Fair Coalitions for Power-Aware Routing in Wireless Networks

Transcription

1 Unversty of Pennsylvana ScholarlyCommons Departmental Papers (ESE) Department of Electrcal & Systems Engneerng February 2007 Far Coaltons for Power-Aware Routng n Wreless Networks Ratul K. Guha Unversty of Pennsylvana Carl A. Gunter Unversty of Illnos Saswat Sarkar Unversty of Pennsylvana, swat@seas.upenn.edu Follow ths and addtonal works at: Recommended Ctaton Ratul K. Guha, Carl A. Gunter, and Saswat Sarkar, "Far Coaltons for Power-Aware Routng n Wreless Networks",. February Copyrght 2007 IEEE. Reprnted n IEEE Transactons on Moble Computng, Volume 6, Issue 2, February 2007, pages Ths materal s posted here wth permsson of the IEEE. Such permsson of the IEEE does not n any way mply IEEE endorsement of any of the Unversty of Pennsylvana's products or servces. Internal or personal use of ths materal s permtted. However, permsson to reprnt/republsh ths materal for advertsng or promotonal purposes or for creatng new collectve works for resale or redstrbuton must be obtaned from the IEEE by wrtng to pubs-permssons@eee.org. By choosng to vew ths document, you agree to all provsons of the copyrght laws protectng t. Ths paper s posted at ScholarlyCommons. For more nformaton, please contact repostory@pobox.upenn.edu.

2 Far Coaltons for Power-Aware Routng n Wreless Networks Abstract Several power-aware routng schemes have been developed for wreless networks under the assumpton that nodes are wllng to sacrfce ther power reserves n the nterest of the network as a whole. But, n several applcatons of practcal utlty, nodes are organzed n groups, and as a result, a node s wllng to sacrfce n the nterest of other nodes n ts group but not necessarly for nodes outsde ts group. Such groups arse naturally as sets of nodes assocated wth a sngle owner or task. We consder the premse that groups wll share resources wth other groups only f each group experences a reducton n power consumpton. Then, the groups may form a coalton n whch they route each other s packets. We demonstrate that sharng between groups has dfferent propertes from sharng between ndvduals and nvestgate far, mutually benefcal sharng between groups. In partcular, we propose a paretoeffcent condton for group sharng based on max-mn farness called far coalton routng. We propose dstrbuted algorthms for computng the far coalton routng. Usng these algorthms, we demonstrate that far coalton routng allows dfferent groups to mutually benefcally share ther resources. Keywords wreless communcaton, algorthm desgn and analyss, energy-aware systems and routng Comments Copyrght 2007 IEEE. Reprnted n IEEE Transactons on Moble Computng, Volume 6, Issue 2, February 2007, pages Ths materal s posted here wth permsson of the IEEE. Such permsson of the IEEE does not n any way mply IEEE endorsement of any of the Unversty of Pennsylvana's products or servces. Internal or personal use of ths materal s permtted. However, permsson to reprnt/republsh ths materal for advertsng or promotonal purposes or for creatng new collectve works for resale or redstrbuton must be obtaned from the IEEE by wrtng to pubs-permssons@eee.org. By choosng to vew ths document, you agree to all provsons of the copyrght laws protectng t. Ths journal artcle s avalable at ScholarlyCommons:

3 206 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 6, NO. 2, FEBRUARY 2007 Far Coaltons for Power-Aware Routng n Wreless Networks Ratul K. Guha, Student Member, IEEE, Carl A. Gunter, Senor Member, IEEE, and Saswat Sarkar, Member, IEEE Abstract Several power-aware routng schemes have been developed for wreless networks under the assumpton that nodes are wllng to sacrfce ther power reserves n the nterest of the network as a whole. But, n several applcatons of practcal utlty, nodes are organzed n groups, and as a result, a node s wllng to sacrfce n the nterest of other nodes n ts group but not necessarly for nodes outsde ts group. Such groups arse naturally as sets of nodes assocated wth a sngle owner or task. We consder the premse that groups wll share resources wth other groups only f each group experences a reducton n power consumpton. Then, the groups may form a coalton n whch they route each other s packets. We demonstrate that sharng between groups has dfferent propertes from sharng between ndvduals and nvestgate far, mutually benefcal sharng between groups. In partcular, we propose a paretoeffcent condton for group sharng based on max-mn farness called far coalton routng. We propose dstrbuted algorthms for computng the far coalton routng. Usng these algorthms, we demonstrate that far coalton routng allows dfferent groups to mutually benefcally share ther resources. Index Terms Wreless communcaton, algorthm desgn and analyss, energy-aware systems and routng. Ç 1 INTRODUCTION WIRELESS networks typcally consst of nodes that must dscharge ncreasngly complex computng and communcaton functonaltes despte rgorous constrants on power, bandwdth, sze, and memory. Sgnfcant progress has been made to mprove hardware to address these needs and much s beng done to develop software that uses technques lke power-optmzng algorthms. Comparatvely less has been done to explot sharng among nodes as a way to address these challenges. Ths s unfortunate, snce sharng can yeld great benefts. A varety of challenges mpede progress: 1. determnng whch resources can be shared, 2. decdng when to share resources, as sharng would evdently nvolve a cost, 3. decdng wth whom to share resources, and 4. determnng how to share resources. Often, groups of nodes rather than ndvdual nodes are basc enttes n the sharng mechansm. The resource expendture/utlzaton of the group as a whole s more mportant than that of a sngle node or the entre network. Groups are often formed on the bass of membershp n an organzaton or a shared task. For example, employees of an. R.K. Guha s wth the Multmeda & Networkng Lab, Department of Electrcal and Systems Engneerng, Unversty of Pennsylvana, Room 306 Moore Buldng, 200 South 33rd Street, Phladelpha, PA E-mal: rguha@seas.upenn.edu.. C.A. Gunter s wth the Department of Computer Scence, Sebel Center, 201 N. Goodwn, Urbana, IL E-mal: cgunter@cs.uuc.edu.. S. Sarkar s wth the Department of Electrcal and Systems Engneerng, Unversty of Pennsylvana, Room 354 Moore Buldng, 200 South 33rd Street, Phladelpha, PA E-mal: swat@ee.upenn.edu. Manuscrpt receved 18 Sept. 2004; revsed 14 Feb. 2006; accepted 14 June 2006; publshed onlne 14 Dec For nformaton on obtanng reprnts of ths artcle, please send e-mal to: tmc@computer.org, and reference IEEECS Log Number TMC organzaton A may carry wearable computers that belong to A. When these devces form an ad hoc network, they may share resources wth other devces wth the objectve of mnmzng the total resource consumed by the devces n A, rather than that of all devces n the network. Thus, the devces belongng to an organzaton form a natural group. Wearable computers nvolved n one dstrbuted computaton may form a group. In a sensor network, dfferent groups would consst of sensors that montor dfferent attrbutes such as temperature, pressure, wldlfe presence, etc. Sensors can also be deployed n the same area by dfferent organzatons, e.g., sesmc sensors can be deployed n the ocean by two dfferent agences. Then, sensors belongng to each agency wll consttute a group. In the above cases, the resource consumed by groups s more mportant than that consumed by ndvdual nodes as the dstrbuted computaton can be performed and the attrbutes can be measured even when some members fal. The research n ths case must nvestgate ssues pertnent to the sharng of resources from the perspectve of groups. A group s an ntermngled set of nodes havng a purpose n common. We do not consder the motvaton behnd the group formaton, but nvestgate the sharng of resources among dfferent groups. The crtcal resource we focus on s power. Nodes n wreless networks are powered by battery, and sze lmtatons compel the usage of low lfetme batteres. Ths calls for judcous consumpton of battery power. Normally, communcaton consumes sgnfcantly hgher power than other operatons. Nodes share power by routng each others packets, and t s well-known that multhop routng substantally decreases the overall power consumpton of the network [34]. We address the research challenges that arse when nodes decde to route each others packets wth the sole objectve of reducng the power consumpton of ther groups /07/$20.00 ß 2007 IEEE Publshed by the IEEE CS, CASS, ComSoc, IES, & SPS

