Power Optiml Routing in Wireless Networks Rjit Mnohr nd Ann Scglione ECE, Cornell University Abstrct Reducing power consumption nd incresing bttery life of nodes in n d-hoc network requires n integrted power control nd routing strtegy. Power optiml routing selects the multi-hop links tht require the minimum totl power cost for dt trnsmission under constrint on the link qulity. This pper studies optiml power routing under the constrint of fixed end-to-end probbility of error nd compres the power optiml routes obtined with this criterion with those from the more commonly used fixed per hop error rte constrint. The comprison is crried out by looking t the properties of the power optiml grph, formed by the union of ll the power optiml routes. The pper lso provides lgorithms to determine the power optiml routes. Index Terms Sensor Networks, Power Control, Routing. I. INTRODUCTION In multi-hop networks link relibility, vilbility, dely nd, lst but not lest, the bttery life of ech node re ll entngled through unique vrible, the power spent on ech bit trnsferred. Power control issues hve been ddressed for quite some time in the literture, especilly in the context of cellulr networks [12]. While the dependence between multiple ccess control nd power control is lso evident in cellulr networks, the trde-mrk of multi-hop networks is the interdependence between routing nd power control. In [6] the optiml trnsmission rdius in multi-hop wireless networks ws derived under the constrint tht ll nodes trnsmit the sme power, which ws lter relxed in [7]. In ddition to drining the bttery of the node, since wireless link is brodcst mechnism, incresing the power used to trnsmit pcket might cuse other side-effects such s interference with other nodes in the network. Therefore, it is importnt to determine the minimum power necessry to route pcket, nd recent work in d-hoc network hs focused on the problem of optimized routing tht minimize the totl pth power consumption, see e.g. [11], [15], [10]. Given the per hop trnsmit power for pcket nd its route, the power cost of pth is defined to be the sum of the per-hop trnsmit power long the pth. Routes tht minimize the power cost under qulity of service constrint re sid to be power optiml. The union of the optiml multi-hop pths between ll pirs of nodes for specific optimlity criterion form grph, which we will refer to s the power optiml grph. The power optiml grph is the ensemble of optimum physicl (single nd multi-hop) links supporting peer-to-peer trnsmission for ech pir of nodes optimlly. Clerly the power cost of ech route depends on the optimlity criterion chosen nd nturl This work ws supported in prt by the Multidisciplinry University Reserch Inititive (MURI) under the Office of Nvl Reserch Contrct N00014-00-1-0564. This work ws supported by NSF Creer Awrd No. CCR-0133635 question tht rises is how the optiml grph behves s function of the optimlity criterion. A study of the properties of the power-optimum grph for fixed per hop error rte is in [1] where it ws shown tht the power optiml grph with fixed per hop error rte constrint hs no crossing edges. As will be clrified in Section II, the problem with using per hop error rte constrint is tht the qulity of the end-to-end connection is not gurnteed. As result, pths with lrger number of hops will produce not only n incresed dely but lso n incresed error rte. This pper derives solution for the power optimum route problem under n end-to-end error rte constrint nd crries comprtive study of the power optiml grphs obtined with those obtined with per hop error rte constrint. Note tht we do not consider multiple ccess issues s we ssume n idel scenrio where ll trnsmissions re scheduled to occur t different times (or over-different bnds simultneously), similrly to [4]. The symbol error rte expression is clculted ccordingly, without considering the effect of multi-ccess interference. We lso do not investigte dely constrints. The interesting conclusion of our study is tht the endto-end fixed error rte power optiml grph lwys contins the per-hop fixed error rte power optiml grph, with some dditionl edges which