Optimal Local Topology Knowledge for Energy Efficient Geographical Routing in Sensor Networks

Transcription

1 Optmal Local Topology Knowledge for Energy Effcent Geographcal Routng n Sensor Networks Tommaso Meloda, Daro Pompl, Ian F. Akyldz Broadband and Wreless Networkng Laboratory School of Electrcal and Computer Engneerng Georga Insttute of Technology Atlanta, GA e-mal: {tommaso,daro,an}@ece.gatech.edu Abstract Snce sensor networks can be composed of a very large number of, the developed protocols for these networks must be scalable. Moreover, these protocols must be desgned to prolong the battery lfetme of the. Typcal exstng routng technques for ad hoc networks are known not to scale well. On the other hand, the so-called geographcal routng algorthms are known to be scalable but ther energy effcency has never been extensvely and comparatvely studed. For ths reason, a novel analytcal framework s ntroduced. In a geographcal routng algorthm, the packets are forwarded by a node to ts neghbor based on ther respectve postons. The proposed framework allows to analyze the relatonshp between the energy effcency of the routng tasks and the extenson of the range of the topology knowledge for each node. The leadng forwardng rules for geographcal routng are compared n ths framework, and the energy effcency of each of them s studed. Moreover Partal Topology Knowledge Forwardng, a new forwardng scheme, s ntroduced. A wder topology knowledge can mprove the energy effcency of the routng tasks but can ncrease the cost of topology nformaton due to sgnalng packets that each node must transmt and receve to acqure ths nformaton, especally n networks wth hgh moblty. The problem of determnng the optmal Knowledge Range for each node to make energy effcent geographcal routng decsons s tackled by Integer Lnear Programmng. It s demonstrated that the problem s ntrnscally localzed,.e., a lmted knowledge of the topology s suffcent to take energy effcent forwardng decsons, and that the proposed forwardng scheme outperforms the others n typcal applcaton scenaros. For onlne soluton of the problem, a probe-based dstrbuted protocol whch allows each node to effcently select ts topology knowledge, s ntroduced and shown to converge to a near-optmal soluton very fast. Index Terms Wreless Sensor Networks, Mathematcal programmng/optmzaton, Poston Based routng, Topology Control. I. INTRODUCTION Recent advances n wreless communcatons and electroncs are pavng the way for the deployment of low-cost, lowpower networks of untethered and unattended sensors and actuators. Sensor networks [] dffer from tradtonal ad hoc networks n many aspects. The number of n a sensor network can be several orders of magntude hgher than n ad hoc networks, and the deployment of s usually denser. Moreover, sensor are lmted n power, computatonal capactes and memory, and they may not have global dentfcaton (ID) because of the very large number of and the accordng overhead. Because of the above constrants, sensor networks protocols and algorthms must possess self-organzng capabltes,.e. sensors must be able to cooperate n order to organze and perform networkng tasks n an effcent way. The prmary desgn constrants of these algorthms are: energy effcency, scalablty and localzaton. It has been ponted out n [] that the mproved energy effcency can be obtaned by desgnng protocols and algorthms wth a cross-layer approach,.e., by takng nto account nteractons among dfferent layers of the communcaton process so that the overall energy expendture can be mnmzed. In ths paper we consder dependences between physcal and network layers wth the objectve to perform energy effcent routng tasks. All networkng tasks, such as routng, should perform well for wreless networks wth an arbtrary number of. A scalable algorthm performs well n a large network. The noton of scalablty for an algorthm s strctly related to that of localzaton: n a scalable algorthm each node exchanges nformaton only wth ts neghbors (localzed nformaton exchange) n a very large wreless network []. In a localzed routng algorthm, each node decdes on the next hop based only on the poston of tself, of ts neghbors, and of the destnaton node. As a result, the local node behavor tres to acheve global network objectves such as mnmum latency, mnmum energy consumpton, etc. On the other hand, n a non-localzed routng algorthm a node mantans an accurate descrpton of the overall network topology to compute the next hop, so that a global objectve can be maxmzed. The routng problem becomes then equal to the shortest path problem f the hop count s used as the global performance metrc or the shortest weghted path f power [] or cost [][] lnk metrcs are used. It has been shown n [][] that the routng protocols whch do not use geographcal locaton nformaton are not scalable, e.g, AODV (Ad hoc on-demand Dstance Vector), DSDV (Destnaton Sequenced Dstance Vector) or DSR (Dynamc Source Routng). On the other hand, the recent avalablty of small, nexpensve and low-power GPS (Global Postonng System) recevers, together wth technques whch can deduce relatve sensor coordnates from sgnal strengths [] encourage people to deploy Geographcal Routng [] (also Poston

