1 Energy Conservng Routng n Wreless Ad-hoc Networks Jae-Hwan Chang and Leandros Tassulas Department of Electrcal and Computer Engneerng & Insttute for Systems Research Unversty of Maryland at College ark College ark, MD Abstract An ad-hoc network of wreless statc nodes s consdered as t arses n a rapdly deployed, sensor based, montorng system. Informaton s generated n certan nodes and needs to reach a set of desgnated gateway nodes. Each node may adjust ts power wthn a certan range that determnes the set of possble one hop away neghbors. Traffc forwardng through multple hops s employed when the ntended destnaton s not wthn mmedate reach. The nodes have lmted ntal amounts of energy that s consumed n dfferent rates dependng on the power level and the ntended recever. We propose algorthms to select the routes and the correspondng power levels such that the tme untl the batteres of the nodes dran-out s maxmzed. The algorthms are local and amenable to dstrbuted mplementaton. When there s a sngle power level, the problem s reduced to a maxmum flow problem wth node capactes and the algorthms converge to the optmal soluton. When there are multple power levels then the achevable lfetme s close to the optmal (that s computed by lnear programmng) most of the tme. It turns out that n order to maxmze the lfetme, the traffc should be routed such that the energy consumpton s balanced among the nodes n proporton to ther energy reserves, nstead of routng to mnmze the absolute consumed power. Keywords energy-senstve routng, wreless ad-hoc networks, sensor networks montorng nodes gateways g.. A mult-hop wreless ad-hoc network s depcted where the nodes are randomly dstrbuted and the nformaton generated at the montorng nodes are to be delvered to the gateway nodes. I. C INTRODUCTION ONSIDER a group of wreless statc nodes randomly dstrbuted n a regon as n g., where each node has a lmted battery energy supply used manly for the transmsson of data. Assume that at each node some type of nformaton s generated as t montors the data such as sound or vbraton n ts vcnty usng the sensor, and the nformaton needs to be delvered to a set of gateway nodes. These wreless nodes are assumed to have the capablty of packet forwardng,.e., relayng an ncomng packet to one of ts neghborng nodes, and the transmtted energy level can be adjusted to a level approprate for the recever to be able to receve the data correctly f the recever s wthn the transmsson range. Upon or before a new arrval of nformaton ether generated at the node tself or forwarded from the other nodes, routng decson has to be made so that the node knows whch of ts neghborng nodes to forward ts data to. Note that the routng decson and the transmsson energy level selecton are ntrnscally connected n ths powercontrolled ad-hoc network snce the power level wll be adjusted dependng on the locaton of the next hop node. An example scenaro for ths type of wreless ad-hoc network may nclude a wreless sensor network where the sensors gather repared through collaboratve partcpaton n the Advanced Telecommuncatons/Informaton Dstrbuton Research rogram (ATIR) Consortum sponsored by the U.S. Army Research Laboratory under the ederated Laboratory rogram, Cooperatve Agreement DAAL acoustc, magnetc, or sesmc nformaton and send the nformaton to ts gateway node whch has more processng power for further processng of the nformaton or has larger transmsson range for the delvery of the nformaton to a possbly larger network for retreval by a remote user. Most of the prevous works on routng n wreless ad-hoc networks deal wth the problem of fndng and mantanng correct routes to the destnaton durng moblty and changng topology , , . In , , the authors presented a smply mplementable algorthm whch guarantees strong connectvty and assumes lmted node range. Shortest path algorthm s used n ths strongly connected backbone network. However, the route may not be the mnmum energy soluton due to possble omsson of the optmal lnks at the tme of the backbone connecton network calculaton. In , the authors developed a dynamc routng algorthm for establshng and mantanng connectonorented sessons whch uses the dea of predctve re-routng to cope wth the unpredctable topology changes. Some other routng algorthms n moble wreless networks can be found n , , , , whch, as the majorty of routng protocols n moble ad-hoc networks do, use shortest-path routng where the number of hops s the path length. The problem of mnmum energy routng has been addressed before n , , , , , , , and . The approach n those works was to mnmze the total consumed en-