symptoticlly, s the end-to-end error rte required ɛ 0, tend to vnish. The pper is orgnized s follows: in Section II we introduce the power optiml routing with constnt end-to-end constrint, providing generl bounds tht llow to simplify the solution of the problem for rbitrry error rte (SER) expressions. Specific SER models re introduced in Section III. The properties of the power optiml grph re studied in Section IV nd in Section V we provide distributed lgorithm to clculte the power optiml pths. All the proofs for the Theorems nd Lemms in the following cn be found in [16] nd will be omitted for brevity. II. CONSTANT END-TO-END ERROR RATE We consider the cse when pcket is trnsmitted from source to its destintion long multiple hops where there is some probbility of error per hop tht depends on the distnce between the hops nd the trnsmit power. In existing work, the ssumption mde is tht if the received power (signl-tonoise rtio) P recv γ where γ is some constnt, then the pcket is successfully received; otherwise the pcket is lost. The symbol error rte (SER) is monotoniclly decresing function of P recv therefore, for ech hop P recv cn be mde lrge enough tht the SER for the hop stisfies SER SER(γ) (1)
Since ny trnsmission will use the minimum mount of power required to meet the necessry error rte, this mens tht SER = SER(γ). Assuming tht errors per hop re independent, this implies tht the end-to-end error rte SER e2e is given by SER e2e =1 (1 SER(γ)) N (2) where N is the number of hops long the pth. SER e2e is monotoniclly incresing with N. Insted of using routing scheme nd power cost metric tht llows the end-to-end error rte to increse with hop count, we exmine the effect of constrining the end-to-end error rte to be constnt. In the following section, we formulte this optimiztion problem. A. Optimiztion Problem Let X =(X 0, X 1,..., X N )benn-hop pth from node X 0 to X N tht psses through nodes X 1, X 2,..., X N 1 in tht order. Let P i be the power llocted for the hop (X i 1,X i ), nd let SER i be the corresponding symbol error rte for the hop (i =1, 2,..., N). Assuming independent errors per hop, the end-to-end SER is given by SER e2e =1 N (1 SER i ) (3) We define the power cost function s follows: PC (ɛ; X) = min P 1,...,P N P i given SER e2e ɛ (4) We will ssume tht we re interested in the cse when the quntity ɛ is much smller thn one, i.e., when there is very low probbility of error. A quick nlysis shows tht the optimiztion problem (4) leds to equtions tht do not hve simple closed form solutions in terms of the power per hop nd the power cost, even for the simplest models for the SER. We formulte our pproximte power cost metric by the following equtions: PC(ɛ; X) = min P 1,...,P N P i given SER i ɛ (5) We cn justify this pproximtion by the following result: Theorem 1: Assuming ɛ 0 nd ɛ+4ɛ 2 < 1/ 2, nd given the power cost metrics PC nd PC defined by equtions (4) nd (5) respectively, we hve: PC(ɛ +4ɛ 2 ; X) PC (ɛ; X) PC(ɛ; X) The proof of Theorem 1 only relies on the power cost metric being minimiztion problem where the error rte constrint is met with equlity t the solution. In prticulr, it does not depend on the expression for SER i, or the fct tht metric being minimized is the sum of the power per hop. For the rest of this pper, we use PC s our power cost metric becuse it will led to nlyticl expressions for the power cost of pth.while this is n pproximtion, Theorem 1 provides wy of bounding the error E(ɛ; X) = PC(ɛ; X) PC (ɛ; X) since E(ɛ; X) PC(ɛ; X) PC(ɛ +4ɛ 2 ; X) (6) When we compute the power cost with specific models for SER, we will use this expression to show tht the pproximtion error is O(ɛ), nd thus smll compred to the power cost of the pth. Error correction mechnisms (both ARQ nd FEC) cn be esily included in our frmework with minor chnges s discussed in [16]. III. ERROR RATE MODELS To study the optiml power lloction strtegy for pth, we exmine two different models for SER. Using these models, we derive nlyticl expressions for PC(ɛ; X). These closed form expressions will be used in Section V to derive lgorithms for computing power optiml pths. A. Time-invrint Attenution