4 GUHA ET AL.: FAIR COALITIONS FOR POWER-AWARE ROUTING IN WIRELESS NETWORKS 207 Fg. 1. In (a) and (b), we show two dfferent routngs, where node a consttutes Group A and node b consttutes Group B. Both groups need to send traffc to the access pont (AP). In (a), the farther node a routes ts traffc to b and b sends to AP. So, the routng n (a) reduces the power cost of a, but ncreases that for b. In (b), each node routes drectly to AP and there s no reducton n power costs for both groups. In (c), nodes a 1 and a 2 consttute Group A and b 1 consttutes Group B. Here, a 1 can send ts traffc through b 1 and b 1 can, n turn, send through a 2. Ths could result n a decrease n the total power for Groups A and B as aganst the case when the groups route to AP ndependently. We now enumerate some of these challenges. The nodes n a group share power by routng each other s packets to common destnatons. Groups are sad to form coaltons 1 when they route each other s packets. The frst challenge s to determne whch groups would form coaltons. Presumably, a precondton for formng coaltons among groups s that each group communcates the same amount of nformaton to the chosen destnatons whle consumng less power after the coalton s formed. Whether or not the precondton s satsfed depends on the routng n the coalton and the number of possble routes can be an exponental functon of the number of nodes n the groups. There need not even exst a routng that reduces the power consumpton of each group. Fgs. 1a and 1b show that, f each group conssts of a sngle node, then groups do not mutually beneft from the coalton, but ths no longer holds f the groups consst of two or more nodes (Fg. 1c). The challenge then s to answer whether there exsts at least one jont routng that makes the coalton mutually benefcal. The next challenge s to compute such a jont routng. We wll show n Secton 3.3 that the routng that mnmzes the total power consumpton of all groups may not result n mutually benefcal coaltons, as t may ncrease the power consumpton of some groups. The beneft ncurred by a group due to the coalton operaton s the decrease n ts power consumpton after t jons the coalton. We need to determne a routng that shares the beneft equtably. A smplstc approach s to nsst that the groups each get the same beneft, but ths can be wasteful f one group can gan beneft wthout harmng the others. A max-mn far [1] routng uses the followng strategy for a par of groups: Determne the greatest mnmum beneft to be ganed by ether of the two groups when sharng and maxmze the beneft of the other group so long as the changes do not reduce ths mnmum. Ths strategy can be generalzed to multple groups. The challenge now s to compute a maxmn far power aware coalton routng. Fnally, the network topology s dynamc snce nodes move and the transmsson condton n the lnks sgnfcantly change over tme. Thus, the benefts obtaned 1. Even after formng a coalton, dfferent groups mantan ther separate denttes, assocatons wth ther ndvdual organzatons, and dscharge ther ndvdual responsbltes. The coalton operaton just allows jont routng. through coalton and, hence, the decsons to reman n coalton change wth tme. When the topology changes, even f the coalton operaton remans mutually benefcal, the max-mn far power aware coalton routng may change. We therefore need a dstrbuted and dynamc algorthm that seamlessly adapts the computatons n the event of topology change. In Secton 2, we survey the relevant lterature. In Secton 3, we provde a mathematcal framework for a coalton of two groups. Ths secton presents several dstnctve propertes of coalton routngs. For example, a max-mn far power aware coalton routng exhbts mportant characterstcs that do not hold for max-mn far allocaton of other resources such as bandwdth. We show that the max-mn far coalton routng s guaranteed to attan the desred mnmum benefts for each group should the coalton be feasble. In Secton 4, we present a polynomal complexty algorthm for computng the far coalton routng. Ths algorthm needs to solve a lnear program at a central processor, whch requres the knowledge of the global topology. In Secton 5, we present a dstrbuted computng scheme whch allows the routng to be computed va smple teratve computatons and message exchanges at each partcpatng node. In Secton 6, we generalze the framework and the computaton algorthms for a coalton among multple groups n more general networks and also consder more general models for power consumpton and sgnal propagaton. These coalton routng algorthms provde foundatons for developng operatonal protocols. The desgn of such protocols would requre deployment of mechansms to enforce group routngs, e.g., securty checks. In Secton 6, we brefly dscuss some of these ssues. Refer to the Appendx for all proofs. 2 RELATED WORK The exstng research on effcent utlzaton of power n wreless networks can be classfed nto the followng broad categores: The frst maxmzes the lfetme of any gven node through optmum battery dscharge strategy [6], [19]. The second vares the transmsson power levels of nodes so as to control the network topology as desred [8], [14], [23], [25], [32]. The thrd reduces the power consumpton by optmzng several parameters at the MAC layer [11], [21], [22], [31]. The last maxmzes the lfetme of the network by balancng the power consumpton of dfferent nodes [3], [4], [17]. Another prevalent approach s to route n accordance wth a power-based metrc rather than a dstance metrc [34]. However, the common feature of the exstng research s that the basc entty s a node. The performance of the network s ether quantfed n terms of the aggregate performance of the nodes or that of the bottleneck node. Hou et al. [10] propose a polynomal tme algorthm to compute lexcographc max-mn (LMM) far rate allocaton and show that ths rate allocaton attans the LMM node lfetmes. The dstnctve feature of our work s that the basc entty s a group rather than a sngle node, and the operatons are coaltons. The performance objectve we consder s farness and the ssues sgnfcantly dffer due to the choce of the basc entty. We are concerned about the

5 208 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 6, NO. 2, FEBRUARY 2007 performance of each group rather than the network as a whole. Relayng and cachng strateges have been proposed for node cooperaton when a node decdes to relay the requests of other nodes based on ts selfsh nterests [24], [30]. Our research s complementary snce we assume that a group of nodes decde to route the packets of other groups based on the nterest of the group as a whole. We present an algorthm that obtans a specfc pareto optmal objectve, the max-mn far operatng pont. 3 MATHEMATICAL FRAMEWORK FOR COALITION OF GROUPS 3.1 Power Model We frst present the mathematcal model we use for power consumpton [7], [33]. Let the transmtted energy per bt be E t. The receved energy depends on the dstance between the transmtter and the recever and on other phenomena lke refracton (e.g., through walls), dffracton (e.g., around buldngs), reflecton (e.g., on ground and objects), scatterng, and absorpton. The collectve varaton due to these phenomena s referred to as shadowng [26]. The receved energy at a dstance d s then E t 1 d, where 2 6 and represents the lnk attenuaton due to shadowng. For smplfcaton, we assume that does not change wth tme and s the same for all lnks [7], [33] and we relax these assumptons n Secton 6.3. We assume that the nose level s the same at all nodes. Let E rx be the energy per bt requred to mantan the SNR necessary for successful decodng at the recever. Then, for successful communcaton, a node must transmt each bt at energy E tx, where E tx 1 d E rx. The power consumed by a transmttng node then s of the form K 1 þ K 0 re rx d, where K 0 s a constant, r s the node s data rate, and K 1 s the node s dle power consumpton. The node dsspates power K 1 even f t does not transmt or receve any traffc. Let constant K ¼ K 0 E rx : The MAC and the physcal layers determne K 1, K, and. For example, s hgher for obstructed paths wthn buldngs. Unless otherwse stated, we wll use ¼ 4, whch corresponds to the path-loss n closed areas; however, all analyss hold for any 0. Nodes may exchange control packets for transmttng data packets; the control packet exchange depends on the MAC protocol, e.g., IEEE uses RTS, CTS packets. The energy consumed n exchangng control packets determne the constant K 0. The lnear relaton between transmsson power and data rate mplctly assumes that the expected number of control packets exchanged per data packet does not depend on the data rate. But, for example, n IEEE , the expected number of control packets exchanged per data packet ncreases wth an ncrease n data rates due to an ncrease n collsons of RTS, CTS. Thus, strctly speakng, the dependence s not lnear. But, the naccuracy due to the lnear assumpton s neglgble except when the energy consumed n transmttng the control packets s comparable to that for transmttng data packets (Fg. 2). Snce the sze of each control packet s sgnfcantly less than that of a data packet, ths happens only when the expected number of control packets exchanged per data packet s very hgh, whch happens only at very hgh data rates. Usually, n order to Fg. 2. We consder a network wth 10 nodes such that all nodes are n each other s transmsson range and share a sngle channel of capacty 11 Mbps. Node transmts data to node ð þ 1Þ%10 at network layer rate r. The MAC protocol s IEEE We plot the power consumed by node 1 as a functon of r. The power ncludes the power consumed n transmttng both control and data packets. avod excessve energy consumpton n retransmttng control packets, the system does not operate at these data rates. Thus, most power-aware routng schemes assume ths lnear dependence, e.g., [3], [4], [7], [17]. 3.2 Formulaton for a Sngle Group We consder a network wth M ext ponts. We denote the set of ext ponts (EP) as e ¼ðe 1 ;...; e M Þ. We model the network nodes as a Weghted Drected Graph GhV;E;e;W, where V s the node set for the group, E s the edge set, e s the ext pont set, and W denotes the edge weghts whch are postve real numbers. Every node v 2 V has at least one path to an ext pont and the outdegree of each ext pont s 0. Hence, the ext ponts act as a snk for data traffc. The node set V and the ext ponts are defned through ther coordnates n the eucldean plane. The dstance dðv; v 0 Þ s the dstance between node v 2 V and node v 0 2 V [ e. If ðv; v 0 Þ2E; weght wðv; v 0 Þ¼dðv; v 0 Þ 4 and wðv; v 0 Þ2W. The edge set E s usually determned at the MAC and physcal layers and can be arbtrary except that the ext ponts only have ncomng edges. We now descrbe an example edge set. When the node rados have lmtatons on maxmum transmsson power for each bt, then an acceptable SNR level can be mantaned at the recever only f the dstance from the transmtter s below a certan maxmum value, whch s referred to as the transmsson range ðdþ. In such networks, a drected edge exsts from v 2 V to v 0 2 V [ e f dðv; v 0 Þ <D. Orgn functon O : V!< defnes the traffc orgnatng at a node v 2 V for each ext pont (e )ne. The graph G and the orgn functons are gven. Let the traffc on an edge ðv; v 0 Þ ntended for ext pont e be r ðv; v 0 Þ2<. If ðv; v 0 Þ 62 E, then rðv; v 0 Þ¼0. The total outgong P traffc from a node v for ext pont e s then v 0 2V [fe g r ðv; v 0 Þ, whch s the load on node v, L ðvþ. The sum of the ncomng traffc and the orgnatng traffc at a node must equal the extng traffc. Thus, 8 and 8v 2 V, X r ðv; v 0 Þ¼O ðvþþx r ðv 00 ;vþ¼l ðvþ: ð1þ v 0 2V [fe g v 00 2V