2 ergy to reach the destnaton, whch mnmzes the energy consumed per unt flow or packet. If all the traffc s routed though through the mnmum energy path to the destnaton the nodes n that path wll be dran-out of batteres quckly whle other nodes, whch perhaps wll be more power hungry f traffc s forwarded through them, wll reman ntact. Instead of tryng to mnmze the consumed energy, the performance objectve of maxmzng the lfetme of the system, whch s equvalent to maxmzng the tme to network partton has been consdered. In , the problem of maxmzng the tme to network partton was reported as N-complete. In  we dentfed the maxmum lfetme problem as a lnear programmng problem. Therefore, t s solvable n polynomal tme. The work n  consdered the sngle destnaton verson of the problem, whle here we extend the problem to the multcommodty case, where each commodty has a ts own set of destnatons. In our study the topology of the network s statc and the routng accounts to fndng the traffc splts that balance optmally the energy consumpton. Hence the results are applcable to networks whch are ether statc, lke the sensor networks we mentoned earler, or whose topology changes slowly enough such that there s enough tme for optmally balancng the traffc n the perods between successve topology changes. Ths paper s organzed as follows: In secton II, the problem s formulated. In secton III, we propose a class of flow augmentaton algorthms that use the shortest cost path. In secton IV, we extend the flow redrecton algorthm to cover the multcommodty case. In secton V, random graphs are generated n order to evaluate the performances of these algorthms. nally n secton VI, some concludng remarks are made. II. ROUTING OR THE MAXIMUM SYSTEM LIETIME The wreless ad-hoc network n consderaton s modeled as a drected graph G(N A) where N s the set of all nodes and A s the set of all drected lnks ( j) where j 2 N. Let S be the set of all nodes that can be reached by node wth a certan power level n ts dynamc range. We assume that lnk ( j) exsts f and only f j 2 S. Let each node have the ntal battery energy E, and let Q (c) be the rate at whch nformaton s generated at node belongng to commodty c 2 C, where C s the set of all commodtes. Assume that the transmsson energy requred for node to transmt an nformaton unt to ts neghborng node j s e j, and the rate at whch nformaton of commodty c s transmtted from node to node j s called the flow j. urther, let Q and q j be the aggregate flows of all commodtes,.e., and Q = X c2c q j = X c2c Q (c) j : We are gven, for each commodty c, a set of orgn nodes O (c) where the nformaton s generated,.e., O (c) = f j Q (c) > 0 2 N g and a set of destnaton nodes D (c) among whch any node can be reached n order for the nformaton transfer of commodty c be consdered done. The lfetme of node under a gven flow q = fq j g s gven by T (q) = E e j j2s c2c : j Now, let us defne the system lfetme under flow q as the length of tme untl the frst battery dran-out among all nodes n N, whch s the same as the mnmum lfetme over all nodes,.e., T sys (q) =mn 2N T (q) =mn 2N E ej j2s c2c : q j (c) Our goal s to fnd the flow that maxmzes the system lfetme under the flow conservaton condton. The problem can be wrtten as follows: Maxmze T sys (q) = mn 2N s.t. E ej j2s c2c j j N 8j 2 S 8c 2 C j + Q (c) = 8 k 2 N ; 8c D(c) 2 C: j: 2Sj k2s (6) g.2 llustrates the flow conservaton condton for commodty c at node, and t should be noted that the condton apples to each commodty separately. In the followng we show that the problem s a lnear programmng problem. The problem of maxmzng the system lfetme, gven the nformaton generaton rates Q (c) at the set of orgn nodes O (c) and the set of destnaton nodes D (c) for each commodty c, s equvalent to the followng lnear programmng problem: s.t. j: 2Sj Maxmze T ^q (c) j N 8j 2 S 8c 2 C e j ^ j E 8 2 N c2c ^ j + TQ (c) = ^ k 8 2 8c N;D(c) 2 C (0) j2s k2s where ^ j = T j s the amount of nformaton of commodty c transmtted from node to node j untl tme T.
3 j j Q (c) q (c) k g. 2. The conservaton of flow condton at node for each commodty c requres that the sum of nformaton generaton rate and the total ncomng flow must equal the total outgong flow. The lnear program gven above can be vewed as a varaton of the conventonal maxmum flow problem wth node capactes. If the transmtted power level at each node s fxed regardless of ts next hop node,.e., f there s no power control, k e j = e 8j 2 S () and the problem s equvalent to the maxmum flow problem wth node capactes gven by X X j2s c2c ^ j E =e 8 2 N: (2) When the capacty of a a node s a fxed quantty as n (2) then the problem can be converted to a lnk capacty verson by replacng the node wth two nodes and a lnk havng the same capacty, and the max-flow-mn-cut theorem can be used. However, n our problem, unlke the above, the amount of resource (or energy n ths case) whch a unt flow consumes depends on the energy expendture to the next hop node. Therefore, t s not trval to fnd the mn-cut nodes, and even f they were found the traffc splt at the nodes must also be dentfed. III. LOW AUGMENTATION ALGORITHMS In ths secton, we propose a class of flow augmentaton (A) algorthms whch use the shortest cost path. The general descrpton of the algorthm s gven n the followng. At each