantyng the same maxmum connected node set as when all nodes are transmttng wth full power. In contrast to prevous approaches that rely on knowng and sharng the global poston nformaton of the nodes n the network, our algorthm s a dstrbuted algorthm that reles solely on local nformaton, usng drectonal nformaton of ncomng sgnals from neghborng nodes. We show the valdty of our algorthm both theoretcally and va smulaton. We show that the routes n the multhop network are effcent n power consumpton. We gve an approxmaton scheme n whch the power consumpton of each route can be made arbtrarly close to the optmal by carefully choosng the parameters. Our work s smlar to [], n that we have the same goal as them - of desgnng locaton-based, dstrbuted topology-control algorthm that ncreases network lfetme. We desgned our algorthm wth the followng objectves n mnd: () Each node n the multhop wreless network must use local nformaton only for determnng ts transmsson radus and hence ts operatonal power. The local decsons must be made n such a way that they collectvely guarantee the node connectvty n the global topology just as f all nodes were operatng at full power; () As n [] our algorthm must mnmze power consumpton by fndng mnmum power paths, and thus ndrectly ncreases network lfetme; () Our algorthm must fnd a topology wth small node degree, so that nterference s mnmal and hence throughput s suffcent. () Our algorthm must be smple and effcent so that t s sutable for small and moble (sensor) nodes. () Fnally, our algorthm must make very few assumptons about the rado propagaton model and/or on the hardware of each node (e.g. non-avalablty of Global Postonng System). We descrbe and analyze our cone-based topology control algorthm, whch meets these objectves. Our algorthm s desgned specfcally for multhop wreless ad hoc networks deployed on a -dmensonal surface. It conssts of two phases, whch are summarzed as follows: Startng wth a small radus, each node (denoted by Node u) broadcasts a neghbor-dscovery message. Each recevng node acknowledges ths broadcast message. Node u records all acknowledgments and the drectons they came from. (We assume that the node can determne the drecton of the sender when recevng a message.) It then determnes whether there s at least one neghbor n every cone of ff degrees, centered on Node u. In ths frst phase, Node u contnues the neghbor dscoverng process by ncreasng ts transmsson radus (operatonal power) untl ether the above condton s met or the maxmum transmsson power P s reached. We prove that, for ff smaller than or equal to ß=, the algorthm guarantees maxmum connected node set. For smaller angles we also can guarantee good mnmum power routes. In the second phase, the algorthm performs a redundant edge removal process wthout mpactng the connectvty. Ths phase s desgned to reduce the node degrees, whch helps n reducng nterference and enhancng throughput []. Redundant edge removal s carred out wthout deteroratng the mnmum power routes of the network. Our work s dfferent from Rodoplu and Meng [] n the followng way: Frst, our algorthm guarantees that the maxmum connected set of nodes for the network wll always be found. Second, our algorthm s computatonally less demandng, and we do not need to specfy a deployment regon, whch s an mportant consderaton for the case when nodes regularly change deployment regon. Thrd, our algorthm does not need exact locaton nformaton but only drectonal nformaton. Ths can be a factor when cost of nodes s a consderaton. Forth, our algorthm s not coupled wth any rado propagaton model. Due to the large nfluence of envronmental factor on rado frequency communcatons rado propagaton models can be notorously naccurate. Fnally, ffth our algorthm s able to gve a worst-case analyss for both, the mnmum power routes and the maxmum node degrees n the network. The rest of our paper s organzed as follows. In secton II we descrbe our network model and the assumptons we make about the envronment. In Secton III we descrbe the cone-based topology control algorthm n detal. In Secton IV we prove the correctness of our algorthm. In Secton V we demonstrates that our algorthm s compettve wth respect to mnmum energy path metrc. In Secton VI we present the results from our performance evaluaton of the algorthm. Fnally we conclude n Secton VII. II. MODEL We are gven a set V of n nodes (ponts n the plane). A node conssts of a power supply entty, a processor and local memory to perform smple local computatons, and a rado communcaton unt to send and receve messages. A node does not know ts poston.