6 GUHA ET AL.: FAIR COALITIONS FOR POWER-AWARE ROUTING IN WIRELESS NETWORKS 209 Traffc routng s an jejm-dmensonal vector ~r whose components satsfy (1). The components of ~r are the traffcs on the correspondng edges. Under routng ~r, a node v spends power N ~r ðvþ and N ~r ðvþ ¼K 1 þ K X X r ðv; v 0 Þdðv; v 0 Þ 4 ; v 0 2V [fe g where the constants K 1 and K are defned n Secton 3.1. Dfferent nodes may have dfferent energy lmtatons. Thus, we assume that, for each node v, the average power consumpton s upper bounded by BðvÞ. Hence, K 1 þ K X X r ðv; v 0 Þdðv; v 0 Þ 4 BðvÞ: v 0 2V [fe g The power expendture of a group P ~r s then the total power consumed by all nodes n the group,.e., P ~r ¼ X N v2v ~rðvþ. The group optmal power expendture P opt s the mnmum value of P ~r over all possble ~r and can be obtaned by routng the traffc over the mnmum weght path from any node v 2 V to each ext pont e 2 e for the weghts W. 2 Such mnmum weght paths can be computed by well-known algorthms lke Djkstra, Bellman ford, etc. Let v 0 be the next hop node to v n such a path. If N optðvþ s the power spent by a node v under optmal routng, then N opt ðvþ ¼K 1 þ K X P opt ¼ X N opt ðvþ: v2v L ðvþdðv; v 0 Þ4 and 3.3 Coalton of Groups We have descrbed the termnology and the equatons for a group of nodes. Now, consder two groups of nodes, A and B. Let ther node sets be V a and V b, respectvely. Let ther group optmal power expendtures before formng a coalton be Popt a and P opt b. Next, we consder a combned network wth Groups A and B jontly routng to the ext ponts. Dependng on the network scenaro, each group may route to one or more ext ponts. For example, when groups correspond to an organzaton, they could route to ther own ext pont. On the other hand, n sensor networks, each group could route to multple ext ponts. These scenaros consttute specfc cases of our model. The vertex set V for the combned network s then V a [ V b. The edge set E jont can be determned from V and the MAC and physcal layer consderatons. For example, E jont can be obtaned usng the transmsson range D,.e., drected edge ðv; v 0 Þ2E jont for any v 2 V a [ V b and v 0 2 V a [ V b [ e f dðv; v 0 Þ <D. Also, E jont s a superset of the edge sets of each group. Agan, for any ðv; v 0 Þ2E jont, weght wðv; v 0 Þ¼dðv; v 0 Þ 4. The orgn functons for all the nodes reman the same. Any vector n R MjEjont j whose components are nonnegatve and satsfy (1) s a routng n the jont network and wll be referred to as a coalton routng. Note that rðv; v 0 Þ¼0 f ðv; v 0 Þ 62 E jont. For an 2. Here, the weght of a path s the sum of the weghts of the lnks n the path. ð2þ Fg. 3. Groups A ða 1 ;a 2 Þ and B ðb 1 ;b 2 Þ route to the ext pont EP. Each node sends 1 Mbps. arbtrary coalton routng ~r, we now evaluate the power expendture for each node. Let J~r a and Jb ~r be the total power expendture for nodes n Groups A and B, respectvely, under routng ~r. Then; J a ~r ¼ X v2v a N ~r ðvþ and J b ~r ¼ X v2v b N ~r ðvþ: Defnton 1. Group beneft under coalton routng ~r s the dfference between the power spent by the group under ndvdual optmal routng before mergng and the power spent by the group for coalton routng ~r. The group benefts form the beneft vector ~B ~r, where ~B ~r ðb a ~r ;Bb ~r Þ;Ba ~r ¼ P a opt Ja ~r and Bb ~r ¼ P b opt Jb ~r. The dea behnd combnng two groups s to reduce the total power each group was spendng ntally. Dependng on the system, group coalton may ntroduce some addtonal operatonal cost and groups would want to beneft over and above ths cost. Let t be the beneft below whch groups wll not be wllng to enter nto a coalton. The value of t would depend on group polces and the overhead for the coalton. Defnton 2. A coalton s useful wth a routng ~r f mnðb~r a ;Bb ~r Þt. Defton 3. A coalton s useful f t s useful wth some routng ~r. We wll present an algorthm to compute such a routng ~r f one exsts. Defnton 4. A mnmal coalton routng s a coalton routng that results n the optmal or the mnmal total power expendture for Groups A and B combned. Next, we llustrate the combnaton of two groups wth an example. Consder Fg. 3, n whch Groups A and B route to a sngle ext pont. Each node generates traffc at the rate of 1 Mbps. Let K ¼ 1, p K 1 ¼ 0. Optmal power expendture for Group A s 2 4 þ ffffff 4 p 2 ¼ 20 and, for Group B, s 1 4 þ ffffffffffffffffff 4 4: For the mnmal power coalton pffffff routng shown, power expendture for A s 1 4 þ 2ð 2 Þ 4 ¼ 9 and

7 210 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 6, NO. 2, FEBRUARY 2007 Fg. 5. Far coalton routng when each node sends 1 Mbps. The numbers next to the lnks are the rates. Fg. 4. Beneft vectors under mnmal coalton. for B s 2ð1Þ 4 p þ ffffffffffffffffff 4 1:25 3:6. The beneft for Group A s 20 9 ¼ 11 and for B s 19 3:6 ¼ 15:4 and both the components are postve. Consder now that node b 2 has a hgher load to send, e.g., 5 Mbps. Ths wll be relayed through a 2 n the coalton routng of Fg. 3. Node a 2 wll have a hgh power consumpton (24) and the beneft of Group A wll be negatve ( 5). Ths demonstrates that the mnmal coalton routng may not beneft each group. Defnton 5. A feasble beneft vector s one that results from a coalton routng ~r. The set of all feasble beneft vectors s the feasble beneft regon. 3.4 Propertes of the Feasble Beneft Regon Theorem 1. The set of feasble beneft vectors s convex and closed. We now demonstrate that dfferent feasble beneft vectors can lead to dsparate benefts for the groups. For the mnmal coalton routng, we can fnd the power expendture for each node,.e., N opt ðvþ for each v 2 V a [ V b. Further, let Jopt a and Jb opt be the powers spent by nodes of Groups A and B, respectvely, under the mnmal coalton routng. J a opt ¼ X v2v a N opt ðvþ and J b opt ¼ X v2v b N opt ðvþ: Note agan that the subscrpt opt to J refers to the mnmal coalton routng for nodes of Groups A and B combned. The beneft vector ~L correspondng to the mnmal coalton routng s then ðl a opt ;Lb opt Þ, where La opt ¼ Popt a Ja opt and Lb opt ¼ P opt b Jb opt. Let K ¼ 1 and let there be a sngle ext pont. The vector ~L s plotted n Fg. 4 for dfferent random placements of nodes. Each group has 20 nodes unformly dstrbuted over a square of sde 100m and the network s fully connected,.e., each node can drectly transmt to every other node. If the beneft vector s n the frst quadrant (both coordnates are postve), then the groups mutually beneft from beng merged; otherwse, one of the groups s a loser. Most pars of groups beneft from a mnmal coalton, but there are many nstances n whch only one group benefts. Even when a par of groups mutually benefts, there s often some dsproporton n the extent of beneft, wth one group gettng somewhat more than the other. Ths motvates the far allocaton of benefts. 3.5 Max-Mn Far Beneft Vector Defnton 6. A feasble beneft vector B ~r s max-mn far f, for all, B~r cannot be ncreased whle mantanng feasblty wthout decreasng B j ~r for some group j, for whch Bj ~r B ~r. Corollary 1. The max-mn far beneft vector exsts and s unque. The corollary follows as a consequence of Theorem 1 and results from [28]. Defnton 7. A far coalton routng s a coalton routng that results n a max-mn far beneft vector. Mnmum component property. If ~r s a far coalton routng, then mnðb~r a ;Bb ~r ÞmnðBa ~r 1 ;B b ~r 1 Þ for any other coalton routng ~r 1. Ths property follows from the defnton of the max-mn far beneft vector. In Fg. 3, the max-mn far beneft vector when K ¼ 1 and M ¼ 1 s (11.9,11.9). Ths s acheved when node b 2 sends 0.78 Mbps to a 2 and 0.22 Mbps drectly to EP lke n Fg. 5. Proposton 1. Let ~r be a far coalton routng. Then, mnðb~r a ;Bb ~r Þ0. Thus, a coalton does not ncrease the power consumpton of any group f far coalton routng s used. Theorem 2. A coalton wll be useful f and only f t s useful wth a far coalton routng ~r. Theorem 2 presents a necessary and a suffcent condton for decdng whether the coalton would be useful. Theorem 3. For two groups, the max-mn far beneft vector has equal components. Theorem 3 wll be used n developng an effcent algorthm for computng a far coalton routng for two groups. Note that, for other resource allocaton problems, e.g., bandwdth allocaton, the max-mn far vector need not have equal components even for two contenders (Fg. 6) [5]. Fg. 6. Consder two sessons (a,c) and (b,d). The numbers next to the lnks are the lnk bandwdths. The max-mn far bandwdth for sesson (a,c) and (b,d) are 3 and 1, respectvely.