The first model is deterministic power model where we ssume tht the receive power of link is ttenuted by timeinvrint quntity. This ttenution coefficient cn be given physicl interprettion by setting it to d α /, where d is the distnce between the trnsmitter nd receiver, nd α>2 nd re constnts. We cn ssume tht the expression for SER of link is given by: SER i = be Pi/i (7) where P i is the trnsmit power nd i is the time-invrint ttenution coefficient of the link. Eqution (7) is sufficiently prmeterized to be ble to provide bound for the probbility of error for most digitl modultions tht re detected optimlly in the presence of dditive Gussin noise [12]. Theorem 2: Under the ssumptions of Theorem 1 nd the link SER ssumption given by eqution (7), the optiml power cost PC(ɛ; X) for pth X =(X 0,X 1,...,X n ) is obtined when 1) SER j = ɛ j N ( i 2) P j = j log(b/ɛ) + log( ) N i/ j ) 3) E(ɛ; X) log(1 + 4ɛ) N j where j, SER j, nd P j re the ttenution coefficient, link error rte, nd link power lloction respectively for link (X j 1,X j ). B. Lrge nd Smll-Scle Fding In wireless mobile scenrio the received power is ffected by mny more fctors thn the mere distnce nd it is common to represent the received power s doubly stochstic rndom vrible, with long term nd short term vritions [12]. The trnsmit power is ttenuted by two fctors: G L (t) cused by lrge-scle fding, nd G S (t) cused by smll-scle fding. To ccount for the effects of lrge nd smll-scle fding, the time-vrying SER of link is given by the rndom process SER i = be Ωτ i (8)
where nd b re constnts, nd Ω τi is the received power, whose sttistics re function of position nd time. As is stndrd, we ssume log-norml distribution for the lrgescle fding coefficient. For the smll scle fding severl distributions hve been introduced [13] nd using (8) the corresponding verge SER for given lrge scle fding prmeter is the chrcteristic function of the smll scle fding density. For exmple, for the Nkgmi m-distribution the expected SER i forlinkisgivenby[13]: SER i = b(1 + P i / i ) m (9) where i is the contribution from the slow-vrying lrge-scle fding coefficient nd P i is the verge trnsmit power. The Nkgmi m-distribution cptures the intermedite ground between strong line-of-sight nd non line-of-sight systems. Note, in fct, tht Ryleigh fding is specil cse of Nkgmi fding when m =1while the deterministic cse is obtined s m. It is not difficult to generlize our power cost expressions nd power optiml routing to these scenrios: Theorem 3: Under the ssumptions of Theorem 1 nd the link SER ssumption given by eqution (9), the optiml power cost PC(ɛ; X) for pth X =(X 0,X 1,...,X n ) is obtined when m/(m+1) j 1) SER j = ɛ N ( m/(m+1) i ( N ) ) 1/m 2) P j = j (b/ɛ) 1/m ( i/ j ) m/(m+1) 1 ) (m+1)/m 3) E(ɛ; X) 4ɛ ( N m (b/ɛ)1/m m/(m+1) i where j, SER j, nd P j re the lrge-scle ttenution coefficient, link error rte, nd link power lloction respectively for link (X j 1,X j ). IV. PROPERTIES OF POWER OPTIMAL PATHS In this section we discuss some of the consequences of dopting n end-to-end SER constrint for power optimiztion. In prticulr, we compre our results to those reported by [1], which ssumes tht the mount of power required for link (X i 1,X i ) is given by P i = log(b/ɛ) dα i (10) where nd α>2 re constnts, nd d i = X i 1 X i is the distnce between points X i 1 nd X i. This model ssumes tht the link SER is constnt (= ɛ), nd the ttenution coefficient i is given by d α i /. Under this model, the power cost of using link does not depend on the pth under considertion, unlike the link power lloction in Theorems 2 nd 3. The power cost using this model, which we denote KC(ɛ; X), is given by d α N i KC(ɛ; X) = log(b/ɛ) = log(b/ɛ) i (11) where d i = X i 1 X i is the distnce between points X i 1 nd X i. Note tht since ɛ only ppers in the fctor in front of this prticulr power cost metric, the best pth to route pcket ccording to KC will not depend on ɛ [1]. We compre the properties of power optiml pths obtined by the metric introduced in Theorem 2 with those obtined using the metric from eqution (11). A. Compring Power Cost Metrics If we use eqution (10) to determine the