3 A node s able to send a broadcast message wth arbtrary power p. It s called broadcast because the sendng node has no control over the drecton n whch the message s transmtted. Nodes can vary ther broadcast power, but not beyond a maxmum power P, that s» p» P. We assume the exstence of an underlyng MAC layer that resolves nterference problems. For example, f node u broadcasts wth power p, the nodes that can receve node u s broadcast message (the set N) wll acknowledge (wth another broadcast message) to node u. After havng receved acknowledgments of all nodes n N, node u knows the set N. The assumpton to have a relable broadcast s not needed for the correctness of our algorthm, but t smplfes the presentaton. We assume that the rado communcaton unt s able to determne the drecton of the sender when recevng a message. Thus, f two nodes u; v exchange a broadcast/acknowledge message par, both of them know whch drecton the other node s, that s, node u knows that node v s n drecton ρ, and node v knows that node u s n drecton ρ + ß, wth wth» ρ<ß. Technques to estmate drecton wthout postonng nformaton are avalable, and dscussed n the IEEE antenna and propagaton communty as the Angle-of-Arrval (AOA) problem. It can be accomplshed by usng more than one drectonal antenna. We refer to []. If the rado communcaton unt s not capable to conclude the drecton of a message, we can alternatvely supply a node wth a more abundant global postonng unt, and calculate drectons from postons pggybacked to messages. Compared wth [], we have a weak physcal rado propagaton model. We assume that the envronment s not obstructed, and that the nodes are homogeneous. More formally, we assume that the power p s a unform and nondecreasng, but unknown functon of the dstance d. Due to unformty, f a node u can reach node v wth power p p, then node v can also reach node u wth power p p. In other words, a node u can fgure out how much power s needed to communcate wth node v but cannot deduce the dstance of v. Power models lke Rcan are ntutvely appealng, but t s very dffcult to determne the model parameters such as the local mean of the scattered power and the power of domnant component precsely as ths requres physcally solatng the drect wave from the scattered components. In order to keep our system smple and easy to deploy, we decded aganst models that are unduly complex. For an excellent dscusson on the applcablty of other power models we refer to Secton of []. III. ALGORITHM Our algorthm has two phases. In the frst phase we descrbe a decentralzed scheme that bulds a connected graph upon our node network by lettng nodes fnd close neghbor nodes n dfferent drectons. The second phase mproves the performance by elmnatng non-effcent edges n the communcaton graph. The algorthm s smple and does not need any complcated operatons. The algorthm s also dstrbuted and wthout synchronzaton. The two phases are merely for the ease of descrpton. The frst phase of the algorthm: Each node u beacons wth growng power p, ntally p = ffl. If node u dscovers a new neghbor node v, node u wll put v nto ts local set of neghbors N (u). Node u wll contnue to grow the transmsson power untl the neghbor set N (u) s bg enough such that, for any cone wth angle ff there s at least one neghbor v N (u), or untl node u hts the maxmum transmsson power P. The termnaton crteron can be easly determned. For a gven node u, each neghbor v N (u) covers a cone, as n Fgure. If the unon of these cones cover the whole ß angle, node u goes to phase. Fg.. Coverage determnaton Node u mght use heurstcs n order to optmze conflcts on the lower MAC layer. For example, node u wll grow the transmsson power so that exactly one new neghbor s expected to acknowledge, gven the probablty dstrbuton of the nodes n the plane. Moreover, node u mght nclude meta-nformaton n ts broadcast, n order to prevent already establshed neghbors to answer agan, or n order to fnd new neghbors n a specfc drecton, where no neghbor has been found yet. These optmzatons are not essental for the correctness of our algorthm;