8 GUHA ET AL.: FAIR COALITIONS FOR POWER-AWARE ROUTING IN WIRELESS NETWORKS FAIR COALITION ALGORITHM (FC) 4.1 Descrpton We show that the far coalton routng and the assocated beneft vector can be computed by solvng the followng lnear program: FC: Maxmze Z: Subject to: K 1 þ K X X v 0 2V a [V b [fe g X v 0 2V a [V b [fe g Z B~r a 0; Z B~r b 0; r ðv; v 0 Þdðv; v 0 Þ 4 BðvÞ 8v 2 V a [ V b ; ð3þ r ðv; v 0 Þ X r ðv 00 ;vþ¼o ðvþ8 v; v 0 2 V a [ V b and : v 00 2V a [V b The power consumpton of each node s constraned n (3) and the flows are balanced n (4). Let Z be the objectve functon value obtaned from FC. Theorem 4. The routng ~r obtaned as a soluton of FC s a far coalton routng. Proof: Let mnbenð~rþ ¼mnðB~r a ;Bb ~r Þ. From Theorem 3 and the mnmum component property, any feasble routng that attans the maxmum value of mnbenð~rþ s a far coalton routng ~r. Thus, FC computes the far coalton routng. tu The ext pont can solve FC to compute the far coalton routng and the max-mn far beneft. The lnear program nvolves ðm þ 1ÞjV a [ V b jþ2 constrants and MjE jont jþ1 varables. Hence, the max-mn far beneft vector and the far coalton routng are polynomal complexty computable [13]. For solvng FC, an ext pont needs to know the edge set E jont and the dstances between the nodes. Intally, the nodes nform the ext pont, ther ncdent edges, and the dstances from ther neghbors, and later they nform the ext pont only when these change. The MAC and the physcal layers of a node v determne ts ncdent edges ðv; v 0 Þ and ðv 0 ;vþ n E jont. Nodes can learn the dstances from ther neghbors by power measurements and postonng algorthms, some of whch do not need GPS [2]. 4.2 Smulaton Results We nvestgate the effcacy of far coalton routng through smulatons usng MATLAB. We evaluate the benefts attaned by dfferent coalton routng schemes. We also consder other performance attrbutes, such as network lfetme, end-to-end path lengths, addtonal power consumpton for provdng farness, etc. We consder a network wth one ext pont ðm ¼ 1Þ and a coalton of two groups. Nodes of both groups are dstrbuted n a square of sde 100m. Each node generates traffc at the rate of 1 Mbps. The value of K depends on the choce of the wreless nterface, ð4þ and ts effect s to scale our measurements. Thus, wthout loss of generalty, we consder K ¼ 1. We wll later menton detals for a specfc nterface. Note that the beneft values do not depend on K 1. We consder a dfferent number of nodes, dfferent dstrbutons of nodes, dfferent locatons of the ext pont, dfferent szes of the groups, dfferent dstances between groups, and report averages over 100 random topologes n each case. We frst consder a fully connected network,.e., each node can transmt drectly to every other node. We assume that both groups have an equal number of nodes, the ext pont s at the center of the square, and all nodes are unformly dstrbuted n the square. In Fg. 7a, we plot the beneft values as a functon of the number of nodes. As proved before, the max-mn far beneft vector wll have equal components. We plot the average values of the maxmum component of the beneft vector of the mnmal coalton routng (max-opt), the mnmum component of the beneft vector of the mnmal coalton routng (mn-opt), and the max-mn far beneft (max-mn). As expected, the max-mn group beneft s between the maxmum and the mnmum components of the beneft vector of the mnmal coalton routng. Benefts ntally ncrease and later decrease wth an ncrease n the number of nodes. Ths can be explaned as follows: Power consumpton n a routng scheme decreases f the dstance between consecutve nodes n a path decreases. Ths holds even f such a decrease ncreases the number of hops. Ths s because the power consumed n any routng s proportonal to 1) the expectaton of the fourth power of the dstance n each hop and 2) the number of hops. When the number of nodes s small, each group has a small number of nodes and, thus, jont routngs allow packet transmssons across hops that are sgnfcantly shorter than those n the ndvdually optmal routngs n each group. Thus, jont routngs have substantally lower power consumpton. Ths effect becomes more pronounced wth an ncrease n the number of nodes for a moderate number of nodes. But, when the number of nodes becomes really large, each group has a large number of nodes and the hop dstances and, hence, the power consumptons n the ndvdual optmal routngs become small as well. 3 Thus, the benefts of jont routng decrease. Nevertheless, the beneft values are stll consderable even for networks wth 200 nodes. In Fg. 7b, we consder a dfferent path loss exponent, ¼ 2, whch arses n open envronments. Here, the trends are smlar to Fg. 7a, but the benefts are somewhat smaller. Ths s because the reducton n power consumpton due to the reducton n hop-dstances dðv; v 0 Þ obtaned by the jont routngs are less for ¼ 2 than for ¼ 4, as the power consumed n a lnk ðv; v 0 Þ s proportonal to dðv; v 0 Þ. We now revert to the closed envronment, ¼ 4, and compare the lfetme of the network attaned under dfferent coalton routng schemes. The network lfetme 3. Recently, Zhao et al. [35] proved that when nodes are unformly dstrbuted and ther number n becomes large, the network transfers ðn=lognþ amount of data before any node des. In other words, the data transferred by a network n ts lfetme becomes arbtrarly large wth an ncrease n n. Ths happens because of a reducton n the dstance between consecutve nodes n the routes. Although Zhao et al. do not consder networks wth groups, ther result s consstent wth our observaton.

9 212 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 6, NO. 2, FEBRUARY 2007 Fg. 7. Performance of coalton routngs n networks consstng of two groups of equal szes and nodes unformly dstrbuted n a square of sze 100m. (a) Benefts n closed envronments. (b) Benefts n open envronments. (c) Maxmum node power. (d) Network lfetme. (e) Farness overhead. (f) Hop delay performance. can be defned n dfferent ways, e.g., t can be consdered as the tme requred for a certan fracton of nodes to de, the frst tme nstant at whch the network s dsconnected, etc. [3], [4], [34]. The lfetme of a network for all these metrcs s governed by the power consumpton of the nodes that spend hgh power and de faster than others. Thus, n Fg. 7c, we plot the quantty ð X þ x Þ= X, where X s the mean power over all nodes and x s the standard devaton. Note that ths quantty s a measure of the statstcal maxmum of the power spent by any node. Far coalton routng has a lower value of ths quantty as compared to the mnmal. Ths happens because the mnmal coalton routng derves ts advantages by routng a sgnfcant amount of traffc through a few nodes. We therefore expect that far coalton routng wll have a hgher lfetme under most metrcs (.e., all metrcs that depend on the power consumpton of the nodes that consume more power than others). To demonstrate that ths s ndeed the case, we choose a partcular noton of lfetme; namely, the tme requred for a certan fracton (e.g., 5 percent) of nodes to de. We assume that all nodes have the same ntal energy. In Fg. 7d, we plot the rato between the lfetmes of the network under the far and the mnmal coalton routngs that are computed when all nodes are functonal. We also plot the rato of the lfetmes of the group wth the mnmum lfetme under far coalton and the group wth the mnmum lfetme under mnmal coalton routngs. Consstent wth our expectaton, the rato s always above 1. Fg. 7e plots the total powers spent under the mnmal and far coalton routngs and ther dfference. Ths dfference can be looked upon as the cost for provdng farness. Here, K 1 ¼ 0. The average cost s modest (18 percent) consderng the beneft (46 percent) 4 obtaned and the farness acheved. In Fg. 7f, we plot the average number of hops traversed by each packet before t reaches the ext pont. We notce that, on an average, the far and mnmal coalton routngs use a smlar number of hops. The hop count affects the average end-to-end delay experenced by packets. But, the delay also depends on other factors such as nterference. The detaled nvestgaton of the delay and nterference ssues n coalton routng s beyond the scope of ths paper. We now evaluate the benefts for dfferent dstrbutons of nodes, dfferent locatons of the ext pont, dfferent szes of the groups, and dfferent dstances between groups. But, the trends and the conclusons reman the same as n the prevous cases. Fg. 8a shows the results for unequal group szes. One group s four tmes as large as the other. The nodes are stll unformly dstrbuted. The smaller group has a lesser beneft under the mnmal coalton routng n ths case. The remanng trends are the same as for groups wth equal szes. We now nvestgate the effect of clustered topologes on the beneft values (Fgs. 8b and 8c). Both groups have an equal number of nodes. In Fg. 8b, nodes of each group are normally dstrbuted wth a varance of 25 around the respectve group centrods that are unformly dstrbuted. The group wth the centrod closer to the ext pont has negatve beneft under the mnmal coalton routng and zero beneft under the far coalton routng. The group closer to the ext pont loses after coalton when the 4. The cost percent s obtaned from Fg. 7e. The beneft percent s wth respect to the total power consumed pror to the coalton and s obtaned from Fg. 7a and Fg. 7e.