power for hop, then the SER per hop cn be s high s ɛ, thereby incresing the totl end-to-end SER. The metric KC will therefore underestimte the mount of power required to trnsmit pcket long pth with n error tht does not exceed ɛ. Exmining the expressions in Theorem 2, we conclude tht ) PC(ɛ; X) = j (log(b/ɛ) + log( i / j ) (12) KC(ɛ; X) (13) with equlity holding if nd only if we re considering one-hop pth. If we tret p j = j / N i s probbility distribution, observe tht ( N ) PC(ɛ; X) = i )(log(b/ɛ)+ p i log 1/p i ( KC(ɛ; X) 1+ log N ) (14) log(b/ɛ) with equlity holding when p 1 = p 2 = = p N. When exmining long pths with equidistnt hops, using simple dditive cost function will underestimte the power cost by fctor tht grows logrithmiclly with the number of hops compred to metric tht keeps the end-to-end error bounded. B. Compring Optiml Pths The purpose of computing the power cost metric is twofold. First, given pth, it determines the mount of power required to trnsmit pcket long tht pth s well s the mount of power necessry per hop. Secondly, the power cost metric llows us to compre two pths between source nd destintion to determine which pth would require less power for trnsmission. Section IV-A showed tht the KC nd PC metrics might differ substntilly when determining the power necessry to trnsmit pcket long given pth. In this section, we will show tht the KC nd PC metrics might select different pths. We cn demonstrte tht PC-optiml pths cn differ from KC-optiml pths by mens of simple exmple. Consider the scenrio in Figure 1 with three points A, B, nd C. Without loss of generlity, we cn fix the distnce between A nd C to be one unit. Let d 1 be the distnce between A nd B, nd let d 2 be the distnce between B nd C, s shown in Figure 1. We consider optiml pths between points A nd C. We cn clculte the boundry between the regions where one-hop pth (A, C) is optiml nd where two-hop pth (A, B, C) is optiml s function of d 1 nd d 2 ccording the KC nd the PC metrics. Obviously d 1 nd d 2 re constrined by the tringle inequlity d 1 + d 2 1. Also, if d 1 1 or d 2 1,
A d 1 θ B 1 d 2 C 1 0.9 0.8 0.7 0.6 eps=1e-3: SER Upper Bound Boundry (U2) eps=1e-3: SER Lower Bound Boundry (L2) eps=3e-2: SER Upper Bound Boundry (U) eps=3e-2: SER Lower Bound Boundry (L) Fesible Region Boundry (F) 1-hop optiml region Fig. 1. Compring pths in three node network 0.5 0.4 U L U2 L2 then it is cler tht one-hop pth is optiml for both PC nd KC. Therefore we restrict our ttention to the region specified by the constrints 0 d 1 1, 0 d 2 1, nd d 1 + d 2 1. These regions re shown in Figure 2, where the horizontl nd verticl xes re d α 1 nd d α 2 respectively, with α =2nd b =0.5. The points below curve F do not stisfy the tringle inequlity nd re therefore infesible. All the other lines show the boundry between one-hop pth nd two-hop pth ccording to different power cost metrics. Curve L corresponds to PC(ɛ +4ɛ 2 ; ) nd curve U corresponds to PC(ɛ; ), for ɛ =3 10 2. The true boundry for PC is between these two lines by Theorem 1. Curve K corresponds to KC(ɛ; ). This plot shows tht for ll points between curves U nd K, the optiml pth selected bsed on KC differs from the optiml selected bsed on PC. C. Dependence of Pths on Error Rte As noted bove, the optiml routes in terms of power cost ccording to metric KC will not depend on the vlue of ɛ. In this section we show tht PC-optiml pths chnge if we chnge ɛ. Consider the exmple shown in Figure 1 with the sme prmeters s in Section IV-B. Figure 3 is similr to Figure 2, except we show the boundry between one-hop nd two-hop pths for different vlues of ɛ. It is evident tht the region between curves U nd L2 corresponds to cses where onehop pth is optiml for ɛ =3 10 2, wheres two-hop pth is optiml if ɛ =10 3. Exmining the eqution for the power cost metric in (12) the term log(b/ɛ) becomes more prominent s ɛ 0. Therefore in the limit s ɛ 0, the difference in power cost between pths obtined with the PC metric nd those obtined with the KC metric will be negligible. Fig. 2. 