2 Based Routng) algorthms whch are becomng most promsng scalable solutons for crtcally power-constraned sensor networks. For these reasons ths paper deals wth the nteractons between topology control [] and energy effcent geographcal routng. The queston we try to answer s How extensve should be the Local Knowledge of the global topology n each sensor node, so that an energy effcent geographcal routng can be guaranteed?. Ths queston s clearly related to the degree of localzaton of the routng scheme. If each sensor node could have the complete knowledge of the topology, t could then compute the global optmal next hop whch mnmzes the energy expendture. However, the process of acqurng complete topology nformaton has a cost,.e., energy spent to exchange the sgnalng traffc. We develop an analytcal framework to capture the tradeoff between what we call the topology nformaton cost, whch ncreases wth the Knowledge Range of each node, and the communcaton cost, whch decreases when the knowledge becomes more complete. We apply ths analytcal framework to dfferent poston based forwardng schemes [][][][][] and demonstrate by usng Monte Carlo smulatons that a lmted knowledge s suffcent to make energy effcent routng decsons. Wth respect to exstng lterature on geographcal routng, we try to better defne the terms localzed and neghbor. A neghbor for a certan sensor node s another node whch falls nto ts topology Knowledge Range, denoted as KR n what follows. Our man contrbutons are: ) We ntroduce a novel analytcal framework to evaluate the energy expendture of geographcal routng algorthms [] for sensor networks; ) We gve an Integer Lnear Programmng (ILP) formulaton of the topology Knowledge Range optmzaton problem; ) We provde a detaled comparson of the leadng exstng forwardng schemes [][][][][] and ntroduce a new scheme called Partal Topology Knowledge Forwardng (); ) For the on-lne soluton of the problem we ntroduce the PRobe-bAsed Dstrbuted protocol for knowledge range adjustment (PRADA), whch allows the network to select near-optmal Knowledge Ranges n a dstrbuted way. The remander of the paper s organzed as follows. In Secton II we revew the forwardng schemes [][][][][] for geographcal routng and other related work. In Secton III we state the problem and n Secton IV we provde a mathematcal formulaton of the optmzaton problem. In Secton V we ntroduce the dstrbuted protocol for Knowledge Range adjustment and n Secton VI we show numercal results obtaned usng the above analytcal framework. Fnally, n Secton VII we conclude the paper. II. RELATED WORK Frst we descrbe the exstng poston based forwardng rules whch wll be utlzed n the remander of the paper. Fg.. Dfferent Forwardng Schemes A. Forwardng Rules In a localzed routng scheme, the node S (Fg. ) whch currently holds a message, knows only the poston of ts neghbors,.e., the wthn ts Knowledge Range, and the destnaton node D. Defnton : Gven a sender node S, and a destnaton node D, theprogress of a generc node X, neghbor of S, s defned as the orthogonal projecton of the lne connectng S and X onto the lne connectng S and D. Takag and Klenrock [] proposed the frst geographcal routng scheme based on the noton of progress. In ther Most Forward wthn Radus () scheme [], the packet s forwarded to the neghbor whose progress s maxmum, e.g., the node M, whose progress s Sm, n Fg.. Note that though node G s closer to the destnaton, ts progress Sg s smaller than Sm. Hou and L [] dscuss the Nearest Forward Progress () method whch selects the neghbor wth the mnmum progress wthn the Knowledge Range of S, e.g., the node N n Fg., whose progress s Sn. Fnn [] proposes the Greedy Routng Scheme (), based on the geographcal dstance where the node S selects from ts neghbors the closest one to ts destnaton, e.g., G n Fg.. In the so-called Routng method [], the message s forwarded to a neghbor, e.g., C n Fg., such that the drecton SC s the closest to the drecton SD,.e., the angle CSD s mnmum. In the so-called Random Progress Forwardng () method [] a random next hop s selected among the wthn the Knowledge Range. Let us now ntroduce the followng Defnton : Gven a sender node S and a destnaton node D, the advance of a generc neghborng node X s defned as the dstance between S and D mnus the dstance between X and D. A suffcent condton for a geographcal routng scheme to be loop free s that only next hop wth postve advance can be selected. Accordng to Defnton, a generc neghbor has a postve advance wth respect to a sender node f t s closer than the sender to the destnaton. When a routng scheme selects next hop only f they have postve advance, then

3 Fg.. Counterexample on the noton of progress the overall path s guaranteed to be loop free. On the other hand, a postve progress for each next hop s not a suffcent condton for a routng scheme to be loop free, as can be nferred from the counterexample n Fg., where three, A, B and a destnaton node D are shown. A s a possble next hop for B and vce versa, snce both A and B have postve progress wth respect to each other (Ak >, Bh > ). However, ths does not avod loops. Both node could choose the other as next hop, thus generatng a loop. Conversely, loops are avoded when the postve advance crteron s used as a necessary condton for a node to be the next hop. Referrng agan to the example n Fg., when a postve advance s a necessary condton for a node to be next hop, A s feasble next hop for B, but not vce versa, snce A s closer than B to the destnaton (AD < BD). Snce postve advance s a stronger condton, and guarantees loop free paths, we