10 GUHA ET AL.: FAIR COALITIONS FOR POWER-AWARE ROUTING IN WIRELESS NETWORKS 213 Fg. 8. Beneft values for coalton routngs for dfferent network scenaros. (a) Skewed Group szes. (b) Effect of clustered groups. (c) Clustered topology. (d) Ext pont at end. (e) Intergroup dstance. (f) Lower transmsson range D. mnmal coalton routng s used, but not when the far coalton routng s used. Here, the benefts of the far coalton routng starts decreasng for a much larger number of nodes than n the unform dstrbuton case (Fg. 7a), as the topology becomes pervasve only for a much larger number of nodes. For example, when the number of nodes n the network s 400, the beneft reduces by 25 percent as compared to the beneft n a network wth 200 nodes. In Fg. 8c, we consder a network wth two clusters of equal szes, but now the clusters nclude an equal number of nodes from both groups. The nodes n each cluster are normally dstrbuted wth a varance of 25 around the respectve group centrods that are unformly dstrbuted. Here, both groups obtan postve benefts under far coalton. We now nvestgate the case when the ext pont s at the edge of the square. We consder two dfferent dstrbutons of nodes: 1) unform (Fg. 8d) and 2) normal (Fg. 8e). For unform dstrbuton, the trends are smlar to the case wth the ext pont at the center (Fg. 7a). But, snce all nodes are now n the same sde of the ext pont, the paths to the ext pont contan a larger number of nodes of both groups and, hence, the benefts are hgher. For normal dstrbuton, the nodes of each group are normally dstrbuted around the centrod of the group wth a varance of 25. The centrods are equdstant from the ext pont and at a dstance d from each other, where d s a measure of the separaton between the groups. In Fg. 8e, we plot the benefts as a functon of d. The benefts decrease as d ncreases as then fewer nodes from one group can route the packets of the other group due to the larger separaton between the groups. We now relax the assumpton that the network s fully connected and assume that each node can transmt drectly to only nodes wthn dstance D. We nvestgate the effect of dfferent transmsson ranges D on the benefts n Fg. 8f. The network has 20 nodes n each group, but the characterstcs are otherwse smlar to that consdered n Fg. 7a. Lower values of D wll result n fewer edges n the network. The beneft ncreases sgnfcantly wth an ncrease n D for lower values of D as more and more nodes can be ncluded n potental routes to the ext pont. Note that the maxmum possble p dstance between any two nodes n ths network s 100 ffffff p 2. A slght drop can be notced when D s around 10 ffffff 2. Ths s because the power consumpton of the group optmal decreases by a smaller amount p than that of the far coalton routng. When D exceeds 18 ffff 2, the curves level off. The transmsson range s now hgh enough to nclude those nodes whch would have been a part of the coalton routng n the fully connected case. For the Lucent b Ornoco card, a rate of 1 Mbps n closed envronment corresponds to 15dBm of output power [18]. The constant K s then roughly 5: W=Mbt m 4. For any value of K 1, ths translates to a beneft of 30 Watts for a group wth 10 nodes for the unform case wth equal group szes. It s also worthwhle to note that the CPU tme to compute FC, for any of the above topologes, was not more than 0.5 seconds on a 700Mhz/256MB RAM laptop usng a smplex algorthm mplementaton [9]. 5 DISTRIBUTED IMPLEMENTATION The algorthm n Secton 4.1 for computng the far coalton routng requres a centralzed computaton at the ext pont. Though the smplest soluton, t wll not be computatonally tractable when the ext ponts have capablty smlar to the

11 214 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 6, NO. 2, FEBRUARY 2007 nodes themselves. Consder, for example, a sensor network where a group of sensors communcate ther measurements to a common node whch, n turn, transmts to, say, a satellte. Here, we would not want to overwhelm the relay node wth the lnear programmng computaton. Furthermore, when nodes move, the edge set E changes. For example, when a node can drectly transmt to only nodes wthn ts transmsson range D, then lnks between two nodes wll be created (cease to exst) when one moves n to (out of) the transmsson range of another. Fnally, the power consumed for transmsson of each bt n a lnk wll change wth a change n the dstance between the ncdent nodes. The traffc generaton rate of each node wll also change wth tme. Due to these changes, the coalton may no longer be useful or may start beng useful or the far coalton routng may change. Thus, FC must be solved every tme such changes occur. Rather than havng the ext pont repeat the entre computaton n every such nstance, t s benefcal to have a dstrbuted mplementaton where every node performs some smple teratve computatons and the values seamlessly converge to the max-mn far soluton. Based on the new max-mn far soluton, the groups can determne whether the coalton s useful (Theorem 3) and use the far coalton routng f they reman n or jon the coalton. Now, we present an teratve approach to compute a far coalton routng for two groups. Ths has been motvated by recently proposed solutons for optmzaton problems n other resource allocaton settngs [12], [29]. Let Z n and ~r n denote the correspondng quanttes n teraton n, where Z 0 and ~r 0 can be arbtrarly chosen. The ntal choces need not satsfy any of the constrants. Thus, each node can select the ntal values of the loads for each of ts outgong edges wthout any coordnaton wth the other nodes. Smlarly, Z 0 s selected at an ext pont. Now, we defne some ndcators. The beneft ndcator of a group s 1 f Z n s more than the group beneft. ( a n ¼ 0; f Z n þ J a ~r n P a opt ; 1; f Z n þ J a ~r n >P a opt : ( b n ¼ 0; f Z n þ J b ~r n P b opt ; 1; f Z n þ J b ~r n >P b opt : We now outlne the rate update mechansm for the traffc ntended for each of the M ext ponts. Node congeston c v n; s the dfference between the outgong and the sum of the orgnatng and ncomng traffc at node v for ext pont. From (4), c v n; ¼ X r n; ðv; v 0 Þ O ðvþþ X! r n; ðv 00 ;vþ : v 0 2V a [V b [fe g v 00 2V a [V b The node congeston ndcator for node v for traffc drected to ext pont s 8 0 f cv n; ¼ 0; >< s v n; ¼ 1 f c v n; >: > 0; 1 f c v n; < 0: Traffc for ext pont at node v s consdered balanced, lghtly loaded, or heavly loaded as s v n; s 0; 1 and 1, respectvely. For the ext pont, s e n ¼ 0. The power level ndcator at node v, t v n s set to 1 f the current power consumpton exceeds the lmt BðvÞ and 0 otherwse. Hence, t v n 8 ¼ < 0 f K 1 þ K P P v 0 2V a [V b [fe r g ðv; v 0 Þdðv; v 0 Þ 4 BðvÞ; : 1 f K 1 þ K P P v 0 2V a [V b [fe g r ðv; v 0 Þdðv; v 0 Þ 4 >BðvÞ: We present an teratve approach usng the above ndcators. Note that s v n; and tv n can be updated at node v usng the ncomng rates n the prevous teraton. Now, the updates of a n and b n requre a knowledge of the total power beng spent by the nodes of a group. We wll dscuss how to acqure ths nformaton n a dstrbuted manner. Let f n g be the step-szes that satsfy lm n!1 n ¼ 0 and P 1 n¼1 n ¼1. For example, n ¼ 1=n satsfes the condtons. Each node v updates ts outgong traffc n edges ðv; v 0 Þ2E jont as follows. ½Š þ denotes the projecton on ½0; 1Þ. r nþ1; ðv; v 0 Þ¼ h r n; ðv; v 0 Þ n r nþ1; ðv; v 0 Þ¼ h r n; ðv; v 0 Þ n s v n; sv0 s v n; sv0 Trvally, r nþ1; ðv; v 0 Þ¼0 f ðv; v 0 Þ¼E jont. The ext pont updates Z as follows: n; þþdðv; v0 Þ 4 ðt v n þ a n Þ f v 2 V a : þ n; þþdðv; v0 Þ 4 ðt v n þ b n Þ f v 2 V b : þ Z nþ1 ¼½Z n þ n ð1 ð a n þ b n ÞÞŠ þ : Theorem 5. For all >1, the teratve procedure stated above wll converge to the max-mn far beneft vector and far coalton routng, rrespectve of the ntal choce of the terates. Snce the convergence guarantees n Theorem 5 hold rrespectve of the ntal choce of the terates, the procedure converges to the far allocatons even after changes n E jont and the power consumed n the lnks. Now, we outlne a dstrbuted scheme to mplement the teratons. Assume that we have a spannng tree connectng nodes of each group to any one of the ext ponts. Refer to Fg. 9a. Each leaf node L sends a power packet (PP) upstream that contans the power expended by L. Each node of a group adds all the power values n the PP arrvng from ts downstream branches, adds ts own power expendture to the sum, and sends a PP upstream wth the resultng power value. Usng these group powers, the ext pont determnes a nþ1 and b nþ1 and updates Z n. The ext pont communcates a n and b n to each group through congeston ndcator packet CP and the nodes can use these to update ther rates. The PP and CP can be separate packets or they can be pggybacked on the data and acknowledgement packets. We now evaluate the convergence tme of the dstrbuted mplementaton. We consder a fully connected network