teraton, each orgn node o 2 O (c) of commodty c calculates the shortest cost path to ts destnaton nodes n D (c). Then the flow s augmented by an amount of Q (c) on the shortest cost path, where s the augmentaton step sze. After the flow augmentaton, the shortest cost paths are recalculated and the procedures are repeated untl any node 2 N runs out of ts ntal total energy E. As a result of the algorthm, we obtan the flow whch wll be used at each node to properly splt ncomng traffc. Our objectve s to fnd the best lnk cost functon whch wll lead to the maxmzaton of the system lfetme. There are three parameters to consder n calculatng the lnk cost c j for lnk ( j). One s the energy expendture for unt flow transmsson over the lnk, e j, the second s the ntal energy E, and S the thrd s the resdual energy at the transmttng node whch s denoted by E. A good canddate for the flow augmentng path should consume less energy and should avod nodes wth small resdual energy snce we would lke to maxmze the mnmum lfetme of all nodes. In , each of these were separately consdered, whch falls short of optmzng the system lfetme. Obvously, both of these can t be optmzed at the same tme, whch means there s a tradeoff between the two. In the begnnng when all the nodes have plenty of energy, the mnmum total consumed energy path s better off, whereas towards the end avodng the small resdual energy node becomes more mportant. Therefore, the lnk cost functon should be such that when the nodes have plenty of resdual energy, the energy expendture term s emphaszed, whle f the resdual energy of a node becomes small the resdual energy term should be more emphaszed. Wth the above n mnd, the lnk cost c j s proposed to be c j = e x j E;x2 E x3 (3) where x, x 2, and x 3 are nonnegatve weghtng factors for each tem. Note that f fx x 2 x 3 g = f0 0 0g then the shortest cost path s the mnmum hop path, and f t s f 0 0g then the shortest cost path s the mnmum transmtted energy path. If x 2 = x 3 then normalzed resdual energy s used, whle f x 3 = 0 then the absolute resdual energy s used. Let s refer to the algorthm as A(x x 2 x 3 ) n the rest of the paper ndcatng the parameters, and the meanngs of the parameters are summarzed n Table I for reference. The path cost s computed by the summaton of the lnk costs on the path, and the algorthm can be mplemented wth any exstng shortest path algorthms ncludng the dstrbuted Bellman-ord algorthm, whch wll be used n our smulaton. IV. LOW REDIRECTION ALGORITHM In ths secton, we extend the flow redrecton (R) algorthm to the multcommodty case. Ths algorthm s based on the followng observaton. If we have a sngle orgn and a sngle destnaton or f we have multple orgns and destnatons but wthout any constrants on the nformaton generaton rates, then under the optmal flow, the mnmum lfetme of every path from the orgn to the destnaton wth postve flow s the same. Note that the latter case can be converted to a sngle orgn and a sngle destnaton verson by addng a super orgn and a super destnaton connected to the orgns and the destnatons respectvely wth zero energy expendture lnks. The above fact can be shown as follows. Assume that the flow s optmal,.e., mnmum lfetme over all nodes s maxmzed. If we further assume that the mnmum lfetmes of the paths wth postve flow to the destnaton are not all dentcal then there s a set of path(s) wth postve flow whose mnmum lfetme s the shortest. We can always ncrease the mnmum
4 TABLE I THE MEANINGS O THE ARAMETERS IN THE ALGORITHM A. A(x x 2 x 3 ) A(0 0 0) A( 0 0) A( x x) A( 0) Meanng Mnmum hop path Mnmum transmtted energy path Normalzed resdual energy s used Absolute resdual energy s used lfetme of ths set of path(s), whch s also the system lfetme, by redrectng an arbtrarly small amount of flow to the paths whose lfetme s longer than these paths such that the mnmum lfetme of the latter path after the redrecton s stll longer than the system lfetme before the redrecton. Ths contradcts our assumpton that the flow s optmal. In ths algorthm we redrect a porton of each commodty flow at every node n a way that the mnmum lfetme of every path wth postve flow from the node to the destnaton wll ncrease or at least wll stay the same. In the followng, we descrbe the mplementaton of R. Let s use an magnary super destnaton node d ~ (c) where d ~ (c) 2 S d and e d d ~ (c) = 0 for all d 2 D (c). Let the ntal flow be such that from o 2 O (c) to d ~ (c) the mnmum total transmtted energy path s used wth a flow value of Q (c) o. Note that any path to the destnaton can be used as the ntal flow. Each node 2 N ; D (c) redrects ts outgong flow of commodty c by subtractng (c) from the flow of a certan path to d ~ (c) and by addng the same amount to the flow of another path to d ~ (c). It s possble that the flow can be re-routed to a dfferent destnaton node n D (c). The steps to be taken at each node 2 N ; D (c) for each commodty c are as follows:. (Determne the Two aths) Determne the two paths to the destnaton whch are to be nvolved n the redrecton. 