assume a postve advance as a necessary condton for a node to be the next hop n what follows. In other words, a node must choose the next hop among the wthn ts Knowledge Range and wth postve advance wth respect to the destnaton node, for all the consdered forwardng schemes. B. Other Related Work Here we revew related work on geographcal routng, whch consttutes the background of our work. An excellent survey on poston based routng technques for ad hoc networks s gven n [], []. The methods to determne absolute and relatve coordnates for network,.e., on locaton update technques are revewed n []. Most of the pror research assumes that can work ether n greedy mode or n recovery mode. In the greedy mode, the node that currently holds the message can forward t towards the destnaton. The recovery mode s entered when a node fals to forward a message n the greedy mode, snce none of ts neghbors s a feasble next hop. Usually ths occurs because the node observes a vod regon between tself and the destnaton. For example the Greedy Permeter Stateless Routng (GPSR), ntroduced n [], makes greedy forwardng decsons (as n Secton II.A). When a packet reaches a concave node, the GPSR tres to recover by routng around the permeter of the vod regon. Recovery mechansms, whch allow a packet to be forwarded to the destnaton when a concave node s reached, are out of the scope of our paper. Here we assume that the packet s drectly forwarded to the destnaton whenever such a node s reached. The so-called Trajectory Based Forwardng (TBF) s proposed n [] where the packet s forwarded along a predefned parametrc curve encoded n the packet at the source. Several localzed algorthms for power, cost and power-cost effcent routng are proposed and ther effcency s analyzed n []. Scalablty propertes of dfferent ad hoc routng technques, such as flat, herarchcal and geographcal routng are dscussed n []. A topology control algorthm called GAF gven n [] dentfes, based on poston nformaton, that are equvalent from a routng perspectve and adaptvely turns unnecessary off n order to mantan a constant level of performance. A taxonomy of locaton systems s gven n [] for ubqutous computng applcatons ncludng locaton sensng technques and propertes as well as a survey of commercally avalable locaton systems. In [] t s shown how to derve poston nformaton for all usng Angle of Arrval (AOA) capabltes, when only a fracton of the have postonng capabltes. Fnally a dstrbuted locaton servce (GLS) s descrbed n [], where a node sends ts poston updates to ts locaton servers wthout knowng ther actual denttes. Ths nformaton s then used by the other n the network to perform geographcal routng operatons. III. PROBLEM SETUP Frst we descrbe the Neghborhood Dscovery Protocol whch allows each node to gather nformaton about ts neghborhood. We then ntroduce the network model and defne some notons. The network model s followed by the energy effcency model. Fnally we develop a new forwardng scheme called Partal Topology Knowledge Forwardng (). Let us consder the followng Neghborhood Dscovery Protocol. Node S n Fg. perodcally sends a Neghborhood Dscovery packet, called ND-packet, to gather nformaton about ts neghbor, at a power level that allows the packet to be receved by all wthn ts chosen Knowledge Range (KR n Fg. ). As a result, N, N and N receve the ND-packet whle other do not. Then, the whch receved the ND-packet reply wth a Locaton Update packet, called LUpacket. Ths contans the geographcal poston of the node. Now the queston we are tryng to answer s what should the Knowledge Range (KR) of each node be n the network so that the energy requred by the network to perform the routng tasks s mnmzed. It s ntutve that ncreasng the KR may result n more effcent routng decsons. However, ths comes wth the penalty that more energy s needed to exchange sgnalng traffc. A. Network Model The network of sensor s represented as (V, D), where V = {v,v,.., v N } s a fnte set of n a fntedmenson terran, wth N = V, and D s the matrx whose

4 Fg.. Neghborhood Dscovery Protocol element (, j) contans the value of the dstance between v and v j. We assocate each node k wth ts Knowledge Range, r k, based on the Neghborhood Dscovery protocol as explaned above. Thus, the array R = [r,r,.., r N ] descrbes the KRs of all n the network. Let S be the set of traffc sources and D the set of destnaton. We defne P = {(s, d) : s S,d D} as the set of source-destnaton connectons. The nformaton rate of each connecton s descrbed by the traffc matrx P =[p j ], where p sd represents the average nformaton rate (bts/s) between a source node s S and a destnaton node d D. Defnton : A loop-free Forwardng Rule F, gven a node v, ts KR r k and a destnaton node v d, assocates the node v wth another node v n n V\{v }, n such a way that the path {v,v n,..., v d } obtaned by recursvely applyng the rule s composed of dstnct. Defnton : v n s called next hop of node v towards v d wth KR r, accordng to F, whch we ndcate wth v n = lv F (v d,r ). Note that for the sake of smplcty we wll refer to a generc node v n as n n what follows. We wll also omt the ndex F. Thus, lv F (v d,r ) s referred to as l (d, r ). Gven the set of KRs of all R, therulef nduces paths among any possble source-destnaton par n the network. Thus, F : R x sd j (R) () where x sd j (R)= ff the lnk between node and node j s part of the path between node s