12 GUHA ET AL.: FAIR COALITIONS FOR POWER-AWARE ROUTING IN WIRELESS NETWORKS 215 Fg. 9. Dstrbuted mplementaton. (a) Exchange of PP and CP. Crcles and pentagons denote the two groups. Let the power spent by nodes a, b, and c be 1, 2, and 3, respectvely. The PPs sent by a, b, and c have power values 1, 2, and 6, respectvely. (b) Convergence for the dstrbuted computaton. Here, ¼ 2;500 and n ¼ 1=n; 8 n, Z 0 ¼ 10 7, and ~r 0 ¼ ~0. Durng teraton 2,000, nodes change ther postons. After teraton 2,400, nodes change ther postons one by one (by 10 percent) tll teraton 6,000. After teraton 6,100, the transmsson rates change (by 5 percent). wth 10 nodes n each group, where the nodes are unformly dstrbuted n a square of sde 100m and one ext pont s at the center. Each node generates traffc at the rate of 1 Mbps. We assume that the sze of each CP and PP packet s 15 bytes. The CP and the PP packets traverse a total of 12 hops per teraton. Now, f the transmsson rate n each lnk s 11 Mbps, then each teraton consumes approxmately 0.13 mllseconds. Here, K ¼ 1, ¼ 2;500, and n ¼ 1=n, 8n, Z 0 ¼ 10 7, and ~r 0 ¼ ~0. The beneft Z n converges to the max-mn far beneft value of 5: n 1,000 teratons whch consume 130 mllseconds (Fg. 9b). In general, the ntal convergence tme wll depend on how far the ntal guess s from the optmal. We next demonstrate that the recomputatons that result from ncremental changes n topology and traffc generaton rates converge much faster. We assume that, durng teraton number 2,000 (.e., after the ntal convergence), all nodes select new locatons the new locatons are also unformly dstrbuted. The power consumptons n the lnks now change due to the topology rearrangement, but Z n converges to the new max-mn far value n 400 teratons, whch consumed 50 mllseconds. The convergence s faster as compared to the ntal convergence because only the node postons were changed, whle ther traffc generaton rates remaned the same. Thereafter, between teratons 2,400 and 6,000, nodes change ther postons one by one. If a node s current x-coordnate (y-coordnate) s x, then t selects ts new x-coordnate (y-coordnate) unformly wthn ½0:9x ; 1:1x Š ð½0:9y ; 1:1z ŠÞ. On an average, 60 teratons ( 8 ms) are requred for convergence for each change. Fnally, between teraton 6,100 and 8,100, the nodes change ther traffc generaton rates one by one. If a node s current generaton rate s OðÞ, then ts new rate s unformly dstrbuted wthn ½0:95OðÞ; 1:05OðÞŠ. Now, on average after each change, Z n converges to the new max-mn far value n 20 teratons ( 3 ms). Groups jon or reman n the coalton f and only f the new max-mn far beneft Z n exceeds the mnmum requred beneft t (Theorem 3), and they use the correspondng far coalton routng whenever they are n a coalton. To prevent routng nstablty and oscllatons, the groups evaluate the coalton formaton decson and alter the routng only when 1) the current value of Z n substantally dffers from that at the prevous decson epoch and 2) Z n remans at ts current value for some tme whch ensures convergence. Determnaton of these necessary devatons and tme duratons and also the securty mechansms requred to enforce the coalton formaton decsons and the far coalton routng consttute separate research topcs and are beyond the scope of the current work. However, we brefly dscuss some of the securty ssues n Secton DISCUSSION AND GENERALIZATIONS We now descrbe how the framework we have proposed and the analytcal results we have obtaned can be generalzed to nclude several addtonal features of practcal relevance. 6.1 Multgroup Far Coalton Algorthm We now nvestgate the max-mn far beneft vector and far coalton routng when multple ðnþ groups attempt to form a coalton. Defnton 6 also defnes the max-mn far beneft vector n ths case. Ths case s sgnfcantly dfferent from the two group case dscussed earler. Let Popt be the mnmum possble power spent by Group to route to the ext ponts before jonng the coalton. Also, let J~r be the power spent by nodes of Group under coalton routng ~r. The beneft for Group s then B~r 8I ¼ 1...n wth B~r ¼ P opt J ~r. The beneft vector for the coalton routng s ~B ~r ðb~r 1 ;B2 ~r...bn ~r Þ. We menton some mportant propertes of far coalton routng for multple group coalton. Proposton 2. Consder three groups, A, B, and C. Consder three separate coaltons (A, B), (B, C), and (A, B, C). If the parwse coaltons (A, B) and (B, C) are mutually benefcal for each group (.e., the beneft for each group under some coalton routng s postve), then the coalton (A, B, C) s benefcal for each group. A counterexample presented n Fg. 10 shows that the converse s not true. The components of the max-mn far beneft vector need not be equal when more than two groups combne. Refer to Fg. 11, where each node generates 1 Mbps. Here, M ¼ 1 and K ¼ 1.

13 216 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 6, NO. 2, FEBRUARY 2007 Fg. 10. All three groups, A, B, and C, wll beneft f routng together but no two taken at a tme wll be mutually benefcal. Now, we present the multgroup FC algorthm. Ths algorthm solves a sequence of lnear programs. Note that solvng a sngle lnear program s not suffcent snce the components of the max-mn far beneft allocaton need not be equal n ths general case. Let I ¼f1...ng, INC refer to the ndvdual node constrants (3), and LF refer to the load flow condton (4) generalzed to multple groups. Stage 1 : Maxmze: Z: Subject to:z B~r 8 2 I ~r satsfes INC and LF. Let Z1 be the objectve value and ~r 1 be the routng obtaned from above. Let equal ¼ft : B t ¼ Z ~r 1 g. 1 Substage 1 For each k 2 equal, Maxmze: B~r k : Subject to:b~r Z I nfkg satsfes INC and LF. Let ~r k be the routng correspondng to the kth maxmzaton 8k 2 equal. Let e 1 ¼fn : B n ~r n ¼ Z1 g. Stage 2 : Maxmze: Z: Subject to:z B~r 8 2 I n e 1 B~r Z e 1 ~r satsfes INC and LF. Let Z2 be the objectve value and ~r 2 be the routng obtaned from above. Let equal ¼ft : B t ¼ Z ~r 2 g. 2 Substage 2 For each k 2 equal Maxmze: B~r k : Subject to:b~r Z I n e 1 nfkg B~r Z e 1 ~r satsfes INC and LF. Let ~r j be the routng correspondng to the jth maxmzaton 8j 2 equal. Let e 2 ¼fn : B n ~r n ¼ Z2 g. Smlarly n the th step. Stage : Maxmze: Z: Subject to:z B~r 8 2 I n e 1 n e 2...n e 1 Fg. 11. D s the transmsson range. The max-mn far beneft for the star and the crcle group s 14D 4 =16, whle that for the pentagon group s zero. Fg. 12. We average the mnmum, second mnmum, and the largest component over 100 topologes. B ~r Z t 8 2 e t 8t ¼ 1...ð 1Þ ~r satsfes INC and LF. Theorem 6. The routng ~r obtaned as a soluton of multgroup FC s a far coalton routng. Fg. 12 shows benefts for far coalton routng for three equal szed groups spread over a square of sde 100m. Here, M ¼ 1 and K ¼ Recevng Power We have so far assumed that a node does not consume any power when t s recevng nformaton. We now relax ths assumpton and assume that the recevng power of a node s proportonal to the ncomng traffc rate. The total power expendture of a node v s the sum of the power spent to transmt load P L ðvþ and to receve load P ðl ðvþ O ðvþþ. Thus, N ~r ðvþ ¼K 1 þ K X X r ðv; v 0 Þdðv; v 0 Þ 4 v 0 2V [fe g þ K X 0 ðl ðvþ O ðvþþ ¼ K 1 þ K X X r ðv; v 0 Þdðv; v 0 Þ 4 v 0 2V [fe g þ K X X 0 r ðv 00 ;vþðfrom ð1þþ: J a ~r ¼ X v2v a N ~r ðvþ: Smlarly; J b ~r ¼ X v2v b N ~r ðvþ: v 00 2V The max-mn far beneft vector and the far coalton routng can be computed by substtutng the expressons for J~r a, Jb ~r n FC wth the above.5 The dstrbuted algorthm remans smlar except for the rate update strategy whch needs to be modfed. We descrbe the update strategy for r nþ1 ðv; v 0 Þ when v 2 V a and ðv; v 0 Þ2E jont : The update strategy for r nþ1; ðv; v 0 Þ when v 2 V b and ðv; v 0 Þ2E jont can be obtaned by nterchangng a wth b n the followng: 5. Now, Popt a (P opt b ) can stll be obtaned by routng the traffc usng the mnmum weght path n Group a ðbþ, but the weght of a lnk ðv; v 0 Þ s now Kdðv; v 0 Þ 4 þ K 0 nstead of dðv; v 0 Þ 4. Ths happens snce we assume that the recevng power depends only on the receved rate.

14 GUHA ET AL.: FAIR COALITIONS FOR POWER-AWARE ROUTING IN WIRELESS NETWORKS 217 r nþ1; ðv; v 0 Þ¼ h r n; ðv; v 0 Þ n ðs v n; r nþ1; ðv; v 0 Þ¼ h r n; ðv; v 0 Þ n s v n; f v; v 0 2 V a : r nþ1; ðv; v 0 Þ¼ h r n; ðv; v 0 Þ n ðs v n; f v 2 V a ;v 2 V b : sv0 n; þ dðv; v0 Þ 4 a n Þ f v 2 V a ;v 0 2 e: þ sv0 n; þ K0 =K þ dðv; v 0 Þ 4 sv0 n; þ dðv; v0 Þ 4 a n þ b n Þ The convergence guarantees n Theorem 5 hold. þ a n 6.3 Generalzed Propagaton Model We frst consder a smple generalzaton where ðv; v 0 Þs are dfferent for dfferent lnks, but do not change wth tme. Ths happens when the envronment s statc. Now, for successful communcaton to v 0, a node v must transmt each bt at energy E tx, where E tx ðv; v 0 Þ 1 dðv; v 0 Þ E rx : The power consumed by node v under routng ~r s then K 1 þ K P v 0 2V [feg rðv; v0 Þðv; v 0 Þdðv; v 0 Þ. Thus, dðv; v 0 Þ must now be replaced wth ðv; v 0 Þdðv; v 0 Þ everywhere (note that ðv; v 0 Þdðv; v 0 Þ can be obtaned by measurng the sgnal strength at recever v 0 ). The framework remans the same other than ths change, and all analytcal guarantees hold. We next consder the case that the envronment and, hence, ðv; v 0 Þ changes wth tme for each lnk ðv; v 0 Þ. 6 The tme duraton durng whch ðv; v 0 Þ does not change for a lnk ðv; v 0 Þ s referred to as the coherence tme of the lnk. Coherence tmes are large when nodes move around slowly, e.g., when the maxmum node velocty v max s lower than 5 m/s, the coherence tme s c=ðv max fþ ¼ ð Þ=ð5 2: Þ¼25 ms [26, p. 165]. Here, f s the center frequency of the sgnal and c s the speed of lght. The far coalton routng can now be recomputed every tme ðv; v 0 Þ changes. Snce the dstrbuted algorthm converges fast n the presence of ncremental changes, the rate allocaton can seamlessly adapt to changes n ðv; v 0 Þ. However, f ðv; v 0 Þ changes rapdly, statstcal nformaton must be used to determne the lnk rates and the transmsson powers. Specfcally, the transmsson powers and the routng p can ffffffffffffffffffffffffffffffffffffffffffffffffff be determned assumng that ðv; v 0 Þ¼Eðv; ½ v 0 ÞŠþ2 Var½ðv; pffffffffffffffffffffffffffffffffffffffffffffffffff v 0 ÞŠ, as wth a hgh probablty, ðv; v 0 ÞEðv; ½ v 0 ÞŠþ2 Var½ðv; v 0 ÞŠ. 6.4 Trust Issues We assume that members of a group trust one another and are wllng to jontly route packets to save power n the nterest of the group as a whole. We assume that when groups agree to form a coalton, they trust one another to use the far coalton routng. There s related work [20] on how to detect cheatng n whch one or more partes do not support ther agreed routng rules. Nodes n a group can use securty schemes to ensure that they route for other 6. When the envronment s not statc, ðv; v 0 Þ s modeled as a random varable whose logarthm s normally dstrbuted wth mean zero and a varance of 5-12dB dependng on the envronment [26]. þ Fg. 13. IPSec Costs. nodes n the same group and n groups that are partcpatng n the coalton. Wthn a group, one can dentfy trusted members wth publc key certfcates and thereafter establsh a symmetrc key for authentcatng ndvdual packets. Dfferent groups can be authentcated va thrd party publc key repostory. Ths can prevent nodes from masqueradng as nodes of some other group that s already a part of an actve coalton. Ths leads to a natural queston as to what s the cost ncurred to enforce group routng. We tested whether ths ncurs sgnfcant addtonal power f t s done wth IPSec tunnels [16] between neghborng nodes. To get an dea of the processng overhead, we let a Dell L400 laptop runnng Wndows 2000 generate constant bt rate UDP traffc over an b network. The payload rate was fxed at 4 Mbps. For varous securty parameters, we measured the tme for the laptop to de down. 7 Fg. 13 shows the results for three cases averaged over fve runs of the experment. The frst column shows that the laptop battery ded n 95 mnutes after sendng 2861MB of data n plantext. Header overhead accounts for the rate of Mbps to send 4 Mbps of payload. Authentcaton used null-encrypted ESP [15] wth SHA1 for message authentcaton codes; encrypton used ESP wth SHA1 and 3DES. Encrypton has a sgnfcant effect on power, but t s not really needed to enforce group routng. We can assume that nodes encrypt end-to-end and do not need hop-by-hop encrypton. Hence, t s possble to enforce group routng effcently wth only modest power costs by usng authentcaton wth null encrypton. Thus, t s clearly worthwhle to use group routng. IPSec s a suffcently effcent enforcement mechansm when the number of nodes s less than 50. Ths s because each node s lkely to route to only a few others. Thus, about tunnels are requred and these can all use null encrypton. There are technques that work effcently for larger groups (see, for example, the IETF documents from the Multcast Securty workng group, msec) but these seem unnecessary f the nodes are laptops. For sensor networks, a more specalzed securty protocol may be necessary. A comprehensve desgn of securty mechansms s beyond the focus of ths paper. 7. Before each experment, the laptop was charged fully from a completely dead battery to nullfy battery memory and hysteress and was subsequently swtched off for 2 hours to elmnate heatng-related dscrepances.