4 they are subject of future work. For node u, let p(u) be the mnmum power to fnd a neghbor n every cone wth angle ff, or p(u) = P. Dependng on our goal we wll later specfy ff to be at most ß= (correctness only) or at most ß= (compettve performance). If a node u wth maxmum transmsson power P has a cone C = [ρ; ρ + ff] wthout any node n N (u), then node u wll decrease ts transmsson power, back to the mnmum power p such that there s no cone wthout a neghbor that has a neghbor when transmttng wth maxmum power P. The algorthm s symmetrc, that s, f node u wants node v to be n ts neghbor-set, then node v also needs to put node u n ts neghbor-set. From the algorthm descrpton of phase we conclude: Fact III.: For each node u and for each angle ρ (» ρ < ß), f there s a node v n the cone C = [ρ; ρ + ff] when sendng wth maxmum power P, then there s a node v n the cone C when sendng wth mnmum power p(u). Because of the smple nature of the frst phase of the algorthm there s room for mprovement. The second phase of the algorthm: If node u has two neghbor nodes v; w N (u), such that the power needed to send from u to w drectly s not less than the total power to send va v, we can remove w from N (u). More formally, f there are two nodes v; w wth v; w N (u) and w N (v), and p(u; v) +p(v; w)» p(u; w), then we remove node w from N (u). Ths mprovement gves us less neghbors, whle keepng all the best routes. We can determne two neghbors v; w for whch ths basc power nequalty holds by some smple local exchange of the transmsson powers, or, f dstances and power model are known, by a smple local computaton step wthout any message exchange. It s beleved that, from a performance pont of vew, a node should have as few neghbors as possble. Thus we mght consder removng nodes from our neghborhood even though a drect transmsson uses less power than an ndrect. One good canddate for removal s a neghbor node v that s n great dstance of the sendng node u, snce, whenever u transmts to dstant neghbor v, many other nodes experence nterference. We extend the frst dea n the followng way: If there are two nodes v; w wth v; w N (u) and w N (v) and p(u; v)» p(u; w), and p(u; v) +p(v; w)» q p(u; w), then we remove w from N (u) (and by symmetry also u from N (w)). If there s more than one node v that satsfes the power nequalty for node w, we chose the node wth mnmum p(u; v). By traversng the neghbor nodes wth ncreasng power dstance (wth tes broken by dentfer), we make sure that the edge (u; v) wll stay. Note that f constant q =we only remove edges that use more power than an ndrect path. From the algorthm descrpton of phase we conclude: Fact III.: For each node u, f there was a neghbor node w N (u) after the frst phase of the algorthm, there s a neghbor v N (u) after the second phase of the algorthm such that p(u; v) +p(v; w)» q p(u; w), for a constant q. Note that after phase of the algorthm Fact III. s not necessarly true anymore. Let us sum up ths secton. We have presented an algorthm that, startng from a set of nodes V, bulds an undrected graph G = (V;E) such that there s an edge e =(u; v) f and only f v N (u) (and because of symmetry also u N (v)). Ths graph G has several advantageous propertes, whch wll be proven n the next two sectons of ths paper: ffl If ff» ß=, the graph G wll be connected f t was connected when all nodes broadcast wth maxmum power P. ffl For a reasonable class of power cost functons and for ff» ß= we wll show that the graph G has very good power consumpton, n fact wthn an arbtrarly small constant factor of the optmal (acheved by a much more complcated algorthm). ffl The degree of any node can be bounded by a constant, for q. IV. CORRECTNESS In ths secton we wll prove that an angle ff» ß= s suffcent to make the graph G connected. Defnton IV.: A path p of nodes s an ordered set (u ;u, :::;u k ) of nodes such that there s an edge between consecutve nodes: e = (u ;u + ) for = ;:::;k wth e E. A graph s connected f there s a path from any node to any other node n the graph. Defnton IV.: The dstance of two nodes s ther Eucldean dstance n the plane. Let G = (V;E) be the graph constructed by our algorthm. On the other hand, let G = (V;E ) be the connecton graph when all nodes always beacon wth maxmum power P. Theorem IV.: We have ff» ß=. Let G be connected. Then graph G wll be connected. Proof: We prove the frst phase of the algorthm by contradcton. Assume that graph G s not connected, whle G s. Then there exsts a least a par of nodes such that there s no path between the par. Let the nodes u,v be the par wth mnmum power to beacon each other, that s p(u; v)» p(u ;v ) for any par of nodes u,v wthout a path. Snce G s connected we know that p(u; v)» P.