and node d wth the gven choce R of ranges, when we apply the forwardng rule F. B. Energy Model An accurate model for energy consumpton per bt at the physcal layer s E = E trans elec + βd α + E rec elec () where Eelec trans s the energy utlzed by transmtter electroncs (PLLs, VCOs, bas currents, etc) and dgtal processng. Ths energy s ndependent of dstance; s the energy utlzed by recever electroncs, and E rec elec βd α accounts for the radated power necessary to transmt over a dstance d between source and destnaton. As n [], we assume that E trans elec = E rec elec = E elec () Thus the overall expresson for E n eq., whch we refer to as lnk metrc hereafter, smplfes to E = E elec + βd α () Accordng to ths lnk metrc, the topology nformaton cost for node v s expressed as: C INF (r )=[L D βr α +(N (r )+) L D E elec + + m ζ (r ) L U βd α m +N (r ) L U E elec ] T M () wth α s the path loss ( α ); β s a constant [Joule/(bts m α )]; L D s the length of neghborhood dscovery packets [bts]; L U s the length of locaton update packets [bts]; E elec s the energy needed by the transcever crcutry to transmt or receve one bt [Joule/bts]; N (r ) s the number of neghbors of node v when ts Knowledge Range s r ; ζ (r ) s the set contanng the ndces of the n range r of node ; T M s the perod between two consecutve neghborhood dscovery messages [sec]; The expresson βr α represents the energy needed to transmt one bt at dstance r ; thus L D E elec +L D βr α s the energy needed for node to transmt the ND-packet n ts Knowledge Range, where as each of the N (r ) n ts KR spends only L D E elec to receve the ND-packet. By addng these two components we obtan the frst lne of eq.. Then, each of the N (r ) transmts an LU-packet. The energy expendture has agan a constant factor, L U E elec, plus a factor, L U βd α m ), whch depends on the dstance between the transmttng node v m and node v. Moreover, v spends L U E elec to receve each of the N (r ) LU-packets. By addng all these components, and dvdng by T M, whch depends on the moblty rate of the n the network, we obtan the fnal expresson for C INF. In other words, C INF s the average energy (measured n watts) whch s needed to allow node v to obtan topology nformaton wthn the range r. The communcaton cost for node v can be computed from: wth C COM (R) = (s,d) Π (R) [βd α l (d,r ) +E elec] p sd () Π (R) ={(s, d) s.t. x sd j =for at least one j} () The set Π (R) contans all source-destnaton pars whose path ncludes v as a transt node, as well as those for whch v s the source. Thus, n eq. we sum over all the connectons n whch v s a transmttng node. Note that each term has a dstancendependent component E elec (the energy needed to transmt

5 and receve one bt), and a dstance dependent component, d α l (d,r ), whch represents the α-th power of the dstance between node v and v l(d,r ), ts next hop towards v d when ts KR s r. Every term s then multpled by the average bt rate of the communcaton p sd. Thus, C COM (R) s measured n watts and represents the average energy expendture for all the communcatons node v s nvolved n. We can now state the total cost for node v as: C TOT (R) =C COM (R)+C INF (r ),. () Note that whle the nformaton cost of each node only depends on ts own KR, the communcaton cost depends on the KRs of all nvolved n the communcaton process. C. Partal Topology Knowledge Forwardng () Here we descrbe a novel forwardng scheme called Partal Topology Knowledge Forwardng (). Ths s essentally a shortest weghted path routng scheme wth a power lnk metrc. Consder a node S whch must forward a message to a gven destnaton D. Gven ts KR, S knows the poston of all nsde ths range and the poston of the destnaton node. The topologcal vew of S s consttuted by node D and by all the n the KR wth postve advance wth respect to D, so that the loop freedom condton holds. To evaluate the next hop towards the destnaton node, a lnk metrc of E elec + βd α j, accordng to eq., s assumed to be the cost of the lnk between each node par v and v j. A shortest weghted path algorthm (such as Bellman-Ford s) s executed to calculate the path towards the destnaton. The message s forwarded to the frst node N n ths shortest path. The node N calculates, n ts turn, the optmal path towards the destnaton D, but ths tme accordng to ts own KR. Ths can actually result n a very dfferent path beng chosen by N compared to the path calculated by S. It s easy to see the exstng trade-off between the communcaton cost and the nformaton cost for ths scheme. Note that, unlke the forwardng schemes descrbed n Secton II.A, ths s not a greedy scheme. Ths scheme s more localzed the smaller the KR of each node becomes. However, we wll demonstrate by usng realstc models that small KRs are chosen when energy effcency s the major concern. IV. INTEGER LINEAR PROGRAMMING FORMULATION Our objectve s to select the vector of Knowledge Ranges (KR) R whch mnmzes the energy expendture of the overall network, gven the set of connectons P and a Forwardng Rule F: P : mn R CTOT = (C COM + C INF ) () V Here we gve an Integer Lnear Programmng (ILP) formulaton of the problem. To lnearze an nherently non-lnear problem we consder dscrete values of the Knowledge Ranges. The granularty of ths quantzaton can be whatever, but obvously fner-graned transmsson ranges ncrease the complexty of the problem. Each varable r, r r max assumes one out of the k max