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 Infesible region F U L 2-hop optiml region Kumr Boundry (K) eps=3e-2: SER Upper Bound Boundry (U) eps=3e-2: SER Lower Bound Boundry (L) Fesible Region Boundry (F) 1-hop optiml region 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Boundry of 1-hop v/s 2-hop pths, three points. K Fig. 3. 0.3 0.2 0.1 Infesible region F 2-hop optiml region 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Boundry of 1-hop v/s 2-hop pths, three points. D. The Grph of Power Optiml Pths Given two points, we cn determine the pth between them tht minimizes the power cost. Let G(ɛ) =(V,E(ɛ)) be the directed grph formed with vertex set V being the set of nodes in the network, nd edge (x, y) E(ɛ) just when the edge is prt of some power optiml pth between two nodes in the network. This is referred to s the grph of power optiml pths [1]. It is cler tht if we use the KC metric, then the grph will not be function of ɛ. 1) Crossings in Optiml Pths: Power optiml pths computed ccording to KC hve the property tht two power optiml pths will never cross ech other [1]. Unfortuntely, this property no longer holds when we use the PC metric. Consider the sme tringle s shown in Figure 1, except we let the side AC hve distnce d 3. From geometry, we know tht d 2 3 = d 2 1 + d 2 2 2d 1 d 2 cos θ Consider the cse θ = π/2, α =2, nd d 1 = d 2 = d. We know tht d 3 = 2d. PC(ɛ;(A, C)) = 2d2 log(b/ɛ) PC(ɛ;(A, B, C)) = PC(ɛ;(A, C)) + 2d2 log 2 Therefore, we conclude tht in this cse t lest it is cheper to send pcket directly from A to C insted of vi B. Ifwe pick fourth point D tht is the reflection of B in AC (in the cse we re considering, this mkes ABCD squre), then by symmetry we know tht the power optiml pth from B to D is the one-hop pth (B,D). Hence, there re two power optiml pths (A, C) nd (B,D) tht cross ech other. 2) KC optiml nd PC optiml grphs: In the previous section we showed tht the grph of PC-optiml pths exhibits crossings. In this section we show n inclusion property relting the grph of KC-optiml pths with the grph of PCoptiml pths. The min result we will estblish in this section is tht every edge in the grph of KC-optiml pths lso occurs in the grph of PC-optiml pths for ll vlues of ɛ. Lemm 1: A one-hop pth tht is power optiml by the KC metric is strictly power optiml by the PC metric. We cn lso see this property in Figure 2, where the PC metric lwys picks one-hop pth whenever the KC metric does.
Theorem 4: Let G =(V,E) be grph of power optiml pths by the KC metric, nd let G (ɛ) =(V,E (ɛ)) be grph of power optiml pths by the PC(ɛ; ) metric. Then E E (ɛ). Given two nodes, there my be multiple pths between them tht re optiml in terms of power cost. This implies tht the grph of power optiml pths ccording to either power cost metric need not be unique. As Lemm 1 gurntees strict optimlity in terms of the PC metric, the result from Theorem 4 holds regrdless of which grph is chosen for either power cost metric. This lso implies tht the union of ll possible KC optiml grphs is subgrph of the grph of PC optiml pths. 3) Asymptotic Properties of Optiml Pths: As noted in Section IV-C, we expect tht the difference in the cost of pths chosen by the PC metric nd KC metric to be negligible s ɛ 0 becuse the term log(b/ɛ) will dominte, reducing the difference between the numericl vlue of PC nd KC by equtions (13) nd (14). In this section we exmine the behvior of the optiml pths themselves s ɛ 0. Theorem 5: Let X be PC(ɛ; )-optiml pth nd let X be KC-optiml pth between points A nd B. Then either X is lso KC-optiml, or there exists n ɛ > 0 such tht PC(ɛ ; X ) PC(ɛ ; X) (15) Theorem 5 sttes tht for suitble choice of ɛ, thepc(ɛ; ) criterion picks pth tht is lso optiml by the KC criterion. Therefore, s ɛ 0, the pths chosen by PC will converge to those chosen by KC. V. COMPUTING POWER OPTIMAL PATHS Computing the optiml pths between points in network using the KC metric is simple tsk. The reson for this is tht the cost of link between two nodes is simply log(b/ɛ), where is the ttenution coefficient of the link. Therefore, we cn crete cost mtrix M N N where entry m i,j the ttenution coefficient of the one-hop pth between node i nd node j in the network. The power optiml pths re esily