5 Let d := p (u; v), that s wth power p(u; v) one can reach dstance d. The algorthm has gven node u mnmum transmsson power p(u). Snce there s no edge e =(u; v) we have p (u) <d. The remander of the proof s geometrc. Let w be a neghbor node of u. We construct a trangle of the nodes u; v; w, such as n Fgure. b < d u w a = d c > d Fg.. Trangle u; v; w A basc trangle result s c = a + b ab cos fl. We have a = d(u; v) = p (u; v) = d, b = d(u; w) = p (u; w)» p (u) < d, and c = d(v; w) = p (v; w) p (u; v) d. We are nterested n the angle fl whch s on the opposte of sde c. We get mmedately: cos fl = a + b c ab» b db < : and thus fl>ß= ff=. Therefore there s no node v N (u) n the cone C = [ ff=; +ff=]; ths contradcts Fact III.. By symmetry the same holds for v. The second phase of the algorthm does not destroy connectvty snce we only remove an edge (u; w) when we made sure that there are edges (u; v) E and (v; w) E. V. COMPETITIVE ANALYSIS In ths secton we wll show that our algorthm s not only correct (results n a connected graph), but also that the routes that can be found n the graph are very power effcent. In ths secton, we need to make stronger assumptons on the power model. Defnton V.: A drect transmsson from node u to node v costs power p = p(u; v), where p s a functon of the dstance d = d(u; v). Any functon p(d) s elgble as long as cd x» p(d)» czd x, for parameters c; x; z, all ndependent of d, and wth z, and x. Defnton V.: The power consumpton P of route r = k (s = u ;u ;:::;u k = t) s C(r) = p(u = ;u + ). v Let G be the graph when all nodes transmt wth maxmum power P,as defned n Defnton IV.. For gven source node s and snk node t, let r Λ be a route such that C(r Λ )» C(r), for any elgble route r n G. Then route r Λ s a mnmum power route n G. After our algorthm has done the neghborhood detecton as descrbed n Facts III. and III., a routng algorthm s appled that fnds mnmum power routes n the graph G. In other words, nodes keep tables that tell them to whch neghbor they should send n order to route a message to a gven destnaton node. These tables are generally small snce the geometry of the plane can be used []. (Usually nodes n a destnaton regon wll be sent to the same neghbor.) We drectly get: Defnton V.: Let G be the graph constructed by our algorthm, as defned n Defnton IV.. For gven source node s and snk node t, let ^r be a route such that C(^r)» C(r), for any elgble route r n G. Then route ^r s a mnmum power route n G. Lemma V.: We are gven a trangle where angle fl ß=. Then a x + b x» c x for x. Proof: We have cos fl». Wth c = a + b ab cos fl, we get a + b» c. From a= sn ff = b= sn f = c= sn fl we know that» a; b» c and we drectly get a x + b x» a c x + b c x =(a + b )c x» c x. Lemma V.: We have a trangle wth nodes A; B; C, edges a; b; c and angles ff; f; fl. Let b < c and b < a and ff» ß=. Then fl ß=, and a<c. Proof: Wth a= sn ff = b= sn f and b<awe know that f<ff. Wth ff» ß= we get fl = ß ff f ß=, and wth a= sn ff = c= sn fl we get a<c. Lemma V.: We have a trangle wth nodes A; B; C, edges a; b; c and angles ff; f; fl. Let a» b < c and ff» ß=. Then a x + b x» c x ( + sn(ff=)) for x. Proof: We know that fl (ß ff)=. Wth c = a + b ab cos fl and a» b<c,weget a + b <c +c cos fl» c ( + sn(ff=)): We use the same method as n Lemma V. to extend ths result for x. Theorem V.: We have ff» ß=. Let s be a source node and t be a snk node. Let C(^r) resp. C(r Λ ) be the mnmum power routes n G resp. G,asnDefntons V. and V.. Then C(^r)» C(r Λ )zq( + sn(ff=)), for z from the rado model V. and q from Fact III.. Proof: Frst we consder phase of the algorthm: The mnmum power route r Λ s an ordered set of nodes r Λ = (s = u ;u ;:::;u k = t), where p(u ;u + )» P for all =;:::;k.