2. (Calculate (c) ) Calculate the amount (c) of redrecton. 3. (Redrect the low) roperly ncrement and decrement the flows of the two paths determned above by an amount of (c). The frst step of the algorthm at each node for commodty c, (Determne the Two aths), s descrbed n more detal. The goal of ths step s to dentfy the ascent drecton. We wll need two dfferent path calculatons for each commodty. Let s frst form a subnetwork G (c) (N A(c) ) of G(N A) where A (c) A conssts only of edges wth postve flow,.e., Let (c) A (c) = f( j)jq(c) j > 0 ( j) 2 Ag: (4) be the set of all paths n G (c) ) from node (N A(c) to any of the destnaton nodes n D (c). or a path p 2 (c), defne ts path length L p (q) under flow q as a vector whose elements are the lfetmes of all the nodes n the path before reachng any of ts destnaton nodes D (c). or example, f path p 2 (c) startng from node traverses nodes j j 2 j m before reachng any node n D (c), then L p (q) = [T (q) T j (q) T j2 (q) T j m (q)]. The length of path p, L p (q), s sad to be shorter (longer) than the length of path p 0, L p 0(q), f the smallest element of L p (q) s smaller (larger) than that of L p 0(q). We compare the smallest element frst snce t s the mnmum lfetme of all nodes n the path. In case they are the same, the next smallest elements are compared, and so on. If there are more than one smallest elements wth the same value then each one s compared separately. Usng ths so-called lexcographcal orderng, the shortest length path from any node to the destnaton s defned. We modfy the dstance comparson part of the Bellman-ord algorthm  to obtan the shortest length paths dstrbutvely. Let s denote the shortest length path n G (c) (N A(c) ) from node to the destnaton node d ~ (c) by sp (c) (). Note that the shortest path, sp (c) (), passes through the node whch has the mnmum lfetme of all downstream nodes of node. The other path calculaton s to fnd the longest length path n G(N A) usng the same path length vector. If two path lengths are the same, choose the one wth less number of elements n the path length vector. The longest length path s the path whose mnmum lfetme s the longest. Let s denote the longest length path n G(N A) from node to the destnaton node d ~ (c) by lp (c) (). Note that the longest length path s the path whch, n some sense, has the largest capacty snce we wll need to assgn more flows to ths path than any other path n order to make the mnmum lfetme of the path to equal the mnmum lfetme of the other paths. Let g denote the next hop node of node from whch path the flow wll be subtracted, and let t denote the next hop node of node to whch path the flow wll be added, where g 2 S, the gver, and t 2 S, the taker, are to be carefully chosen among the neghbors of node. Note that notatons such as g (c) and t (c) could be used, but we use g and t nstead for smplcty snce there s no ambguty. Dependng on whether or not the lfetme of node, T (q), s the mnmum n the shortest length path sp (c) (), two dfferent measures are taken. If T (q) mn[l sp (c) ()(q)] then the lfetme of node s the mnmum over all nodes n the subnetwork consstng of node and all ts downstream nodes, and hence we would lke to ncrease the lfetme of node. Ths can be acheved f we redrect a flow at node to the drecton where the requred transmsson energy per nformaton unt s smaller. In other words, f we choose a flow passng node g we redrect the flow to node t where e t <e g. Ths can be done n many dfferent ways. One such choce wll be to redrect the flow whose energy expendture to the next hop s the maxmum to the drecton of mnmum energy expendture to the next hop,.e., g = argmax j : j2s j >0 e j
5 and t = argmn j : j2s e j : (6) Another possblty for the taker s to choose the longest length path whose path length s the longest among all the next hop nodes that has smaller energy expendture than e g,.e., t = argmax j : j2s ej <eg L lp (c) (j)(q) where MAX denotes the maxmum n the lexcographcal orderng. The gver doesn t have to be the node wth the maxmum energy expendture. In fact we can choose any node wth nonmnmum energy expendture. In our algorthm, all these possbltes are used alternately. On the other hand, f T (q) > mn[l sp (c) ()(q)] then we would lke to ncrease the lfetme of the mnmum lfetme node n the path sp (c) () by redrectng some of the flow to another path snce the lfetme of that node s the mnmum lfetme over all nodes n the subnetwork consstng of node and ts downstream nodes. The gver g s the next hop node of node n the shortest length path sp (c) (), and the taker can be ether the node whose longest length path s the longest,.e., t = argmax j2s L lp (c) (j)(q) or the node wth the mnmum energy expendture whose longest length path s longer than the shortest length path of node,.e., t = argmn j : j2s mn[l lp (c) (j) (q)]>mn[l sp (c) () (q)] e j : In