obtined by using n ll-points shortest-pth lgorithm on the mtrix M (see e.g., [3]). Insted, if we solve the end-to-end optimiztion problem using the PC metric, the cost of hop depends on the pth tht uses the hop by Theorems 2 nd 3. Therefore the mtrix formultion described bove will not compute the correct pths. In this section we describe lgorithms tht will compute the PC-optiml pths for grph tht re generliztions of stndrd shortest-pth lgorithms. For the generliztion, we rely on the existence of two pth cost functions c nd d. One of these, c, will correspond to the power cost of the pth, nd the other will be suitbly defined uxiliry function. We require certin properties of this pir of cost functions. Property P1. Both cost functions re ssumed to stisfy reversibility criteri, nmely the cost of pth (X 0,...,X n ) is the sme s the cost of pth (X n,...,x 0 ). Property P2. Let X = (A, X 1,...,X n 1,B) nd Y = (A, Y 1,...,Y m 1,B) be two pths between nodes A nd B. Further, let X =(A, X 1,...,X n 1,B,Z 1,...,Z l 1,C) nd Y = (A, Y 1,...,Y m 1,B,Z 1,...,Z l 1,C) be the extensions of pths X nd Y by the sme set of hops to node C. The cost functions c nd d re ssumed to stisfy the following property: (c(x) < c(y)) (d(x) < d(y)) (c(x ) < c(y )) (d(x ) < d(y )) In other words, if pth X between two nodes hs lower cost function in terms of both functions c nd d thn pth Y, then ll extensions of pth X to third node will hve lower cost (in terms of both c nd d) when compred with extensions to Y. This property llows us to discrd pth Y from further considertion since it will never be subpth of n optiml pth between node A nd ny other node in the network. Note tht while this property is stted for the cse when both pths re extended by n l-hop pth, it cn be estblished by proving it for one-hop extensions nd then pplying induction. We sy tht pth X domintes pth Y. A pth X is sid to be fesible if it is not dominted by ny other pth. Property P3. We ssume tht subpth will lwys hve lower cost (using both c nd d cost functions) thn the originl pth. Given these cost functions, we now describe lgorithms for computing optiml pths in network. We will instntite these lgorithms using different cost functions to solve the power optiml route computtion problem ccording to the power cost metrics of Theorems 2 nd 3. The correctness of our lgorithms depends on the following result. We use the nottion p XY to denote pth from X to Y. Lemm 2: Let S k be the set of ll fesible pths tht hve t most k hops. Then every l-hop subpth in S k is contined in S l. We now estblish the following lemm tht cn be used to construct set S l+m given sets S l nd S m. Lemm 3: The following procedure cn be used to construct S l+m given S l nd S m : 1) Initilize S l+m to S l. 2) For every pir of pths p AB S l nd q BC S m ) Plce the conctented pth r AC in set S l+m if it is not dominted by n existing pth from A to C in S l+m. b) Eliminte ll pths from A to C tht re in S l+m tht re dominted by r AC. We use the nottion S l S m to denote the opertion specified in Lemm 3. Given Lemm 3, the lgorithm for computing optiml pths is strightforwrd. We provide two techniques for computing optiml pths. Algorithm 1 (Single step): The following lgorithm computes S N in vrible S, where N is the number of nodes in the network. 1 S 1 {(i, j): 1 i, j N,i j} 2 S S 1 3 for i 1 to N 1 4 S S S 1 A fster technique for computing S N is shown below.
Algorithm 2 (Doubling): The following lgorithm computes S N in vrible S, where N is the number of nodes in the network. 