dscrete, equdstant values n the set {r,r,.., r kmax }, wth r k r k = r, k s.t. k k max, wth r =and r max = r kmax. We refer to the set of ndces {,,.., k max } as R. We ntroduce the followng notatons and varables: r(k) s the k-th Knowledge Range; r α (k) s the α-th power of the k-th KR; N (k) s the number of neghbors for node v when t selects the k-th KR f j dk =ff, accordng to F, node v j s next hop for node v, when v d s destnaton, and the the k-th Range s chosen; a k j =ff node v j s n the k-th KR of node v ; d α j s the α-th power of the dstance between v and v j. We ntroduce the followng routng varables: x sd j =ff lnk j s part of the path between v s and v d. The assgnment varables are: y k =ff node v uses k-th Knowledge Range. We refer to the varables y k as Knowledge Range ndces. We can now express the problem as: P: Optmal Topology Knowledge Ranges Problem: Mnmze: Subject to: j V j V C TOT = V (C COM + C INF ) () y k =, () k R (x sd sj x sd js) =, s S, d Ds.t. s d; () (x sd dj x sd jd) =, s S, d Ds.t. s d; () j V (x sd j x sd j )=, s S, d D, V s.t. s d, s, d; () x sd j (y k f j dk ), s S, d D,, j V; k R () x sd sj = (ys k f sj dk ), s S, d D, j Vs.t. s d () k R C INF =(L N β k R(y k r α (k)+( k R (y k N (k)) + ) L N E elec + m V(L U β d α m + L U E elec ) (y k a m (k))), V. () T M C COM k R = s S (x sd j p sd ( E elec d D j V +β d α j)), V. () The constrant () mposes the exstence of a sngle Knowledge Range ndex dfferent from zero for each node. The constrants ()()() express conservaton of flows [], whle

6 the constrants ()() mpose that paths are bult accordng to the forwardng rule defned by the nput parameters f j dk. Fnally the constrants () and () express the nformaton and communcaton cost wth the Knowledge Range ndex notaton, respectvely. Note that gven a forwardng rule F, expressed by the f j dk parameters, the assgnment of the routng (x sd j ) varables s completely dependent on the choce of Knowledge Ranges (y k varables). Once the values of the y k varables have been selected, the set X = {x sd j } defnes the path from source to destnaton for any connecton n P. V. PRADA: A DISTRIBUTED PROTOCOL FOR TOPOLOGY KNOWLEDGE ADJUSTMENT The soluton of the ILP problem s not feasble n a practcal settng due to ts complexty and centralzed nature. Here we ntroduce the PRobe-bAsed Dstrbuted protocol for knowledge range adjustment PRADA, whch determnes the KRs on-lne n a dstrbuted way. The objectve of PRADA s to allow network to select stable and effcent topology Knowledge Ranges (KRs). Ths global target s acheved through local decsons and by means of probe packets exchanged among the. The man dea behnd PRADA s to allow each node to adjust ts KR accordng to the feedback nformaton t receves from neghborng nvolved n the same multhop connectons. A quck convergence to a near-optmal soluton and robustness are the key features of PRADA. To trade off between the topology nformaton cost and the communcaton cost, each node whch s part of the path of a partcular connecton (as a source or a transt node), perodcally probes ts possble KRs. For each of them the node evaluates the ncrease/decrease n energy expendture when selected that KR could affect the network operaton. To clearly understand the ratonale behnd PRADA we pont out that whle the nformaton cost of each node only depends on ts KR, the communcaton cost depends on the KRs of all nvolved n the communcaton process. Thus, the communcaton cost must be montored wth probe packets. PRADA s executed at each node v that has an actve role n the network, as a source or a transt node, n a certan set of connectons P. For each connecton p k n ths set, v selects the next hop lv F (vd k,r probe), where vd k s the destnaton node of the k-th connecton, accordng to the selected forwardng rule F and to ts current KR. Perodcally, each actve node selects a certan KR to be probed, dfferent from the current one, n the dscrete set of possble KRs. We refer to the selected KR as r probe and to the current KR as r current. Then the node calculates: C TOT (r probe )=C INF (r probe )+ c p (r probe) () p P where c p (r probe) s the cost of the transmssons along the path from v to the destnaton of the connecton p, wth KR r probe. Ths way, the node can calculate the communcaton cost, from the node tself to all the destnatons, plus the nformaton cost that ths new KR r probe would cause. If C TOT (r probe ) <C TOT (r current ), the value of the KR s updated (r current = r probe ). Fg.. Structures of Probe Packet and of Incremental Cost Record Table Let us descrbe the felds of the probe packets to explan how ths nformaton s obtaned. As shown n Fg., a probe packet has fve felds. The frst two contan the geographcal coordnates of the source and the destnaton. The thrd contans a parameter called Cumulatve Communcaton Cost and the fourth contans the value r probe of KR. The last feld s a one-bt flag, whch s equal to f the packet s on the forward path towards the destnaton, or equal to f t s on the reverse path. The cumulatve communcaton cost feld, ntalzed to when the packet s created, s updated hop-by-hop by addng the ncremental communcaton cost,.e., the communcaton cost necessary to reach the next hop, to the communcaton cost stored n the packet. Ths way, the partal cumulatve communcaton costs are computed hop-by-hop along the path from the sender to the destnaton. Algorthm PRADA