fact, t suffces to fnd any node whose path length of the longest length path s longer than that of the shortest length path of node. In our algorthm, all these possbltes are used alternately. Gven the two nodes g and t, the flow of the path composed of ( g) and sp (c) (g) wll be re-routed to the path composed of ( t) and lp (c) (t). The second step of the algorthm at each node for commodty c, (Calculate (c) ), s descrbed n more detal. The am of ths step s to determne the amount of redrecton that guarantees monotonc non-decrease of the system lfetme. The constrants that (c) should meet are as follows. rst, t should be less than or equal to the flow n the path of gver node,.e., g and (2) jk for each lnk (j k) n the path sp (c) (g). urthermore, none of the lfetmes should become shorter than the currently mnmum lfetme of the subnetwork consstng of node and all ts downstream nodes snce ths wll lead us to the opposte drecton to that of our objectve. If T (q) mn[l sp (c) ()(q)] then none of the lfetmes n the path lp (c) (t) should become shorter than T (q),.e., e jk (c) T j (q) + E j T (22) (q) for each lnk (j k) n the path lp (c) (t). On the other hand f T (q) > mn[l sp (c) ()(q)] then we need to consder two thngs. rst, none of the lfetmes n the path lp (c) (t) should become shorter than the mnmum lfetme of the path sp (c) (),.e., e jk (c) T j (q) + (23) E j mn[l sp (c) ()(q)] for each lnk (j k) n the path lp (c) (t). Second, f e t >e g then the lfetme of node may decrease due to the redrecton, but t should not become shorter than the mnmum lfetme of the path sp (c) (),.e., (e T (q) + t ; e g ) (c) : (24) E mn[l sp (c) ()(q)] nally, the value of (c) should be chosen among the values that meet all the constrants stated above. We could ether use the maxmum (c) that meets all the constrants or just a fracton of t. To avod possble oscllatons and for faster convergence, we choose the half of the maxmum (c) that meets all the constrants except and (2). The thrd step of the algorthm at each node for commodty c, (Redrect the low), s descrbed n more detal. Subtractng (c) from the path sp (c) (g) s smple. We made sure that (c) s less than or equal to what s avalable n each edge n the path n and (2). Addng (c) to the path lp (c) (t) s also smple, but there s a possblty that one or more loops of postve flow value can be formed. These loops should be removed n order to avod unnecessary energy consumpton and to ensure that the path ndeed leads to the destnaton. After addng (c) to the path lp (c) (t), the formaton of one or possbly more loops s checked and the loop(s) are removed lnk by lnk along the path. or nstance, f lnk (j k) n the path lp (c) (t) s checked by calculatng the shortest hop dstance from node k to node j n the subnetwork G (c) (N A(c) ) of G(N A). If the dstance s fnte then at least one loop exsts. Remove the loop flow and then repeat the procedure untl all loops nvolvng the lnk (j k) s removed and then proceed to the next lnk. In the followng t s shown that R can have arbtrarly poor performance. or the performance comparson, let s denote the maxmum system lfetme obtaned usng algorthm X by Tsys X, and the optmum system lfetme by Tsys, opt and the rato between these two values s denoted by R T sys X X = (25) Tsys opt
6 Q =.000 E =.0 Q =.000 E =.0 e 5 = δ / 2 q 5 =0.328 q 2 =0.33 q 3 =0.338 e 5 = δ / 2 q 5 =0.97 q 2 =0.00 q 3 =0.00 e 2 = δ e 3 = 2 δ e 2 = δ e 3 = 2 δ E 5 =.0 q 4 =0.003 q 52 =0.97 e 52 = + 3 δ E 2 =.0 e 23 = + δ e 24 = + 2 δ E 3 =.0 e = 34 e 4 = / δ q 54 =0.328 q 24 =0.33 q 34 =0.338 T =2.96, =, 2, 3, 5 (a) q 23 =0.980 E 5 =.0 q 4 =0.00 e 52 = + 3 δ E 2 =.0 e 23 = + δ e 24 = + 2 δ E 3 =.0 e = 34 e 4 = / δ T =.0, =, 2, 3, 5 (b) e 54 = + 4 δ e 54 = + 4 δ q 34 =0.990 g. 3. An example showng local optmum convergence of R for arbtrary postve constant. The flow values and lfetmes correspond to the case when =0:0 : (a) global optmum wth Tsys opt 2:95; (b) local optmum wth Tsys R :0. whch wll be used throughout the paper as the performance measure. An example showng the convergence to a local optmum s gven n g.3, where a sngle commodty s orgnated from node and s destned for node 4. s an arbtrary constant whle the values of flows and lfetmes n the fgure are for the case when = 0:0. When = 0:0, the optmum s Tsys opt 2:95, but the maxmum system lfetme obtaned by R s Tsys R =:0. We can verfy that R R can be as small as /3 as approaches zero. In the worst case R R can be shown to be arbtrarly small by expandng the network n a smlar fashon. or example, the rato R R reaches /4 f we expand the network n g.3 by addng node 6 wth e 6 = =3, e 65 =+5, and e 64 =+6. V. ERORMANCE COMARISON THROUGH SIMULATION In ths secton, random graphs are generated n order to evaluate the performances of the proposed algorthms. The performances are compared wth that of mnmum transmtted energy (MTE) routng algorthm n order to see how much we gan n terms of the system lfetme compared