6 In ths proof we wll show that our algorthm fnds a path r = (s = u ;u ;u ;:::;u l ;u ;u ;u ;:::;u l ; u ;:::;u k ;u k ;u k ;:::;u l k k ;u k = t); where u n r s the same node as u n r Λ. We focus on the path between u and u +. Let l = l, and for convenence u = u, for any. Let us construct our path (u ;u ;u ;:::;ul +). We dstngush the followng cases. Case : Nodes u j and u + are neghbors n the graph G. Then l = j. Case : Nodes u j and u + are not neghbors. Snce there s a neghbor n each cone [ρ ff=;ρ + ff=] we know that node u j has a neghbor node uj+ such that the angle at node u j (n the trangle uj ;uj+ ;j + ) s less than ff=. Case a: If d(u j ;u +) < d(u j+ ;u + ), we know by Lemma V. that the angle at u j+ s at least ß=, and that node u j+ s strctly closer to u + than node u j was. Snce there are only a fnte number of nodes we wll eventually arrve at node u +, or get nto one of the other cases. Case b: If d(u j ;u +) d(u j+ ;u + ), we know that node u + s a neghbor of node u j+. Thus l = j +. Fgure shows an example of a path from u to u +, where we have a seres of cases a, followed by a sngle case b. u u u u l- u u + Fg.. Path from u to u + Let us calculate the cost of our path from u to u +. C(u ;u + ) = p(u ;u )+p(u ;u )+::: +p(u l ;u l )+p(ul ;u + ): By usng our rado model (Defnton V.) we know that p(d)» czd x, thus C(u ;u + )» czd x (u ;u )+czdx (u ;u )+::: +czd x (u l ;u l )+czdx (u l ;u + ): We know that all nodes except u l are of case a, therefore we can apply Lemma V. repeatedly, and get C(u ;u + )» czd x (u ;u l )+czdx (u l ;u +): u l By Lemma V. we know that C(u ;u + )» czd x (u ;u + )( + sn(ff=)): From the rado model (Defnton V.) we know that cd x» p(d), thus C(u ;u + )» p(u ;u + )z( + sn(ff=)): We can use the same analyss for all the peces of the optmal path. Wth fact V. we get C(^r)» C(r)» C(r Λ )z( + sn(ff=)): Wth Fact III., phase of the algorthm mght replace an edge wth two edges such that the total power consumpton s at most multpled wth a factor q. The Theorem follows drectly. Corollary V.: Let z = q =. In order to guarantee paths that use at most +ffl of the power of the optmal paths we need ff» arcsn(ffl=), whch s roughly ff» ffl. The second phase of the algorthm already helps to arrve at a sparse graph, as you can see n the smulaton secton. More formally: Theorem V.9: Let q of phase (confer Fact III.) be not less than. Then the degree at any node s at most. Proof: For node u, let v; w be two nodes n the neghborhood of u. Because of symmetry, ether () v N (w) and w N (v), or () both v = N (w) and w = N (v). Case (): If there are three nodes u; v; w such that they all are n each other s neghborhood, then phase wll at least remove the edge wth maxmum power between them. The largest angle fl n the trangle u; v; w s at least ß=. Therefore the edge that uses most power s at least the same sze as the other two, and q would remove that edge. Case (): We have p(u; v)» p(v) <p(v; w) >p(w) p(u; w). Therefore the sde opposte of node u s the largest n the trangle u; v; w, and the angle at node u s the largest,.e. larger than ß=. In both cases any two nodes v; w n the neghborhood of u have at least angle ß=. There cannot be more than neghbors. VI. SIMULATION RESULTS AND EVALUATION We measure the mpact of our topology control on the network through smulatons. To compare the performance of our algorthm wth pror work n topology control, we would also want to smulate topology control algorthms n the lterature. In a multhop wreless network, each node s expected to potentally send and receve messages from many nodes. Therefore an mportant requrement of such