begn randomly select r probe r current for each p k P do v lv F (vd k,r probe): probe packet end for wat for return packets C TOT f (C TOT (r probe )=C INF (r probe ) <C TOT r current = r probe end f end (r probe )+ p P c p (r probe) (r current )) then After choosng a KR r probe, for each of the connectons n P the node sends a probe packet to the relevant next hop and wats for ts return. When a node receves a probe packet on the forward path, t looks nto the Incremental Cost Record table to check f t already knows the ncremental communcaton cost needed to reach ths destnaton. If t does, there s no need to forward the probe packet to the destnaton. The probe packet s sent back wth the updated nformaton and the path bt s set to reverse. If t does not, the packet s forwarded to the next hop towards the destnaton

7 Scenaro Scenaro Scenaro Terran (mxm) (mxm) (mxm) KRs (,,,,)m (,,,,)m (,,..,)m α vares L D bts bts bts L U bts bts bts T M s vares s E elec vares pj/bt nj/bt β pj/bt/m α pj/bt/m α pj/bt/m α traffc Kbt/s Kbt/s Kbt/s.... TABLE I PARAMETERS OF THE MODEL USED FOR SIMULATIONS. n order to evaluate the communcaton cost. The packet s forwarded untl a node wth nformaton for that destnaton or the destnaton tself s reached. The pseudocode n ths page (Algorthm ) descrbes the operatons performed by a node v whch executes PRADA. In order to reach stablty, we choose to update the KR only f the movng average of the communcaton cost for the last N probe values gathered s lower than the cost of the current range. In the experments we assume that all the KRs are probed wth the same probablty. More sophstcated strateges can also be mplemented n order to selectvely scan the KRs, amed at savng transmsson power, e.g. by avodng values of KR that are not lkely to brng any beneft and provdng a better estmate of the cost. Fg.. Scenaro - for the mplemented forwardng schemes, E elec = J/bt Protocol Cost VI. PERFORMANCE EVALUATION We mplemented the forwardng schemes descrbed n Secton II-A, gven n Secton III-C and PRADA, gven n Secton V. We further mplemented the ILP problem n AMPL [] and solved t wth the CPLEX [] solver. We are partcularly nterested n scenaros, such as those encountered n sensor networks applcatons, where the densty of s very hgh. However, due to the computatonal complexty of the problem we nvestgate, and to the large amount of the nput data, a state-of-the-art workstaton can fnd the optmal soluton wth CPLEX for networks wth at most. Thus, we consder small geographcal areas n order to take nto account the effects of hgh node denstes on the problem. The model depends on several nput parameters, and on the approprate choce of these parameters whch are hghly dependent on the technology and on the target applcatons. Our choce for these parameters was motvated by the model presented n []. However we also vary these parameters n order to study ther relevant effects on the network performance. Moreover, we beleve that a realstc tunng of these parameters must be aded by real hardware mplementaton of the consdered protocols. We present smulaton results for the scenaros llustrated n Table I. In Scenaro, all are sources wth Kbt/s flows drected towards a unque snk node. In Fg. we show. Fg.. Scenaro - Cost wth PRADA for the mplemented forwardng schemes, E elec = J/bt the optmal cost (the mnmum of the objectve functon of problem P, eq. ), wth ncreasng number of for all the mplemented forwardng schemes (descrbed n Sectons II-A and III-C). The value chosen for the parameter E elec s J/bt []. Note that the confdence ntervals are not shown for the sake of clarty. Snce the area of the terran s very lmted, mult-hop s often not energy effcent, whch leads source to drectly transmt to the destnaton. For ths reason, many forwardng schemes show smlar performance. In Fg. we show the total cost for all the mplemented forwardng schemes n Scenaro obtaned by applyng PRADA wth N probe =. In Fg. we compare the optmal cost obtaned for wth three dfferent approaches for the soluton of the optmzaton problem, wth % confdence ntervals. The problem s solved wth CPLEX (optmal soluton), wth a greedy local search heurstc, and by applyng the dstrbuted protocol PRADA ntroduced n Secton V. CPLEX fnds the optmal

8 .. Greedy Local Search Cost PRADA Cost Confdence Interval (Optmal) Confdence Interval (Greedy Local Search) Confdence Interval (PRADA) Fg.. Scenaro - Comparson of for wth dfferent approaches, E elec = J/bt Fg.. Scenaro - for the mplemented forwardng schemes, E elec = J/bt N=. Dstrbuton of Kowledge Ranges x N=. N=. N=. Knowledge Range [meters] Fg.. Scenaro - Dstrbuton of values of Knowledge Range, E elec = J/bt Fg.. Scenaro - for the mplemented forwardng schemes, E elec = J/bt soluton for mxed nteger problems by usng a branch and bound algorthm. The greedy local search heurstc bascally scans the one after another and selects for each of them the KR whch mnmzes the cost; the process s repeated perodcally untl the stablty s reached. Results obtaned wth PRADA are also gven where the PRADA curve s very close to those obtaned wth CPLEX and wth the greedy local search heurstc. Ths behavor, as wll be shown, becomes more evdent when the problem becomes more localzed. In Fg. we show the dstrbuton of the values of the KRs n Scenaro, wth N =,, and. The average KR s, n ths Scenaro, below. meters, and t s easy to see that most ether have a