to the conventonal mnmum transmtted energy routng algorthm. Comparson s also made wth the maxmum resdual energy path (MRE) routng algorthm proposed n , where the path length was a vector whose elements were the lnk costs gven by c j =(E ; e j ) ; : (26) The lexcographcal orderng was used n comparson of the two length vectors. The dea was to augment the flow on the path whose mnmum resdual energy after the flow augmentaton wll be the largest. It has been shown n  that MTE can perform arbtrarly bad by an example. In the followng example, t s shown that the mnmum hop (MH) routng can perform arbtrarly bad. g.4 (a) shows the optmal soluton and g.4 (b) shows the mnmum hop soluton. The rato between the system lfetme obtaned by MH and the optmal soluton s R MH =. As>0 approaches zero, R MH approaches zero. Note that the example scenaro s possble snce n a wreless envronment, path loss s proportonal to the square of the dstance n free space and n hgher orders n urban area, whch makes multhop transmsson less energy consumng then a sngle hop counterpart n many cases. Let there be 20 nodes randomly dstrbuted n a square of sze 5 by 5. Assume that the transmsson range of each node s lmted by 2.5,.e., j 2 S f and only f d j 2:5, where d j s the dstance between node and node j. The energy expendture per unt nformaton transmsson from node to j s assumed to be gven by e j = :0 0 ;8 f d j 0:025 ( dj (27) f 0:025 <d 2:5 )4 j 2:5: Note that there may be cases where no path s avalable between the orgns and the destnatons, although t s very rare n our settng. We smply dscard these cases to assume the connectvty. Two dfferent scenaros are smulated: ) sngle commodty case where nformaton generated at 5 orgn nodes need to reach any one of two destnaton nodes; ) multcommodty case where each of the 5 orgn nodes has ts own sngle desgnated destnaton node. rst of all, A(x x 2 x 3 ) s smulated to fnd the best parameters x, x 2, and x 3. Let node have ntal energy of E = f s even and E = 2 f s odd. In the sngle commodty case, the orgn nodes are gven by O = f g and assume the nformaton generaton rates are Q = Q 2 = Q 3 = 2 and
7 = /δ E 2 = q e 2 = δ e 23 = δ 2 = q 23 = T = /δ Q = e = 3 E = (a) E = 2 average R A x 0.9 A(,x,0) A(,x,x) A(0,x,0) A(0,x,x) A(x,x,0) A(x,x,x) g. 5. The average performances of A(x x 2 x 3 ). e 2 = δ e 23 = δ T = Q = E = e 3 = q 3 = worst case R A A(,x,0) A(,x,x) A(0,x,0) A(0,x,x) A(x,x,0) A(x,x,x) (b) g. 4. An example showng that MH can have arbtrarly poor performance where >0 s a postve constant : (a) optmum system lfetme Tsysx opt = =; (b) system lfetme obtaned by MH, T MH sys =: x g. 6. The worst case performances of A(x x 2 x 3 ). Q 4 = Q 5 =. The destnaton nodes are D = f9 20g. In the multcommodty case O () = fg and D () = f +5g for = wth Q () =2for = 2 3 and Q () =for =4 5. gs.5 and 6 show the results for sngle commodty case when = 0:0. Multcommodty case results are not shown here snce they were smlar to the sngle commodty case. In all cases, A( x x) was the best n terms of both average and worst case performance. It should be noted that even wth x =, R A( x x) was always over 0.8 of the optmal and about 0.98 of the optmal on the average. A(0 x 0) and A(0 x x) was the worst wth average performance of about 0.2, whch means that by consderng only the resdual energy wthout takng the energy expendture nto account the system lfetme can t be mproved much. It s better than the MH soluton but consderably worse than all the others whch consders the energy expendture term. The results also suggest that we use the normalzed resdual energy nstead of the absolute resdual energy, whch can be more clearly seen n g.6 by comparng A( x x) wth A( x 0). gs. 7 and 8 plot the average and the worst case performance of the best A( x x) for varous values of. We could observe that as got smaller, the performance was better. Note that the worst case of A( 50 50) when =0:00 was The curves weren t monotoncally ncreasng, but we can see that for smaller t s so up to a larger x than the curves of larger. Ths phenomenon can be best explaned as follows. Whle the shortest cost path may ndeed be the optmal drecton for the flow augmentaton, t s only so for a certan amount of flow. As soon as gets larger than ths amount, monotoncty of the convergence breaks. A somewhat smlar behavor, though not exactly dentcal, can be found n many optmzaton methods usng descent drecton , where a procedure called lne search s done to guarantee monotonc convergence. Let s compare the performances of the other algorthms. In both sngle commodty and multcommodty case, let each node have ntal energy E = and assume that the nformaton generaton rate at each orgn node o 2 O (c) s Q (c) o = for each commodty. The sngle commodty case results are presented frst. Be-