7 network s strong connectvty. Besdes strong connectvty, the most mportant desgn metrc of multhop wreless networks s perhaps energy effcency. As t drectly mpact the network lfetme. As far as we know, among the topology control algorthms n the lterature [], [], [], [], [], only Rodoplu and Meng s algorthm [] attempts to optmze for energy effcency subject to mantanng strong network connectvty. The work n [], [], [] tres to maxmze network throughput. Ther algorthms do not guarantee strong connectvty. Ramanathan and Rosales-Han [] has consdered optmzng for the mnmax transmsson power n centralzed algorthms, however ther dstrbuted heurstc algorthms do not guarantee strong connectvty. Therefore, we only compare wth []. We refer to ther algorthm as R&M. We refer to our basc algorthm as PhaseOnly, and to our complete algorthm wth ConeBased. Sometmes we gve the parameter ff, the sze of the angle of the cone. As a reference, we also compare wth the no topology control case where each node always uses the maxmum transmsson radus for broadcastng a packet (MaxPower). For example, the AODV [9] route request packet s sent usng neghbor broadcast. Uncast packet only needs to use the mnmum power to reach a gven next hop. The use of maxmum transmsson radus for broadcast packets s the only way to avod unnecessary partton f no topology control s used. A. Smulaton Envronment Our topology control algorthm s mplemented n ns- [], usng the wreless extenson developed at Carnege Mellon []. Our smulaton s done for a network of nodes wth WaveLAN-I rados. The nodes are placed unformly at random n a rectangular regon of by meters. There has been some work on realstc topology generaton such as [], []. However, ther work has the Internet n mnd. Snce large multhop wreless networks such as sensor networks are deployed automatcally, we beleve unform random assumpton s vald n most such networks. We assume the two-ray propagaton model for terrestral communcatons. It has a =d transmt roll-off []. The model has been shown to be close to realty n many envronment settngs []. The carrer frequency s 9MHz, and the transmsson raw bandwdth MHz. We assume omn-drectonal antennas wth db gan,and the antenna s placed : meter above a node. The receve threshold s 9dBW. The carrer sense threshold s dbw and the capture threshold s db. These parameters smulate the 9MHz Lucent WaveLAN DSSS rado nterface. In order to smulate the effect of power control, we made changes to the physcal layer of the ns- smulaton code. Specfcally, for every neghbor broadcast packet, a node s transmsson power uses the fnal transmsson power of ts neghbor dscovery process of each topology control algorthm. For every uncast packet, a node s transmsson power uses the mnmum power for the source to reach the destnaton, as determned durng the neghbor dscovery process. A node s energy reserve s then subtracted by the approprate amount for any transmsson and recepton. To smulate nterference and collson, we choose the WaveLAN-I CSMA/CA MAC protocol. Snce topology control s ndependent of routng, a routng protocol s needed. We choose AODV n our smulaton. Other protocols to dssemnate applcaton data wthout an explct routng protocol n sensor network can also be used [], []. Snce [] optmzes for mnmum energy path metrc, we modfy the ns- AODV mplementaton wth the mnmum energy path metrc nstead of usng the current shortest path metrc. To smulate the network applcaton traffc, we use the followng applcaton scenaro: All nodes perodcally send UDP traffc to the master data collecton ste stuated at the boundary of the network. Ths applcaton scenaro has also been used n []. Network traffc characterstcs has been studed extensvely n the telephony network and the Internet [], []. Although our applcaton traffc scenaro s not vald n those settngs, t does represent a set of envronment montorng sensor applcatons. In ths settng, sensors perodcally transmt data to the data collecton ste. The data collecton ste wll analyze the data for nterestng events. B. Analyss of the Resultng Topology of Dfferent Topology Control Algorthms Before we move on to smulate dfferent topology control algorthms, we would lke to understand the characterstcs of the resultng topology of dfferent topology control algorthms. Fgure shows the topology generated by dfferent topology control algorthms. The average node degree of each topology s shown n Table