15 218 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 6, NO. 2, FEBRUARY CONCLUSIONS We have studed the problem of formng coaltons between groups of nodes wth the ntent of savng power. We found that an applcaton of max-mn far technques to ths problem yelds an effcent and balanced approach whch we call far coalton routng. We developed a theory and algorthms for far coalton routng. We have carred out a range of smulatons that demonstrate that far coalton routng s practcal and benefcal n common cases. APPENDIX Proof of Theorem 1. Consder two possble coalton routngs between A and B. Let P~r a 1 and P~r b 1 be the powers expended for routng ~r 1 by Groups A and B, respectvely, and P~r a 2 and P~r b 2 smlarly for routng ~r 2. The benefts vector for routng ~r 1 s ðpopt a P ~r a 1 ;Popt b P ~r b 1 Þ and for routng ~r 2 s ðpopt a P ~r a 2 ;Popt b P ~r b 2 Þ. Consder a new routng that sends fracton of traffc through routng ~r 1 and 1 fracton through routng ~r 2. Snce P ~r s a lnear functon of ~r, we have the new power expendture as P~r a 1 þð1 ÞP~r a 2 for Group A and P~r b 1 þð1 ÞP~r b 2 for Group B. The beneft vector for the new routng s then Popt a ðp ~r a 1 þð1 ÞP~r a 2 ;Popt b P ~r b 1 þð1 ÞP~r b 2 Þ ; whch s Popt a P ~r a 1 ;Popt b P ~r b 1 þð1 Þ Popt a P ~r a 2 ;Popt b P ~r b 2 : Hence, the set of feasble beneft vectors s convex. tu Proof for Proposton 1. Let ~r be a far coalton routng and mnðb~r a ;Bb ~r Þ < 0. Consder the routng ~r 1 n whch each group uses ts group optmal. Then, B a ~r 1 ¼ 0B b ~r 1 ¼ 0 and mnðb~r a ;Bb ~r Þ < mnðba ~r 1 ;B b ~r 1 Þ. Thus, from the mnmum component property, ~r s not a far coalton routng, whch s a contradcton. tu Proof for Theorem 2. Let ðb~r a ;Bb ~r Þ be the beneft vector under far coalton routng ~r. If the mnmum s greater that t, then all other components are also greater than t. Hence, ~r wll result n a useful coalton. Now, we prove the only f condton usng contradcton. Let the mnmum component of the max-mn far beneft vector be less that t. Also, suppose a routng ~r 1 exsts such that mnðb a ~r 1 ;B b ~r 1 Þt. Thus, ~r s not a far coalton routng from the mnmum component property. Ths s a contradcton. tu Proof for Theorem 3. Consder two Groups A and B. Let ~r be a far coalton routng. Suppose that B~r a >Bb ~r. From Proposton 1, B~r a 0 and Bb ~r 0. Thus, Ba ~r > 0 snce A benefts from the coalton t sends traffc to at least one node n B. Now, consder a coalton routng ~r n whch Group A sends fracton of traffc through the jont routng ~r and 1 fracton of traffc through ts group optmal, 0 <<1. B routes as n ~r. Clearly, ~r s feasble. Now, consder the lnks ðv; v 0 Þ from Group A nodes (v 2 V a ) to Group B nodes (v 2 V b ) n the jont routng. Snce, n the optmal routng, nodes n A do not route ther traffc through the nodes n B, for each such ðv; v 0 Þ, P P r ðv; v 0 Þ P rðv; v0 Þ and for some v 2 V a and v 0 2 V b, r ðv; v 0 Þ < P rðv; v0 Þ. Hence, J~r b <J b ~r. Now, Bb ~r ¼ Popt b Jb ~r and B b ¼ P ~r b opt Jb : Snce J ~r b <J ~r b ~r, Bb >B ~r b ~r for any 2ð0; 1Þ: Snce B~r a >Bb ~r, when s suffcently close to 1, B a r ~ >Bb r ~, but then ~r does not satsfy the mnmum component property. Ths s a contradcton.tu We wll use the followng concepts n provng Theorem 5. Consder a convex and contnuous functon f defned on a convex set F R k. Then, a vector w 0 2 R k s called a subgradent of f at a pont y 0 2 F f t satsfes fðyþ fðy 0 Þðw 0, y y 0 Þ8y 2 F. An nteror pont y 0 of F s the mnmum pont of f n F f and only f ~0 belongs to the set of subgradents at y 0. Proof for Theorem 5. Let gðvþ ¼ P ðp v r ðv; v 0 Þ O 0 ðvþ P v r ðv 00 ;vþþ and 00 zðvþ ¼K 1 þ K X X r ðv; v 0 Þdðv; v 0 Þ 4 BðvÞ: v 0 2V a [V b [fe g P: Maxmze : F ð~r; ZÞ ¼Z sð~r; ZÞ where sð~r; ZÞ ¼ X ðjgðvþj þ maxð0;zðvþþþ þ maxð0;z B~r a Þ v2v a [V b þ maxð0;z B~r b Þ: Let Q ~ ð~r; ZÞ. Let Q ~ ð~r ;Z Þ be the optmal soluton and U be the optmal value of F ð~r; ZÞ. We prove ths n two steps. In the frst step, we prove that P has the same soluton as FC for >1. In the second step, we prove that the routng obtaned by the teratve approach converges to the optmal soluton of P,.e., lm n!1 k~r n ~r k¼0, where ~r n s the routng obtaned n the nth teraton and k ~Xk denotes the norm of ~X,.e., f X ðx 1 ;x 2...Þ, then p k ~Xk ¼ ffffffffffffffffffffffffffffffffffffffffffff x 2 1 þ x The result follows. Step 1. Select Q ~ such that sð QÞ ~ > 0. For such Q, ~ there always exsts a component of the subgradent that s less than or equal to 1 and 1 s less than 0. Therefore, ~0 does not belong to the set of subgradents. Hence, Q ~ cannot be an optmal soluton for P. Therefore, all solutons of P nvolve Q ~ for whch sð QÞ¼0. ~ Also, for sð QÞ¼0, ~ the value of the objectve functon of FC and P are equal. Therefore, for >1, any optmal soluton of P s an optmal soluton of FC. Step 2. Choose an arbtrary >0. Let 0 ¼ =2. For any 0 > 0, defne D 0 as D 0 ¼f Q ~ : Fð QÞU ~ 0 g. From Theorem 27.2 [27], t follows that there exsts an ¼ ð 0 Þ > 0 such that n D Q ~ : k Q ~ Q ~ o k 0 : ð5þ Consder n, for whch ~ Q n 62 D. Therefore, F ð ~ Q n Þ < U : The update equatons at the nodes of Groups A and B can be compactly stated as ~Q nþ1 ðv; v 0 ¼½ ~ Q n ðv; v 0 Þþ n ~ n Š þ ;