KR equal to (that s, they prefer to know nothng about ther neghborhood and drectly transmt to destnaton) or they try to know far (, meters) to use them as ntermedate relays. As a result, t s ether effcent to drectly transmt to destnaton or use at most one ntermedate node as relay. By decreasng the E elec parameter, we decrease the weght of the component n energy expendture (lnk metrc n eq. ) whch s ndependent of the dstance. It becomes more energy effcent to select mult hop paths, snce the overall dstance ndependent part of the energy expendture ncreases wth the number of hops. We would obtan the same effect by ncreasng the area of the terran, but we would have a less dense terran. It can be nferred by comparng Fgures,, and that the more mult hop paths are energy effcent (low values for E elec ), the more (Secton III-C) outperforms the other schemes. In the above fgures, the values for E elec are,, and J/bt respectvely. For E elec = J/bt, the cost obtaned wth PRADA s optmal, as can be seen from Fg.. When the dstance ndependent term E elec n eq. becomes small compared to the area of the terran, mult hop paths become more energy effcent. When ths occurs, by selectng KRs whch are opt-

9 . x Fg.. Scenaro - for the mplemented forwardng schemes, E elec = J/bt. Fg.. Scenaro - for the mplemented forwardng schemes, T M =.s... x Greedy Local Search Cost PRADA Cost Confdence Interval (Optmal) Confdence Interval (Greedy Local Search) Confdence Interval (PRADA) Fg.. Scenaro - Comparson of for wth dfferent approaches, E elec = J/bt N= N= N= N=.... Dstrbuton of Knowledge Ranges Knowledge Ranges [meters] Fg.. Scenaro - Dstrbuton of values of Knowledge Range, E elec = J/bt mal only locally, as PRADA does, we obtan globally optmal solutons, because the problem becomes more localzed when E elec decreases. In Fg. we demonstrate that t s more energy effcent to select near (KRs are meters), as E elec decreases. Ths s partcularly true when the densty of the ncreases. In Scenaro, all are sources wth Kbt/s flows drected towards a unque snk node. In Fg. we report optmal costs wth ncreasng number of for all the mplemented forwardng schemes (Secton II-A). Agan, (Secton III-C) performs better than the other forwardng schemes. More greedy schemes such as Nearest Forward Progress () and Most Forward wthn Radus (), both descrbed n Secton II-A, consume more energy. In Fg. we gve optmal paths for all the consdered forwardng schemes n a Smulaton wth. Fg. shows the total cost for all the mplemented forwardng schemes n Scenaro obtaned by applyng PRADA wth N probe =.Fg and are almost dentcal, whch s explctly shown by Fg. where we compare the results obtaned for wth the three dfferent optmzaton approaches (CPLEX, greedy local search, PRADA). In Fg. we depct the nformaton cost (eq. ) and the communcaton cost (eq. ) for, agan wth the three dfferent approaches. The communcaton cost s shown to hghly exceed the nformaton cost when relatvely hgh data rate flows must be supported. In Fg. we show the average value of the Knowledge Range wth ncreasng number of for all the proposed schemes. It s obvous that a very lmted knowledge of the topology s needed n average, less than meters. In Fgures and we show the average convergence dynamcs of PRADA to the optmal soluton wth and. At every step, any sensor node selects and probes randomly one of ts KRs. For, after steps we obtan a near-optmal soluton. In Fg. we assume a lower moblty rate, thus, we set T M =. As can be seen n Fg., for lower rates of moblty even more evdently outperforms the other schemes. A more extended

10 .... (a) (b). Greedy Local Search Cost PRADA Cost Confdence Interval (Optmal) Confdence Interval (Greedy Local Search) Confdence Interval (PRADA) Fg.. Scenaro - Comparson of for wth dfferent approaches, T M =.s. (c) (d).. Cost of Communcaton (Optmal) Cost of Informaton (Optmal) Cost of Communcaton (Greedy Local Search) Cost of Informaton (Greedy Local Search) Cost of Communcaton (PRADA) Cost of Informaton (PRADA) (e) (f) Fg.. Optmal Routng Trees wth dfferent Routng schemes - Scenaro,. Fg.. Scenaro - nformaton cost and communcaton cost for, T M =.s.. Protocol Cost.. Average Optmal Knowledge Range... KR [meters] Fg.. Scenaro - Cost wth PRADA for the mplemented forwardng schemes, T M =.s. Fg.. Scenaro - Average KR wth dfferent forwardng schemes, T M =.s

11 ... Convergence of Total Cost Average Cost wth PRADA after n steps... Average Optmal Knowledge Range.... KR [meters] steps. Fg.. Scenaro - Convergence of PRADA wth,, T M =.s Fg.. Scenaro - Average KR wth dfferent forwardng schemes, T M = Convergence of Total Cost Average Cost wth PRADA after n steps steps Fg.. Scenaro - Convergence of PRADA wth,, T M =.s.. local topology knowledge brngs benefts n terms of energy to the scheme whch best explots ths nformaton. Ths s confrmed by Fg. that shows how the average KRs ncrease n general, and partcularly for whch s able by ts nature to better take advantage of a more extended knowledge. Stll, the extenson of local knowledge of the topology s very lmted compared to the terran dmensons. In Scenaro, Kbt/s traffc flows are smultaneously generated by sensor n the network towards a snk node, but the terran s bgger (mxm). Fgures and report optmal cost wth ncreasng number of for all the mplemented forwardng schemes wth α =and α =, respectvely. For hgh values of the parameter α the optmal