I. The average degree μ d of the multhop wreless networks should not be too large because a large μ d typcally mples that a node has to communcate wth other dstant nodes drectly. Ths ncreases nterference and collson, and would waste energy. The average degree μ d should not be too small ether because that tends to ncrease the overall network energy consumpton as longer paths have to be taken. So we beleve the average node degree s an mportant performance metrc for multhop wreless network topology. Other metrcs lke k-connectvty and regular structure are also mportant. Those metrcs wll be our future research. The average node degree of our PhaseOnly ncreases as the ff

8 (a) Phase Only ff =ß= (b) Phase Only ff = ß= (c) Cone Based ff =ß= (d) Cone Based ff = ß= (e) R&M [] (f) Max Power Fg.. Topology Graph of dfferent topology control algorthms parameter decreases. We remark that the boundary node contrbutes more to the average degree statstcs. Ths s because, n tryng to cover the maxmum angle, t tends to nvolve more dstant nodes. The average node degree of the nner nodes are much less than the average (as shown n Fgure ). We remark that the average node degree can be reduced f we know the boundary of the network. It s true that our PhaseOnly topology has a much hgher μd than R&M. However, n the envronment where there s only drectonal nformaton, PhaseOnly works whle R&M does not. Our ConeBased algorthm mplements the redundant

9 edge removal as descrbed n Secton III wth q =. As shown n both Table I and Fgure, t generates smlar low degree topology graph as R&M algorthm. C. Network Performance Analyss of Dfferent Topology Control Algorthms Number of Nodes Alve Average Number of Neghbors Cone Based R & M [] Phase Only Max Power Tme Fg.. Network lfetme Cone Based R & M [] Phase Only Max Power Tme Fg.. Average node degree over tme We would lke to measure the network performance usng dfferent topology control algorthms. We partcularly care about network lfetme n the multhop wreless networks envronment. We measure the network lfetme as the number of nodes stll alve over tme. We also want to understand how the network topology evolves over tme. We only smulated a statc network. If moblty s low, a proactve approach to reconfgure the network topology may be used. If the moblty s hgh, an on-demand approach to reconfgure the network topology may be the only vable way to keep the reconfguraton control traffc low. How to make the topology control algorthm deal wth moblty effcently s our future research. In our smulaton, we do not smulate the process of adjustng to the rght transmsson radus. It s adjusted to the rght transmsson radus mmedately after AODV detects a node that has faled. As can be seen from Fgure, our ConeBased algorthm performs as good as the R&M algorthm, whle usng only drectonal nformaton.. They both perform sgnfcantly better than MaxPower. From Fgure, we see that when % of the MaxPower nodes are dead, both ConeBased and R&M stll have around % percent of nodes alve. Our PhaseOnly algorthm performs not as good as our ConeBased algorthm and the R&M algorthm, but t performs much better than no topology control case. When % of the MaxPower nodes are dead, PhaseOnly stll has more than % of nodes alve. It s nterestng to see that some constant number of nodes stay alve for all the topology control algorthms except MaxPower. The reason s that, when a node s parttoned from the rest of the network, f ts lower layer receves an AODV route request packet whch s a broadcast packet, t wll be sent wth zero transmsson range due to topology control. However, MaxPower wll stll be broadcastng wth maxmum radus snce t has a pre-confgured transmsson power. Fgure shows how the network topology evolves over tme. It s nterestng to note that the topology control algorthms tend to mantan the same average node degree for the remanng alve nodes as nodes de over tme. The average node degree decreases notceably only when the network has less than % nodes alve. Snce MaxPower