8 average R A Average performance of A(,x,x) λ=0.00 λ=0.005 λ=0.0 TABLE II THE ERORMANCE COMARISON O THE ALGORITHMS IN THE SINGLE COMMODITY CASE. Algorthm X avg R X mn R X rfr X > 0:9g MTE % R % MRE % A( ) % A( 50 50) % x g. 7. The average performance of A( x x) for varous values of. : Orgn : Destnaton MTE T =.36 sys R MTE = 0.22 T = () (3) (4) Worst case performance of A(,x,x) worst case R A λ=0.00 λ=0.005 λ=0.0 (6) T = 5.62 (6)(0) T 5 =220.4 =36.66 T = T 5 =35.65 (2) T 4 =.36 g. 9. An example showng the soluton by MTE for sngle commodty case where nodes through 5 are the orgn nodes, and nodes 9 and 20 are the destnaton nodes where any one of the two nodes need to be reached x g. 8. The worst case performance of A( x x) for varous values of. fore gong nto the statstcs, let s compare the algorthms by an example graph, where orgn nodes are gven by O = f g and destnaton nodes are gven by D = f9 20g. gs.9, 0, and show the solutons of MTE, R, and A( 50 50) wth =0:00, respectvely. The true optmum s Tsys opt =6:3. One can observe that the advantage of our algorthms over MTE les n the fact that the traffc s more spread out. The system lfetme obtaned by A( 50 50) and R were more than four tmes as long as that of MTE n ths example, and both were close to the optmal. The performances of the algorthms are presented n Table II, and n g.2 average and worst cases of the algorthms are compared. Note that = 0:00 was used for MRE and A. or each algorthm a total of 200 randomly generated graphs were smulated. R A( 50 50) was always over 0.99 of the optmal,.e., even n the worst case. A( ) s performance was comparable to MRE s. Whle the average of R MTE was about 0.730, the average system lfetme of R, MRE, and A( x x) for x were above 0.95 of the optmal. R R and R MRE were over 0.9 n about 90 % of the case whle that of MTE was so n only 33 % of the case. The worst case of R MTE, R R and R MRE were 0.837, , and 0.80, respectvely. Although t was shown earler that both MTE and R can perform arbtrarly bad n the worst case, smulaton results were n favor of R. The average gan n the system lfetme obtaned by the proposed algorthms were between 49 % and 55 % compared wth MTE. In the multcommodty case, commodty 2 C where C = f g s assumed to be generated at node and ts destnaton node s node +5 among 20 randomly dstrbuted nodes. gs.3, 4, and 5 show examples of multcommodty case solutons by MTE, R, and A( 50 50) wth =0:00 respectvely, where only the aggregate flows are depcted. In ths example, the optmal system lfetme s Tsys opt =7:8, and the system lfetme obtaned by R and A( 50 50) were more than one and a half tmes as long as that of MTE, where both were close to the optmal. In the multcommodty case, the performances of the algorthms gven n Table III and g.6 showed smlar behavor to the sngle commodty case. = 0:00 was used for MRE and A( 50 50). R A( 50 50) was always over 0.99 of the optmal,.e., ncludng the worst case, and agan A( ) s performance was comparable to MRE s. Whle the average R MTE was , those of R R and R MRE were
9 : Orgn : Destnaton R T = 6.00 sys R R = 0.95 (6) T 6 =29.83 T = (6)(0) T = 6.0 T 0 = 6.0 T 3 =34.35 T = T = T = (3) (4) T 5 = 6.0 = 6.0 T = T 7 =34.36 T 8 =34.37 = 6.00 T 5 =34.34 (2) T = T = () T 7 = g. 0. An example showng the soluton by R for sngle commodty case where nodes through 5 are the orgn nodes, and nodes 9 and 20 are the destnaton nodes where any one of the two nodes need to be reached. : Orgn : Destnaton A T sys = 6.29 R =.00 A T = () T 3 =33.98 (3) (4) T = T 8 =37.7 TABLE III THE ERORMANCE COMARISON O THE ALGORITHMS IN THE MULTICOMMODITY CASE. Algorthm X avg R X mn R X rfr X > 0:9g MTE % R % MRE % A( ) % A( 50 50) % (4) T 4 = 4.7 T 5 = 8.25 T 5 = 4.7 : Orgn : Destnaton MTE T sys = 4.7 R MTE = 0.60 T (0) 2 = 23. (2) = T 0 = T () = (6) T 4 = T 6 = 39.0 T 9 = T 8 = 7.48 (3) T 3 = 5.00 T 9 = 9.5 (6) T = T = T = (6)(0) T = (6) T = = 6.29 T 5 =33.25 g. 3. An example showng the soluton by MTE for multcommodty case where nodes through 5 are the orgn nodes and nodes 6 through 20 are the correspondng destnaton nodes, respectvely. T 6 = 9.60 T = 6.30 T = = 6.29 (2) T 4 = 6.29 g.. An example showng the soluton by A( 50 50) when =0:00 for sngle commodty case where nodes through 5 are the orgn nodes, and nodes 9 and 20 are the destnaton nodes where any one of the two nodes need to be reached. R worst case average MTE R MRE A(,,) A(,50,50) Algorthms g. 2. The comparson of average and worst case performances of all three algorthms are made n the sngle commodty case. and , respectvely. R R and R MRE were over 0.9 n 54 % and 69 % of the case respectvely, whle that of MTE was so n only 25 % of the case. The worst cases of R MTE, R R, and R MRE were 0.220, , and respectvely. Whle the performances of MTE, R, and MRE deterorated compared wth sngle commodty case, the performance of A( 50 50) was stll very close to the optmal. The average gan n the system lfetme obtaned by the proposed algorthms were between 40 % and 62 % compared wth MTE. VI. CONCLUSION In power-controlled wreless ad-hoc networks, battery energy at network nodes s a very lmted resource that needs to be utlzed effcently. One of the conventonal routng objectves was to mnmze the total consumed energy n reachng the destnaton. However, the conventonal approach may dran out the batteres of certan paths whch may dsable further nformaton delvery even though there are many nodes wth plenty of energy. Therefore, we formulated the routng problem wth the objectve of maxmzng the system lfetme gven the sets of orgn and destnaton nodes and the nformaton generaton rates at the orgn nodes, and proposed a class of flow augmentaton algorthms and a flow redrecton algorthm whch balance the energy consumpton rates among the nodes n proporton to