1 S {(i, j): 1 i, j N,i j} 2 for i 1 to lg N 3 S S S Once ll the cndidte pths re computed, we cn pick the optiml pth from A to B ccording to metric c by picking the lest cost pth (ccording to c) mong the cndidtes in S N. Notice tht both lgorithms shown bove turn into the stndrd mtrix-bsed lgorithms for shortest pth computtion if c nd d re the sme cost function tht is dditive. Finlly, we present the cost functions c nd d for the two power cost metrics derived in Section III. In both cses, c(x) will lwys be the power cost PC(ɛ; X). Theorem 6: The following two cost functions stisfy properties (P 1) (P 3). ) c(x) = j (log(b/ɛ) + log( i / j ) d(x) = j where j is the ttenution coefficient of the jth hop in X. Theorem 7: The following two cost functions stisfy properties (P 1) (P 3), ssuming tht (b/ɛ) > 1. (m+1)/m c(x) = (b/ɛ) 1/m m/(m+1) d(x) = m/(m+1) j j j where j is the ttenution coefficient of the jth hop in pth X nd m is the Nkgmi m-prmeter. Since the definition of c(x) in Theorem 6 is the sme s the power cost of pth s defined by Theorem 2, we cn instntite Algorithms 1 nd 2 using the cost functions from Theorem 6 to compute the power optiml pths when we consider time-invrint ttenution. Since the definition of c(x) in Theorem 7 is the sme s the power cost of pth s defined by Theorem 3, we cn instntite Algorithms 1 nd 2 using the cost functions from Theorem 7 to compute the power optiml pths when we consider lrge nd smll-scle fding. fixed per hop constrint. Finlly, we provided n lgorithm to compute power optiml routes. There re importnt issues tht this pper does not ddress: the fixed end-to end error rte constrint does not incorporte mechnism to prevent congestion t specific nodes. In ddition, since power optiml routing does not uniformly distribute trffic, it ends up drining the resources of some nodes more thn others. Future investigtions will be directed towrds evluting the impct of power optiml routing on the network lifetime [9], [10]. REFERENCES [1] S. Nrynswmy, V. Kwdi, R. S. Sreenivs, nd P. R. Kumr, Power Control in Ad-Hoc Networks: Theory, Architecture, Algorithm nd Implementtion of the COMPOW protocol, Proc. of Europen Wireless 2002. Next Genertion Wireless Networks: Technologies, Protocols, Services nd Applictions, pp. 156 162, Feb. 25-28, 2002. Florence, Itly. [2] J.-H. Chng nd L. Tssiuls, Energy conserving routing in wireless d-hoc networks, in Proc. of IEEE INFOCOM 2000, vol. 1, pp. 22 31, 2000. [3] T.H. Cormen, C.E. Leiserson, nd R.L. Rivest. Introduction to Algorithms, MIT Press, 1990. [4] T. ElBtt, A. Ephremides, Joint Scheduling nd Power Control for Wireless Ad-hoc Networks, INFOCOM 2001. [5] T.J. Kwon nd M. Gerl, Clustering with power control, IEEE MIL- COM, vol. 2, pp. 1424 1428, Nov 1999. [6] L. Kleinrock nd J. Silvester, Optimum trnsmission rdii pcket rdio networks or why six is mgic number, Proc. IEEE Ntionl Telecommunictions Conference, pp.4.3.1 4.3.6, Dec. 1978. [7] T. Hou nd V. Li Trnsmission Rnge Control in Multihop Pcket Rdio Networks, IEEE Trnsctions on Communictions, vol. 34, No. 1, pp. 38 44, Jn. 1986. [8] S. Singh, M. Woo, nd C.S. Rghvendr, Power wre routing in mobile d hoc networks, in Proc. of MOBICOM, 1998. [9] J.H. Chng nd L. Tssiuls, Mximum Lifetime Routing In Wireless Sensor Networks submitted to the ACM/IEEE Trnsctions on Networking. [10] J.-H. Chng nd L. Tssiuls, Energy Conserving Routing in Wireless Ad-hoc Networks, Proc. IEEE INFOCOM 2000, pp. 22 31, Tel Aviv, Isrel, Mr. 2000. [11] T. ElBtt, S. Krishnmurthy, D. Connors nd S. Do Power Mngement for Throughput Enhncement in Wireless Ad-Hoc Networks, Proc. IEEE ICC, 2000. [12] T. Rppport, Wireless Communictions, Prentice Hll, 1999. [13] M. K. Simon, M. S. Alouini, Digitl communiction over Fding chnnels, Wiley Interscience, 2000. [14] V. Rodoplu nd T. H. Meng, Minimum energy mobile wireless networks, IEEE Journl on Selected Ares in Communictions, vol. 17, pp. 1333 1344, Aug. 1999. [15] R. Rmnthn nd R. Rosles-Hin, Topology Control of Multihop Wireless Networks using Trnsmit Power Adjustment, Proc.IEEE INFOCOM 00, 2000. [16] R. Mnohr nd A. Scglione, Power Optiml Routing in Wireless Networks. Cornell Computer Systems Technicl Report CSL-TR-2003-1028, Jnury 2003. Avilble t http://www.csl.cornell.edu/publictions.html VI. DISCUSSION AND CONCLUSIONS This pper describes the dditionl cost nd complexity necessry to gurntee the sme error rte cross ll pths in multi-hop network nd compres the power optiml routes obtined with this criterion to the power optiml routing obtined with fixed per hop error rte constrint. We exmined the power cost of pths under two different models for the symbol error rte of link, nd provided severl results tht relte the routes obtined with our criterion with those obtined from