do not respond to topologcal changes, the average node degree wll decrease quckly over tme. We also collected throughput statstcs at the end of our smulaton. Our ConeBased algorthm and the R&M algorthm acheve tmes the throughput of the Max- Power. Our basc PhaseOnly algorthm acheves tmes the throughput of the MaxPower. The throughput statstcs show that t s undesrable to transmt over large radus. Ths wll ncrease energy consumpton and also cause unnecessary nterference. Increased nterference wll result n decreased throughput. VII. CONCLUSION The lfetme of a wreless network operatng on battery power s crtcal to ts usefulness. Network lfetme can be

10 Phase Only Cone Based R&M [] Max Power Average ff =ß= ff = ß= ff =ß= ff = ß= Node Degree TABLE I AVERAGE DEGREE OF DIFFERENT TOPOLOGY CONTROL ALGORITHMS ncreased by effcently managng the power-consumpton n each ndvdual node belongng to the network. In ths paper we descrbe a dstrbuted cone-based topology control algorthm that determnes the mnmal power consumpton operatng pont for each node n a multhop wreless ad hoc network. Our algorthm s unque n that t requres only local reachablty nformaton to determne the node power-consumpton that guarantees a maxmum connected node set. Runnng on every node n the wreless mode, our algorthm uses n-exact drecton nformaton about the locaton of neghborng nodes for makng operatng pont decsons. The result s an approxmaton scheme that s able to brng the total power consumed for each route arbtrarly close to optmal. We prove our algorthm theoretcally and present results obtaned va extensve ns- based smulatons that show ts valdty. We focus prmarly on the statc ad hoc multhop network topology case, leavng the case of moble nodes and changng network topology to future research. [] THE VINT PROJECT, The ucb/lbnl/vnt network smulator-ns (verson ), [] THE CMU MONARCH GROUP, Wreless and moblty extensons to ns-, October 9. [] E.W. Zegura, K. Calvert, and S. Bhattacharjee, How to model and nternetwork, n Proc. IEEE Infocom,. [] K. Calvert, M. Doar, and E.W. Zegura, Modelng nternet topology, IEEE Communcatons, June. [] T.S. Rappaport, Wreless communcatons: prncples and practce, Prentce Hall,. [] C. Intanagonwwat, R. Govndan, and D. Estrn, Drected dffuson: a scalable and robust communcaton paradgm for sensor networks, n Proc. ACM/IEEE MOBICOM,. [] W. Henzelman, J. Kulk, and H. Balakrshnan, Adaptve protocols for nformaton dssemnaton n wreless sensor networks, n Proc. ACM/IEEE MOBICOM, 9. [] V. Paxson and S. Floyd, Wde-area traffc: the falure of Posson modelng, IEEE/ACM Transactons on Networkng, vol., no., pp.,. [] V. Paxson and S. Floyd, Why we don t know how to smulate the nternet, Proceedngs of the Wnter Smulaton Conference, pp.,. REFERENCES [] L. Hu, Topology control for multhop packet rado networks, IEEE Trans. on Communcatons, vol., no., October. [] R. Ramanathan and R. Rosales-Han, Topology control of multhop wreless networks usng transmt power adjustment, n Proc. IEEE Infocom, March. [] V. Rodoplu and T. H. Meng, Mnmum energy moble wreless networks, IEEE J. Selected Areas n Communcatons, vol., no., August 9. [] T.-C. Hou and V. O. K. L, transmsson range control n multhop rado networks, IEEE Trans. on Communcatons, vol., no., pp., January 9. [] H. Takag and L. Klenrock, Optmal transmsson ranges for randomly dstrbuted packet rado termnals, IEEE Trans. on Communcatons, vol., March 9. [] W. R. Henzelman, A. Chandrakasan, and H. Balakrshnan, Energy-effcent communcaton protocol for wreless mcrosensor networks, n Proc. IEEE Hawa Int. Conf. on System Scences, January, pp.. [] K. Krzman, T. E. Bedka, and T.S. Rappaport, Wreless poston locaton: fundamentals, mplementaton strateges, and source of error, n IEEE th Vehcular Technology Conference,, vol.. [] B. Karp and H.T. Kung, Greedy permeter stateless routng (gpsr) for wreless networks, n Proc. ACM/IEEE MOBICOM,. [9] C. Perkns, Ad hoc on demand dstance vector(aodv) routng, Internet Draft, draft-etf-manet-aodv-.txt, October 9.