10 0 = 4.46 (4) T 4 T 5 T 5 : Orgn : Destnaton R T sys R = 0.89 R T 7 = T 7 T (0) 2 = (2) = T 0 = 37.7 T () T 4 = (6) T = T T = (3) T 3 T 9 T 6 (6) T 3 = T T 8 = 37.7 g. 4. An example showng the soluton by R for multcommodty case where nodes through 5 are the orgn nodes and nodes 6 through 20 are the correspondng destnaton nodes, respectvely. T = (4) T 4 T = T 5 : Orgn : Destnaton A T = 7.79 sys R A =.00 T 7 = 49.9 T 7 T (0) 2 = (2) = T 0 = T () T 4 = 44.4 (6) T 6 = T T 8 = (3) T 3 T 9 T 6 (6) T 3 T T 8 = g. 5. An example showng the soluton by A( 50 50) when =0:00 for multcommodty case where nodes through 5 are the orgn nodes and nodes 6 through 20 are the correspondng destnaton nodes, respectvely. R worst case average MTE R MRE A(,,) A(,50,50) Algorthms g. 6. The comparson of average and worst case performances of all three algorthms are made n the multcommodty case. ther energy reserves. The proposed algorthms are local and amenable to dstrbuted mplementaton and showed close to the optmal performance most of the tme, sgnfcantly mprovng the system lfetme, that s, as much as 60 % on the average over the conventonal mnmum transmtted energy routng. REERENCES  Denns J. Baker and Anthony Ephremdes, The archtectural organzaton of a moble rado network va a dstrbuted algorthm, IEEE Transactons on Communcatons, vol. COM-29, no., pp , Jan. 98.  Dmtr Bertsekas and Robert Gallager, Data Networks, rentce-hall, Inc., 2nd edton, 987.  J.-H. Chang and L. Tassulas, Routng for maxmum system lfetme n wreless ad-hoc networks, n roceedngs of 37-th Annual Allerton Conference on Communcaton, Control, and Computng, Montcello, IL, Sept  Wa-Ka Chen, Theory of Nets: lows n Networks, Wley, 990.  T. Cormen, C. Leserson, and R. Rvest, Introducton to Algorthms, McGraw-Hll and MIT ress, 990.  Anthony Ephremdes, Jeffrey E. Weselther, and Denns J. Baker, A desgn concept for relable moble rado networks wth frequency hoppng sgnalng, roceedngs of the IEEE, vol. 75, no., pp , Jan  M. Ettus, System capacty, latency, and power consumpton n multhoprouted SS-CDMA wreless networks, n roceedngs of IEEE Rado and Wreless Conference (RAWCON) 98, Colorado Sprngs, CO, Aug. 998, pp  R.G. Gallager,.A. Humblet, and.m. Spra, A dstrbuted algorthm for mnmum weght spannng trees, Tech. Rep. LIDS--906-A, Lab. Inform. Decson Syst., Massachusetts Inst. of Technol., Cambrdge, MA, Oct  D. Johnson and D. Maltz, Dynamc source routng n ad hoc wreless networks, Moble Computng, 996.  Teresa H. Meng and Volkan Rodoplu, Dstrbuted network protocols for wreless communcaton, n roceedngs of the 998 IEEE Internatonal Symposum on Crcuts and Systems, ISCAS 98, Monterey, CA, June 998, vol. 4, pp  A. Mchal and A. Ephremdes, A dstrbuted routng algorthm for supportng connecton-orented servce n wreless networks wth tmevaryng connectvty, n roceedngs Thrd IEEE Symposum on Computers and Communcatons, ISCC 98, Athens, Greece, June 998, pp  S. Murthy and J.J. Garca-Luna-Aceves, An effcent routng protocol for wreless networks, ACM Moble Networks and Applcatons Journal, Specal Issue on Routng n Moble Communcaton Networks, 996.  Stephen G. Nash and Arela Sofer, Lnear and Nonlnear rogrammng, McGraw-Hll, 996.  Vncent D. ark and M. Scott Corson, A hghly dstrbuted routng algorthm for moble wreless networks, n roc. IEEE INOCOM 97, Kobe, Japan, 997.  C. erkns and. Bhagwat, Hghly dynamc destnaton-sequenced dstance vector routng (DSDV) for moble computers, n ACM SIGCOMM, Oct  Volkan Rodoplu and Teresa H. Meng, Mnmum energy moble wreless networks, n roceedngs of the 998 IEEE Internatonal Conference on Communcatons, ICC 98, Atlanta, GA, June 998, vol. 3, pp  Tmothy Shepard, Decentralzed channel management n scalable multhop spread spectrum packet rado networks, Tech. Rep. MIT/LCS/TR- 670, Massachusetts Insttute of Technology Laboratory for Computer Scence, July 995.  S. Sngh, M. Woo, and C.S. Raghavendra, ower-aware routng n moble ad hoc networks, n roceedngs of ourth Annual ACM/IEEE Internatonal Conference on Moble Computng and Networkng, Dallas, TX, Oct. 998, pp The vews and conclusons contaned n ths document are those of the authors and should not be nterpreted as representng the offcal polces, ether expressed or mpled, of the Army Research Laboratory or the U.S. Government.