cost decreases as the node densty ncreases, whle for low values of α the ncrease n the amount of traffc overcomes the postve effect of a hgher node densty. Agan, n all the experments of Scenaro s shown to perform better than any other scheme. Ths s more evdent agan when mult-hop paths are energy effcent, that s, when α s hgher (the dstance dependent part of the cost has a hgher weght). Agan more greedy schemes, such as Nearest Forward Progress () [] and Most Forward wthn Radus () [], both descrbed n Secton II-A, are shown to lead to hgher energy consumptons.... Fg.. Scenaro - for the mplemented forwardng schemes, T M = VII. CONCLUSIONS AND FUTURE WORK In ths paper we solve the problem how to determne optmal local topology knowledge for energy effcent geographcal routng for sensor networks. We gve an Integer Lnear Programmng Formulaton of the problem whch consttutes a framework for the analyss of the energy effcency of dfferent forwardng schemes. We show that only a lmted local topology knowledge s needed to take energy effcent routng decsons. We ntroduce a dstrbuted protocol called PRADA whch quckly acheves a near-optmal soluton. Future research wll nclude the extenson of the model, prmarly to nclude features such as battery and bandwdth

12 Fg.. α = Fg.. α = Scenaro - for the mplemented forwardng schemes, Scenaro - for the mplemented forwardng schemes, constrants for the. Moreover, the consdered schemes wll be mplemented n a tool smulatng all layers of the communcaton task to evaluate the effect of the sgnalng traffc. ACKNOWLEDGMENTS The authors would lke to thank Zhaosong Lu of the School of Industral and Systems Engneerng at the Georga Insttute of Technology, for hs suggestons to formulate the ILP Problem. REFERENCES [] D. Estrn, R. Govndan, J. Hedemann, S. Kumar, Next Century Challenges: Scalable Coordnaton n Sensor Networks, Proc. IEEE/ACM Mobcom, Seattle, pp. -. [] V. Rodoplu, T. Meng, Mnmum Energy Moble Wreless Networks, IEEE Journal of Selected Aresas In Communcatons, Vol., no., August, pp. -. [] S.M. Woo, S. Sngh, C. S. Raghavendra, Power-Aware Routng n Moble Ad Hoc Networks, Proc. of IEEE/ACM Mobcom, pp. -. [] J. Chang, L. Tassulas, Energy Conservng Routng n Wreless Ad-hoc Networks,, Proc. of IEEE Infocom, pp. -. [] J. L, J. Jannott, D. De Couto, D. Karger, R. Morrs, A Scalable Locaton Servce for Geographc Ad Hoc Routng, Proc. IEEE/ACM Mobcom, pp. -. [] R. Jan, A. Pur, R. Sengupta, Geographcal Routng Usng Partal Informaton for Wreless Ad Hoc Networks, IEEE Personal Communcatons, Feb., pp. -. [] J. Hghtower and G. Borrello, Locaton Systems for Ubqutous Computng, IEEE Computers, Aug., pp. -. [] R. Ramanathan, R. Rosales-Han, Topology Control of Multhop Wreless Networks usng Transmt Power Adjustement, Proc. IEEE Infocom. [] H. Takag, L. Klenrock, Optmal Transmsson Ranges for Randomly Dstrbuted Packet Rado Termnals, IEEE Transactons on Communcatons, Vol., no.,, pp. -. [] T.C. Hou, V.O.K. L, Transmsson Range Control n multhop packet rado networks, IEEE Transactons on Communcatons, Vol., no., pp. -. [] G.G. Fnn, Routng and Addressng Problems n Large Metropoltan- Scale Internetworks, ISI res. rep ISU/RR- -, Mar.. [] E. Kranaks, H. Sngh, J. Urruta, routng on geometrc networks, Proceedngs of the th Canadan Conference on Computatonal Geometry, Vancouver, Canada, August. [] R. Nelson, L. Klenrock, The spatal capacty of a slotted ALOHA multhop packet rado network wth capture, IEEE Transactons on Communcatons, Vol., no.,, pp. -. [] I. Stojmenovc, Poston-Based Routng n Ad Hoc Networks, IEEE Communcatons Magazne, Vol., No., July, -. [] S. Gordano, I. Stojmenovc, L. Blazevc, Poston Based Routng Algorthms for Ad Hoc Networks: A Taxonomy, Ad Hoc Wreless Networkng, Kluwer, [] I. Stojmenovc, Locaton Updates for Effcent Routng n ad hoc network, Handbook of Wreless Networks and Moble Computng, Wley,, -. [] B. Karp, H. T. Kung, GPSR: Greedy Permeter Stateless Routng for Wreless Networks, Proc. of ACM/IEEE MobCom. [] D. Nculescu, B. Nath, Trajectory based forwardng and ts applcatons, Tech. Rep. DCS-TR-, Rutgers Unversty, June. [] I. Stojmenovc, X. Ln, Power-Aware Localzed Routng n wreless networks, IEEE Transactons on Parallel and Dstrbuted Systems, Vol., No., November, -. [] X. Hong, K. Xu, M. Gerla, Scalable Routng Protocols for Moble Ad Hoc Networks, IEEE Network, July/August. [] Y. Xu, J. Hedemann, D. Estrn, Geography-nformed Energy Conservaton for Ad Hoc Routng, Proc. of ACM/IEEE Mobcom, Rome, Italy, July. [] D. Nculescu, B. Nath, Localzed postonng n ad hoc networks, Elsever s Journal of Ad Hoc Networks, Specal Issue on Sensor Network Protocols and Applcatons. Vol., Issues -, Pages -, September. [] R. K. Ahuja, T. L. Magnant, J. B. Orln, Network Flows: Theory, Algorthms, and Applcatons, Prentce Hall, February. [] R. Fourer, D. M. Gay, B. W. Kernghan AMPL: A Modelng Language for Mathematcal Programmng, Duxbury Press / Brooks/Cole Publshng Company,. [] [] W. B. Henzelman, A. P. Chandrakasan, H. Balakrshnan, An Applcaton-Specfc Protocol Archtecture for Wreless Mcrosensor Networks, IEEE Transactons on Wreless Communcatons, Vol., No., October. [] I. Akyldz, W. Su, Y. Sankarasubramanam, E. Cayrc, Wreless Sensor Networks: A Survey, Computer Networks (Elsever) Journal, pp. -, March. [] Z.J. Haas, Desgn Methodologes for Adaptve and Multmeda Networks, IEEE Communcatons Magazne, Vol., no., November, pp -.