16 GUHA ET AL.: FAIR COALITIONS FOR POWER-AWARE ROUTING IN WIRELESS NETWORKS 219 where ~ n s the subgradent of F ð~r; ZÞ. It follows from the defnton of subgradents that ð~ n ; ~ Q n ~ Q ÞFð ~ Q n Þ U < : Now, k~ n kt, where qffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff T ¼ 2 2 ð1 þ L 4 Þ 2 N 2 þð2 þ 1Þ 2 ; L s the maxmum dstance between any two nodes, and N s the total number of nodes n the network. k ~ Q nþ1 ~ Q k 2 ¼k½ ~ Q n þ n ~ n Š þ ~ Q k 2 k ~ Q n þ n ~ n ~ Q k 2 ¼k Q ~ n Q ~ k 2 þ n 2 k~ nk 2 þ 2 n ð~ n ; Q ~ n Q ~ Þ < k Q ~ n Q ~ k 2 þ T 2 n 2 2 n: Snce n! 0, n =T 2 when n s suffcently large. For all such n, k ~ Q nþ1 ~ Q k 2 < k ~ Q n ~ Q k 2 n : Suppose there exsts a N 0 < 1 such that ~ Q n 62 D for all n N 0. Therefore, there exsts N N 0 such that (6) holds for all n N. Addng the nequaltes obtaned from (6) for n ¼ N to N þ m, we obtan k Q ~ Nþmþ1 Q ~ k 2 < k Q ~ N Q ~ k 2 XNþm n ; n¼n whch mples that k Q ~ Nþmþ1 Q ~ k! 1 as m!1 snce P 1 1 n ¼1. Ths s not possble snce k ~ Q N þmþ1 ~ Q k0: Hence, the supposton was ncorrect. Hence, there exsts a sequence n 1; <n 2; <... such that ~ Qn; 2 D for all I ¼ 1; 2;... Let 1 ¼ n 1;. Snce n! 0, there exsts 2 such that n mnð 0 =T ; =T 2 Þ; 8 n n 2;. Let 0 ¼ maxð 1 ; 2 Þ. Consder the followng cases: Case 1. n ¼ n j; for some j 0. Here, ~ Qn 2 D and, from (5), t follows that k ~ Q n ~ Q k 0 <. Case 2. n ¼ n j; þ 1 for some j 0. Then, Thus, ~Q n ¼ ~ Q nj; þ1 ¼½ ~ Q nj; þ nj; ~ nj; Š þ : k Q ~ n Q ~ nj; k¼k½ Q ~ nj; þ nj; ~ nj; Š þ Q ~ nj; k k Q ~ nj; þ nj; ~ nj; Q ~ nj; k ¼ nj; k~ nj; ku nj; 0 : From the above, and snce k ~ Q nj; ~ Q k 0 (Case 1), we get k ~ Q n ~ Q kk ~ Q nj; ~ Q kþk ~ Q n ~ Q nj; k 0 þ 0 ¼ 2 0 ¼ : Case 3. n j; þ 1 <n<n jþ1; for some j 0. Also, ~Q n 0 62 D 8n j; <n 0 <n jþ1;. From (6), t follows that k Q ~ n 0 þ1 Q ~ k < k Q ~ n 0 Q ~ k. Thus, k Q ~ n Q ~ k < k Q ~ nj; þ1 ~Q k: Snce k Q ~ nj; þ1 Q ~ k (Case 2), k Q ~ n Q ~ k. ð6þ From Cases 1, 2, and 3, t follows that k Q ~ n Q ~ k 8n n 0 ;: Snce s arbtrary, lm n!1 k Q ~ n Q ~ k¼0, and snce Q ~ ð~r; ZÞ, we have lm n!1 k~r n ~r k¼0. tu Proof for Proposton 2. Consder the jont routng ~r 1 under whch 1) A and B jontly route to the ext ponts wthout usng any node n C and both groups have postve benefts and 2) C routes optmally to the ext pont wthout usng nodes of Groups A and B. Under ~r 1, Group C has zero beneft and Groups A and B have postve benefts. Such ~r 1 exsts because the coalton between A and B s mutually benefcal. Now, usng ~r 1, we construct a coalton routng ~r that wll make benefts of all three groups postve. Snce the coalton between B and C s mutually benefcal, at least one node n C can send traffc through at least one node n B. Let b1 and c1 be such a node par. Let c1 send fracton of ts traffc to b1, where >0, and 1 fracton of ts traffc usng ts group optmal. Now, for any >0under ~r, the beneft of Group C wll be greater than that under ~r 1 (as nodes n C route less traffc under ~r than under ~r 1 ) and, hence, postve. Also, the benefts of Groups A and B under ~r s less than that under ~r 1, as nodes n Groups A and B route more traffc under ~r than ~r 1. But, can be sutably reduced to keep the benefts of Groups A and B postve. Hence, a routng ~r exsts under whch all three groups have a postve beneft. Proof for Theorem 6. Consder a feasble beneft vector ~B ~r such that there exsts subsets y 1 ;y 2...y k such that for k n, y 1 [...y k ¼f1...ng and the followng condtons hold: 1. B~r ¼ Bj ~r f, j 2 y m for each m 2f1...kg. 2. B~r >Bj ~r f 2 y m and j 2 y m 1 for each m 2f2...kg. 3. For any 2 y m, whle mantanng feasblty, B~r cannot be ncreased wthout reducng B j ~r for some j 2 y 1 [...y m. Then, ~B ~r s a max-mn far beneft vector. Each stage of the lnear program has a feasble soluton. Let the program yeld a routng ~r and termnate at stage k. Clearly, ~B r ~ s feasble. Note that e 1 [...[ e k ¼f1...ng. We wll show that ~B r ~ satsfes the above propertes wth y 1 ¼ e 1 ;...;y k ¼ e k. Note that B ¼ Z ~r m 8 2 e m and 1 m k. Also, Z1 <Z 2...<Z k. Thus, Propertes 1 and 2 hold. Let Property 3 not hold. Then, there exsts a routng ~r 1 such that B >Z ~r m for some 2 e m and B j ~r 1 B j for each j 2fe ~r 1 [...[ e m g. Case A. Let B j ~r 1 Zm for each j 2fe mþ1 [...[ e k g, but then ~r 1 s a feasble soluton of a substage of stage m and, therefore, 62 e m. Case B. Let B j ~r 1 <Zm for some j 2fe mþ1 [...[ e k g. Then, we have two feasble beneft vectors ~B ~r1 and ~B r ~ such that B j ~r 1 B j for each j 2fe ~r 1 [...e m g, B ~r 1 >Zm, and B j >Z ~r m for each j 2fe mþ1 [...e k g. Let AðÞ ~ ¼ ~B r ~ þð1 Þ ~B ~r1 for 0 <<1. Now, from Theorem 1, AðÞ ~ s a feasble beneft vector. For each >0, A j ðþ B j for each j 2fe ~r 1 [...e m g and A ðþ >Zm. For close to 1, Aj ðþ >Zm for each j 2fe mþ1 [...e k g. Let 0 be one such. Then, lke n

17 220 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 6, NO. 2, FEBRUARY 2007 Case A, Að~ 0 Þ s a feasble soluton of a substage of stage m and 62 e m. Ths s a contradcton and, thus, Property C also holds. tu ACKNOWLEDGMENTS The authors apprecate comments from Ron Brachman and Bob Hckok. Ths research was supported n part by grants NSF EIA , NSF ANI NSF, NCR , CNS , and ONR N REFERENCES [1] D. Bertsekas and R. Gallager, Data Networks. Prentce Hall, [2] S. Capkun, M. Hamd, and J.P. Hubaux, GPS-Free Postonng n Moble Ad Hoc Networks, Proc. Hawa Int l Conf. System Scences, [3] J. Chang and L. Tassulas, Routng for Maxmum System Lfetme n Wreless Ad Hoc Networks, Proc. 37th Ann. Allerton Conf. Comm., Control and Computng, [4] J. Chang and L. Tassulas, Energy Conservng Routng n Wreless Ad-Hoc Networks, Proc. IEEE INFOCOM, [5] S. Chen and K. Nahrstedt, Maxmn Far Routng n Connecton- Orented Networks, Proc. Euro-Parallel and Dstrbuted Systems Conf., [6] C.F. Chassern and R.R. 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Guha receved the BTech degree n electrcal engneerng from the Indan Insttute of Technology, New Delh, Inda, n He receved the MEng degree n electrcal and systems engneerng from the Unversty of Pennsylvana n He s currently actng as a research assstant n the Multmeda and Networkng Lab at the Unversty of Pennsylvana, Phladelpha. Hs research nterests are n wreless networks, computer securty, and dstrbuted control. He s a student member of the IEEE. Carl A. Gunter receved the BA degree from the Unversty of Chcago n 1979 and the PhD degree from the Unversty of Wsconsn at Madson n He s currently a professor at the Unversty of Illnos at Urbana-Champagn, where he s the drector of the Illnos Securty Lab. Hs research nterests are n securty, networks, programmng languages, and software engneerng. He s a senor member of the IEEE and the IEEE Computer Socety. Saswat Sarkar (S 98, M 00) receved the MEng degree n electrcal communcaton engneerng from the Indan Insttute of Scence n 1996 and the PhD degree n electrcal and computer engneerng from the Unversty of Maryland, College Park, n She s currently an assstant professor n the Department of Electrcal and Systems Engneerng at the Unversty of Pennsylvana. Her research nterests are n resource allocaton and performance analyss n communcaton networks. She receved the Motorola gold medal for beng the best masters student n the dvson of electrcal scences at the Indan Insttute of Scence and a US Natonal Scence Foundaton (NSF) Faculty Early Career Development Award n She has been an assocate edtor of the IEEE Transactons on Wreless Communcatons snce She s a member of the IEEE.. For more nformaton on ths or any other computng topc, please vst our Dgtal Lbrary at